Split (1)

train · 11.7k rows

text stringlengths 204 3.13k
of its two arguments will multiply the value of the function by 4. However, the monotonic function that simply adds 1.0 to f (i.e., F( f ) 1) is not homogeneous at all. Thus, ¼ except in special cases, homothetic functions do not possess the homogeneity properties of their underlying functions. Homothetic functions, however, do preserve one nice x1x2 þ ¼ þ ¼ 1 f 18Because a limiting case of a monotonic transformation is to leave the function unchanged, all homogeneous functions are also homothetic. Chapter 2: Mathematics for Microeconomics 57 feature of homogeneous functions—that the implicit trade-offs implied by the function depend only on the ratio of the two variables being traded, not on their absolute levels. To show this, remember that Equation 2.23 showed that for a two-variable function of f (x1, x2) the implicit trade-off between the two variables required to keep the form y the value of the function constant is given by ¼ dx2 dx1 ¼ % f1 f2 : If we assume that f is homogeneous of degree k, its partial derivatives will be homogeneous of degree k – 1; therefore, we can write this trade-off as: tk tk 1f1ð 1f2ð : % % ¼ ¼ % f1ð f2ð Now let t dx2 dx1 ¼ % tx1, tx2Þ tx1, tx2Þ tx1, tx2Þ tx1, tx2Þ 1/x2 so Equation 2.124 becomes f1ð f2ð which shows that the trade-offs implicit in f depend only on the ratio of x1 to x2. If we apply any monotonic transformation F (with F 0 > 0) to the original homogeneous function f, the trade-offs implied by the new homothetic function F[ f (x1, x2)] are unchanged x1=x2, 1 Þ x1=x2, 1 Þ dx2 dx1 ¼ % (2:124) (2:125), dx2 dx1 ¼ % x1=x2, 1 x1=x2, 1 F0f1ð F0f2ð Þ Þ At many places in this book we will ﬁnd it instructive to discuss some theoretical results with
two-dimensional graphs, and Equation 2.126 can be used to focus our attention on the ratios of the key variables rather than on their absolute levels. ¼ % (2:126) : f1ð f2ð x1=x2, 1 Þ x1=x2, 1 Þ EXAMPLE 2.12 Cardinal and Ordinal Properties In applied economics it is sometimes important to know the exact numerical relationship among variables. For example, in the study of production, one might wish to know precisely how much extra output would be produced by hiring another worker. This is a question about the ‘‘cardinal’’ (i.e., numerical) properties of the production function. In other cases, one may only care about the order in which various points are ranked. In the theory of utility, for example, we assume that people can rank bundles of goods and will choose the bundle with the highest ranking, but that there are no unique numerical values assigned to these rankings. Mathematically, ordinal properties of functions are preserved by any monotonic transformation because, by deﬁnition, a monotonic transformation preserves order. Usually, however, cardinal properties are not preserved by arbitrary monotonic transformations. These distinctions are illustrated by the functions we examined in Example 2.11. There we studied monotonic transformations of the function k f x1x2Þ x1, x2Þ ¼ ð ð by considering various values of the parameter k. We showed that quasi-concavity (an ordinal property) was preserved for all values of k. Hence when approaching problems that focus on maximizing or minimizing such a function subject to linear constraints we need not worry about precisely which transformation is used. On the other hand, the function in Equation 2.127 is concave (a cardinal property) only for a narrow range of values of k. Many monotonic transformations destroy the concavity of f. (2:127) 58 Part 1: Introduction The function in Equation 2.127 also can be used to illustrate the difference between homogeneous and homothetic functions. A proportional increase in the two arguments of f would yield f t2kx1x2 ¼ Hence the degree of homogeneity for this function depends on k—that is, the degree of homogeneity is not preserved independently of which monotonic transformation is used. Alternatively, the function in Equation 2.127 is homothetic
because tx1, tx2Þ ¼ ð : x1, x2Þ ð (2:128) t2kf dx2 dx1 ¼ % f 1 f 2 ¼ % kxk kxk 1 1 xk % 2 1 1xk 2 ¼ % % x2 x1 : (2:129) That is, the trade-off between x2 and x1 depends only on the ratio of these two variables and is unaffected by the value of k. Hence homotheticity is an ordinal property. As we shall see, this property is convenient when developing graphical arguments about economic propositions. QUERY: How would the discussion in this example be changed if we considered monotonic transformations of the form f (x1, x2, k) k for various values of k? x1x2 þ ¼ Integration Integration is another of the tools of calculus that ﬁnds a number of applications in microeconomic theory. The technique is used both to calculate areas that measure various economic outcomes and, more generally, to provide a way of summing up outcomes that occur over time or across individuals. Our treatment of the topic here necessarily must be brief; therefore, readers desiring a more complete background should consult the references at the end of this chapter. Antiderivatives Formally, integration is the inverse of differentiation. When you are asked to calculate the integral of a function, f (x), you are being asked to ﬁnd a function that has f (x) as its derivative. If we call this ‘‘antiderivative’’ F(x), this function is supposed to have the property that dF x Þ ð dx ¼ F0 x ð Þ ¼ f x ð : Þ If such a function exists, then we denote it as F x ð Þ ¼ ð dx: f x ð Þ (2:130) (2:131) The precise reason for this rather odd-looking notation will be described in detail later. First, let’s look at a few examples. If f (x) x then ¼ F x ð Þ ¼ dx f x ð Þ ¼ x dx x2 2 þ ¼ C, (2:132) ð where C is an arbitrary ‘‘constant of integration’’ that disappears on differentiation. The
correctness of this result can be easily veriﬁed: ð F0 x ð Þ ¼ d x2=2 ð dx þ C Þ x 0 þ ¼ ¼ x: (2:133) Chapter 2: Mathematics for Microeconomics 59 Calculating antiderivatives Calculation of antiderivatives can be extremely simple, or difﬁcult, or agonizing, or impossible, depending on the particular f (x) speciﬁed. Here we will look at three simple methods for making such calculations, but, as you might expect, these will not always work. 1. Creative guesswork. Probably the most common way of ﬁnding integrals (antiderivatives) is to work backward by asking ‘‘What function will yield f (x) as its derivative?’’ Here are a few obvious examples: x2 dx xn dx ¼ ¼ C, x3 3 þ xn 1 þ n þ C, 1 þ ax2 ð bx dx c Þ þ ¼ þ ex dx ex C, þ ¼ ax3 3 þ bx2 2 þ cx C, þ (2:134) ax ln a þ C, ln x ðj ¼ jÞ þ C, ¼ dx dx ¼ x ln x x % þ C: Þ ax dx 1 x ( ) ln You should use differentiation to check that all these obey the property that F 0(x) ¼ f (x). Notice that in every case the integral includes a constant of integration because antiderivatives are unique only up to an additive constant, which would become zero on differentiation. For many purposes, the results in Equation 2.134 (or trivial generalizations of them) will be sufﬁcient for our purposes in this book. Nevertheless, here are two more methods that may work when intuition fails. 2. Change of variable. A clever redeﬁnition of variables may sometimes make a function much easier to integrate. For example, it is not at all obvious what the integral of x2, then dy 2x/(1 x2) is. But, if we let y 2xdx and 1 þ ¼ 2x 1 ð þ ¼ þ 1 y ð x2 dx ¼ dy ln y ðj ¼ jÞ ¼ 1 l
n ðj þ x2 : jÞ (2:135) The key to this procedure is in breaking the original function into a term in y and a term in dy. It takes a lot of practice to see patterns for which this will work. 3. Integration by parts. A similar method for ﬁnding integrals makes use of the identity vdu for any two functions u and v. Integration of this differential yields udv duv ¼ þ duv uv ¼ ð ¼ ð u dv þ ð v du or u dv uv ¼ ð % ð v du: (2:136) Here the strategy is to deﬁne functions u and v in a way that the unknown integral on the left can be calculated by the difference between the two known expressions on the right. For example, it is by no means obvious what the integral of xex is. But we can deﬁne u ex). Hence we now have exdx (thus, v x (thus, du dx) and dv ¼ ¼ ¼ xex dx ð u dv uv ¼ v du ¼ % ð ¼ ð ex dx x 1 ex Þ % þ C: ¼ ð (2:137) % ð ¼ xex 60 Part 1: Introduction Again, only practice can suggest useful patterns in the ways in which u and v can be deﬁned. Deﬁnite integrals The integrals we have been discussing thus far are ‘‘indeﬁnite’’ integrals—they provide only a general function that is the antiderivative of another function. A somewhat different, although related, approach uses integration to sum up the area under a graph of a function over some deﬁned interval. Figure 2.5 illustrates this process. We wish to know b. One way to do this would be to the area under the function f (x) from x partition the interval into narrow slivers of x (Dx) and sum up the areas of the rectangles shown in the ﬁgure. That is: a to x ¼ ¼ area under f x ð Þ, f ð Dxi; xiÞ i X (2:138) where the notation is intended to indicate that the height of each rectangle is approx
imated by the value of f (x) for a value of x in the interval. Taking this process to the limit by shrinking the size of the D x intervals yields an exact measure of the area we want and is denoted by: area under dx: f x ð Þ (2:139) This then explains the origin of the oddly shaped integral sign—it is a stylized S, indicating ‘‘sum.’’ As we shall see, integrating is a general way of summing the values of a continuous function over some interval. FIGURE 2.5 Deﬁnite Integrals Show the Areas Under the Graph of a Function Deﬁnite integrals measure the area under a curve by summing rectangular areas as shown in the graph. The dimension of each rectangle is f(x)dx. f(x) f(x) a b x Chapter 2: Mathematics for Microeconomics 61 Fundamental theorem of calculus Evaluating the integral in Equation 2.139 is simple if we know the antiderivative of f (x), say, F(x). In this case we have area under dx 2:140) That is, all we need do is calculate the antiderivative of f (x) and subtract the value of this function at the lower limit of integration from its value at the upper limit of integration. This result is sometimes termed the fundamental theorem of calculus because it directly ties together the two principal tools of calculus—derivatives and integrals. In Example 2.13, we show that this result is much more general than simply a way to measure areas. It can be used to illustrate one of the primary conceptual principles of economics—the distinction between ‘‘stocks’’ and ‘‘ﬂows.’’ EXAMPLE 2.13 Stocks and Flows The deﬁnite integral provides a useful way for summing up any function that is providing a continuous ﬂow over time. For example, suppose that net population increase (births minus 1,000e0.02t. Hence the net deaths) for a country can be approximated by the function f(t) population change is growing at the rate of 2 percent per year—it is 1,000 new people in year 0, 1,020 new people in the ﬁrst year, 1,041 in the second year, and so forth. Suppose we wish
to know how much in total the population will increase within 50 years. This might be a tedious calculation without calculus, but using the fundamental theorem of calculus provides an easy answer: ¼ increase in population ¼ ¼ 50 t ¼ f dt t ð Þ ð 0 t ¼ 1,000e0:02t 0:02 t 50 ¼ 1,000e0:02tdt ð 0 ¼ 1,000e 0:02 % 50,000 50 F t ð 0 Þ!!!! 85,914 ¼ ¼ (2:141) ¼ t 50 0 ¼!!!! j b F(a)]. Hence the a indicates that the expression is to be evaluated as F(b) [where the notation conclusion is that the population will grow by nearly 86,000 people over the next 50 years. Notice how the fundamental theorem of calculus ties together a ‘‘ﬂow’’ concept, net population increase (which is measured as an amount per year), with a ‘‘stock’’ concept, total population (which is measured at a speciﬁc date and does not have a time dimension). Note also that the 86,000 calculation refers only to the total increase between year 0 and year 50. To know the actual total population at any date we would have to add the number of people in the population at year 0. That would be similar to choosing a constant of integration in this speciﬁc problem. % ¼ 0.1q2 Now consider an application with more economic content. Suppose that total costs for a particular ﬁrm are given by C(q) 500 (where q represents output during some period). Here the term 0.1q2 represents variable costs (costs that vary with output), whereas the 500 ﬁgure represents ﬁxed costs. Marginal costs for this production process can be found 0.2q—hence marginal costs are increasing with q through differentiation—MC and ﬁxed costs drop out on differentiation. What are the total costs associated with producing, say, q ¼ 0.1(100)2 0 to 100 to get total variable cost: ¼ 1,500. An alternative way would be to integrate marginal cost over the range 100? One way to answer this question is to use the total cost function directly: C(100) d
C(q)/dq 500 þ ¼ ¼ ¼ þ variable cost 100 q ¼ ¼ ð 0 q ¼ 0:2q dq 0:1q2 ¼ 100 0 ¼!!!! 1,000 0 % ¼ 1,000; (2:142) 62 Part 1: Introduction to which we would have to add ﬁxed costs of 500 (the constant of integration in this problem) to get total costs. Of course, this method of arriving at total cost is much more cumbersome than just using the equation for total cost directly. But the derivation does show that total variable cost between any two output levels can be found through integration as the area below the marginal cost curve—a conclusion that we will ﬁnd useful in some graphical applications. QUERY: How would you calculate the total variable cost associated with expanding output from 100 to 110? Explain why ﬁxed costs do not enter into this calculation. Differentiating a deﬁnite integral Occasionally we will wish to differentiate a deﬁnite integral—usually in the context of seeking to maximize the value of this integral. Although performing such differentiations can sometimes be rather complex, there are a few rules that should make the process easier. 1. Differentiation with respect to the variable of integration. This is a trick question, but instructive nonetheless. A deﬁnite integral has a constant value; hence its derivative is zero. That is: d b a f Ð x ð dx dx Þ 0: ¼ (2:143) The summing process required for integration has already been accomplished once we write down a deﬁnite integral. It does not matter whether the variable of integration is x or t or anything else. The value of this integrated sum will not change when the variable x changes, no matter what x is (but see rule 3 below). 2. Differentiation with respect to the upper bound of integration. Changing the upper bound of integration will obviously change the value of a deﬁnite integral. In this case, we must make a distinction between the variable determining the upper bound of integration (say, x) and the variable of integration (say, t). The result then is a simple application of the fundamental theorem of calculus. For example: dt t Þ d x a f ð dx Ð d x F ½ ð Þ % dx ¼ F a
ð Þ2:144) where F(x) is the antiderivative of f (x). By referring back to Figure 2.5 we can see why this conclusion makes sense—we are asking how the value of the deﬁnite integral changes if x increases slightly. Obviously, the answer is that the value of the integral increases by the height of f (x) (notice that this value will ultimately depend on the speciﬁed value of x). If the upper bound of integration is a function of x, this result can be generalized using the chain rule dx dt t Þ d g F ½ ð ð x ÞÞ % dx ¼ F a Þ( ð d F ½ x g ð ð dx ÞÞ( f dg x Þ ð dx ¼ ¼ ¼ f g ð x ð g0 x, Þ ð ÞÞ (2:145) where, again, the speciﬁc value for this derivative would depend on the value of x assumed. Finally, notice that differentiation with respect to a lower bound of integration just changes the sign of this expression: d b g ð x Ð dt t Þ f ð Þ dx d b F ½ ð F Þ % dx ¼ x g ð ð ÞÞ( dF x g ð ð dx ÞÞ ¼ % f x g ð ð ÞÞ g0 x ð : Þ ¼ % (2:146) Chapter 2: Mathematics for Microeconomics 63 3. Differentiation with respect to another relevant variable. In some cases we may wish to integrate an expression that is a function of several variables. In general, this can involve multiple integrals, and differentiation can become complicated. But there is one simple case that should be mentioned. Suppose that we have a function of two variables, f (x, y), and that we wish to integrate this function with respect to the variable x. The speciﬁc value for this integral will obviously depend on the value of y, and we might even ask how that value changes when y changes. In this case, it is possible to ‘‘differentiate through the integral sign’’ to obtain a result. That is: d b a f Ð dx x, y ð dy �
� ¼ b ð a x, y f yð Þ dx: (2:147) This expression shows that we can ﬁrst partially differentiate f (x, y) with respect to y before proceeding to compute the value of the deﬁnite integral. Of course, the resulting value may still depend on the speciﬁc value that is assigned to y, but often it will yield more economic insights than the original problem does. Some further examples of using deﬁnite integrals are found in Problem 2.8. Dynamic Optimization Some optimization problems that arise in microeconomics involve multiple periods.19 We are interested in ﬁnding the optimal time path for a variable or set of variables that succeeds in optimizing some goal. For example, an individual may wish to choose a path of lifetime consumptions that maximizes his or her utility. Or a ﬁrm may seek a path for input and output choices that maximizes the present value of all future proﬁts. The particular feature of such problems that makes them difﬁcult is that decisions made in one period affect outcomes in later periods. Hence one must explicitly take account of this interrelationship in choosing optimal paths. If decisions in one period did not affect later periods, the problem would not have a ‘‘dynamic’’ structure—one could just proceed to optimize decisions in each period without regard for what comes next. Here, however, we wish to explicitly allow for dynamic considerations. The optimal control problem Mathematicians and economists have developed many techniques for solving problems in dynamic optimization. The references at the end of this chapter provide broad introductions to these methods. Here, however, we will be concerned with only one such method that has many similarities to the optimization techniques discussed earlier in this chapter—the optimal control problem. The framework of the problem is relatively simple. A decision-maker wishes to ﬁnd the optimal time path for some variable x(t) over a speciﬁed time interval [t0, t1]. Changes in x are governed by a differential equation: dx t ð Þ dt 2:148) where the variable c(t) is used to ‘‘control’’ the change in x(t). In each period, the decision-maker derives value from x and c according to the function f[x(t), c(t
), t] and his or 19Throughout this section we treat dynamic optimization problems as occurring over time. In other contexts, the same techniques can be used to solve optimization problems that occur across a continuum of ﬁrms or individuals when the optimal choices for one agent affect what is optimal for others. The material in this section will be used in only a few places in the text, but is provided here as a convenient reference. 64 Part 1: Introduction t1 t0 to optimize, t her goal ( Þ to ‘‘endpoint’’ constraints on the variable x. These might be written as x(t0) x(t1) dt. Often this problem will also be subject x0 and, c ð Þ x1. ¼ x ð Ð t t f ½ Notice how this problem is ‘‘dynamic.’’ Any decision about how much to change x this period will affect not only the future value of x, but it will also affect future values of the outcome function f. The problem then is how to keep x(t) on its optimal path. ¼ Economic intuition can help to solve this problem. Suppose that we just focused on the function f and chose x and c to maximize it at each instant of time. There are two difﬁculties with this ‘‘myopic’’ approach. First, we are not really free to ‘‘choose’’ x at any time. Rather, the value of x will be determined by its initial value x0 and by its history of changes as given by Equation 2.148. A second problem with this myopic approach is that it disregards the dynamic nature of the problem by forgetting to ask how this period’s decisions affect the future. We need some way to reﬂect the dynamics of this problem in a single period’s decisions. Assigning the correct value (price) to x at each instant of time will do just that. Because this implicit price will have many similarities to the Lagrange multipliers studied earlier in this chapter, we will call it l(t). The value of l is treated as a function of time because the importance of x can obviously change over time. The maximum principle Now let’s look at the decision-maker’s problem at a single point in time. He or she must be concerned with both the current value of
the objective function f [x(t), c(t), t] and with the implied change in the value of x(t). Because the current value of x(t) is given by l(t)x(t), the instantaneous rate of change of this value is given by dt Þ( k t ð Þ ¼ dx t ð Þ dt þ x t ð Þ dk t ð dt Þ, (2:149) and so at any time t a comprehensive measure of the value of concern20 to the decisionmaker is dk ð dt Þ : (2:150) This comprehensive value represents both the current beneﬁts being received and the instantaneous change in the value of x. Now we can ask what conditions must hold for x(t) and c(t) to optimize this expression.21 That is: @H @c ¼ @H @x ¼ f c þ f x þ kgc ¼ kgx þ 0 or t dk Þ ð dt ¼ fc ¼ % or 0 kgc; f x þ kgx ¼ % t dk ð dt Þ : (2:151) These are then the two optimality conditions for this dynamic problem. They are usually referred to as the maximum principle. This solution to the optimal control problem was ﬁrst proposed by the Russian mathematician L. S. Pontryagin and his colleagues in the early 1960s. Although the logic of the maximum principle can best be illustrated by the economic applications we will encounter later in this book, a brief summary of the intuition behind them may be helpful. The ﬁrst condition asks about the optimal choice of c. It suggests 20We denote this current value expression by H to suggest its similarity to the Hamiltonian expression used in formal dynamic optimization theory. Usually the Hamiltonian expression does not have the ﬁnal term in Equation 2.150, however. 21Notice that the variable x is not really a choice variable here—its value is determined by history. Differentiation with respect to x can be regarded as implicitly asking the question: ‘‘If x(t) were optimal, what characteristics would it have?’’ Chapter 2: Mathematics for Microeconomics 65 that, at the margin, the gain from c in terms of the function f must be balanced by the losses from c in terms of the
value of its ability to change x. That is, present gains must be weighed against future costs. The second condition relates to the characteristics that an optimal time path of x(t) should have. It implies that, at the margin, any net gains from more current x (either in terms of f or in terms of the accompanying value of changes in x) must be balanced by changes in the implied value of x itself. That is, the net current gain from more x must be weighed against the declining future value of x. EXAMPLE 2.14 Allocating a Fixed Supply As an extremely simple illustration of the maximum principle, assume that someone has inherited 1,000 bottles of wine from a rich uncle. He or she intends to drink these bottles over the next 20 years. How should this be done to maximize the utility from doing so? Suppose that this person’s utility function for wine is given by u[c(t)] utility from wine drinking exhibits diminishing marginal utility goal is to maximize ¼ u0 > 0, u00 < 0 ð ln c(t). Hence the. This person’s Þ 20 ð 0 u ½ t c ð Þ( dt ¼ 20 ð 0 ln c t ð Þ dt: (2:152) Let x(t) represent the number of bottles of wine remaining at time t. This series is constrained by x(0) 0. The differential equation determining the evolution of x(t) takes ¼ the simple form:22 1,000 and x(20) ¼ dx t c ð Þ dt ¼ % ð t : Þ (2:153) That is, each instant’s consumption just reduces the stock of remaining bottles. The current value Hamiltonian expression for this problem is H t ln c ð ¼ Þ þ k c ½% t ð Þ( þ x t ð Þ dk dt, and the ﬁrst-order conditions for a maximum are 0, @H @c ¼ @H @x ¼ k 1 c % dk dt ¼ ¼ 0: (2:154) (2:155) The second of these conditions requires that l (the implicit value of wine) be constant over time. This makes intuitive sense: Because consuming a bottle of wine always reduces the available stock by one bottle, any solution where the value of wine differed over
� þ x t ð Þ dk t ð dt Þ, and the maximum principle requires that @H @c ¼ @H @x ¼ 0 and k ¼ dt e( dk dt ¼ 0: (2:157) (2:158) (2:159) (2:160) Hence we can again conclude that the implicit value of the wine stock (l) should be constant over time (call this constant k) and that t ð Thus, optimal wine consumption should fall over time to compensate for the fact that future 0.1 and g consumption is being discounted in the consumer’s mind. If, for example, we let d k or c (2:161) t c ð Þ ¼ e% ¼ Þ( % % % ½ g k1= ð 1 Þ edt= ð g 1 Þ: dt g 1 1 (‘‘reasonable’’ values, as we will show in later chapters), then ¼ % Now we must do a bit more work in choosing k to satisfy the endpoint constraints. We want c t ð Þ ¼ k% 0:5e% 0:05t (2:162) ¼ 20 ð 0 dt c t ð Þ ¼ 20 ð 0 k% 0:5e% 0:05t dt 20k% ¼ % 20 0:5e% 0:05t 0!!!! ¼ 1,000: Þ ¼ Finally, then, we have the optimal consumption plan as ¼ % % 0:5 20k% 1 e% ð 1 12:64k% 0:5 c t ð Þ, 79e% 0:05t: (2:163) (2:164) Chapter 2: Mathematics for Microeconomics 67 This consumption plan requires that wine consumption start out fairly high and decrease at a continuous rate of 5 percent per year. Because consumption is continuously decreasing, we must use integration to calculate wine consumption in any particular year (x) as follows: x x consumption in year x dt,580 79e% 0:05t dt 1,580e% 0:05t ¼ % 0:05 e% ð x 1 % Þ ð % 0:05x e2:165) 1, consumption is approximately 77 bottles in this �
��rst year. Consumption then decreases If x smoothly, ending with approximately 30 bottles being consumed in the 20th year. ¼ QUERY: Our ﬁrst illustration was just an example of the second in which d 0. Explain how alternative values of these parameters will affect the path of optimal wine consumption. Explain your results intuitively (for more on optimal consumption over time, see Chapter 17). ¼ ¼ g Mathematical Statistics In recent years microeconomic theory has increasingly focused on issues raised by uncertainty and imperfect information. To understand much of this literature, it is important to have a good background in mathematical statistics. Therefore, the purpose of this section is to summarize a few of the statistical principles that we will encounter at various places in this book. Random variables and probability density functions A random variable describes (in numerical form) the outcomes from an experiment that is subject to chance. For example, we might ﬂip a coin and observe whether it lands heads or tails. If we call this random variable x, we can denote the possible outcomes (‘‘realizations’’) of the variable as: 1 0 if coin is heads, if coin is tails. x ¼, Notice that, before the ﬂip of the coin, x can be either 1 or 0. Only after the uncertainty is resolved (i.e., after the coin is ﬂipped) do we know what the value of x is.23 Discrete and continuous random variables The outcomes from a random experiment may be either a ﬁnite number of possibilities or a continuum of possibilities. For example, recording the number that comes up on a single die is a random variable with six outcomes. With two dice, we could either record the sum of the faces (in which case there are 12 outcomes, some of which are more likely than others) or we could record a two-digit number, one for the value of each die (in which case there would be 36 equally likely outcomes). These are examples of discrete random variables. Alternatively, a continuous random variable may take on any value in a given range of real numbers. For example, we could view the outdoor temperature tomorrow as a 23Sometimes random variables are denoted by ~x to make a distinction between variables whose outcome is subject to random chance and (nonrandom) algebraic variables. This notational device can be useful for keeping track of what is random and what is not in a particular problem, and we will use it
in some cases. When there is no ambiguity, however, we will not use this special notation. 68 Part 1: Introduction þ 50!C to continuous variable (assuming temperatures can be measured ﬁnely) ranging from, say, 50!C. Of course, some of these temperatures would be unlikely to occur, but % in principle the precisely measured temperature could be anywhere between these two bounds. Similarly, we could view tomorrow’s percentage change in the value of a particu1,000%. Again, of lar stock index as taking on all values between course, percentage changes around 0% would be considerably more likely to occur than would be the extreme values. 100% and, say, % þ Probability density functions For any random variable, its probability density function (PDF) shows the probability that each speciﬁc outcome will occur. For a discrete random variable, deﬁning such a function poses no particular difﬁculties. In the coin ﬂip case, for example, the PDF [denoted by f (x)] would be given by :5, 0:5: (2:166) Þ ¼ For the roll of a single die, the PDF would be: ¼ 1 2 3 4 5 6 1=6, 1=6, 1=6, 1=6, 1=6, 1=6 Notice that in both these cases the probabilities speciﬁed by the PDF sum to 1.0. This is because, by deﬁnition, one of the outcomes of the random experiment must occur. More generally, if we denote all the outcomes for a discrete random variable by xi for i 1, …, n, then we must have: ¼ ¼ ¼ ¼ ¼ ¼ (2:167) ¼ n 1 i X ¼ f xiÞ ¼ ð 1: (2:168) For a continuous random variable we must be careful in deﬁning the PDF concept. Because such a random variable takes on a continuum of values, if we were to assign any nonzero value as the probability for a speciﬁc outcome (i.e., a temperature of 25.53470!C), we could quickly have sums of probabilities that are inﬁnitely large. þ Hence for a continuous random variable we deﬁne the PDF f (x) as a function
with the property that the probability that x falls in a particular small interval dx is given by the area of f (x)dx. Using this convention, the property that the probabilities from a random experiment must sum to 1.0 is stated as follows: þ1 ð %1 dx f x ð Þ ¼ 1:0: (2:169) A few important PDFs Most any function will do as a PDF provided that f (x) 0 and the function sums (or integrates) to 1.0. The trick, of course, is to ﬁnd functions that mirror random experiments that occur in the real world. Here we look at four such functions that we will ﬁnd useful in various places in this book. Graphs for all four of these functions are shown in Figure 2.6. + FIGURE 2.6 Four Common Probability Density Functions Chapter 2: Mathematics for Microeconomics 69 Random variables that have these PDFs are widely used. Each graph indicates the expected value of the PDF shown. f(xa) Binomial b x a + b 2 (b) Uniform f(x) f(x) p 1 − p f(x) λ  1/ 2π √ 1/λ x (c) Exponential 0 (d) Normal x 1. Binomial distribution. This is the most basic discrete distribution. Usually x is assumed to take on only two values, 1 and 0. The PDF for the binomial is given by: f f x x ¼ ð ¼ ð where p: p, The coin ﬂip example is obviously a special case of the binomial where p (2:170) 0.5. ¼ 2. Uniform distribution. This is the simplest continuous PDF. It assumes that the possible values of the variable x occur in a deﬁned interval and that each value is equally likely. That is for a x - - b; for x < a or x > b: (2:171) 70 Part 1: Introduction Notice that here the probabilities integrate to 1.0: þ1 ð %1 dx dx :0: (2:172) 3. Exponential distribution. This is a continuous distribution for which the probabilities decrease at a smooth exponential rate as x increases. Formally: kx ke% 0 f x ð Þ ¼, if x > 0, 0, if x - (2:173)
where l is a positive constant. Again, it is easy to show that this function integrates to 1.0: dx f x ð Þ ¼ 1 ð 0 þ1 ð %1 ke% kx dx kx e% 1 Þ ¼ 1:0: (2:174) 4. Normal distribution. The Normal (or Gaussian) distribution is the most important in mathematical statistics. Its importance stems largely from the central limit theorem, which states that the distribution of any sum of independent random variables will increasingly approximate the Normal distribution as the number of such variables increases. Because sample averages can be regarded as sums of independent random variables, this theorem says that any sample average will have a Normal distribution no matter what the distribution of the population from which the sample is selected. Hence it may often be appropriate to assume a random variable has a Normal distribution if it can be thought of as some sort of average. The mathematical form for the Normal PDF is f x ð Þ ¼ 1 2pp e% x2=2, (2:175) ﬃﬃﬃﬃﬃ and this is deﬁned for all real values of x. Although the function may look complicated, a few of its properties can be easily described. First, the function is symmetric around zero (because of the x2 term). Second, the function is asymptotic to zero as x 0. This becomes large or small. Third, the function reaches its maximal value at x 0:4. Finally, the graph of this function has a general ‘‘bell shape’’— value is 1 a shape used throughout the study of statistics. Integration of this function is relatively tricky (although easy in polar coordinates). The presence of the constant 1 2pp is needed if the function is to integrate to 1.0. % ﬃﬃﬃﬃﬃ 2pp ¼, % ﬃﬃﬃﬃﬃ Expected value The expected value of a random variable is the numerical value that the random variable might be expected to have, on average.24 It is the ‘‘center of gravity’’ of the PDF. For a discrete random variable that takes on the values x1, x2, …, xn, the expected value is deﬁned as xi f xiÞ ð (
2:176) 24The expected value of a random variable is sometimes referred to as the mean of that variable. In the study of sampling this can sometimes lead to confusion between the expected value of a random variable and the separate concept of the sample arithmetic average. Chapter 2: Mathematics for Microeconomics 71 That is, each outcome is weighted by the probability that it will occur, and the result is summed over all possible outcomes. For a continuous random variable, Equation 2.176 is readily generalized as E x ð Þ ¼ þ1 ð %1 xf x ð Þ dx: (2:177) Again, in this integration, each value of x is weighted by the probability that this value will occur. The concept of expected value can be generalized to include the expected value of any function of a random variable [say, g (x)]. In the continuous case, for example, we would write þ1 g E g x ð ½ Þ( ¼ ð %1 As a special case, consider a linear function y x f x ð Þ ð Þ dx: ax þ ¼ b. Then E y ð Þ ¼ E ax ð þ b Þ ¼ þ1 ð %1 ax ð f b Þ x ð Þ þ dx a ¼ þ1 ð %1 xf x ð Þ dx þ b þ1 ð %1 f x ð Þ dx aE x ð Þ þ b: ¼ (2:178) (2:179) Sometimes expected values are phrased in terms of the cumulative distribution function (CDF) F(x), deﬁned as dt: (2:180) ð %1 That is, F(x) represents the probability that the random variable t is less than or equal to x. Using this notation, the expected value of x can be written as E x ð Þ ¼ þ1 ð %1 xdF x ð Þ (2:181) Because of the fundamental theorem of calculus, Equation 2.181 and Equation 2.177 mean exactly the same thing. EXAMPLE 2.15 Expected Values of a Few Random Variables The expected values of each of the random variables with the simple PDFs introduced earlier are easy to calculate. All these expected values are indicated on the graphs of the functions’ PDFs in Figure
2.6. 1. Binomial. In this case: (2:182) 72 Part 1: Introduction For the coin ﬂip case (where p ¼ this random variable is, as you might have guessed, one half. 0.5), this says that E(x) ¼ 0.5—the expected value of p ¼ 2. Uniform. For this continuous random variable dx ¼ 2 x2 b ð % a b a ¼ 2 b2 b ð % a2 2:183) Þ!!!! Again, as you might have guessed, the expected value of the uniform distribution is precisely halfway between a and b. 3. Exponential. For this case of declining probabilities: E x ð Þ ¼ 1 ð 0 xke% kxdx xe% ¼ % kx 1 k % kx e% 1 0 ¼ 1 k, (2:184)!!!! where the integration follows from the integration by parts example shown earlier in this chapter (Equation 2.137). Notice here that the faster the probabilities decline, the lower is the 20. 0.5 then E(x) expected value of x. For example, if l 0. A 0.05 then E(x) 4. Normal. Because the Normal PDF is symmetric around zero, it seems clear that E(x) 2, whereas if l ¼ ¼ formal proof uses a change of variable integration by letting u ¼ xdx): ¼ ¼ x2/2 (du ¼ ¼ 1 2pp xe% x2=2 dx 1 2pp ¼ þ1 ð %1 ﬃﬃﬃﬃﬃ e% u du ¼ þ1 ð %1 2pp ½% 1 ﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃ x2=2 e% þ1 (! %1!!! ﬃﬃﬃﬃﬃ 1 2pp ½ 0 ¼ 0 % ( ¼ 0: (2:185) Of course, the expected value of a normally distributed random variable (or of any random variable) may be altered by a linear transformation, as shown in Equation 2.179. þ ax b, then E (y) QUERY: A linear
transformation changes a random variable’s expected value in a predictable way—if y b. Hence for this transformation [say, h(x)] we have ¼ E[h(x)] h[E(x)]. Suppose instead that x were transformed by a concave function, say, g (x) with g0 > 0 and g00 < 0. How would E[ g (x)] compare with g [E(x)]? Note: This is an illustration of Jensen’s inequality, a concept we will pursue in detail in Chapter 7. See also Problem 2.14. aE(x) þ ¼ ¼ Variance and standard deviation The expected value of a random variable is a measure of central tendency. On the other hand, the variance of a random variable [denoted by r2 x or Var(x)] is a measure of dispersion. Speciﬁcally, the variance is deﬁned as the ‘‘expected squared deviation’’ of a random variable from its expected value. Formally: x Var ð Þ ¼ r2 x ¼ E x ½ð % E x ð 2 ÞÞ ( ¼ þ1 ð %1 x ð E x ð ÞÞ % 2f x ð Þ dx: (2:186) Somewhat imprecisely, the variance measures the ‘‘typical’’ squared deviation from the central value of a random variable. In making the calculation, deviations from the expected value are squared so that positive and negative deviations from the expected value will both contribute to this measure of dispersion. After the calculation is made, the squaring process can be reversed to yield a measure of dispersion that is in the original units in which the random variable was measured. This square root of the variance is. The wording of the term called the standard deviation and is denoted as rxð¼ Þ r2 x p ﬃﬃﬃﬃﬃﬃ Chapter 2: Mathematics for Microeconomics 73 effectively conveys its meaning: sx is indeed the typical (‘‘standard’’) deviation of a random variable from its expected value. When a random variable is subject to a linear transformation, its variance and stan- dard deviation will be changed in a fairly obvious way. If y ax ¼ þ
b, then r2 y ¼ þ1 ð %1 ax ½ b E ax ð % þ þ 2f b Þ( x ð Þ dx ¼ þ1 ð %1 a2 x ½ E x ð % 2f Þ( dx x ð Þ ¼ a2r2 x: (2:187) Hence addition of a constant to a random variable does not change its variance, whereas multiplication by a constant multiplies the variance by the square of the constant. Therefore, it is clear that multiplying a variable by a constant multiplies its standard deviation by that constant: sax ¼ asx. EXAMPLE 2.16 Variances and Standard Deviations for Simple Random Variables Knowing the variances and standard deviations of the four simple random variables we have been looking at can sometimes be useful in economic applications. 1. Binomial. The variance of the binomial can be calculated by applying the deﬁnition in its discrete analog: n r2 xi % p p 1 xiÞ ¼ ð ð p2 p % ÞÞ p2 1 ð i X ¼ 1 % ¼ ð. One implication of this result is that a binomial variable has the p 1 Þ % ð 0:25 and 0.5, ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:5. Because of the relatively ﬂat parabolic shape of p(1 – p), modest deviations of p Hence rx ¼ largest variance and standard deviation when p rx ¼ from 0.5 do not change this variance substantially. in which case r2 (2:188 % % % Þð p p Þ ð ) 2. Uniform. Calculating the variance of the uniform distribution yields a mildly interesting result: r2 & dx 12!!!! (2:189) This is one of the few places where the number 12 has any use in mathematics other than in measuring quantities of oranges or doughnuts. 3. Exponential. Integrating the variance formula for the exponential is relatively laborious. k2 and 1=k. Hence the mean and standard deviation are the same for the exponential distri- Fortunately, the result is simple; for
the exponential, it turns out that r2 rx ¼ bution—it is a ‘‘one-parameter distribution.’’ x ¼ % 1 4. Normal. The integration can also be burdensome in this case. But again the result is simple: For the Normal distribution, r2 1. Areas below the Normal curve can be readily calculated, and tables of these are available in any statistics text. Two useful facts about the Normal PDF are: rx ¼ x ¼ 1 þ ð 1 % dx f x ð Þ, 0:68 and 2 þ ð 2 % dx f x ð Þ, 0:95: (2:190) 74 Part 1: Introduction That is, the probability is approximately two thirds that a Normal variable will be within ±1 standard deviation of the expected value, and ‘‘most of the time’’ (i.e., with probability 0.95) it will be within ±2 standard deviations. Standardizing the Normal. If the random variable x has a standard Normal PDF, it will have an expected value of 0 and a standard deviation of 1. However, a simple linear transformation can be used to give this random variable any desired expected value (m) and standard deviation (s). Consider the transformation y m. Now sx y E ð Þ ¼ rE x ð Þ þ l ¼ l and Var y ð Þ ¼ r2 Var(x) r2: ¼ (2:191) ¼ þ r2 y ¼. N(m, s)) by using z Reversing this process can be used to ‘‘standardize’’ any Normally distributed random variable (y) with an arbitrary expected value (m) and standard deviation (s) (this is sometimes denoted ( y – m)/s. For example, SAT scores (y) are distributed Normally as y with an expected value of 500 points and a standard deviation of 100 points (i.e., y N(500, 100)). Hence z ( y – 500)/100 has a standard Normal distribution with expected value 0 and standard deviation 1. Equation 2.190 shows that approximately 68 percent of all scores lie between 400 and 600 points and 95 percent of all scores lie between 300 and 700 points. ¼ ¼. QUERY: Suppose that the random variable x is distributed uniformly along the interval [0, 12
]. What are the mean and standard deviation of x? What fraction of the x distribution is within ±1 standard deviation of the mean? What fraction of the distribution is within ±2 standard deviations of the expected value? Explain why this differs from the fractions computed for the Normal distribution. Covariance Some economic problems involve two or more random variables. For example, an investor may consider allocating his or her wealth among several assets, the returns on which are taken to be random. Although the concepts of expected value, variance, and so forth carry over more or less directly when looking at a single random variable in such cases, it is also necessary to consider the relationship between the variables to get a complete picture. The concept of covariance is used to quantify this relationship. Before providing a deﬁnition, however, we will need to develop some background. Consider a case with two continuous random variables, x and y. The PDF for these two variables, denoted by f (x, y), has the property that the probability associated with a set of outcomes in a small area (with dimensions dxdy) is given by f (x, y)dxdy. To be a proper PDF, it must be the case that: þ1 þ1 0 and f x, y ð Þ + f x, y ð Þ ¼ dx dy 1: (2:192) ð %1 The single variable measures we have already introduced can be developed in this twovariable context by ‘‘integrating out’’ the other variable. That is, ð %1 E x ð Þ ¼ x Var ð Þ ¼ þ1 þ1 ð %1 þ1 ð %1 þ1 ð %1 ð %1 xf x, y ð Þ dy dx and x ½ E x ð Þ ( % 2f ð x, y Þ dy dx: (2:193) Chapter 2: Mathematics for Microeconomics 75 In this way, the parameters describing the random variable x are measured over all possible outcomes for y after taking into account the likelihood of those various outcomes. In this context, the covariance between x and y seeks to measure the direction of association between the variables. Speciﬁcally the covariance between x and y [denoted as Cov(x, y)] is deﬁned as Cov x, y ð Þ ¼ þ
1 þ1 ð %1 ð %1 x ½ E x ð Þ(½ y % E y ð Þ( f x, y ð Þ % dx dy: (2:194) The covariance between two random variables may be positive, negative, or zero. If values of x that are greater than E(x) tend to occur relatively frequently with values of y that are greater than E( y) (and similarly, if low values of x tend to occur together with low values of y), then the covariance will be positive. In this case, values of x and y tend to move in the same direction. Alternatively, if high values of x tend to be associated with low values for y (and vice versa), the covariance will be negative. Two random variables are deﬁned to be independent if the probability of any particular value of, say, x is not affected by the particular value of y that might occur (and vice versa).25 In mathematical terms, this means that the PDF must have the property that f (x, y) g (x)h( y)—that is, the joint PDF can be expressed as the product of two single variable PDFs. If x and y are independent, their covariance will be zero: ¼ Cov x, y ð Þ ¼ þ1 þ1 ð %1 ð %1 x ½ E x ð Þ(½ y E y g Þ( x ð h y ð Þ Þ ð % % dx dy þ1 ð %1 ¼ x ½ E x ð g Þ( x ð Þ % dx ) þ1 ð %1 y ½ y E ð h Þ( y Þ ð % dy ¼ 0 ) 0 ¼ 0: (2:195) The converse of this statement is not necessarily true, however. A zero covariance does not necessarily imply statistical independence. Finally, the covariance concept is crucial for understanding the variance of sums or differences of random variables. Although the expected value of a sum of two random variables is (as one might guess) the sum of their expected values: E x ð þ y Þ ¼ þ1 þ1 ð %1 ð %1 x ð y f x, y ð Þ Þ þ dx dy þ1 ð %1 ¼ xf x, y
ð Þ dy dx þ þ1 ð %1 yf x, y ð Þ dx dy 2:196) the relationship for the variance of such a sum is more complicated. Using the deﬁnitions we have developed yields 25A formal deﬁnition relies on the concept of conditional probability. The conditional probability of an event B given that A has occurred (written P(B\|A) is deﬁned as P(B\|A) P(B). In this case, P(A and B) P(A); B and A are deﬁned to be independent if P(B\|A) P(A) Æ P(B). P(A and B) ¼ ¼ ¼ ¼ 76 Part 1: Introduction Var y x ð þ Þ ¼ ¼ ¼ þ1 þ1 ð %1 þ1 ð %1 þ1 ð %1 þ1 ð %1 þ1 ð ð %1 %1 Var 2f x, y ð Þ Þ ( dx dy ( 2f x, y ð Þ dx dy, y ð Þ % þ dx dy Var y 2Cov x Var( y). The variance of Hence if x and y are independent, then Var(x the sum will be greater than the sum of the variances if the two random variables have a positive covariance and will be less than the sum of the variances if they have a negative covariance. Problems 2.14–2.16 provide further details on some of the statistical results that are used in microeconomic theory. Var(x) y) þ ¼ þ (2:197) SUMMARY Despite the formidable appearance of some parts of this chapter, this is not a book on mathematics. Rather, the intention here was to gather together a variety of tools that will be used to develop economic models throughout the remainder of the text. Material in this chapter will then be useful as a handy reference. One way to summarize the mathematical tools introduced in this chapter is by stressing again the economic lessons that these tools illustrate: • Using mathematics provides a convenient, shorthand way for economists to develop their models. Implications of various economic assumptions can be studied in a simpliﬁed setting through the use of such mathematical tools. • The mathematical concept of the derivatives of a function is widely used in economic models because economists are often interested
in how marginal changes in one variable affect another variable. Partial derivatives are especially useful for this purpose because they are deﬁned to represent such marginal changes when all other factors are held constant. • The mathematics of optimization is an important tool for the development of models that assume that economic agents rationally pursue some goal. In the unconstrained case, the ﬁrst-order conditions state that any activity that contributes to the agent’s goal should be expanded up to the point at which the marginal contribution of further expansion is zero. In mathematical terms, the ﬁrst-order condition for an optimum requires that all partial derivatives be zero. • Most economic optimization problems involve constraints on the choices agents can make. In this case the ﬁrst-order conditions for a maximum suggest that each activity be operated at a level at which the ratio of the marginal beneﬁt of the activity to its marginal cost is the same for all activities actually used. This common marginal beneﬁt–marginal cost ratio is also equal to the Lagrange multiplier, which is often introduced to help solve constrained optimization problems. The Lagrange multiplier can also be interpreted as the implicit value (or shadow price) of the constraint. • The implicit function theorem is a useful mathematical device for illustrating the dependence of the choices that result from an optimization problem on the parameters of that problem (e.g., market prices). The envelope theorem is useful for examining how these optimal choices change when the problem’s parameters (prices) change. • Some optimization problems may involve constraints that are inequalities rather than equalities. Solutions to these problems often illustrate ‘‘complementary slackness.’’ That is, either the constraints hold with equality and their related Lagrange multipliers are nonzero, or the constraints are strict inequalities and their related Lagrange multipliers are zero. Again this illustrates how the Lagrange multiplier implies something about the ‘‘importance’’ of constraints. • The ﬁrst-order conditions shown in this chapter are only the necessary conditions for a local maximum or minimum. One must also check second-order Chapter 2: Mathematics for Microeconomics 77 conditions that require that certain curvature conditions be met. various ways of differentiating integrals play an important role in the theory of optimizing behavior. • Certain types of functions occur in many economic problems. Quasi-concave functions (those functions for which the level curves
by substitution and by using the Lagrange multiplier method. ¼ xy. Find the maximum value for f if x and y are constrained to sum to 1. Solve this problem in two ways: 2.4 The dual problem to the one described in Problem 2.3 is minimize x subject to xy þ y ¼ 0:25: 78 Part 1: Introduction Solve this problem using the Lagrangian technique. Then compare the value you get for the Lagrange multiplier with the value you got in Problem 2.3. Explain the relationship between the two solutions. 2.5 The height of a ball that is thrown straight up with a certain force is a function of the time (t) from which it is released given by f (t) 40t (where g is a constant determined by gravity). 0.5gt2 ¼ % þ a. How does the value of t at which the height of the ball is at a maximum depend on the parameter g? b. Use your answer to part (a) to describe how maximum height changes as the parameter g changes. c. Use the envelope theorem to answer part (b) directly. d. On the Earth g 32, but this value varies somewhat around the globe. If two locations had gravitational constants that dif- fered by 0.1, what would be the difference in the maximum height of a ball tossed in the two places? ¼ 2.6 A simple way to model the construction of an oil tanker is to start with a large rectangular sheet of steel that is x feet wide and 3x feet long. Now cut a smaller square that is t feet on a side out of each corner of the larger sheet and fold up and weld the sides of the steel sheet to make a traylike structure with no top. a. Show that the volume of oil that can be held by this tray is given by V t x ð 2t 3x 2t 3tx2 8t2x 4t3: % ¼ Þ ¼ b. How should t be chosen to maximize V for any given value of x? c. Is there a value of x that maximizes the volume of oil that can be carried? d. Suppose that a shipbuilder is constrained to use only 1,000,000 square feet of steel sheet to construct an oil tanker. This constraint can be represented by the equation 3x2 – 4t2 1,000,000 (because the builder can return the cut-out squares for credit
). How does the solution to this constrained maximum problem compare with the solutions described in parts (b) and (c)? % ¼ ð Þ þ % 2.7 Consider the following constrained maximization problem: where k is a constant that can be assigned any speciﬁc value. maximize subject to y k x1 þ x1 % 5 lnx2 x2 ¼ 0, ¼ % 10, this problem can be solved as one involving only equality constraints. a. Show that if k b. Show that solving this problem for k c. If the x’s in this problem must be non-negative, what is the optimal solution when k 1. 4 requires that x1 ¼ % either intuitively or using the methods outlined in the chapter.) ¼ ¼ 4? (This problem may be solved ¼ d. What is the solution for this problem when k for part (a)? ¼ 20? What do you conclude by comparing this solution with the solution Note: This problem involves what is called a quasi-linear function. Such functions provide important examples of some types of behavior in consumer theory—as we shall see. 2.8 Suppose that a ﬁrm has a marginal cost function given by MC (q) q 1. ¼ þ a. What is this ﬁrm’s total cost function? Explain why total costs are known only up to a constant of integration, which repre- sents ﬁxed costs. b. As you may know from an earlier economics course, if a ﬁrm takes price ( p) as given in its decisions then it will produce 15? MC(q). If the ﬁrm follows this proﬁt-maximizing rule, how much will it produce when p that output for which p Assuming that the ﬁrm is just breaking even at this price, what are ﬁxed costs? ¼ ¼ c. How much will proﬁts for this ﬁrm increase if price increases to 20? d. Show that, if we continue to assume proﬁt maximization, then this ﬁrm’s proﬁts can be expressed solely as a function of the price it receives for its output. Chapter 2: Mathematics for Microeconomics 79 e. Show that the increase in proﬁts from p 15 to p derived in part (d
); and (ii) by integrating the inverse marginal cost function [MC % Explain this result intuitively using the envelope theorem. ¼ ¼ ¼ 20 can be calculated in two ways: (i) directly from the equation 20. p – 1] from p 15 to p 1 ( p) ¼ ¼ Analytical Problems 2.9 Concave and quasi-concave functions Show that if f (x1, x2) is a concave function then it is also a quasi-concave function. Do this by comparing Equation 2.114 (deﬁning quasi-concavity) with Equation 2.98 (deﬁning concavity). Can you give an intuitive reason for this result? Is the converse of the statement true? Are quasi-concave functions necessarily concave? If not, give a counterexample. 2.10 The Cobb–Douglas function One of the most important functions we will encounter in this book is the Cobb–Douglas function: ¼ ð where a and b are positive constants that are each less than 1. y x1Þ a b, x2Þ ð a. Show that this function is quasi-concave using a ‘‘brute force’’ method by applying Equation 2.114. b. Show that the Cobb–Douglas function is quasi-concave by showing that any contour line of the form y positive constant) is convex and therefore that the set of points for which y > c is a convex set. c (where c is any ¼ c. Show that if a b > 1 then the Cobb–Douglas function is not concave (thereby illustrating again that not all quasi- þ concave functions are concave). Note: The Cobb–Douglas function is discussed further in the Extensions to this chapter. 2.11 The power function Another function we will encounter often in this book is the power function: 1 (at times we will also examine this function for cases where d can be negative, too, in which case we will use xd/d to ensure that the derivatives have the proper sign). xd, y ¼ where 0 the form y d ¼ a. Show that this function is concave (and therefore also, by the result of Problem 2.9, quasi-concave). Notice that the d 1 ¼ is a special case and
that the function is ‘‘strictly’’ concave only for d < 1. b. Show that the multivariate form of the power function y f x1, x2Þ ¼ ð x1Þ ð ¼ d d x2Þ þ ð is also concave (and quasi-concave). Explain why, in this case, the fact that f12 ¼ cavity especially simple. f21 ¼ 0 makes the determination of con- c. One way to incorporate ‘‘scale’’ effects into the function described in part (b) is to use the monotonic transformation g x1, x2Þ ¼ ð yg d x1Þ ¼ ½ð x2Þ d g, ( þ ð where g is a positive constant. Does this transformation preserve the concavity of the function? Is g quasi-concave? 2.12 Proof of the envelope theorem in constrained optimization problems Because we use the envelope theorem in constrained optimization problems often in the text, proving this theorem in a simple case may help develop some intuition. Thus, suppose we wish to maximize a function of two variables and that the value of this function also depends on a parameter, a: f (x1, x2, a). This maximization problem is subject to a constraint that can be written as: g (x1, x2, a) 0. a. Write out the Lagrangian expression and the ﬁrst-order conditions for this problem. b. Sum the two ﬁrst-order conditions involving the x’s. c. Now differentiate the above sum with respect to a—this shows how the x’s must change as a changes while requiring that ¼ the ﬁrst-order conditions continue to hold. 80 Part 1: Introduction d. As we showed in the chapter, both the objective function and the constraint in this problem can be stated as functions of a: f (x1(a), x2(a), a), g (x1(a), x2(a), a) 0. Differentiate the ﬁrst of these with respect to a. This shows how the value of the objective changes as a changes while keeping the x’s at their optimal values. You should have terms that involve the x’s and a single term in @f/@
a. ¼ e. Now differentiate the constraint as formulated in part (d) with respect to a. You should have terms in the x’s and a single term in @g/@a. f. Multiply the results from part (e) by l (the Lagrange multiplier), and use this together with the ﬁrst-order conditions from part (c) to substitute into the derivative from part (d). You should be able to show that df x1ð ð a, x2ð a Þ da, a Þ Þ @f @a þ k @g @a ; ¼ which is just the partial derivative of the Lagrangian expression when all the x’s are at their optimal values. This proves the envelope theorem. Explain intuitively how the various parts of this proof impose the condition that the x’s are constantly being adjusted to be at their optimal values. g. Return to Example 2.8 and explain how the envelope theorem can be applied to changes in the fence perimeter P—that is, how do changes in P affect the size of the area that can be fenced? Show that in this case the envelope theorem illustrates how the Lagrange multiplier puts a value on the constraint. 2.13 Taylor approximations Taylor’s theorem shows that any function can be approximated in the vicinity of any convenient point by a series of terms involving the function and its derivatives. Here we look at some applications of the theorem for functions of one and two variables. a. Any continuous and differentiable function of a single variable, f (x), can be approximated near the point a by the formula Þð 0:5f 00 a Þ þ x a Þð 2 a Þ terms in f 000, f 0000,... : ð Using only the ﬁrst three of these terms results in a quadratic Taylor approximation. Use this approximation together with the deﬁnition of concavity given in Equation 2.85 to show that any concave function must lie on or below the tangent to the function at point a. Þ ¼ % % þ ð b. The quadratic Taylor approximation for any function of two variables, f (x, y), near the point (a, b) is given by f x, y f 2ð 2f 12ð Use this approximation to show that any concave function (as de�
��ned by Equation 2.98) must lie on or below its tan- Þ þ f 11ð f 1ð a, b a Þ þ 2 Þð a, b x Þð x a, b ð f 22ð a, b a:5 þ Þð % % % % þ % Þð Þð gent plane at (a, b). 2.14 More on expected value Because the expected value concept plays an important role in many economic theories, it may be useful to summarize a few more properties of this statistical measure. Throughout this problem, x is assumed to be a continuous random variable with PDF f (x). a. ( Jensen’s inequality) Suppose that g (x) is a concave function. Show that E[g (x)] g (x) at the point E(x). This tangent will have the form c and d are constants. þ dx + g (x) for all values of x and c - g[E(x)]. Hint: Construct the tangent to g[E(x)] where c dE(x) þ ¼ b. Use the procedure from part (a) to show that if g (x) is a convex function then E[ g (x)] c. Suppose x takes on only non-negative values—that is, 0 x. Use integration by parts to show that g[E(x)]. + - -( dx, where F(x) is the cumulative distribution function for x [that is, F(x) d. (Markov’s inequality) Show that if x takes on only positive values then the following inequality holds: x 0 f t ð dt Þ (. ¼ Ð Hint: E x ð Þ ¼ 0 xf 1 x ð Þ dx ¼ Ð t 0 xf Ð dx x ð Þ þ t xf 1 x ð Þ dx : Chapter 2: Mathematics for Microeconomics 81 2x% e. Consider the PDF f (x) 3 for x 1. Show that this is a proper PDF. 2. Calculate F(x) for this PDF. 3. Use the results of part (c) to calculate E(x) for this PDF. 4. Show that Markov’s inequality holds for this function. + ¼ 1. f. The concept of conditional expected value is useful in
some economic problems. We denote the expected value of x conditional on the occurrence of some event, A, as E(x\|A). To compute this value we need to know the PDF for x given that A has occurred [denoted by f (x\|A)]. With this notation, E A dx. Perhaps the easiest way to understand A Þ these relationships is with an example. Let þ1 %1 x ð x ð Þ ¼ xf j j Ð f x ð Þ ¼ x2 3 for 1 % - x - 2: 0. 1 % - x - 2, and call this event A. What is f (x\|A)? 1. Show that this is a proper PDF. 2. Calculate E(x). 3. Calculate the probability that 4. Consider the event 0 5. Calculate E(x\|A). 6. Explain your results intuitively. - - x 2.15 More on variances The deﬁnition of the variance of a random variable can be used to show a number of additional results. a. Show that Var(x) b. Use Markov’s inequality (Problem 2.14d) to show that if x can take on only non-negative values, ¼ % E(x2) [E(x)]2. P x ½ð lxÞ + % k ( - r2 x k2 : This result shows that there are limits on how often a random variable can be far from its expected value. If k result also says that hs this ¼ P x ½ð lxÞ + % hr ( - 1 h2 : Therefore, for example, the probability that a random variable can be more than two standard deviations from its expected value is always less than 0.25. The theoretical result is called Chebyshev’s inequality. c. Equation 2.197 showed that if two (or more) random variables are independent, the variance of their sum is equal to the sum of their variances. Use this result to show that the sum of n independent random variables, each of which has expected value m and variance s2, has expected value nm and variance ns2. Show also that the average of these n random variables (which is also a random variable) will have expected value m and variance s2/n. This is sometimes called the law of large numbers—that is, the variance of an average shrinks
down as more independent variables are included. d. Use the result from part (c) to show that if x1 and x2 are independent random variables each with the same expected value 0.5. and variance, the variance of a weighted average of the two X How much is the variance of this sum reduced by setting k properly relative to other possible values of k? 1 is minimized when k (1 – k)x2, 0 kx1 þ ¼ - - ¼ k e. How would the result from part (d) change if the two variables had unequal variances? 2.16 More on covariances Here are a few useful relationships related to the covariance of two random variables, xl and x2. a. Show that Cov(x1, x2) 0, E(x1x2) E(x1)E(x2). That is, the expected value of a product of two random variables is the product of these variables’ expected E(x1x2) – E(x1)E(x2). An important implication of this is that if Cov(x1, x2) ¼ ¼ ¼ values. b. Show that Var bx2Þ ¼ c. In Problem 2.15d we looked at the variance of X 2abCov x1Þ þ ð kx1 þ ð 1 0.5 changed by considering cases where Cov x2Þ þ ð ¼ ax1 þ ð a2Var b2Var minimized for k ¼. x1, x2Þ ð x2 k 0 % Þ x1, x2Þ 6¼ ð 1. Is the conclusion that this variance is k - 0? 82 Part 1: Introduction d. The correlation coefﬁcient between two random variables is deﬁned as Explain why 1 % Corr ð x1, x2Þ - - p 1 and provide some intuition for this result. e. Suppose that the random variable y is related to the random variable x by the linear equation y x1, x2Þ ¼ Corr ð Var Cov x1, x2Þ ð x2Þ Var x1Þ ð ð ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : a þ ¼ bx. Show that y, x Cov ð x Var ð Here b is sometimes called the (theoretical) regression coefﬁcient of y on x. With actual data, the sample analog of this ¼ b Þ Þ : expression is the ordinary least squares (OLS) regression coefﬁcient. SUGGESTIONS FOR FURTHER READING Dadkhan, Kamran. Foundations of Mathematical and Computational Economics. Mason, OH: Thomson/SouthWestern, 2007. This is a good introduction to many calculus techniques. The book shows how many mathematical questions can be approached using popular software programs such as Matlab or Excel. Dixit, A. K. Optimization in Economic Theory, 2nd ed. New York: Oxford University Press, 1990. A complete and modern treatment of optimization techniques. Uses relatively advanced analytical methods. Hoy, Michael, John Livernois, Chris McKenna, Ray Rees, and Thanasis Stengos. Mathematics for Economists, 2nd ed. Cambridge, MA: MIT Press, 2001. A complete introduction to most of the mathematics covered in microeconomics courses. The strength of the book is its presentation of many worked-out examples, most of which are based on microeconomic theory. Luenberger, David G. Microeconomic Theory. New York: McGraw Hill, Inc., 1995. This is an advanced text with a variety of novel microeconomic concepts. The book also has ﬁve brief but useful mathematical appendices. Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green. Microeconomic Theory. New York: Oxford University Press, 1995. Encyclopedic treatment of mathematical microeconomics. Extensive mathematical appendices cover relatively high-level topics in analysis. Samuelson, Paul A. Foundations of Economic Analysis. Cambridge, MA: Harvard University Press, 1947. Mathematical Appendix A. A basic reference. Mathematical Appendix A provides an advanced treatment of necessary and sufﬁcient conditions for a maximum. Silberberg, E., and W. Suen. The Stract
are of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGraw-Hill, 2001. A mathematical microeconomics text that stresses the observable predictions of economic theory. The text makes extensive use of the envelope theorem. Simon, Carl P., and Lawrence Blume. Mathematics for Economists. New York: W. W. Norton, 1994. A useful text covering most areas of mathematics relevant to economists. Treatment is at a relatively high level. Two topics discussed better here than elsewhere are differential equations and basic point-set topology. Sydsaeter, K., A. Strom, and P. Berck. Economists’ Mathematical Manual, 4th ed. Berlin, Germany: Springer-Verlag, 2005. An indispensable tool for mathematical review. Contains 35 chapters covering most of the mathematical tools that economists use. Discussions are brief, so this is not the place to encounter new concepts for the ﬁrst time. Taylor, Angus E., and W. Robert Mann. Advanced Calculus, 3rd ed. New York: John Wiley, 1983, pp. 183–95. A comprehensive calculus text with a good discussion of the Lagrangian technique. Thomas, George B., and Ross L. Finney. Calculus and Analytic Geometry, 8th ed. Reading, MA: Addison-Wesley, 1992. Basic calculus text with excellent coverage of differentiation techniques. SECOND-ORDER CONDITIONS AND MATRIX ALGEBRA EXTENSIONS The second-order conditions described in Chapter 2 can be written in compact ways by using matrix algebra. In this extension, we look brieﬂy at that notation. We return to this notation at a few other places in the extensions and problems for later chapters. Matrix algebra background The extensions presented here assume some general familiarity with matrix algebra. A succinct reminder of these principles might include: 1. An n form k matrix, A, is a rectangular array of terms of the * A ¼ aij ¼ -. a11 a21... an1 2 6 6 6 4 a12 a22 ) ) ) ) ) ) a1k a2k 3 : an2 ank 7 7 7 5 ¼ ¼ 1, n; j ) ) ) Here i 1, k. Matrices can be added, subtracted, or multiplied providing their dimensions are conformable. 2. If n k, then A is a square matrix. A square matrix is n
aji. The identity matrix, In, is an n symmetric if aij ¼ square matrix where aij ¼ 3. The determinant of a square matrix (denoted by \|A\|) is a scalar (i.e., a single term) found by suitably multiplying together all the terms in the matrix. If A is 2 j and aij ¼ 0 if i 1 if i þ j. 6¼ ¼ ¼ 2, * A j¼ j a11a22 % a21a12: Example: If A 1 5 3 2 ¼ & A j¼ j 4. The inverse of an n n matrix, A% 1, such that * then ’ 2 15 % ¼ % 13: n square matrix, A, is another n * ¼ umns of A, where p 1, n. If A is 2 2, then the ﬁrst leading principal minor is a11 and the second is a11a22 % a21a12. n square matrix, A, is positive deﬁnite if all its 6. An n leading principal minors are positive. The matrix is negative deﬁnite if its principal minors alternate in sign starting with a minus.1 * 7. A particularly useful symmetric matrix is the Hessian matrix formed by all the second-order partial derivatives of a function. If f is a continuous and twice differentiable function of n variables, then its Hessian is given by f12 f22 ) ) ) ) ) ) f1n f2n 3 : H f ð Þ ¼ 2 f11 f21... fn1 6 6 6 4 7 7 7 5 ideas, we can now examine again some of the second-order conditions derived in Chapter 2. Using these notational ) ) ) fnn fn2 E2.1 Concave and convex functions A concave function is one that is always below (or on) any tangent to it. Alternatively, a convex function is always above (or on) any tangent. The concavity or convexity of any function is determined by its second derivative(s). For a function of a single variable, f (x), the requirement is straightforward. Using the Taylor approximation at any point (x0) f x0 þ ð dx Þ ¼ f ð þ f 0 dx x0Þ x0Þ þ ð higher-order
terms. þ f 00 x0ð dx2 2 Þ Assuming that the higher-order terms are 0, we have * f x0 þ ð dx Þ - f x0Þ þ ð f 0 dx x0Þ ð A Æ A–1 In. ¼ Not every square matrix has an inverse. A necessary and sufﬁcient condition for the existence of A–1 is that \|A\| 0. 5. The leading principal minors of an n n square matrix A are the series of determinants of the ﬁrst p rows and col- * 6¼ if f 00ð x0Þ - 0 and f x0 þ ð dx Þ + f x0Þ þ ð f 0 dx x0Þ ð 1If some of the determinants in this deﬁnition are 0 then the matrix is said to be positive semideﬁnite or negative semideﬁnite. 84 Part 1: Introduction 0. x0Þ + ð x0Þ + ð 0 and (locally) convex if f 00 0. Because the expressions on the right of these if f 00 inequalities are in fact the equation of the tangent to the function at x0, it is clear that the function is (locally) concave if f 00 x0Þ ð Extending this intuitive idea to many dimensions is cumbersome in terms of functional notation but relatively simple when matrix algebra is used. Concavity requires that the Hessian matrix be negative deﬁnite, whereas convexity requires that this matrix be positive deﬁnite. As in the single variable case, these conditions amount to requiring that the function move consistently away from any tangent to it no matter what direction is taken.2 If f (x1, x2) is a function of two variables, the Hessian is given by f11 f21 H ¼ & f12 f22 : ’ This is negative deﬁnite if f11 < 0 and f11 f12 % f21 f12 > 0, which is precisely the condition described in Equation 2.98. Generalizations to functions of three or more variables follow the same matrix pattern. Example 1 For the health status function in Chapter 2 (Equation 2.29), the Hessian is given by 2 % 0 H ¼ & 0 2 %,
’ and so the function will be concave, providing H2 ¼ ¼ a a ð ab 1 ð % 1 2 b 2y2b x2a : 2y2b x2a b a % % Þ % a2b2x2a % 2 2y2b % % % b < 1. That is, in production This condition clearly holds if a function terminology, the function must exhibit diminishing returns to scale to be concave. Geometrically, the function must turn downward as both inputs are increased together. þ E2.2 Maximization As we saw in Chapter 2, the ﬁrst-order conditions for an unconstrained maximum of a function of many variables requires ﬁnding a point at which the partial derivatives are zero. If the function is concave it will be below its tangent plane at this point; therefore, the point will be a true maximum.3 Because the health status function is concave, for example, the ﬁrst-order conditions for a maximum are also sufﬁcient. E2.3 Constrained maxima When the x’s in a maximization or minimization problem are subject to constraints, these constraints have to be taken into account in stating second-order conditions. Again, matrix algebra provides a compact (if not intuitive) way of denoting these conditions. The notation involves adding rows and columns of the Hessian matrix for the unconstrained problem and then checking the properties of this augmented matrix. and the ﬁrst and second leading principal minors are Speciﬁcally, we wish to maximize 2 < 0 and H1 ¼ % H2 Hence the function is concave. 4 > 0: 0 ¼ Example 2 The Cobb–Douglas function xayb where a, b (0, 1) is used to illustrate utility functions and production functions in many places in this text. The ﬁrst- and second-order derivatives of the function are 2 fx ¼ fy ¼ fxx ¼ fyy ¼ axa 1yb, % bxayb 1, % 2yb, xa 1 a a % Þ ð 2: xayb 1 b b % Þ ð % % Hence the Hessian for this function is H : ’ The ﬁrst leading principal minor of this Hessian is ¼ % &
Hb be positive deﬁnite—that is, all the leading principal ( % minors of Hb (except the ﬁrst) should be negative. Example The Lagrangian expression for the constrained health status problem (Example 2.6) is + 2x1 % and the bordered Hessian for this problem is 4x2 þ k 1 ð ¼ % þ 5 x2 1 þ x2 2 þ % x1 % x2, Þ Hb The second leading principal minor here is 4 Hb2, ¼ % and the third is 2 Hb3 % % % thus, the leading principal minors of the Hb have the required pattern and the point, 3 0 x2 ¼ is a constrained maximum. 1, x1 ¼ 0, Example In the optimal fence problem (Example 2.7), the bordered Hessian is and Hb, Hb2 ¼ % Hb3 ¼ 8, thus again the leading principal minors have the sign pattern required for a maximum. E2.4 Quasi-concavity If the constraint g is linear, then the second-order conditions explored in Extension 2.3 can be related solely to the shape of the function to be optimized, f. In this case the constraint can be written as g x1, ð..., xn c b1x1 % b2x2 % ) ) ) % bnxn ¼ % 0, Þ ¼ 6Notice that the ﬁrst leading principal minor of Hb is 0. Using the conditions, it is clear that the bordered Hessian Hb and the matrix 0 f1 f2 fn f1 f11 f21 fn1 f2 f12 f22 fn2 fn f1n f2n fnn ) ) ) ) ) ) 3 7 7 5 H0 2 ¼ 6 6 4 have the same leading principal minors except for a (positive) constant of proportionality.7 The conditions for a maximum of f subject to a linear constraint will be satisﬁed provided H 0 1)H 0 follows the same sign conventions as Hb—that is, ( must be negative deﬁnite. A function f for which H 0 does follow this pattern is called quasi-concave. As we shall see, f has the property that the set of points x for which f (x) c (where c is any constant
) is convex. For such a function, the necessary conditions for a maximum are also sufﬁcient. % + Example For the fences problem, f (x, y) xy and H 0 is given by ¼ Thus, 0 y x y 0 1 H0 ¼ 2 4 x 1 0 : 3 5 y2 < 0, 2xy > 0, H 02 ¼ % H 03 ¼ and the function is quasi-concave.8 Example More generally, if f is a function of only two variables, then quasi-concavity requires that H 02 ¼ % H 03 ¼ % 2< 0 and f22 f 2 f1ð Þ f11 f 2 2 % 1 þ 2f1 f2 f12 > 0, which is precisely the condition stated in Equation 2.114. Hence we have a fairly simple way of determining quasiconcavity. References Simon, C. P., and L. Blume. Mathematics for Economists. New York: W. W. Norton, 1994. Sydsaeter, R., A. Strom, and P. Berck. Economists’ Mathematical Manual, 3rd ed. Berlin, Germany: Springer-Verlag, 2000. 7This can be shown by noting that multiplying a row (or a column) of a matrix by a constant multiplies the determinant by that constant. 8Because f (x, y) xy is a form of a Cobb–Douglas function that is not concave, this shows that not every quasi-concave function is concave. Notice that a monotonic function of f (such as f 1/3) could be concave, however. ¼ This page intentionally left blank Choice and Demand PART TWO Chapter 3 Preferences and Utility Chapter 4 Utility Maximization and Choice Chapter 5 Income and Substitution Effects Chapter 6 Demand Relationships among Goods In Part 2 we will investigate the economic theory of choice. One goal of this examination is to develop the notion of demand in a formal way so that it can be used in later sections of the text when we turn to the study of markets. A more general goal of this part is to illustrate the approach economists use for explaining how individuals make choices in a wide variety of contexts. Part 2 begins with a description of the way economists model individual preferences, which are usually referred to by the formal term utility. Chapter 3 shows how economists are able to conceptualize utility in a mathematical
way. This permits an examination of the various exchanges that individuals are willing to make voluntarily. The utility concept is used in Chapter 4 to illustrate the theory of choice. The fundamental hypothesis of the chapter is that people faced with limited incomes will make economic choices in such a way as to achieve as much utility as possible. Chapter 4 uses mathematical and intuitive analyses to indicate the insights that this hypothesis provides about economic behavior. Chapters 5 and 6 use the model of utility maximization to investigate how individuals will respond to changes in their circumstances. Chapter 5 is primarily concerned with responses to changes in the price of a commodity, an analysis that leads directly to the demand curve concept. Chapter 6 applies this type of analysis to developing an understanding of demand relationships among different goods. 87 This page intentionally left blank CHAPTER THREE Preferences and Utility In this chapter we look at the way in which economists characterize individuals’ preferences. We begin with a fairly abstract discussion of the ‘‘preference relation,’’ but quickly turn to the economists’ primary tool for studying individual choices—the utility function. We look at some general characteristics of that function and a few simple examples of speciﬁc utility functions we will encounter throughout this book. Axioms of Rational Choice One way to begin an analysis of individuals’ choices is to state a basic set of postulates, or axioms, that characterize ‘‘rational’’ behavior. These begin with the concept of ‘‘preference’’: An individual who reports that ‘‘A is preferred to B’’ is taken to mean that all things considered, he or she feels better off under situation A than under situation B. The preference relation is assumed to have three basic properties as follows. I. Completeness. If A and B are any two situations, the individual can always specify exactly one of the following three possibilities: 1. 2. 3. ‘‘A is preferred to B,’’ ‘‘B is preferred to A,’’ or ‘‘A and B are equally attractive.’’ Consequently, people are assumed not to be paralyzed by indecision: They completely understand and can always make up their minds about the desirability of any two alternatives. The assumption also rules out the possibility that an individual can report both that A is preferred to B and that B is preferred to A. II. Transitivity. If an individual
reports that ‘‘A is preferred to B’’ and ‘‘B is preferred to C,’’ then he or she must also report that ‘‘A is preferred to C.’’ This assumption states that the individual’s choices are internally consistent. Such an assumption can be subjected to empirical study. Generally, such studies conclude that a person’s choices are indeed transitive, but this conclusion must be modiﬁed in cases where the individual may not fully understand the consequences of the choices he or she is making. Because, for the most part, we will assume choices are fully informed (but see the discussion of uncertainty in Chapter 7 and elsewhere), the transitivity property seems to be an appropriate assumption to make about preferences. III. Continuity. If an individual reports ‘‘A is preferred to B,’’ then situations suitably ‘‘close to’’ A must also be preferred to B. This rather technical assumption is required if we wish to analyze individuals’ responses to relatively small changes in income and prices. The purpose of the assumption is to rule out certain kinds of discontinuous, knife-edge preferences that pose problems for a mathematical development of the theory of choice. Assuming 89 90 Part 2: Choice and Demand continuity does not seem to risk missing types of economic behavior that are important in the real world (but see Problem 3.14 for some counterexamples). Utility Given the assumptions of completeness, transitivity, and continuity, it is possible to show formally that people are able to rank all possible situations from the least desirable to the most.1 Following the terminology introduced by the nineteenth-century political theorist Jeremy Bentham, economists call this ranking utility.2 We also will follow Bentham by saying that more desirable situations offer more utility than do less desirable ones. That is, if a person prefers situation A to situation B, we would say that the utility assigned to option A, denoted by U(A), exceeds the utility assigned to B, U(B). ¼ ¼ 5 and U(B) 4, or that U(A) 1,000,000 and U(B) Nonuniqueness of utility measures We might even attach numbers to these utility rankings; however, these numbers will not be unique. Any set of numbers we arbitrarily assign that accurately reﬂects the original preference ordering will imply the same set of
choices. It makes no difference whether we say that U(A) 0.5. In both cases the numbers imply that A is preferred to B. In technical terms, our notion of utility is deﬁned only up to an order-preserving (‘‘monotonic’’) transformation.3 Any set of numbers that accurately reﬂects a person’s preference ordering will do. Consequently, it makes no sense to ask ‘‘how much more is A preferred than B?’’ because that question has no unique answer. Surveys that ask people to rank their ‘‘happiness’’ on a scale of 1 to 10 could just as well use a scale of 7 to 1,000,000. We can only hope that a person who reports he or she is a ‘‘6’’ on the scale one day and a ‘‘7’’ on the next day is indeed happier on the second day. Therefore, utility rankings are like the ordinal rankings of restaurants or movies using one, two, three, or four stars. They simply record the relative desirability of commodity bundles. ¼ ¼ This lack of uniqueness in the assignment of utility numbers also implies that it is not possible to compare utilities of different people. If one person reports that a steak dinner provides a utility of ‘‘5’’ and another person reports that the same dinner offers a utility of ‘‘100,’’ we cannot say which individual values the dinner more because they could be using different scales. Similarly, we have no way of measuring whether a move from situation A to situation B provides more utility to one person or another. Nonetheless, as we will see, economists can say quite a bit about utility rankings by examining what people voluntarily choose to do. The ceteris paribus assumption Because utility refers to overall satisfaction, such a measure clearly is affected by a variety of factors. A person’s utility is affected not only by his or her consumption of physical commodities but also by psychological attitudes, peer group pressures, personal experiences, and the general cultural environment. Although economists do have a general interest in examining such inﬂuences, a narrowing of focus is usually necessary. Consequently, a common 1These properties and their connection to representation of preferences by a utility function are discussed in detail in Andreu Mas-Colell, Michael
D. Whinston, and Jerry R. Green, Microeconomic Theory (New York: Oxford University Press, 1995). 2J. Bentham, Introduction to the Principles of Morals and Legislation (London: Hafner, 1848). 3We can denote this idea mathematically by saying that any numerical utility ranking (U ) can be transformed into another set of numbers by the function F providing that F(U ) is order preserving. This can be ensured if F 0(U ) > 0. For example, the ln U. At some places in the text and problems transformation F(U ) we will ﬁnd it convenient to make such transformations to make a particular utility ranking easier to analyze. U2 is order preserving as is the transformation F(U ) ¼ ¼ Chapter 3: Preferences and Utility 91 practice is to devote attention exclusively to choices among quantiﬁable options (e.g., the relative quantities of food and shelter bought, the number of hours worked per week, or the votes among speciﬁc taxing formulas) while holding constant the other things that affect behavior. This ceteris paribus (‘‘other things being equal’’) assumption is invoked in all economic analyses of utility-maximizing choices so as to make the analysis of choices manageable within a simpliﬁed setting. Utility from consumption of goods As an important example of the ceteris paribus assumption, consider an individual’s problem of choosing, at a single point in time, among n consumption goods x1, x2, …, xn. We shall assume that the individual’s ranking of these goods can be represented by a utility function of the form utility U x1, x2, ð ¼..., xn; other things, Þ (3:1) where the x’s refer to the quantities of the goods that might be chosen and the ‘‘other things’’ notation is used as a reminder that many aspects of individual welfare are being held constant in the analysis. Often it is easier to write Equation 3.1 as Or, if only two goods are being considered, as utility U x1, x2, ð ¼..., xn Þ (3:2) utility U x, y ð where it is clear that everything is being held constant (i.e., outside the frame of analysis) except
the goods actually referred to in the utility function. It would be tedious to remind you at each step what is being held constant in the analysis, but it should be remembered that some form of the ceteris paribus assumption will always be in effect. (3:20) ¼ Þ, Arguments of utility functions The utility function notation is used to indicate how an individual ranks the particular arguments of the function being considered. In the most common case, the utility function (Equation 3.2) will be used to represent how an individual ranks certain bundles of goods that might be purchased at one point in time. On occasion we will use other arguments in the utility function, and it is best to clear up certain conventions at the outset. For example, it may be useful to talk about the utility an individual receives from real wealth (W ). Therefore, we shall use the notation utility U W ð : Þ ¼ (3:3) Unless the individual is a rather peculiar, Scrooge-type person, wealth in its own right gives no direct utility. Rather, it is only when wealth is spent on consumption goods that any utility results. For this reason, Equation 3.3 will be taken to mean that the utility from wealth is in fact derived by spending that wealth in such a way as to yield as much utility as possible. Two other arguments of utility functions will be used in later chapters. In Chapter 16 we will be concerned with the individual’s labor–leisure choice and will therefore have to consider the presence of leisure in the utility function. A function of the form will be used. Here, c represents consumption and h represents hours of nonwork time (i.e., leisure) during a particular period. utility U c, h Þ ð ¼ (3:4) 92 Part 2: Choice and Demand In Chapter 17 we will be interested in the individual’s consumption decisions in differ- ent periods. In that chapter we will use a utility function of the form, c1, c2Þ ð where c1 is consumption in this period and c2 is consumption in the next period. By changing the arguments of the utility function, therefore, we will be able to focus on speciﬁc aspects of an individual’s choices in a variety of simpliﬁed settings. utility (3:5) ¼ U In summary, we start our examination of individual behavior with the following deﬁn
ition Utility. Individuals’ preferences are assumed to be represented by a utility function of the form where x1, x2, …, xn are the quantities of each of n goods that might be consumed in a period. This function is unique only up to an order-preserving transformation. U x1, x2, ð..., xn, Þ (3:6) Economic goods In this representation the variables are taken to be ‘‘goods’’; that is, whatever economic quantities they represent, we assume that more of any particular xi during some period is preferred to less. We assume this is true of every good, be it a simple consumption item such as a hot dog or a complex aggregate such as wealth or leisure. We have pictured this convention for a two-good utility function in Figure 3.1. There, all consumption bundles in the shaded area are preferred to the bundle x$, y$ because any bundle in the shaded area provides more of at least one of the goods. By our deﬁnition of ‘‘goods,’’ bundles of goods in the shaded area are ranked higher than x$, y$. Similarly, bundles in the area marked ‘‘worse’’ are clearly inferior to x$, y$ because they contain less of at least one of the goods and no more of the other. Bundles in the two areas indicated by question marks are difﬁcult to compare with x$, y$ because they contain more of one of the goods and less of the other. Movements into these areas involve trade-offs between the two goods. Trades and Substitution Most economic activity involves voluntary trading between individuals. When someone buys, say, a loaf of bread, he or she is voluntarily giving up one thing (money) for something else (bread) that is of greater value to that individual. To examine this kind of voluntary transaction, we need to develop a formal apparatus for illustrating trades in the utility function context. We ﬁrst motivate our discussion with a graphical presentation and then turn to some more formal mathematics. Indifference curves and the marginal rate of substitution Voluntary trades can best be studied using the graphical device of an indifference curve. In Figure 3.2, the curve U1 represents all the alternative combinations of x and y for which an individual is equally well off (remember again that all other arguments of the utility function are held
constant). This person is equally happy consuming, for example, either the combination of goods x1, y1 or the combination x2, y2. This curve representing all the consumption bundles that the individual ranks equally is called an indifference curve. Chapter 3: Preferences and Utility 93 FIGURE 3.1 More of a Good Is Preferred to Less The shaded area represents those combinations of x and y that are unambiguously preferred to the combination x. Ceteris paribus, individuals prefer more of any good rather than less. Combinations, y $ $ identiﬁed by ‘‘?’’ involve ambiguous changes in welfare because they contain more of one good and less of the other. Quantity of y? Worse than x, y y* Preferred to x, y? x* Quantity of Indifference curve. An indifference curve (or, in many dimensions, an indifference surface) shows a set of consumption bundles about which the individual is indifferent. That is, the bundles all provide the same level of utility. The slope of the indifference curve in Figure 3.2 is negative, showing that if the individual is forced to give up some y, he or she must be compensated by an additional amount of x to remain indifferent between the two bundles of goods. The curve is also drawn so that the slope increases as x increases (i.e., the slope starts at negative inﬁnity and increases toward zero). This is a graphical representation of the assumption that people become progressively less willing to trade away y to get more x. In mathematical terms, the absolute value of this slope diminishes as x increases. Hence we have the following deﬁnition Marginal rate of substitution. The negative of the slope of an indifference curve (U1) at some point is termed the marginal rate of substitution (MRS) at that point. That is, where the notation indicates that the slope is to be calculated along the U1 indifference curve. MRS dy dx ¼ % U!!!!, U1 ¼ (3:7) 94 Part 2: Choice and Demand FIGURE 3.2 A Single Indifference Curve The curve U1 represents those combinations of x and y from which the individual derives the same utility. The slope of this curve represents the rate at which the individual is willing to trade x for y while remaining equally well off. This slope (or, more properly, the negative of the slope) is termed the marginal rate of substitution.
In the ﬁgure, the indifference curve is drawn on the assumption of a diminishing marginal rate of substitution. Quantity of y U1 y1 y2 U1 x1 x2 Quantity of x Therefore, the slope of U1 and the MRS tell us something about the trades this person will voluntarily make. At a point such as x1, y1, the person has a lot of y and is willing to trade away a signiﬁcant amount to get one more x. Therefore, the indifference curve at x1, y1 is rather steep. This is a situation where the person has, say, many hamburgers (y) and little to drink with them (x). This person would gladly give up a few burgers (say, 5) to quench his or her thirst with one more drink. At x2, y2, on the other hand, the indifference curve is ﬂatter. Here, this person has a few drinks and is willing to give up relatively few burgers (say, 1) to get another soft drink. Consequently, the MRS diminishes between x1, y1 and x2, y2. The changing slope of U1 shows how the particular consumption bundle available inﬂuences the trades this person will freely make. Indifference curve map In Figure 3.2 only one indifference curve was drawn. The x, y quadrant, however, is densely packed with such curves, each corresponding to a different level of utility. Because every bundle of goods can be ranked and yields some level of utility, each point in Figure 3.2 must have an indifference curve passing through it. Indifference curves are similar to contour lines on a map in that they represent lines of equal ‘‘altitude’’ of utility. In Figure 3.3 several indifference curves are shown to indicate that there are inﬁnitely many in the plane. The level of utility represented by these curves increases as we move in a northeast direction; the utility of curve U1 is less than that of U2, which is less than that of U3. This is because of the assumption made in Figure 3.1: More of a good is preferred to less. As was discussed earlier, there is no unique way to assign numbers to these Chapter 3: Preferences and Utility 95 FIGURE 3.3 There Are Inﬁnitely Many Indifference Curves in the x–y Plane There is an indifference curve
passing through each point in the x–y plane. Each of these curves records combinations of x and y from which the individual receives a certain level of satisfaction. Movements in a northeast direction represent movements to higher levels of satisfaction. Quantity of y U1 U2 U3 Increasing utility U3 U2 U1 Quantity of x utility levels. The curves only show that the combinations of goods on U3 are preferred to those on U2, which are preferred to those on U1. Indifference curves and transitivity As an exercise in examining the relationship between consistent preferences and the representation of preferences by utility functions, consider the following question: Can any two of an individual’s indifference curves intersect? Two such intersecting curves are shown in Figure 3.4. We wish to know if they violate our basic axioms of rationality. Using our map analogy, there would seem to be something wrong at point E, where ‘‘altitude’’ is equal to two different numbers, U1 and U2. But no point can be both 100 and 200 feet above sea level. To proceed formally, let us analyze the bundles of goods represented by points A, B, C, and D. By the assumption of nonsatiation (i.e., more of a good always increases utility), ‘‘A is preferred to B’’ and ‘‘C is preferred to D.’’ But this person is equally satisﬁed with B and C (they lie on the same indifference curve), so the axiom of transitivity implies that A must be preferred to D. But that cannot be true because A and D are on the same indifference curve and are by deﬁnition regarded as equally desirable. This contradiction shows that indifference curves cannot intersect. Therefore, we should always draw indifference curve maps as they appear in Figure 3.3. Convexity of indifference curves An alternative way of stating the principle of a diminishing marginal rate of substitution uses the mathematical notion of a convex set. A set of points is said to be convex if any two points within the set can be joined by a straight line that is contained completely 96 Part 2: Choice and Demand FIGURE 3.4 Intersecting Indifference Curves Imply Inconsistent Preferences Combinations A and D lie on the same indifference curve and therefore are equally desirable. But the axiom of transitivity can be used to show that A is preferred to D. Hence
intersecting indifference curves are not consistent with rational preferences. Quantity of y C D E A U1 B U2 Quantity of x within the set. The assumption of a diminishing MRS is equivalent to the assumption that all combinations of x and y that are preferred or indifferent to a particular combination x$, y$ form a convex set.4 This is illustrated in Figure 3.5a, where all combinations preferred or indifferent to x$, y$ are in the shaded area. Any two of these combinations—say, x1, y1 and x2, y2—can be joined by a straight line also contained in the shaded area. In Figure 3.5b this is not true. A line joining x1, y1 and x2, y2 passes outside the shaded area. Therefore, the indifference curve through x$, y$ in Figure 3.5b does not obey the assumption of a diminishing MRS because the set of points preferred or indifferent to x$, y$ is not convex. Convexity and balance in consumption By using the notion of convexity, we can show that individuals prefer some balance in their consumption. Suppose that an individual is indifferent between the combinations x1, y1 and x2, y2. If the indifference curve is strictly convex, then the combination (x1 þ x2)/2, y2)/2 will be preferred to either of the initial combinations.5 Intuitively, ‘‘well(y1 þ balanced’’ bundles of commodities are preferred to bundles that are heavily weighted toward one commodity. This is illustrated in Figure 3.6. Because the indifference curve is assumed to be convex, all points on the straight line joining (x1, y1) and (x2, y2) are prey2)/2, ferred to these initial points. Therefore, this will be true of the point (x1 þ x2)/2, (y1 þ 4This deﬁnition is equivalent to assuming that the utility function is quasi-concave. Such functions were discussed in Chapter 2, and we shall return to examine them in the next section. Sometimes the term strict quasi-concavity is used to rule out the possibility of indifference curves having linear segments. We generally will assume strict quasi-concavity, but in a few places we will illustrate the complications posed by linear portions of indifference curves. 5In the case in which the indifference curve has
a linear segment, the individual will be indifferent among all three combinations. FIGURE 3.5 The Notion of Convexity as an Alternative Deﬁnition of a Diminishing MRS Chapter 3: Preferences and Utility 97 In (a) the indifference curve is convex (any line joining two points above U1 is also above U1). In (b) this is not the case, and the curve shown here does not everywhere have a diminishing MRS. Quantity of y U1 y1 y* y2 (a) Quantity of y U1 y1 y* y2 (b) x1 x* x2 U1 Quantity of x x1 x* x2 U1 Quantity of x FIGURE 3.6 Balanced Bundles of Goods Are Preferred to Extreme Bundles If indifference curves are convex (if they obey the assumption of a diminishing MRS), then the line joining any two points that are indifferent will contain points preferred to either of the initial combinations. Intuitively, balanced bundles are preferred to unbalanced ones. Quantity of y U1 y1 y1 + y2 2 y2 U1 x1 x1 + x2 2 x2 Quantity of x 98 Part 2: Choice and Demand which lies at the midpoint of such a line. Indeed, any proportional combination of the two indifferent bundles of goods will be preferred to the initial bundles because it will represent a more balanced combination. Thus, strict convexity is equivalent to the assumption of a diminishing MRS. Both assumptions rule out the possibility of an indifference curve being straight over any portion of its length. EXAMPLE 3.1 Utility and the MRS Suppose a person’s ranking of hamburgers ( y) and soft drinks (x) could be represented by the utility function ¼ An indifference curve for this function is found by identifying that set of combinations of x and y for which utility has the same value. Suppose we arbitrarily set utility equal to 10. Then the equation for this indifference curve is utility : yp x ’ ﬃﬃﬃﬃﬃﬃﬃﬃ (3:8) Because squaring this function is order preserving, the indifference curve is also represented by utility 10 ¼ ¼ yp x ’ ﬃﬃﬃﬃﬃﬃﬃﬃ : (3:9) which is easier to graph.
In Figure 3.7 we show this indifference curve; rectangular hyperbola. One way to calculate the MRS is to solve Equation 3.10 for y, it is a familiar 100 y, x ’ ¼ (3:10) 100=x, y ¼ (3:11) FIGURE 3.73Indifference Curve for Utility ¼ yp x ’ ﬃﬃﬃﬃﬃﬃﬃﬃﬃ This indifference curve illustrates the function 10 ¼ implying that this person is willing to trade 4y for an additional x. At point B (20, 5), however, the MRS is 0.25, implying a greatly reduced willingness to trade.. At point A (5, 20), the MRS is 4, yp x ’ ﬃﬃﬃﬃﬃﬃﬃﬃ ¼ U Quantity of y 20 A 12.5 5 C B U = 10 0 5 12.5 20 Quantity of x Chapter 3: Preferences and Utility 99 And then use the deﬁnition (Equation 3.7): MRS dy=dx along U1 ð ¼ % Þ ¼ 100=x2: (3:12) Clearly this MRS decreases as x increases. At a point such as A on the indifference curve with a lot of hamburgers (say, x 20), the slope is steep so the MRS is high: 5, y ¼ ¼ MRS at 5, 20 ð Þ ¼ 100=x2 100=25 4: ¼ ¼ (3:13) Here the person is willing to give up 4 hamburgers to get 1 more soft drink. On the other hand, 5), the slope is ﬂat and the at B where there are relatively few hamburgers (here x MRS is low: 20, y ¼ ¼ MRS at 20, 5 ð Þ ¼ 100=x2 100=400 0:25: ¼ ¼ (3:14) Now he or she will only give up one quarter of a hamburger for another soft drink. Notice also how convexity of the indifference curve U1 is illustrated by this numerical example. Point C is midway between points A and B; at C this person has 12.5 hamburgers
and 12.5 soft drinks. Here utility is given by ¼ which clearly exceeds the utility along U1 (which was assumed to be 10). q ¼ ¼ utility yp x ’ ﬃﬃﬃﬃﬃﬃﬃﬃ 2 12:5 Þ ð ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 12:5, (3:15) QUERY: From our derivation here, it appears that the MRS depends only on the quantity of x consumed. Why is this misleading? How does the quantity of y implicitly enter into Equations 3.13 and 3.14? The Mathematics of Indifference Curves A mathematical derivation of the indifference curve concept provides additional insights about the nature of preferences. In this section we look at a two-good example that ties directly to the graphical treatment provided previously. Later in the chapter we look at the many-good case, but conclude that this more complicated case adds only a few additional insights. The marginal rate of substitution Suppose an individual receives utility from consuming two goods whose quantities are given by x and y. This person’s ranking of bundles of these goods can be represented by a utility function of the form U x, y. Those combinations of the two goods that yield a speciﬁc level of utility, say k, are represented by solutions to the implicit equation k. In Chapter 2 (see Equation 2.23) we showed that the trade-offs implied by U x, y ð such an equation are given by: Þ ¼ Þ ð dy dx jU x, y ð k ¼ % Þ ¼ Ux Uy : (3:16) That is, the rate at which x can be traded for y is given by the negative of the ratio of the ‘‘marginal utility’’ of good x to that of good y. Assuming additional amounts of both goods provide added utility, this trade-off rate will be negative, implying that increases in the quantity of good x must be met by decreases in the quantity of good y to keep utility 100 Part 2: Choice and Demand constant. Earlier we deﬁned the marginal rate of substitution as the negative (or absolute value) of this trade
-off, so now we have: MRS dy dx jU x, y ð ¼ % k ¼ Þ ¼ Ux Uy : (3:17) This derivation helps in understanding why the MRS does not depend speciﬁcally on how utility is measured. Because the MRS is a ratio of two utility measures, the units ‘‘drop out’’ in the computation. For example, suppose good x represents food and that we have chosen a utility function for which an extra unit of food yields 6 extra units of utility (sometimes these units are called utils). Suppose also that y represents clothing and with this utility function each extra unit of clothing provides 2 extra units of utility. In this case it is clear that this person is willing to give up 3 units of clothing (thereby losing 6 utils) in exchange for 1 extra unit of food (thereby gaining 6 utils): MRS dy dx ¼ Ux Uy ¼ 6 utils per unit x 2 utils per unit y ¼ ¼ % 3 units y per unit x: (3:18) Notice that the utility measure used here (utils) drops out in making this computation and what remains is purely in terms of the units of the two goods. This shows that the MRS will be unchanged no matter what speciﬁc utility ranking is used.6 Convexity of Indifference Curves In Chapter 1 we described how economists were able to resolve the water–diamond paradox by proposing that the price of water is low because one more gallon provides relatively little in terms of increased utility. Water is (for the most part) plentiful; therefore, its marginal utility is low. Of course, in a desert, water would be scarce and its marginal utility (and price) could be high. Thus, one might conclude that the marginal utility associated with water consumption decreases as more water is consumed—in formal @ 2U=@x 2) should terms, the second (partial) derivative of the utility function (i.e., Uxx ¼ be negative. seems Intuitively it seems that this commonsense idea should also explain why indifference curves are convex. The fact that people are increasingly less willing to part with good y to get more x (while holding utility constant) to the same phenomenon—that people do not want too much of any one good. Unfortunately, the precise connection between diminishing marginal utility and a diminishing MRS
is complex, even in the two-good case. As we showed in Chapter 2, a function will (by deﬁnition) have convex indifference curves, providing it is quasi-concave. But the conditions required for quasi-concavity are messy, and the assumption of diminishing marginal utility (i.e., negative second-order partial derivatives) will not ensure that they hold.7 Still, as we shall see, there are good reasons for assuming that utility functions (and many other functions used in microeconomics) are quasi-concave; thus, we will not be too concerned with situations in which they are not. to refer 6More formally, let F U x, y ½ ð be any monotonic transformation of the utility function with F0 > 0. With this new utility U ð Þ Þ ) ranking the MRS is given by: MRS @F=@x @F=@y ¼ ¼ U F0 ð U F0ð :Ux Þ :Uy ¼ Þ Ux Uy ; which is the same as the MRS for the original utility function. 7Speciﬁcally, for the function U x, y ð Þ to be quasi-concave the following condition must hold (see Equation 2.114): 2UxyUxUy þ The assumptions that Uxx, Uy y < 0 will not ensure this. One must also be concerned with the sign of the cross partial derivative Uxy. y < 0: x % Uy yU 2 UxxU 2 Chapter 3: Preferences and Utility 101 EXAMPLE 3.2 Showing Convexity of Indifference Curves Calculation of the MRS for speciﬁc utility functions is frequently a good shortcut for showing convexity of indifference curves. In particular, the process can be much simpler than applying the deﬁnition of quasi-concavity, although it is more difﬁcult to generalize to more than two goods. Here we look at how Equation 3.17 can be used for three different utility functions (for more practice, see Problem 3.1). 1. U x, y ð Þ ¼. yp x ’ ﬃﬃﬃﬃﬃﬃﬃﬃ This example just repeats
the case illustrated in Example 3.1. One shortcut to applying Equation 3.17 that can simplify the algebra is to take the logarithm of this utility function. Because taking logs is order preserving, this will not alter the MRS to be calculated. Thus, let U $ x, y ð Þ ¼ ln U x, y ð ½ ) ¼ Þ 0:5 ln x þ 0:5 ln y: (3:19) Applying Equation 3.17 yields MRS @U $=@x @U $=@y ¼ 0:5=x 0:5=y ¼ y x, ¼ (3:20) which seems to be a much simpler approach than we used previously.8 Clearly this MRS is diminishing as x increases and y decreases. Therefore, the indifference curves are convex. 2. U(x, y) x xy y. ¼ þ þ In this case there is no advantage to transforming this utility function. Applying Equation 3.17 yields MRS @U=@x @U=@3:21) Again, this ratio clearly decreases as x increases and y decreases; thus, the indifference curves for this function are convex. 3. U x, y ð Þ ¼ x2 y2 þ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p For this example it is easier to use the transformation U $ x, y ð Þ ¼ U x, y ð ½ ) Þ 2 x2 þ ¼ y2: (3:22) Because this is the equation for a quarter-circle, we should begin to suspect that there might be some problems with the indifference curves for this utility function. These suspicions are conﬁrmed by again applying the deﬁnition of the MRS to yield MRS @U $=@x @U $@y ¼ 2x 2y ¼ x y : ¼ (3:23) For this function, it is clear that, as x increases and y decreases, the MRS increases! Hence the indifference curves are concave, not convex, and this is clearly not a quasi-concave function. QUERY: Does a doubling of x and
y change the MRS in each of these three examples? That is, does the MRS depend only on the ratio of x to y, not on the absolute scale of purchases? (See also Example 3.3.) 8In Example 3.1 we looked at the U would be MRS 100/x2 as calculated before. ¼ ¼ 10 indifference curve. Thus, for that curve, y 100/x, and the MRS in Equation 3.20 ¼ 102 Part 2: Choice and Demand Utility Functions for Speciﬁc Preferences Individuals’ rankings of commodity bundles and the utility functions implied by these rankings are unobservable. All we can learn about people’s preferences must come from the behavior we observe when they respond to changes in income, prices, and other factors. Nevertheless, it is useful to examine a few of the forms particular utility functions might take. Such an examination may offer insights into observed behavior, and (more to the point) understanding the properties of such functions can be of some help in solving problems. Here we will examine four speciﬁc examples of utility functions for two goods. Indifference curve maps for these functions are illustrated in the four panels of Figure 3.8. As should be visually apparent, these cover a few possible shapes. Even greater variety is possible once we move to functions for three or more goods, and some of these possibilities are mentioned in later chapters. FIGURE 3.8 Examples of Utility Functions The four indifference curve maps illustrate alternative degrees of substitutability of x for y. The Cobb– Douglas and constant elasticity of substitution (CES) functions (drawn here for relatively low substitutability) fall between the extremes of perfect substitution (b) and no substitution (c). Quantity of y Quantity of y U2 U1 U0 Quantity of x (a) Cobb-Douglas (b) Perfect substitutes Quantity of y Quantity of y U2 U1 U0 U2 U1 U0 Quantity of x U2 U1 U0 (c) Perfect complements (d) CES Quantity of x Quantity of x U x, y ð b= a ð þ Þ ¼ b : Chapter 3: Preferences and Utility 103 Cobb–Douglas utility Figure 3.8a shows the familiar shape of an indifference curve. One commonly used utility function that generates such curves has the form utility U x, y ð ¼ Þ ¼ xayb, (3:24) where
a and b are positive constants. In Examples 3.1 and 3.2, we studied a particular case of this function for which a ¼ 0.5. The more general case presented in Equation 3.24 is termed a Cobb–Douglas utility function, after two researchers who used such a function for their detailed study of production relationships in the U.S. economy (see Chapter 9). In general, the relative sizes of a and b indicate the relative importance of the two goods to this individual. Because utility is unique only up to a monotonic transformation, it is often convenient to normalize these parameters so that a 1. In this case, utility would be given by ¼ b b þ ¼ xd y1 d % (3:25) d ¼ þ %, 1 b Þ where d a= a ð Perfect substitutes The linear indifference curves in Figure 3.8b are generated by a utility function of the form ¼ Þ (3:26) utility U x, y ð ax by, ¼ þ Þ ¼ where, again, a and b are positive constants. That the indifference curves for this function are straight lines should be readily apparent: Any particular level curve can be calculated by setting U(x, y) equal to a constant that speciﬁes a straight line. The linear nature of these indifference curves gave rise to the term perfect substitutes to describe the implied relationship between x and y. Because the MRS is constant (and equal to a/b) along the entire indifference curve, our previous notions of a diminishing MRS do not apply in this case. A person with these preferences would be willing to give up the same amount of y to get one more x no matter how much x was being consumed. Such a situation might describe the relationship between different brands of what is essentially the same product. For example, many people (including the author) do not care where they buy gasoline. A gallon of gas is a gallon of gas despite the best efforts of the Exxon and Shell advertising departments to convince me otherwise. Given this fact, I am always willing to give up 10 gallons of Exxon in exchange for 10 gallons of Shell because it does not matter to me which I use or where I got my last tankful. Indeed, as we will see in the next chapter, one implication of such a relationship is that I will buy all my gas from the least expensive seller. Because I do not experience a diminishing MRS of Exxon
for Shell, I have no reason to seek a balance among the gasoline types I use. Perfect complements A situation directly opposite to the case of perfect substitutes is illustrated by the L-shaped indifference curves in Figure 3.8c. These preferences would apply to goods that ‘‘go together’’—coffee and cream, peanut butter and jelly, and cream cheese and lox are familiar examples. The indifference curves shown in Figure 3.8c imply that these pairs of goods will be used in the ﬁxed proportional relationship represented by the vertices of the curves. A person who prefers 1 ounce of cream with 8 ounces of coffee will want 2 ounces of cream with 16 ounces of coffee. Extra coffee without cream is of no value to this person, just as extra cream would be of no value without coffee. Only by choosing the goods together can utility be increased. 104 Part 2: Choice and Demand These concepts can be formalized by examining the mathematical form of the utility function that generates these L-shaped indifference curves: utility U x, y ð min ax, by : Þ ¼ Here a and b are positive parameters, and the operator ‘‘min’’ means that utility is given by the smaller of the two terms in the parentheses. In the coffee–cream example, if we let ounces of coffee be represented by x and ounces of cream by y, utility would be given by ¼ ð Þ (3:27) utility U x, y ð min x, 8y ð : ¼ Þ ¼ Now 8 ounces of coffee and 1 ounce of cream provide 8 units of utility. But 16 ounces of coffee and 1 ounce of cream still provide only 8 units of utility because min(16, 8) 8. The extra coffee without cream is of no value, as shown by the horizontal section of the indifference curves for movement away from a vertex; utility does not increase when only x increases (with y constant). Only if coffee and cream are both doubled (to 16 and 2, respectively) will utility increase to 16. ¼ Þ (3:28) More generally, neither of the two goods speciﬁed in the utility function given by by. In this case, the ratio Equation 3.27 will be consumed in superﬂuous amounts if ax of the quantity of good x consumed to that of good y will be a constant given by y x ¼ (3:29)
a b ¼ : Consumption will occur at the vertices of the indifference curves shown in Figure 3.8c. CES utility The three speciﬁc utility functions illustrated thus far are special cases of the more general CES function, which takes the form where d 1, d 6¼ * 0, and utility U x, y ð ¼ Þ ¼ xd d þ yd d, utility U x, y ð ¼ Þ ¼ ln x þ ln y (3:30) (3:31) 0. It is obvious that the case of perfect substitutes corresponds to the limiting when d 1, in Equation 3.30 and that the Cobb–Douglas9 case corresponds to d 0 in case, d Equation 3.31. Less obvious is that the case of ﬁxed proportions corresponds to d in Equation 3.30, but that result can also be shown using a limits argument. ¼ ¼ %1 ¼ ¼ The use of the term elasticity of substitution for this function derives from the notion that the possibilities illustrated in Figure 3.8 correspond to various values for the substitu1/(1 tion parameter, s, which for this function is given by s d). For perfect substi0.10 Because the CES function tutes, then s, and the ﬁxed proportions case has s allows us to explore all these cases, and many cases in between, it will prove useful for illustrating the degree of substitutability present in various economic relationships. ¼ ¼ ¼1 % The speciﬁc shape of the CES function illustrated in Figure 3.8a is for the case d ¼ % 1. That is, utility 1 x% 1 y3:32) 9The CES function could easily be generalized to allow for differing weights to be attached to the two goods. Because the main use of the function is to examine substitution questions, we usually will not make that generalization. In some of the applications of the CES function, we will also omit the denominators of the function because these constitute only a scale factor when d is positive. For negative values of d, however, the denominator is needed to ensure that marginal utility is positive. 10The elasticity of substitution concept is discussed in more detail in connection with production functions in Chapter 9. Chapter 3: Preferences and Utility 105 ¼ % ¼ d) 1/(1 1/2, and, as
the graph shows, these sharply curved For this situation, s indifference curves apparently fall between the Cobb–Douglas and ﬁxed proportion cases. The negative signs in this utility function may seem strange, but the marginal utilities of both x and y are positive and diminishing, as would be expected. This explains why d must appear in the denominators in Equation 3.30. In the particular case of Equation 3.32, utility increases from 0) toward 0 as x and y increase. This is y an odd utility scale, perhaps, but perfectly acceptable and often useful. (when x %1 ¼ ¼ EXAMPLE 3.3 Homothetic Preferences All the utility functions described in Figure 3.8 are homothetic (see Chapter 2). That is, the marginal rate of substitution for these functions depends only on the ratio of the amounts of the two goods, not on the total quantities of the goods. This fact is obvious for the case of the perfect substitutes (when the MRS is the same at every point) and the case of perfect complements (where the MRS is inﬁnite for y/x > a/b, undeﬁned when y/x a/b, and zero when y/x < a/b). For the general Cobb–Douglas function, the MRS can be found as ¼ MRS @U=@x @U=@y ¼ axa 1yb % bxayb % 3:33) which clearly depends only on the ratio y/x. Showing that the CES function is also homothetic is left as an exercise (see Problem 3.12). The importance of homothetic functions is that one indifference curve is much like another. Slopes of the curves depend only on the ratio y/x, not on how far the curve is from the origin. Indifference curves for higher utility are simple copies of those for lower utility. Hence we can study the behavior of an individual who has homothetic preferences by looking only at one indifference curve or at a few nearby curves without fearing that our results would change dramatically at different levels of utility. QUERY: How might you deﬁne homothetic functions geometrically? What would the locus of all points with a particular MRS look like on an individual’s indifference curve map? EXAMPLE 3.4 Nonhomothetic Preferences Although all the indifference curve maps in Figure 3.
8 exhibit homothetic preferences, this need not always be true. Consider the quasi-linear utility function For this function, good y exhibits diminishing marginal utility, but good x does not. The MRS can be computed as utility U x, y ð ¼ Þ ¼ x þ ln y: (3:34) MRS @U=@x @U=@y ¼ 1 1=y ¼ y ’ ¼ (3:35) The MRS diminishes as the chosen quantity of y decreases, but it is independent of the quantity of x consumed. Because x has a constant marginal utility, a person’s willingness to give up y to get one more unit of x depends only on how much y he or she has. Contrary to the homothetic case, a doubling of both x and y doubles the MRS rather than leaving it unchanged. QUERY: What does the indifference curve map for the utility function in Equation 3.34 look like? Why might this approximate a situation where y is a speciﬁc good and x represents everything else? 106 Part 2: Choice and Demand The Many-Good Case All the concepts we have studied thus far for the case of two goods can be generalized to situations where utility is a function of arbitrarily many goods. In this section, we will brieﬂy explore those generalizations. Although this examination will not add much to what we have already shown, considering peoples’ preferences for many goods can be important in applied economics, as we will see in later chapters. If utility is a function of n goods of the form U x1, x2,, then the equation..., xn ð Þ U x1, x2, ð..., xn k Þ ¼ (3:36) deﬁnes an indifference surface in n dimensions. This surface shows all those combinations of the n goods that yield the same level of utility. Although it is probably impossible to picture what such a surface would look like, we will continue to assume that it is convex. That is, balanced bundles of goods will be preferred to unbalanced ones. Hence the utility function, even in many dimensions, will be assumed to be quasi-concave. The MRS with many goods We can study the trades that a person might voluntarily make between any two of these goods (say, x1 and x2) by again using the implicit
function theorem:!!!! MRS dx2 dx1 ¼ % U x1, x2,..., xn ð k ¼ Þ ¼ Ux1 x1, x2, ð Ux2 x1, x2, ð..., xn..., xn : Þ Þ (3:37) The notation here makes the important point that an individual’s willingness to trade x1 for x2 will depend not only on the quantities of these two goods but also on the quantities of all the other goods. An individual’s willingness to trade food for clothing will depend not only on the quantities of food and clothing he or she has but also on how much ‘‘shelter’’ he or she has. In general it would be expected that changes in the quantities of any of these other goods would affect the trade-off represented by Equation 3.37. It is this possibility that can sometimes make it difﬁcult to generalize the ﬁndings of simple twogood models to the many-good case. One must be careful to specify what is being assumed about the quantities of the other goods. In later chapters we will occasionally look at such complexities. However, for the most part, the two-good model will be good enough for developing intuition about economic relationships. SUMMARY In this chapter we have described the way in which economists formalize individuals’ preferences about the goods they choose. We drew several conclusions about such preferences that will play a central role in our analysis of the theory of choice in the following chapters: • If individuals obey certain basic behavioral postulates in their preferences among goods, they will be able to rank all commodity bundles, and that ranking can be represented by a utility function. In making choices, individuals will behave as though they were maximizing this function. • The negative of the slope of an indifference curve is deﬁned as the marginal rate of substitution (MRS). This shows the rate at which an individual would willingly give up an amount of one good (y) if he or she were compensated by receiving one more unit of another good (x). • The assumption that the MRS decreases as x is substituted for y in consumption is consistent with the notion that individuals prefer some balance in their consumption choices. If the MRS is always decreasing, individuals will have strictly convex indifference curves. That is, their utility function will
be strictly quasi-concave. • Utility functions for two goods can be illustrated by an indifference curve map. Each indifference curve contour on this map shows all the commodity bundles that yield a given level of utility. • A few simple functional forms can capture important differences in individuals’ preferences for two (or more) goods. Here we examined the Cobb–Douglas function, the linear function (perfect substitutes), the ﬁxed proportions Chapter 3: Preferences and Utility 107 function (perfect complements), and the CES function (which includes the other three as special cases). • It is a simple matter mathematically to generalize from two-good examples to many goods. And, as we shall see, studying peoples’ choices among many goods can yield many insights. But the mathematics of many goods is not especially intuitive; therefore, we will primarily rely on two-good cases to build such intuition. PROBLEMS 3.1 Graph a typical indifference curve for the following utility functions, and determine whether they have convex indifference curves (i.e., whether the MRS declines as x increases). a. U(x, y) 3x y. ¼ b. U x, y ð c. U x, y ð d. U x, y ð e. U x. þ yp x ’ xp ﬃﬃﬃﬃﬃﬃﬃﬃ þ x2 ﬃﬃﬃ % xy. p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x þ y2 y. 3.2 In footnote 7 we showed that for a utility function for two goods to have a strictly diminishing MRS (i.e., to be strictly quasiconcave), the following condition must hold: 2UxyUxUy þ Use this condition to check the convexity of the indifference curves for each of the utility functions in Problem 3.1. Describe the precise relationship between diminishing marginal utility and quasi-concavity for each case. y < 0 x % UxxU 2 UyyU 2 3.3 Consider the following utility functions: a. U(x, y) b. U(x, y) c. U(x, y) x
y. x2y2. ln x þ ¼ ¼ ¼ ln y. Show that each of these has a diminishing MRS but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude? 3.4 As we saw in Figure 3.5, one way to show convexity of indifference curves is to show that, for any two points (x1, y1) and (x2, y2) on an indifference curve that promises U k, the utility associated with the point ¼ great as k. Use this approach to discuss the convexity of the indifference curves for the following three functions. Be sure to $ x2 ; y1 þ 2 x1 þ 2 # y2 is at least as graph your results. a. U(x, y) b. U(x, y) c. U(x, y) min(x, y). max(x, y). y. x þ ¼ ¼ ¼ 3.5 The Phillie Phanatic (PP) always eats his ballpark franks in a special way; he uses a foot-long hot dog together with precisely half a bun, 1 ounce of mustard, and 2 ounces of pickle relish. His utility is a function only of these four items, and any extra amount of a single item without the other constituents is worthless. 108 Part 2: Choice and Demand a. What form does PP’s utility function for these four goods have? b. How might we simplify matters by considering PP’s utility to be a function of only one good? What is that good? c. Suppose foot-long hot dogs cost $1.00 each, buns cost $0.50 each, mustard costs $0.05 per ounce, and pickle relish costs $0.15 per ounce. How much does the good deﬁned in part (b) cost? d. If the price of foot-long hot dogs increases by 50 percent (to $1.50 each), what is the percentage increase in the price of the good? e. How would a 50 percent increase in the price of a bun affect the price of the good? Why is your answer different from part (d)? f. If the government wanted to raise $1.00 by taxing the goods that PP buys, how should it spread this tax over the four goods so as to minimize the utility cost to PP? 3.6
Many advertising slogans seem to be asserting something about people’s preferences. How would you capture the following slogans with a mathematical utility function? a. Promise margarine is just as good as butter. b. Things go better with Coke. c. You can’t eat just one Pringle’s potato chip. d. Krispy Kreme glazed doughnuts are just better than Dunkin’ Donuts. e. Miller Brewing advises us to drink (beer) ‘‘responsibly.’’ [What would ‘‘irresponsible’’ drinking be?] 3.7 a. A consumer is willing to trade 3 units of x for 1 unit of y when she has 6 units of x and 5 units of y. She is also willing to trade in 6 units of x for 2 units of y when she has 12 units of x and 3 units of y. She is indifferent between bundle (6, 5) and bundle (12, 3). What is the utility function for goods x and y? Hint: What is the shape of the indifference curve? b. A consumer is willing to trade 4 units of x for 1 unit of y when she is consuming bundle (8, 1). She is also willing to trade in 1 unit of x for 2 units of y when she is consuming bundle (4, 4). She is indifferent between these two bundles. Assuming that the utility function is Cobb–Douglas of the form U(x, y) utility function for this consumer? xayb, where a and b are positive constants, what is the ¼ c. Was there a redundancy of information in part (b)? If yes, how much is the minimum amount of information required in that question to derive the utility function? 3.8 Find utility functions given each of the following indifference curves [deﬁned by U (Æ) a. z ¼ k1=d xa=dyb=d. x2 0:5 b. y c. z ¼ 4 x2 k % % Þ ð 4x x2y ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ�
�ﬃﬃﬃ k Þ ð % 2x ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y4 p p % 0:5x. % y2 2x. % k]: ¼ Analytical Problems 3.9 Initial endowments Suppose that a person has initial amounts of the two goods that provide utility to him or her. These initial amounts are given by x and y. a. Graph these initial amounts on this person’s indifference curve map. b. If this person can trade x for y (or vice versa) with other people, what kinds of trades would he or she voluntarily make? What kinds would not be made? How do these trades relate to this person’s MRS at the point x, y c. Suppose this person is relatively happy with the initial amounts in his or her possession and will only consider trades that? Þ ð increase utility by at least amount k. How would you illustrate this on the indifference curve map? Chapter 3: Preferences and Utility 109 3.10 Cobb–Douglas utility Example 3.3 shows that the MRS for the Cobb–Douglas function is given by U x, y ð Þ ¼ xayb MRS ¼ a b : y x # $ a. Does this result depend on whether a b. For commodity bundles for which y b 1? Does this sum have any relevance to the theory of choice? þ x, how does the MRS depend on the values of a and b? Develop an intuitive expla- ¼ ¼ nation of why, if a > b, MRS > 1. Illustrate your argument with a graph. c. Suppose an individual obtains utility only from amounts of x and y that exceed minimal subsistence levels given by x0, y0. In this case, Is this function homothetic? (For a further discussion, see the Extensions to Chapter 4.) U x, y ð Þ ¼ x ð % x0 a y ð �
� y0 b Þ % 3.11 Independent marginal utilities Two goods have independent marginal utilities if @ 2U @y@x ¼ @ 2U @x@y ¼ 0: Show that if we assume diminishing marginal utility for each good, then any utility function with independent marginal utilities will have a diminishing MRS. Provide an example to show that the converse of this statement is not true. 3.12 CES utility a. Show that the CES function a xd d þ b yd d is homothetic. How does the MRS depend on the ratio y/x? b. Show that your results from part (a) agree with our discussion of the cases d Douglas). 1 (perfect substitutes) and d 0 (Cobb– ¼ ¼ c. Show that the MRS is strictly diminishing for all values of d < 1. d. Show that if x e. Calculate the MRS for this function when y/x y, the MRS for this function depends only on the relative sizes of a and b. 0.9 and y/x 1.1 for the two cases d ¼ clude about the extent to which the MRS changes in the vicinity of x y? How would you interpret this geometrically? ¼ ¼ ¼ 0.5 and d 1. What do you con- ¼ ¼ % 3.13 The quasi-linear function Consider the function U(x, y) some useful properties. þ ¼ x ln y. This is a function that is used relatively frequently in economic modeling as it has a. Find the MRS of the function. Now, interpret the result. b. Conﬁrm that the function is quasi-concave. c. Find the equation for an indifference curve for this function. d. Compare the marginal utility of x and y. How do you interpret these functions? How might consumers choose between x and y as they try to increase their utility by, for example, consuming more when their income increases? (We will look at this ‘‘income effect’’ in detail in the Chapter 5 problems.) e. Considering how the utility changes as the quantities of the two goods increase, describe some situations where this func- tion might be useful. 110 Part 2: Choice and Demand x1, x2,..., xn 3.14 Preference relations The formal study of preferences uses a general vector notation. A bundle of n
commodities is denoted by the vector, and a preference relation (C) is deﬁned over all potential bundles. The statement x1 C x2 means that x Þ bundle x1 is preferred to bundle x2. Indifference between two such bundles is denoted by x1 ¼ ð The preference relation is ‘‘complete’’ if for any two bundles the individual is able to state either x1 C x2, x2 C x1, or x1 + x2. The relation is ‘‘transitive’’ if x1 C x2 and x2 C x3 implies that x1 C x3. Finally, a preference relation is ‘‘continuous’’ if for any bundle y such that y C x, any bundle suitably close to y will also be preferred to x. Using these deﬁnitions, discuss whether each of the following preference relations is complete, transitive, and continuous. x2. + a. Summation preferences: This preference relation assumes one can indeed add apples and oranges. Speciﬁcally, x1 C x2 if n and only if x1 i > x2 i. If x1 i ¼ i, x1 x2 x2. + 1 i ¼ P 1 goods). If x1 b. Lexicographic preferences: In this case the preference relation is organized as a dictionary: If x1 of the amounts of the other n goods). The lexicographic preference relation then continues in this way throughout the entire list of goods. 2, x1 C x2 (regardless of the amounts of the other n 1 and x1 x2 2 > x2 1 ¼ % % 1 > x2 1, x1, x2 (regardless 2 c. Preferences with satiation: For this preference relation there is assumed to be a consumption bundle (x$) that provides complete ‘‘bliss.’’ The ranking of all other bundles is determined by how close they are to x$. That is, x1 C x2 if and only if \| x1 x$\| < \| x2 x$\| where... xi 2 2. x$ % % j % j¼ 2 Þ x$1 xi x$x 1 % ð ﬃﬃﬃﬃﬃﬃ�
�ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ & & xi 2 % xi n % x$n þ þ þ % % q 3.15 The beneﬁt function In a 1992 article David G. Luenberger introduced what he termed the beneﬁt function as a way of incorporating some degree of cardinal measurement into utility theory.11 The author asks us to specify a certain elementary consumption bundle and then measure how many replications of this bundle would need to be provided to an individual to raise his or her utility level to a : Suppose also that the particular target. Suppose there are only two goods and that the utility target is given by U $ x, y Þ elementary consumption bundle is given by x0, y0, is that value of a for which U ax0, ay0 ð. Then the value of the beneﬁt function, b U $ Þ Þ ð a. Suppose utility is given by U x, y b. Using the utility function from part (a), calculate the beneﬁt function for x0 ¼ c. The beneﬁt function can also be deﬁned when an individual has initial endowments of the two goods. If these initial b. Calculate the beneﬁt function for x0 ¼ 1, y0 ¼ y0 ¼ 0. Explain why
your results differ from those in part (a). xby1 % U $. Þ ¼ Þ ¼ 1. ð ð ð ax0, y < U $) or negative (when > U $). Develop a graphical description of these two possibilities, and explain how the nature of the elementary U $. In this situation the ‘‘beneﬁt’’ can be either positive (when U x, y ð Þ is given by that value of a which satisﬁes the equation endowments are given by x, y, then b U $, x, y U x ð þ U x, y ð bundle may affect the beneﬁt calculation. ay0 Þ ¼ þ Þ ð Þ d. Consider two possible initial endowments, x1, y1 and x2, y2. Explain both graphically and intuitively why x2 x1 þ 2 y2 y1 þ 2 b U $, ð Þ the initial endowments.), < 0:5b U $, x1, y1 % þ & 0:5b U $, x2, y2. (Note: This shows that the beneﬁt function is concave in % & SUGGESTIONS FOR FURTHER READING Aleskerov, Fuad, and Bernard Monjardet. Utility Maximization, Choice, and Preference. Berlin: Springer-Verlag, 2002. Kreps, David M. A Course in Microeconomic Theory. Princeton, NJ: Princeton University Press, 1990. A complete study of preference theory. Covers a variety of threshold models and models of ‘‘context-dependent’’ decision making. Chapter 1 covers preference theory in some detail. Good discussion of quasi-concavity. Jehle, G. R., and P. Theory, 2nd ed. Boston: Addison Wesley/Longman, 2001. J. Reny. Advanced Microeconomic Kreps, David M. Notes on the Theory of Choice. London: Westview Press, 1988. Chapter 2 has a good proof of the existence of utility functions when basic axioms of rationality hold. Good discussion of the foundations of preference theory. Most of the focus of the book is on utility in uncertain situations. 11Luenberger, David G. ‘‘Beneﬁ
t Functions and Duality.’’ Journal of Mathematical Economics 21: 461–81. The presentation here has been simpliﬁed considerably from that originally presented by the author, mainly by changing the direction in which ‘‘beneﬁts’’ are measured. Chapter 3: Preferences and Utility 111 Mas-Colell, Andrea, Michael D. Whinston, and Jerry R. Green. Microeconomic Theory. New York: Oxford University Press, 1995. Chapters 2 and 3 provide a detailed development of preference relations and their representation by utility functions. ‘‘The Development of Utility Theory.’’ Journal of Stigler, G. Political Economy 59, pts. 1–2 (August/October 1950): 307–27, 373–96. A lucid and complete survey of the history of utility theory. Has many interesting insights and asides. EXTENSIONS SPECIAL PREFERENCES The utility function concept is a general one that can be adapted to a large number of special circumstances. Discovery of ingenious functional forms that reﬂect the essential aspects of some problem can provide a number of insights that would not be readily apparent with a more literary approach. Here we look at four aspects of preferences that economists have tried to model: (1) threshold effects, (2) quality, (3) habits and addiction, and (4) second-party preferences. In Chapters 7 and 17, we illustrate a number of additional ways of capturing aspects of preferences. E3.1 Threshold effects The model of utility that we developed in this chapter implies an individual will always prefer commodity bundle A to bundle B, provided U(A) > U(B). There may be events that will cause people to shift quickly from consuming bundle A to consuming B. In many cases, however, such a lightning-quick response seems unlikely. People may in fact be ‘‘set in their ways’’ and may require a rather large change in circumstances to change what they do. For example, individuals may not have especially strong opinions about what precise brand of toothpaste they choose and may stick with what they know despite a proliferation of new (and perhaps better) brands. Similarly, people may stick with an old favorite TV show even though it has declined in quality. One way to capture such behavior is to assume individuals make decisions as though they faced thresholds of preference. In such a situation, commodity bundle A might be chosen over B only when
(i) U A Þ ð > U B ‹!; Þ þ ð where! is the threshold that must be overcome. With this speciﬁcation, indifference curves then may be rather thick and even fuzzy, rather than the distinct contour lines shown in this chapter. Threshold models of this type are used extensively in marketing. The theory behind such models is presented in detail in Aleskerov and Monjardet (2002). There, the authors consider a number of ways of specifying the threshold so that it might depend on the characteristics of the bundles being considered or on other contextual variables. Alternative fuels Vedenov, Dufﬁeld, and Wetzstein (2006) use the threshold idea to examine the conditions under which individuals will shift from gasoline to other fuels (primarily ethanol) for powering their cars. The authors point out that the main disadvantage of using gasoline in recent years has been the excessive price volatility of the product relative to other fuels. They conclude that switching to ethanol blends is efﬁcient (especially during periods of increased gasoline price volatility), provided that the blends do not decrease fuel efﬁciency. E3.2 Quality Because many consumption items differ widely in quality, economists have an interest in incorporating such differences into models of choice. One approach is simply to regard items of different quality as totally separate goods that are relatively close substitutes. But this approach can be unwieldy because of the large number of goods involved. An alternative approach focuses on quality as a direct item of choice. Utility might in this case be reﬂected by, ¼ (ii) utility U q, Q Þ ð where q is the quantity consumed and Q is the quality of that consumption. Although this approach permits some examinait encounters difﬁculty tion of quality–quantity trade-offs, when the quantity consumed of a commodity (e.g., wine) consists of a variety of qualities. Quality might then be deﬁned as an average (see Theil,1 1952), but that approach may not be appropriate when the quality of new goods is changing rapidly (e.g., as in the case of personal computers). A more general approach (originally suggested by Lancaster, 1971) focuses on a well-deﬁned set of attributes of goods and assumes that those attributes provide utility. If a good q provides two such attributes, a1
and a2, then utility might be written as utility U q, a1ð q ½ q, a2ð Þ ) Þ, ¼ (iii) and utility improvements might arise either because this individual chooses a larger quantity of the good or because a given quantity yields a higher level of valuable attributes. Personal computers This is the practice followed by economists who study demand in such rapidly changing industries as personal computers. In this case it would clearly be incorrect to focus only on the quantity of personal computers purchased each year 1Theil also suggests measuring quality by looking at correlations between changes in consumption and the income elasticities of various goods. because new machines are much better than old ones (and, presumably, provide more utility). For example, Berndt, Griliches, and Rappaport (1995) ﬁnd that personal computer quality has been increasing about 30 percent per year over a relatively long period, primarily because of improved attributes such as faster processors or better hard drives. A person who spends, say, $2,000 for a personal computer today buys much more utility than did a similar consumer 5 years ago. E3.3 Habits and addiction Because consumption occurs over time, there is the possibility that decisions made in one period will affect utility in later periods. Habits are formed when individuals discover they enjoy using a commodity in one period and this increases their consumption in subsequent periods. An extreme case is addiction (be it to drugs, cigarettes, or Marx Brothers movies), where past consumption signiﬁcantly increases the utility of present consumption. One way to portray these ideas mathematically is to assume that utility in period t depends on consumption in period t and the total of all previous consumption of the habit-forming good (say, X): utility Ut xt, yt, st ð, Þ ¼ (iv) where Chapter 3: Preferences and Utility 113 E3.4 Second-party preferences Individuals clearly care about the well-being of other individuals. Phenomena such as making charitable contributions or making bequests to children cannot be understood without recognizing the interdependence that exists among people. Second-party preferences can be incorporated into the utility function of person i, say, by utility ¼ Ui xi, yi, Uj, (vi) where Uj is the utility of someone else. % & If @Ui /@Uj > 0 then this person will engage in altruistic behavior, whereas if @Ui /@U
i (2001) show that smoking can be approached as a rational, although time-inconsistent,2 choice. 2For more on time inconsistency, see Chapter 17. References Aleskerov, Fuad, and Bernard Monjardet. Utility Maximization, Choice, and Preference. Berlin: Springer-Verlag, 2002. Becker, Gary S. The Economic Approach to Human Behavior. Chicago: University of Chicago Press, 1976. ____. A Treatise on the Family. Cambridge, MA: Harvard University Press, 1981. Becker, Gary S., Michael Grossman, and Kevin M. Murphy. ‘‘An Empirical Analysis of Cigarette Addiction.’’ American Economic Review (June 1994): 396–418. Bergstrom, Theodore C. ‘‘Economics in a Family Way.’’ Journal of Economic Literature (December 1996): 1903–34. Berndt, Ernst R., Zvi Griliches, and Neal J. Rappaport. ‘‘Econometric Estimates of Price Indexes for Personal Computers in the 1990s.’’ Journal of Econometrics (July 1995): 243–68. Gruber, Jonathan, and Botond Koszegi. ‘‘Is Addiction ‘Rational’? Theory and Evidence.’’ Quarterly Journal of Economics (November 2001): 1261–303. 114 Part 2: Choice and Demand Lancaster, Kelvin J. Consumer Demand: A New Approach. New York: Columbia University Press, 1971. Stigler, George J., and Gary S. Becker. ‘‘De Gustibus Non Est Disputandum.’’ American Economic Review (March 1977): 76–90. Theil, Henri. ‘‘Qualities, Prices, and Budget Enquiries.’’ Review of Economic Studies (April 1952): 129–47. Vedenov, Dmitry V., James A. Dufﬁeld, and Michael E. Wetzstein. ‘‘Entry of Alternative Fuels in a Volatile U.S. Gasoline Market.’’ Journal of Agricultural and Resource Economics (April 2006): 1–13. This page intentionally left blank CHAPTER FOUR Utility Maximization and Choice In this chapter we examine the basic model of choice that economists use to explain individuals’ behavior. That model assumes that individuals who are constrained by limited incomes will
behave as though they are using their purchasing power in such a way as to achieve the highest utility possible. That is, individuals are assumed to behave as though they maximize utility subject to a budget constraint. Although the speciﬁc applications of this model are varied, as we will show, all are based on the same fundamental mathematical model, and all arrive at the same general conclusion: To maximize utility, individuals will choose bundles of commodities for which the rate of trade-off between any two goods (the MRS) is equal to the ratio of the goods’ market prices. Market prices convey information about opportunity costs to individuals, and this information plays an important role in affecting the choices actually made. Utility maximization and lightning calculations Before starting a formal study of the theory of choice, it may be appropriate to dispose of two complaints noneconomists often make about the approach we will take. First is the charge that no real person can make the kinds of ‘‘lightning calculations’’ required for utility maximization. According to this complaint, when moving down a supermarket aisle, people just grab what is available with no real pattern or purpose to their actions. Economists are not persuaded by this complaint. They doubt that people behave randomly (everyone, after all, is bound by some sort of budget constraint), and they view the lightning calculation charge as misplaced. Recall, again, Friedman’s pool player from Chapter 1. The pool player also cannot make the lightning calculations required to plan a shot according to the laws of physics, but those laws still predict the player’s behavior. So too, as we shall see, the utility-maximization model predicts many aspects of behavior even though no one carries around a computer with his or her utility function programmed into it. To be precise, economists assume that people behave as if they made such calculations; thus, the complaint that the calculations cannot possibly be made is largely irrelevant. Still, in recent times economists have increasingly tried to model some of the behavioral complications that arise in the actual decisions people make. We look at some of these complications in a variety of problems throughout this book. Altruism and selﬁshness A second complaint against our model of choice is that it appears to be extremely selﬁsh; no one, according to this complaint, has such solely self-centered goals. Although economists are probably more ready to accept self-interest as a motivating force than are other, 117 118 Part 2: Choice
and Demand more Utopian thinkers (Adam Smith observed, ‘‘We are not ready to suspect any person of being deﬁcient in selﬁshness’’1), this charge is also misplaced. Nothing in the utilitymaximization model prevents individuals from deriving satisfaction from philanthropy or generally ‘‘doing good.’’ These activities also can be assumed to provide utility. Indeed, economists have used the utility-maximization model extensively to study such issues as donating time and money to charity, leaving bequests to children, or even giving blood. One need not take a position on whether such activities are selﬁsh or selﬂess because economists doubt people would undertake them if they were against their own best interests, broadly conceived. An Initial Survey The general results of our examination of utility maximization can be stated succinctly as follows Utility maximization. To maximize utility, given a ﬁxed amount of income to spend, an individual will buy those quantities of goods that exhaust his or her total income and for which the psychic rate of trade-off between any two goods (the MRS) is equal to the rate at which the goods can be traded one for the other in the marketplace. That spending all one’s income is required for utility maximization is obvious. Because extra goods provide extra utility (there is no satiation) and because there is no other use for income, to leave any unspent would be to fail to maximize utility. Throwing money away is not a utility-maximizing activity. The condition specifying equality of trade-off rates requires a bit more explanation. Because the rate at which one good can be traded for another in the market is given by the ratio of their prices, this result can be restated to say that the individual will equate the MRS (of x for y) to the ratio of the price of x to the price of y (px / py). This equating of a personal trade-off rate to a market-determined trade-off rate is a result common to all individual utility-maximization problems (and to many other types of maximization problems). It will occur again and again throughout this text. A numerical illustration To see the intuitive reasoning behind this result, assume that it were not true that an individual had equated the MRS to the ratio of the prices of goods. Speciﬁcally,
suppose that the individual’s MRS is equal to 1 and that he or she is willing to trade 1 unit of x for 1 unit of y and remain equally well off. Assume also that the price of x is $2 per unit and of y is $1 per unit. It is easy to show that this person can be made better off. Suppose this person reduces x consumption by 1 unit and trades it in the market for 2 units of y. Only 1 extra unit of y was needed to keep this person as happy as before the trade—the second unit of y is a net addition to well-being. Therefore, the individual’s spending could not have been allocated optimally in the ﬁrst place. A similar method of reasoning can be used whenever the MRS and the price ratio px / py differ. The condition for maximum utility must be the equality of these two magnitudes. 1Adam Smith, The Theory of Moral Sentiments (1759; reprint, New Rochelle, NY: Arlington House, 1969), p. 446. Chapter 4: Utility Maximization and Choice 119 The Two-Good Case: A Graphical Analysis This discussion seems eminently reasonable, but it can hardly be called a proof. Rather, we must now show the result in a rigorous manner and, at the same time, illustrate several other important attributes of the maximization process. First we take a graphic analysis; then we take a more mathematical approach. Budget constraint Assume that the individual has I dollars to allocate between good x and good y. If px is the price of good x and py is the price of good y, then the individual is constrained by pxx pyy I: " þ (4:1) That is, no more than I can be spent on the two goods in question. This budget constraint is shown graphically in Figure 4.1. This person can afford to choose only combinations of x and y in the shaded triangle of the ﬁgure. If all of I is spent on good x, it will buy I/px units of x. Similarly, if all is spent on y, it will buy I / py px / py. This slope shows units of y. The slope of the constraint is easily seen to be 1, then 2 units of y how y can be traded for x in the market. If px ¼ will trade for 1 unit of x. 2 and py ¼ # FIGURE 4.1 The
Individual’s Budget Constraint for Two Goods Those combinations of x and y that the individual can afford are shown in the shaded triangle. If, as we usually assume, the individual prefers more rather than less of every good, the outer boundary of this triangle is the relevant constraint where all the available funds are spent either on x or on y. The slope of this straight-line boundary is given by –px / py. Quantity of y I py I = pxx + pyy 0 Quantity of x I px 120 Part 2: Choice and Demand First-order conditions for a maximum This budget constraint can be imposed on this person’s indifference curve map to show the utility-maximization process. Figure 4.2 illustrates this procedure. The individual would be irrational to choose a point such as A; he or she can get to a higher utility level just by spending more of his or her income. The assumption of nonsatiation implies that a person should spend all his or her income to receive maximum utility. Similarly, by reallocating expenditures, the individual can do better than point B. Point D is out of the question because income is not large enough to purchase D. It is clear that the position of maximum utility is at point C, where the combination x%, y% is chosen. This is the only point on indifference curve U2 that can be bought with I dollars; no higher utility level can be bought. C is a point of tangency between the budget constraint and the indifference curve. Therefore, at C we have slope of budget constraint ¼ ¼ px # p y ¼ slope of indifference curve dy dx!!!! constant U ¼ or px p y ¼ # dy dx U ¼!!!! constant ¼ MRS of x for y ð Þ : (4:2) (4:3) FIGURE 4.2 A Graphical Demonstration of Utility Maximization Point C represents the highest utility level that can be reached by the individual, given the budget constraint. Therefore, the combination x%, y% is the rational way for the individual to allocate purchasing power. Only for this combination of goods will two conditions hold: All available funds will be spent, and the individual’s psychic rate of trade-off (MRS) will be equal to the rate at which the goods can be traded in the market ( px/py). Quantity of y U1 U2 U3 B y* A D C I
= pxx + pyy 0 x* U1 U3 U2 Quantity of x Chapter 4: Utility Maximization and Choice 121 FIGURE 4.3 Example of an Indifference Curve Map for Which the Tangency Condition Does Not Ensure a Maximum If indifference curves do not obey the assumption of a diminishing MRS, not all points of tangency (points for which MRS C is inferior to many other points that can also be purchased with the available funds. In order that the necessary conditions for a maximum (i.e., the tangency conditions) also be sufﬁcient, one usually assumes that the MRS is diminishing; that is, the utility function is strictly quasi-concave. px/py) may truly be points of maximum utility. In this example, tangency point # Quantity of y U1 U2 U3 A I = pxx + pyy C B U3 U2 U1 Quantity of x Our intuitive result is proved: For a utility maximum, all income should be spent, and the MRS should equal the ratio of the prices of the goods. It is obvious from the diagram that if this condition is not fulﬁlled, the individual could be made better off by reallocating expenditures. Second-order conditions for a maximum The tangency rule is only a necessary condition for a maximum. To see that it is not a sufﬁcient condition, consider the indifference curve map shown in Figure 4.3. Here, a point of tangency (C) is inferior to a point of nontangency (B). Indeed, the true maximum is at another point of tangency (A). The failure of the tangency condition to produce an unambiguous maximum can be attributed to the shape of the indifference curves in Figure 4.3. If the indifference curves are shaped like those in Figure 4.2, no such problem can arise. But we have already shown that ‘‘normally’’ shaped indifference curves result from the assumption of a diminishing MRS. Therefore, if the MRS is assumed to be always diminishing, the condition of tangency is both a necessary and sufﬁcient condition for a maximum.2 Without this assumption, one would have to be careful in applying the tangency rule. 2As we saw in Chapters 2 and 3, this is equivalent to assuming that the utility function is quasi-concave. Because we will usually assume quasi-concavity, the necessary
conditions for a constrained utility maximum will also be sufﬁcient. 122 Part 2: Choice and Demand FIGURE 4.4 Corner Solution for Utility Maximization With the preferences represented by this set of indifference curves, utility maximization occurs at E, where 0 amounts of good y are consumed. The ﬁrst-order conditions for a maximum must be modiﬁed somewhat to accommodate this possibility. Quantity of y U1 U2 U3 E x* Quantity of x Corner solutions The utility-maximization problem illustrated in Figure 4.2 resulted in an ‘‘interior’’ maximum, in which positive amounts of both goods were consumed. In some situations individuals’ preferences may be such that they can obtain maximum utility by choosing to consume no amount of one of the goods. If someone does not like hamburgers, there is no reason to allocate any income to their purchase. This possibility is reﬂected in Figure 4.4. There, utility is maximized at E, where x 0; thus, any point on the budget constraint where positive amounts of y are consumed yields a lower utility than does point E. Notice that at E the budget constraint is not precisely tangent to the indifference curve U2. Instead, at the optimal point the budget constraint is ﬂatter than U2, indicating that the rate at which x can be traded for y in the market is lower than the individual’s psychic trade-off rate (the MRS). At prevailing market prices the individual is more than willing to trade away y to get extra x. Because it is impossible in this problem to consume negative amounts of y, however, the physical limit for this process is the X-axis, along which purchases of y are 0. Hence as this discussion makes clear, it is necessary to amend the ﬁrst-order conditions for a utility maximum a bit to allow for corner solutions of the type shown in Figure 4.4. Following our discussion of the general n-good case, we will use the mathematics from Chapter 2 to show how this can be accomplished. x% and y ¼ ¼ The n-Good Case The results derived graphically in the case of two goods carry over directly to the case of n goods. Again it can be shown that for an interior utility maximum, the MRS between any two goods must equal the ratio of the prices of these goods. To study this more general case, however,
it is best to use some mathematics. Chapter 4: Utility Maximization and Choice 123 First-order conditions With n goods, the individual’s objective is to maximize utility from these n goods: subject to the budget constraint3 utility U, x1, x2,..., xnÞ ð ¼ I p1x1 þ p2x2 þ ( ( ( þ ¼ pnxn or (4:4) (4:5) p2x2 # ( ( ( # Following the techniques developed in Chapter 2 for maximizing a function subject to a constraint, we set up the Lagrangian expression pnxn ¼ p1x1 # (4:6) # 0: I + U x1, x2,..., xnÞ þ ð k(I p1x1 # p2x2 # ( ( ( # pnxn). ¼ # Setting the partial derivatives of + (with respect to x1, x2,..., xn and l) equal to 0 yields n 1 equations representing the necessary conditions for an interior maximum: (4:7) þ @+ @x1 ¼ @+ @x2 ¼... @+ @xn ¼ @+ @k ¼ @U @x1 # @U @x2 # kp1 ¼ 0, kp2 ¼ 0, @U @xn # kpn ¼ 0, I p1x1 # p2x2 # ( ( ( # pnxn ¼ # 0: (4:8) These n (see Examples 4.1 and 4.2 to be convinced that such a solution is possible). 1 equations can, in principle, be solved for the optimal x1, x2,..., xn and for l þ Equations 4.8 are necessary but not sufﬁcient for a maximum. The second-order conditions that ensure a maximum are relatively complex and must be stated in matrix terms (see the Extensions to Chapter 2). However, the assumption of strict quasi-concavity (a diminishing MRS in the two-good case) is sufﬁcient to ensure that any point obeying Equation 4.8 is in fact a true maximum. Implications of ﬁrst-order conditions The ﬁrst-order conditions represented by Equation 4.
8 can be rewritten in a variety of instructive ways. For example, for any two goods, xi and xj, we have @U=@xi @U=@xj ¼ pi pj : (4:9) In Chapter 3 we showed that the ratio of the marginal utilities of two goods is equal to the marginal rate of substitution between them. Therefore, the conditions for an optimal allocation of income become MRS xi for xjÞ ¼ ð pi pj : (4:10) 3Again, the budget constraint has been written as an equality because, given the assumption of nonsatiation, it is clear that the individual will spend all available income. 124 Part 2: Choice and Demand This is exactly the result derived graphically earlier in this chapter; to maximize utility, the individual should equate the psychic rate of trade-off to the market trade-off rate. Interpreting the Lagrange multiplier Another result can be derived by solving Equations 4.8 for l: k ¼ @U=@x1 p1 @U=@x2 p2 ¼ @U=@xn pn ¼ ( ( ( ¼ (4:11) These equations state that, at the utility-maximizing point, each good purchased should yield the same marginal utility per dollar spent on that good. Therefore, each good should have an identical (marginal) beneﬁt-to-(marginal)-cost ratio. If this were not true, one good would promise more ‘‘marginal enjoyment per dollar’’ than some other good, and funds would not be optimally allocated. Although the reader is again warned against talking conﬁdently about marginal utility, what Equation 4.11 says is that an extra dollar should yield the same ‘‘additional utility’’ no matter which good it is spent on. The common value for this extra utility is given by the Lagrange multiplier for the consumer’s budget constraint (i.e., by l). Consequently, l can be regarded as the marginal utility of an extra dollar of consumption expenditure (the marginal utility of ‘‘income’’). One ﬁnal way to rewrite the necessary conditions for a maximum is pi ¼ @U=@xi k (4:12) for every good i that is bought. To interpret this expression, remember (from Equation 4.11)
that the Lagrange multiplier, l, represents the marginal utility value of an extra dollar of income, no matter where it is spent. Therefore, the ratio in Equation 4.12 compares the extra utility value of one more unit of good i to this common value of a marginal dollar in spending. To be purchased, the utility value of an extra unit of a good must be worth, in dollar terms, the price the person must pay for it. For example, a high price for good i can only be justiﬁed if it also provides a great deal of extra utility. At the margin, therefore, the price of a good reﬂects an individual’s willingness to pay for one more unit. This is a result of considerable importance in applied welfare economics because willingness to pay can be inferred from market reactions to prices. In Chapter 5 we will see how this insight can be used to evaluate the welfare effects of price changes, and in later chapters we will use this idea to discuss a variety of questions about the efﬁciency of resource allocation. Corner solutions The ﬁrst-order conditions of Equations 4.8 hold exactly only for interior maxima for which some positive amount of each good is purchased. As discussed in Chapter 2, when corner solutions (such as those illustrated in Figure 4.4) arise, the conditions must be modiﬁed slightly.4 In this case, Equations 4.8 become @+ @xi ¼ @U @xi # kpi " 0 i ð ¼ 1,..., n) (4:13) 4Formally, these conditions are called the Kuhn–Tucker conditions for nonlinear programming. Chapter 4: Utility Maximization and Choice 125 and, if then @+ @xi ¼ @U @xi # kpi < 0, 0: xi ¼ To interpret these conditions, we can rewrite Equation 4.14 as pi > @U=@xi k : (4:14) (4:15) (4:16) Hence the optimal conditions are as before, except that any good whose price (pi) exceeds its marginal value to the consumer will not be purchased (xi ¼ 0). Thus, the mathematical results conform to the commonsense idea that individuals will not purchase goods that they believe are not worth the money. Although corner solutions do not provide a major focus for our analysis in this book, the reader should keep in mind
the possibilities for such solutions arising and the economic interpretation that can be attached to the optimal conditions in such cases. EXAMPLE 4.1 Cobb–Douglas Demand Functions As we showed in Chapter 3, the Cobb–Douglas utility function is given by where, for convenience,5 we assume a ¼ values of x and y for any prices ( px, py) and income (I). Setting up the Lagrangian expression þ Þ ¼ 1. We can now solve for the utility-maximizing xayb, U x, y ð b yields the ﬁrst-order conditions + xayb + k I ð # ¼ pxx # pyy) @+ @x ¼ @+ @y ¼ @+ @k ¼ axa # 1yb bxayb # 1 kpx ¼ kpy ¼ # # 0, 0, I # pxx # pyy ¼ 0: Taking the ratio of the ﬁrst two terms shows that ay bx ¼ px py, or pyy b a ¼ pxx where the ﬁnal equation follows because a þ Equation 4.21 into the budget constraint gives b ¼ ¼ a 1 # a pxx, 1. Substitution of this ﬁrst-order condition in I ¼ pxx þ pyy ¼ pxx þ a 1 # a pxx ¼ pxx pxx; (4:22) 5As we discussed in Chapter 3, the exponents in the Cobb–Douglas utility function can always be normalized to sum to 1 b) is a monotonic transformation. because U1/(a þ (4:17) (4:18) (4:19) (4:20) (4:21) 126 Part 2: Choice and Demand solving for x yields x% ¼ aI px, and a similar set of manipulations would give y% ¼ bI py. (4:23) (4:24) These results show that an individual whose utility function is given by Equation 4.17 will always choose to allocate a proportion of his or her income to buying good x (i.e., px x/I a) and b proportion to buying good y ( pyy/I b). Although this feature of the Cobb–Douglas function often makes it easy to work out simple problems, it does suggest that the function
has limits in its ability to explain actual consumption behavior. Because the share of income devoted to particular goods often changes signiﬁcantly in response to changing economic conditions, a more general functional form may provide insights not provided by the Cobb–Douglas function. We illustrate a few possibilities in Example 4.2, and the general topic of budget shares is taken up in more detail in the Extensions to this chapter. ¼ ¼ Numerical example. First, however, let’s look at a speciﬁc numerical example for the Cobb– Douglas case. Suppose that x sells for $1 and y sells for $4 and that total income is $8. Succinctly then, assume that px ¼ 0.5, so that this individual splits his or her income equally between these two goods. Now the demand Equations 4.23 and 4.24 imply 8. Suppose also that a 1, py ¼ 4, I ¼ ¼ ¼ b x% y% aI=px ¼ bI=py ¼ 0:5I=px ¼ 0:5I=py ¼ ¼ ¼ and, at these optimal choices, 0:5 0:5 8 ð 8 ð =1 Þ =4 Þ 4, 1, ¼ ¼ (4:25) 4 ¼ ð Notice also that we can compute the value for the Lagrange multiplier associated with this income allocation by using Equation 4.19: utility (4:26) 1 ð ¼ ¼ 2: 0:5 Þ 0:5 Þ x0:5y0:5 0:5 axa # 1yb=px ¼ k ¼ 0:5 4 ð This value implies that each small change in income will increase utility by approximately one fourth of that amount. Suppose, for example, that this person had 1 percent more income ($8.08). In this case he or she would choose x 1.01, and utility would be 4.04 and y 4.040.5 Æ 1.010.5 2.02. Hence a $0.08 increase in income increased utility by 0.02, as predicted by the fact that l 0.25. (4:27) 0:25:5=1 Þ ¼ QUERY: Would a change in py affect the quantity of x demanded in Equation 4.23? Explain your
answer mathematically. Also develop an intuitive explanation based on the notion that the share of income devoted to good y is given by the parameter of the utility function, b. EXAMPLE 4.2 CES Demand To illustrate cases in which budget shares are responsive to economic circumstances, let’s look at three speciﬁc examples of the CES function. Case 1: d ¼ 0.5. In this case, utility is U x, y ð Þ ¼ x0:5 þ y0:5. (4:28) Chapter 4: Utility Maximization and Choice 127 Setting up the Lagrangian expression yields the following ﬁrst-order conditions for a maximum: x0:5 + ¼ þ y0:5 k I ð þ # pxx pyy Þ # @+/@x @+/@y @+/@k 0:5x# 0:5y# 0:5 0:5 ¼ ¼ I pxx # kpx ¼ kpy ¼ pyy ¼ # # # 0, 0, 0: ¼ Division of the ﬁrst two of these shows that (4:29) (4:30) ¼ By substituting this into the budget constraint and doing some messy algebraic manipulation, we can derive the demand functions associated with this utility function: ð y=x 0:5 Þ px=py: (4:31) x% y% 1 I=px½ I=py½ 1 ¼ ¼, px=pyÞ* : py=pxÞ* þ ð þ ð (4:32) (4:33) ¼ 1/[1 Price responsiveness. In these demand functions notice that the share of income spent on, say, good x—that is, px x / I ( px/py)]—is not a constant; it depends on the price ratio px/py. The higher the relative price of x, the smaller the share of income spent on that good. In other words, the demand for x is so responsive to its own price that an increase in the price reduces total spending on x. That the demand for x is price responsive can also be illustrated by comparing the implied exponent on px in the demand function given by Equation 4.32 ( 2) to that from Equation 4.23 ( 1). In Chapter 5 we will discuss this
observation more fully when we examine the elasticity concept in detail. þ # # Case 2: d 5 21. Alternatively, let’s look at a demand function with less substitutability6 than the Cobb–Douglas. If d 1, the utility function is given by ¼ # and it is easy to show that the ﬁrst-order conditions for a maximum require U x, y ð Þ ¼ # 1 x# # y# 1, y=x px=pyÞ ¼ ð 0:5: (4:34) (4:35) Again, substitution of this condition into the budget constraint, together with some messy algebra, yields the demand functions x% y% ¼ I=px½ 1 I=py½ 1 þ ð 0:5 0:5, . py=pxÞ px=pyÞ (4:36) ¼ þ ð That these demand functions are less price responsive can be seen in two ways. First, now the ( py/px)0.5]—responds positively to share of income spent on good x—that is, px x /I increases in px. As the price of x increases, this individual cuts back only modestly on good x; thus, total spending on that good increases. That the demand functions in Equation 4.36 are less price responsive than the Cobb–Douglas is also illustrated by the relatively small implied exponents of each good’s own price ( 0.5). 1/[1 ¼ þ # 6One way to measure substitutability is by the elasticity of substitution, which for the CES function is given by s Here d the CES function in connection with the theory of production in Chapter 9. 0 (the Cobb–Douglas) implies s d). 0.5. See also the discussion of 0.5 implies s 1 implies s 1, and d ¼ # 1/(1 2, d ¼ ¼ ¼ ¼ ¼ ¼ # 128 Part 2: Choice and Demand Case 3: d 5 2‘. This is the important case in which x and y must be consumed in ﬁxed proportions. Suppose, for example, that each unit of y must be consumed together with exactly 4 units of x. The utility function that represents this situation is U x, y ð Þ ¼ min x, 4y ð : Þ
(4:37) In this situation, a utility-maximizing person will choose only combinations of the two goods for which x 4y; that is, utility maximization implies that this person will choose to be at a vertex of his or her L-shaped indifference curves. Because of the shape of these indifference curves, calculus cannot be used to solve this problem. Instead, one can adopt the simple procedure of substituting the utility-maximizing condition directly into the budget constraint: ¼ I pxx pyy pxx py þ ¼ þ ¼ x 4 ¼ ð px þ x: 0:25pyÞ Hence and similar substitutions yield x% ¼ px þ I 0:25py, (4:38) (4:39) : ¼ y% py I 4px þ In this case, the share of a person’s budget devoted to, say, good x rises rapidly as the price of x increases because x and y must be consumed in ﬁxed proportions. For example, if we use the values assumed in Example 4.1 ( px ¼ 8), Equations 4.39 and 4.40 would predict 4, I x% 1, and, as before, half of the individual’s income would be spent on each good. If ¼ we instead use px ¼ 2/3, and this person spends two 4, and I ¼ thirds [ px x / I 2/3] of his or her income on good x. Trying a few other numbers suggests that the share of income devoted to good x approaches 1 as the price of x increases.7 1, py ¼ 8, then x% 2, py ¼ (2 Æ 8/3)/8 ¼ 8/3, y% (4:40) 4, y% ¼ ¼ ¼ ¼ ¼ QUERY: Do changes in income affect expenditure shares in any of the CES functions discussed here? How is the behavior of expenditure shares related to the homothetic nature of this function? Indirect Utility Function Examples 4.1 and 4.2 illustrate the principle that it is often possible to manipulate the ﬁrst-order conditions for a constrained utility-maximization problem to solve for the optimal values of x1, x2,..., xn. These optimal values in general will depend on the prices of all the goods and on the individual’
s income. That is, 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 (4:41) In the next chapter we will analyze in more detail this set of demand functions, which show the dependence of the quantity of each xi demanded on p1, p2,..., pn and I. Here 7These relationships for the CES function are pursued in more detail in Problem 4.9 and in Extension E4.3. Chapter 4: Utility Maximization and Choice 129 we use the optimal values of the x’s from Equation 4.42 to substitute in the original utility function to yield maximum utility U V ¼ ¼ ½ p1,... pn, I, x%2ð x%1ð Þ : p1, p2,..., pn, I Þ ð p1,... pn, I,..., x%nð Þ p1,... pn, I Þ* (4:42) (4:43) In words, because of the individual’s desire to maximize utility given a budget constraint, the optimal level of utility obtainable will depend indirectly on the prices of the goods being bought and the individual’s income. This dependence is reﬂected by the indirect utility function V. If either prices or income were to change, the level of utility that could be attained would also be affected. Sometimes, in both consumer theory and many other contexts, it is possible to use this indirect approach to study how changes in economic circumstances affect various kinds of outcomes, such as utility or (later in this book) ﬁrms’ costs. The Lump Sum Principle Many economic insights stem from the recognition that utility ultimately depends on the income of individuals and on the prices they face. One of the most important of these is the so-called lump sum principle that illustrates the superiority of taxes on a person’s general purchasing power to taxes on speciﬁc goods. A related insight is that general income grants to low-income people will raise utility more than will a similar amount of money
spent subsidizing speciﬁc goods. The intuition behind these results derives directly from the utility-maximization hypothesis; an income tax or subsidy leaves the individual free to decide how to allocate whatever ﬁnal income he or she has. On the other hand, taxes or subsidies on speciﬁc goods both change a person’s purchasing power and distort his or her choices because of the artiﬁcial prices incorporated in such schemes. Hence general income taxes and subsidies are to be preferred if efﬁciency is an important criterion in social policy. The lump sum principle as it applies to taxation is illustrated in Figure 4.5. Initially this person has an income of I and is choosing to consume the combination x%, y%. A tax on good x would raise its price, and the utility-maximizing choice would shift to combination x1, y1. Tax collections would be t Æ x1 (where t is the tax rate imposed on good x). Alternatively, an income tax that shifted the budget constraint inward to I0 would also collect this same amount of revenue.8 But the utility provided by the income tax (U2) exceeds that provided by the tax on x alone (U1). Hence we have shown that the utility burden of the income tax is smaller. A similar argument can be used to illustrate the superiority of income grants to subsidies on speciﬁc goods. EXAMPLE 4.3 Indirect Utility and the Lump Sum Principle In this example we use the notion of an indirect utility function to illustrate the lump sum principle as it applies to taxation. First we have to derive indirect utility functions for two illustrative cases. Case 1: Cobb–Douglas. In Example 4.1 we showed that for the Cobb–Douglas utility function with a 0.5, optimal purchases are b ¼ ¼ 8Because I t)x1 þ ¼ income tax also passes through the point x1, y1. pyy1, we have I 0 ( px þ ¼ I – tx1 ¼ pxx1 þ pyy1, which shows that the budget constraint with an equal-size 130 Part 2: Choice and Demand x% y% ¼ ¼ I 2px I 2py, : Thus, the indirect utility function in this case is V ð px, py, I U x%, y% ð Þ ¼ x% Þ ¼ ð
0:5 Þ ð y% 0:5 Þ ¼ I x p0:5 2p0:5 y : (4:44) (4:45) Notice that when px ¼ ¼ that we calculated before for this situation. 1, py ¼ 4, and I 8 we have V 8/(2 Æ 1 Æ 2) ¼ ¼ 2, which is the utility Case 2: Fixed proportions. In the third case of Example 4.2 we found that (4:46) (4:47) I 0:25py, x% y% ¼ ¼ px þ I 4px þ : py Thus, in this case indirect utility is given by V ð px; py; I x%; 4y% min ð Þ ¼ x% ¼ Þ ¼ I 0:25py px þ I 0:25py ; 4y% 4 4px þ ¼ ¼ px þ 8, indirect utility is given by V py ¼ with px ¼ before. 1, py ¼ 4, and I ¼ 4, which is what we calculated ¼ The lump sum principle. Consider ﬁrst using the Cobb–Douglas case to illustrate the lump sum principle. Suppose that a tax of $1 were imposed on good x. Equation 4.45 shows that indirect utility in this case would fall from 2 to 1.41 [ 2 with the tax, total tax collections will be $2. Therefore, an equal-revenue income tax would reduce 6/(2 Æ 1 Æ 2)]. Thus, the income tax is a clear net income to $6, and indirect utility would be 1.5 [ improvement in utility over the case where x alone is taxed. The tax on good x reduces utility for two reasons: It reduces a person’s purchasing power, and it biases his or her choices away from good x. With income taxation, only the ﬁrst effect is felt and so the tax is more efﬁcient.9 8/(2 Æ 20.5 Æ 2)]. Because this person chooses x% ¼ ¼ ¼ The ﬁxed-proportions case supports this intuition. In that case, a $1 tax on good x would reduce indirect utility from 4 to 8/3 [ 8/3 and tax collections þ would be $8/3. An
income tax that collected $8/3 would leave this consumer with $16/3 in net income, and that income would yield an indirect utility of V 1)]. Hence after-tax utility is the same under both the excise and income taxes. The reason the lump sum principle does not hold in this case is that with ﬁxed-proportions utility, the excise tax does not distort choices because preferences are so rigid. 1)]. In this case x% (16/3)/(1 8/3 [ 8/(2 þ ¼ ¼ ¼ ¼ QUERY: Both indirect utility functions illustrated here show that a doubling of income and all prices would leave indirect utility unchanged. Explain why you would expect this to be a property of all indirect utility functions. That is, explain why the indirect utility function is homogeneous of degree zero in all prices and income. 9This discussion assumes that there are no incentive effects of income taxation—probably not a good assumption. Chapter 4: Utility Maximization and Choice 131 FIGURE 4.5 The Lump Sum Principle of Taxation A tax on good x would shift the utility-maximizing choice from x%, y% to x1, y1. An income tax that collected the same amount would shift the budget constraint to I 0. Utility would be higher (U2) with the income tax than with the tax on x alone (U1). Quantity of y y1 y* y2 I′ U3 I U2 U1 x1 x2 x* Quantity of x Expenditure Minimization In Chapter 2 we pointed out that many constrained maximum problems have associated ‘‘dual’’ constrained minimum problems. For the case of utility maximization, the associated dual minimization problem concerns allocating income in such a way as to achieve a given utility level with the minimal expenditure. This problem is clearly analogous to the primary utility-maximization problem, but the goals and constraints of the problems have been reversed. Figure 4.6 illustrates this dual expenditureminimization problem. There, the individual must attain utility level U2; this is now the constraint in the problem. Three possible expenditure amounts (E1, E2, and E3) are shown as three ‘‘budget constraint’’ lines in the ﬁgure. Expenditure level E1 is clearly too small to achieve U2; hence it cannot solve the dual problem. With expenditures given by E3, the individual can reach U
2 (at either of the two points B or C), but this is not the minimal expenditure level required. Rather, E2 clearly provides just enough total expenditures to reach U2 (at point A), and this is in fact the solution to the dual problem. By comparing Figures 4.2 and 4.6, it is obvious that both the primary utility-maximization approach and the dual expenditure-minimization approach yield the same solution (x%, y%); they are simply alternative ways of viewing the same process. Often the expenditure-minimization approach is more useful, however, because expenditures are directly observable, whereas utility is not. 132 Part 2: Choice and Demand FIGURE 4.6 The Dual ExpenditureMinimization Problem The dual of the utility-maximization problem is to attain a given utility level (U2) with minimal expenditures. An expenditure level of E1 does not permit U2 to be reached, whereas E3 provides more spending power than is strictly necessary. With expenditure E2, this person can just reach U2 by consuming x% and y%. Quantity of y B E3 y* A E2 E1 C U2 x* Quantity of x A mathematical statement More formally, the individual’s dual expenditure-minimization problem is to choose x1, x2,..., xn to minimize total expenditures E p1x1 þ p2x2 þ ( ( ( þ ¼ ¼ pnxn, (4:48) subject to the constraint!U ¼ ¼ utility : x1, x2,..., xnÞ U ð The optimal amounts of x1, x2,..., xn chosen in this problem will depend on the prices of the various goods ( p1, p2,..., pn) and on the required utility level!U. If any of the prices were to change or if the individual had a different utility ‘‘target,’’ then another commodity bundle would be optimal. This dependence can be summarized by an expenditure function. (4:49 Expenditure function. The individual’s expenditure function shows the minimal expenditures necessary to achieve a given utility level for a particular set of prices. That is, minimal expenditures E ð ¼ p1, p2,..., pn, U). (4:50) Chapter 4: Utility Maximization and Choice 133 This de�
��nition shows that the expenditure function and the indirect utility function are inverse functions of one another (compare Equations 4.43 and 4.50). Both depend on market prices but involve different constraints (income or utility). In the next chapter we will see how this relationship is useful in allowing us to examine the theory of how individuals respond to price changes. First, however, let’s look at two expenditure functions. EXAMPLE 4.4 Two Expenditure Functions There are two ways one might compute an expenditure function. The ﬁrst, most straightforward method would be to state the expenditure-minimization problem directly and apply the Lagrangian technique. Some of the problems at the end of this chapter ask you to do precisely that. Here, however, we will adopt a more streamlined procedure by taking advantage of the relationship between expenditure functions and indirect utility functions. Because these two functions are inverses of each other, calculation of one greatly facilitates the calculation of the other. We have already calculated indirect utility functions for two important cases in Example 4.3. Retrieving the related expenditure functions is simple algebra. Case 1: Cobb–Douglas utility. Equation 4.45 shows that the indirect utility function in the two-good, Cobb–Douglas case is px, py, I V ð Þ ¼ I x p0:5 2p0:5 y : (4:51) If we now interchange the role of utility (which we will now treat as the utility ‘‘target’’ denoted by U) and income (which we will now term ‘‘expenditures,’’ E, and treat as a function of the parameters of this problem), then we have the expenditure function E px, py, U 2p0:5 x p0:5 y U: (4:52) ð Checking this against our former results, now we use a utility target of U and py ¼ are $8 ( ¼ the dual expenditure-minimization problem are formally identical. 2 with, again, px ¼ 1 4. With these parameters, Equation 4.52 shows that the required minimal expenditures 2 Æ 10.5 Æ 40.5 Æ 2). Not surprisingly, both the primal utility-maximization problem and Þ ¼ ¼ Case 2: Fixed proportions. For the ﬁxed-proportions case, Equation
4.47 gave the indirect utility function as px, py, I V ð Þ ¼ px þ I 0:25py : (4:53) If we again switch the role of utility and expenditures, we quickly derive the expenditure function: 0:25pyÞ A check of the hypothetical values used in Example 4.3 ( px ¼ it would cost $8 [ 0.25 Æ 4) Æ 4] to reach the utility target of 4. px, py, U px þ Þ ¼ ð U: 1, py ¼ (1 E ð ¼ þ (4:54) 4, U ¼ 4) again shows that Compensating for a price change. These expenditure functions allow us to investigate how a person might be compensated for a price change. Speciﬁcally, suppose that the price of good y were to increase from $4 to $5. This would clearly reduce a person’s utility, so we might ask what amount of monetary compensation would mitigate the harm. Because the expenditure function allows utility to be held constant, it provides a direct estimate of this amount. Speciﬁcally, in the Cobb–Douglas case, expenditures would have to be increased from $8 to 2 Æ 1 Æ 50.5 Æ 2) to provide enough extra purchasing power to precisely compensate for $8.94 ( ¼ 134 Part 2: Choice and Demand this price increase. With ﬁxed proportions, expenditures would have to be increased from $8 to $9 to compensate for the price increase. Hence the compensations are about the same in these simple cases. There is one important difference between the two examples, however. In the ﬁxedproportions case, the $1 of extra compensation simply permits this person to return to his or her previous consumption bundle (x 4 for this rigid person. In the Cobb–Douglas case, however, this person will not use the extra compensation to revert to his or her previous consumption bundle. Instead, utility maximization will require that 2, the $8.94 be allocated so that x but this person will economize on the now more expensive good y. In the next chapter we will pursue this analysis of the welfare effects of price changes in much greater detail. 0.894. This will still provide a utility level of U 1). That is the only way to restore utility to U 4
.47 and y 4, y ¼ ¼ ¼ ¼ ¼ ¼ QUERY: How should a person be compensated for a price decrease? What sort of compensation would be required if the price of good y fell from $4 to $3? Properties of Expenditure Functions Because expenditure functions are widely used in applied economics, it is important to understand a few of the properties shared by all such functions. Here we look at three properties. All these follow directly from the fact that expenditure functions are based on individual utility maximization. 1. Homogeneity. For both of the functions illustrated in Example 4.4, a doubling of all prices will precisely double the value of required expenditures. Technically, these expenditure functions are ‘‘homogeneous of degree one’’ in all prices.10 This is a general property of expenditure functions. Because the individual’s budget constraint is linear in prices, any proportional increase in both prices and purchasing power will permit the person to buy the same utility-maximizing commodity bundle that was chosen before the price increase. In Chapter 5 we will see that, for this reason, demand functions are homogeneous of degree zero in all prices and income. 2. Expenditure functions are nondecreasing in prices. This property can be succinctly summarized by the mathematical statement @E @pi + 0 for every good i: (4:55) This seems intuitively obvious. Because the expenditure function reports the minimum expenditure necessary to reach a given utility level, an increase in any price must increase this minimum. More formally, suppose the price of one good increases and that all other prices stay the same. Let A represent the bundle of goods purchased before the price increase and B the bundle purchased after the price increase. Clearly bundle B costs more after the price increase than it did previously. The only change between the two situations is an increase in one of the prices; therefore, spending on that good increases and all other spending stays the same. However, we also know that, before the price increase, bundle A cost less than bundle B because A was the expenditure-minimizing bundle. Hence actual expenditures when B is chosen after 10As described in Chapter 2, the function f (x1, x2,..., xn) is said to be homogeneous of degree k if f (tx1, tx2,..., txn) tkf (x1, x2,..., xn). In
this case, k 1. ¼ ¼ Chapter 4: Utility Maximization and Choice 135 FIGURE 4.7 Expenditure Functions Are Concave in Prices At p%1 this person spends E then expenditures would be given by Epseudo. Because his or her consumption patterns will likely change as p1 changes, actual expenditures will be less than this.. If he or she continues to buy the same set of goods as p1 changes, Þ p%1,... ð E( p1,...) E( p1,...) pseudo E E( p1,...) E( p1,...) p1 the price increase must exceed those on A before the price increase. A similar chain of logic could be used to show that a decrease in price should cause expenditures to decrease (or possibly stay the same). 3. Expenditure functions are concave in prices. In Chapter 2 we discussed concave functions, which are deﬁned as functions that always lie below tangents to them. Although the technical mathematical conditions that describe such functions are complicated, it is relatively simple to show how the concept applies to expenditure functions by considering the variation in a single price. Figure 4.7 shows an individual’s expenditures as a function of the single price, p1. At the initial price, p%1, this person’s expenditures are given by E p%1,.... Now consider prices higher or lower than p%1. If this person continued to buy the same bundle of goods, expenditures would increase or decrease linearly as this price changed. This would give rise to the pseudo-expenditure function Epseudo in the ﬁgure. This line shows a level of expenditures that would allow this person to buy the original bundle of goods despite the changing value of p1. If, as seems more likely, this person adjusted his or her purchases as p1 changed, we know (because of expenditure minimization) that actual expenditures would be less than these pseudo-amounts. Hence the actual expenditure function, E, will lie everywhere below Epseudo and the function will be concave.11 The concavity of the expenditure function is a useful property for a number of applications, especially those related to the substitution effect from price changes (see Chapter 5). Þ ð 11One result of concavity is that fii ¼ @ 2E=@p 2 i " 0. This is

Subsets and Splits

No community queries yet

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