Split (1)

train · 11.7k rows

text stringlengths 204 3.13k
precisely what Figure 4.7 shows. 136 Part 2: Choice and Demand SUMMARY In this chapter we explored the basic economic model of utility maximization subject to a budget constraint. Although we approached this problem in a variety of ways, all these approaches led to the same basic result. consumption of some goods is zero. In this case, the ratio of marginal utility to price for such a good will be below the common marginal beneﬁt–marginal cost ratio for goods actually bought. • To reach a constrained maximum, an individual should spend all available income and should choose a commodity bundle such that the MRS between any two goods is equal to the ratio of those goods’ market prices. This basic tangency will result in the individual equating the ratios of the marginal utility to market price for every good that is actually consumed. Such a result is common to most constrained optimization problems. • The tangency conditions are only the ﬁrst-order conditions for a unique constrained maximum, however. To ensure that these conditions are also sufﬁcient, the individual’s indifference curve map must exhibit a diminishing MRS. In formal terms, the utility function must be strictly quasi-concave. • The tangency conditions must also be modiﬁed to allow for corner solutions in which the optimal level of PROBLEMS • A consequence of the assumption of constrained utility maximization is that the individual’s optimal choices will depend implicitly on the parameters of his or her budget constraint. That is, the choices observed will be implicit functions of all prices and income. Therefore, utility will also be an indirect function of these parameters. • The dual to the constrained utility-maximization problem is to minimize the expenditure required to reach a given utility target. Although this dual approach yields the same optimal solution as the primal constrained maximum problem, it also yields additional insight into the theory of choice. Speciﬁcally, this approach leads to expenditure functions in which the spending required to reach a given utility target depends on goods’ market prices. Therefore, expenditure functions are, in principle, measurable. 4.1 Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies (t) and soda (s), and these provide him a utility of utility U t, s ð Þ ¼ ¼ tsp : a. If Twinkies cost $0.10 each and soda costs $0.25
per cup, how should Paul spend the $1 his mother gives him to maximize ﬃﬃﬃﬃ his utility? b. If the school tries to discourage Twinkie consumption by increasing the price to $0.40, by how much will Paul’s mother have to increase his lunch allowance to provide him with the same level of utility he received in part (a)? 4.2 a. A young connoisseur has $600 to spend to build a small wine cellar. She enjoys two vintages in particular: a 2001 French Bordeaux (wF) at $40 per bottle and a less expensive 2005 California varietal wine (wC) priced at $8. If her utility is U wF, wCÞ ¼ ð F w1=3 w2=3 C, then how much of each wine should she purchase? b. When she arrived at the wine store, our young oenologist discovered that the price of the French Bordeaux had fallen to $20 a bottle because of a decrease in the value of the euro. If the price of the California wine remains stable at $8 per bottle, how much of each wine should our friend purchase to maximize utility under these altered conditions? c. Explain why this wine fancier is better off in part (b) than in part (a). How would you put a monetary value on this utility increase? Chapter 4: Utility Maximization and Choice 137 4.3 a. On a given evening, J. P. enjoys the consumption of cigars (c) and brandy (b) according to the function How many cigars and glasses of brandy does he consume during an evening? (Cost is no object to J. P.) b. Lately, however, J. P. has been advised by his doctors that he should limit the sum of glasses of brandy and cigars consumed to 5. How many glasses of brandy and cigars will he consume under these circumstances? U c, b ð Þ ¼ 20c # c2 þ 18b # 3b2: 4.4 a. Mr. Odde Ball enjoys commodities x and y according to the utility function Maximize Mr. Ball’s utility if px ¼ $3, py ¼ than U. Why will this not alter your results? U x, y ð Þ ¼ x2 y2 : þ ﬃﬃﬃ�
�ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ $4, and he has $50 to spend. Hint: It may be easier here to maximize U2 rather p b. Graph Mr. Ball’s indifference curve and its point of tangency with his budget constraint. What does the graph say about Mr. Ball’s behavior? Have you found a true maximum? 4.5 Mr. A derives utility from martinis (m) in proportion to the number he drinks: Þ ¼ Mr. A is particular about his martinis, however: He only enjoys them made in the exact proportion of two parts gin (g) to one part vermouth (v). Hence we can rewrite Mr. A’s utility function as U m ð m: U m ð Þ ¼ U ð g, v Þ ¼ min g 2 %, v : & a. Graph Mr. A’s indifference curve in terms of g and v for various levels of utility. Show that, regardless of the prices of the two ingredients, Mr. A will never alter the way he mixes martinis. b. Calculate the demand functions for g and v. c. Using the results from part (b), what is Mr. A’s indirect utility function? d. Calculate Mr. A’s expenditure function; for each level of utility, show spending as a function of pg and pv. Hint: Because this problem involves a ﬁxed-proportions utility function, you cannot solve for utility-maximizing decisions by using calculus. 4.6 Suppose that a fast-food junkie derives utility from three goods—soft drinks (x), hamburgers (y), and ice cream sundaes (z)— according to the Cobb–Douglas utility function Þ ¼ Suppose also that the prices for these goods are given by px ¼ I ¼ a. Show that, for z 8. U x, y, z ð x0:5y0:5 1 ð 1, py ¼ z 0:5: þ Þ 4, and pz ¼ 8 and that this consumer’s income is given by choice that results in z > 0 (even for a fractional z) reduces utility from this optimum. ¼ 0, maximization of
utility results in the same optimal choices as in Example 4.1. Show also that any b. How do you explain the fact that z c. How high would this individual’s income have to be for any z to be purchased? 0 is optimal here? ¼ 4.7 The lump sum principle illustrated in Figure 4.5 applies to transfer policy and taxation. This problem examines this application of the principle. 138 Part 2: Choice and Demand a. Use a graph similar to Figure 4.5 to show that an income grant to a person provides more utility than does a subsidy on good x that costs the same amount to the government. b. Use the Cobb–Douglas expenditure function presented in Equation 4.52 to calculate the extra purchasing power needed to increase this person’s utility from U 2 to U 3. ¼ ¼ c. Use Equation 4.52 again to estimate the degree to which good x must be subsidized to increase this person’s utility from 3. How much would this subsidy cost the government? How would this cost compare with the cost calcu- U 2 to U lated in part (b)? ¼ ¼ d. Problem 4.10 asks you to compute an expenditure function for a more general Cobb–Douglas utility function than the one 0.3, a ﬁgure close to used in Example 4.4. Use that expenditure function to re-solve parts (b) and (c) here for the case a the fraction of income that low-income people spend on food. ¼ e. How would your calculations in this problem have changed if we had used the expenditure function for the ﬁxed- proportions case (Equation 4.54) instead? 4.8 Two of the simplest utility functions are: 1. Fixed proportions: U 2. Perfect substitutes: U min x x, y ð x, y ð Þ ¼ Þ ¼ þ x, y ½ y. * a. For each of these utility functions, compute the following: • Demand functions for x and y • Indirect utility function • Expenditure function b. Discuss the particular forms of these functions you calculated—why do they take the speciﬁc forms they do? 4.9 Suppose that we have a utility function involving two goods that is linear of the form U(x, y) expenditure function for this utility function. Hint: The expenditure function will have kinks at various price
ratios. ax ¼ þ by. Calculate the Analytical Problems 4.10 Cobb–Douglas utility In Example 4.1 we looked at the Cobb–Douglas utility function U(x, y) few more attributes of that function. ¼ xay1 # a, where 0 a " " 1. This problem illustrates a a. Calculate the indirect utility function for this Cobb–Douglas case. b. Calculate the expenditure function for this case. c. Show explicitly how the compensation required to offset the effect of an increase in the price of x is related to the size of the exponent a. 4.11 CES utility The CES utility function we have used in this chapter is given by U x, y ð Þ ¼ xd d þ yd d : a. Show that the ﬁrst-order conditions for a constrained utility maximum with this function require individuals to choose goods in the proportion d 1= ð 1 Þ # : x y ¼ px py! b. Show that the result in part (a) implies that individuals will allocate their funds equally between x and y for the Cobb– Douglas case (d ¼ 0), as we have shown before in several problems. Chapter 4: Utility Maximization and Choice 139 c. How does the ratio pxx/pyy depend on the value of d? Explain your results intuitively. (For further details on this function, see Extension E4.3.) d. Derive the indirect utility and expenditure functions for this case and check your results by describing the homogeneity properties of the functions you calculated. 4.12 Stone–Geary utility Suppose individuals require a certain level of food (x) to remain alive. Let this amount be given by x0. Once x0 is purchased, individuals obtain utility from food and other goods (y) of the form 1. U x, y ð x Þ ¼ ð # ayb, x0Þ where a b þ ¼ a. Show that if I > px x0 then the individual will maximize utility by spending a(I – px x0) pxx0 on good x and b(I – px x0) þ on good y. Interpret this result. b. How do the ratios px x/I and py y/I change as income increases in this problem? (See also Extension E4.2 for more on this utility function.) 4.13 CES indirect utility and expenditure
functions In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure functions. Suppose utility is given by [in this function the elasticity of substitution s a. Show that the indirect utility function for the utility function just given is xd yd 1=d Þ þ Þ ¼ ð U x, y ð 1/(1 – d)]. ¼ V I ð pr x þ 1=r, pr # yÞ ¼ 1 – s. where r d/(d – 1) ¼ ¼ b. Show that the function derived in part (a) is homogeneous of degree zero in prices and income. c. Show that this function is strictly increasing in income. d. Show that this function is strictly decreasing in any price. e. Show that the expenditure function for this case of CES utility is given by f. Show that the function derived in part (e) is homogeneous of degree one in the goods’ prices. g. Show that this expenditure function is increasing in each of the prices. h. Show that the function is concave in each price. E V pr x þ ð 1=r: pr yÞ ¼ 4.14 Altruism Michele, who has a relatively high income I, has altruistic feelings toward Soﬁa, who lives in such poverty that she essentially has no income. Suppose Michele’s preferences are represented by the utility function Þ ¼ where c1 and c2 are Michele and Soﬁa’s consumption levels, appearing as goods in a standard Cobb–Douglas utility function. Assume that Michele can spend her income either on her own or Soﬁa’s consumption (through charitable donations) and that $1 buys a unit of consumption for either (thus, the ‘‘prices’’ of consumption are p1 ¼ a. Argue that the exponent a can be taken as a measure of the degree of Michele’s altruism by providing an interpretation of p2 ¼ 1). U1 c1, c2 ð a c1 # 1 ca 2, ¼ ¼ extremes values a 0 and a 1. What value would make her a perfect altruist (regarding others the same as oneself )? b. Solve for Michele’s optimal choices and demonstrate how they change with a. c. Solve for Michele’s
optimal choices under an income tax at rate t. How do her choices change if there is a charitable deduction (so income spent on charitable deductions is not taxed)? Does the charitable deduction have a bigger incentive effect on more or less altruistic people? 140 Part 2: Choice and Demand d. Return to the case without taxes for simplicity. Now suppose that Michele’s altruism is represented by the utility function U1ð c1, U2Þ ¼ 1 U a c1 a 2, # which is similar to the representation of altruism in Extension E3.4 to the previous chapter. According to this speciﬁcation, Michele cares directly about Soﬁa’s utility level and only indirectly about Soﬁa’s consumption level. 1. Solve for Michele’s optimal choices if Soﬁa’s utility function is symmetric to Michele’s: U2ð 2. Repeat the previous analysis assuming Soﬁa’s utility function is U2(c2) your answer with part (b). Is Michele more or less charitable under the new speciﬁcation? Explain. c2, U1Þ ¼ 1. Compare 2 U a c1 a # c2. ¼ SUGGESTIONS FOR FURTHER READING Barten, A. P., and Volker Bo¨hm. ‘‘Consumer Theory.’’ In K. J. Arrow and M. D. Intriligator, Eds., Handbook of Mathematical Economics, vol. II. Amsterdam: North-Holland, 1982. Sections 10 and 11 have compact summaries of many of the concepts covered in this chapter. Deaton, A., and J. Muelbauer. Economics and Consumer Behavior. Cambridge, UK: Cambridge University Press, 1980. Section 2.5 provides a nice geometric treatment of duality concepts. Dixit, A. K. Optimization in Economic Theory. Oxford, UK: Oxford University Press, 1990. Chapter 2 provides several Lagrangian analyses focusing on the Cobb–Douglas utility function. Hicks, J. R. Value and Capital. Oxford, UK: Clarendon Press, 1946. Chapter II and the Mathematical Appendix provide some early suggestions of the importance of the expenditure function. Luenberger, D. G. Microeconomic Theory. New York: McGraw Hill, 1992. In Chapter 4 the author
shows several interesting relationships between his ‘‘Beneﬁt Function’’ (see Problem 3.15) and the more standard expenditure function. This chapter also offers insights on a number of unusual preference structures. Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. Oxford, UK: Oxford University Press, 1995. Chapter 3 contains a thorough analysis of utility and expenditure functions. Samuelson, Paul A. Foundations of Economic Analysis. Cambridge, MA: Harvard University Press, 1947. Chapter V and Appendix A provide a succinct analysis of the ﬁrst-order conditions for a utility maximum. The appendix provides good coverage of second-order conditions. Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGraw-Hill, 2001. A useful, although fairly difﬁcult, treatment of duality in consumer theory. Theil, H. Theory and Measurement of Consumer Demand. Amsterdam: North-Holland, 1975. Good summary of basic theory of demand together with implications for empirical estimation. BUDGET SHARES EXTENSIONS The nineteenth-century economist Ernst Engel was one of the ﬁrst social scientists to intensively study people’s actual spending patterns. He focused speciﬁcally on food consumption. His ﬁnding that the fraction of income spent on food decreases as income increases has come to be known as Engel’s law and has been conﬁrmed in many studies. Engel’s law is such an empirical regularity that some economists have suggested measuring poverty by the fraction of income spent on food. Two other interesting applications are: (1) the study by Hayashi (1995) showing that the share of income devoted to foods favored by the elderly is much higher in two-generation households than in one-generation households; and (2) ﬁndings by Behrman (1989) from less-developed countries showing that people’s desires for a more varied diet as their incomes increase may in fact result in reducing the fraction of income spent on particular nutrients. In the remainder of this extension we look at some evidence on budget shares (denoted by si ¼ pi xi /I ) together with a bit more theory on the topic. funds on food. Other important variations in the
table include the declining share of income spent on health-care needs and the much larger share of income devoted to retirement plans by higher-income people. Interestingly, the shares of income devoted to shelter and transportation are relatively constant over the range of income shown in the table; apparently, high-income people buy bigger houses and cars. The variable income shares in Table E4.1 illustrate why the Cobb–Douglas utility function is not useful for detailed empirical studies of household behavior. When utility is given by U (x, y) 1), the implied demand ¼ equations are x b bI/py. Therefore, ¼ a and b, ¼ ¼ and budget shares are constant for all observed income levels and relative prices. Because of this shortcoming, economists have investigated a number of other possible forms for the utility function that permit more ﬂexibility. (i) þ ¼ ¼ xayb (where a aI/px and y sx ¼ sy ¼ pxx=I pyy=I E4.1 The variability of budget shares Table E4.1 shows some recent budget share data from the United States. Engel’s law is clearly visible in the table: As income increases families spend a smaller proportion of their E4.2 Linear expenditure system A generalization of the Cobb–Douglas function that incorporates the idea that certain minimal amounts of each good TABLE E4.1 BUDGET SHARES OF U.S. HOUSEHOLDS, 2008 $10,000–$14,999 $40,000–$49,999 Over $70,000 Annual Income Expenditure Item Food Shelter Utilities, fuel, and public services Transportation Health insurance Other health-care expenses Entertainment (including alcohol) Education Insurance and pensions Other (apparel, personal care, other housing expenses, and misc.) 15.7 23.1 11.2 14.1 5.3 2.6 4.6 2.3 2.2 18.9 13.4 21.2 8.6 17.8 4.0 2.8 5.2 1.2 8.5 17.3 11.8 19.3 5.8 16.8 2.6 2.3 5.8 2.6 14.6 18.4 Consumer Expenditure Report, 2008, Bureau of Labor Statistics website: http://www.bls.gov. 142 Part 2: Choice and Demand must be bought by an individual (x0, y0) is
– 1). 1 1 1= ½ 1= ½ þ ð þ ð K K py=pxÞ px=pyÞ,, * * (vii) where K ¼ ¼ 0 and so K The homothetic nature of the CES function is shown by the fact that these share expressions depend only on the price ratio, px/py. Behavior of the shares in response to changes in relative prices depends on the value of the parameter K. For the Cobb–Douglas case, d sy ¼ 1/2. When d > 0, substitution possibilities are great and K < 0. In this case, Equation vii shows that sx and px/py move in opposite directions. If px/py increases, the individual substitutes y for x to such an extent that sx decreases. Alternatively, if d < 0, then substitution possibilities are limited, K > 0, and sx and px/py move in the same direction. In this case, an increase in px/py causes only minor substitution of y for x, and sx actually increases because of the relatively higher price of good x. 0 and sx ¼ ¼ North American free trade CES demand functions are most often used in large-scale computer models of general equilibrium (see Chapter 13) that economists use to evaluate the impact of major economic changes. Because the CES model stresses that shares respond to changes in relative prices, it is particularly appropriate for looking at innovations such as changes in tax policy or in international trade restrictions, where changes in relative prices are likely. One important area of such research has been on the impact of the North American Free Trade Agreement for Canada, Mexico, and the United States. In general, these models ﬁnd that all the countries involved might be expected to gain from the agreement, but that Mexico’s gains may be the greatest because it is experiencing the greatest change in relative prices. Kehoe and Kehoe (1995) present a number of computable equilibrium models that economists have used in these examinations.1 E4.4 The almost ideal demand system An alternative way to study budget shares is to start from a speciﬁc expenditure function. This approach is especially convenient because the envelope theorem shows that budget shares can be derived directly from expenditure functions through logarithmic differentiation (for more details, see Chapter 5): E4.3 CES utility In Chapter 3 we introduced the CES utility function @ ln E U xvi)
" 1, d for d 0. The primary use of this function is to illustrate alternative substitution possibilities (as reﬂected in the value of the parameter d). Budget shares implied by this utility 6¼ px, py, V ð @ ln px Þ 1 px, py, V E ð xpx E ¼ sx: ¼ ¼ ( Þ @E @px ( @px @ ln px (viii) 1The research on the North American Free Trade Agreement is discussed in more detail in the Extensions to Chapter 13. Deaton and Muellbauer (1980) make extensive use of this relationship to study the characteristics of a particular class of expenditure functions that they term an almost ideal demand system (AIDS). Their expenditure function takes the form 2 ð þ Þ ¼ ln E a2 ln py px, py, V a0 þ þ 0:5b1ð 0:5b3ð a1 ln px þ ln pxÞ ln pyÞ This form approximates any expenditure function. For the function to be homogeneous of degree one in the prices, the parameters of the function must obey the constraints a1 þ 1, b1 þ 0. Using the results of c2 ¼ 0, and c1 þ b3 ¼ Equation viii shows that, for this function, b2 ln px ln py x pc 2 Vc 0 pc 1 y : 0, b2 þ b2 ¼ a2 ¼ (ix) þ þ 2 sx ¼ sy ¼ a1 þ a2 þ b1 ln px þ b2 ln px þ b2 ln py þ b3 ln py þ c1Vc0 pc 1 c2Vc0 pc 1 x pc 2 y, x pc 2 y : (x) Notice that, given the parameter restrictions, sx þ 1. Making use of the inverse relationship between indirect utility and expenditure functions and some additional algebraic manipulation will put these budget share equations into a simple form suitable for econometric estimation: sy ¼ sx ¼ sy ¼ b2 ln py þ b3 ln py þ where p is an index of prices deﬁned by b1 ln px þ b2 ln px þ a1 þ a2 þ E=p E=p c1ð c2
ð,, Þ Þ ln p ¼ a0 þ þ a1 ln px þ b2 ln px ln py þ a2 ln py þ 0:5b3ð 0:5b1ð 2: ln pyÞ 2 ln pxÞ (xi) (xii) In other words, the AIDS share equations state that budget shares are linear in the logarithms of prices and in total real expenditures. In practice, simpler price indices are often substituted for the rather complex index given by Equation xii, although there is some controversy about this practice (see the Extensions to Chapter 5). Chapter 4: Utility Maximization and Choice 143 British expenditure patterns Deaton and Muellbauer apply this demand system to the study of British expenditure patterns between 1954 and 1974. They ﬁnd that food and housing have negative coefﬁcients of real expenditures, implying that the share of income devoted to these items decreases (at least in Britain) as people get richer. The authors also ﬁnd signiﬁcant relative price effects in many of their share equations, and prices have especially large effects in explaining the share of expenditures devoted to transportation and communication. In applying the AIDS model to real-world data, the authors also encounter a variety of econometric difﬁculties, the most important of which is that many of the equations do not appear to obey the restrictions necessary for homogeneity. Addressing such issues has been a major topic for further research on this demand system. References Behrman, Jere R. ‘‘Is Variety the Spice of Life? Implications for Caloric Intake.’’ Review of Economics and Statistics (November 1989): 666–72. Deaton, Angus, and John Muellbauer. ‘‘An Almost Ideal Demand System.’’ American Economic Review (June 1980): 312–26. Hyashi, Fumio. ‘‘Is the Japanese Extended Family Altruistically Linked? A Test Based on Engel Curves.’’ Journal of Political Economy (June 1995): 661–74. Kehoe, Patrick J., and Timothy J. Kehoe. Modeling North American Economic Integration. London: Kluwer Academic Publishers, 1995. Oczkowski, E., and N. E. Philip. ‘‘Household Expenditure Patterns and Access to Consumer Goods
in a Transitional Economy.’’ (June 1994): 165–83. Journal of Economic Development Stone, R. ‘‘Linear Expenditure Systems and Demand Analy- sis.’’ Economic Journal (September 1954): 511–27. This page intentionally left blank CHAPTER FIVE Income and Substitution Effects In this chapter we will use the utility-maximization model to study how the quantity of a good that an individual chooses is affected by a change in that good’s price. This examination allows us to construct the individual’s demand curve for the good. In the process we will provide a number of insights into the nature of this price response and into the kinds of assumptions that lie behind most analyses of demand. Demand Functions As we pointed out in Chapter 4, in principle it will usually be possible to solve the necessary conditions of a utility maximum for the optimal levels of x1, x2, …, xn (and l, the Lagrange multiplier) as functions of all prices and income. Mathematically, this can be expressed as n demand functions1 of the form x!1 ¼ x!2 ¼... x!n ¼ x1ð x2ð xnð p1, p2,..., pn, I p1, p2,..., pn, I p1, p2,..., pn, I, Þ, Þ : Þ (5:1) If there are only two goods, x and y (the case we will usually be concerned with), this notation can be simpliﬁed a bit as x! y! ¼ ¼ x y ð ð px, py, I px, py, I, Þ : Þ (5:2) Once we know the form of these demand functions and the values of all prices and income, we can ‘‘predict’’ how much of each good this person will choose to buy. The notation stresses that prices and income are ‘‘exogenous’’ to this process; that is, these are parameters over which the individual has no control at this stage of the analysis. Changes in the parameters will, of course, shift the budget constraint and cause this person to make different choices. That question is the focus of this chapter and the next. Speciﬁcally, in this
chapter we will be looking at the partial derivatives @x/@I and @x/@px for 1Sometimes the demand functions in Equation 5.1 are referred to as Marshallian demand functions (after Alfred Marshall) to differentiate them from the Hicksian demand functions (named for John Hicks) we will encounter later in this chapter. The difference between the two concepts derives from whether income or utility enters the functions. For simplicity, throughout this text the term demand functions or demand curves will refer to the Marshallian concept, whereas references to Hicksian (or ‘‘compensated’’) demand functions and demand curves will be explicitly noted. 145 146 Part 2: Choice and Demand any arbitrary good x. Chapter 6 will carry the discussion further by looking at ‘‘crossprice’’ effects of the form @x/@py for any arbitrary pair of goods x and y. Homogeneity A ﬁrst property of demand functions requires little mathematics. If we were to double all prices and income (indeed, if we were to multiply them all by any positive constant), then the optimal quantities demanded would remain unchanged. Doubling all prices and income changes only the units by which we count, not the ‘‘real’’ quantity of goods demanded. This result can be seen in a number of ways, although perhaps the easiest is through a graphic approach. Referring back to Figures 4.1 and 4.2, it is clear that doubling px, py, and I does not affect the graph of the budget constraint. Hence x!, y! will still be the combination that is chosen. In algebraic terms, pxx I is the same con2I. Somewhat more technically, we can write this result as saying 2pyy straint as 2pxx that, for any good xi, x!i ¼ Þ ¼ for any t > 0. Functions that obey the property illustrated in Equation 5.3 are said to be homogeneous of degree 0.2 Hence we have shown that individual demand functions are homogeneous of degree 0 in all prices and income. Changing all prices and income in the same proportions will not affect the physical quantities of goods demanded. This result shows that (in theory) individuals’ demands will not be affected by a ‘‘pure’’ inﬂation during which all prices and incomes increase proportionally. They will continue to demand the same bundle of goods.
Of course, if an inﬂation were not pure (i.e., if some prices increased more rapidly than others), this would not be the case. tp1, tp2,..., tpn, tI p1, p2,..., pn, I xið xið (5:3) pyy þ ¼ ¼ þ Þ EXAMPLE 5.1 Homogeneity Homogeneity of demand is a direct result of the utility-maximization assumption. Demand functions derived from utility maximization will be homogeneous, and, conversely, demand functions that are not homogeneous cannot reﬂect utility maximization (unless prices enter directly into the utility function itself, as they might for goods with snob appeal). If, for example, an individual’s utility for food (x) and housing ( y) is given by utility U x, y ð ¼ Þ ¼ x0:3y0:7, (5:4) then it is a simple matter (following the procedure used in Example 4.1) to derive the demand functions x! y! ¼ ¼ 0:3I px 0:7I py, : (5:5) These functions obviously exhibit homogeneity because a doubling of all prices and income would leave x! and y! unaffected. If the individual’s preferences for x and y were reﬂected instead by the CES function U x, y ð Þ ¼ x0:5 þ y0:5, (5:6) t kf (x1, x2, …, xn) for any t > 0. The most common cases of homogeneous functions are k 2More generally, as we saw in Chapters 2 and 4, a function f (x1, x2, …, xn) is said to be homogeneous of degree k if f (tx1, tx2, …, txn) 1. If f is homogeneous of degree 0, then doubling all its arguments leaves f unchanged in value. If f is homogeneous of degree 1, then doubling all its arguments will double the value of f. 0 and k ¼ ¼ ¼ Chapter 5: Income and Substitution Effects 147 then (as shown in Example 4.2) the demand functions are given by x! y! ¼ 1 ¼ 1 1 px=py 1 py
=px!&!& I px I py, : þ þ (5:7) As before, both of these demand functions are homogeneous of degree 0; a doubling of px, py, and I would leave x! and y! unaffected. QUERY: Do the demand functions derived in this example ensure that total spending on x and y will exhaust the individual’s income for any combination of px, py, and I? Can you prove that this is the case? Changes in Income As a person’s purchasing power increases, it is natural to expect that the quantity of each good purchased will also increase. This situation is illustrated in Figure 5.1. As expenditures increase from I1 to I2 to I3, the quantity of x demanded increases from x1 to x2 to x3. Also, the quantity of y increases from y1 to y2 to y3. Notice that the budget lines I1, I2, As income increases from I1 to I2 to I3, the optimal (utility-maximizing) choices of x and y are shown by the successively higher points of tangency. Observe that the budget constraint shifts in a parallel way because its slope (given by px/py) does not change. ’ Quantity of y U3 U2 U1 y3 y2 y1 I3 I2 I1 x1 x2 x3 U3 U2 U1 Quantity of x FIGURE 5.1 Effect of an Increase in Income on the Quantities of x and y Chosen 148 Part 2: Choice and Demand FIGURE 5.2 An Indifference Curve Map Exhibiting Inferiority In this diagram, good z is inferior because the quantity purchased decreases as income increases. Here, y is a normal good (as it must be if there are only two goods available), and purchases of y increase as total expenditures increase. Quantity of y y 3 y 2 y1 U3 U2 U1 I1 I2 I3 z 3 z 2 z1 Quantity of z and I3 are all parallel, reﬂecting that only income is changing, not the relative prices of x and y. Because the ratio px/py stays constant, the utility-maximizing conditions also require that the MRS stay constant as the individual moves to higher levels of satisfaction. Therefore, the MRS is the same at point (x3, y3) as at (x1, y1
). Normal and inferior goods In Figure 5.1, both x and y increase as income increases—both @x/@I and @y/@I are positive. This might be considered the usual situation, and goods that have this property are called normal goods over the range of income change being observed. For some goods, however, the quantity chosen may decrease as income increases in some ranges. Examples of such goods are rotgut whiskey, potatoes, and secondhand clothing. A good z for which @z/@I is negative is called an inferior good. This phenomenon is illustrated in Figure 5.2. In this diagram, the good z is inferior because, for increases in income in the range shown, less of z is chosen. Notice that indifference curves do not have to be ‘‘oddly’’ shaped to exhibit inferiority; the curves corresponding to goods y and z in Figure 5.2 continue to obey the assumption of a diminishing MRS. Good z is inferior because of the way it relates to the other goods available (good y here), not because of a peculiarity unique to it. Hence we have developed the following deﬁnitions Inferior and normal goods. A good xi for which @xi/@I < 0 over some range of income changes is an inferior good in that range. If @xi/@I 0 over some range of income variation, then the good is a normal (or ‘‘noninferior’’) good in that range. ( Chapter 5: Income and Substitution Effects 149 Changes in a Good’s Price The effect of a price change on the quantity of a good demanded is more complex to analyze than is the effect of a change in income. Geometrically, this is because changing a price involves changing not only one of the intercepts of the budget constraint but also its slope. Consequently, moving to the new utility-maximizing choice entails not only moving to another indifference curve but also changing the MRS. Therefore, when a price changes, two analytically different effects come into play. One of these is a substitution effect: Even if the individual were to stay on the same indifference curve, consumption patterns would be allocated so as to equate the MRS to the new price ratio. A second effect, the income effect, arises because a price change necessarily changes an individual’s ‘‘real’’ income. The individual cannot stay on
the initial indifference curve and must move to a new one. We begin by analyzing these effects graphically. Then we will provide a mathematical development. ¼ þ Graphical analysis of a decrease in price Income and substitution effects are illustrated in Figure 5.3. This individual is initially maximizing utility (subject to total expenditures, I ) by consuming the combination x!, y!. The initial budget constraint is I pyy. Now suppose that the price of x decreases to p2 x. The new budget constraint is given by the equation I pyy in Figure 5.3. p1 xx p2 xx þ ¼ I ð It is clear that the new position of maximum utility is at x!!, y!!, where the new budget line is tangent to the indifference curve U2. The movement to this new point can be viewed as being composed of two effects. First, the change in the slope of the budget constraint would have motivated a move to point B, even if choices had been conﬁned to those on the original indifference curve U1. The dashed line in Figure 5.3 has the same p2 but is drawn to be tangent to U1 xx slope as the new budget constraint because we are conceptually holding ‘‘real’’ income (i.e., utility) constant. A relatively lower price for x causes a move from x!, y! to B if we do not allow this individual to be made better off as a result of the lower price. This movement is a graphic demonstration of the substitution effect. The additional move from B to the optimal point x!!, y!! is analytically identical to the kind of change exhibited earlier for changes in income. Because the price of x has decreased, this person has a greater ‘‘real’’ income and can afford a utility level (U2) that is greater than that which could previously be attained. If x is a normal good, more of it will be chosen in response to this increase in purchasing power. This observation explains the origin of the term income effect for the movement. Overall then, the result of the price decrease is to cause more x to be demanded. pyy ¼ þ Þ It is important to recognize that this person does not actually make a series of choices from x!, y! to B and then to x!!, y!!. We never observe point B; only the two optimal positions are reﬂected in observed behavior. However, the
notion of income and substitution effects is analytically valuable because it shows that a price change affects the quantity of x that is demanded in two conceptually different ways. We will see how this separation offers major insights in the theory of demand. Graphical analysis of an increase in price If the price of good x were to increase, a similar analysis would be used. In Figure 5.4, the x to p2 budget line has been shifted inward because of an increase in the price of x from p1 x. The movement from the initial point of utility maximization (x!, y!) to the new point (x!!, y!!) can be decomposed into two effects. First, even if this person could stay on the initial indifference curve (U2), there would still be an incentive to substitute y for x and move along U2 to point B. However, because purchasing power has been reduced by the 150 Part 2: Choice and Demand FIGURE 5.3 Demonstration of the Income and Substitution Effects of a Decrease in the Price of x x to p2 When the price of x decreases from p1 x, the utility-maximizing choice shifts from x!, y! to x!!, y!!. This movement can be broken down into two analytically different effects: ﬁrst, the substitution effect, involving a movement along the initial indifference curve to point B, where the MRS is equal to the new price ratio; and second, the income effect, entailing a movement to a higher level of utility because real income has increased. In the diagram, both the substitution and income effects cause more x to be bought when its price decreases. Notice that point I/py is the same as before the price change; this is because py has not changed. Therefore, point I/py appears on both the old and new budget constraints. Quantity of y U1 U2 I py y** y* B x* xB x** Substitution eﬀect Income eﬀect Total increase in x I = px 1x + pyy I = p2 xx + pyy U2 U1 Quantity of x increase in the price of x, he or she must move to a lower level of utility. This movement is again called the income effect. Notice in Figure 5.4 that both the income and substitution effects work in the same direction and cause the quantity of x demanded to be reduced in response to an increase in its price. Effects of price changes
for inferior goods Thus far we have shown that substitution and income effects tend to reinforce one another. For a price decrease, both cause more of the good to be demanded, whereas for a price increase, both cause less to be demanded. Although this analysis is accurate for the case of normal (noninferior) goods, the possibility of inferior goods complicates the story. In this case, income and substitution effects work in opposite directions, and the combined result of a price change is indeterminate. A decrease in price, for example, will always cause an individual to tend to consume more of a good because of the substitution Chapter 5: Income and Substitution Effects 151 FIGURE 5.4 Demonstration of the Income and Substitution Effects of an Increase in the Price of x When the price of x increases, the budget constraint shifts inward. The movement from the initial utility-maximizing point (x!, y!) to the new point (x!!, y!!) can be analyzed as two separate effects. The substitution effect would be depicted as a movement to point B on the initial indifference curve (U2). The price increase, however, would create a loss of purchasing power and a consequent movement to a lower indifference curve. This is the income effect. In the diagram, both the income and substitution effects cause the quantity of x to decrease as a result of the increase in its price. Again, the point I/py is not affected by the change in the price of x. Quantity of y U2 U1 B I py y** y* x** xB x* Income eﬀect Substitution eﬀect Total reduction in x I = px 2x + pyy I = px 1x + pyy U2 U1 Quantity of x effect. But if the good is inferior, the increase in purchasing power caused by the price decrease may cause less of the good to be bought. Therefore, the result is indeterminate: The substitution effect tends to increase the quantity of the inferior good bought, whereas the (perverse) income effect tends to reduce this quantity. Unlike the situation for normal goods, it is not possible here to predict even the direction of the effect of a change in px on the quantity of x consumed. Giffen’s paradox If the income effect of a price change is strong enough, the change in price and the resulting change in the quantity demanded could actually move in the same direction. Legend 152 Part 2: Choice and Demand has
it that the English economist Robert Giffen observed this paradox in nineteenthcentury Ireland: When the price of potatoes rose, people reportedly consumed more of them. This peculiar result can be explained by looking at the size of the income effect of a change in the price of potatoes. Potatoes were not only inferior goods but they also used up a large portion of the Irish people’s income. Therefore, an increase in the price of potatoes reduced real income substantially. The Irish were forced to cut back on other luxury food consumption to buy more potatoes. Even though this rendering of events is historically implausible, the possibility of an increase in the quantity demanded in response to an increase in the price of a good has come to be known as Giffen’s paradox.3 Later we will provide a mathematical analysis of how Giffen’s paradox can occur. A summary Hence our graphical analysis leads to the following conclusions. Substitution and income effects. The utility-maximization hypothesis suggests that, for normal goods, a decrease in the price of a good leads to an increase in quantity purchased because: (1) the substitution effect causes more to be purchased as the individual moves along an indifference curve; and (2) the income effect causes more to be purchased because the price decrease has increased purchasing power, thereby permitting movement to a higher indifference curve. When the price of a normal good increases, similar reasoning predicts a decrease in the quantity purchased. For inferior goods, substitution and income effects work in opposite directions, and no deﬁnite predictions can be made. The Individual’s Demand Curve Economists frequently wish to graph demand functions. It will come as no surprise to you that these graphs are called ‘‘demand curves.’’ Understanding how such widely used curves relate to underlying demand functions provides additional insights to even the most fundamental of economic arguments. To simplify the development, assume there are only two goods and that, as before, the demand function for good x is given by x! x ð ¼ px, py, I : Þ The demand curve derived from this function looks at the relationship between x and px while holding py, I, and preferences constant. That is, it shows the relationship x! x ð ¼ px, py, I, Þ (5:8) where the bars over py and I indicate that these determinants of demand are being held constant. This construction is shown in Figure 5.5. The graph shows utility
-maximizing choices of x and y as this individual is presented with successively lower prices of good x (while holding py and I constant). We assume that the quantities of x chosen increase from x 0 to x 00 to x 000 as that good’s price decreases from p 0x to p00 to p000x. Such an 3A major problem with this explanation is that it disregards Marshall’s observation that both supply and demand factors must be taken into account when analyzing price changes. If potato prices increased because of the potato blight in Ireland, then supply should have become smaller; therefore, how could more potatoes possibly have been consumed? Also, because many Irish people were potato farmers, the potato price increase should have increased real income for them. For a detailed discussion of these and other fascinating bits of potato lore, see G. P. Dwyer and C. M. Lindsey, ‘‘Robert Giffen and the Irish Potato,’’ American Economic Review (March 1984): 188–92. FIGURE 5.5 Construction of an Individual’s Demand Curve Chapter 5: Income and Substitution Effects 153 In (a), the individual’s utility-maximizing choices of x and y are shown for three different prices of x ð The demand curve is drawn on the assumption that py, I, and preferences remain constant as px varies.. In (b), this relationship between px and x is used to construct the demand curve for x. p0x, p00x, and p000x Þ Quantity of y per period I /py I = px′ x + pyy I = px″ x + pyy I = px′′′ x + pyy U3 U2 U1 x ′ x ″ x′′′ Quantity of x per period (a) Individual’s indifference curve map px px′ px″ px′′′ x( px, py, I) x′ x″ x′′′ Quantity of x per period (b) Demand curve 154 Part 2: Choice and Demand assumption is in accord with our general conclusion that, except in the unusual case of Giffen’s paradox, @x/@px is negative. In Figure 5.5b, information about the utility-maximizing choices of good x is transferred to a demand curve with px on the vertical axis and sharing the same horizontal axis as Figure 5.
5a. The negative slope of the curve again reﬂects the assumption that @x/@px is negative. Hence we may deﬁne an individual demand curve as follows Individual demand curve. An individual demand curve shows the relationship between the price of a good and the quantity of that good purchased by an individual, assuming that all other determinants of demand are held constant. The demand curve illustrated in Figure 5.5 stays in a ﬁxed position only so long as all other determinants of demand remain unchanged. If one of these other factors were to change, then the curve might shift to a new position, as we now describe. Shifts in the demand curve Three factors were held constant in deriving this demand curve: (1) income, (2) prices of other goods (say, py), and (3) the individual’s preferences. If any of these were to change, the entire demand curve might shift to a new position. For example, if I were to increase, the curve would shift outward (provided that @x/@I > 0, i.e., provided the good is a ‘‘normal’’ good over this income range). More x would be demanded at each price. If another price (say, py) were to change, then the curve would shift inward or outward, depending precisely on how x and y are related. In the next chapter we will examine that relationship in detail. Finally, the curve would shift if the individual’s preferences for good x were to change. A sudden advertising blitz by the McDonald’s Corporation might shift the demand for hamburgers outward, for example. As this discussion makes clear, one must remember that the demand curve is only a two-dimensional representation of the true demand function (Equation 5.8) and that it is stable only if other things do stay constant. It is important to keep clearly in mind the difference between a movement along a given demand curve caused by a change in px and a shift in the entire curve caused by a change in income, in one of the other prices, or in preferences. Traditionally, the term an increase in demand is reserved for an outward shift in the demand curve, whereas the term an increase in the quantity demanded refers to a movement along a given curve caused by a fall in px. EXAMPLE 5.2 Demand Functions and Demand Curves To be able to graph a demand curve from a given demand
function, we must assume that the preferences that generated the function remain stable and that we know the values of income and other relevant prices. In the ﬁrst case studied in Example 5.1, we found that and 0:3I px x ¼ 0:7I py : y ¼ (5:9) Chapter 5: Income and Substitution Effects 155 If preferences do not change and if this individual’s income is $100, these functions become 30 px 70 py, ; x y ¼ ¼ (5:10) or ¼ ¼ which makes clear that the demand curves for these two goods are simple hyperbolas. An increase in income would shift both of the demand curves outward. Notice also, in this case, that the demand curve for x is not shifted by changes in py and vice versa. pxx pyy 30, 70, For the second case examined in Example 5.1, the analysis is more complex. For good x, we know that (5:11) 100 and ¼ (5:12) x 1 px=py I px, 1! so to graph this in the px – x plane we must know both I and py. If we again assume I let py ¼ 1, then Equation 5.11 becomes ¼ þ " & x 100, p2 x þ which, when graphed, would also show a general hyperbolic relationship between price and quantity consumed. In this case the curve would be relatively ﬂatter because substitution effects are larger than in the Cobb–Douglas case. From Equation 5.11, we also know that px ¼ and @x @I ¼ 1 px=py 1 px & " > 0 1! þ @x @py ¼ I px þ ð 2 > 0, pyÞ (5:13) thus increases in I or py would shift the demand curve for good x outward. QUERY: How would the demand functions in Equations 5.10 change if this person spent half of his or her income on each good? Show that these demand functions predict the same x 100 as does Equation 5.11. Use a numerical consumption at the point px ¼ ¼ example to show that the CES demand function is more responsive to an increase in px than is the Cobb–Douglas demand function. 1, py ¼ 1, I Compensated (HICKSIAN) Demand
Curves and Functions In Figure 5.5, the level of utility this person gets varies along the demand curve. As px decrease, he or she is made increasingly better off, as shown by the increase in utility from U1 to U2 to U3. The reason this happens is that the demand curve is drawn on the assumption that nominal income and other prices are held constant; hence a decline in px makes this person better off by increasing his or her real purchasing power. Although 156 Part 2: Choice and Demand this is the most common way to impose the ceteris paribus assumption in developing a demand curve, it is not the only way. An alternative approach holds real income (or utility) constant while examining reactions to changes in px. The derivation is illustrated in Figure 5.6, where we hold utility constant (at U2) while successively reducing px. As px decreases, the individual’s nominal income is effectively reduced, thus preventing any increase in utility. In other words, the effects of the price change on purchasing power are ‘‘compensated’’ to constrain the individual to remain on U2. Reactions to changing prices include only substitution effects. If we were instead to examine effects of increases in px, income compensation would be positive: This individual’s income would have to be increased to permit him or her to stay on the U2 indifference curve in response to the price increases. We can summarize these results as follows. FIGURE 5.6 Construction of a Compensated Demand Curve The curve xc shows how the quantity of x demanded changes when px changes, holding py and utility constant. That is, the individual’s income is ‘‘compensated’’ to keep utility constant. Hence xc reﬂects only substitution effects of changing prices. Quantity of y Slope = px′ py Slope = px″ py Slope = px′′′ py U2 x* x″ x ′′′ Quantity of x (a) Individual’s indifference curve map px px′ px″ px′′′ xc( px, py, U) x* x″ x** Quantity of x (b) Compensated demand curve Chapter 5: Income and Substitution Effects 157 Compensated demand curve. A compensated demand curve shows the relationship between the price of a good and the quantity purchased on the
assumption that other prices and utility are held constant. Therefore, the curve (which is sometimes termed a Hicksian demand curve after the British economist John Hicks) illustrates only substitution effects. Mathematically, the curve is a two-dimensional representation of the compensated demand function xc xc ð ¼ px, py, U : Þ (5:14) Notice that the only difference between the compensated demand function in Equation 5.14 and the uncompensated demand functions in Equations 5.1 or 5.2 is whether utility or income enters the functions. Hence the major difference between compensated and uncompensated demand curves is whether utility or income is held constant in constructing the curves. Shephard’s lemma Many facts about compensated demand functions can be easily proven by using a remarkable result from duality theory called Shephard’s lemma (named for R. W. Shephard, who pioneered the use of duality theory in production and cost functions—see Chapters 9 and 10). Consider the dual expenditure minimization problem discussed in Chapter 4. The Lagrangian expression for this problem was + pyy pxx ¼ k ½ The solution to this problem yields the expenditure function E(px, py, U). We can apply the envelope theorem to this function by noting that its derivative with respect to one of the good’s prices can be interpreted by differentiating the Lagrangian expression in Equation 5.15: Þ ’ þ þ ð * (5:15) x, y U U : @E ð px, py, U @px Þ @+ @px ¼ xc ð ¼ px, py, U : Þ (5:16) That is, the compensated demand function for a good can always be found from the expenditure function by differentiation with respect to that good’s price. To see intuitively why such a derivative is a compensated demand function notice ﬁrst that both the expenditure function and the compensated demand function depend on the same variables (px, py, and U)—the value of a derivative will always depend on the same variables that enter into the original function. Second, because we are differentiating a minimized function, we are assured that any change in prices will be met by a series of adjustments in quantities bought that will continue to minimize the expenditures needed to reach a given utility level. Finally, changes in the price of a good will affect expenditures roughly in proportion to the
quantity of that good being bought—that is precisely what Equation 5.16 says. One of the many insights that can be derived from Shephard’s lemma concerns the slope of the compensated demand curve. In Chapter 4 we showed that the expenditure function must be concave in prices. In mathematical terms, @ 2E x < 0. Taking account of Shephard’s lemma, however, implies that: px, py, V =@p 2 Þ ð @ 2E ð px, py, V @p 2 x Þ ¼ @E @ ½ ð px, py, V @px =@px* Þ ¼ @x c ð px, py, V @px Þ < 0: (5:17) Hence the compensated demand curve must have a negative slope. The ambiguity that arises when substitution and income effects work in opposite directions for Marshallian 158 Part 2: Choice and Demand demand curves does not arise in the case of compensated demand curves because they involve only substitution effects. Relationship between compensated and uncompensated demand curves This relationship between the two demand curve concepts is illustrated in Figure 5.7. At p00x the curves intersect because at that price the individual’s income is just sufﬁcient to attain utility level U2 (compare Figures 5.5 and Figure 5.6). Hence x00 is demanded under either demand concept. For prices below p00x, however, the individual suffers a compensating reduction in income on the curve xc that prevents an increase in utility arising from the lower price. Assuming x is a normal good, it follows that less x is demanded at p00x along xc than along the uncompensated curve x. Alternatively, for a price above p00x (such, income compensation is positive because the individual needs some help to as p00xÞ remain on U2. Again, assuming x is a normal good, at p0x more x is demanded along xc than along x. In general, then, for a normal good the compensated demand curve is somewhat less responsive to price changes than is the uncompensated curve. This is because the latter reﬂects both substitution and income effects of price changes, whereas the compensated curve reﬂects only substitution effects. The choice between using compensated or uncompensated demand curves in economic analysis is largely a matter of convenience. In most empirical work,
uncompensated (or Marshallian) demand curves are used because the data on prices and nominal FIGURE 5.7 Comparison of Compensated and Uncompensated Demand Curves The compensated (xc) and uncompensated (x) demand curves intersect at p00x because x00 is demanded under each concept. For prices above p00x, the individual’s purchasing power must be increased with the compensated demand curve; thus, more x is demanded than with the uncompensated curve. For prices below p00x, purchasing power must be reduced for the compensated curve; therefore, less x is demanded than with the uncompensated curve. The standard demand curve is more price-responsive because it incorporates both substitution and income effects, whereas the curve xc reﬂects only substitution effects. px px′ px″ px′′′ x( px, py, I) xc( px, py, U) x′ x* x″ x** x′′′ Quantity of x Chapter 5: Income and Substitution Effects 159 incomes needed to estimate them are readily available. In the Extensions to Chapter 12 we will describe some of these estimates and show how they might be used for practical policy purposes. For some theoretical purposes, however, compensated demand curves are a more appropriate concept because the ability to hold utility constant offers some advantages. Our discussion of ‘‘consumer surplus’’ later in this chapter offers one illustration of these advantages. EXAMPLE 5.3 Compensated Demand Functions In Example 3.1 we assumed that the utility function for hamburgers (y) and soft drinks (x) was given by utility U x, y ð ¼ Þ ¼ x0:5y0:5, (5:18) and in Example 4.1 we showed that we can calculate the Marshallian demand functions for such utility functions as px, py, I px, py, I x ð y ð 0:5I px 0:5I py : Þ ¼ Þ ¼ (5:19) In Example 4.4 we found that 2p0:5 E compensated demand functions as: px, py, U x p0:5 Þ ¼ ð y U. Thus, we can now use Shephard’s the expenditure function in this case is given by lemma to calculate the xc yc ð ð px
, py, U px, py, U @E @E ð ð px, py, U @px px, py, U @py Þ Þ ¼ ¼ Þ ¼ Þ ¼ p’ x 0:5 p0:5 y U p0:5 x p’ 0:5 y U: (5:20) Sometimes indirect utility, V, is used in these compensated demand functions rather than U, but this does not change the meaning of the expressions—these demand functions show how an individual reacts to changes in prices while holding utility constant. ¼ 2; Equations 5.19 predict x Although py did not enter into the uncompensated demand function for good x, it does enter into the compensated function: Increases in py shift the compensated demand curve for x outward. The two demand concepts agree at the assumed initial point px ¼ 8, and 1, py ¼ 1 at this point, as do Equations 5.20. For px > 1 or 4, y U ¼ px < 1, the demands differ under the two concepts, however. If, say, px ¼ 4, then the 2, 1, y uncompensated functions predict x ¼ y 2. The reduction in x resulting from the increase in its price is smaller with the compensated demand function than it is with the uncompensated function because the former concept adjusts for the negative effect on purchasing power that comes about from the price increase. 1, whereas the compensated functions predict x 4, I ¼ ¼ ¼ ¼ ¼ This example makes clear the different ceteris paribus assumptions inherent in the two demand concepts. With uncompensated demand, expenditures are held constant at I 2 and so the increase in px from 1 to 4 results in a loss of utility; in this case, utility decreases from 2 to 1. In the compensated demand case, utility is held constant at U 2. To keep utility constant, expenditures must increase to E 16 to offset the effects of the price increase. ¼ QUERY: Are the compensated demand functions given in Equations 5.20 homogeneous of degree 0 in px and py if utility is held constant? Would you expect that to be true for all compensated demand functions? 4(2) 4(2) ¼ þ ¼ ¼ 160 Part 2: Choice and Demand A Mathematical Development of Response to Price Changes Up to this point we have largely
relied on graphical devices to describe how individuals respond to price changes. Additional insights are provided by a more mathematical approach. Our basic goal is to examine the partial derivative @x/@px—that is, how a change in the price of a good affects its purchase, ceteris paribus for the usual Marshallian demand curve. In the next chapter, we take up the question of how changes in the price of one commodity affect purchases of another commodity. Direct approach Our goal is to use the utility-maximization model to learn something about how the demand for good x changes when px changes; that is, we wish to calculate @x/@px. The direct approach to this problem makes use of the ﬁrst-order conditions for utility maximization. Differentiation of these n 1 equations, which eventually can be solved for the derivative we seek.4 Unfortunately, obtaining this solution is cumbersome and the steps required yield little in the way of economic insights. Hence we will instead adopt an indirect approach that relies on the concept of duality. In the end, both approaches yield the same conclusion, but the indirect approach is much richer in terms of the economics it contains. 1 equations yields a new system of n þ þ Indirect approach To begin our indirect approach,5 we will assume (as before) there are only two goods (x and y) and focus on the compensated demand function, xc(px, py, U ), and its relationship to the ordinary demand function, x(px, py, I). By deﬁnition we know that xc ð px, py, U x ½ Þ ¼ px, py, E px, py, U : Þ* ð (5:21) This conclusion was already introduced in connection with Figure 5.7, which showed that the quantity demanded is identical for the compensated and uncompensated demand functions when income is exactly what is needed to attain the required utility level. Equation 5.21 is obtained by inserting that expenditure level into the demand function, x(px, py, I). Now we can proceed by partially differentiating Equation 5.21 with respect to px and recognizing that this variable enters into the ordinary demand function in two places. Hence and rearranging terms yields @xc @px ¼ @x @px þ @x @E & @E @px, @x @px ¼ @xc @px ’ @x @E & @E @px
: (5:22) (5:23) 4See, for example, Paul A. Samuelson, Foundations of Economic Analysis (Cambridge, MA: Harvard University Press, 1947), pp. 101–3. 5The following proof was ﬁrst made popular by Phillip J. Cook in ‘‘A ‘One Line’ Proof of the Slutsky Equation,’’ American Economic Review 62 (March 1972): 139. Chapter 5: Income and Substitution Effects 161 The substitution effect Consequently, the derivative we seek has two terms. Interpretation of the ﬁrst term is straightforward: It is the slope of the compensated demand curve. But that slope represents movement along a single indifference curve; it is, in fact, what we called the substitution effect earlier. The ﬁrst term on the right of Equation 5.23 is a mathematical representation of that effect. The income effect The second term in Equation 5.23 reﬂects the way in which changes in px affect the demand for x through changes in necessary expenditure levels (i.e., changes in purchasing power). Therefore, this term reﬂects the income effect. The negative sign in Equation 5.23 shows the direction of the effect. For example, an increase in px increases the expenditure level that would have been needed to keep utility constant (mathematically, @E/@px > 0). But because nominal income is held constant in Marshallian demand, these extra expenditures are not available. Hence x (and y) must be reduced to meet this shortfall. The extent of the reduction in x is given by @x/@E. On the other hand, if px decreases, the expenditure level required to attain a given utility also decreases. The decline in x that would normally accompany such a decrease in expenditures is precisely the amount that must be added back through the income effect. Notice that in this case the income effect works to increase the amount of x. The Slutsky equation The relationships embodied in Equation 5.23 were ﬁrst discovered by the Russian economist Eugen Slutsky in the late nineteenth century. A slight change in notation is required to state the result the way Slutsky did. First, we write the substitution effect as substitution effect @xc @px ¼ ¼ constant U ¼ (5:24) @x @px# # # # to indicate movement along a
single indifference curve. For the income effect, we have income effect @x @E & @E @px ¼ ’ @x @I & @E @px, ¼ ’ (5:25) because changes in income or expenditures amount to the same thing in the function x(px, py, I). The second term of the income effect can be interpreted using Shephard’s lemma. xc. Consequently, the entire income effect is given by That is, @E/@px ¼ income effect xc @x @I : ¼ ’ (5:26) Final form of the Slutsky equation Bringing together Equations 5.24–5.26 allows us to assemble the Slutsky equation in the form in which it was originally derived: @x ð px, py, I @px Þ ¼ substitution effect income effect þ ¼ where we have made use of the fact that x(px, py, I) point. ¼ x @x @I (5:27) constant ’ U ¼ xc(px, py, V ) at the utility-maximizing @x @px # # # # 162 Part 2: Choice and Demand This equation allows a more deﬁnitive treatment of the direction and size of substitution and income effects than was possible with a graphic analysis. First, as we have just shown, the substitution effect (and the slope of the compensated demand curve) is always negative. This result derives both from the quasi-concavity of utility functions (a diminishing MRS) and from the concavity of the expenditure function. We will show the negativity of the substitution effect in a somewhat different way in the ﬁnal section of this chapter. The sign of the income effect (–x@x/@I) depends on the sign of @x/@I. If x is a normal good, then @x/@I is positive and the entire income effect, like the substitution effect, is negative. Thus, for normal goods, price and quantity always move in opposite directions. For example, a decrease in px increases real income, and because x is a normal good, purchases of x increase. Similarly, an increase in px reduces real income and so purchases of x decrease. Overall, then, as we described previously using a graphic analysis, substitution and income effects work in the same direction to yield a negatively sloped demand
curve. In the case of an inferior good, @x/@I < 0 and the two terms in Equation 5.27 have different signs. Hence the overall impact of a change in the price of a good is ambiguous—it all depends on the relative sizes of the effects. It is at least theoretically possible that, in the inferior good case, the second term could dominate the ﬁrst, leading to Giffen’s paradox (@x/@px > 0). EXAMPLE 5.4 A Slutsky Decomposition The decomposition of a price effect that was ﬁrst discovered by Slutsky can be nicely illustrated with the Cobb–Douglas example studied previously. In Example 5.3, we found that the Marshallian demand function for good x was px, py, I x ð Þ ¼ 0:5I px (5:28) and that the compensated demand function for this good was Þ ¼ Hence the total effect of a price change on Marshallian demand can be found by differentiating Equation 5.28: ð xc px, py, U p’ x 0:5 p0:5 y U: (5:29) @x ð px, py, I @px Þ ¼ ’ 0:5I p2 x : (5:30) We wish to show that this is the sum of the two effects that Slutsky identiﬁed. To derive the substitution effect we must ﬁrst differentiate the compensated demand function from Equation 5.29: substitution effect @xc ð px, py, U @px Þ ¼ ¼ ’ 0:5p’ x 1:5 pyU: Now in place of U we use indirect utility: V ð px, py, I substitution effect 0:5p’ x ¼ ’ Þ ¼ p0:5 1:5 y V 0:5Ip’ x 0:5 0:5 : p’ y 2 x I: 0:25p’ ¼ ’ (5:31) (5:32) Calculation of the income effect in this example is considerably easier. Applying the results from Equation 5.27, we have income effect x ¼ ’ @x @I ¼ ’ 0:5I px$ & % 0:5
px ¼ ’ 0:25I p2 x : (5:33) A comparison of Equation 5.30 with Equations 5.32 and 5.33 shows that we have indeed this demand function into substitution and income decomposed the price derivative of Chapter 5: Income and Substitution Effects 163 components. Interestingly, the substitution and income effects are of precisely the same size. This, as we will see in later examples, is one of the reasons that the Cobb–Douglas is a special case. 4 to x The well-worn numerical example we have been using also demonstrates this decomposition. When the price of x increases from $1 to $4, the (uncompensated) demand for x decreases from x 2. That 2 to x 4 to x ¼ decline of 50 percent is the substitution effect. The further 50 percent decrease from x ¼ 1 represents reactions to the decline in purchasing power incorporated in the Marshallian demand function. This income effect does not occur when the compensated demand notion is used. 1, but the compensated demand for x decreases only from x ¼ ¼ ¼ ¼ QUERY: In this example, the individual spends half of his or her income on good x and half on good y. How would the relative sizes of the substitution and income effects be altered if the exponents of the Cobb–Douglas utility function were not equal? Demand Elasticities Thus far in this chapter we have been examining how individuals respond to changes in prices and income by looking at the derivatives of the demand function. For many analytical questions this is a good way to proceed because calculus methods can be directly applied. However, as we pointed out in Chapter 2, focusing on derivatives has one major disadvantage for empirical work: The sizes of derivatives depend directly on how variables are measured. That can make comparisons among goods or across countries and time periods difﬁcult. For this reason, most empirical work in microeconomics uses some form of elasticity measure. In this section we introduce the three most common types of demand elasticities and explore some of the mathematical relations among them. Again, for simplicity we will look at a situation where the individual chooses between only two goods, although these ideas can be easily generalized. Marshallian demand elasticities Most of the commonly used demand elasticities are derived from the Marshallian demand function x(px, py, I). Speciﬁcally, the following deﬁnitions are used. Price elasticity of demand.
This measures the proportionate change in quantity demanded in response to a proportionate change in a good’s own price. Mathematically, ex, px Þ ð ex, px ¼ Dx=x Dpx=px ¼ Dx Dpx & px x ¼ @x px, py, I @px & & ’ px x : (5:34) 2. Income elasticity of demand (ex, I ). This measures the proportionate change in quantity demanded in response to a proportionate change in income. In mathematical terms, ex, I ¼ Dx=x DI=I ¼ Dx DI & ex, py Þ ð I x ¼ @x px, py, I @I & & ’ I x : (5:35) 3. Cross-price elasticity of demand. This measures the proportionate change in the quantity of x demanded in response to a proportionate change in the price of some other good (y): ex, py ¼ Dx=x Dpy=py ¼ Dx Dpy & py x ¼ @x px, py, I @py & & ’ py x : (5:36) 164 Part 2: Choice and Demand Notice that all these deﬁnitions use partial derivatives, which signiﬁes that all other determinants of demand are to be held constant when examining the impact of a speciﬁc variable. In the remainder of this section we will explore the own-price elasticity deﬁnition in some detail. Examining the cross-price elasticity of demand is the primary topic of Chapter 6. Price elasticity of demand The (own-) price elasticity of demand is probably the most important elasticity concept in all of microeconomics. Not only does it provide a convenient way of summarizing how people respond to price changes for a wide variety of economic goods, but it is also a central concept in the theory of how ﬁrms react to the demand curves facing them. As you probably already learned in earlier economics courses, a distinction is usually made between cases of elastic demand (where price affects quantity signiﬁcantly) and inelastic demand (where the effect of price is small). One mathematical complication in making these ideas precise is that the price elasticity of demand itself is negative6 because, except in the unlikely case of Giffen�
�s paradox, @x/@px is negative. The dividing line between 1, changes in x and px are of large and small responses is generally set at ’ the same proportionate size. That is, a 1 percent increase in price leads to a decrease of 1 percent in quantity demanded. In this case, demand is said to be ‘‘unit-elastic.’’ Alternatively, if ex, px < 1, then quantity changes are proportionately larger than price changes, and we say that demand is ‘‘elastic.’’ For example, if ex, px ¼ ’ 3, each 1 percent increase in price leads to a decrease of 3 percent in quantity demanded. Finally, if ex, px > 1, then demand is inelastic, and quantity changes are proportionately smaller than price 0:3, for example, means that a 1 percent increase in price changes. A value of ex, px ¼ ’ leads to a decrease in quantity demanded of 0.3 percent. In Chapter 12 we will see how aggregate data are used to estimate the typical individual’s price elasticity of demand for a good and how such estimates are used in a variety of questions in applied microeconomics. 1. If ex, px ¼ ’ ’ ’ Price elasticity and total spending The price elasticity of demand determines how a change in price, ceteris paribus, affects total spending on a good. The connection is most easily shown with calculus: @ ð px & x Þ @px ¼ px & @x @px þ x x ex, px þ ð 1 : Þ ¼ (5:37) Þ ð 1 ’ ’ 0 > ex, px > 1. If Thus, the sign of this derivative depends on whether ex, px is larger or smaller than demand is inelastic, the derivative is positive and price and total spending move in the same direction. Intuitively, if price does not affect quantity demanded very much, then quantity stays relatively constant as price changes and total spending reﬂects mainly those price movements. This is the case, for example, with the demand for most agricultural products. Weather-induced changes in price for speciﬁc crops usually cause total spending on those crops to move in the same direction. On the other hand, if, reactions to a price
change are so large that the effect on demand is elastic total spending is reversed: An increase in price causes total spending to decrease (because quantity decreases a lot), and a decrease in price causes total spending to increase (quan, total spending is contity increases signiﬁcantly). For the unit-elastic case stant no matter how price changes. ex, px ¼ ’ ex, px < ð 1 Þ ’ 1 Þ ð 6Sometimes economists use the absolute value of the price elasticity of demand in their discussions. Although this is mathemati1.2 may sometimes report the price elasticcally incorrect, such usage is common. For example, a study that ﬁnds that ex,px ¼ ’ ity of demand as ‘‘1.2.’’ We will not do so here, however. Chapter 5: Income and Substitution Effects 165 Compensated price elasticities Because some microeconomic analyses focus on the compensated demand function, it is also useful to deﬁne elasticities based on that concept. Such deﬁnitions follow directly from their Marshallian counterparts Let the compensated demand function be given by xc( px, py, U). Then we have the following deﬁnitions. 1. Compensated own-price elasticity of demand. This elasticity measures the proportionate compensated change in quantity demanded in response to a proportionate change in a good’s own price: exc, pxÞ ð exc, px ¼ Dxc=xc Dpx=px ¼ Dxc Dpx & px xc ¼ px xc : & ’ (5:38) @xc px, py, U @px & exc, px Þ 2. Compensated cross-price elasticity of demand. This measures the proportionate compensated change in quantity demanded in response to a proportionate change in the price of another good: ð exc, py ¼ Dxc=xc Dpy=py ¼ Dxc Dpy & py xc ¼ @xc px, py, U @py & & ’ py xc : (5:39) Whether these price elasticities differ much from their Marshallian counterparts depends on the importance of income effects in the overall demand for good x. The precise connection between the two can be shown by multiplying the Sluts
ky result from Equation 5.27 by the factor px/x: px x & @x @px ¼ ex, px ¼ px x & @xc @px ’ px x & x @x @I ¼ & exc, px ’ sxex, I, (5:40) pxx/I is the share of total income devoted to the purchase of good x. where sx ¼ Equation 5.40 shows that compensated and uncompensated own-price elasticities of demand will be similar if either of two conditions hold: (1) The share of income devoted to good x (sx) is small; or (2) the income elasticity of demand for good x(ex, I) is small. Either of these conditions serves to reduce the importance of the income compensation used in the construction of the compensated demand function. If good x is unimportant in a person’s budget, then the amount of income compensation required to offset a price change will be small. Even if a good has a large budget share, if demand does not react strongly to changes in income, then the results of either demand concept will be similar. Hence there will be many circumstances where one can use the two price elasticity concepts more or less interchangeably. Put another way, there are many economic circumstances in which substitution effects constitute the most important component of price responses. Relationships among demand elasticities There are a number of relationships among the elasticity concepts that have been developed in this section. All these are derived from the underlying model of utility maximization. Here we look at three such relationships that provide further insight on the nature of individual demand. Homogeneity. The homogeneity of demand functions can also be expressed in elasticity terms. Because any proportional increase in all prices and income leaves quantity 166 Part 2: Choice and Demand demanded unchanged, the net sum of all price elasticities together with the income elasticity for a particular good must sum to zero. A formal proof of this property relies on Euler’s theorem (see Chapter 2). Applying that theorem to the demand function x(px, py, I) and remembering that this function is homogeneous of degree 0 yields 0 px & ¼ @x @px þ py & @x @py þ @x @I & I & (5:41) If we divide Equation 5.41 by x then we obtain 0 ex,px þ ex,py þ as intuition suggests
. This result shows that the elasticities of demand for any good cannot follow a completely ﬂexible pattern. They must exhibit a sort of internal consistency that reﬂects the basic utility-maximizing approach on which the theory of demand is based. ex, I, (5:42) ¼ Engel aggregation. In the Extensions to Chapter 4 we discussed the empirical analysis of market shares and took special note of Engel’s law that the share of income devoted to food decreases as income increases. From an elasticity perspective, Engel’s law is a statement of the empirical regularity that the income elasticity of demand for food is generally found to be considerably less than 1. Because of this, it must be the case that the income elasticity of all nonfood items must be greater than 1. If an individual experiences an increase in his or her income, then we would expect food expenditures to increase by a smaller proportional amount; but the income must be spent somewhere. In the aggregate, these other expenditures must increase proportionally faster than income. A formal statement of this property of income elasticities can be derived by differentipyy) with respect to income while px x ating the individual’s budget constraint (I treating the prices as constants: ¼ þ 1 px & ¼ @x @I þ py & @y @I : A bit of algebraic manipulation of this expression yields 1 px & ¼ @x @I & xI xI þ py & @y @I & yI yI ¼ sxex, I þ syey, I; (5:43) (5:44) here, as before, si represents the share of income spent on good i. Equation 5.44 shows that the weighted average on income elasticities for all goods that a person buys must be 1. If we knew, say, that a person spent a quarter of his or her income on food and the income elasticity of demand for food were 0.5, then the income elasticity of demand for (1 – 0.25 Æ 0.5)/0.75]. Because food is an everything else must be approximately 1.17 [ important ‘‘necessity,’’ everything else is in some sense a ‘‘luxury.’’ ¼ Cournot aggregation. The eighteenth-century French economist Antoine Cournot provided one of the ﬁrst mathematical analyses
of price changes using calculus. His most important discovery was the concept of marginal revenue, a concept central to the proﬁt-maximization hypothesis for ﬁrms. Cournot was also concerned with how the change in a single price might affect the demand for all goods. Our ﬁnal relationship shows that there are indeed connections among all of the reactions to the change in a single price. We begin by differentiating the budget constraint again, this time with respect to px: @I @px ¼ 0 px & ¼ @x @px þ x py & þ @y @px : Chapter 5: Income and Substitution Effects 167 Multiplication of this equation by px/I yields @x px & @px & sxex, px þ 0 0 ¼ ¼ so the ﬁnal Cournot result is px I þ px I & sx þ x x x þ & syey, px, py & @y @px & px I & y y, sxex, px þ syey, px ¼ ’ sx: (5:45) (5:46) This equation shows that the size of the cross-price effect of a change in the price of x on the quantity of y consumed is restricted because of the budget constraint. Direct, ownprice effects cannot be totally overwhelmed by cross-price effects. This is the ﬁrst of many connections among the demands for goods that we will study more intensively in the next chapter. Generalizations. Although we have shown these aggregation results only for the case of two goods, they are easily generalized to the case of many goods. You are asked to do just that in Problem 5.11. A more difﬁcult issue is whether these results should be expected to hold for typical economic data in which the demands of many people are combined. Often economists treat aggregate demand relationships as describing the behavior of a ‘‘typical person,’’ and these relationships should in fact hold for such a person. But the situation may not be that simple, as we will show when discussing aggregation later in the book. EXAMPLE 5.5 Demand Elasticities: The Importance of Substitution Effects In this example we calculate the demand elasticities implied by three of the utility functions we have been using. Although the possibilities incorporated in these functions are too simple to reﬂect how
this utility ¼ ¼ ¼ function are px, py, I x ð Þ ¼ 1 pxð I þ I, pxp’ 1 y Þ : y px, py, I 1 pyð As you might imagine, calculating elasticities directly from these functions can take some time. Here we focus only on the own-price elasticity and make use of the result (from Problem 5.6) that the ‘‘share elasticity’’ of any good is given by 1 x pyÞ p’ Þ ¼ þ ð In this case, esx, px ¼ @sx @px & px sx ¼ 1 þ ex, px : pxx I ¼ sx ¼ 1 1 pxp’ y, 1 þ so the share elasticity is more easily calculated and is given by (5:49) esx, px ¼ @sx @px & px sx ¼ 1 p’ y ’ pxp’ 2 & px 1 pxp’ ’ y Þ 1 ¼ 1 pxp’ y pxp Because the units in which goods are measured are rather arbitrary in utility theory, we might as well deﬁne them so that initially px ¼ ex, px ¼ py, in which case7 we get esx, px ’ 1 1 ’ (5:51) ¼ ’ 1:55:50) ’ 1 þ Hence demand is more elastic in this case than in the Cobb–Douglas example. The reason for this is that the substitution effect is larger for this version of the CES utility function. This can be shown by again applying the Slutsky equation (and using the facts that ex,I ¼ ex, px þ Þ ¼ ’ which is twice the size of the substitution effect for the Cobb–Douglas. sxex, I ¼ ’ 1 and sx ¼ exc, px ¼ (5:52) 0.5): 1 ð 0:5 1:5 þ 1, Case 3: CES (s 0.5; d ¼ 1). U(x, y) 1 – x’ y’ 1. ¼ ’ ¼ ’ Referring back to Example 4.2, we can see that the share of good x implied by
this utility function is given by 7Notice that this substitution must be made after differentiation because the deﬁnition of elasticity requires that we change only px while holding py constant. (5:53) (5:54) (5:55) Chapter 5: Income and Substitution Effects 169 so the share elasticity is given by sx ¼ 1 þ 1 p0:5 y p’ x 0:5, esx, px ¼ @sx @px & px sx ¼ 1:5 0:5p0:5 y p’ x p0:5 y p’ x px p0:5 y p’ x 2 & Þ If we again adopt the simpliﬁcation of equal prices, we can compute the own-price elasticity as 1 ¼ ’ 1 ð þ þ þ 0:5 0:5 0:5 1 ð Þ 0:5 0:5p0:5 y p’ x p0:5 y p’ 1 x : ex, px ¼ and the compensated price elasticity as ex, px þ exc, px ¼ esx, px ’ 0:5 2 ’ 1 1 ¼ 0:75 ¼ ’ sxex, I ¼ ’ 0:75 0:5 1 ð þ 0:25: Þ ¼ ’ Thus, for this version of the CES utility function, the own-price elasticity is smaller than in Case 1 and Case 2 because the substitution effect is smaller. Hence the main variation among the cases is indeed caused by differences in the size of the substitution effect. If you never want to work out this kind of elasticity again, it may be helpful to make use of the general result that 1 exc, px ¼ ’ð You may wish to check out that this formula works in these three examples (with sx ¼ 0.5 and 1, 2, 0.5, respectively), and Problem 5.9 asks you to show that this result is generally true. s Because all these cases based on the CES utility function have a unitary income elasticity, the own-price elasticity can be computed from the compensated price elasticity by simply adding –sx to the ﬁgure computed in Equation 5.56. sxÞ (5:56) r: ’ ¼
QUERY: Why is it that the budget share for goods other than x (i.e., 1 compensated own-price elasticities in this example? ’ sx) enters into the Consumer Surplus An important problem in applied welfare economics is to devise a monetary measure of the utility gains and losses that individuals experience when prices change. One use for such a measure is to place a dollar value on the welfare loss that people experience when a market is monopolized with prices exceeding marginal costs. Another application concerns measuring the welfare gains that people experience when technical progress reduces the prices they pay for goods. Related applications occur in environmental economics (measuring the welfare costs of incorrectly priced resources), law and economics (evaluating the welfare costs of excess protections taken in fear of lawsuits), and public economics (measuring the excess burden of a tax). To make such calculations, economists use empirical data from studies of market demand in combination with the theory that underlies that demand. In this section we will examine the primary tools used in that process. Consumer welfare and the expenditure function The expenditure function provides the ﬁrst component for the study of the price/welfare connection. Suppose that we wished to measure the change in welfare that an individual experiences if the price of good x increases from p0 x. Initially this person requires p0 expenditures of E to reach a utility of U0. To achieve the same utility once x, py, U0Þ x to p1 ð 170 Part 2: Choice and Demand p1. x, py, U0Þ the price of x increases, he or she would require spending of at least E Therefore, to compensate for the price increase, this person would require a compensation (formally called a compensating variation8 or CV) of ð CV E p1 x, py, U 0Þ ’ E p0 : x, py, U 0Þ ð ð ¼ (5:57) This situation is shown graphically in the top panel of Figure 5.8. This ﬁgure shows the quantity of the good whose price has changed on the horizontal axis and spending on all other goods (in dollars) on the vertical axis. Initially, this person consumes the combination x0, y0 and obtains utility of U0. When the price of x increases, he or she would be forced to move to combination x2, y2 and suffer a loss in utility. If he or she were compensated with extra purchasing power of amount
CV, he or she could afford to remain on the U0 indifference curve despite the price increase by choosing combination x1, y1. The distance CV, therefore, provides a monetary measure of how much this person needs to be compensated for the price increase. Using the compensated demand curve to show CV Unfortunately, individuals’ utility functions and their associated indifference curve maps are not directly observable. But we can make some headway on empirical measurement by determining how the CV amount can be shown on the compensated demand curve in the bottom panel of Figure 5.8. Shephard’s lemma shows that the compensated demand function for a good can be found directly from the expenditure function by differentiation: xc ð px, py, U Þ ¼ @E ð px, py, U @px Þ : (5:58) Hence the compensation described in Equation 5.57 can be found by integrating across a sequence of small increments to price from p0 x to p1 x: p1 x CV ¼ p1 x ð p0 x @E ð px, py, U0Þ @px dpx ¼ ð p0 x xc px, py, U0Þ ð dpx (5:59) while holding py and utility constant. The integral deﬁned in Equation 5.59 has a geometric interpretation, which is shown in the lower panel of Figure 5.8: It is the shaded area to the left of the compensated demand curve and bounded by p0 x. Thus, the welfare cost of this price increase can also be illustrated using changes in the area below the compensated demand curve. x and p1 The consumer surplus concept There is another way to look at this issue. We can ask how much this person would be willing to pay for the right to consume all this good that he or she wanted at the market price of p0 x rather than doing without the good completely. The compensated demand curve in the bottom panel of Figure 5.8 shows that if the price of x increased to p2 x, this person’s consumption would decrease to zero, and he or she would require an amount of compensation equal to area p2 x to accept the change voluntarily. Therefore, the right xAp0 8Some authors deﬁne the compensating variation as the amount of income that must be given to this person to permit him or. This expresher to increase utility from U1
to U0 given the new price of good x, that is, CV. Some authors also sion is equivalent to the one given in Equation 5.57 because, by assumption, E look at CV from the point of view of the budget of a ‘‘social planner’’ who must make these compensations rather than from the point of view of the consumer who receives them. In that case, the CV illustrated would be negative. p1 p1 E x, py, U0Þ ’ x, py, U1Þ ð ¼ ð p1 p0 x, py, U1Þ E x, py, U0Þ ¼ E ð ð Chapter 5: Income and Substitution Effects 171 FIGURE 5.8 Showing Compensating Variation If the price of x increases from p0 indifference curve. Integration shows that CV can also be represented by the shaded area below the compensated demand curve in panel (b). x, this person needs extra expenditures of CV to remain on the U0 x to p1 Spending on other goods ($) E(px 1,...,U0) CV E( px 0,...,U0) E(px 1,...,U0) y1 y2 y0 U0 E(px 0,...,U0) U1 x2 x1 x0 (a) Indifference curve map Quantity of x Price 2 px 1 px 0 px B A xc( px,...,U0) x1 x0 (b) Compensated demand curve Quantity of x 172 Part 2: Choice and Demand FIGURE 5.9 Welfare Effects of Price Changes and the Marshallian Demand Curve The usual Marshallian (nominal income constant) demand curve for good x is x( px,…). Further, xc(…, U0) and xc(…, U1) denote the compensated demand curves associated with the utility levels experienced when p0 x is bounded by the similar areas to the left of the compensated demand curves. Hence for small changes in price, the area to the left of the Marshallian demand curve is a good measure of welfare loss. x, respectively, prevail. The area to the left of x( px, …) between p0 x and p1 x and p1 px 1 px 0 px C B A D x(
px,... ) xc(...,U0) xc(...,U1) x1 x0 Quantity of x per period to consume x0 at a price of p0 x is worth this amount to this individual. It is the extra beneﬁt that this person receives by being able to make market transactions at the prevailing market price. This value, given by the area below the compensated demand curve and above the market price, is termed consumer surplus. Looked at in this way, the welfare problem caused by an increase in the price of x can be described as a loss in consumer surplus. When the price increases from p0 x, the consumer surplus ‘‘triangle’’ decreases in size from p2 x to p2 x. As the ﬁgure makes clear, that is simply another way of describing the welfare loss represented in Equation 5.59. x to p1 xAp0 xBp1 Welfare changes and the Marshallian demand curve Thus far our analysis of the welfare effects of price changes has focused on the compensated demand curve. This is in some ways unfortunate because most empirical work on demand actually estimates ordinary (Marshallian) demand curves. In this section we will show that studying changes in the area below a Marshallian demand curve may in fact be a good way to measure welfare losses. Consider the Marshallian demand curve x( px, …) illustrated in Figure 5.9. Initially this consumer faces the price p0 x and chooses to consume x0. This consumption yields a utility level of U0, and the initial compensated demand curve for x [i.e., xc(px, py, U0)] also passes through the point x0, p0 x (which we have labeled point A). When price increases to p1 x, the Marshallian demand for good x decreases to x1 (point C on the demand curve), and this person’s utility also decreases to, say, U1. There is another compensated demand curve associated with this lower level of utility, and it also is shown in Figure 5.9. Both Chapter 5: Income and Substitution Effects 173 the Marshallian demand curve and this new compensated demand curve pass through point C. The presence of a second compensated demand curve in Figure 5.9 raises an intriguing conceptual question. Should we measure the welfare loss from the price increase as we did in Figure 5.8 using the compensating variation (
CV) associated with the initial comarea p1, or should we perhaps use this new compensated pensated demand curve ð demand curve and measure the welfare loss as area p1 x? A potential rationale for using the area under the second curve would be to focus on the individual’s situation after the price increase (with utility level U1). We might ask how much he or she would now be willing to pay to see the price return to its old, lower levels.9 The answer to this would be given by area p1 x. Therefore, the choice between which compensated demand curve to use boils down to choosing which level of utility one regards as the appropriate target for the analysis. xBAp0 xÞ xCDp0 xCDp0 Luckily, the Marshallian demand curve provides a convenient compromise between these two measures. Because the size of the area between the two prices and below the Marshallian curve is smaller than that below the compensated demand curve based on U0 but larger than that below the curve based on U1, it does seem an attractive middle ground. Hence this is the measure of welfare losses we will primarily use throughout this book. area p1 ð xCAp0 xÞ Consumer surplus. Consumer surplus is the area below the Marshallian demand curve and above market price. It shows what an individual would pay for the right to make voluntary transactions at this price. Changes in consumer surplus can be used to measure the welfare effects of price changes. We should point out that some economists use either CV or EV to compute the welfare effects of price changes. Indeed, economists are often not clear about which measure of welfare change they are using. Our discussion in the previous section shows that if income effects are small, it really does not make much difference in any case. EXAMPLE 5.6 Welfare Loss from a Price Increase These ideas can be illustrated numerically by returning to our old hamburger/soft drink example. Let’s look at the welfare consequences of an unconscionable price increase for soft drinks (good x) from $1 to $4. In Example 5.3, we found that the compensated demand for good x was given by xc ð px, py, V Þ ¼ Vp0:5 y p0:5 x : Hence the welfare cost of the price increase is given by CV ¼ 4 ð 1 xc ð px, py, V dpx ¼ Þ 4 ð 1 Vp0
:5 y p’ x 0:5 dpx ¼ 2Vp0:5 y p0:5 x (5:60) (5:61) 4 px¼ : 1 px¼ # # # # 9This alternative measure is termed the equivalent variation (EV). More formally, EV. Again, some authors use a different deﬁnition of EV as being the income necessary to restore utility given the old prices, that is, EV, these deﬁnitions are equivalent. p1 x, py, U1Þ ’ p0 x, py, U1Þ. But because E ¼ E E E E E ð ð p0 x, py, U0Þ ’ ð p0 x, py, U1Þ ð ¼ p0 x, py, U0Þ ¼ ð p1 x, py, U1Þ ð 174 Part 2: Choice and Demand If we use the values we have been assuming throughout this gastronomic feast (V then 2, py ¼ ¼ 4), CV 2 2 2 0::5 Þ 8: ’ This ﬁgure would be cut in half (to 4) if we believed that the utility level after the price increase (V 1) were the more appropriate utility target for measuring compensation. If instead we had used the Marshallian demand function ¼ ¼ ¼ Þ & & & & (5:62) the loss would be calculated as px, py, I x ð Þ ¼ 1 0:5Ip’ x, loss ¼ 4 ð 1 x ð px, py, I dpx ¼ Þ 4 ð 1 Thus, with I ¼ 8, this loss is 0:5Ip’ 1 x dpx ¼ 0:5I ln px 4 1 : # # # # loss 4 4 ln ð Þ ’ ¼ 4 ln 1 ð Þ ¼ 4 4 ln ð Þ ¼ 4 1:39 ð Þ ¼ 5:55, (5:63) (5:64) (5:65) which seems a reasonable compromise between the two alternative measures based on the compensated demand functions. QUERY: In this problem, none of the demand curves has a ﬁnite price at which
demand goes to precisely zero. How does this affect the computation of total consumer surplus? Does this affect the types of welfare calculations made here? Revealed Preference and The Substitution Effect The principal unambiguous prediction that can be derived from the utility-maximation model is that the slope (or price elasticity) of the compensated demand curve is negative. We have shown this result in two ways. The ﬁrst proof was based on the quasi-concavity of utility functions, that is, because any indifference curve must exhibit a diminishing MRS, any change in a price will induce a quantity change in the opposite direction when moving along that indifference curve. A second proof derives from Shephard’s lemma— because the expenditure function is concave in prices, the compensated demand function (which is the derivative of the expenditure function) must have a negative slope. Again utility is held constant in this calculation as one argument in the expenditure function. To some economists, the reliance on a hypothesis about an unobservable utility function represented a weak foundation on which to base a theory of demand. An alternative approach, which leads to the same result, was ﬁrst proposed by Paul Samuelson in the late 1940s.10 This approach, which Samuelson termed the theory of revealed preference, deﬁnes a principle of rationality that is based on observed behavior and then uses this principle to approximate an individual’s utility function. In this sense, a person who follows Samuelson’s principle of rationality behaves as if he or she were maximizing a proper utility function and exhibits a negative substitution effect. Because Samuelson’s approach provides additional insights into our model of consumer choice, we will brieﬂy examine it here. 10Paul A. Samuelson, Foundations of Economic Analysis (Cambridge, MA: Harvard University Press, 1947). FIGURE 5.10 Demonstration of the Principle of Rationality in the Theory of Revealed Preference Chapter 5: Income and Substitution Effects 175 With income I1 the individual can afford both points A and B. If A is selected, then A is revealed preferred to B. It would be irrational for B to be revealed preferred to A in some other price-income conﬁguration. Quantity of y ya yb C A B I2 I1 I3 xa xb Quantity of x Graphical approach The principle of rationality in the theory of revealed preference is as follows:
Consider two bundles of goods, A and B. If, at some prices and income level, the individual can afford both A and B but chooses A, we say that A has been ‘‘revealed preferred’’ to B. The principle of rationality states that under any different price–income arrangement, B can never be revealed preferred to A. If B is in fact chosen at another price–income conﬁguration, it must be because the individual could not afford A. The principle is illustrated in Figure 5.10. Suppose that, when the budget constraint is given by I1, point A is chosen even though B also could have been purchased. Then A has been revealed preferred to B. If, for some other budget constraint, B is in fact chosen, then it must be a case such as that represented by I2, where A could not have been bought. If B were chosen when the budget constraint is I3, this would be a violation of the principle of rationality because, with I3, both A and B can be bought. With budget constraint I3, it is likely that some point other than either A or B (say, C) will be bought. Notice how this principle uses observable reactions to alternative budget constraints to rank commodities rather than assuming the existence of a utility function itself. Also notice how the principle offers a glimpse of why indifference curves are convex. Now we turn to a formal proof. Negativity of the substitution effect Suppose that an individual is indifferent between two bundles, C (composed of xC and yC) and D (composed of xD and yD). Let pC x, pC y be the prices at which bundle C is chosen and pD y the prices at which bundle D is chosen. x, pD 176 Part 2: Choice and Demand Because the individual is indifferent between C and D, it must be the case that when C was chosen, D cost at least as much as C: pC y yC + A similar statement holds when D is chosen: pC x xC þ pC x xD þ pC y yD: pD x xD þ pD y yD + pD x xC þ pD y yC: Rewriting these equations gives pC x ð pD x ð xC ’ xD ’ xDÞ þ xCÞ þ pC y ð pD y ð yC ’
yD ’ yDÞ + yCÞ + 0, 0: Adding these together yields (5:66) (5:67) (5:68) (5:69) (5:70) xC ’ Now suppose that only the price of x changes; assume that pC xDÞ þ ð yC ’ ð pC x ’ pD x Þð pC y ’ pD y Þð 0: yDÞ + pD y. Then y ¼ xC ’ But Equation 5.71 says that price and quantity move in the opposite direction when utility is held constant (remember, bundles C and D are equally attractive). This is precisely a statement about the nonpositive nature of the substitution effect: xDÞ + (5:71) 0: ð pC x ’ pD x Þð @xc @x @px# # # We have arrived at the result by an approach that does not require the existence of a # quasi-concave utility function. constant + ¼ px, py, V @px (5:72) ¼ 0: Þ ð U SUMMARY In this chapter, we used the utility-maximization model to study how the quantity of a good that an individual chooses responds to changes in income or to changes in that good’s price. The ﬁnal result of this examination is the derivation of the familiar downward-sloping demand curve. In arriving at that result, however, we have drawn a wide variety of insights from the general economic theory of choice. • Proportional changes in all prices and income do not shift the individual’s budget constraint and therefore do not change the quantities of goods chosen. In formal terms, demand functions are homogeneous of degree 0 in all prices and income. • When purchasing power changes (i.e., when income increases with prices remaining unchanged), budget constraints shift and individuals will choose new commodity bundles. For normal goods, an increase in purchasing power causes more to be chosen. In the case of inferior goods, however, an increase in purchasing power causes less to be purchased. Hence the sign of @xi/@I could be either positive or negative, although @xi/@I 0 is the most common case. ( • A decrease in the price of a good causes substitution and income effects that, for a normal good, cause more of
the good to be purchased. For inferior goods, however, substitution and income effects work in opposite directions, and no unambiguous prediction is possible. • Similarly, an increase in price induces both substitution and income effects that, in the normal case, cause less to be demanded. For inferior goods the net result is again ambiguous. • Marshallian demand curves represent two-dimensional depictions of demand functions for which only the ownprice varies—other prices and income are held constant. Changes in these other variables will usually shift the position of the demand curve. The sign of the slope of the Marshallian demand curve is theoretically ambiguous because substitution and income effects may work in opposite directions. The Slutsky equation permits a formal study of this ambiguity. px, py, I @px * ) @x ð Þ Chapter 5: Income and Substitution Effects 177 deﬁned for movements along the compensated demand curve. • There are many relationships among demand elasticities. Some of the more important ones are (1) ownprice elasticities determine how a price change affects total spending on a good; (2) substitution and income effects can be summarized by the Slutsky equation in elasticity form; and (3) various aggregation relations hold among elasticities—these show how the demands for different goods are related. • Welfare effects of price changes can be measured by changing areas below either compensated or Marshallian demand curves. Such changes affect the size of the consumer surplus that individuals receive from being able to make market transactions. • The negativity of the substitution effect is the most basic conclusion from demand theory. This result can be shown using revealed preference theory and so does not require assuming the existence of a utility function. • Compensated (or Hicksian) demand functions show how quantities demanded are functions of all prices and utility. The compensated demand function for a good can be generated by partially differentiating the expenditure function with respect to that good’s price (Shephard’s lemma). • Compensated (or Hicksian) demand curves represent two-dimensional depictions of compensated demand functions for which only the own-price varies—other prices and utility are held constant. The sign of the slope of the compensated demand curve @xc px, py, U @px Þ ð ) * is unambiguously negative because of the quasiconcavity of utility functions or the related concavity of the expenditure function. • Demand elasticities are often used in empirical work to summarize how individuals react to changes
in prices and income. The most important such elasticity is the (own-) price elasticity of demand, ex, px. This measures the proportionate change in quantity in response to a 1 percent change in price. A similar elasticity can be PROBLEMS 5.1 Thirsty Ed drinks only pure spring water, but he can purchase it in two different-sized containers: 0.75 liter and 2 liter. Because the water itself is identical, he regards these two ‘‘goods’’ as perfect substitutes. a. Assuming Ed’s utility depends only on the quantity of water consumed and that the containers themselves yield no utility, express this utility function in terms of quantities of 0.75-liter containers (x) and 2-liter containers (y). b. State Ed’s demand function for x in terms of px, py, and I. c. Graph the demand curve for x, holding I and py constant. d. How do changes in I and py shift the demand curve for x? e. What would the compensated demand curve for x look like in this situation? 5.2 David N. gets $3 per week as an allowance to spend any way he pleases. Because he likes only peanut butter and jelly sandwiches, he spends the entire amount on peanut butter (at $0.05 per ounce) and jelly (at $0.10 per ounce). Bread is provided free of charge by a concerned neighbor. David is a particular eater and makes his sandwiches with exactly 1 ounce of jelly and 2 ounces of peanut butter. He is set in his ways and will never change these proportions. a. How much peanut butter and jelly will David buy with his $3 allowance in a week? b. Suppose the price of jelly were to increase to $0.15 an ounce. How much of each commodity would be bought? c. By how much should David’s allowance be increased to compensate for the increase in the price of jelly in part (b)? d. Graph your results in parts (a) to (c). e. In what sense does this problem involve only a single commodity, peanut butter and jelly sandwiches? Graph the demand curve for this single commodity. f. Discuss the results of this problem in terms of the income and substitution effects involved in the demand for jelly. 5.3 As deﬁned in Chapter 3, a utility function is homothetic if any straight line through the origin cuts all indifference curves
at points of equal slope: The MRS depends on the ratio y/x. 178 Part 2: Choice and Demand a. Prove that, in this case, @x/@I is constant. b. Prove that if an individual’s tastes can be represented by a homothetic indifference map then price and quantity must move in opposite directions; that is, prove that Giffen’s paradox cannot occur. 5.4 As in Example 5.1, assume that utility is given by utility U ð ¼ x, y Þ ¼ x0:3y0:7: a. Use the uncompensated demand functions given in Example 5.1 to compute the indirect utility function and the expendi- ture function for this case. b. Use the expenditure function calculated in part (a) together with Shephard’s lemma to compute the compensated demand function for good x. c. Use the results from part (b) together with the uncompensated demand function for good x to show that the Slutsky equa- tion holds for this case. 5.5 Suppose the utility function for goods x and y is given by utility U x, y ð ¼ Þ ¼ xy þ y: a. Calculate the uncompensated (Marshallian) demand functions for x and y, and describe how the demand curves for x and y are shifted by changes in I or the price of the other good. b. Calculate the expenditure function for x and y. c. Use the expenditure function calculated in part (b) to compute the compensated demand functions for goods x and y. Describe how the compensated demand curves for x and y are shifted by changes in income or by changes in the price of the other good. 5.6 Over a three-year period, an individual exhibits the following consumption behavior: Year 1 Year 2 Year 3 px 3 4 5 py Is this behavior consistent with the axioms of revealed preference? 5.7 Suppose that a person regards ham and cheese as pure complements—he or she will always use one slice of ham in combination with one slice of cheese to make a ham and cheese sandwich. Suppose also that ham and cheese are the only goods that this person buys and that bread is free. a. If the price of ham is equal to the price of cheese, show that the own-price elasticity of demand for ham is the cross-price elasticity of demand for
ham with respect to the price of cheese is also 0.5. 0.5 and that ’ b. Explain why the results from part (a) reﬂect only income effects, not substitution effects. What are the compensated price elasticities in this problem? c. Use the results from part (b) to show how your answers to part (a) would change if a slice of ham cost twice the price of a slice of cheese. d. Explain how this problem could be solved intuitively by assuming this person consumes only one good—a ham and cheese sandwich. 5.8 Show that the share of income spent on a good x is sx ¼ d ln E d ln px, where E is total expenditure. ’ Chapter 5: Income and Substitution Effects 179 Analytical Problems 5.9 Share elasticities In the Extensions to Chapter 4 we showed that most empirical work in demand theory focuses on income shares. For any good, x, the income share is deﬁned as sx ¼ pxx/I. In this problem we show that most demand elasticities can be derived from corresponding share elasticities. ex, px þ ex a. Show that the elasticity of a good’s budget share with respect to income this conclusion with a few numerical examples. esx, I ¼ ð @sx=@I b. Show that the elasticity of a good’s budget share with respect to its own price 1. Again, interpret this ﬁnding with a few numerical examples. I=sxÞ & esx, px ¼ ð is equal to ex, I – 1. Interpret @sx=@px & px=sxÞ is equal to c. Use your results from part (b) to show that the ‘‘expenditure elasticity’’ of good x with respect to its own price & ½ * x @ 1=x =@px & Þ py=sxÞ is also equal to ex, px þ px & px, px ¼ ð the elasticity of a good’s budget share with respect d. Show that @sx=@py & esx, py ¼ ð pk 1= by sx ¼ yp’ 1, where k ð Hint: This problem can be simpliﬁed by assuming px
� þ a. Calculate the income effect for each good. Also calculate the income elasticity of demand for each good. b. Calculate the substitution effect for each good. Also calculate the compensated own-price elasticity of demand for each good. c. Show that the Slutsky equation applies to this function. d. Show that the elasticity form of the Slutsky equation also applies to this function. Describe any special features you observe. 180 Part 2: Choice and Demand 5.13 The almost ideal demand system The general form for the expenditure function of the almost ideal demand system (AIDS) is given by ln E ð p1,..., pn, U a0 þ Þ ¼ n 1 i X ¼ ai ln pi gij ln pi ln pj þ Ub0 pbk k, k 1 i Y ¼ For analytical ease, assume that the following restrictions apply: gij ¼ gji, n 1 i X ¼ ai ¼ 1, and n n gij ¼ 1 j X ¼ 1 k X ¼ bk ¼ 0: a. Derive the AIDS functional form for a two-goods case. b. Given the previous restrictions, show that this expenditure function is homogeneous of degree 1 in all prices. This, along with the fact that this function resembles closely the actual data, makes it an ‘‘ideal’’ function. c. Using the fact that sx ¼ d ln E d ln px (see Problem 5.8), calculate the income share of each of the two goods. 5.14 Price indifference curves Price indifference curves are iso-utility curves with the prices of two goods on the X- and Y-axes, respectively. Thus, they have the following general form: ( p1, p2)\| v( p1, p2, I) v0. ¼ a. Derive the formula for the price indifference curves for the Cobb–Douglas case with a b. What does the slope of the curve show? c. What is the direction of increasing utility in your graph? b ¼ ¼ 0.5. Sketch one of them. SUGGESTIONS FOR FURTHER READING Cook, P. J. ‘‘A ‘One Line’ Proof of the Slutsky Equation.’’ American Economic Review 62 (March
1972): 139. Samuelson, Paul A. Foundations of Economic Analysis. Cambridge, MA: Harvard University Press, 1947, chapter 5. Clever use of duality to derive the Slutsky equation; uses the in Chapter 5 but with rather complex same method as notation. Fisher, F. M., and K. Shell. The Economic Theory of Price Indices. New York: Academic Press, 1972. Complete, technical discussion of the economic properties of various price indexes; describes ‘‘ideal’’ indexes based on utilitymaximizing models in detail. Luenberger, D. G. Microeconomic Theory. New York: McGraw Hill, 1992. Pages 147–151 provide a concise summary of how to state the Slutsky equations in matrix notation. Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green. Microeconomic Theory. New York: Oxford University Press, 1995. Chapter 3 covers much of the material in this chapter at a somewhat higher level. Section I on measurement of the welfare effects of price changes is especially recommended. Provides a complete analysis of substitution and income effects. Also develops the revealed preference notion. Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGraw-Hill, 2001. Provides an extensive derivation of the Slutsky equation and a lengthy presentation of elasticity concepts. Sydsaetter, K., A. Strom, and P. Berck. Economist’s Mathematical Manual. Berlin, Germany: Springer-Verlag, 2003. Provides a compact summary of elasticity concepts. The coverage of elasticity of substitution notions is especially complete. Varian, H. Microeconomic Analysis, 3rd ed. New York: W. W. Norton, 1992. Formal development of preference notions. Extensive use of expenditure functions and their relationship to the Slutsky equation. Also contains a nice proof of Roy’s identity. DEMAND CONCEPTS AND THE EVALUATION OF PRICE INDICES EXTENSIONS In Chapters 4 and 5 we introduced a number of related demand concepts, all of which were derived from the underlying model of utility maximization. Relationships among these various concepts are summarized in Figure E5.1. We have already looked at most of the links in the table formally. We have not yet discussed the mathematical relationship between indirect utility functions and Marshallian demand
functions (Roy’s identity), and we will do that below. All the entries in the table make clear that there are many ways to learn something about the relationship between individuals’ welfare and the prices they face. In this extension we will explore some of look at how the these approaches. Speciﬁcally, we will concepts can shed light on the accuracy of the consumer price index (CPI), the primary measure of inﬂation in the United States. We will also look at a few other price index concepts. The CPI is a ‘‘market basket’’ index of the cost of living. Researchers measure the amounts that people consume of a set of goods in some base period (in the two-good case these base-period consumption levels might be denoted by x0 and y0) and then use current price data to compute the changing price of this market basket. Using this procedure, the cost of p0 the market basket initially would be I0 ¼ yy0 and the p1 p1 cost in period 1 would be I1 ¼ yy0. The change in the xx0 þ cost of living between these two periods would then be measured by I1/I0. Although this procedure is an intuitively plausible way of measuring inﬂation and market basket p0 xx0 þ FIGURE E5.1 Relationships among Demand Concepts Primal Dual Maximize U(x, y) s.t. I = pxx + pyy Minimize E(x, y) s.t. U = U(x, y) Indirect utility function U* = V(px, py, I) Inverses Expenditure function E* = E(px, py, U) Roy’s identity Shephard’s lemma Marshallian demand ∂V ∂px ∂V ∂I x(px, py, I) = − Compensated demand xc(px, py, U) = ∂E ∂px 182 Part 2: Choice and Demand price indices are widely used, such indices have many shortcomings. E5.1 Expenditure functions and substitution bias Market basket price indices suffer from ‘‘substitution bias.’’ Because the indices do not permit individuals to make substitutions in the market basket in response to changes in relative prices, they will tend to overstate the welfare losses that people incur from increasing prices. This exaggeration
is illustrated in Figure E5.2. To achieve the utility level U0 initially requires expenditures of E0, resulting in a purchase of the basket x0, y0. If px/py decrease, the initial utility level can now be obtained with expenditures of E1 by altering the consumption bundle to x1, y1. Computing the expenditure level needed to continue consuming x0, y0 exaggerates how much extra purchasing power this person needs to restore his or her level of wellbeing. Economists have extensively studied the extent of this substitution bias. Aizcorbe and Jackman (1993), for example, ﬁnd that this difﬁculty with a market basket index may exaggerate the level of inﬂation shown by the CPI by approximately 0.2 percent per year. E5.2 Roy’s identity and new goods bias When new goods are introduced, it takes some time for them to be integrated into the CPI. For example, Hausman (1999, 2003) states that it took more than 15 years for cell phones to appear in the index. The problem with this delay is that market basket indices will fail to reﬂect the welfare gains that people experience from using new goods. To measure these costs, Hausman sought to measure a ‘‘virtual’’ price (p!) at which the demand for, say, cell phones would be zero and then argued that the introduction of the good at its market price represented a change in consumer surplus that could be measured. Hence the author was faced with the problem of how to get from the Marshallian demand function for cell phones FIGURE E5.2 Substitution Bias in the CPI Initially expenditures are given by E0, and this individual buys x0, y0. If px/py decreases, utility level U0 can be reached most cheaply by consuming x1, y1 and spending E1. Purchasing x0, y0 at the new prices would cost more than E1. Hence holding the consumption bundle constant imparts an upward bias to CPI-type computations. Quantity of y y0 E0 E1 U0 x0 x1 Quantity of x (which he estimated econometrically) to the expenditure function. To do so he used Roy’s identity (see Roy, 1942). Remember that the consumer’s utility-maximizing problem can be represented by the Lagrangian expression + þ l(I –
pxx – pyy). If we apply the envelope theorem to this expression, we know that U(x, y) ¼ kx px, py, I @U! @px ¼ @U! @I ¼ @+ @px ¼ ’ @+ @I ¼ k: &, ’ Hence the Marshallian demand function is given by px, py, I x ð Þ ¼ ’ @U!=@px @U!=@I : (i) (ii) Using his estimates of the Marshallian demand function, Hausman integrated Equation ii to obtain the implied indirect utility function and then calculated its inverse, the expenditure function (check Figure E5.1 to see the logic of the process). Although this certainly is a roundabout scheme, it did yield large estimates for the gain in consumer welfare from cell phones—a present value in 1999 of more than $100 billion. Delays in the inclusion of such goods into the CPI can therefore result in a misleading measure of consumer welfare. E5.3 Other complaints about the CPI Researchers have found several other faults with the CPI as currently constructed. Most of these focus on the consequences of using incorrect prices to compute the index. For example, when the quality of a good improves, people are made better off, although this may not show up in the good’s price. Throughout the 1970s and 1980s the reliability of color television sets improved dramatically, but the price of a set did not change much. A market basket that included ‘‘one color television set’’ would miss this source of improved welfare. Similarly, the opening of ‘‘big box’’ retailers such as Costco and Home Depot during the 1990s undoubtedly reduced the prices that consumers paid for various goods. But including these new retail outlets into the sample scheme for the CPI took several years, so the index misrepresented what people were actually paying. Assessing the magnitude of error introduced by these cases where incorrect prices are used in the CPI can also be accomplished by using the various demand concepts in Figure E5.1. For a summary of this research, see Moulton (1996). E5.4 Exact price indices In principle, it is possible that some of the shortcomings of price indices such as the CPI might be ameliorated by more careful attention to demand theory. If the expenditure function for the representative consumer were known, for example, it would be possible to
construct an ‘‘exact’’ index for changes in purchasing power that would take commodity substitution into account. To illustrate this, suppose there are only two goods and we wish to know how purchasing power Chapter 5: Income and Substitution Effects 183 has changed between period 1 and period 2. If the expenditure function is given by E(px, py, U), then the ratio I1,2 ¼ E E ð ð p2 x, p2 x, p1 p1 y, U y, U Þ Þ (iii) shows how the cost of attaining the target utility level U has changed between the two periods. If, for example, I1,2 ¼ 1.04, then we would say that the cost of attaining the utility target had increased by 4 percent. Of course, this answer is only a conceptual one. Without knowing the representative person’s utility function, we would not know the speciﬁc form of the expenditure function. But in some cases Equation iii may suggest how to proceed in index construction. Suppose, for example, that the typical person’s preferences could be reprexay1–a. sented by the Cobb–Douglas utility function U(x, y) In this case it is easy to show that the expenditure function the one given in Example 4.4: is a generalization of 1 kpa y U=aa pa x p1 E Insert’ ’ ing this function into Equation iii yields a x p1 y U. ’ px, py I1,2 ¼ k k ð ð a p2 xÞ a p1 xÞ ð ð 1 p2 ’ yÞ 1 p1 ’ yÞ aU aU ¼ a p2 xÞ a p1 xÞ ð ð 1 p2 ’ yÞ 1 p1 ’ yÞ ð ð a a : (iv) Thus, in this case, the exact price index is a relatively simple function of the observed prices. The particularly useful feature of this example is that the utility target cancels out in the construction of the cost-of-living index (as it will anytime the expenditure function is homogeneous in utility). Notice also that the expenditure shares (a and 1 – a) play an important role in the index—the larger a good’s share, the more
important will changes be in that good’s price in the ﬁnal index. E5.5 Development of exact price indices The Cobb–Douglas utility function is, of course, a simple one. Much recent research on price indices has focused on more general types of utility functions and on the discovery of the exact price indices they imply. For example, Feenstra and Reinsdorf (2000) show that the almost ideal demand system described in the Extensions to Chapter 4 implies an exact price index (I) that takes a ‘‘Divisia’’ form: n wiD ln pi (v) I ln ð Þ ¼ 1 i X ¼ (here the wi are weights to be attached to the change in the logarithm of each good’s price). Often the weights in Equation v are taken to be the budget shares of the goods. Interestingly, this is precisely the price index implied by the Cobb–Douglas utility function in Equation iv because I1, 2Þ ¼ ð a ln p2 x þ ð ln a 1 Þ ’ a ln p1 1 x ’ ð ’ aD ln px þ ð 1 ’ a ’ ¼ a ln p2 y ln p1 y Þ D ln py: Þ (vi) In actual applications, the weights would change from period to period to reﬂect changing budget shares. Similarly, changes 184 Part 2: Choice and Demand over several periods would be ‘‘chained’’ together from a number of single-period price change indices. Changing demands for food in China China has one of the fastest growing economies in the world: Its GDP per capita is currently growing at a rate of approximately 8 percent per year. Chinese consumers also spend a large fraction of their incomes on food—approximately 38 percent of total expenditures in recent survey data. One implication of the rapid growth in Chinese incomes, however, is that patterns of food consumption are changing rapidly. Purchases of staples, such as rice or wheat, are declining in relative importance, whereas purchases of poultry, ﬁsh, and processed foods are growing rapidly. An article by Gould and Villarreal (2006) studies these patterns in detail using the AIDS model. They identify a variety of substitution effects across speciﬁc food categories in response to changing relative prices
. Such changing patterns imply that a ﬁxed market basket price index (such as the U.S. Consumer Price Index) would be particularly inappropriate for measuring changes in the cost of living in China and that some alternative approaches should be examined. References Aizcorbe, Ana M., and Patrick C. Jackman. ‘‘The Commodity in CPI Data, 1982–91.’’ Monthly Substitution Effect Labor Review (December 1993): 25–33. Feenstra, Robert C., and Marshall B. Reinsdorf. ‘‘An Exact Price Index for the Almost Ideal Demand System.’’ Economics Letters (February 2000): 159–62. Gould, Brain W., and Hector J. Villarreal. ‘‘An Assessment of the Current Structure of Food Demand in Urban China.’’ Agricultural Economics (January 2006): 1–16. Hausman, Jerry. ‘‘Cellular Telephone, New Products, and the CPI.’’ Journal of Business and Economic Statistics (April 1999): 188–94. Hausman, Jerry. ‘‘Sources of Bias and Solutions to Bias in the Consumer Price Index.’’ Journal of Economic Perspectives (Winter 2003): 23–44. Moulton, Brent R. ‘‘Bias in the Consumer Price Index: What Is the Evidence?’’ Journal of Economic Perspectives (Fall 1996): 159–77. Roy, R. De l’utilite´, contribution a´ la the´orie des choix. Paris: Hermann, 1942. This page intentionally left blank C H A P T E R SIX Demand Relationships among Goods In Chapter 5 we examined how changes in the price of a particular good (say, good x) affect the quantity of that good chosen. Throughout the discussion, we held the prices of all other goods constant. It should be clear, however, that a change in one of these other prices could also affect the quantity of x chosen. For example, if x were taken to represent the quantity of automobile miles that an individual drives, this quantity might be expected to decrease when the price of gasoline increases or increase when air and bus fares increase. In this chapter we will use the utility-maximization model to study such relationships. The Two-Good Case We begin our study of the demand relationship among goods with the two-good case. Unfortunately
, this case proves to be rather uninteresting because the types of relationships that can occur when there are only two goods are limited. Still, the two-good case is useful because it can be illustrated with two-dimensional graphs. Figure 6.1 starts our examination by showing two examples of how the quantity of x chosen might be affected by a change in the price of y. In both panels of the ﬁgure, py has decreased. This has the result of shifting the budget constraint outward from I0 to I1. In both cases, the quantity of good y chosen has also increased from y0 to y1 as a result of the decrease in py, as would be expected if y is a normal good. For good x, however, the results shown in the two panels differ. In (a), the indifference curves are nearly L-shaped, implying a fairly small substitution effect. A decrease in py does not induce a large move along U0 as y is substituted for x. That is, x drops relatively little as a result of the substitution. The income effect, however, reﬂects the greater purchasing power now available, and this causes the total quantity of x chosen to increase. Hence @x/@py is negative (x and py move in opposite directions). In Figure 6.1b this situation is reversed: @x/@py is positive. The relatively ﬂat indifference curves in Figure 6.1a result in a large substitution effect from the fall in py. The quantity of x decreases sharply as y is substituted for x along U0. As in Figure 6.1a, the increased purchasing power from the decrease in py causes more x to be bought, but now the substitution effect dominates and the quantity of x decreases to x1. In this case, x and py then move in the same direction. A mathematical treatment The ambiguity in the effect of changes in py can be further illustrated by a Slutsky-type equation. By using procedures similar to those in Chapter 5, it is fairly simple to show that 187 188 Part 2: Choice and Demand FIGURE 6.1 Differing Directions of Cross-Price Effects In both panels, the price of y has decreased. In (a), substitution effects are small; therefore, the quantity of x consumed increases along with y. Because @x/@py < 0, x and y are gross complements. In (b), substitution effects are large; therefore, the quantity of x chosen decreases. Because @
x/@py > 0, x and y would be termed gross substitutes. Quantity of y Quantity of y I1 I0 I1 I0 y1 y0 y1 y0 U1 U0 U1 U0 x0 x1 Quantity of x x1 x0 Quantity of x (a) Gross complements (b) Gross substitutes @x ð px, py, I @py Þ ¼ substitution effect y & constant% ¼ U ¼ @x @py!!!! ex, py ¼ income effect, (6:1) þ @x @I or, in elasticity terms, exc, py % Notice that the size of the income effect is determined by the share of good y in this person’s purchases. The impact of a change in py on purchasing power is determined by how important y is to this person. syex, I: (6:2) For the two-good case, the terms on the right side of Equations 6.1 and 6.2 have different signs. Assuming that indifference curves are convex, the substitution effect @x/@py\|U constant is positive. If we conﬁne ourselves to moves along one indifference curve, increases in py increase x and decreases in py decrease the quantity of x chosen. However, assuming x is a normal good, the income effect (–y@x/@I or –syex, I) is clearly negative. Hence the combined effect is ambiguous; @x/@py could be either positive or negative. Even in the two-good case, the demand relationship between x and py is rather complex. ¼ EXAMPLE 6.1 Another Slutsky Decomposition for Cross-Price Effects In Example 5.4 we examined the Slutsky decomposition for the effect of a change in the price of x. Now let’s look at the cross-price effect of a change in y prices on x purchases. Remember that the uncompensated and compensated demand functions for x are given by Chapter 6: Demand Relationships among Goods 189 and px, py, I x ð Þ ¼ 0:5I px (6:3) xc px, py, V Vp0:5 y p% x 0:5 : (6:4) ð Þ ¼ As we have pointed out before, the Marshallian demand function in this case yields @x/@py �
� 0; that is, changes in the price of y do not affect x purchases. Now we show that this occurs because the substitution and income effects of a price change are precisely counterbalancing. The substitution effect in this case is given by @xc @py ¼ 0:5Vp% y 0:5 0:5 p% x : constant¼ ¼ (6:5) @x @py! U!!! Substituting for V from the indirect utility function for the substitution effect: V ð ¼ 0:5Ip% y 0:5 0:5 p% x Þ gives a ﬁnal statement Returning to the Marshallian demand function for y @x @py! U!!! 0:25Ip% 1 1 x : y p% constant¼ ¼ (6:6) y ð ¼ 0:5Ip% 1 y Þ to calculate the income effect yields @x @I ¼ %½ 1 0:5Ip% y y % ( & ½ 1 0:5p% x ( ¼ % 1 0:25Ip% y p% 1 x, and combining Equations 6.6 and 6.7 gives the total effect of the change in the price of y as @x @py ¼ 0:25Ip% 1 1 y p% x % 1 0:25Ip% y p% 1 x ¼ 0: (6:7) (6:8) This makes clear that the reason that changes in the price of y have no effect on x purchases in the Cobb–Douglas case is that the substitution and income effects from such a change are precisely offsetting; neither of the effects alone, however, is zero. ¼ 4, I 8, V Returning to our numerical example (px ¼ 2), suppose now that py falls to 1, py ¼ 2. This should have no effect on the Marshallian demand for good x. The compensated demand function in Equation 6.4 shows that the price change would cause the quantity of x demanded to decrease from 4 to 2.83 as y is substituted for x with utility unchanged. However, the increased purchasing power arising from the price decrease precisely reverses this effect. Þ ﬃﬃﬃ QUERY: Why would it be incorrect to argue that if @x/@py ¼ 0, then x and y have no substitution possibilities—that
is, they must be consumed in ﬁxed proportions? Is there any case in which such a conclusion could be drawn? 2p ð¼ ¼ 2 Substitutes and Complements With many goods, there is much more room for interesting relations among goods. It is relatively easy to generalize the Slutsky equation for any two goods xi, xj as @xið p1,..., pn, I @pj Þ ¼ xj @xi @I, U constant% ¼ (6:9) @xi @pj!!!! and again this can be readily translated into an elasticity relation: ec i, j % This says that the change in the price of any good (here, good j) induces income and substitution effects that may change the quantity of every good demanded. Equations 6.9 ei, j ¼ sjei, I: (6:10) 190 Part 2: Choice and Demand and 6.10 can be used to discuss the idea of substitutes and complements. Intuitively, these ideas are rather simple. Two goods are substitutes if one good may, as a result of changed conditions, replace the other in use. Some examples are tea and coffee, hamburgers and hot dogs, and butter and margarine. Complements, on the other hand, are goods that ‘‘go together,’’ such as coffee and cream, ﬁsh and chips, or brandy and cigars. In some sense, ‘‘substitutes’’ substitute for one another in the utility function, whereas ‘‘complements’’ complement each other. There are two different ways to make these intuitive ideas precise. One of these focuses on the ‘‘gross’’ effects of price changes by including both income and substitution effects; the other looks at substitution effects alone. Because both deﬁnitions are used, we will examine each in detail. Gross (Marshallian) substitutes and complements Whether two goods are substitutes or complements can be established by referring to observed price reactions as follows Gross substitutes and complements. Two goods, xi and xj, are said to be gross substitutes if and gross complements if @xi @pj > 0 @xi @pj < 0: (6:11) (6:12) That is, two goods are gross substitutes if
an increase in the price of one good causes more of the other good to be bought. The goods are gross complements if an increase in the price of one good causes less of the other good to be purchased. For example, if the price of coffee increases, the demand for tea might be expected to increase (they are substitutes), whereas the demand for cream might decrease (coffee and cream are complements). Equation 6.9 makes it clear that this deﬁnition is a ‘‘gross’’ deﬁnition in that it includes both income and substitution effects that arise from price changes. Because these effects are in fact combined in any real-world observation we can make, it might be reasonable always to speak only of ‘‘gross’’ substitutes and ‘‘gross’’ complements. Asymmetry of the gross deﬁnitions There are, however, several things that are undesirable about the gross deﬁnitions of substitutes and complements. The most important of these is that the deﬁnitions are not symmetric. It is possible, by the deﬁnitions, for x1 to be a substitute for x2 and at the same time for x2 to be a complement of x1. The presence of income effects can produce paradoxical results. Let’s look at a speciﬁc example. EXAMPLE 6.2 Asymmetry in Cross-Price Effects Suppose the utility function for two goods (x and y) has the quasi-linear form U x, y ð Þ ¼ ln x y: þ (6:13) Chapter 6: Demand Relationships among Goods 191 Setting up the Lagrangian expression yields the following ﬁrst-order conditions: + ln x y I k ð þ þ % pxx pyy Þ % ¼ kpx ¼ kpy ¼ 0, 0, @+ @x ¼ @+ @y ¼ @+ @k ¼ 1 x % % 1 I pxx pyy 0: ¼ % % Moving the terms in l to the right and dividing the ﬁrst equation by the second yields 1 x ¼ px py, (6:14) (6:15) (6:16) (6:17) ¼ Substitution into the budget constraint now permits us to solve
for the Marshallian demand function for y: pxx py: Hence I pxx pyy py þ ¼ þ ¼ pyy: I py : % py y ¼ (6:18) (6:19) This equation shows that an increase in py must decrease spending on good y (i.e., pyy). Therefore, because px and I are unchanged, spending on x must increase. Thus @x @py > 0, (6:20) and we would term x and y gross substitutes. On the other hand, Equation 6.19 shows that spending on y is independent of px. Consequently, @y @px ¼ 0 (6:21) and, looked at in this way, x and y would be said to be independent of each other; they are neither gross substitutes nor gross complements. Relying on gross responses to price changes to deﬁne the relationship between x and y would therefore run into ambiguity. QUERY: In Example 3.4, we showed that a utility function of the form given by Equation 6.13 is not homothetic: The MRS does not depend only on the ratio of x to y. Can asymmetry arise in the homothetic case? Net (Hicksian) Substitutes and Complements Because of the possible asymmetries involved in the deﬁnition of gross substitutes and complements, an alternative deﬁnition that focuses only on substitution effects is often used. 192 Part 2: Choice and Demand Net substitutes and complements. Goods xi and xj are said to be net substitutes if and net complements if > 0 constant U ¼ @xi @pj!!!!! < 0: constant ¼ @xi @pj! U!!!! (6:22) (6:23) These deﬁnitions1 then look only at the substitution terms to determine whether two goods are substitutes or complements. This deﬁnition is both intuitively appealing (because it looks only at the shape of an indifference curve) and theoretically desirable (because it is unambiguous). Once xi and xj have been discovered to be substitutes, they stay substitutes, no matter in which direction the deﬁnition is applied. As a matter of fact, the deﬁnitions are symmetric: @xj @pi!!! The substitution effect of a change in pi on
good xj is identical to the substitution effect! of a change in pj on the quantity of xi chosen. This symmetry is important in both theoretical and empirical work.2 @xi @pj!!!!! (6:24) constant constant ¼ ¼ ¼ U U : The differences between the two deﬁnitions of substitutes and complements are easily demonstrated in Figure 6.1a. In this ﬁgure, x and y are gross complements, but they are net substitutes. The derivative @x/@py turns out to be negative (x and y are gross complements) because the (positive) substitution effect is outweighed by the (negative) income effect (a decrease in the price of y causes real income to increase greatly, and, consequently, actual purchases of x increase). However, as the ﬁgure makes clear, if there are only two goods from which to choose, they must be net substitutes, although they may be either gross substitutes or gross complements. Because we have assumed a diminishing MRS, the own-price substitution effect must be negative and, consequently, the crossprice substitution effect must be positive. 1These are sometimes called Hicksian substitutes and complements, named after the British economist John Hicks, who originally developed the deﬁnitions. 2This symmetry is easily shown using Shephard’s lemma. Compensated demand functions can be calculated from expenditure functions by differentiation: p1,..., pn, V xc i ð @E p1,..., pn, V ð @pi Þ. Þ ¼ Hence the substitution effect is given by But now we can apply Young’s theorem to the expenditure function: @xc i @pj ¼ @2E @pj@pi ¼ ¼ Eij. constant U ¼ @xi @pj!!!!! which proves the symmetry. Eij ¼ Eji ¼ @xc j @pi ¼, constant U ¼ @xj @pi!!!! Chapter 6: Demand Relationships among Goods 193 Substitutability with Many Goods Once the utility-maximizing model is extended to many goods, a wide variety of demand patterns become possible. Whether a particular pair of goods are net substitutes or net complements is basically a question of a person’s preferences; thus, one might observe all sorts of
into 3See John Hicks, Value and Capital (Oxford, UK: Oxford University Press, 1939), mathematical appendices. There is some debate about whether this result should be called Hicks’ second or third law. In fact, two other laws that we have already seen 0 (negativity of the own-substitution effect); and (2) @xc are listed by Hicks: (1) @xc j =@pi (symmetry of crosssubstitution effects). But he refers explicitly only to two ‘‘properties’’ in his written summary of his results. 4To see this, notice that all substitution effects, sij, could be recorded in an n n matrix. However, symmetry of the effects sji) implies that only those terms on and below the principal diagonal of this matrix may be distinctly different from (sij ¼ each other. This amounts to half the terms in the matrix (n2/2) plus the remaining half of the terms on the main diagonal of the matrix (n/2). i =@pi ) i =@pj ¼ @xc + 194 Part 2: Choice and Demand larger aggregates such as food, clothing, shelter, and so forth. At the most extreme level of aggregates, we might wish to examine one speciﬁc good (say, gasoline), which we might call x, and its relationship to ‘‘all other goods,’’ which we might call y. This is the procedure we have been using in some of our two-dimensional graphs, and we will continue to do so at many other places in this book. In this section we show the conditions under which this procedure can be defended. In the Extensions to this chapter, we explore more general issues involved in aggregating goods into larger groupings. % Composite commodity theorem Suppose consumers choose among n goods but that we are only interested speciﬁcally in one of them—say, x1. In general, the demand for x1 will depend on the individual prices of the other n 1 commodities. But if all these prices move together, it may make sense 2,..., p0 to lump them into a single ‘‘composite commodity,’’ y. Formally, if we let p0 n represent the initial prices of these goods, then we assume that these prices can only vary together. They might all double, or all decrease by 50 percent
, but the relative prices of x2,..., xn would not change. Now we deﬁne the composite commodity y to be total expenditures on x2,..., xn using the initial prices p0 p0 3x3 þ & & & þ 2,..., p0 n: p0 nxn: p0 2x2 þ (6:28) ¼ y This person’s initial budget constraint is given by I p1x1 þ p0 2x2 þ & & & þ p0 nxn ¼ p1x1 þ y: ¼ (6:29) By assumption, all the prices p2, …, pn change in unison. Assume all these prices change by a factor of t (t > 0). Now the budget constraint is I p1x1 þ tp0 2x2 þ & & & þ tp0 nxn ¼ p1x1 þ ¼ ty: (6:30) Consequently, the factor of proportionality, t, plays the same role in this person’s budget constraint as did the price of y (py) in our earlier two-good analysis. Changes in p1 or t induce the same kinds of substitution effects we have been analyzing. As long as p2,..., pn move together, we can therefore conﬁne our examination of demand to choices between buying x1 or buying ‘‘everything else.’’5 Therefore, simpliﬁed graphs that show these two goods on their axes can be defended rigorously as long as the conditions of this ‘‘composite commodity theorem’’ (that all other prices move together) are satisﬁed. Notice, however, that the theorem makes no predictions about how choices of x2,..., xn behave; they need not move in unison. The theorem focuses only on total spending on x2,..., xn, not on how that spending is allocated among speciﬁc items (although this allocation is assumed to be done in a utility-maximizing way). Generalizations and limitations The composite commodity theorem applies to any group of commodities whose relative prices all move together. It is possible to have more than one such commodity if there are several groupings that obey the theorem (i.e., expenditures on ‘�
�food,’’ ‘‘clothing,’’ and so forth). Hence we have developed the following deﬁnition. 5The idea of a composite commodity was also introduced by J. R. Hicks in Value and Capital, 2nd ed. (Oxford, UK: Oxford University Press, 1946), pp. 312–13. Proof of the theorem relies on the notion that to achieve maximum utility, the ratio of the marginal utilities for x2,..., xn must remain unchanged when p2,..., pn all move together. Hence the n-good problem can be reduced to the two-dimensional problem of equating the ratio of the marginal utility from x to that from y to the ‘‘price ratio’’ p1/t. Chapter 6: Demand Relationships among Goods 195 Composite commodity. A composite commodity is a group of goods for which all prices move together. These goods can be treated as a single ‘‘commodity’’ in that the individual behaves as though he or she were choosing between other goods and total spending on the entire composite group. This deﬁnition and the related theorem are powerful results. They help simplify many problems that would otherwise be intractable. Still, one must be rather careful in applying the theorem to the real world because its conditions are stringent. Finding a set of commodities whose prices move together is rare. Slight departures from strict proportionality may negate the composite commodity theorem if cross-substitution effects are large. In the Extensions to this chapter, we look at ways to simplify situations where prices move independently. EXAMPLE 6.3 Housing Costs as a Composite Commodity Suppose that an individual receives utility from three goods: food (x), housing services (y) measured in hundreds of square feet, and household operations (z) as measured by electricity use. If the individual’s utility is given by the three-good CES function utility U x, y6:31) then the Lagrangian technique can be used to calculate Marshallian demand functions for these goods as x y ¼ px þ ¼ py þ z ¼ pz þ 4, and pz ¼ x, y, z, p I pxpy I ﬃﬃﬃﬃﬃﬃﬃﬃﬃ pypx I ﬃ
ﬃﬃﬃﬃﬃﬃﬃﬃ p pz pxp, pxpzp, ﬃﬃﬃﬃﬃﬃﬃﬃﬃ pypz p ﬃﬃﬃﬃﬃﬃﬃﬃﬃ pzpy : þ þ þ 25, 12:5, 25: ¼ ¼ ¼ If initially I 100, px ¼ 1, py ¼ ¼ 1, then the demand functions predict p ﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃ (6:32) (6:33) Hence 25 is spent on food and a total of 75 is spent on housing-related needs. If we assume that housing service prices (py) and household operation prices (pz) always move together, then we can use their initial prices to deﬁne the ‘‘composite commodity’’ housing (h) as Here, we also (arbitrarily) deﬁne the initial price of housing (ph) to be 1. The initial quantity of housing is simply total dollars spent on h: 4y h ¼ þ 1z: (6:34) h 4 12:5 ð Þ þ 1 25 ð Þ ¼ ¼ 75: (6:35) 196 Part 2: Choice and Demand Furthermore, because py and pz always move together, ph will always be related to these prices by pz ¼ Using this information, we can recalculate the demand function for x as a function of I, px, and ph: ph ¼ 0:25py: (6:36) I 4pxph I p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ px php 3 x ¼ px þ ¼ py þ 1, and ph ¼ pxphp ﬃﬃﬃﬃﬃﬃﬃﬃ�
�� þ : (6:37) As before, initially I easily calculated from the budget constraint as h, ‘‘everything’’ other than food. 100, px ¼ ¼ ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1; thus, x, 25. Spending on housing can be most 75 because spending on housing represents ¼ An increase in housing costs. If the prices of y and z were to increase proportionally to py ¼ 4 (with px remaining at 1), then ph would also increase to 4. Equation 6.37 now predicts that the demand for x would decrease to 16, pz ¼ þ and that housing purchases would be given by ﬃﬃﬃ x, 100 3 ¼ 1 4p ¼ 100 7 or, because ph ¼ 4, phh, 100 ¼ 100 7 ¼ 600 7, % h, ¼ 150 7 : (6:38) (6:39) (6:40) Notice that this is precisely the level of housing purchases predicted by the original demand functions for three goods in Equation 6.32. With I 4, these equations can be solved as 16, and pz ¼ 100, px ¼ 1, py ¼ ¼ x, y, z, 100 7 100 28 100 14,,, ¼ ¼ ¼ (6:41) and so the total amount of the composite good ‘‘housing’’ consumed (according to Equation 6.34) is given by h, ¼ 4y, þ 1z, ¼ 150 7 : (6:42) Hence we obtained the same responses to price changes regardless of whether we chose to examine demands for the three goods x, y, and z or to look only at choices between x and the composite good h. QUERY: How do we know that the demand function for x in Equation 6.37 continues to ensure utility maximization? Why is the Lagrangian constrained maximization problem unchanged by making the substitutions represented by Equation 6.36? Chapter 6: Demand Relationships among Goods 197 Home Production, Attributes of Goods, and Implicit Prices Thus far in this chapter we have focused on what economists can learn about the relationships among goods by observing individuals’ changing consumption of these goods in reaction to changes in market
prices. In some ways this analysis skirts the central question of why coffee and cream go together or why ﬁsh and chicken may substitute for each other in a person’s diet. To develop a deeper understanding of such questions, economists have sought to explore activities within individuals’ households. That is, they have devised models of nonmarket types of activities such as parental child care, meal preparation, or do-it-yourself construction to understand how such activities ultimately result in demands for goods in the market. In this section we brieﬂy review some of these models. Our primary goal is to illustrate some of the implications of this approach for the traditional theory of choice. Household production model The starting point for most models of household production is to assume that individuals do not receive utility directly from goods they purchase in the market (as we have been assuming thus far). Instead, it is only when market goods are combined with time inputs by the individual that utility-providing outputs are produced. In this view, raw beef and uncooked potatoes then yield no utility until they are cooked together to produce stew. Similarly, market purchases of beef and potatoes can be understood only by examining the individual’s preferences for stew and the underlying technology through which it is produced. In formal terms, assume as before that there are three goods that a person might purchase in the market: x, y, and z. Purchasing these goods provides no direct utility, but the goods can be combined by the individual to produce either of two home-produced goods: a1 or a2. The technology of this household production can be represented by the production functions f1 and f2 (see Chapter 9 for a more complete discussion of the production function concept). Therefore, a1 ¼ a2 ¼ x, y, z x, y, z f1ð f2ð, Þ, Þ (6:43) and : a1, a2Þ ð The individual’s goal is to choose x, y, z so as to maximize utility subject to the production constraints and to a ﬁnancial budget constraint:6 utility (6:44) ¼ U pxx pyy pzz I: ¼ þ þ (6:45) Although we will not examine in detail the results that can be derived from this general model, two insights that can be drawn from it might be mentioned. First, the model may help clarify the nature of market relationships among goods. Because the
production functions in Equations 6.43 are in principle measurable using detailed data on household operations, households can be treated as ‘‘multiproduct’’ ﬁrms and studied using many of the techniques economists use to study production. A second insight provided by the household production approach is the notion of the ‘‘implicit’’ or ‘‘shadow’’ prices associated with the home-produced goods a1 and a2. Because 6Often household production theory also focuses on the individual’s allocation of time to producing a1 and a2 or to working in the market. In Chapter 16 we look at a few simple models of this type. 198 Part 2: Choice and Demand consuming more a1, say, requires the use of more of the ‘‘ingredients’’ x, y, and z, this activity obviously has an opportunity cost in terms of the quantity of a2 that can be produced. To produce more bread, say, a person must not only divert some ﬂour, milk, and eggs from using them to make cupcakes but may also have to alter the relative quantities of these goods purchased because he or she is bound by an overall budget constraint. Hence bread will have an implicit price in terms of the number of cupcakes that must be forgone to be able to consume one more loaf. That implicit price will reﬂect not only the market prices of bread ingredients but also the available household production technology and, in more complex models, the relative time inputs required to produce the two goods. As a starting point, however, the notion of implicit prices can be best illustrated with a simple model. The linear attributes model A particularly simple form of the household production model was ﬁrst developed by K. J. Lancaster to examine the underlying ‘‘attributes’’ of goods.7 In this model, it is the attributes of goods that provide utility to individuals, and each speciﬁc good contains a ﬁxed set of attributes. If, for example, we focus only on the calories (a1) and vitamins (a2) that various foods provide, Lancaster’s model assumes that utility is a function of these attributes and that individuals purchase various foods only for the purpose of obtaining the calories and vitamins they offer. In mathematical terms, the model assumes that the ‘‘production’’ equations have
the simple form a1 ¼ a2 ¼ a1 xx a2 xx þ þ a1 yy a2 yy þ þ a1 zz, a2 zz, (6:46) x represents the number of calories per unit of food x, a2 where a1 x represents the number of vitamins per unit of food x, and so forth. In this form of the model, there is no actual ‘‘production’’ in the home. Rather, the decision problem is how to choose a diet that provides the optimal mix of calories and vitamins given the available food budget. Illustrating the budget constraints To begin our examination of the theory of choice under the attributes model, we ﬁrst illustrate the budget constraint. In Figure 6.2, the ray 0x records the various combinations of a1 and a2 available from successively larger amounts of good x. Because of the linear production technology assumed in the attributes model, these combinations of a1 and a2 lie along such a straight line, although in more complex models of home production that might not be the case. Similarly, rays of 0y and 0z show the quantities of the attributes a1 and a2 provided by various amounts of goods y and z that might be purchased. If this person spends all his or her income on good x, then the budget constraint (Equation 6.45) allows the purchase of and that will yield x, ¼ I px, a,1 ¼ a1 xx, a,2 ¼ a2 xx, a1 xI px a2 xI px, : ¼ ¼ (6:47) (6:48) 7See K. J. Lancaster, ‘‘A New Approach to Consumer Theory,’’ Journal of Political Economy 74 (April 1966): 132–57. Chapter 6: Demand Relationships among Goods 199 FIGURE 6.2 Utility Maximization in the Attributes Model The points x,, y,, and z, show the amounts of attributes a1 and a2 that can be purchased by buying only x, y, or z, respectively. The shaded area shows all combinations that can be bought with mixed bundles. Some individuals may maximize utility at E, others at E0. a2 a2 x U′0 x E′ 0 a1 y E y U0 z* z a1 This point is recorded as point x, on the 0x ray in
Figure 6.2. Similarly, the points y, and z, represent the combinations of a1 and a2 that would be obtained if all income were spent on good y or good z, respectively. Bundles of a1 and a2 that are obtainable by purchasing both x and y (with a ﬁxed budget) are represented by the line joining x, and y, in Figure 6.2.8 Similarly, the line x,z, represents the combinations of a1 and a2 available from x and z, and the line y,z, shows combinations available from mixing y and z. All possible combinations from mixing the three market goods are represented by the shaded triangular area x,y,z,. Corner solutions One fact is immediately apparent from Figure 6.2: A utility-maximizing individual would never consume positive quantities of all three of these goods. Only the northeast perimeter of the x,y,z, triangle represents the maximal amounts of a1 and a2 available to this person given his or her income and the prices of the market goods. Individuals with a preference toward a1 will have indifference curves similar to U0 and will maximize utility by choosing a point such as E. The combination of a1 and a2 speciﬁed by that point can be obtained by consuming only goods y and z. Similarly, a person with preferences 8Mathematically, suppose a fraction a of the budget is spent on x and (1 – a) on y; then a1 ¼ a2 ¼ aa1 aa2 xx, xx, 1 1 þ ð þ ð % % a1 y y,, a2 y y,. a a Þ Þ The line x,y, is traced out by allowing a to vary between 0 and 1. The lines x,z, and y,z, are traced out in a similar way, as is the triangular area x,y,z,. 200 Part 2: Choice and Demand represented by the indifference curve U00 will choose point E0 and consume only goods x and y. Therefore, the attributes model predicts that corner solutions at which individuals consume zero amounts of some commodities will be relatively common, especially in cases where individuals attach value to fewer attributes (here, two) than there are market goods to choose from (three). If income, prices, or preferences change, then consumption patterns may also change abruptly. Goods that were previously consumed may cease to be bought and goods previously neglected may experience a sign
iﬁcant increase in purchases. This is a direct result of the linear assumptions inherent in the production functions assumed here. In household production models with greater substitutability assumptions, such discontinuous reactions are less likely. SUMMARY In this chapter, we used the utility-maximizing model of choice to examine relationships among consumer goods. Although these relationships may be complex, the analysis presented here provided a number of ways of categorizing and simplifying them. • When there are only two goods, the income and substitution effects from the change in the price of one good (say, py) on the demand for another good (x) usually work in opposite directions. Therefore, the sign of @x/@py is ambiguous: Its substitution effect is positive but its income effect is negative. • In cases of more than two goods, demand relationships can be speciﬁed in two ways. Two goods (xi and xj) are ‘‘gross substitutes’’ if @xi/@pj > 0 and ‘‘gross complements’’ if @xi/@pj < 0. Unfortunately, because these price effects include income effects, they need not be symmetric. That is, @xi/@pj does not necessarily equal @xj/@pi. PROBLEMS • Focusing only on the substitution effects from price changes eliminates this ambiguity because substitution effects are symmetric; that is, @xc j =@pi. Now two goods are deﬁned as net (or Hicksian) substitutes if @xc i =@pj < 0. Hicks’ ‘‘second law of demand’’ shows that net substitutes are more prevalent. i =@pj > 0 and net complements if @xc i =@pj ¼ @xc • If a group of goods has prices that always move in unison, then expenditures on these goods can be treated as a ‘‘composite commodity’’ whose ‘‘price’’ is given by the size of the proportional change in the composite goods’ prices. • An alternative way to develop the theory of choice among market goods is to focus on the ways in which market goods are used in household production to yield utility-providing attributes. This may provide additional insights into relationships among goods. 6.1 Heidi receives utility from two goods, goat’s milk (m) and str
’s demand functions for g and p? d. Once Sarah decides how much to spend on g, how will she allocate those expenditures between b and t? 6.5 Suppose that an individual consumes three goods, x1, x2, and x3, and that x2 and x3 are similar commodities (i.e., cheap kp3, where k < 1—that is, the goods’ prices have a constant relationship to one and expensive restaurant meals) with p2 ¼ another. a. Show that x2 and x3 can be treated as a composite commodity. b. Suppose both x2 and x3 are subject to a transaction cost of t per unit (for some examples, see Problem 6.6). How will this transaction cost affect the price of x2 relative to that of x3? How will this effect vary with the value of t? c. Can you predict how an income-compensated increase in t will affect expenditures on the composite commodity x2 and x3? Does the composite commodity theorem strictly apply to this case? d. How will an income-compensated increase in t affect how total spending on the composite commodity is allocated between x2 and x3? 6.6 Apply the results of Problem 6.5 to explain the following observations: a. It is difﬁcult to ﬁnd high-quality apples to buy in Washington State or good fresh oranges in Florida. b. People with signiﬁcant babysitting expenses are more likely to have meals out at expensive (rather than cheap) restaurants than are those without such expenses. c. Individuals with a high value of time are more likely to ﬂy the Concorde than those with a lower value of time. d. Individuals are more likely to search for bargains for expensive items than for cheap ones. Note: Observations (b) and (d) form the bases for perhaps the only two murder mysteries in which an economist solves the crime; see Marshall Jevons, Murder at the Margin and The Fatal Equilibrium. 6.7 In general, uncompensated cross-price effects are not equal. That is, @xi @pj 6¼ @xj @pi : Use the Slutsky equation to show that these effects are equal if the individual spends a constant fraction of income on each good regardless of relative prices. (This is a generalization of Problem 6.1.) 202 Part 2: Choice and Demand
6.8 Example 6.3 computes the demand functions implied by the three-good CES utility function U x, y. Use the demand function for x in Equation 6.32 to determine whether x and y or x and z are gross substitutes or gross complements. b. How would you determine whether x and y or x and z are net substitutes or net complements? Analytical Problems 6.9 Consumer surplus with many goods In Chapter 5, we showed how the welfare costs of changes in a single price can be measured using expenditure functions and compensated demand curves. This problem asks you to generalize this to price changes in two (or many) goods. a. Suppose that an individual consumes n goods and that the prices of two of those goods (say, p1 and p2) increase. How would you use the expenditure function to measure the compensating variation (CV) for this person of such a price increase? b. A way to show these welfare costs graphically would be to use the compensated demand curves for goods x1 and x2 by assuming that one price increased before the other. Illustrate this approach. c. In your answer to part (b), would it matter in which order you considered the price changes? Explain. d. In general, would you think that the CV for a price increase of these two goods would be greater if the goods were net sub- stitutes or net complements? Or would the relationship between the goods have no bearing on the welfare costs? 6.10 Separable utility A utility function is called separable if it can be written as U1ð x where Ui0 > 0, Ui00 < 0, and U1, U2 need not be the same function. x, y ð Þ ¼ U Þ þ U2ð y, Þ a. What does separability assume about the cross-partial derivative Uxy? Give an intuitive discussion of what word this condition means and in what situations it might be plausible. b. Show that if utility is separable then neither good can be inferior. c. Does the assumption of separability allow you to conclude deﬁnitively whether x and y are gross substitutes or gross complements? Explain. d. Use the Cobb–Douglas utility function to show that separability is not invariant with respect to monotonic transformations. Note: Separable functions are examined in more detail in the Extensions to this chapter. 6.11 Graphing comple
ments Graphing complements is complicated because a complementary relationship between goods (under Hicks’ deﬁnition) cannot occur with only two goods. Rather, complementarity necessarily involves the demand relationships among three (or more) goods. In his review of complementarity, Samuelson provides a way of illustrating the concept with a two-dimensional indifference curve diagram (see the Suggested Readings). To examine this construction, assume there are three goods that a consumer might choose. The quantities of these are denoted by x1, x2, and x3. Now proceed as follows. a. Draw an indifference curve for x2 and x3, holding the quantity of x1 constant at x0 1. This indifference curve will have the customary convex shape. b. Now draw a second (higher) indifference curve for x2, x3, holding x1 constant at x0 h. For this new indifference curve, show the amount of extra x2 that would compensate this person for the loss of x1; call this amount j. Similarly, show that amount of extra x3 that would compensate for the loss of x1 and call this amount k. c. Suppose now that an individual is given both amounts j and k, thereby permitting him or her to move to an even higher x2, x3 indifference curve. Show this move on your graph, and draw this new indifference curve. d. Samuelson now suggests the following deﬁnitions: • If the new indifference curve corresponds to the indifference curve when x1 ¼ • If the new indifference curve provides more utility than when x1 ¼ x0 1 % x0 1 % 2h, goods 2 and 3 are complements. 2h, goods 2 and 3 are independent. 1 % Chapter 6: Demand Relationships among Goods 203 • If the new indifference curve provides less utility than when x1 ¼ graphical deﬁnitions are symmetric. x0 1 % 2h, goods 2 and 3 are substitutes. Show that these e. Discuss how these graphical deﬁnitions correspond to Hicks’ more mathematical deﬁnitions given in the text. f. Looking at your ﬁnal graph, do you think that this approach fully explains the types of relationships that might exist between x2 and x3? 6.12 Shipping the good apples out Details of the analysis suggested in Problems 6.5 and 6.6 were originally worked out by Borcherding
and Silberberg (see the Suggested Readings) based on a supposition ﬁrst proposed by Alchian and Allen. These authors look at how a transaction charge affects the relative demand for two closely substitutable items. Assume that goods x2 and x3 are close substitutes and are subject to a transaction charge of t per unit. Suppose also that good 2 is the more expensive of the two goods (i.e., ‘‘good apples’’ as opposed to ‘‘cooking apples’’). Hence the transaction charge lowers the relative price of the more expensive good =@t > 0 [i.e., (p2 þ (where we use compensated demand functions to eliminate pesky income effects). Borcherding and Silberberg show this result will probably hold using the following steps. t)/(p3 + t) decreases as t increases]. This will increase the relative demand for the expensive good if @ 2=xc xc 3Þ ð a. Use the derivative of a quotient rule to expand @ b. Use your result from part (a) together with the fact that, in this problem, @xc =@t. 2=xc xc 3Þ ð i =@t that the derivative we seek can be written as @xc i =@p2 þ ¼ @xc i =@p3 for i = 2, 3, to show @ xc 2=xc 3Þ @t ð ¼ xc 2 xc 3 s22 x2 þ # s23 x2 % s32 x3 % s33 x3, $ @xc i =@pj. where sij ¼ c. Rewrite the result from part (b) in terms of compensated price elasticities: ec ij ¼ @xc i @pj & pj xc i : d. Use Hicks’ third law (Equation 6.26) to show that the term in brackets in parts (b) and (c) can now be written as [(e22 – e23)(1/p2 – 1/p3) + (e21 – e31)/p3]. e. Develop an intuitive argument about why the expression in part (d) is likely to be positive under the conditions of this problem. Hints: Why is the ﬁrst product in the brackets positive? Why is the second term in brackets likely to be small
? f. Return to Problem 6.6 and provide more complete explanations for these various ﬁndings. SUGGESTIONS FOR FURTHER READING Borcherding, T. E., and E. Silberberg. ‘‘Shipping the Good Apples Out—The Alchian-Allen Theorem Reconsidered.’’ Journal of Political Economy (February 1978): 131–38. Good discussion of the relationships among three goods in demand theory. See also Problems 6.5 and 6.6. Hicks, J. R. Value and Capital, 2nd ed. Oxford, UK: Oxford University Press, 1946. See Chapters I–III and related appendices. Proof of the composite commodity theorem. Also has one of the ﬁrst treatments of net substitutes and complements. Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. New York: Oxford University Press, 1995. Explores the consequences of the symmetry of compensated cross-price effects for various aspects of demand theory. Rosen, S. ‘‘Hedonic Prices and Implicit Markets.’’ Journal of Political Economy (January/February 1974): 34–55. Nice graphical and mathematical treatment of the attribute approach to consumer theory and of the concept of ‘‘markets’’ for attributes. Samuelson, P. A. ‘‘Complementarity—An Essay on the 40th Anniversary of the Hicks-Allen Revolution in Demand Theory.’’ Journal of Economic Literature (December 1977): 1255–89. Reviews a number of deﬁnitions of complementarity and shows the connections among them. Contains an intuitive, graphical discussion and a detailed mathematical appendix. Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGraw-Hill, 2001. Good discussion of expenditure functions and the use of indirect utility functions to illustrate the composite commodity theorem and other results. EXTENSIONS SIMPLIFYING DEMAND AND TWO-STAGE BUDGETING In Chapter 6 we saw that the theory of utility maximization in its full generality imposes rather few restrictions on what might happen. Other than the fact that net cross-substitution effects are symmetric, practically any type of relationship among goods is consistent with the underlying theory. This situation poses problems for
economists who wish to study consumption behavior in the real world—theory just does not provide much guidance when there are many thousands of goods potentially available for study. There are two general ways in which simpliﬁcations are made. The ﬁrst uses the composite commodity theorem from Chapter 6 to aggregate goods into categories within which relative prices move together. For situations where economists are speciﬁcally interested in changes in relative prices within a category of spending (such as changes in the relative prices of various forms of energy), however, this process will not do. An alternative is to assume that consumers engage in a two-stage process in their consumption decisions. First they allocate income to various broad groupings of goods (e.g., food, clothing) and then, given these expenditure constraints, they maximize utility within each of the subcategories of goods using only information about those goods’ relative prices. In that way, decisions can be studied in a simpliﬁed setting by looking only at one category at a time. This process is called two-stage budgeting. In these Extensions, we ﬁrst look at the general theory of twostage budgeting and then turn to examine some empirical examples. E6.1 Theory of two-stage budgeting The issue that arises in two-stage budgeting can be stated succinctly: Does there exist a partition of goods into m nonoverlapping groups (denoted by r 1, m) and a separate budget ¼ (lr) devoted to each category such that the demand functions for the goods within any one category depend only on the prices of goods within the category and on the category’s budget allocation? That is, can we partition goods so that demand is given by xið p1,..., pn, I xi rð 2 for r 1, m? That it might be possible to do this is suggested by comparing the following two-stage maximization problem, Þ ¼ ¼ 2 pi r, IrÞ (i) ð p1,..., pn, I1,..., ImÞ max x1,..., xnÞ x1,..., xn ð U " V, ¼ and s.t. r i X 2 pixi ) Ir, r ¼ 1, m # (ii) max I1,..., Im V, s
S. consumer expenditures into six large groups (i.e., food, clothing, household operation, medical care, transportation, and recreation). Using these aggregates, he concludes that his procedure is much more accurate than assuming two-stage budgeting among these expenditure categories. E6.3 Homothetic functions and energy demand One way to simplify the study of demand when there are many commodities is to assume that utility for certain subcategories of goods is homothetic and may be separated from the demand for other commodities. This procedure was followed by Jorgenson, Slesnick, and Stoker (1997) in their study of energy demand by U.S. consumers. By assuming that demand functions for speciﬁc types of energy are proportional to total spending on energy, the authors were able to concentrate their empirical study on the topic that is of most interest to them: estimating the price elasticities of demand for various types of energy. They conclude that most types of energy (i.e., electricity, natural gas, gasoline) have fairly elastic demand functions. Demand appears to be most responsive to price for electricity. References Blackorby, Charles, Daniel Primont, and R. Robert Russell. Duality, Separability and Functional Structure: Theory and Economic Applications. New York: North Holland, 1978. Diewert, W. Erwin, and Terrence J. Wales. ‘‘Flexible Functional Forms and Tests of Homogeneous Separability.’’ Journal of Econometrics (June 1995): 259–302. Jorgenson, Dale W., Daniel T. Slesnick, and Thomas M. Stoker. ‘‘Two-Stage Budgeting and Consumer Demand for Energy.’’ In Dale W. Jorgenson, Ed., Welfare, vol. 1: Aggregate Consumer Behavior, pp. 475–510. Cambridge, MA: MIT Press, 1997. Lewbel, Arthur. ‘‘Aggregation without Separability: A Standardized Composite Commodity Theorem.’’ American Economic Review (June 1996): 524–43. This page intentionally left blank Uncertainty and Strategy P A R T THREE Chapter 7 Uncertainty Chapter 8 Game Theory This part extends the analysis of individual choice to more complicated settings. In Chapter 7 we look at individual behavior in uncertain situations. A decision is no longer associated with a single outcome but a number of more or less likely ones. We describe why
people generally dislike the risk involved in such situations. We seek to understand the steps they take to mitigate risk, including buying insurance, acquiring more information, and preserving options. Chapter 8 looks at decisions made in strategic situations in which a person’s well-being depends not just on his or her own actions but also on the actions of others and vice versa. This leads to a certain circularity in analyzing strategic decisions, which we will resolve using the tools of game theory. The equilibrium notions we develop in studying such situations are widely used throughout economics. Although this part can be regarded as the natural extension of the analysis of consumer choice from Part 2 to more complicated settings, it applies to a much broader set of decision-makers, including ﬁrms, other organizations, even whole countries. For example, game theory will provide the framework to study imperfect competition among few ﬁrms in Chapter 15. 207 This page intentionally left blank C H A P T E R SEVEN Uncertainty In this chapter we explore some of the basic elements of the theory of individual behavior in uncertain situations. We discuss why individuals do not like risk and the various methods (buying insurance, acquiring more information, and preserving options) they may adopt to reduce it. More generally, the chapter is intended to provide a brief introduction to issues raised by the possibility that information may be imperfect when individuals make utility-maximizing decisions. The Extensions section provides a detailed application of the concepts in this chapter to the portfolio problem, a central problem in ﬁnancial economics. Whether a well-informed person can take advantage of a poorly informed person in a market transaction (asymmetric information) is a question put off until Chapter 18. Mathematical Statistics Many of the formal tools for modeling uncertainty in economic situations were originally developed in the ﬁeld of mathematical statistics. Some of these tools were reviewed in Chapter 2, and in this chapter we will make a great deal of use of the concepts introduced there. Speciﬁcally, four statistical ideas will recur throughout this chapter. • Random variable: A random variable is a variable that records, in numerical form, the possible outcomes from some random event.1 • Probability density function (PDF): A function f (x) that shows the probabilities associated with the possible outcomes from a random variable. • Expected value of a random variable: The outcome of a random variable that will occur ‘‘on average.’’ The expected
value is denoted by E(x). If x is a discrete random n variable with n outcomes, then E i x variable, then E ð. If x is a continuous random • Variance and standard deviation of a random variable: These concepts measure the dispersion of a random variable about its expected value. In the discrete case, x Var ð þ1 %1 ½ R n xi % 1 ½ i ¼ dx. The standard deviation is the square root of the variance. x Þ ; in the continuous case, Var r2 x ¼ x E Þ’ ð Þ ¼ x % 1 xif ¼ x ð dx. 2f ð P þ1 %1 xiÞ xiÞ r2 P 2f xf Þ’ E ð ð Þ R As we shall see, all these concepts will come into play when we begin looking at the decision-making process of a person faced with a number of uncertain outcomes that can be conceptually represented by a random variable. 1When it is necessary to distinguish between random variables and nonrandom variables, we will use the notation ~x to denote the fact that the variable x is random in that it takes on a number of potential randomly determined outcomes. Often, however, it will not be necessary to make the distinction because randomness will be clear from the context of the problem. 209 210 Part 3: Uncertainty and Strategy Fair Gambles and The Expected Utility Hypothesis A ‘‘fair’’ gamble is a speciﬁed set of prizes and associated probabilities that has an expected value of zero. For example, if you ﬂip a coin with a friend for a dollar, the expected value of this gamble is zero because E x ð 0:5 $1 0:5 $1 0, (7.1) Þ þ where wins are recorded with a plus sign and losses with a minus sign. Similarly, a game that promised to pay you $10 if a coin came up heads but would cost you only $1 if it came up tails would be ‘‘unfair’’ because Þ ¼ Þ ¼ ðþ ð% E x ð Þ ¼ 0:5 ðþ $10 Þ þ 0:5 ð% $1 Þ ¼ $4:50: (7.2) This game
can easily be converted into a fair game, however, simply by charging you an entry fee of $4.50 for the right to play. It has long been recognized that most people would prefer not to take fair gambles.2 Although people may wager a few dollars on a coin ﬂip for entertainment purposes, $1 million or they would generally balk at playing a similar game whose outcome was $1 million. One of the ﬁrst mathematicians to study the reasons for this unwillingness % to engage in fair bets was Daniel Bernoulli in the eighteenth century.3 His examination of the famous St. Petersburg paradox provided the starting point for virtually all studies of the behavior of individuals in uncertain situations. þ St. Petersburg paradox In the St. Petersburg paradox, the following gamble is proposed: A coin is ﬂipped until a head appears. If a head ﬁrst appears on the nth ﬂip, the player is paid $2n. This gamble has an inﬁnite number of outcomes (a coin might be ﬂipped from now until doomsday and never come up a head, although the likelihood of this is small), but the ﬁrst few can easily be written down. If xi represents the prize awarded when the ﬁrst head appears on the ith trial, then x1 ¼ $2, x2 ¼ The probability of getting a head for the ﬁrst time on the ith trial is ity of getting (i Equation 7.3 are i; it is the probabil1) tails and then a head. Hence the probabilities of the prizes given in $8,..., xn ¼ $4, x3 ¼ (7.3) 1 2Þ % ð $2n: p1 ¼ Therefore, the expected value of the gamble is inﬁnite:, p2 ¼, p3 ¼,..., pn ¼ 1 2 1 4 1 8 1 2n : pixi ¼ 1 1 þ 1 2i 1=2i ( ( ( ¼1. (7.4) (7.5) 2The gambles discussed here are assumed to yield no utility in their play other than the prizes; hence the observation that many individuals gamble at ‘‘unfair’’ odds is not necessarily a refutation of this statement.
Rather, such individuals can reasonably be assumed to be deriving some utility from the circumstances associated with the play of the game. Therefore, it is possible to differentiate the consumption aspect of gambling from the pure risk aspect. 3The paradox is named after the city where Bernoulli’s original manuscript was published. The article has been reprinted as D. Bernoulli, ‘‘Exposition of a New Theory on the Measurement of Risk,’’ Econometrica 22 (January 1954): 23–36. Chapter 7: Uncertainty 211 Some introspection, however, should convince anyone that no player would pay very much (much less than inﬁnity) to take this bet. If we charged $1 billion to play the game, we would surely have no takers, despite the fact that $1 billion is still considerably less than the expected value of the game. This then is the paradox: Bernoulli’s gamble is in some sense not worth its (inﬁnite) expected dollar value. Expected utility Bernoulli’s solution to this paradox was to argue that individuals do not care directly about the dollar prizes of a gamble; rather, they respond to the utility these dollars provide. If we assume that the marginal utility of wealth decreases as wealth increases, the St. Petersburg gamble may converge to a ﬁnite expected utility value even though its expected monetary value is inﬁnite. Because the gamble only provides a ﬁnite expected utility, individuals would only be willing to pay a ﬁnite amount to play it. Example 7.1 looks at some issues related to Bernoulli’s solution. EXAMPLE 7.1 Bernoulli’s Solution to the Paradox and Its Shortcomings Suppose, as did Bernoulli, that the utility of each prize in the St. Petersburg paradox is given by xiÞ ¼ ð This logarithmic utility function exhibits diminishing marginal utility (i.e., U0 > 0 but U 00 < 0), and the expected utility value of this game converges to a ﬁnite number: : xiÞ ln ð (7.6) U expected utility ¼ ¼ piU xiÞ ð 1 2i 2i ln ð : Þ (7.7 Some manipulation of this expression yields4 the result that the expected utility from this gamble
is 1.39. Therefore, an individual with this type of utility function might be willing to invest resources that otherwise yield up to 1.39 units of utility (a certain wealth of approximately $4 provides this utility) in purchasing the right to play this game. Thus, assuming that the large prizes promised by the St. Petersburg paradox encounter diminishing marginal utility permitted Bernoulli to offer a solution to the paradox. Unbounded utility. Unfortunately, Bernoulli’s solution to the St. Petersburg paradox does not completely solve the problem. As long as there is no upper bound to the utility function, the paradox can be regenerated by redeﬁning the gamble’s prizes. For example, with the logarithmic e2i, in which case utility function, prizes can be set as xi ¼ U (7.8) e2 ln 2i i xiÞ ¼ ð ½ ’ ¼ and the expected utility from the gamble would again be inﬁnite. Of course, the prizes in this redeﬁned gamble are large. For example, if a head ﬁrst appears on the ﬁfth ﬂip, a person would 4Proof: expected utility i 2i ( ¼ 1 1 i X ¼ ln 2 ¼ ln 2 i 2i. 1 1 i X ¼ But the value of this ﬁnal inﬁnite series can be shown to be 2.0. Hence expected utility 2 ln 2 1.39. ¼ ¼ 212 Part 3: Uncertainty and Strategy ¼ $79 trillion, although the probability of winning this would be only 1/25 win e25 0.031. The idea that people would pay a great deal (say, trillions of dollars) to play games with small probabilities of such large prizes seems, to many observers, to be unlikely. Hence in many respects the St. Petersburg game remains a paradox. ¼ QUERY: Here are two alternative solutions to the St. Petersburg paradox. For each, calculate the expected value of the original game. 1. Suppose individuals assume that any probability less than 0.01 is in fact zero. 2. Suppose that the utility from the St. Petersburg prizes is given by U xiÞ ¼ ð! xi 1,000,000 1,000,000, if xi ) if xi > 1,000
,000: The von Neumann–Morgenstern Theorem Among many contributions relevant to Part 3 of our text, in their book The Theory of Games and Economic Behavior, John von Neumann and Oscar Morgenstern developed a mathematical foundation for Bernoulli’s solution to the St. Petersburg paradox.5 In particular, they laid out basic axioms of rationality and showed that any person who is rational in this way would make choices under uncertainty as though he or she had a utility function over money U(x) and maximized the expected value of U(x) (rather than the expected value of the monetary payoff x itself). Although most of these axioms seem eminently reasonable at ﬁrst glance, many important questions about their tenability have been raised.6 We will not pursue these questions here, however. The von Neumann–Morgenstern utility index To begin, suppose that there are n possible prizes that an individual might win by participating in a lottery. Let these prizes be denoted by x1, x2,…, xn, and assume that these have been arranged in order of ascending desirability. Therefore, x1 is the least preferred prize for the individual and xn is the most preferred prize. Now assign arbitrary utility numbers to these two extreme prizes. For example, it is convenient to assign 0, 1, U U x1Þ ¼ ð xnÞ ¼ ð but any other pair of numbers would do equally well.7 Using these two values of utility, the point of the von Neumann–Morgenstern theorem is to show that a reasonable way exists to assign speciﬁc utility numbers to the other prizes available. Suppose that we choose any other prize, say, xi. Consider the following experiment. Ask the individual to state the probability, say, pi, at which he or she would be indifferent between xi with (7.9) 5J. von Neumann and O. Morgenstern, The Theory of Games and Economic Behavior (Princeton, NJ: Princeton University Press, 1944). The axioms of rationality in uncertain situations are discussed in the book’s appendix. 6For a discussion of some of the issues raised in the debate over the von Neumann–Morgenstern axioms, especially the assumption of independence, see C. Gollier, The Economics of Risk and Time (Cambridge, MA
: MIT Press, 2001), chap. 1. 7Technically, a von Neumann–Morgenstern utility index is unique only up to a choice of scale and origin—that is, only up to a ‘‘linear transformation.’’ This requirement is more stringent than the requirement that a utility function be unique up to a monotonic transformation. Chapter 7: Uncertainty 213 % certainty, and a gamble offering prizes of xn with probability pi and x1 with probability pi). It seems reasonable (although this is the most problematic assumption in the (1 von Neumann–Morgenstern approach) that such a probability will exist: The individual will always be indifferent between a gamble and a sure thing, provided that a high enough probability of winning the best prize is offered. It also seems likely that pi will be higher the more desirable xi is; the better xi is, the better the chance of winning xn must be to get the individual to gamble. Therefore, the probability pi measures how desirable the prize xi is. In fact, the von Neumann–Morgenstern technique deﬁnes the utility of xi as the expected utility of the gamble that the individual considers equally desirable to xi: U xiÞ ¼ piÞ ð Because of our choice of scale in Equation 7.9, we have xnÞ þ ð ð piU % U 1 : x1Þ ð U xiÞ ¼ ð pi ( 1 1 0 piÞ ( % þ ð ¼ pi: (7.10) (7.11) By judiciously choosing the utility numbers to be assigned to the best and worst prizes, we have been able to devise a scale under which the utility index attached to any other prize is simply the probability of winning the top prize in a gamble the individual regards as equivalent to the prize in question. This choice of utility indices is arbitrary. Any other two numbers could have been used to construct this utility scale, but our initial choice (Equation 7.9) is a particularly convenient one. Expected utility maximization In line with the choice of scale and origin represented by Equation 7.9, suppose that a utility index pi has been assigned to every prize xi. Notice in particular that p1 ¼ 1, and that the other utility indices range between these extremes. Using these utility indices, we can show that a
‘‘rational’’ individual will choose among gambles based on their expected ‘‘utilities’’ (i.e., based on the expected value of these von Neumann–Morgenstern utility index numbers). 0, pn ¼ As an example, consider two gambles. Gamble A offers x2 with probability a and x3 a). Gamble B offers x4 with probability b and x5 with probability b). We want to show that this person will choose gamble A if and only if the with probability (1 (1 expected utility of gamble A exceeds that of gamble B. Now for the gambles: % % expected utility of A expected utility of B aU bU ¼ ¼ 1 x2Þ þ ð ð x4Þ, x3Þ ð : x5Þ ð (7.12) Substituting the utility index numbers (i.e., p2 is the ‘‘utility’’ of x2, and so forth) gives expected utility of A expected utility of B 1 ap2 þ ð 1 bp4 þ ð a b p3, Þ p5: Þ % % ¼ ¼ (7.13) We wish to show that the individual will prefer gamble A to gamble B if and only if 1 b 1 a % ap2 þ ð p3 > bp4 þ ð Þ To show this, recall the deﬁnitions of the utility index. The individual is indifferent between x2 and a gamble promising x1 with probability (1 p2) and xn with probability p2. We can use this fact to substitute gambles involving only x1 and xn for all utilities in Equation 7.13 (even though the individual is indifferent between these, the assumption that this substitution can be made implicitly assumes that people can see through complex lottery combinations). After a bit of messy algebra, we can conclude that gamble A is p5: Þ (7.14) % % 214 Part 3: Uncertainty and Strategy a)p3, and gamble B is equivalent to a gamble promising xn with probability ap2 + (1 % equivalent to a gamble promising xn with probability bp4 + (1 b)p5. The individual will presumably prefer the gamble with the higher probability of winning the best prize. Consequently, he or she will choose gamble A if and
only if ap2 þ ð p3 > bp4 þ ð Þ But this is precisely what we wanted to show. Consequently, we have proved that an individual will choose the gamble that provides the highest level of expected (von Neumann– Morgenstern) utility. We now make considerable use of this result, which can be summarized as follows. p5: b Þ (7.15 Expected utility maximization. If individuals obey the von Neumann–Morgenstern axioms of behavior in uncertain situations, they will act as though they choose the option that maximizes the expected value of their von Neumann–Morgenstern utility. Risk Aversion Economists have found that people tend to avoid risky situations, even if the situation amounts to a fair gamble. For example, few people would choose to take a $10,000 bet on the outcome of a coin ﬂip, even though the average payoff is 0. The reason is that the gamble’s money prizes do not completely reﬂect the utility provided by the prizes. The utility that people obtain from an increase in prize money may increase less rapidly than the dollar value of these prizes. A gamble that is fair in money terms may be unfair in utility terms and thus would be rejected. In more technical terms, extra money may provide people with decreasing marginal utility. A simple example can help explain why. An increase in income from, say, $40,000 to $50,000 may substantially increase a person’s well-being, ensuring he or she does not have to go without essentials such as food and housing. A further increase from $50,000 to $60,000 allows for an even more comfortable lifestyle, perhaps providing tastier food and a bigger house, but the improvement will likely not be as great as the initial one. Starting from a wealth of $50,000, the individual would be reluctant to take a $10,000 bet on a coin ﬂip. The 50 percent chance of the increased comforts that he or she could have with $60,000 does not compensate for the 50 percent chance that he or she will end up with $40,000 and perhaps have to forgo some essentials. These effects are only magniﬁed with riskier gambles, that is, gambles having even more variable outcomes.8 The person with initial wealth of $50,000 would likely be even more reluctant to take a $
20,000 bet on a coin ﬂip because he or she would face the prospect of ending up with only $30,000 if the ﬂip turned out badly, severely cutting into life’s essentials. The equal chance of ending up with $70,000 is not adequate compensation. On the other hand, a bet of only $1 on a coin ﬂip is relatively inconsequential. Although the person may still decline the bet, he or she would not try hard to do so because his or her ultimate wealth hardly varies with the outcome of the coin toss. Risk aversion and fair bets This argument is illustrated in Figure 7.1. Here W0 represents an individual’s current wealth and U(W ) is a von Neumann–Morgenstern utility index (we will call this a utility 8Often the statistical concepts of variance and standard deviation are used to measure. We will do so at several places later in this chapter. Chapter 7: Uncertainty 215 FIGURE 7.1 Utility of Wealth from Two Fair Bets of Differing Variability If the utility-of-wealth function is concave (i.e., exhibits a diminishing marginal utility of wealth), then this person will refuse fair bets. A 50–50 chance of winning or losing h dollars, for example, yields less expected utility [EU(A)] than does refusing the bet. The reason for this is that winning h dollars means less to this individual than does losing h dollars. Utility U(W0) EU(A) = U(CEA) EU(B) U(W) W0 – 2h W0 + h W0 W0 + h W0 + 2h Wealth (W) CEA function from now on) that reﬂects how he or she feels about various levels of wealth.9 In the ﬁgure, U(W ) is drawn as a concave function of W to reﬂect the assumption of a diminishing marginal utility. Now suppose this person is offered two fair gambles: gamble A, which is a 50–50 chance of winning or losing $h, and gamble B, which is a 50–50 chance of winning or losing $2h. The utility of current wealth is U(W0), which is also the expected value of current wealth because it is certain. The expected utility if he or she participates in gamble A is given by EU(A): EU A �
� Þ ¼ 1 2 U W0 þ ð h Þ þ 1 2 U W0 % ð h, Þ and the expected utility of gamble B is given by EU(B): EU B ð Þ ¼ 1 2 U W0 þ ð 2h Þ þ 1 2 U W0 % ð. 2h Þ (7.16) (7.17) Equation 7.16 shows that the expected utility from gamble A is halfway between the utility h and the utility from favorable outcome W0 + h. from the unfavorable outcome W0 % Likewise, the expected utility from gamble B is halfway between the utilities from unfavorable and favorable outcomes, but payoffs in these outcomes vary more than with gamble A. 9Technically, U(W ) is an indirect utility function because it is the consumption allowed by wealth that provides direct utility. In Chapter 17 we will take up the relationship between consumption-based utility functions and their implied indirect utility of wealth functions. 216 Part 3: Uncertainty and Strategy It is geometrically clear from the ﬁgure that10 U > EU W0Þ ð A Þ ð Therefore, this person will prefer to keep his or her current wealth rather than taking either fair gamble. If forced to choose a gamble, the person would prefer the smaller one (A) to the large one (B). The reason for this is that winning a fair bet adds to enjoyment less than losing hurts. (7.18) > EU B Þ ð : Risk aversion and insurance As a matter of fact, this person might be willing to pay some amount to avoid participating in any gamble at all. Notice that a certain wealth of CEA provides the same expected utility as does participating in gamble A. CEA is referred to as the certainty equivalent of gamble A. The individual would be willing to pay up to W0 % CEA to avoid participating in the gamble. This explains why people buy insurance. They are giving up a small, certain amount (the insurance premium) to avoid the risky outcome they are being insured against. The premium a person pays for automobile collision insurance, for example, provides a policy that agrees to repair his or her car should an accident occur. The widespread use of insurance would seem to imply that aversion to risk is prevalent. In fact, the person in Figure 7.1 would pay even more to avoid taking the larger gamble, B. As an exercise, try to identify the certainty
equivalent CEB of gamble B and the amount the person would pay to avoid gamble B on the ﬁgure. The analysis in this section can be summarized by the following deﬁnition Risk aversion. An individual who always refuses fair bets is said to be risk averse. If individuals exhibit a diminishing marginal utility of wealth, they will be risk averse. As a consequence, they will be willing to pay something to avoid taking fair bets. EXAMPLE 7.2 Willingness to Pay for Insurance To illustrate the connection between risk aversion and insurance, consider a person with a current wealth of $100,000 who faces the prospect of a 25 percent chance of losing his or her $20,000 automobile through theft during the next year. Suppose also that this person’s von Neumann–Morgenstern utility function is logarithmic; that is, U(W ) ln (W ). If this person faces next year without insurance, expected utility will be ¼ no insurance EU ð 0:75U 100,000 ð 0:75 ln 100,000 11:45714: 0:25U 80,000 Þ ð 0:25 ln 80,000 Þ þ þ Þ ¼ ¼ ¼ (7.19) In this situation, a fair insurance premium would be $5,000 (25 percent of $20,000, assuming that the insurance company has only claim costs and that administrative costs are $0). 10Technically this result is a direct consequence of Jensen’s inequality in mathematical statistics. The inequality states that if x is a random variable and f(x) is a strictly concave function of that variable, then E[ f (x)] < f [E(x)]. In the utility context, this means that if utility is concave in a random variable measuring wealth (i.e., if U 0(W) > 0 and U 00(W) < 0), then the expected utility of wealth will be less than the utility associated with the expected value of W. With gamble A, for example, EU(A) < U(W0) because, as a fair gamble, A provides expected wealth W0. Chapter 7: Uncertainty 217 Consequently, if this person completely insures the car, his or her wealth will be $95,000 regardless of whether the car is stolen. In this case then, EU fair insurance ð U 95,000 �
� ð 95,000 ln Þ ð 11:46163: Þ ¼ ¼ ¼ (7.20) This person is made better off by purchasing fair insurance. Indeed, he or she would be willing to pay more than the fair premium for insurance. We can determine the maximum insurance premium (x) by setting EU maximum-premium insurance ð 100,000 U ð ln 100,000 ð 11:45714 Solving this equation for x yields or 100,000 x % ¼ e11:45714; 5,426: x ¼ (7.21) (7.22) (7.23) This person would be willing to pay up to $426 in administrative costs to an insurance company (in addition to the $5,000 premium to cover the expected value of the loss). Even when these costs are paid, this person is as well off as he or she would be when facing the world uninsured. QUERY: Suppose utility had been linear in wealth. Would this person be willing to pay anything more than the actuarially fair amount for insurance? How about the case where utility is a convex function of wealth? Measuring Risk Aversion In the study of economic choices in risky situations, it is sometimes convenient to have a quantitative measure of how averse to risk a person is. The most commonly used measure of risk aversion was initially developed by J. W. Pratt in the 1960s.11 This risk aversion measure, r (W ), is deﬁned as r W U 00 W ð U 0ð W Because the distinguishing feature of risk-averse individuals is a diminishing marginal utility of wealth [U 00(W) < 0], Pratt’s measure is positive in such cases. The measure is invariant with respect to linear transformations of the utility function, and therefore not affected by which particular von Neumann–Morgenstern ordering is used. Þ ¼ % (7.24) Þ Þ ð : Risk aversion and insurance premiums A useful feature of the Pratt measure of risk aversion is that it is proportional to the amount an individual will pay for insurance against taking a fair bet. Suppose the winnings from such a fair bet are denoted by the random variable h (which takes on both 11J. W. Pratt, ‘‘Risk Aversion in the Small and in the Large,’’ Econometrica (January/April 1964

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.