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market price, however. If it faces a downward-sloping demand curve for its product, then more output can be sold only by reducing the good’s price. In this case the revenue obtained from selling one more unit will be less than the price of that unit because to get consumers to take the extra unit, the price of all other units must be lowered. This result can be easily demonstrated. As before, total revenue (R) is the product of the quantity sold (q) times the price at which it is sold (p), which may also depend on q. Using the product rule to compute the derivative, marginal revenue is MR q Þ ¼ ð dR dq ¼ p d ½ q ð dq Þ $ q ( p q $ þ ¼ dp dq : (11:8) Notice that the marginal revenue is a function of output. In general, MR will be different for different levels of q. From Equation 11.8 it is easy to see that if price does not 0), marginal revenue will be equal to price. In change as quantity increases (dp/dq ¼ 376 Part 4: Production and Supply FIGURE 11.1 Marginal Revenue Must Equal Marginal Cost for Proﬁt Maximization Proﬁts, deﬁned as revenues (R) minus costs (C), reach a maximum when the slope of the revenue function (marginal revenue) is equal to the slope of the cost function (marginal cost). This equality is only a necessary condition for a maximum, as may be seen by comparing points q& (a true maximum) and q&& (a local minimum), points at which marginal revenue equals marginal cost. C R q ** q * Output per period q * Output per period Revenues, costs (a) Proﬁts 0 Losses (b) this case we say that the ﬁrm is a price-taker because its output decisions do not inﬂuence the price it receives. On the other hand, if price decreases as quantity increases (dp/dq < 0), marginal revenue will be less than price. A proﬁt-maximizing manager must know how increases in output will affect the price received before making an optimal output decision. If increases in q cause market price to decrease, this must be taken into account. Chapter 11: Prof it Maximization 377 EXAMPLE 11.1 Marginal Revenue from a Linear
Demand Function Suppose a shop selling sub sandwichs (also called grinders, torpedoes, or, in Philadelphia, hoagies) faces a linear demand curve for its daily output over period (q) of the form Solving for the price the shop receives, we have 100 q ¼ % 10p: q % 10 þ p ¼ 10, and total revenues (as a function of q) are given by The sub ﬁrm’s marginal revenue function is pq R ¼ ¼ q2 % 10 þ 10q: MR dR dq ¼ q % 5 þ ¼ 10, (11:9) (11:10) (11:11) (11:12) and in this case MR < p for all values of q. If, for example, the ﬁrm produces 40 subs per day, Equation 11.10 shows that it will receive a price of $6 per sandwich. But at this level of output Equation 11.12 shows that MR is only $2. If the ﬁrm produces 40 subs per day, then total revenue will be $240 ( 40), whereas if it produced 39 subs, then total revenue would be 39) because price will increase slightly when less is produced. Hence the $6.1 $238 ( marginal revenue from the 40th sub sold is considerably less than its price. Indeed, for q 50, marginal revenue is zero (total revenues are a maximum at $250 50), and any further expansion in daily sub output will result in a reduction in total revenue to the ﬁrm. $6 $5 * * ¼ * ¼ ¼ ¼ To determine the proﬁt-maximizing level of sub output, we must know the ﬁrm’s marginal costs. If subs can be produced at a constant average and marginal cost of $4, then Equation MC at a daily output of 30 subs. With this level of output, each sub will 11.12 shows that MR $4) Æ 30]. Although price exceeds average and marginal sell for $7, and proﬁts are $90 [ cost here by a substantial margin, it would not be in the ﬁrm’s interest to expand output. With q % $4.00) Æ 35]. Marginal revenue, not price, is the primary determinant of proﬁt-maximizing behavior. 35, for example,
price will decrease to $6.50 and proﬁts will decrease to $87.50 [ ($6.50 ($7 % ¼ ¼ ¼ ¼ QUERY: How would an increase in the marginal cost of sub production to $5 affect the output decision of this ﬁrm? How would it affect the ﬁrm’s proﬁts? Marginal revenue and elasticity The concept of marginal revenue is directly related to the elasticity of the demand curve facing the ﬁrm. Remember that the elasticity of demand (eq, p) is deﬁned as the percentage change in quantity demanded that results from a 1 percent change in price: eq, p ¼ dq=q dp=p ¼ dq dp $ p q : Now, this deﬁnition can be combined with Equation 11.8 to give MR p þ ¼ q dp dq ¼ $ q p : dp dq eq, p : # (11:13) As long as the demand curve facing the ﬁrm is negatively sloped, then eq, p < 0 and marginal revenue will be less than price, as we have already shown. If demand is elastic 378 Part 4: Production and Supply TABLE 11.1 RELATIONSHIP BETWEEN ELASTICITY AND MARGINAL REVENUE 1 eq, p < % eq, p ¼ % 1 eq, p > % 1 MR > 0 0 MR ¼ MR < 0 % (eq, p < 1), then marginal revenue will be positive. If demand is elastic, the sale of one more unit will not affect price ‘‘very much,’’ and hence more revenue will be yielded by the sale. In fact, if demand facing the ﬁrm is inﬁnitely elastic (eq, p ¼ %1 ), marginal revenue will equal price. The ﬁrm is, in this case, a price-taker. However, if demand is inelastic (eq, p > 1), marginal revenue will be negative. Increases in q can be obtained only through ‘‘large’’ decreases in market price, and these decreases will cause total revenue to decrease. % The relationship between marginal revenue and elasticity is summarized by Table 11.1. Price–marginal cost markup If we assume the ﬁrm wishes to maximize pro�
�ts, this analysis can be extended to illustrate the connection between price and marginal cost. Setting MR MC in Equation 11.13 yields ¼ or, after rearranging, MC ¼ p 1 " þ 1 eq, p # p MC % p 1 eq, p ¼ 1 eq, pj j : ¼ % (11:14) where the last equality holds if demand is downward sloping and thus eq, p < 0. This formula for the percentage ‘‘markup’’ of price over marginal cost is sometimes called the Lerner index after the economist Abba Lerner, who ﬁrst proposed it in the 1930s. The markup depends in a speciﬁc way on the elasticity of demand facing the ﬁrm. First, notice that this demand must be elastic (eq, p < 1) for this formula to make any sense. If demand were inelastic, the ratio in Equation 11.14 would be greater than 1, which is impossible if a positive MC is subtracted from a positive p in the numerator. This simply reﬂects that, when demand is inelastic, marginal revenue is negative and cannot be equated to a positive marginal cost. It is important to stress that it is the demand facing the ﬁrm that must be elastic. This may be consistent with an inelastic market demand for the product in question if the ﬁrm faces competition from other ﬁrms producing the same good. % Equation 11.14 implies that the percentage markup over marginal cost will be higher 1. If the demand facing the ﬁrm is inﬁnitely elastic (perhaps because, and there is no MC). On the other hand, with an elasticity of demand of, say, eq, p ¼ % 2, the closer eq, p is to there are many other ﬁrms producing the same good), then eq, p ¼ %1 markup (p the markup over marginal cost will be 50 percent of price; that is, ( p MC)/p 1/2. % ¼ % ¼ Marginal revenue curve Any demand curve has a marginal revenue curve associated with it. If, as we sometimes assume, the ﬁrm must sell all its output at one price, it is convenient to think of the demand curve facing the ﬁrm as an average revenue curve. That is, the demand curve
shows the revenue per unit (in other words, the price) yielded by alternative output choices. The marginal revenue curve, on the other hand, shows the extra revenue FIGURE 11.2 Market Demand Curve and Associated Marginal Revenue Curve Chapter 11: Prof it Maximization 379 Because the demand curve is negatively sloped, the marginal revenue curve will fall below the demand (‘‘average revenue’’) curve. For output levels beyond q1, MR is negative. At q1, total revenues (p1 Æ q1) are a maximum; beyond this point, additional increases in q cause total revenues to decrease because of the concomitant decreases in price. Price p1 0 D (average revenue) q1 Quantity per period MR provided by the last unit sold. In the usual case of a downward-sloping demand curve, the marginal revenue curve will lie below the demand curve because, according to Equation 11.8, MR < p. In Figure 11.2 we have drawn such a curve together with the demand curve from which it was derived. Notice that for output levels greater than q1, marginal revenue is negative. As output increases from 0 to q1, total revenues (p Æ q) increase. However, at q1 total revenues (p1 Æ q1) are as large as possible; beyond this output level, price decreases proportionately faster than output increases. In Part 2 we talked in detail about the possibility of a demand curve’s shifting because of changes in income, prices of other goods, or preferences. Whenever a demand curve does shift, its associated marginal revenue curve shifts with it. This should be obvious because a marginal revenue curve cannot be calculated without referring to a speciﬁc demand curve. EXAMPLE 11.2 The Constant Elasticity Case In Chapter 5 we showed that a demand function of the form has a constant price elasticity of demand equal to function for this function, ﬁrst solve for p: b. To compute the marginal revenue % apb q ¼ (11:15) 1=b p ¼ 1 a " # q1=b ¼ k q1=b, (11:16) 380 Part 4: Production and Supply where k ¼ (1/a)1/b. Hence and pq R ¼ ¼ kqð 1 þ =b b Þ MR ¼ dR=dq 1 þ b ¼ b kq1=b
b p: 1 þ b ¼ (11:17) ¼ 0.5p. For a more elastic case, suppose b For this particular function, MR is proportional to price. If, for example, eq, p ¼ 2, then 0.9p. The MR curve MR ¼ p;, then MR approaches the demand curve as demand becomes more elastic. Again, if b that is, in the case of inﬁnitely elastic demand, the ﬁrm is a price-taker. For inelastic demand, on the other hand, MR is negative (and proﬁt maximization would be impossible). 10; then MR ¼ %1 ¼ % ¼ % ¼ b QUERY: Suppose demand depended on other factors in addition to p. How would this change the analysis of this example? How would a change in one of these other factors shift the demand curve and its marginal revenue curve? Short-Run Supply by a Price-Taking Firm We are now ready to study the supply decision of a proﬁt-maximizing ﬁrm. In this chapter we will examine only the case in which the ﬁrm is a price-taker. In Part 6 we will look at other cases in considerably more detail. Also, we will focus only on supply decisions in the short run here. Long-run questions concern entry and exit by ﬁrms and are the primary focus of the next chapter. Therefore, the ﬁrm’s set of short-run cost curves is the appropriate model for our analysis. ¼ Proﬁt-maximizing decision Figure 11.3 shows the ﬁrm’s short-run decision. The market price3 is given by P&. Therefore, the demand curve facing the ﬁrm is a horizontal line through P&. This line is labeled MR as a reminder that an extra unit can always be sold by this price-taking ﬁrm P& without affecting the price it receives. Output level q& provides maximum proﬁts because at q& price is equal to short-run marginal cost. The fact that proﬁts are positive can be seen by noting that price at q& exceeds average costs. The ﬁrm earns a proﬁt on each unit sold. If price were below average cost (as is the case for P&&&), the ﬁ
rm would have a loss on each unit sold. If price and average cost were equal, proﬁts would be zero. Notice that at q& the marginal cost curve has a positive slope. This is required if proﬁts are to be a MC on a negatively sloped section of the marginal cost curve, then true maximum. If P this would not be a point of maximum proﬁts because increasing output would yield more in revenues (price times the amount produced) than this production would cost (marginal cost would decrease if the MC curve has a negative slope). Consequently, proﬁt maximization requires both that P MC and that marginal cost increase at this point.4 ¼ ¼ 3We will usually use an uppercase italic P to denote market price here and in later chapters. When notation is complex, however, we will sometimes revert to using a lowercase p. 4Mathematically: because proﬁt maximization requires (the ﬁrst-order condition) p q ð Þ ¼ Pq C, q Þ ð % and (the second-order condition) p 0 q Þ ¼ ð P % MC q Þ ¼ ð 0 q Þ Hence, it is required that MC 0(q) > 0; marginal cost must be increasing. Þ ¼ % MC 0 q ð p 00 ð < 0: Chapter 11: Prof it Maximization 381 FIGURE 11.3 Short-Run Supply Curve for a Price-Taking Firm In the short run, a price-taking ﬁrm will produce the level of output for which SMC example, the ﬁrm will produce q&. The SMC curve also shows what will be produced at other prices. For prices below SAVC, however, the ﬁrm will choose to produce no output. The heavy lines in the ﬁgure represent the ﬁrm’s short-run supply curve. P. At P&, for ¼ Market price P ** P * = MR P * Ps 0 SMC SAC SAVC q * q * q ** Quantity per period The ﬁrm’s short-run supply curve The positively sloped portion of the short-run marginal cost curve is the short-run supply curve for this price-taking ﬁrm. That curve shows how much the ﬁrm will produce
for every possible market price. For example, as Figure 11.3 shows, at a higher price of P&& the ﬁrm will produce q&& because it is in its interest to incur the higher marginal costs entailed by q&&. With a price of P&&&, on the other hand, the ﬁrm opts to produce less (q&&&) because only a lower output level will result in lower marginal costs to meet this lower price. By considering all possible prices the ﬁrm might face, we can see by the marginal cost curve how much output the ﬁrm should supply at each price. The shutdown decision. For low prices we must be careful about this conclusion. Should market price fall below Ps (the ‘‘shutdown price’’), the proﬁt-maximizing decision would be to produce nothing. As Figure 11.3 shows, prices less than Ps do not cover average variable costs. There will be a loss on each unit produced in addition to the loss of all ﬁxed costs. By shutting down production, the ﬁrm must still pay ﬁxed costs but avoids the losses incurred on each unit produced. Because, in the short run, the ﬁrm cannot close down and avoid all costs, its best decision is to produce no output. On the other hand, a price only slightly above Ps means the ﬁrm should produce some output. Although proﬁts may be negative (which they will be if price falls below short-run average total costs, the case at P&&&), the proﬁt-maximizing decision is to continue production as long as variable costs are covered. Fixed costs must be paid in any case, and any price that covers variable costs will provide revenue as an offset to 382 Part 4: Production and Supply the ﬁxed costs.5 Hence we have a complete description of this ﬁrm’s supply decisions in response to alternative prices for its output. These are summarized in the following deﬁnition Short-run supply curve. The ﬁrm’s short-run supply curve shows how much it will produce at various possible output prices. For a proﬁt-maximizing ﬁrm that takes the price of its output as given, this curve consists of the positively sloped segment of the ﬁrm’s
short-run marginal cost above the point of minimum average variable cost. For prices below this level, the ﬁrm’s proﬁtmaximizing decision is to shut down and produce no output. Of course, any factor that shifts the ﬁrm’s short-run marginal cost curve (such as changes in input prices or changes in the level of ﬁxed inputs used) will also shift the short-run supply curve. In Chapter 12 we will make extensive use of this type of analysis to study the operations of perfectly competitive markets. EXAMPLE 11.3 Short-Run Supply In Example 10.5 we calculated the short-run total-cost production function as function for the Cobb–Douglas SC v, w, q, k1Þ ¼ ð vk1 þ wq1=bk% 1 a=b, (11:18) where k1 is the level of capital input that is held constant in the short run.6 Short-run marginal cost is easily computed as SMC v, w, q, k1Þ ¼ ð @SC @q ¼ w b qð 1 % b Þ a=b =bk% 1 : (11:19) Notice that short-run marginal cost increases in output for all values of q. Short-run proﬁt maximization for a price-taking ﬁrm requires that output be chosen so that market price (P) is equal to short-run marginal cost: SMC w b ¼ qð 1 % b Þ a=b =bk% 1 P, ¼ and we can solve for quantity supplied as b= ð 1 % b Þ % ka= 1 1 % b Þ ð P b= ð 1 % b Þ: q ¼ w b " # (11:20) (11:21) This supply function provides a number of insights that should be familiar from earlier economics courses: (1) The supply curve is positively sloped—increases in P cause the ﬁrm to 5Some algebra may clarify matters. We know that total costs equal the sum of ﬁxed and variable costs, and that proﬁts are given by p If q ¼ 0, then variable costs and revenues are 0, and thus The ﬁrm will produce something only if p > SC ¼ SFC þ SVC
, R SC ¼ % P q $ % ¼ SFC % SVC: p ¼ % SFC: % SFC. But that means that p q > SVC or p > SVC=q: $ 6Because capital input is held constant, the short-run cost function exhibits increasing marginal cost and will therefore yield a unique proﬁt-maximizing output level. If we had used a constant returns-to-scale production function in the long run, there would have been no such unique output level. We discuss this point later in this chapter and in Chapter 12. Chapter 11: Prof it Maximization 383 produce more because it is willing to incur a higher marginal cost;7 (2) the supply curve is shifted to the left by increases in the wage rate, w—that is, for any given output price, less is supplied with a higher wage; (3) the supply curve is shifted outward by increases in capital input, k1—with more capital in the short run, the ﬁrm incurs a given level of short-run marginal cost at a higher output level; and (4) the rental rate of capital, v, is irrelevant to short-run supply decisions because it is only a component of ﬁxed costs. Numerical example. We can pursue once more the numerical example from Example 10.5, where a 80. For these speciﬁc parameters, the supply function is 0.5, v 3, w b ¼ ¼ ¼ ¼ p1 $ ¼ 40 P w ¼ 40P 12 ¼ 10P 3 : $ (11:22) 12, and k1 ¼ w k1Þ 0:5 $ ð % 1 1 % q ¼ $ That this computation is correct can be checked by comparing the quantity supplied at various prices with the computation of short-run marginal cost in Table 10.2. For example, if P 12, 40 will be supplied, and Table 10.2 shows that this then the supply function predicts that q 24, an output level of 80 would will agree with the P be supplied, and again Table 10.2 shows that when q 6) would cause less to be produced (q ¼ SMC rule. If price were to double to P 24. A lower price (say, P ¼ 80, SMC 20). ¼ ¼ ¼ ¼ ¼ Before adopting Equation 11.
22 as the supply curve in this situation, we should also check 0 SMC rule? From Equation 11.18 we know that short-run variable costs are the ﬁrm’s shutdown decision. Is there a price where it would be more proﬁtable to produce q than to follow the P given by ¼ ¼ and so SVC ¼ wq 1=bk% 1 a=b SVC q ¼ wqð 1 % b Þ a=b =bk% 1 : (11:23) (11:24) ¼ ¼ A comparison of Equation 11.24 with Equation 11.19 shows that SVC/q < SMC for all values of q provided that b < 1. Thus, in this problem there is no price low enough such that, by following the P SMC rule, the ﬁrm would lose more than if it produced nothing. ¼ In our numerical example, consider the case P 10. Total revenue would be R 3. With such a low price, the ﬁrm would opt 255 for q ¼ 225. Although the situation is dismal (see Table 10.1). Hence proﬁts would be p ¼ for the ﬁrm, it is better than opting for q 0. If it produces nothing, it avoids all variable (labor) costs but still loses 240 in ﬁxed costs of capital. By producing 10 units of output, its revenues 15) and contribute 15 to offset slightly the loss of cover variable costs (R ﬁxed costs. 30, and total short-run costs would be SC ¼ % SVC SC 30 15 % % % ¼ ¼ ¼ ¼ ¼ R QUERY: How would you graph the short-run supply curve in Equation 11.22? How would the curve be shifted if w rose to 15? How would it be shifted if capital input increased to k1 ¼ 100? How would the short-run supply curve be shifted if v fell to 2? Would any of these changes alter the ﬁrm’s determination to avoid shutting down in the short run? Proﬁt Functions Additional insights into the proﬁt-maximization process for a price-taking ﬁrm8 can be obtained by looking at the proﬁt function. This function shows the ﬁrm’s (
maximized) proﬁts as depending only on the prices that the ﬁrm faces. To understand the logic of its construction, remember that economic proﬁts are deﬁned as 7In fact, the short-run elasticity of supply can be read directly from Equation 11.21 as b/(1 8Much of the analysis here would also apply to a ﬁrm that had some market power over the price it received for its product, but we will delay a discussion of that possibility until Part 5. b). % 384 Part 4: Production and Supply p Pq C P f k, l vk wl: (11:25) ¼ % Þ % ð ¼ Only the variables k and l [and also q f (k, l)] are under the ﬁrm’s control in this expression. The ﬁrm chooses levels of these inputs to maximize proﬁts, treating the three prices P, v, and w as ﬁxed parameters in its decision. Looked at in this way, the ﬁrm’s maximum proﬁts ultimately depend only on these three exogenous prices (together with the form of the production function). We summarize this dependence by the proﬁt function Proﬁt function. The ﬁrm’s proﬁt function shows its maximal proﬁts as a function of the prices that the ﬁrm faces: P, v, w P ð Þ ¼ max k, l k, l p ð Þ ¼ Pf max k, l ½ k, l ð Þ % vk wl : ( % (11:26) In this deﬁnition we use an upper case P to indicate that the value given by the function is the maximum proﬁts obtainable given the prices. This function implicitly incorporates the form of the ﬁrm’s production function—a process we will illustrate in Example 11.4. The proﬁt function can refer to either long-run or short-run proﬁt maximization, but in the latter case we would need also to specify the levels of any inputs that are ﬁxed in the short run. Properties of the proﬁt function As for the other optimized functions we have already looked at,
the proﬁt function has a number of properties that are useful for economic analysis. 1. Homogeneity. A doubling of all the prices in the proﬁt function will precisely double proﬁts—that is, the proﬁt function is homogeneous of degree 1 in all prices. We have already shown that marginal costs are homogeneous of degree 1 in input prices; hence a doubling of input prices and a doubling of the market price of a ﬁrm’s output will not change the proﬁt-maximizing quantity it decides to produce. However, because both revenues and costs have doubled, proﬁts will double. This shows that with pure inﬂation (where all prices rise together) ﬁrms will not change their production plans, and the levels of their proﬁts will just keep up with that inﬂation. 2. Proﬁt functions are nondecreasing in output price, P. This result seems obvious—a ﬁrm could always respond to an increase in the price of its output by not changing its input or output plans. Given the deﬁnition of proﬁts, they must increase. Hence if the ﬁrm changes its plans, it must be doing so to make even more proﬁts. If proﬁts were to decrease, the ﬁrm would not be maximizing proﬁts. 3. Proﬁt functions are nonincreasing in input prices, v, and w. Again, this feature of the proﬁt function seems obvious. A proof is similar to that used above in our discussion of output prices. 4. Proﬁt functions are convex in output prices. This important feature of proﬁt functions says that the proﬁts obtainable by averaging those available from two different output prices will be at least as large as those obtainable from the average9 of the two prices. Mathematically, P P1, v, w ð Þ þ 2 P2, v, w P ð P Þ + P1 þ 2 " P2, v, w : # (11:27) 9Although we only discuss a simple averaging of prices here, it is clear that with convexity a condition similar to Equation 11.27 holds for any weighted average price P
1. P2 where 0 1 t t tP1 þ ð ¼ % Þ,, Chapter 11: Prof it Maximization 385 The intuitive reason for this convexity is that, when ﬁrms can freely adapt their decisions to two different prices, better results are possible than when they can make only one set P2)/2 of choices in response to the single average price. More formally, let P3 ¼ and let qi, ki, li represent the proﬁt-maximizing output and input choices for these various prices. Then (P1 þ P ð P3, v, w Þ - P3q3 % vk3 % wl3 ¼, - P1q3 % P1q1 % vk3 % 2 vk1 % 2 wl3 wl1 P P1, v, w ð P Þ þ 2 þ P2q3 % P2q2 % þ P2, v, w ð Þ wl3 wl2 vk3 % 2 vk2 % 2, (11:28) which proves Equation 11.27. The key step is Equation 11.28. Because (q1, k1, l1) is the proﬁt-maximizing combination of output and inputs when the market price is P1, it must generate as much proﬁt as any other choice, including (q3, k3, l3). By similar reasoning, the proﬁt from (q2, k2, l2) is at least as much as that from (q3, k3, l3) when the market price is P2. The convexity of the proﬁt function has many applications to topics such as price stabilization. Envelope results Because the proﬁt function reﬂects an underlying process of unconstrained maximization, we may also apply the envelope theorem to see how proﬁts respond to changes in output and input prices. This application of the theorem yields a variety of useful results. Speciﬁcally, using the deﬁnition of proﬁts shows that @P ð P, v, w @P Þ q ð ¼ P, v, w, Þ @P ð P, v, w @v Þ
k P, v, w ð, Þ ¼ % @P ð P, v, w @w Þ l ð ¼ % P, v, w : Þ (11:29) (11:30) (11:31) Again, these equations make intuitive sense: A small change in output price will increase proﬁts in proportion to how much the ﬁrm is producing, whereas a small increase in the price of an input will reduce proﬁts in proportion to the amount of that input being used. The ﬁrst of these equations says that the ﬁrm’s supply function can be calculated from its proﬁt function by partial differentiation with respect to the output price.10 The second and third equations show that input demand functions11 can also be derived from the proﬁt functions. Because the proﬁt function itself is homogeneous of degree 1, all the functions described in Equations 11.29–11.31 are homogeneous of degree 0. That is, a doubling of both output and input prices will not change the input levels that the ﬁrm chooses, nor will this change the ﬁrm’s proﬁt-maximizing output level. All these ﬁndings also have short-run analogs, as will be shown later with a speciﬁc example. 10This relationship is sometimes referred to as ‘‘Hotelling’s lemma’’—after the economist Harold Hotelling, who discovered it in the 1930s. 11Unlike the input demand functions derived in Chapter 10, these input demand functions are not conditional on output levels. Rather, the ﬁrm’s proﬁt-maximizing output decision has already been taken into account in the functions. Therefore, this demand concept is more general than the one we introduced in Chapter 10, and we will have much more to say about it in the next section. 386 Part 4: Production and Supply Producer surplus in the short run In Chapter 5 we discussed the concept of ‘‘consumer surplus’’ and showed how areas below the demand curve can be used to measure the welfare costs to consumers of price changes. We also showed how such changes in welfare could be captured in the individual’s expenditure function. The process of measuring the welfare effects of price changes for �
�rms is similar in short-run analysis, and this is the topic we pursue here. However, as we show in the next chapter, measuring the welfare impact of price changes for producers in the long run requires a different approach because most such long-term effects are felt not by ﬁrms themselves but rather by their input suppliers. In general, it is this long-run approach that will prove more useful for our subsequent study of the welfare impacts of price changes. Because the proﬁt function is nondecreasing in output prices, we know that if P2 > P1 then P ð P2,... P P1,... ð Þ, Þ + and it would be natural to measure the welfare gain to the ﬁrm from the price change as welfare gain P P2,... ð Þ % P P1,... ð Þ : ¼ (11:32) Figure 11.4 shows how this value can be measured graphically as the area bounded by the two prices and above the short-run supply curve. Intuitively, the supply curve shows the minimum price that the ﬁrm will accept for producing its output. Hence when market If price increases from P1 to P2, then the increase in the ﬁrm’s proﬁts is given by area P2ABP1. At a price of P1, the ﬁrm earns short-run producer surplus given by area PsCBP1. This measures the increase in short-run proﬁts for the ﬁrm when it produces q1 rather than shutting down when price is Ps or below. Market price P2 P1 Ps SMC A B C q1 q2 q FIGURE 11.4 Changes in Short-Run Producer Surplus Measure Firm Proﬁts Chapter 11: Prof it Maximization 387 price increases from P1 to P2, the ﬁrm is able to sell its prior output level (q1) at a higher q1) for which, at the margin, it likewise price and also opts to sell additional output (q2 % earns added proﬁts on all but the ﬁnal unit. Hence the total gain in the ﬁrm’s proﬁts is given by area P2 ABP1. Mathematically, we can make use of
the envelope results from the previous section to derive welfare gain P P2,... ð Þ % P P1,... ð Þ ¼ ¼ P2 ð P1 @P @P dP ¼ P2 ð P1 dP: P q ð Þ (11:33) Thus, the geometric and mathematical measures of the welfare change agree. Using this approach, we can also measure how much the ﬁrm values the right to produce at the prevailing market price relative to a situation where it would produce no output. If we denote the short-run shutdown price as Ps (which may or may not be a price of zero), then the extra proﬁts available from facing a price of P1 are deﬁned to be producer surplus: producer surplus P ð ¼ P1,... P Ps,... ð Þ ¼ Þ % P1 ð Ps dP: q P ð Þ (11:34) This is shown as area P1BCPs deﬁnition. in Figure 11.4. Hence we have the following formal Producer surplus. Producer surplus is the extra return that producers earn by making transactions at the market price over and above what they would earn if nothing were produced. It is illustrated by the size of the area below the market price and above the supply curve. In this deﬁnition, we have made no distinction between the short run and the long run, although our development thus far has involved only short-run analysis. In the next chapter, we will see that the same deﬁnition can serve dual duty by describing producer surplus in the long run, so using this generic deﬁnition works for both concepts. Of course, as we will show, the meaning of long-run producer surplus is different from what we have studied here. One more aspect of short-run producer surplus should be pointed out. Because the vk1; that is, ﬁrm produces no output at its shutdown price, we know that II(PS,…) proﬁts at the shutdown price are solely made up of losses of all ﬁxed costs. Therefore, ¼ % producer surplus P1,... P ð P1,... P ð P PS,... ð Þ % Þ vk1�
� ¼ Þ % ð% ¼ ¼ P P1,... ð Þ þ vk1: (11:35) That is, producer surplus is given by current proﬁts being earned plus short-run ﬁxed costs. Further manipulation shows that magnitude can also be expressed as producer surplus PS,... P Þ % ð wl1 þ vk1 % In words, a ﬁrm’s short-run producer surplus is given by the extent to which its revenues exceed its variable costs—this is, indeed, what the ﬁrm gains by producing in the short run rather than shutting down and producing nothing. P1,... P ð P1q1 % Þ vk1 ¼ P1q1 % (11:36) ¼ ¼ wl1: 388 Part 4: Production and Supply EXAMPLE 11.4 A Short-Run Proﬁt Function These various uses of the proﬁt function can be illustrated with the Cobb–Douglas production kalb and because we treat capital as ﬁxed at k1 in the function we have been using. Because q short run, it follows that proﬁts are ¼ % To ﬁnd the proﬁt function we use the ﬁrst-order conditions for a maximum to eliminate l from this expression: ¼ p Pka 1 l b vk1 % w l: (11:37) @p @l ¼ so bPka bPka 1 " 1= b 1 % ð Þ : # (11:38) (11:39) We can simplify the process of substituting this back into the proﬁt equation by letting A : Making use of this shortcut, we have w ¼ ð bPka 1 Þ ’ P ð P, v, w, k1Þ ¼ Pka 1 Ab= ð wA1= b 1 Þ % % Þ Pka 1 1 " wb= b ð % vk1 % A 1 w % ÞP1= 1 ð % # b 1 % wA1= ð b % 1 % b bb= ð b 1 Þ % ¼ ¼ Þka= ð 1 1 % b Þ vk
1: % b ð 1 Þ % vk1 (11:40) Though admittedly messy, this solution is what was promised—the ﬁrm’s maximal proﬁts are expressed as a function of only the prices it faces and its technology. Notice that the ﬁrm’s ﬁxed costs (vk1) enter this expression in a simple linear way. The prices the ﬁrm faces determine the extent to which revenues exceed variable costs; then ﬁxed costs are subtracted to obtain the ﬁnal proﬁt number. Because it is always wise to check that one’s algebra is correct, let’s try out the numerical 12, and k1 ¼ 80, we know that at a 20. Hence 0. The ﬁrm will just break even at a 0.5, v 12 the ﬁrm will produce 40 units of output and use labor input of l example we have been using. With a price of P proﬁts will be p price of P % 12. Using the proﬁt function yields 12 Æ 20 12 Æ 40 3 Æ 80 3, v, w, k1Þ ¼ P ð P 12, 3, 12, 80 ð Þ ¼ 0:25 $ 1 12% 122 $ 80 3 $ % $ 80 ¼ 0: (11:41) Thus, at a price of 12, the ﬁrm earns 240 in proﬁts on its variable costs, and these are precisely offset by ﬁxed costs in arriving at the ﬁnal total. With a higher price for its output, the ﬁrm earns positive proﬁts. If the price falls below 12, however, the ﬁrm incurs short-run losses.12 Hotelling’s lemma. We can use the proﬁt function in Equation 11.40 together with the envelope theorem to derive this ﬁrm’s short-run supply function: P, v, w, k1Þ ¼ q ð b= ð b % 1 Þ @P @P ¼ w b " # ka= 1 ð b 1 % P b= ð 1 % b Þ, Þ (11:42) which is precisely the short-run
supply function that we calculated in Example 11.3 (see Equation 11.21). 12In Table 10.2 we showed that if q 40, then SAC ¼ ¼ 12. Hence zero proﬁts are also indicated by P 12 ¼ ¼ SAC. Chapter 11: Prof it Maximization 389 Producer surplus. We can also use the supply function to calculate the ﬁrm’s short-run producer surplus. To do so, we again return to our numerical example: a 12, 3, w and k1 ¼ 10P/3 and the shutdown price is zero. Hence at a price of P ¼ 80. With these parameters, the short-run supply relationship is q 12, producer surplus is 0.5, v ¼ ¼ ¼ ¼ b ¼ producer surplus 12 ¼ ð 0 10P 3 dP ¼ 10P 2 6 12 0 ¼!!!! 240: (11:43) This precisely equals short-run proﬁts at a price of 12 (p vk1 240). If price were to rise to (say) 15, then producer surplus would increase to 375, 0) plus short-run ﬁxed costs ( ¼ which would still consist of 240 in ﬁxed costs plus total proﬁts at the higher price (P 3 Æ 80 ¼ ¼ ¼ 135). ¼ QUERY: How is the amount of short-run producer surplus here affected by changes in the rental rate for capital, v? How is it affected by changes in the wage, w? Proﬁt Maximization and Input Demand Thus far, we have treated the ﬁrm’s decision problem as one of choosing a proﬁt-maximizing level of output. But our discussion throughout has made clear that the ﬁrm’s outin fact, determined by the inputs it chooses to use, a relationship that is put is, summarized by the production function q f(k, l). Consequently, the ﬁrm’s economic proﬁts can also be expressed as a function of only the inputs it uses: ¼ p k, l ð Þ ¼ Pq C q ð % Þ ¼ P f k, l ð vk wl : Þ þ Þ % ð (11:44) View
ed in this way, the proﬁt-maximizing ﬁrm’s decision problem becomes one of choosing the appropriate levels of capital and labor input.13 The ﬁrst-order conditions for a maximum are @p @k ¼ @p @l ¼ P P @f @k % @f @l % 0, v ¼ 0: w ¼ (11:45) (11:46) These conditions make the intuitively appealing point that a proﬁt-maximizing ﬁrm should hire any input up to the point at which the input’s marginal contribution to revenue is equal to the marginal cost of hiring the input. Because the ﬁrm is assumed to be a price-taker in its hiring, the marginal cost of hiring any input is equal to its market price. The input’s marginal contribution to revenue is given by the extra output it produces (the marginal product) times that good’s market price. This demand concept is given a special name as follows Marginal revenue product. The marginal revenue product is the extra revenue a ﬁrm receives when it uses one more unit of an input. In the price-taking14 case, MRPl ¼ Pfl and MRPk ¼ Pfk. 13Throughout our discussion in this section, we assume that the ﬁrm is a price-taker; thus, the prices of its output and its inputs can be treated as ﬁxed parameters. Results can be generalized fairly easily in the case where prices depend on quantity. 14If the ﬁrm is not a price-taker in the output market, then this deﬁnition is generalized by using marginal revenue in place of price. That is, MRPl ¼ MR Æ MPl. A similiar derivation holds for capital input. @R/@q Æ @q/@l @R/@l ¼ ¼ 390 Part 4: Production and Supply Hence proﬁt maximization requires that the ﬁrm hire each input up to the point at which its marginal revenue product is equal to its market price. Notice also that the proﬁt-maximizing Equations 11.45 and 11.46 also imply cost minimization because RTS w/v. fl/fk ¼ ¼ Second-order conditions Because the proﬁt function in Equation
11.44 depends on two variables, k and l, the secondorder conditions for a proﬁt maximum are somewhat more complex than in the singlevariable case we examined earlier. In Chapter 2 we showed that, to ensure a true maximum, the proﬁt function must be concave. That is, p kk ¼ f kk < 0, p ll ¼ f ll < 0, (11:47) and f 2 kl > 0: p2 kl ¼ (11:48) p kkp ll % f kk f ll % Therefore, concavity of the proﬁt relationship amounts to requiring that the production function itself be concave. Notice that diminishing marginal productivity for each input is not sufﬁcient to ensure increasing marginal costs. Expanding output usually requires the ﬁrm to use more capital and more labor. Thus, we must also ensure that increases in capital input do not raise the marginal productivity of labor (and thereby reduce marginal cost) by a large enough amount to reverse the effect of diminishing marginal productivity of labor itself. Therefore, Equation 11.47 requires that such cross-productivity effects be relatively small—that they be dominated by diminishing marginal productivities of the inputs. If these conditions are satisﬁed, then marginal costs will increase at the proﬁt-maximizing choices for k and l, and the ﬁrst-order conditions will represent a local maximum. Input demand functions In principle, the ﬁrst-order conditions for hiring inputs in a proﬁt-maximizing way can be manipulated to yield input demand functions that show how hiring depends on the prices that the ﬁrm faces. We will denote these demand functions by capital demand labor demand ¼ k l, P, v, w Þ ð : P, v, w Þ (11:49) ð ¼ Notice that, contrary to the input demand concepts discussed in Chapter 10, these demand functions are ‘‘unconditional’’—that is, they implicitly permit the ﬁrm to adjust its output to changing prices. Hence these demand functions provide a more complete picture of how prices affect input demand than did the contingent demand functions introduced in Chapter 10. We have already shown that these input demand functions can also be derived from the proﬁt function through differentiation; in Example 11
.5, we show that process explicitly. First, however, we will explore how changes in the price of an input might be expected to affect the demand for it. To simplify matters, we look only at labor demand, but the analysis of the demand for any other input would be the same. In general, we conclude that the direction of this effect is unambiguous in all cases—that is, @l/@w 0 no matter how many inputs there are. To develop some intuition for this result, we begin with some simple cases., Single-input case One reason for expecting @l/@w to be negative is based on the presumption that the marginal physical product of labor decreases as the quantity of labor employed increases. A Chapter 11: Prof it Maximization 391 P Æ MPl: decrease in w means that more labor must be hired to bring about the equality w A decrease in w must be met by a decrease in MPl (because P is ﬁxed as required by the ceteris paribus assumption), and this can be brought about by increasing l. That this argument is strictly correct for the case of one input can be shown as follows. With one input, Equation 11.44 is the sole ﬁrst-order condition for proﬁt maximization, rewritten here in a slightly different form: ¼ Pfl % w F l, w, P ð ¼ Þ ¼ 0: (11:50) where F is just a shorthand we will use to refer to the left side of Equation 11.50. If w changes, the optimal value of l must adjust so that this condition continues to hold, which deﬁnes l as an implicit function of w. Applying the rule for ﬁnding the derivative of an implicit function in Chapter 2 (Equation 2.23 in particular) gives dl dw ¼ % @F=@w @F=@l ¼ w Pfll, 0, (11:51) where the ﬁnal inequality holds because the marginal productivity of labor is assumed to be diminishing ( fll, 0). Hence we have shown that, at least in the single-input case, a ceteris paribus increase in the wage will cause less labor to be hired. Two-input case For the case of two (or more) inputs, the story is more complex. The assumption of a diminishing marginal physical product of labor can be misleading
here. If w falls, there will not only be a change in l but also a change in k as a new cost-minimizing combination of inputs is chosen. When k changes, the entire fl function changes (labor now has a different amount of capital to work with), and the simple argument used previously cannot be made. First we will use a graphic approach to suggest why, even in the two-input case, @l/@w must be negative. A more precise, mathematical analysis is presented in the next section. Substitution effect In some ways, analyzing the two-input case is similar to the analysis of the individual’s response to a change in the price of a good that was presented in Chapter 5. When w falls, we can decompose the total effect on the quantity of l hired into two components. The ﬁrst of these components is called the substitution effect. If q is held constant at q1, then there will be a tendency to substitute l for k in the production process. This effect is illustrated in Figure 11.5a. Because the condition for minimizing the cost of producing q1 w/v, a fall in w will necessitate a movement from input combination requires that RTS A to combination B. And because the isoquants exhibit a diminishing RTS, it is clear from the diagram that this substitution effect must be negative. A decrease in w will cause an increase in labor hired if output is held constant. ¼ Output effect It is not correct, however, to hold output constant. It is when we consider a change in q (the output effect) that the analogy to the individual’s utility-maximization problem breaks down. Consumers have budget constraints, but ﬁrms do not. Firms produce as much as the available demand allows. To investigate what happens to the quantity of output produced, we must investigate the ﬁrm’s proﬁt-maximizing output decision. A change in w, because it changes relative input costs, will shift the ﬁrm’s expansion path. Consequently, all the ﬁrm’s cost curves will be shifted, and probably some output level 392 Part 4: Production and Supply FIGURE 11.5 The Substitution and Output Effects of a Decrease in the Price of a Factor When the price of labor falls, two analytically different effects come into play. One of these, the substitution effect, would cause
more labor to be purchased if output were held constant. This is shown as a movement from point A to point B in (a). At point B, the cost-minimizing condition (RTS w/v) is satisﬁed for the new, lower w. This change in w/v will also shift the ﬁrm’s expansion path and its marginal cost curve. A normal situation might be for the MC curve to shift downward in response to a decrease in w as shown in (b). With this new curve (MC 0) a higher level of output (q2) will be chosen. Consequently, the hiring of labor will increase (to l2), also from this output effect. ¼ k per period Price k1 k2 A C B P q2 q1 MC MC′ l1 l2 l per period q1 q2 Output per period (a) The isoquant map (b) The output decision other than q1 will be chosen. Figure 11.5b shows what might be considered the ‘‘normal’’ case. There, the fall in w causes MC to shift downward to MC 0. Consequently, the proﬁtmaximizing level of output rises from q1 to q2. The proﬁt-maximizing condition (P ¼ MC) is now satisﬁed at a higher level of output. Returning to Figure 11.5a, this increase in output will cause even more l to be demanded as long as l is not an inferior input (see below). The result of both the substitution and output effects will be to move the input choice to point C on the ﬁrm’s isoquant map. Both effects work to increase the quantity of labor hired in response to a decrease in the real wage. The analysis provided in Figure 11.5 assumed that the market price (or marginal revenue, if this does not equal price) of the good being produced remained constant. This would be an appropriate assumption if only one ﬁrm in an industry experienced a fall in unit labor costs. However, if the decline were industry wide, then a slightly different analysis would be required. In that case, all ﬁrms’ marginal cost curves would shift outward, and hence the industry supply curve (which as we will see in the next chapter is the sum of ﬁrm’s individual supply curves) would shift also
. Assuming that output demand is downward sloping, this will lead to a decline in product price. Output for the industry and for the typical ﬁrm will still increase and (as before) more labor will be hired, but the precise cause of the output effect is different (see Problem 11.11). Cross-price effects We have shown that, at least in simple cases, @l/@w is unambiguously negative; substitution and output effects cause more labor to be hired when the wage rate falls. From Figure 11.5 it should be clear that no deﬁnite statement can be made about how capital usage responds to the wage change. That is, the sign of @k/@w is indeterminate. In the simple two-input case, a fall in the wage will cause a substitution away from capital; that Chapter 11: Prof it Maximization 393 is, less capital will be used to produce a given output level. However, the output effect will cause more capital to be demanded as part of the ﬁrm’s increased production plan. Thus, substitution and output effects in this case work in opposite directions, and no deﬁnite conclusion about the sign of @k/@w is possible. A summary of substitution and output effects The results of this discussion can be summarized by the following principle Substitution and output effects in input demand. When the price of an input falls, two effects cause the quantity demanded of that input to rise: 1. the substitution effect causes any given output level to be produced using more of the input; and 2. the fall in costs causes more of the good to be sold, thereby creating an additional output effect that increases demand for the input. Conversely, when the price of an input rises, both substitution and output effects cause the quantity demanded of the input to decline. We now provide a more precise development of these concepts using a mathematical approach to the analysis. A mathematical development Our mathematical development of the substitution and output effects that arise from the change in an input price follows the method we used to study the effect of price changes in consumer theory. The ﬁnal result is a Slutsky-style equation that resembles the one we derived in Chapter 5. However, the ambiguity stemming from Giffen’s paradox in the theory of consumption demand does not occur here. We start with a reminder that we have two concepts of demand for any input (say, labor): (1) the conditional demand
for labor, denoted by lc(v, w, q); and (2) the unconditional demand for labor, which is denoted by l(P, v, w). At the proﬁt-maximizing choice for labor input, these two concepts agree about the amount of labor hired. The two concepts also agree on the level of output produced (which is a function of all the prices): P, v, w l ð l c v, w, q ð ð Þ ¼ P, v, w ÞÞ : (11:52) Differentiation of this expression with respect to the wage (and holding the other prices constant) yields @l P, v, w ð @w Þ ¼ @l c v, w, q ð @w Þ þ @l c v, w, q ð @q Þ $ @q P, v, w ð @w Þ : (11:53) Thus, the effect of a change in the wage on the demand for labor is the sum of two components: a substitution effect in which output is held constant; and an output effect in which the wage change has its effect through changing the quantity of output that the ﬁrm opts to produce. The ﬁrst of these effects is clearly negative—because the production function is quasi-concave (i.e., it has convex isoquants), the output-contingent demand for labor must be negatively sloped. Figure 11.5b provides an intuitive illustration of why the output effect in Equation 11.53 is negative, but it can hardly be called a proof. The particular complicating factor is the possibility that the input under consideration (here, labor) may be inferior. Perhaps oddly, inferior inputs also have negative output effects, but for rather 394 Part 4: Production and Supply arcane reasons that are best relegated to a footnote.15 The bottom line, however, is that Giffen’s paradox cannot occur in the theory of the ﬁrm’s demand for inputs: Input demand functions are unambiguously downward sloping. In this case, the theory of proﬁt maximization imposes more restrictions on what might happen than does the theory of utility maximization. In Example 11.5 we show how decomposing input demand into its substitution and output components can yield useful insights into how changes in input prices affect �
�rms. EXAMPLE 11.5 Decomposing Input Demand into Substitution and Output Components To study input demand we need to start with a production function that has two features: (1) The function must permit capital–labor substitution (because substitution is an important part of the story); and (2) the production function must exhibit increasing marginal costs (so that the second-order conditions for proﬁt maximization are satisﬁed). One function that satisﬁes these conditions is a three-input Cobb–Douglas function when one of the inputs is held k0.25l0.25g0.5, where k and l are the familiar capital and labor ﬁxed. Thus, let q inputs and g is a third input (size of the factory) that is held ﬁxed at g 16 (square meters?) for ¼ 4k0.25l0.25. We assume that all our analysis. Therefore, the short-run production function is q the factory can be rented at a cost of r per square meter per period. To study the demand for (say) labor input, we need both the total cost function and the proﬁt function implied by this production function. Mercifully, your author has computed these functions for you as f (k, l, g) ¼ ¼ ¼ and C v, w, r, q ð Þ ¼ q2v 0:5w 0:5 8 16r þ P P, v, w, r ð Þ ¼ 2P 2v% 0:5w% 0:5 16r: % (11:54) (11:55) As expected, the costs of the ﬁxed input ( g) enter as a constant in these equations, and these costs will play little role in our analysis. Envelope results. Labor-demand relationships can be derived from both of these functions through differentiation: and l c v, w, r, q ð Þ ¼ @C @w ¼ q 2v 0:5w% 16 0:5 l P, v, w, r ð Þ ¼ @P @w ¼ P 2v% 0:5w% 1:5: (11:56) (11:57) These functions already suggest that a change in the wage has a larger effect on total labor demand than
it does on contingent labor demand because the exponent of w is more negative in the total demand equation. That is, the output effect must also play a role here. To see that directly, we turn to some numbers. 15In words, an increase in the price of an inferior reduces marginal cost and thereby increases output. But when output increases, less of the inferior input is hired. Hence the end result is a decrease in quantity demanded in response to an increase in price. A formal proof makes extensive use of envelope relationships. The output effect equals @l c @q $ @q @w ¼ @l c @q $ @ 2P @w @P ¼ @l c @q $ % @l @P 2 @lc @q " # @q @P ¼ % $ 2 @lc @q " # @ 2P @P 2, $ ¼ % # where the ﬁrst step holds by Equation 11.52, the second by Equation 11.29, the third by Young’s theorem and Equation 11.31, the fourth by Equation 11.52, and the last by Equation 11.29. But the convexity of the proﬁt function in output prices implies the last factor is positive, so the whole expression is clearly negative. " Chapter 11: Prof it Maximization 395 Numerical example. Let’s start again with the assumed values that we have been using in several previous examples: v 60. Let’s ﬁrst calculate what output the ﬁrm 3, w will choose in this situation. To do so, we need its supply function: 12, and P ¼ ¼ ¼ q P, v, w, r ð Þ ¼ @P @P ¼ 4Pv% 0:5w% 0:5: (11:58) ¼ ¼ With this function and the prices we have chosen, the ﬁrm’s proﬁt-maximizing output level is 40. With these prices and an output level of 40, both of the demand functions (surprise) q 50. Because the RTS here is given by k/l, we also know that predict that the ﬁrm will hire l k/l w/v; therefore, at these prices k ¼ Suppose now that the wage rate rises to w 27 but that the other prices remain unchanged. The �
��rm’s supply function (Equation 11.58) shows that it will now produce q 26.67. The rise in the wage shifts the ﬁrm’s marginal cost curve upward, and with a constant output price, this causes the ﬁrm to produce less. To produce this output, either of the labor-demand functions can be used to show that the ﬁrm will hire l 133.3 because of the large reduction in output. 14.8. Hiring of capital will also fall to k 200. ¼ ¼ ¼ ¼ ¼ We can decompose the fall in labor hiring from l 40 even though the wage rose, Equation 11.56 shows that it would have used l 14.8 into substitution and output effects by using the contingent demand function. If the ﬁrm had continued to produce q 33.33. Capital input would have increased to k 300. Because we are holding output constant at its initial level of q 40, these changes represent the ﬁrm’s substitution effects in response to the higher wage. 50 to l ¼ ¼ ¼ ¼ ¼ ¼ The decline in output needed to restore proﬁt maximization causes the ﬁrm to cut back on its output. In doing so it substantially reduces its use of both inputs. Notice in particular that, in this example, the rise in the wage not only caused labor usage to decline sharply but also caused capital usage to fall because of the large output effect. QUERY: How would the calculations in this problem be affected if all ﬁrms had experienced the rise in wages? Would the decline in labor (and capital) demand be greater or smaller than found here? SUMMARY In this chapter we studied the supply decision of a proﬁtmaximizing ﬁrm. Our general goal was to show how such a ﬁrm responds to price signals from the marketplace. In addressing that question, we developed a number of analytical results. • To maximize proﬁts, the ﬁrm should choose to produce that output level for which marginal revenue (the revenue from selling one more unit) is equal to marginal cost (the cost of producing one more unit). • If a ﬁrm is a price-taker, then its output decisions do not affect the price of its output; thus, marginal revenue is given by this price. If the ﬁ
rm faces a downwardsloping demand for its output, however, then it can sell more only at a lower price. In this case marginal revenue will be less than price and may even be negative. • Marginal revenue and the price elasticity of demand are related by the formula MR P 1 " where P is the market price of the ﬁrm’s output and eq, p is the price elasticity of demand for its product., # ¼ þ 1 eq, p • The supply curve for a price-taking, proﬁt-maximizing ﬁrm is given by the positively sloped portion of its marginal cost curve above the point of minimum average variable cost (AVC). If price falls below minimum AVC, the ﬁrm’s proﬁt-maximizing choice is to shut down and produce nothing. • The ﬁrm’s reactions to changes in the various prices it faces can be studied through use of its proﬁt function, P(P, v, w). That function shows the maximum proﬁts 396 Part 4: Production and Supply that the ﬁrm can achieve given the price for its output, the prices of its input, and its production technology. The proﬁt function yields particularly useful envelope results. Differentiation with respect to market price yields the supply function, whereas differentiation with respect to any input price yields (the negative of ) the demand function for that input. • Short-run changes in market price result in changes to the ﬁrm’s short-run proﬁtability. These can be measured graphically by changes in the size of producer surplus. The proﬁt function can also be used to calculate changes in producer surplus. • Proﬁt maximization provides a theory of the ﬁrm’s derived demand for inputs. The ﬁrm will hire any input up to the point at which its marginal revenue product to its per-unit market price. Increases in the price of an input will induce substitution and output effects that cause the ﬁrm to reduce hiring of that input. is just equal PROBLEMS 11.1 John’s Lawn Mowing Service is a small business that acts as a price-taker (i.e., MR mowing is $20 per acre. John’s costs are given by ¼ P ). The prevailing market
price of lawn the number of acres John chooses to cut a day. total cost 0:1q 2 ¼ 10q 50, þ þ where q ¼ a. How many acres should John choose to cut to maximize proﬁt? b. Calculate John’s maximum daily proﬁt. c. Graph these results, and label John’s supply curve. 11.2 Universal Widget produces high-quality widgets at its plant in Gulch, Nevada, for sale throughout the world. The cost function for total widget production (q) is given by total cost 0.25q2. ¼ 2PA) and Lapland (where the Widgets are demanded only in Australia (where the demand curve is given by qA ¼ demand curve is given by qL ¼ qL. If Universal Widget can control the quantities supplied to each market, how many should it sell in each location to maximize total proﬁts? What price will be charged in each location? 4PL); thus, total demand equals q qA þ 100 100 % ¼ % 11.3 The production function for a ﬁrm in the business of calculator assembly is given by lp, 2 q ¼ where q denotes ﬁnished calculator output and l denotes hours of labor input. The ﬁrm is a price-taker both for calculators (which sell for P) and for workers (which can be hired at a wage rate of w per hour). ﬃﬃ a. What is the total cost function for this ﬁrm? b. What is the proﬁt function for this ﬁrm? c. What is the supply function for assembled calculators [q(P, w)]? d. What is this ﬁrm’s demand for labor function [l(P, w)]? e. Describe intuitively why these functions have the form they do. 11.4 The market for high-quality caviar is dependent on the weather. If the weather is good, there are many fancy parties and caviar sells for $30 per pound. In bad weather it sells for only $20 per pound. Caviar produced one week will not keep until the next week. A small caviar producer has a cost function given by Chapter 11: Prof it Maximization 397 0.5q2 C ¼ 5q þ þ 100, where q is
weekly caviar production. Production decisions must be made before the weather (and the price of caviar) is known, but it is known that good weather and bad weather each occur with a probability of 0.5. a. How much caviar should this ﬁrm produce if it wishes to maximize the expected value of its proﬁts? b. Suppose the owner of this ﬁrm has a utility function of the form utility pp, ¼ where p is weekly proﬁts. What is the expected utility associated with the output strategy deﬁned in part (a)? ﬃﬃﬃ c. Can this ﬁrm owner obtain a higher utility of proﬁts by producing some output other than that speciﬁed in parts (a) and (b)? Explain. d. Suppose this ﬁrm could predict next week’s price but could not inﬂuence that price. What strategy would maximize expected proﬁts in this case? What would expected proﬁts be? 11.5 The Acme Heavy Equipment School teaches students how to drive construction machinery. The number of students that the 10 min(k, l )r, where k is the number of backhoes the ﬁrm rents per week, l is the school can educate per week is given by q number of instructors hired each week, and g is a parameter indicating the returns to scale in this production function. ¼ a. Explain why development of a proﬁt-maximizing model here requires 0 < g < 1. b. Supposing g c. If v 500, and P d. If the price students are willing to pay rises to P e. Graph Acme’s supply curve for student slots, and show that the increase in proﬁts calculated in part (d) can be plotted on 600, how many students will Acme serve and what are its proﬁts? 0.5, calculate the ﬁrm’s total cost function and proﬁt function. 900, how much will proﬁts change? ¼ 1000, w ¼ ¼ ¼ ¼ that graph. 11.6 Would a lump-sum proﬁts tax affect the proﬁt-maximizing quantity of output? How about a proportional tax on pro�
�ts? How about a tax assessed on each unit of output? How about a tax on labor input? 11.7 This problem concerns the relationship between demand and marginal revenue curves for a few functional forms. a. Show that, for a linear demand curve, the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price. b. Show that, for any linear demand curve, the vertical distance between the demand and marginal revenue curves is where b (< 0) is the slope of the demand curve. 1/b Æ q, % c. Show that, for a constant elasticity demand curve of the form q aPb, the vertical distance between the demand and marginal revenue curves is a constant ratio of the height of the demand curve, with this constant depending on the price elasticity of demand. ¼ d. Show that, for any downward-sloping demand curve, the vertical distance between the demand and marginal revenue curves at any point can be found by using a linear approximation to the demand curve at that point and applying the procedure described in part (b). e. Graph the results of parts (a)–(d) of this problem. 11.8 How would you expect an increase in output price, P, to affect the demand for capital and labor inputs? a. Explain graphically why, if neither input is inferior, it seems clear that a rise in P must not reduce the demand for either factor. b. Show that the graphical presumption from part (a) is demonstrated by the input demand functions that can be derived in the Cobb–Douglas case. c. Use the proﬁt function to show how the presence of inferior inputs would lead to ambiguity in the effect of P on input demand. 398 Part 4: Production and Supply Analytical Problems 11.9 A CES proﬁt function With a CES production function of the form q s)(g g)(v1 P(P, v, w) % s)g/(1 % KP1/(1 w1 % % % s ¼ þ lq kq ¼ ð 1), where s þ g=q a whole lot of algebra is needed to compute the proﬁt function as Þ ¼ r) and K is a constant. 1/(1 % a. If you are a glutton for punishment (or if your instructor is), prove that the proﬁt function takes this form. Perhaps the easiest way to do so
is to start from the CES cost function in Example 10.2. b. Explain why this proﬁt function provides a reasonable representation of a ﬁrm’s behavior only for 0 < g < 1. c. Explain the role of the elasticity of substitution (s) in this proﬁt function. d. What is the supply function in this case? How does s determine the extent to which that function shifts when input prices change? e. Derive the input demand functions in this case. How are these functions affected by the size of s? 11.10 Some envelope results Young’s theorem can be used in combination with the envelope results in this chapter to derive some useful results. a. Show that @l(P, v, w)/@v @k(P, v, w)/@w. Interpret this result using substitution and output effects. b. Use the result from part (a) to show how a unit tax on labor would be expected to affect capital input. c. Show that @q/@w d. Use the result from part (c) to discuss how a unit tax on labor input would affect quantity supplied. @l/@P. Interpret this result. ¼ % ¼ 11.11 Le Chaˆ telier’s Principle Because ﬁrms have greater ﬂexibility in the long run, their reactions to price changes may be greater in the long run than in the short run. Paul Samuelson was perhaps the ﬁrst economist to recognize that such reactions were analogous to a principle from physical chemistry termed the Le Chaˆtelier’s Principle. The basic idea of the principle is that any disturbance to an equilibrium (such as that caused by a price change) will not only have a direct effect but may also set off feedback effects that enhance the response. In this problem we look at a few examples. Consider a price-taking ﬁrm that chooses its inputs to maximize a proﬁt vk. This maximization process will yield optimal solutions of the general function of the form P(P, v, w) form q&(P, v, w), l&(P, v, w), and k&(P, v, w). If we constrain capital input to be ﬁxed at k in the short run, this ﬁrm’s short-run responses can be represented by
qs Pf (k, l) and ls wl ¼ % % a. Using the deﬁnitional relation q&(P, v, w) P, w, k Þ ð P, w, k ð qs(P, w, k&(P, v, w)), show that. Þ ¼ @q& @P ¼ @q s @P þ 2 : % @k& @P " # @k& @v Do this in three steps. First, differentiate the deﬁnitional relation with respect to P using the chain rule. Next, differentiate the deﬁnitional relation with respect to v (again using the chain rule), and use the result to substitute for @q s=@k in the initial derivative. Finally, substitute a result analogous to part (c) of Problem 11.10 to give the displayed equation. b. Use the result from part (a) to argue that @q&=@P @q s=@P. This establishes Le Chaˆtelier’s Principle for supply: Long-run supply responses are larger than (constrained) short-run supply responses. + c. Using similar methods as in parts (a) and (b), prove that Le Chaˆtelier’s Principle applies to the effect of the wage on labor @l s=@w, demand. That is, starting from the deﬁnitional relation l &(P, v, w) implying that long-run labor demand falls more when wage goes up than short-run labor demand (note that both of these derivatives are negative). l s(P, w, k&(P, v, w)), show that @l&=@w ¼, d. Develop your own analysis of the difference between the short- and long-run responses of the ﬁrm’s cost function [C (v, w, q)] to a change in the wage (w). 11.12 More on the derived demand with two inputs The demand for any input depends ultimately on the demand for the goods that input produces. This can be shown most explicitly by deriving an entire industry’s demand for inputs. To do so, we assume that an industry produces a homogeneous D(P), where good, Q, under constant returns to scale using only capital and labor. The demand function for Q is given by Q
we imagine are deciding between remaining as separate ﬁrms or having GM acquire Fisher Body and thus become one (larger) ﬁrm. xF, xGÞ ¼ Let the total surplus that the units generate together be S G, where xF and xG are the investments ð undertaken by the managers of the two units before negotiating, and where a unit of investment costs $1. The parameter a measures the importance of GM’s manager’s investment. Show that, according to the property rights model worked out in the Extensions, it is efﬁcient for GM to acquire Fisher Body if and only if GM’s manager’s investment is important enough, in particular, if a > x1/2 F þ ax1/2 3p. ﬃﬃﬃ SUGGESTIONS FOR FURTHER READING Hart, O. Firms, Contracts, and Financial Structure. Oxford, UK: Oxford University Press, 1995. Samuelson, P. A. Foundations of Economic Analysis. Cambridge, MA: Harvard University Press, 1947. Discusses the philosophical issues addressed by alternative theories of the ﬁrm. Derives further results for the property rights theory discussed in the Extensions. Hicks, J. R. Value and Capital, 2nd ed. Oxford, UK: Oxford University Press, 1947. The Appendix looks in detail at the notion of factor complementarity. Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. New York: Oxford University Press, 1995. Provides an elegant introduction to the theory of production using vector and matrix notation. This allows for an arbitrary number of inputs and outputs. Early development of the proﬁt function idea together with a nice discussion of the consequences of constant returns to scale for market equilibrium. Pages 36–46 have extensive applications of Le Chaˆtelier’s Principle (see Problem 11.11). Sydsaeter, K., A. Strom, and P. Berck. Economists’ Mathematical Manual, 3rd ed. Berlin: Springer-Verlag, 2000. Chapter 25 offers formulas for a number of proﬁt and factor demand functions. Varian, H. R. Microeconomic Analysis, 3rd ed. New York: W. W. Norton, 1992. Includes an entire chapter on the proﬁt function
. Varian offers a novel approach for comparing short- and long-run responses using Le Chaˆtelier’s Principle. BOUNDARIES OF THE FIRM EXTENSIONS Chapter 11 provided fairly straightforward answers to the questions of what determines the boundaries of a ﬁrm and its objectives. The ﬁrm is identiﬁed by the production function f (k, l ) it uses to produce its output, and the ﬁrm makes its input and output decisions to maximize proﬁt. Ronald Coase, winner of the Nobel Prize in economics in 1991, was the ﬁrst to point out (back in the 1930s) that the nature of the ﬁrm is a bit more subtle than that. The ﬁrm is one way to organize the economic transactions necessary for output to be produced and sold, transactions including the purchase of inputs, ﬁnancing of investment, advertising, management, and so forth. But these transactions could also be conducted in other ways: Parties could sign long-term contracts or even just trade on a spot market; see Coase (1937). There is a sense in which ﬁrms and spot markets are not just different ways of organizing transactions but polar opposites. Moving a transaction within a ﬁrm is tantamount to insulating the transaction from short-term market forces, eliminating price signals, by placing it inside a more durable institution. This presents a puzzle. Economists are supposed to love markets—why are they then so willing to take the existence of ﬁrms for granted? On the other hand, if ﬁrms are so great, why is there not just one huge ﬁrm that controls the whole economy, removing all transactions from the market? Clearly, a theory is needed to explain why there are ﬁrms of intermediate sizes, and why these sizes vary across different industries and even across different ﬁrms in the same industry. To make the ideas in the Extensions concrete, we will couch the discussion in terms of the classic case of Fisher Body and General Motors (GM) mentioned at the beginning of Chapter 11. Recall that Fisher Body was the main supplier of auto bodies to GM, which GM would assemble with other auto parts into a car that it then sold to consumers. At ﬁrst the ﬁrms operated separately, but GM acquired Fisher Body in 1926 after a series of
supply disruptions. We will narrow the broad question of where ﬁrm boundaries should be set down to the question of whether it made economic sense for GM and Fisher Body to merge into a single ﬁrm. E11.1 Common features of alternative theories A considerable amount of theoretical and empirical research continues to be directed toward the fundamental question of the nature of the ﬁrm, but it is fair to say that it has not pro- vided a ‘‘ﬁnal answer.’’ Reﬂecting this uncertainty, the Extensions present two different theories that have been proposed as alternatives to the neoclassical model studied in Chapter 11. The ﬁrst is the property rights theory associated with the work of Sanford Grossman, Oliver Hart, and John Moore. The second is the transactions cost theory associated with the work of Oliver Williamson, co-winner of the Nobel Prize in economics in 2009.1 The theories share some features. Both acknowledge that if all markets looked like the supply–demand model encountered in principles courses—where a large number of suppliers and buyers trade a commodity anonymously—that would be the most efﬁcient way to organize transactions, leaving no role for ﬁrms. However, it is unrealistic to assume that all markets look that way. Three factors often present—uncertainty, complexity, and specialization—lead markets to look more like negotiations among a few market participants. We can see how these three factors would have operated in the GM–Fisher Body example. The presence of uncertainty and complexity would have made it difﬁcult for GM to sign contracts years in advance for auto bodies. Such contracts would have to specify how the auto bodies should be designed, but successful design depends on the vagaries of consumer taste, which are difﬁcult to predict (after all, large tail ﬁns were popular at one point in history) and hard to specify in writing. The best way to cope with uncertainty and complexity may be for GM to negotiate the purchase of auto bodies at the time they are needed for assembly rather than years in advance at the signing of a long-term contract. The third factor, specialization, leads to obvious advantages. Auto bodies that are tailored to GM’s styling and other technical requirements would be more valuable than ‘‘generic’’ ones. But specialization has the drawback of limiting GM to a small set of suppliers rather than buying auto
bodies as it would an input on a competitive commodity market. Markets exhibiting these three factors—uncertainty, complexity, and specialization—will not involve the sale of perfect long-term contracts in a competitive equilibrium with large numbers of suppliers and demanders. Rather, they will often involve few parties, perhaps just two, negotiating often not far 1Seminal articles on the property rights theory are Grossman and Hart (1986) and Hart and Moore (1990). See Williamson (1979) for a comprehensive treatment of the transactions cost theory. Gibbons (2005) provides a good summary of these and other alternatives to the neoclassical model. 402 Part 4: Production and Supply in advance of when the input is required. This makes the alternative theories of the ﬁrm interesting. If the alternative theories merely compared ﬁrms to perfectly competitive markets, markets would always end up ‘‘winning’’ in the comparison. Instead, ﬁrms are compared to negotiated sales, a more subtle comparison without an obvious ‘‘winner.’’ We will explore the subtle comparisons offered by the two different theories next. E11.2 Property rights theory To make the analysis of this alternative theory as stark as possible, suppose that there are just two owner-managers, one who runs Fisher Body and one who runs GM. Let S(xF, xG) be the total surplus generated by the transaction between Fisher Body and GM, the sum of both ﬁrm’s proﬁts (Fisher Body from its sale of auto bodies to GM and GM from its sale of cars to consumers). Instead of being a function of capital and labor or input and output prices, we now put those factors aside and just write surplus as a function of two new variables: the investments made by Fisher Body (xF) and GM (xG). The surplus function subtracts all production costs (just as the producer surplus concept from Chapter 11 did) but does not subtract the cost of the investments xF and xG. The investments are sunk before negotiations between them over the transfer of the auto bodies. The investments include, for example, any effort made by Fisher Body’s manager to improve the precision of its metal-cutting dies and to reﬁne the shapes to GM’s speciﬁcations, as well as the effort expended by GM’s manager in designing and marketing the car and tailoring
its assembly process to use the bodies. Both result in a better car model that can be sold at a higher price and that generates more proﬁt (not including the investment effort). For simplicity, assume one unit of investment costs a manager $1, implying that investment level xF costs Fisher Body’s manager xF dollars and that the marginal cost of investment for both parties is 1. Before computing the equilibrium investment levels under various ownership structures, as a benchmark we will compute the efﬁcient investment levels. The efﬁcient levels maximize total surplus minus investment costs, S xF, xGÞ % ð xF % xG: (i) The ﬁrst-order conditions for maximization of this objective are @S @xF ¼ @S @xG ¼ 1: (ii) The efﬁcient investment levels equalize the total marginal beneﬁt with the marginal cost. Next, let’s compute equilibrium investment levels under various ownership structures. Assume the investments are too complicated to specify in a contract before they are undertaken. So too is the speciﬁcation of the auto bodies themselves. Instead, starting with the case in which Fisher Body and GM are separate ﬁrms, they must bargain over the terms of trade of the auto bodies (prices, quantities, nature of the product) when they are needed. There is a large body of literature on how to model bargaining (we will touch on this a bit more in Chapter 13 when we introduce Edgeworth boxes and contract curves). To make the analysis as simple as possible, we will not solve for all the terms of the bargain but will just assume that they come to an agreement to split any gains from the transaction equally.2 Because cars cannot be produced without auto bodies, no surplus is generated if parties do not consummate a deal. Therefore, the gain from bargaining is the whole surplus, S(xF, xG). The investment expenditures are not part of the negotiation because they were sunk =2 in before. Fisher Body and GM each end up with S equilibrium from bargaining. xF, xGÞ ð To solve for equilibrium investments, subtract Fisher Body’s cost of investment from its share of the bargaining gains, yielding the objective function 1 2 S xF, xGÞ % ð Taking the ﬁrst-order condition with respect to xF and
rearranging yields the condition (iii) xF: 1 2 @S @xF" ¼ 1: (iv) # The left side of Equation iv is the marginal beneﬁt to Fisher Body from additional investment: Fisher Body receives its bargaining share, half, of the surplus. The right side is the marginal cost, which is 1 because investment xF is measured in dollar terms. As usual, the optimal choice (here investment) equalizes marginal beneﬁt and marginal cost. A similar condition characterizes GM’s investment decision: 1 2 @S @xG" ¼ 1: (v) # In sum, if Fisher Body and GM are separate ﬁrms, investments are given by Equations iv and v. If instead GM acquires Fisher Body so they become one ﬁrm, the manager of the auto body subsidiary is now in a worse bargaining position. He or she can no longer extract half of the bargaining surplus by threatening not to use Fisher Body’s assets to produce bodies for GM; the assets are all under GM’s control. To make the point as clear as possible, assume that Fisher Body’s manager obtains no bargaining surplus; GM obtains all of it. Without the prospect of a return, the manager will not undertake any investment, implying xF ¼ 0. On the other hand, because GM’s manager now obtains the whole surplus S(xF, xG), the objective function determining his or her investment is now 1 2 S xF, xGÞ % ð xG: (vi) 2This is a special case of so-called Nash bargaining, an inﬂuential bargaining theory developed by the same John Nash behind Nash equilibrium. yielding ﬁrst-order condition @S @xG ¼ 1: (vii) When both parties were in separate ﬁrms, each had less than efﬁcient investment incentives (compare the ﬁrst-order conditions in the efﬁcient outcome in Equation ii with Equations iv and v) because they only obtain half the bargaining surplus. Combining the two units under GM’s ownership further dilutes Fisher Body’s investment incentives, reducing its 0, but boosts GM’s, so investment all the way down to xF ¼ that GM’s ﬁrst-order
condition resembles the efﬁcient one. Intuitively, asset ownership gives parties more bargaining power, and this bargaining power in turn protects the party from having the returns from their investment appropriated by the other party in bargaining.3 Of course there is only so much bargaining power to go around. A shift of assets from one party to another will increase one’s bargaining power at the expense of the other’s. Therefore, a trade-off is involved in merging two units into one; the merger only makes economic sense under certain conditions. If GM’s investment is much more important for surplus, then it will be efﬁcient to allocate ownership over all the assets to GM. If both units’ investments are roughly equally important, then maintaining both parties’ bargaining power by apportioning some of the assets to each might be a good idea. If Fisher Body’s investment is the most important, then having Fisher Body acquire GM may produce the most efﬁcient structure. More speciﬁc recommendations would depend on functional forms, as will be illustrated in the following numerical example. E11.3 Numerical example For a simple numerical example of the property rights theory, x1/2 let S G. The ﬁrst-order condition for the ð efﬁcient level of Fisher Body’s investment is xF, xGÞ ¼ x1/2 F þ 1 1/2 x% F ¼ 2 1/4. Likewise, x&G ¼ implying x&F ¼ ing the investment costs is 1/2. 1, 1/4. Total surplus subtract- If Fisher Body and GM remain separate ﬁrms, half the surplus from each party’s investment is ‘‘held up’’ by the other party. Fisher Body’s ﬁrst-order condition is Chapter 11: Prof it Maximization 403 the integrated ﬁrm bargaining surplus. The manager of obtains all the bargaining surplus and invests at the efﬁcient level, x&G ¼ 1/4. Overall, total surplus subtracting investment costs is 1/4. Combining the ﬁrms decreases Fisher Body’s investment and increases GM’s, but the net effect is to make them jointly worse off; therefore, the ﬁrms should remain separate. If
GM’s investment were more important than Fisher Body’s, ax1/2 merging them could be efﬁcient. Let S G, where a allows the impact of GM’s investment on surplus to vary. One of the problems at the end of this chapter asks you to show that having GM’s manager own all assets is more efﬁcient than keeping the ﬁrms separate for high enough a, in particular, a > xF, xGÞ ¼ ð x1/2 F þ 3p. ﬃﬃﬃ E11.4 Transaction cost theory Next, turn to the second alternative theory of the ﬁrm—the it shares transaction cost theory. As discussed previously, many common elements with the property rights theory, but there are subtle differences. With the property rights theory, the main beneﬁt of restructuring the ﬁrm was to get the right incentives for investments made before bargaining. With the transaction cost theory, the main beneﬁt is to reduce haggling costs at the time of bargaining. Let hF be a costly action undertaken by Fisher Body at the time of bargaining that increases its bargaining power at the expense of GM. We loosely interpret this action as ‘‘haggling,’’ but more concretely it could be a costly signal such as was seen in the Spence education signaling game in Chapter 8, or it could represent bargaining delay or an input supplier strike. GM can take a similar haggling action, hG. Rather than ﬁxing the bargaining shares at 1/2 each, we now assume a(hF, hG) is the share accruing to Fisher Body and 1 a(hF, hG) is the share accruing to GM, where a is between 0 and 1 and is increasing in hF and decreasing in hG. For simplicity, assume that the marginal cost for one unit of the haggling action is $1, implying a haggling level of hF costs Fisher Body hF dollars and of hG costs GM hG dollars. To abstract from some of the bargaining issues in the previous theory, assume that investments are made at the time of bargaining rather than beforehand, so that in principle they can be set at the efﬁcient levels x&F and x&G satisfying Equation ii. % 1/2
x% F ¼ 1 4 1/16. Likewise, xG ¼ 1: 1/16. Thus, parties are implying xF ¼ underinvesting relative to the efﬁcient outcome. Total surplus subtracting investment costs is only 3/8. If GM acquires Fisher Body, the manager of the auto body 0) because he or she obtains no unit does not invest (xF ¼ 3The appropriation of the returns from one party’s investment by the other party in bargaining is called the hold-up problem, referring to the colorful image of a bandit holding up a citizen at gunpoint. Nothing illegal is happening here; hold up is just a feature of bargaining. hG ¼ The efﬁcient outcome is for investments to be set at x&F and x&G and for parties not to undertake any haggling actions: hF ¼ 0. Haggling does not generate any more total surplus but rather reallocates it from one party to another. If Fisher Body and GM are separate ﬁrms, they will undertake some of these actions, much like the prisoners were led to ﬁnk on each other in equilibrium of the Prisoners’ Dilemma in Chapter 8 when it would have been better for the two of them to remain silent. Fisher Body’s objective function determining its equilibrium level of haggling is a hF, hGÞ½ x&F, x&GÞ % ð ð x&G( % x&F % (viii) hF, S 404 Part 4: Production and Supply where it is assumed the parties naturally would agree on the investments maximizing their joint surplus. Fisher Body’s ﬁrst-order condition is, after rearranging, @a @xF ½ x&F, x&GÞ % S ð x&F % x&G( ¼ 1: Similarly, GM will have ﬁrst-order condition @a @xG ½ S x&F, x&GÞ % ð x&F % x&G( ¼ 1: (ix) (x) The main point to take away from these somewhat complicated conditions is that both parties will engage in some wasteful haggling if they remain separate. If instead GM acquires Fisher Body and they become one ﬁrm, assume this enables GM to authorize what
a separate supplier. Masten (1984) found similar results in the aerospace industry. Anderson and Schmittlein (1984) found that proxies for complexity and specialization could help explain why some electronic components were sold by sales representatives employed by the manufacturers themselves and some by independent operators. References Anderson, E., and D. C. Schmittlein. ‘‘Integration of the Sales Force: An Empirical Examination.’’ Rand Journal of Economics (Autumn 1984): 385–95. Coase, R. H. ‘‘The Nature of the Firm.’’ Economica (Novem- ber 1937): 386–405. Gibbons, R. ‘‘Four Formal(izable) Theories of the Firm?’’ Journal of Economic Behavior and Organization (October 2005): 200–45. Hart, O. Firms, Contracts, and Financial Structure. Oxford, compared with total surplus when the ﬁrms remain separate, UK: Oxford University Press, 1995. S x&F, x&GÞ % ð x&F % x&G % hF % hG: (xii) The trade-offs involved in different ﬁrm structures are apparent from a comparison of these equations: Giving GM the unilateral authority to make the investment decision avoids any haggling costs but may result in inefﬁcient investment levels. Whether it is more efﬁcient to keep the ﬁrms separate or to merge the two units together and have one manager control them depends on the signiﬁcance of the investment Masten, S. E. ‘‘The Organization of Production: Evidence from the Aerospace Industry.’’ Journal of Law and Economics (October 1984): 403–17. Monteverde, K., and D. J. Teece. ‘‘Supplier Switching Costs and Vertical Integration in the Automobile Industry.’’ Bell Journal of Economics (Spring 1982): 206–13. Williamson, O. ‘‘Transaction Cost Economics: The Governance of Contractual Relations.’’ Journal of Law and Economics (October 1979): 233–61. This page intentionally left blank Competitive Markets P A R T FIVE Chapter 12 The Partial Equilibrium Competitive Model Chapter 13 General Equilibrium and Welfare In Parts 2 and 4 we developed models to explain the demand for goods by utility-
maximizing individuals and the supply of goods by proﬁt-maximizing ﬁrms. In the next two parts we will bring together these strands of analysis to discuss how prices are determined in the marketplace. The discussion in this part concerns competitive markets. The principal characteristic of such markets is that ﬁrms behave as price-takers. That is, ﬁrms are assumed to respond to market prices, but they believe they have no control over these prices. The primary reason for such a belief is that competitive markets are characterized by many suppliers; therefore, the decisions of any one of them indeed has little effect on prices. In Part 6 we will relax this assumption by looking at markets with only a few suppliers (perhaps only one). For these cases, the assumption of pricetaking behavior is untenable; thus, the likelihood that ﬁrms’ actions can affect prices must be taken into account. Chapter 12 develops the familiar partial equilibrium model of price determination in competitive markets. The principal result is the Marshallian ‘‘cross’’ diagram of supply and demand that we ﬁrst discussed in Chapter 1. This model illustrates a ‘‘partial’’ equilibrium view of price determination because it focuses on only a single market. In the concluding sections of the chapter we show some of the ways in which such models are applied. A speciﬁc focus is on illustrating how the competitive model can be used to judge the welfare consequences for market participants of changes in market equilibria. Although the partial equilibrium competitive model is useful for studying a single market in detail, it is inappropriate for examining relationships among markets. To capture such cross-market effects requires the development of ‘‘general’’ equilibrium models—a topic we take up in Chapter 13. There we show how an entire economy can be viewed as a system of interconnected competitive markets that determine all prices simultaneously. We also examine how welfare consequences of various economic questions can be studied in this model. 407 This page intentionally left blank C H A P T E R TWELVE The Partial Equilibrium Competitive Model In this chapter we describe the familiar model of price determination under perfect competition that was originally developed by Alfred Marshall in the late nineteenth century. That is, we provide a fairly complete analysis of the supply–demand mechanism as it applies to a single market. This is perhaps the most widely used model for the study of price determination. Market Demand
In Part 2 we showed how to construct individual demand functions that illustrate changes in the quantity of a good that a utility-maximizing individual chooses as the market price and other factors change. With only two goods (x and y) we concluded that an individual’s (Marshallian) demand function can be summarized as quantity of x demanded x(px, py, I). ¼ (12:1) Now we wish to show how these demand functions can be added up to reﬂect the 1, n) to represent demand of all individuals in a marketplace. Using a subscript i (i each person’s demand function for good x, we can deﬁne the total demand in the market as ¼ market demand for X n ¼ 1 i X ¼ xið : px, py, IiÞ (12:2) Notice three things about this summation. First, we assume that everyone in this marketplace faces the same prices for both goods. That is, px and py enter Equation 12.2 without person-speciﬁc subscripts. On the other hand, each person’s income enters into his or her own speciﬁc demand function. Market demand depends not only on the total income of all market participants but also on how that income is distributed among consumers. Finally, observe that we have used an uppercase X to refer to market demand—a notation we will soon modify. The market demand curve Equation 12.2 makes clear that the total quantity of a good demanded depends not only on its own price but also on the prices of other goods and on the income of each person. To construct the market demand curve for good X, we allow px to vary while holding py and the income of each person constant. Figure 12.1 shows this construction for the case where there are only two consumers in the market. For each potential price of x, 409 410 Part 5: Competitive Markets FIGURE 12.1 Construction of a Market Demand Curve from Individual Demand Curves A market demand curve is the ‘‘horizontal sum’’ of each individual’s demand curve. At each price the quantity demanded in the market is the sum of the amounts each individual demands. For example, at p$x the demand in the market is x$1 þ x$2 ¼ X$. px px px px* x1 x1 x1
* x2 x2* x2 X* X X (a) Individual 1 (b) Individual 2 (c) Market demand the point on the market demand curve for X is found by adding up the quantities demanded by each person. For example, at a price of p$x, person 1 demands x$1 and person 2 demands x$2. The total quantity demanded in this two-person market is the x$2). Therefore, the point p$x, X$ is one point on sum of these two amounts (X$ the market demand curve for X. Other points on the curve are derived in a similar way. Thus, the market demand curve is a ‘‘horizontal sum’’ of each individual’s demand curve.1 x$1 þ ¼ Shifts in the market demand curve The market demand curve summarizes the ceteris paribus relationship between X and px. It is important to keep in mind that the curve is in reality a two-dimensional representation of a many-variable function. Changes in px result in movements along this curve, but changes in any of the other determinants of the demand for X cause the curve to shift to a new position. A general increase in incomes would, for example, cause the demand curve to shift outward (assuming X is a normal good) because each individual would choose to buy more X at every price. Similarly, an increase in py would shift the demand curve to X outward if individuals regarded X and Y as substitutes, but it would shift the demand curve for X inward if the goods were regarded as complements. Accounting for all such shifts may sometimes require returning to examine the individual demand functions that constitute the market relationship, especially when examining situations in which the distribution of income changes and thereby raises some incomes while reducing others. To keep matters straight, economists usually reserve the term change in quantity demanded for a movement along a ﬁxed demand curve in response to a change in px. Alternatively, any shift in the position of the demand curve is referred to as a change in demand. 1Compensated market demand curves can be constructed in exactly the same way by summing each individual’s compensated demand. Such a compensated market demand curve would hold each person’s utility constant. Chapter 12: The Partial Equilibrium Competitive Model 411 EXAMPLE 12.1 Shifts in Market Demand These ideas can be illustrated with a simple set of linear demand functions. Suppose individual 1’s demand for
oranges (x, measured in dozens per year) is given by2 x1 ¼ 10 2px þ 0:1I1 þ & 0:5py, where px ¼ I1 ¼ py ¼ price of oranges (dollars per dozen), individual 1’s income (in thousands of dollars), price of grapefruit (a gross substitute for oranges—dollars per dozen). Individual 2’s demand for oranges is given by x2 ¼ Hence the market demand function is 17 px þ 0:05I2 þ & 0:5py: X px, py, I1, I2Þ ¼ ð x1 þ x2 ¼ 27 3px þ 0:1I1 þ 0:05I2 þ & py: (12:3) (12:4) (12:5) Here the coefﬁcient for the price of oranges represents the sum of the two individuals’ coefﬁcients, as does the coefﬁcient for grapefruit prices. This reﬂects the assumption that orange and grapefruit markets are characterized by the law of one price. Because the individuals have differing coefﬁcients for income, however, the demand function depends on each person’s income. To graph Equation 12.5 as a market demand curve, we must assume values for I1, I2, and py (because the demand curve reﬂects only the two-dimensional relationship between x and px). If I1 ¼ 4, then the market demand curve is given by 20, and py ¼ 40, I2 ¼ X 27 3px þ 4 & þ 1 þ 4 ¼ ¼ 36 & 3px, (12:6) which is a simple linear demand curve. If the price of grapefruit were to increase to py ¼ the curve would, assuming incomes remain unchanged, shift outward to 6, then þ whereas an income tax that took 10 (thousand dollars) from individual 1 and transferred it to individual 2 would shift the demand curve inward to þ ¼ & & ¼ X 27 3px þ 4 1 6 38 3px, (12:7) X 27 3px þ 3 & þ ¼ 1:5 4 þ ¼ 35:5 3px & (12:8) because individual 1 has a larger marginal effect of income changes on orange purchases. All these changes
shift the demand curve in a parallel way because, in this linear case, none of them affects either individual’s coefﬁcient for px. In all cases, an increase in px of 0.10 (ten cents) would cause X to decrease by 0.30 (dozen per year). QUERY: For this linear case, when would it be possible to express market demand as a linear I2)? Alternatively, suppose the individuals had differing coefﬁcients function of total income (I1 þ for py. Would that change the analysis in any fundamental way? Generalizations Although our construction concerns only two goods and two individuals, it is easily generalized. Suppose there are n goods (denoted by xi, i 1, n. Assume also that there are m individuals in society. Then the jth individual’s demand for 1, n) with prices pi, i ¼ ¼ 2This linear form is used to illustrate some issues in aggregation. It is difﬁcult to defend this form theoretically, however. For example, it is not homogeneous of degree 0 in all prices and income. 412 Part 5: Competitive Markets the ith good will depend on all prices and on Ij, the income of this person. This can be denoted by where i 1, n and j ¼ ¼ xi, j ¼ xi, jð, p1,..., pn, IjÞ (12:9) 1, m. Using these individual demand functions, market demand concepts are provided by the following deﬁnition Market demand. The market demand function for a particular good (Xi) is the sum of each individual’s demand for that good: Xið p1,..., pn, I1,..., ImÞ ¼ m 1 j X ¼ xi, jð : p1,..., pn, IjÞ (12:10) The market demand curve for Xi is constructed from the demand function by varying pi while holding all other determinants of Xi constant. Assuming that each individual’s demand curve is downward sloping, this market demand curve will also be downward sloping. Of course, this deﬁnition is just a generalization of our previous discussion, but three features warrant repetition. First, the functional representation of Equation 12.10 makes clear that the demand for
Xi depends not only on pi but also on the prices of all other goods. Therefore, a change in one of those other prices would be expected to shift the demand curve to a new position. Second, the functional notation indicates that the demand for Xi depends on the entire distribution of individuals’ incomes. Although in many economic discussions it is customary to refer to the effect of changes in aggregate total purchasing power on the demand for a good, this approach may be a misleading simpliﬁcation because the actual effect of such a change on total demand will depend on precisely how the income changes are distributed among individuals. Finally, although they are obscured somewhat by the notation we have been using, the role of changes in preferences should be mentioned. We have constructed individuals’ demand functions with the assumption that preferences (as represented by indifference curve maps) remain ﬁxed. If preferences were to change, so would individual and market demand functions. Hence market demand curves can clearly be shifted by changes in preferences. In many economic analyses, however, it is assumed that these changes occur so slowly that they may be implicitly held constant without misrepresenting the situation. A simpliﬁed notation Often in this book we look at only one market. To simplify the notation, in these cases we use QD to refer to the quantity of the particular good demanded in this market and P to denote its market price. As always, when we draw a demand curve in the Q–P plane, the ceteris paribus assumption is in effect. If any of the factors mentioned in the previous section (e.g., other prices, individuals’ incomes, or preferences) should change, the Q–P demand curve will shift, and we should keep that possibility in mind. When we turn to consider relationships among two or more goods, however, we will return to the notation we have been using up until now (i.e., denoting goods by x and y or by xi). Chapter 12: The Partial Equilibrium Competitive Model 413 Elasticity of market demand When we use this notation for market demand, we will also use a compact notation for the price elasticity of the market demand function: price elasticity of market demand eQ, P ¼ ¼ @QDð P, P 0, I @P P QD, Þ ’ (12:11) where the notation is intended as a reminder that the demand for Q depends on many factors other than its own price, such as the prices
of other goods (P 0) and the incomes of all potential demanders (I). These other factors are held constant when computing the own-price elasticity of market demand. As in Chapter 5, this elasticity measures the proportionate response in quantity demanded to a 1 percent change in a good’s price. Market demand is also characterized by whether demand is elastic (eQ, P < 1) or inelastic (0 > eQ, P > 1). Many of the other concepts examined in Chapter 5, such as the crossprice elasticity of demand or the income elasticity of demand, also carry over directly into the market context:3 & & cross-price elasticity of market demand income elasticity of market demand ¼ ¼ @QDð @QDð P, P 0, I @P 0 P, P 0, I @I P 0 QD I QD, : Þ Þ ’ ’ (12:12) Given these conventions about market demand, we now turn to an extended examination of supply and market equilibrium in the perfectly competitive model. Timing of the Supply Response In the analysis of competitive pricing, it is important to decide the length of time to be allowed for a supply response to changing demand conditions. The establishment of equilibrium prices will be different if we are talking about a short period during which most inputs are ﬁxed than if we are envisioning a long-run process in which it is possible for new ﬁrms to enter an industry. For this reason, it has been traditional in economics to discuss pricing in three different time periods: (1) very short run, (2) short run, and (3) long run. Although it is not possible to give these terms an exact chronological deﬁnition, the essential distinction being made concerns the nature of the supply response that is assumed to be possible. In the very short run, there is no supply response: The quantity supplied is ﬁxed and does not respond to changes in demand. In the short run, existing ﬁrms may change the quantity they are supplying, but no new ﬁrms can enter the industry. In the long run, new ﬁrms may enter an industry, thereby producing a ﬂexible supply response. In this chapter we will discuss each of these possibilities. Pricing in the Very Short Run In the very short run, or the market period, there
is no supply response. The goods are already ‘‘in’’ the marketplace and must be sold for whatever the market will bear. In this situation, price acts only as a device for rationing demand. Price will adjust to clear the market of the quantity that must be sold during the period. Although the market price 3In many applications, market demand is modeled in per capita terms and treated as referring to the ‘‘typical person.’’ In such applications it is also common to use many of the relationships among elasticities discussed in Chapter 5. Whether such aggregation across individuals is appropriate is discussed brieﬂy in the Extensions to this chapter. 414 Part 5: Competitive Markets FIGURE 12.2 Pricing in the Very Short Run When quantity is ﬁxed in the very short run, price acts only as a device to ration demand. With quantity ﬁxed at Q$, price P1 will prevail in the marketplace if D is the market demand curve; at this price, individuals are willing to consume exactly that quantity available. If demand should shift upward to D 0, the equilibrium market price would increase to P2. Price D P2 P1 D′ S D′ D S Q * Quantity per period may act as a signal to producers in future periods, it does not perform such a function in the current period because current-period output is ﬁxed. Figure 12.2 depicts this situation. Market demand is represented by the curve D. Supply is ﬁxed at Q$, and the price that clears the market is P1. At P1, individuals are willing to take all that is offered in the market. Sellers want to dispose of Q$ without regard to price (suppose that the good in question is perishable and will be worthless if it is not sold in the very short run). Hence P1, Q$ is an equilibrium price–quantity combination. If demand should shift to D 0, then the equilibrium price would increase to P2 but Q$ would stay ﬁxed because no supply response is possible. The supply curve in this situation is a vertical straight line at output Q$. The analysis of the very short run is not particularly useful for many markets. Such a theory may adequately represent some situations in which goods are perishable or must be sold on a given day, as is the case in auctions. Indeed, the study of auctions provides a number of insights about the informational problems
involved in arriving at equilibrium prices, which we take up in Chapter 18. But auctions are unusual in that supply is ﬁxed. The far more usual case involves some degree of supply response to changing demand. It is presumed that an increase in price will bring additional quantity into the market. In the remainder of this chapter, we will examine this process. Before beginning our analysis, we should note that increases in quantity supplied need not come only from increased production. In a world in which some goods are durable (i.e., last longer than a single period), current owners of these goods may supply them in increasing amounts to the market as price increases. For example, even though the supply of Rembrandts is ﬁxed, we would not want to draw the market supply curve for these paintings as a vertical line, such as that shown in Figure 12.2. As the price of Rembrandts increases, individuals and museums will become increasingly willing to part with them. From a market point of view, therefore, the supply curve for Rembrandts will have an upward slope, even though no new production takes place. A similar analysis would Chapter 12: The Partial Equilibrium Competitive Model 415 follow for many types of durable goods, such as antiques, used cars, vintage baseball cards, or corporate shares, all of which are in nominally ‘‘ﬁxed’’ supply. Because we are more interested in examining how demand and production are related, we will not be especially concerned with such cases here. Short-Run Price Determination In short-run analysis, the number of ﬁrms in an industry is ﬁxed. These ﬁrms are able to adjust the quantity they produce in response to changing conditions. They will do this by altering levels of usage for those inputs that can be varied in the short run, and we shall investigate this supply decision here. Before beginning the analysis, we should perhaps state explicitly the assumptions of this perfectly competitive model Perfect competition. A perfectly competitive market is one that obeys the following assumptions. 1. There are a large number of ﬁrms, each producing the same homogeneous product. 2. Each ﬁrm attempts to maximize proﬁts. 3. Each ﬁrm is a price-taker: It assumes that its actions have no effect on market price. 4. Prices are assumed to be known by all market participants—information is perfect. 5. Transactions
are costless: Buyers and sellers incur no costs in making exchanges (for more on this and the previous assumption, see Chapter 18). Throughout our discussion we continue to assume that the market is characterized by a large number of demanders, each of whom operates as a price-taker in his or her consumption decisions. Short-run market supply curve In Chapter 11 we showed how to construct the short-run supply curve for a single proﬁtmaximizing ﬁrm. To construct a market supply curve, we start by recognizing that the quantity of output supplied to the entire market in the short run is the sum of the quantities supplied by each ﬁrm. Because each ﬁrm uses the same market price to determine how much to produce, the total amount supplied to the market by all ﬁrms will obviously depend on price. This relationship between price and quantity supplied is called a shortrun market supply curve. Figure 12.3 illustrates the construction of the curve. For simplicity assume there are only two ﬁrms, A and B. The short-run supply (i.e., marginal cost) curves for ﬁrms A and B are shown in Figures 12.3a and 12.3b. The market supply curve shown in Figure 12.3c is the horizontal sum of these two curves. For example, at a price of P1, ﬁrm A is willing to supply qA 1. Therefore, at this price the total supply in the market is given by Q1, which is equal to qA qB 1. The other points on the curve are constructed in an identical way. Because each ﬁrm’s supply curve has a positive slope, the market supply curve will also have a positive slope. The positive slope reﬂects the fact that short-run marginal costs increase as ﬁrms attempt to increase their outputs. 1 and ﬁrm B is willing to supply qB 1 þ Short-run market supply More generally, if we let qi(P, v, w) represent the short-run supply function for each of the n ﬁrms in the industry, we can deﬁne the short-run market supply function as follows. 416 Part 5: Competitive Markets FIGURE 12.3 Short-Run Market Supply Curve The supply (marginal cost) curves of two ﬁrms are shown in (a) and (
b). The market supply curve (c) is the horizontal sum of these curves. For example, at P1 ﬁrm A supplies qA 1, and total market supply is given by Q1 ¼ 1, ﬁrm B supplies qB qA 1 þ qB 1. P P1 P P SA SB S 1q A q A 1q B q B (a) Firm A (b) Firm B (c) The market Q1 Total output per period Short-run market supply function. The short-run market supply function shows total quantity supplied by each ﬁrm to a market: P, v, w QSð Þ ¼ n 1 i X ¼ P, v, w qið : Þ (12:13) Notice that the ﬁrms in the industry are assumed to face the same market price and the same prices for inputs.4 The short-run market supply curve shows the two-dimensional relationship between Q and P, holding v and w (and each ﬁrm’s underlying technology) constant. The notation makes clear that if v, w, or technology were to change, the supply curve would shift to a new location. Short-run supply elasticity One way of summarizing the responsiveness of the output of ﬁrms in an industry to higher prices is by the short-run supply elasticity. This measure shows how proportional changes in market price are met by changes in total output. Consistent with the elasticity concepts developed in Chapter 5, this is deﬁned as follows Short-run elasticity of supply (es, P). eS, P ¼ percentage change in Q supplied percentage change in P @QS @P ’ P QS : ¼ (12:14) 4Several assumptions that are implicit in writing Equation 12.13 should be highlighted. First, the only one output price (P) enters the supply function—implicitly ﬁrms are assumed to produce only a single output. The supply function for multiproduct ﬁrms would also depend on the prices of the other goods these ﬁrms might produce. Second, the notation implies that input prices (v and w) can be held constant in examining ﬁrms’ reactions to changes in the price of their output. That is, ﬁrms are assumed to be price-takers for inputs—their hiring
decisions do not affect these input prices. Finally, the notation implicitly assumes the absence of externalities—the production activities of any one ﬁrm do not affect the production possibilities for other ﬁrms. Models that relax these assumptions will be examined at many places later in this book. Chapter 12: The Partial Equilibrium Competitive Model 417 Because quantity supplied is an increasing function of price (@QS=@P > 0s), the supply elasticity is positive. High values for eS, P imply that small increases in market price lead to a relatively large supply response by ﬁrms because marginal costs do not increase steeply and input price interaction effects are small. Alternatively, a low value for eS, P implies that it takes relatively large changes in price to induce ﬁrms to change their output levels because marginal costs increase rapidly. Notice that, as for all elasticity notions, computation of eS, P requires that input prices and technology be held constant. To make sense as a market response, the concept also requires that all ﬁrms face the same price for their output. If ﬁrms sold their output at different prices, we would need to deﬁne a supply elasticity for each ﬁrm. EXAMPLE 12.2 A Short-Run Supply Function In Example 11.3 we calculated the general short-run supply function for any single ﬁrm with a two-input Cobb–Douglas production function as b= 1 & b Þ ð ka= ð 1 1 & b Þ P b= ð b 1 & Þ: (12:15) qið 3, w P, v, w & w b Þ ¼! " 12, and k1 ¼ ¼ If we let a function b ¼ ¼ 0.5, v ¼ 80, then this yields the simple, single-ﬁrm supply P, v, w 12 qið 10P 3 : (12:16) ¼ Now assume that there are 100 identical such ﬁrms and that each ﬁrm faces the same market prices for both its output and its input hiring. Given these assumptions, the short-run market supply function is given by Þ ¼ P, v, w QSð 12 Þ ¼ ¼ qi ¼ 100 1 i X ¼ 1 i X ¼ 100 10P 1,
000P 3 : (12:17) 3 ¼ Thus, at a price of (say) P supplying 40 units. We can compute the short-run elasticity of supply in this situation as 12, total market supply will be 4,000, with each of the 100 ﬁrms ¼ eS, P ¼ @QSð P, v, w @P P QS ¼ 1; 000 3 Þ ’ P ’ 1; 000P=3 ¼ 1; (12:18) this might have been expected, given the unitary exponent of P in the supply function. Effect of an increase in w. If all the ﬁrms in this marketplace experienced an increase in the wage they must pay for their labor input, then the short-run supply curve would shift to a new position. To calculate the shift, we must return to the single ﬁrm’s supply function (Equation 12.15) and now use a new wage, say, w 15. If none of the other parameters of the problem have changed (the ﬁrm’s production function and the level of capital input it has in the short run), the supply function becomes ¼ and the market supply function is P, v, w qið 15 Þ ¼ ¼ 8P 3 P, v, w QSð 15 ¼ Þ ¼ 8P 3 ¼ 800P 3 : 100 1 i X ¼ (12:19) (12:20) ¼ 12, now this industry will supply only QS ¼ 3,200, with each ﬁrm 32. In other words, the supply curve has shifted upward because of the increase in 1. Thus, at a price of P producing qi ¼ the wage. Notice, however, that the price elasticity of supply has not changed—it remains eS, P ¼ QUERY: How would the results of this example change by assuming different values for the weight of labor in the production function (i.e., for a and b)? 418 Part 5: Competitive Markets Equilibrium price determination We can now combine demand and supply curves to demonstrate the establishment of equilibrium prices in the market. Figure 12.4 shows this process. Looking ﬁrst at Figure 12.4b, we see the market demand curve D (ignore D 0 for the moment) and the short-run supply curve S. The two
curves intersect at a price of P1 and a quantity of Q1. This price– quantity combination represents an equilibrium between the demands of individuals and the costs of ﬁrms. The equilibrium price P1 serves two important functions. First, this price acts as a signal to producers by providing them with information about how much should be produced: To maximize proﬁts, ﬁrms will produce that output level for which marginal costs are equal to P1. In the aggregate, production will be Q1. A second function of the price is to ration demand. Given the market price P1, utility-maximizing individuals will decide how much of their limited incomes to devote to buying the particular good. At a price of P1, total quantity demanded will be Q1, and this is precisely the amount that will be produced. Hence we deﬁne equilibrium price as follows Equilibrium price. An equilibrium price is one at which quantity demanded is equal to quantity supplied. At such a price, neither demanders nor suppliers have an incentive to alter their economic decisions. Mathematically, an equilibrium price P$ solves the equation or, more compactly, P$, P 0, I QDð P$, v, w QSð Þ Þ ¼ P$ QDð Þ ¼ P$ QSð : Þ (12:21) (12:22) FIGURE 12.4 Interactions of Many Individuals and Firms Determine Market Price in the Short Run Market demand curves and market supply curves are each the horizontal sum of numerous components. These market curves are shown in (b). Once price is determined in the market, each ﬁrm and each individual treat this price as a ﬁxed parameter in their decisions. Although individual ﬁrms and persons are important in determining price, their interaction as a whole is the sole determinant of price. This is illustrated by a shift in an individual’s demand curve to d 0. If only one individual reacts in this way, market price will not be affected. However, if everyone exhibits an increased demand, market demand will shift to D 0; in the short run, price will increase to P2. Price Price SMC SAC D′ D P 2 P 1 Price S d′ d D′ D d′ d q 1 q 2 Output per period Q 1 Q 2 Total output per period q 1 q 2 q 1′ Quantity demanded per period (a)
A typical firm (b) The market (c) A typical individual Chapter 12: The Partial Equilibrium Competitive Model 419 The deﬁnition given in Equation 12.22 makes clear that an equilibrium price depends on the values of many exogenous factors, such as incomes or prices of other goods and of ﬁrms’ inputs. As we will see in the next section, changes in any of these factors will likely result in a change in the equilibrium price required to equate quantity supplied to quantity demanded. The implications of the equilibrium price (P1) for a typical ﬁrm and a typical individual are shown in Figures 12.4a and 12.4c, respectively. For the typical ﬁrm the price P1 will cause an output level of q1 to be produced. The ﬁrm earns a small proﬁt at this particular price because short-run average total costs are covered. The demand curve d (ignore d 0 for the moment) for a typical individual is shown in Figure 12.4c. At a price of P1, this individual demands q1. By adding up the quantities that each individual demands at P1 and the quantities that each ﬁrm supplies, we can see that the market is in equilibrium. The market supply and demand curves provide a convenient way of making such a summation. Market reaction to a shift in demand The three panels in Figure 12.4 can be used to show two important facts about short-run market equilibrium: the individual’s ‘‘impotence’’ in the market and the nature of short-run supply response. First, suppose that a single individual’s demand curve were to shift outward to d 0, as shown in Figure 12.4c. Because the competitive model assumes there are many demanders, this shift will have practically no effect on the market demand curve. Consequently, market price will be unaffected by the shift to d 0, that is, price will remain at P1. Of course, at this price, the person for whom the demand curve has shifted will consume slightly more (q 01), as shown in Figure 12.4c. But this amount is a tiny part of the market. If many individuals experience outward shifts in their demand curves, the entire market demand curve may shift. Figure 12.4b shows the new demand curve D 0. The new equilibrium point will be at P2, Q2; at this point,
supply–demand balance is re-established. Price has increased from P1 to P2 in response to the demand shift. Notice also that the quantity traded in the market has increased from Q1 to Q2. The increase in price has served two functions. First, as in our previous analysis of the very short run, it has acted to ration demand. Whereas at P1 a typical individual demanded q 01, at P2 only q 02 is demanded. The increase in price has also acted as a signal to the typical ﬁrm to increase production. In Figure 12.4a, the ﬁrm’s proﬁt-maximizing output level has increased from q1 to q2 in response to the price increase. That is what we mean by a short-run supply response: An increase in market price acts as an inducement to increase production. Firms are willing to increase production (and to incur higher marginal costs) because the price has increased. If market price had not been permitted to increase (suppose that government price controls were in effect), then ﬁrms would not have increased their outputs. At P1 there would now be an excess (unﬁlled) demand for the good in question. If market price is allowed to increase, a supply–demand equilibrium can be re-established so that what ﬁrms produce is again equal to what individuals demand at the prevailing market price. Notice also that, at the new price P2, the typical ﬁrm has increased its proﬁts. This increasing proﬁtability in the short run will be important to our discussion of long-run pricing later in this chapter. Shifts in Supply and Demand Curves: a Graphical Analysis In previous chapters we established many reasons why either a demand curve or a supply curve might shift. These reasons are brieﬂy summarized in Table 12.1. Although most of these merit little additional explanation, it is important to note that a change in the 420 Part 5: Competitive Markets TABLE 12.1 REASONS FOR SHIFTS IN DEMAND OR SUPPLY CURVES Demand Curves Shift Because Supply Curves Shift Because Incomes change Input prices change Prices of substitutes or complements change Technology changes Preferences change Number of producers changes number of ﬁrms will shift the short-run market supply curve (because the sum in Equation 12.13 will be over a different number of
ﬁrms). This observation allows us to tie together short-run and long-run analysis. It seems likely that the types of changes described in Table 12.1 are constantly occurring in real-world markets. When either a supply curve or a demand curve does shift, equilibrium price and quantity will change. In this section we investigate graphically the relative magnitudes of such changes. In the next section we show the results mathematically. Shifts in supply curves: Importance of the shape of the demand curve Consider ﬁrst a shift inward in the short-run supply curve for a good. As in Example 12.2, such a shift might have resulted from an increase in the prices of inputs used by ﬁrms to produce the good. Whatever the cause of the shift, it is important to recognize that the effect of the shift on the equilibrium level of P and Q will depend on the shape of the demand curve for the product. Figure 12.5 illustrates two possible situations. The demand curve in Figure 12.5a is relatively price elastic; that is, a change in price substantially affects quantity demanded. For this case, a shift in the supply curve from S to S 0 will cause equilibrium price to increase only moderately (from P to P 0), whereas quantity decreases sharply (from Q to Q 0). Rather than being ‘‘passed on’’ in higher prices, the In (a) the shift upward in the supply curve causes price to increase only slightly while quantity decreases sharply. This results from the elastic shape of the demand curve. In (b) the demand curve is inelastic; price increases substantially, with only a slight decrease in quantity. Price D P′ P S′ S S′ S Price D P′ P S′ S Q′ Q Q per period Q′ Q Q per period (a) Elastic demand (b) Inelastic demand FIGURE 12.5 Effect of a Shift in the Short-Run Supply Curve Depends on the Shape of the Demand Curve Chapter 12: The Partial Equilibrium Competitive Model 421 increase in the ﬁrms’ input costs is met primarily by a decrease in quantity (a movement down each ﬁrm’s marginal cost curve) and only a slight increase in price. This situation is reversed when the market demand curve is inelastic. In Figure 12.5b a shift in the supply curve causes equilibrium price to increase substantially while quantity is little changed. The reason
for this is that individuals do not reduce their demands much if prices increase. Consequently, the shift upward in the supply curve is almost entirely passed on to demanders in the form of higher prices. Shifts in demand curves: Importance of the shape of the supply curve Similarly, a shift in a market demand curve will have different implications for P and Q, depending on the shape of the short-run supply curve. Two illustrations are shown in Figure 12.6. In Figure 12.6a the supply curve for the good in question is inelastic. In this situation, a shift outward in the market demand curve will cause price to increase substantially. On the other hand, the quantity traded increases only slightly. Intuitively, what has happened is that the increase in demand (and in Q) has caused ﬁrms to move up their steeply sloped marginal cost curves. The concomitant large increase in price serves to ration demand. Figure 12.6b shows a relatively elastic short-run supply curve. Such a curve would occur for an industry in which marginal costs do not increase steeply in response to output increases. For this case, an increase in demand produces a substantial increase in Q. However, because of the nature of the supply curve, this increase is not met by great cost increases. Consequently, price increases only moderately. These examples again demonstrate Marshall’s observation that demand and supply simultaneously determine price and quantity. Recall his analogy from Chapter 1: Just as it is impossible to say which blade of a scissors does the cutting, so too is it impossible to attribute price solely to demand or to supply characteristics. Rather, the effect of FIGURE 12.6 Effect of a Shift in the Demand Curve Depends on the Shape of the Short-Run Supply Curve In (a), supply is inelastic; a shift in demand causes price to increase greatly, with only a small concomitant increase in quantity. In (b), on the other hand, supply is elastic; price increases only slightly in response to a demand shift. D′ D Price P′ P S Price D′ D S P′ P D′ S D Q Q′ Q per period Q Q′ (a) Inelastic supply (b) Elastic supply S D′ D Q per period 422 Part 5: Competitive Markets shifts in either a demand curve or a supply curve will depend on the shapes of both curves. Mathematical Model of Market Equilibrium A general mathematical model of the supply–demand process can further illuminate
the comparative statics of changing equilibrium prices and quantities. Suppose that the demand function is represented by D, Þ QD ¼ P, a ð where a is a parameter that allows us to shift the demand curve. It might represent consumer income, prices of other goods (this would permit the tying together of supply and demand in several related markets), or changing preferences. In general we expect that @D=@P Da may have any sign, depending precisely on what the parameter a means. Using this same procedure, we can write the supply relationship as DP < 0, but @D=@a (12:23) ¼ ¼ P, b, Þ ð where b is a parameter that shifts the supply curve and might include such factors as input prices, technical changes, or (for a multiproduct ﬁrm) prices of other potential outputs. Here @S=@P Sb may have any sign. The model is closed by requiring that, in equilibrium,5 SP > 0, but @S=@b QS ¼ (12:24) ¼ ¼ S QD ¼ QS: (12:25) To analyze the effect of a small change in one of the exogenous parameters (a or b) on market equilibrium requires a bit of calculus.6 Suppose we are interested in the impact of a shift in demand (a) while keeping the supply function ﬁxed (i.e., holding b constant). Differentiation of the demand and supply functions yields: dQD da ¼ dQS da ¼ Þ P, a dD ð da P, b Þ ð da ¼ dS ¼ SP dP da : DP dP da þ Da (12:26) Notice that the only effect on supply here occurs through the impact of market price— the exogenous factors in the supply function are held constant. Maintenance of market equilibrium for this shift in demand requires that dQD da ¼ dQS da : (12:27) 5The model could be further modiﬁed to show how the equilibrium quantity supplied is to be allocated among the ﬁrms in the industry. If, for example, the industry is composed of n identical ﬁrms, then the output of any one of them would be given by Q n : q ¼ In the short run with n ﬁxed this would add little to our analysis.
In the long run, however, n must also be determined by the model as we show later in this chapter. 6This type of analysis is usually called comparative statics analysis because we are comparing two equilibrium positions but are not especially concerned with the ‘‘dynamics’’ of how the market moves from one equilibrium to the other. Chapter 12: The Partial Equilibrium Competitive Model 423 Hence we can solve for the change in equilibrium price as or, after a bit of algebra, DP dP da þ Da SP ¼ dP da (12:28) : DP (12:29) dP da ¼ Da SP & Because the denominator of this expression is positive, the overall sign of dP=da will depend only on the sign of Da—that is, on how the change of the exogenous factor a affects demand. For example, if a represents consumer income, we would expect Da to be positive and thus dP=da would be positive. That is, an increase in income would be expected to increase equilibrium price. On the other hand, if a represented the price of a (gross) complement, we would expect Da to be negative and dP=da would also be negative. An increase in the price of a complementary good would be expected to reduce P. It would be a simple matter to repeat the steps in Equations 12.27–12.29 to derive a similar expression for how a shift in supply (b) would affect the equilibrium price. An elasticity interpretation Further algebraic manipulation of Equation 12.29 yields a more useful comparative statics result. Multiplying both sides of that equation by a/P gives eP,a ¼ dP da ’ a P ¼ Dað SP & a=Q DPÞ ’ DP ’ Da SP & Þ P=Q ¼ a P eQ,a eS,P & eQ,P : (12:30) ¼ ð Because all the elasticities in this equation may be available from empirical studies, this equation can be a convenient way to make rough estimates of the effects of various events on equilibrium prices. As an example, suppose again that a represents consumer income and that there is interest in predicting how an increase in income affects the equilibrium price of, say, automobiles. Suppose empirical data suggest that eQ, I ¼ 3.0 and 1.0. eQ, P ¼ & Substituting these ﬁg
ures into Equation 12.30 yields 1.2 (these ﬁgures are from Table 12.3; see Extensions) and assume that eS, P ¼ eQ, a ¼ eP,a ¼ ¼ eQ,a eS,P & 3:0 2:2 ¼ eQ,P ¼ 1:36: 3:0 1:0 1:2 Þ & ð& (12:31) Therefore, the empirical elasticity estimates suggest that each 1 percent increase in consumer incomes results in a 1.36 percent increase in the equilibrium price of automobiles. Estimates of other kinds of shifts in supply or demand can be similarly modeled by using the type of calculus-based approach provided in Equations 12.26–12.29. EXAMPLE 12.3 Equilibria with Constant Elasticity Functions An even more complete analysis of supply–demand equilibrium can be provided if we use speciﬁc functional forms. Constant elasticity functions are especially useful for this purpose. Suppose the demand for automobiles is given by P, I QDð Þ ¼ 0:1P& 1:2 I3; (12:32) 424 Part 5: Competitive Markets here price (P) is measured in dollars, as is real family income (I). The supply function for automobiles is P, w QSð Þ ¼ 6,400Pw& 0:5, (12:33) where w is the hourly wage of automobile workers. Notice that the elasticities assumed here 1). If the values for the 3.0, and eS, P ¼ are those used previously in the text (eQ, P ¼ & ‘‘exogenous’’ variables I and w are $20,000 and $25, respectively, then demand–supply equilibrium requires 1.2, eQ, I ¼ QD ¼ ¼ 0:1P& Qs ¼ 1:2I3 ¼ ð 6,400Pw& 8 3 1011 0:5 P& Þ 1,280P 1:2 ¼ or or P2:2 8 3 1011 ¼ ð =1,280 Þ ¼ 6:25 3 108 P$ Q$ ¼ ¼ 9,957, 1;280 P$ ’ ¼ 12,745,000: (12:34) (12:35) Hence
the initial equilibrium in the automobile market has a price of nearly $10,000 with approximately 13 million cars being sold. A shift in demand. A 10 percent increase in real family income, all other factors remaining constant, would shift the demand function to QD ¼ ð 1:06 3 1012 (12:36) P& Þ 1:2 and, proceeding as before, or P 2:2 1:06 3 1012 ¼ ð =1,280 Þ ¼ 8:32 3 108 P$ Q$ ¼ ¼ 11,339, 14,514,000: (12:37) (12:38) As we predicted earlier, the 10 percent increase in real income made car prices increase by nearly 14 percent. In the process, quantity sold increased by approximately 1.77 million automobiles. A shift in supply. An exogenous shift in automobile supply as a result, say, of changing auto workers’ wages would also affect market equilibrium. If wages were to increase from $25 to $30 per hour, the supply function would shift to 30 ð returning to our original demand function (with I 6,400P Qsð P, w Þ ¼ or P2:2 8 3 1011 ¼ ð 0:5 & Þ ¼ 1,168P; $20,000) then yields ¼ =1,168 Þ 6:85 3 108 ¼ P$ Q$ ¼ ¼ 10,381, 12,125,000: (12:39) (12:40) (12:41) Therefore, the 20 percent increase in wages led to a 4.3 percent increase in auto prices and to a decrease in sales of more than 600,000 units. Changing equilibria in many types of markets can be approximated by using this general approach together with empirical estimates of the relevant elasticities. QUERY: Do the results of changing auto workers’ wages agree with what might have been predicted using an equation similar to Equation 12.30? Chapter 12: The Partial Equilibrium Competitive Model 425 Long-Run Analysis We saw in Chapter 10 that, in the long run, a ﬁrm may adapt all its inputs to ﬁt market conditions. For long-run analysis, we should use the ﬁrm’s long-run cost curves. A proﬁtmaximizing ﬁrm that is a price-taker will produce
the output level for which price is equal to long-run marginal cost (MC). However, we must consider a second and ultimately more important inﬂuence on price in the long run: the entry of entirely new ﬁrms into the industry or the exit of existing ﬁrms from that industry. In mathematical terms, we must allow the number of ﬁrms, n, to vary in response to economic incentives. The perfectly competitive model assumes that there are no special costs of entering or exiting from an industry. Consequently, new ﬁrms will be lured into any market in which (economic) proﬁts are positive. Similarly, ﬁrms will leave any industry in which proﬁts are negative. The entry of new ﬁrms will cause the short-run industry supply curve to shift outward because there are now more ﬁrms producing than there were previously. Such a shift will cause market price (and industry proﬁts) to decrease. The process will continue until no ﬁrm contemplating entry would be able to earn a proﬁt in the industry.7 At that point, entry will cease and the industry will have an equilibrium number of ﬁrms. A similar argument can be made for the case in which some of the ﬁrms are suffering short-run losses. Some ﬁrms will choose to leave the industry, and this will cause the supply curve to shift to the left. Market price will increase, thus restoring proﬁtability to those ﬁrms remaining in the industry. ¼ MC (which is required for proﬁt maximization) and P Equilibrium conditions To begin with we will assume that all the ﬁrms in an industry have identical cost functions; that is, no ﬁrm controls any special resources or technologies.8 Because all ﬁrms are identical, the equilibrium long-run position requires that each ﬁrm earn exactly zero economic proﬁts. In graphic terms, the long-run equilibrium price must settle at the low point of each ﬁrm’s long-run average total cost curve. Only at this point do the two equilibrium conditions P AC (which is required for zero proﬁt) hold. It is important to emphasize, however, that these two equilibrium conditions have rather different origins
. Proﬁt maximization is a goal of ﬁrms. Therefore, the P MC rule derives from the behavioral assumptions we have made about ﬁrms and is similar to the output decision rule used in the short run. The zero-proﬁt condition is not a goal for ﬁrms; ﬁrms obviously would prefer to have large, positive proﬁts. The long-run operation of the market, however, forces all ﬁrms to accept a level of zero economic proﬁts (P AC) because of the willingness of ﬁrms to enter and to leave an industry in response to the possibility of making supranormal returns. Although the ﬁrms in a perfectly competitive industry may earn either positive or negative proﬁts in the short run, in the long run only a level of zero proﬁts will prevail. Hence we can summarize this analysis by the following deﬁnition Long-run competitive equilibrium. A perfectly competitive market is in long-run equilibrium if there are no incentives for proﬁt-maximizing ﬁrms to enter or to leave the market. This will occur when (a) the number of ﬁrms is such that P AC and (b) each ﬁrm operates at the low point of its long-run average cost curve. MC ¼ ¼ 7Remember that we are using the economists’ deﬁnition of proﬁts here. These proﬁts represent a return to the owner of a business in excess of that which is strictly necessary to stay in the business. 8If ﬁrms have different costs, then low-cost ﬁrms can earn positive long-run proﬁts, and such extra proﬁts will be reﬂected in the price of the resource that accounts for the ﬁrm’s low costs. In this sense the assumption of identical costs is not restrictive because an active market for the ﬁrm’s inputs will ensure that average costs (which include opportunity costs) are the same for all ﬁrms. See also the discussion of Ricardian rent later in this chapter. 426 Part 5: Competitive Markets Long-Run Equilibrium: Constant Cost Case To discuss long-run pricing in detail
, we must make an assumption about how the entry of new ﬁrms into an industry affects the prices of ﬁrms’ inputs. The simplest assumption we might make is that entry has no effect on the prices of those inputs—perhaps because the industry is a relatively small hirer in its various input markets. Under this assumption, no matter how many ﬁrms enter (or leave) this market, each ﬁrm will retain the same set of cost curves with which it started. This assumption of constant input prices may not be tenable in many important cases, which we will look at in the next section. For the moment, however, we wish to examine the equilibrium conditions for a constant cost industry. Initial equilibrium Figure 12.7 demonstrates long-run equilibrium in this situation. For the market as a whole (Figure 12.7b), the demand curve is given by D and the short-run supply curve by SS. Therefore, the short-run equilibrium price is P1. The typical ﬁrm (Figure 12.7a) will produce output level q1 because, at this level of output, price is equal to short-run marginal cost (SMC). In addition, with a market price of P1, output level q1 is also a long-run equilibrium position for the ﬁrm. The ﬁrm is maximizing proﬁts because price is equal to long-run marginal costs (MC). Figure 12.7a also implies our second long-run equilibrium property: Price is equal to long-run average costs (AC). Consequently, economic proﬁts are zero, and there is no incentive for ﬁrms either to enter or to leave the industry. Therefore, the market depicted in Figure 12.7 is in both short-run and long-run equilibrium. FIGURE 12.7 Long-Run Equilibrium for a Perfectly Competitive Industry: Constant Cost Case An increase in demand from D to D 0 will cause price to increase from P1 to P2 in the short run. This higher price will create proﬁts in the industry, and new ﬁrms will be drawn into the market. If it is assumed that the entry of these new ﬁrms has no effect on the cost curves of the ﬁrms in the industry, then new ﬁrms will continue to enter until price is pushed back down to P1.
At this price, economic proﬁts are zero. Therefore, the long-run supply curve (LS) will be a horizontal line at P1. Along LS, output is increased by increasing the number of ﬁrms, each producing q1. Price P 2 P 1 Price D′ D SMC MC AC SS SS′ SS SS′ LS D′ D q1 q2 Quantity per period Q1 Q2 Q3 Total quantity per period (a) A typical firm (b) Total market Chapter 12: The Partial Equilibrium Competitive Model 427 Firms are in equilibrium because they are maximizing proﬁts, and the number of ﬁrms is stable because economic proﬁts are zero. This equilibrium will tend to persist until either supply or demand conditions change. Responses to an increase in demand Suppose now that the market demand curve in Figure 12.7b shifts outward to D 0. If SS is the relevant short-run supply curve for the industry, then in the short run, price will increase to P2. The typical ﬁrm, in the short run, will choose to produce q2 and will earn proﬁts on this level of output. In the long run, these proﬁts will attract new ﬁrms into the market. Because of the constant cost assumption, this entry of new ﬁrms will have no effect on input prices. New ﬁrms will continue to enter the market until price is forced down to the level at which there are again no pure economic proﬁts. Therefore, the entry of new ﬁrms will shift the short-run supply curve to SS0, where the equilibrium price (P1) is re-established. At this new long-run equilibrium, the price–quantity combination P1, Q3 will prevail in the market. The typical ﬁrm will again produce at output level q1, although now there will be more ﬁrms than in the initial situation. Inﬁnitely elastic supply We have shown that the long-run supply curve for the constant cost industry will be a horizontal straight line at price P1. This curve is labeled LS in Figure 12.7b. No matter what happens to demand, the twin equilibrium conditions of zero long-run proﬁts (because free entry is assumed) and proﬁt maximization will ensure that no
price other than P1 can prevail in the long run.9 For this reason, P1 might be regarded as the ‘‘normal’’ price for this commodity. If the constant cost assumption is abandoned, however, the long-run supply curve need not have this inﬁnitely elastic shape, as we show in the next section. EXAMPLE 12.4 Inﬁnitely Elastic Long-Run Supply Handmade bicycle frames are produced by a number of identically sized ﬁrms. Total (long-run) monthly costs for a typical ﬁrm are given by C q ð Þ ¼ q3 & 20q2 þ 100q þ 8,000; (12:42) where q is the number of frames produced per month. Demand for handmade bicycle frames is given by QD ¼ 2,500 3P, & (12:43) where QD is the quantity demanded per month and P is the price per frame. To determine the long-run equilibrium in this market, we must ﬁnd the low point of the typical ﬁrm’s average cost curve. Because AC C q ð Þ q ¼ ¼ q2 & 20q þ 100 8,000 q þ (12:44) 9These equilibrium conditions also point out what seems to be, somewhat imprecisely, an ‘‘efﬁcient’’ aspect of the long-run equilibrium in perfectly competitive markets: The good under investigation will be produced at minimum average cost. We will have much more to say about efﬁciency in the next chapter. 428 Part 5: Competitive Markets and MC @C q Þ ð @q ¼ ¼ 3q2 40q & þ 100 (12:45) and because we know this minimum occurs where AC q2 & 20q þ 100 8,000 þ q ¼ or ¼ 3q2 MC, we can solve for this output level: 40q 100 þ þ 2q2 20q & 8,000 q, ¼ (12:46) which has a convenient solution of q 20. With a monthly output of 20 frames, each producer has a long-run average and marginal cost of $500. This is the long-run equilibrium price of bicycle frames (handmade frames cost a bundle, as any cyclist can attest). With P $500, Equation 12.43
shows QD ¼ 1,000. Therefore, the equilibrium number of ﬁrms is 50. When each of these 50 ﬁrms produces 20 frames per month, supply will precisely balance what is demanded at a price of $500. ¼ ¼ If demand in this problem were to increase to QD ¼ 3,000 3P, & (12:47) then we would expect long-run output and the number of frames to increase. Assuming that entry into the frame market is free and that such entry does not alter costs for the typical bicycle maker, the long-run equilibrium price will remain at $500 and a total of 1,500 frames per month will be demanded. That will require 75 frame makers, so 25 new ﬁrms will enter the market in response to the increase in demand. QUERY: Presumably, the entry of frame makers in the long run is motivated by the short-run proﬁtability of the industry in response to the increase in demand. Suppose each ﬁrm’s shortrun costs were given by SC 20,000. Show that short-run proﬁts are zero þ when the industry is in long-term equilibrium. What are the industry’s short-run proﬁts as a result of the increase in demand when the number of ﬁrms stays at 50? 1,500q 50q2 ¼ & Shape of the Long-Run Supply Curve Contrary to the short-run situation, long-run analysis has little to do with the shape of the (long-run) marginal cost curve. Rather, the zero-proﬁt condition centers attention on the low point of the long-run average cost curve as the factor most relevant to long-run price determination. In the constant cost case, the position of this low point does not change as new ﬁrms enter the industry. Consequently, if input prices do not change, then only one price can prevail in the long run regardless of how demand shifts—the long-run supply curve is horizontal at this price. Once the constant cost assumption is abandoned, this need not be the case. If the entry of new ﬁrms causes average costs to rise, the long-run supply curve will have an upward slope. On the other hand, if entry causes average costs to decline, it is even possible for the long-run supply curve to be negatively sloped.
We shall now discuss these possibilities. Increasing cost industry The entry of new ﬁrms into an industry may cause the average costs of all ﬁrms to increase for several reasons. New and existing ﬁrms may compete for scarce inputs, thus driving up their prices. New ﬁrms may impose ‘‘external costs’’ on existing ﬁrms (and on themselves) in the form of air or water pollution. They may increase the demand for Chapter 12: The Partial Equilibrium Competitive Model 429 FIGURE 12.8 An Increasing Cost Industry Has a Positively Sloped LongRun Supply Curve Initially the market is in equilibrium at P1, Q1. An increase in demand (to D0) causes price to increase to P2 in the short run, and the typical ﬁrm produces q2 at a proﬁt. This proﬁt attracts new ﬁrms into the industry. The entry of these new ﬁrms causes costs for a typical ﬁrm to increase to the levels shown in (b). With this new set of curves, equilibrium is re-established in the market at P3, Q3. By considering many possible demand shifts and connecting all the resulting equilibrium points, the long-run supply curve (LS) is traced out. Price Price SMC MC P 3 AC P 2 P 1 SMC MC AC Price D P 2 P 3 P 1 D′ SS SS′ LS D′ D q 1 Output q 2 per period q 3 Output per period Q 1 Q 2 Q 3 Output per period (a) Typical firm before entry (b) Typical firm after entry (c) The market tax-ﬁnanced services (e.g., police forces, sewage treatment plants), and the required taxes may show up as increased costs for all ﬁrms. Figure 12.8 demonstrates two market equilibria in such an increasing cost industry. The initial equilibrium price is P1. At this price the typical ﬁrm produces q1, and total industry output is Q1. Suppose now that the demand curve for the industry shifts outward to D 0. In the short run, price will rise to P2 because this is where D 0 and the industry’s short-run supply curve (SS) intersect. At this price the typical ﬁrm will produce q2 and will earn a substantial proﬁt. This
proﬁt then attracts new entrants into the market and shifts the short-run supply curve outward. Suppose that this entry of new ﬁrms causes the cost curves of all ﬁrms to increase. The new ﬁrms may compete for scarce inputs, thereby driving up the prices of these inputs. A typical ﬁrm’s new (higher) set of cost curves is shown in Figure 12.8b. The new long-run AC), and at this price Q3 is equilibrium price for the industry is P3 (here P3 ¼ demanded. We now have two points (P1, Q1 and P3, Q3) on the long-run supply curve. All other points on the curve can be found in an analogous way by considering all possible shifts in the demand curve. These shifts will trace out the long-run supply curve LS. Here LS has a positive slope because of the increasing cost nature of the industry. Observe that the LS curve is ﬂatter (more elastic) than the short-run supply curves. This indicates the greater ﬂexibility in supply response that is possible in the long run. Still, the curve is upward sloping, so price increases with increasing demand. This situation is probably common; we will have more to say about it in later sections. MC ¼ Decreasing cost industry Not all industries exhibit constant or increasing costs. In some cases, the entry of new ﬁrms may reduce the costs of ﬁrms in an industry. For example, the entry of new ﬁrms may provide a larger pool of trained labor from which to draw than was previously available, thus reducing the costs associated with the hiring of new workers. Similarly, the entry of new ﬁrms may provide a ‘‘critical mass’’ of industrialization, which permits the development of more efﬁcient transportation and communications networks. Whatever 430 Part 5: Competitive Markets FIGURE 12.9 A Decreasing Cost Industry Has a Negatively Sloped Long-Run Supply Curve In (c), the market is in equilibrium at P1, Q1. An increase in demand to D0 causes price to increase to P2 in the short run, and the typical ﬁrm produces q2 at a proﬁt. This proﬁt attracts new ﬁrms to the industry. If the entry of
these new ﬁrms causes costs for the typical ﬁrm to decrease, a set of new cost curves might look like those in (b). With this new set of curves, market equilibrium is re-established at P3, Q3. By connecting such points of equilibrium, a negatively sloped long-run supply curve (LS) is traced out. Price P2 P1 AC SMC MC Price Price D′ SS SMC MC AC P3 D LS P2 P1 P3 q1 q2 Output per period q3 Output per period D Q1 Q2 (a) Typical firm before entry (b) Typical firm after entry (c) The market SS′ LS D′ Output Q3 per period the exact reason for the cost reductions, the ﬁnal result is illustrated in the three panels of Figure 12.9. The initial market equilibrium is shown by the price–quantity combination P1, Q1 in Figure 12.9c. At this price the typical ﬁrm produces q1 and earns exactly zero in economic proﬁts. Now suppose that market demand shifts outward to D 0. In the short run, price will increase to P2 and the typical ﬁrm will produce q2. At this price level, positive proﬁts are being earned. These proﬁts cause new entrants to come into the market. If this entry causes costs to decline, a new set of cost curves for the typical ﬁrm might resemble those shown in Figure 12.9b. Now the new equilibrium price is P3; at this price, Q3 is demanded. By considering all possible shifts in demand, the long-run supply curve, LS, can be traced out. This curve has a negative slope because of the decreasing cost nature of the industry. Therefore, as output expands, price falls. This possibility has been used as the justiﬁcation for protective tariffs to shield new industries from foreign competition. It is assumed (only occasionally correctly) that the protection of the ‘‘infant industry’’ will permit it to grow and ultimately to compete at lower world prices. Classiﬁcation of long-run supply curves Thus, we have shown that the long-run supply curve for a perfectly competitive industry may assume a variety of shapes. The principal determinant of the shape is the way in which the entry of ﬁrms into the industry affects all �
��rms’ costs. The following deﬁnitions cover the various possibilities Constant, increasing, and decreasing cost industries. An industry supply curve exhibits one of three shapes. Constant cost: Entry does not affect input costs; the long-run supply curve is horizontal at the longrun equilibrium price. Increasing cost: Entry increases input costs; the long-run supply curve is positively sloped. Decreasing cost: Entry reduces input costs; the long-run supply curve is negatively sloped. Chapter 12: The Partial Equilibrium Competitive Model 431 Now we show how the shape of the long-run supply curve can be further quantiﬁed. Long-Run Elasticity of Supply The long-run supply curve for an industry incorporates information on internal ﬁrm adjustments to changing prices and changes in the number of ﬁrms and input costs in response to proﬁt opportunities. All these supply responses are summarized in the following elasticity concept Long-run elasticity of supply. The long-run elasticity of supply (eLS,P) records the proportionate change in long-run industry output in response to a proportionate change in product price. Mathematically, eLS, P ¼ percentage change in Q percentage change in P ¼ @QLS @P ’ P QLS : (12:48) The value of this elasticity may be positive or negative depending on whether the industry exhibits increasing or decreasing costs. As we have seen, eLS,P is inﬁnite in the constant cost case because industry expansions or contractions can occur without having any effect on product prices. Empirical estimates It is obviously important to have good empirical estimates of long-run supply elasticities. These indicate whether production can be expanded with only a slight increase in relative price (i.e., supply is price elastic) or whether expansions in output can occur only if relative prices increase sharply (i.e., supply is price inelastic). Such information can be used to assess the likely effect of shifts in demand on long-run prices and to evaluate alternative policy proposals intended to increase supply. Table 12.2 presents several long-run supply elasticity estimates. These relate primarily (although not exclusively) to natural resources because economists have devoted considerable attention to the implications of increasing demand for the prices of such resources. As the table makes clear, these estimates vary widely depending on the spatial and geological properties of the particular resources involved. All the
estimates, however, suggest that supply does respond positively to price. Comparative Statics Analysis of Long-Run Equilibrium Earlier in this chapter we showed how to develop a simple comparative statics analysis of changing short-run equilibria in competitive markets. By using estimates of the long-run elasticities of demand and supply, exactly the same sort of analysis can be conducted for the long run as well. For example, the hypothetical auto market model in Example 12.3 might serve equally well for long-run analysis, although some differences in interpretation might be required. Indeed, in applied models of supply and demand it is often not clear whether the author intends his or her results to reﬂect the short run or the long run, and some care must be taken to understand how the issue of entry is being handled. 432 Part 5: Competitive Markets TABLE 12.2 SELECTED ESTIMATES OF LONG-RUN SUPPLY ELASTICITIES Agricultural acreage Corn Cotton Wheat Aluminum Chromium Coal (eastern reserves) Natural gas (U.S. reserves) Oil (U.S. reserves) Urban housing Density Quality 0.18 0.67 0.93 Nearly inﬁnite 0–3.0 15.0–30.0 0.20 0.76 5.3 3.8 SOURCES: Agricultural acreage—M. Nerlove, ‘‘Estimates of the Elasticities of Supply of Selected Agricultural Commodities,’’ Journal of Farm Economics 38 (May 1956): 496–509. Aluminum and chromium—estimated from U.S. Department of Interior, Critical Materials Commodity Action Analysis (Washington, DC: U.S. Government Printing Ofﬁce, 1975). Coal—estimated from M. B. Zimmerman, ‘‘The Supply of Coal in the Long Run: The Case of Eastern Deep Coal,’’ MIT Energy Laboratory Report No. MITEL 75–021 (September 1975). Natural gas—based on estimate for oil (see text) and J. D. Khazzoom, ‘‘The FPC Staff’s Econometric Model of Natural Gas Supply in the United States,’’ The Bell Journal of Economics and Management Science (Spring 1971): 103–17. Oil—E. W. Erickson, S. W. Millsaps, and R. M. Spann, ‘‘Oil Supply and Tax In
centives,’’ Brookings Papers on Economic Activity 2 (1974): 449–78. Urban housing—B. A. Smith, ‘‘The Supply of Urban Housing,’’ Journal of Political Economy 40 (August 1976): 389–405. Industry structure One aspect of the changing long-run equilibria in a perfectly competitive market that is obscured by using a simple supply–demand analysis is how the number of ﬁrms varies as market equilibria change. Because—as we will see in Part 6—the functioning of markets may in some cases be affected by the number of ﬁrms, and because there may be direct public policy interest in entry and exit from an industry, some additional analysis is required. In this section we will examine in detail determinants of the number of ﬁrms in the constant cost case. Brief reference will also be made to the increasing cost case, and some of the problems for this chapter examine that case in more detail. Shifts in demand Because the long-run supply curve for a constant cost industry is inﬁnitely elastic, analyzing shifts in market demand is particularly easy. If the initial equilibrium industry output is Q0 and if q$ represents the output level for which the typical ﬁrm’s long-run average cost is minimized, then the initial equilibrium number of ﬁrms (n0) is given by n0 ¼ Q0 q$. (12:49) A shift in demand that changes equilibrium output to Q1 will, in the long run, change the equilibrium number of ﬁrms to n1 ¼ Q1 q$, (12:50) Chapter 12: The Partial Equilibrium Competitive Model 433 and the change in the number of ﬁrms is given by n1 & n0 ¼ Q0 : Q1 & q$ (12:51) That is, the change in the equilibrium number of ﬁrms is completely determined by the extent of the demand shift and by the optimal output level for the typical ﬁrm. Changes in input costs Even in the simple constant cost industry case, analyzing the effect of an increase in an input price (and hence an upward shift in the inﬁnitely elastic long-run supply curve) is relatively complicated. First, to calculate the decrease in industry output, it is necessary to know both the extent to which minimum average cost is increased
by the input price increase and how such an increase in the long-run equilibrium price affects total quantity demanded. Knowledge of the typical ﬁrm’s average cost function and of the price elasticity of demand permits such a calculation to be made in a straightforward way. But an increase in an input price may also change the minimum average cost output level for the typical ﬁrm. Such a possibility is illustrated in Figure 12.10. Both the average and marginal costs have been shifted upward by the input price increase, but because average cost has shifted up by a relatively greater extent than the marginal cost, the typical ﬁrm’s optimal output level has increased from q$0 to q$1. If the relative sizes of the shifts in cost curves were reversed, however, the typical ﬁrm’s optimal output An increase in the price of an input will shift average and marginal cost curves upward. The precise effect of these shifts on the typical ﬁrm’s optimal output level (q$) will depend on the relative magnitudes of the shifts. Average and marginal costs MC1 MC0 AC1 AC0 q0* *q1 Output per period FIGURE 12.10 An Increase in an Input Price May Change Long-Run Equilibrium Output for the Typical Firm 434 Part 5: Competitive Markets level would have decreased.10 Taking account of this change in optimal scale, Equation 12.51 becomes n1 & n0 ¼ Q1 q$1 & Q0 q$0, (12:52) and a number of possibilities arise. If q$1 ( q$0, the decrease in quantity brought about by the increase in market price will deﬁnitely cause the number of ﬁrms to decrease. However, if q$1 < q$0, then the result will be indeterminate. Industry output will decrease, but optimal ﬁrm size also will decrease, thus the ultimate effect on the number of ﬁrms depends on the relative magnitude of these changes. A decrease in the number of ﬁrms still seems the most likely outcome when an input price increase causes industry output to decrease, but an increase in n is at least a theoretical possibility. EXAMPLE 12.5 Increasing Input Costs and Industry Structure An increase in costs for bicycle frame makers will alter the equilibrium described in Example 12.4, but the precise effect on market structure will depend on how costs increase. The
effects of an increase in ﬁxed costs are fairly clear: The long-run equilibrium price will increase and the size of the typical ﬁrm will also increase. This latter effect occurs because an increase in ﬁxed costs increases AC but not MC. To ensure that the equilibrium condition for AC MC holds, output (and MC) must also increase. For example, if an increase in shop rents causes the typical frame maker’s costs to increase to ¼ C(q) q3 20q2 100q 11,616, (12:53) ¼ þ & it is an easy matter to show that MC 22. Therefore, the increase in rent has AC when q increased the efﬁcient scale of bicycle frame operations by 2 bicycle frames per month. At q 22, the long-run average cost and the marginal cost are both 672, and that will be the longrun equilibrium price for frames. At this price ¼ ¼ ¼ þ ¼ so there will be room in the market now for only 22 ( 22) ﬁrms. The increase in ﬁxed costs resulted not only in an increase in price but also in a signiﬁcant reduction in the number of frame makers (from 50 to 22). 484 ¼ ) & QD ¼ 2,500 3P 484, (12:54) Increases in other types of input costs may, however, have more complex effects. Although a complete analysis would require an examination of frame makers’ production functions and their related input choices, we can provide a simple illustration by assuming that an increase in some variable input prices causes the typical ﬁrm’s total cost function to become C(q) q3 & ¼ 8q2 þ 100q þ 4,950. (12:55) 10A mathematical proof proceeds as follows. Optimal output q is deﬁned such that $ Differentiating both sides of this expression by (say) v yields AC v, w, q$ ð Þ ¼ MC ð v, w, q$ : Þ but @AC=@q$ ¼ 0 because average costs are minimized. Manipulating terms, we obtain @AC @v þ @AC @q$ ’ @q$ @v ¼ @MC @v þ @MC @q$ ’ @q$ @v ; @q$ @v ¼
� inputs may not be indifferent about the level of production in a particular industry, however. In the constant cost case, of course, input prices are assumed to be independent of the level of production on the presumption that inputs can earn the same amount in alternative occupations. But in the increasing cost case, entry will bid up some input prices and suppliers of these inputs will be made better off. Consideration of these price effects leads to the following alternative notion of producer surplus Producer surplus. Producer surplus is the extra return that producers make by making transactions at the market price over and above what they would earn if nothing were produced. It is illustrated by the size of the area below the market price and above the supply curve. Although this is the same deﬁnition we introduced in Chapter 11, the context is now different. Now the ‘‘extra returns that producers make’’ should be interpreted as meaning ‘‘the higher prices that productive inputs receive.’’ For short-run producer surplus, the gainers from market transactions are ﬁrms that are able to cover ﬁxed costs and possibly 436 Part 5: Competitive Markets earn proﬁts over their variable costs. For long-run producer surplus, we must penetrate back into the chain of production to identify who the ultimate gainers from market transactions are. It is perhaps surprising that long-run producer surplus can be shown graphically in much the same way as short-run producer surplus. The former is given by the area above the long-run supply curve and below equilibrium market price. In the constant cost case, long-run supply is inﬁnitely elastic, and this area will be zero, showing that returns to inputs are independent of the level of production. With increasing costs, however, longrun supply will be positively sloped and input prices will be bid up as industry output expands. Because this notion of long-run producer surplus is widely used in applied analysis (as we show later in this chapter), we will provide a formal development. Ricardian rent Long-run producer surplus can be most easily illustrated with a situation ﬁrst described by David Ricardo in the early part of the nineteenth century.11 Assume there are many parcels of land on which a particular crop might be grown. These range from fertile land (low costs of production) to poor, dry land (high costs). The long-run supply curve for the crop is constructed as follows. At low
prices only the best land is used. As output increases, higher-cost plots of land are brought into production because higher prices make it proﬁtable to use this land. The long-run supply curve is positively sloped because of the increasing costs associated with using less fertile land. Market equilibrium in this situation is illustrated in Figure 12.11. At an equilibrium price of P$, owners of both the low-cost and the medium-cost ﬁrms earn (long-run) profits. The ‘‘marginal ﬁrm’’ earns exactly zero economic proﬁts. Firms with even higher costs stay out of the market because they would incur losses at a price of P$. Proﬁts earned by the intramarginal ﬁrms can persist in the long run, however, because they reﬂect a return to a unique resource—low-cost land. Free entry cannot erode these proﬁts even over the long term. The sum of these long-run proﬁts constitutes long-run producer surplus, as given by area P$EB in Figure 12.11d. Equivalence of these areas can be shown by recognizing that each point in the supply curve in Figure 12.11d represents minimum average cost for some ﬁrm. For each such ﬁrm, P AC represents proﬁts per unit of output. Total long-run proﬁts can then be computed by summing over all units of output.12 & 11See David Ricardo, The Principles of Political Economy and Taxation (1817; reprinted London: J. M. Dent and Son, 1965), chap. 2 and chap. 32. 12More formally, suppose that ﬁrms are indexed by i (i In the long-run equilibrium, Q$ also the inverse of the supply function (competitive price as a function of quantity supplied) is given by P ACi and P$ the indexing of ﬁrms, price is determined by the highest cost ﬁrm in the market: P Now, in long-run equilibrium, proﬁts for ﬁrm i are given by 1,…, n) from lowest to highest cost and that each ﬁrm produces q$. n$q$ (where n$ is the equilibrium number of ﬁrms and Q$ is total industry output).
Suppose P (Q). Because of P (n$q$). P (Q$) P (iq$) ¼ ¼ ¼ ¼ ¼ ¼ ¼ and total proﬁts are given by which is the shaded area in Figure 12.11d. pi ¼ ð P$ & ACiÞ q$, p ¼ n$ 0 ð n$ ¼ 0 ð pi di n$ P$ 0 ð n$ ¼ ð ACiÞ & q$ di P$q$ di & n$ 0 ð ACiq$ di ¼ ¼ P$n$q$ & 0 ð Q$ iq$ P ð q$ di Þ P$Q$ & 0 ð P Q Þ ð dQ, Chapter 12: The Partial Equilibrium Competitive Model 437 FIGURE 12.11 Ricardian Rent Owners of low-cost and medium-cost land can earn long-run proﬁts. Long-run producers’ surplus represents the sum of all these rents—area P$EB in (d). Usually Ricardian rents will be capitalized into input prices. Price P * Price P * MC AC Price P * MC AC (a) Low-cost firm q * Quantity per period q * (b) Medium-cost firm Quantity per period MC AC Price P * B E S D q * (c) Marginal firm Quantity per period (d) The market Q * Quantity per period Capitalization of rents The long-run proﬁts for the low-cost ﬁrms in Figure 12.11 will often be reﬂected in prices for the unique resources owned by those ﬁrms. In Ricardo’s initial analysis, for example, one might expect fertile land to sell for more than an untillable rock pile. Because such prices will reﬂect the present value of all future proﬁts, these proﬁts are said to be ‘‘capitalized’’ into inputs’ prices. Examples of capitalization include such disparate phenomena as the higher prices of nice houses with convenient access for commuters, the high value of rock and sport stars’ contracts, and the lower value of land near toxic waste sites. Notice that in all these cases it is market demand that determines rents—these rents are not traditional input costs that indicate forgone
opportunities. Input supply and long-run producer surplus It is the scarcity of low-cost inputs that creates the possibility of Ricardian rent. If lowcost farmland were available at inﬁnitely elastic supply, there would be no such rent. More generally, any input that is ‘‘scarce’’ (in the sense that it has a positively sloped supply curve to a particular industry) will obtain rents in the form of earning a higher return than would be obtained if industry output were zero. In such cases, increases in output 438 Part 5: Competitive Markets not only raise ﬁrms’ costs (and thereby the price for which the output will sell) but also generate factor rents for inputs. The sum of all such rents is again measured by the area above the long-run supply curve and below equilibrium price. Changes in the size of this area of long-run producer surplus indicate changing rents earned by inputs to the industry. Notice that, although long-run producer surplus is measured using the market supply curve, it is inputs to the industry that receive this surplus. Empirical measurements of changes in long-run producer surplus are widely used in applied welfare analysis to indicate how suppliers of various inputs fare as conditions change. The ﬁnal sections of this chapter illustrate several such analyses. Economic Efﬁciency and Welfare Analysis Long-run competitive equilibria may have the desirable property of allocating resources ‘‘efﬁciently.’’ Although we will have far more to say about this concept in a general equilibrium context in Chapter 13, here we can offer a partial equilibrium description of why the result might hold. Remember from Chapter 5 that the area below a demand curve and above market price represents consumer surplus—the extra utility consumers receive from choosing to purchase a good voluntarily rather than being forced to do without it. Similarly, as we saw in the previous section, producer surplus is measured as the area below market price and above the long-run supply curve, which represents the extra return that productive inputs receive rather than having no transactions in the good. Overall then, the area between the demand curve and the supply curve represents the sum of consumer and producer surplus: It measures the total additional value obtained by market participants by being able to make market transactions in this good. It seems clear that this total area is maximized at the competitive market equilibrium. A graphic proof Figure 12.12 shows a simpli�
��ed proof. Given the demand curve (D) and the long-run supply curve (S), the sum of consumer and producer surplus is given by distance AB for the ﬁrst unit produced. Total surplus continues to increase as additional output is produced—up to the competitive equilibrium level, Q$. This level of production will be achieved when price is at the competitive level, P$. Total consumer surplus is represented by the light shaded area in the ﬁgure, and total producer surplus is noted by the darker shaded area. Clearly, for output levels less than Q$ (say, Q1), total surplus would be reduced. One sign of this misallocation is that, at Q1, demanders would value an additional unit of output at P1, whereas average and marginal costs would be given by P2. Because P1 > P2, total welfare would clearly increase by producing one more unit of output. A transaction that involved trading this extra unit at any price between P1 and P2 would be mutually beneﬁcial: Both parties would gain. The total welfare loss that occurs at output level Q1 is given by area FEG. The distribution of surplus at output level Q1 will depend on the precise (nonequilibrium) price that prevails in the market. At a price of P1, consumer surplus would be reduced substantially to area AFP1, whereas producers might gain because producer surplus is now P1 FGB. At a low price such as P2 the situation would be reversed, with producers being much worse off than they were initially. Hence the distribution of the welfare losses from producing less than Q$ will depend on the price at which transactions are FIGURE 12.12 Competitive Equilibrium and Consumer/Producer Surplus Chapter 12: The Partial Equilibrium Competitive Model 439 At the competitive equilibrium (Q$), the sum of consumer surplus (shaded lighter gray) and producer surplus (shaded darker) is maximized. For an output level Q1 < Q$, there is a deadweight loss of consumer and producer surplus that is given by area FEG. Price A P1 P * P2 B F G E S D 0 Q1 Q * Quantity per period conducted. However, the size of the total loss is given by FEG, regardless of the price settled upon.13 A mathematical proof Mathematically, we choose Q to maximize consumer surplus producer surplus [U(Q) PQ] PQ & dQ dQ 3 5
12.59 ð Þ where U(Q) is the utility function of the representative consumer and P(Q) is the longAC run supply relation. In long-run equilibria along the long-run supply curve, P(Q) MC. Maximization of Equation 12.59 with respect to Q yields ¼ U 0(Q) P(Q) AC MC, ¼ ¼ so maximization occurs where the marginal value of Q to the representative consumer is equal to market price. But this is precisely the competitive supply–demand equilibrium because the demand curve represents consumers’ marginal valuations, whereas the supply curve reﬂects marginal (and, in long-term equilibrium, average) cost. ¼ ¼ (12:60) 13Increases in output beyond Q$ also clearly reduce welfare. 440 Part 5: Competitive Markets Applied welfare analysis The conclusion that the competitive equilibrium maximizes the sum of consumer and producer surplus mirrors a series of more general economic efﬁciency ‘‘theorems’’ we will examine in Chapter 13. Describing the major caveats that attach to these theorems is best delayed until that more extended discussion. Here we are more interested in showing how the competitive model is used to examine the consequences of changing economic conditions on the welfare of market participants. Usually such welfare changes are measured by looking at changes in consumer and producer surplus. In the ﬁnal sections of this chapter, we look at two examples. EXAMPLE 12.6 Welfare Loss Computations Use of consumer and producer surplus notions makes possible the explicit calculation of welfare losses from restrictions on voluntary transactions. In the case of linear demand and supply curves, this computation is especially simple because the areas of loss are frequently triangular. For example, if demand is given by QD ¼ 10 & P (12:61) and supply by P 2, QS ¼ then market equilibrium occurs at the point P$ 3 would create a gap between what demanders are willing to pay (PD ¼ 7) and what suppliers require (PS ¼ 5). The welfare loss from restricting transactions is given by a triangle with a base of 2 ( 5) and a height of 1 (the difference between Q$ and Q). Hence the welfare loss is $1 if P is measured in dollars per unit and Q is measured in units. More generally, the loss will be measured in the units in which P Æ Q is measured. 4
to purchase Q4, there will be a shortage given by Q4 & Welfare evaluation The welfare consequences of this price-control policy can be evaluated by comparing consumer and producer surplus measures prevailing under this policy with those that would have prevailed in the absence of controls. First, the buyers of Q1 gain the consumer surplus given by area P3CEP1 because they can buy this good at a lower price than would exist in an uncontrolled market. This gain reﬂects a pure transfer from producers out of the amount of producer surplus that would exist without controls. What current consumers have gained from the lower price, producers have lost. Although this transfer does not represent a loss of overall welfare, it does clearly affect the relative well-being of the market participants. Second, the area AE 0C represents the value of additional consumer surplus that would have been attained without controls. Similarly, the area CE0E reﬂects additional producer surplus available in the uncontrolled situation. Together, these two areas (i.e., area AE 0E) represent the total value of mutually beneﬁcial transactions that are prevented by the government policy of controlling price. This is, therefore, a measure of the pure welfare costs of that policy. Disequilibrium behavior The welfare analysis depicted in Figure 12.13 also suggests some of the types of behavior that might be expected as a result of the price-control policy. Assuming that observed market outcomes are generated by Q(P1Þ ¼ min ½ QD(P1), QS(P1)], (12:65) suppliers will be content with this outcome, but demanders will not because they will be forced to accept a situation of excess demand. They have an incentive to signal their dissatisfaction to suppliers through increasing price offers. Such offers may not only tempt existing suppliers to make illegal transactions at higher than allowed prices but may also encourage new entrants to make such transactions. It is this kind of activity that leads to the prevalence of black markets in most instances of price control. Modeling the resulting transactions is difﬁcult for two reasons. First, these may involve non–price-taking behavior because the price of each transaction must be individually negotiated rather than set by ‘‘the market.’’ Second, nonequilibrium transactions will often involve imperfect information. Any pair of market participants will usually not know what other transactors are doing, although such actions may affect their welfare by changing the options available. Some progress has
been made in modeling such disequilibrium behavior using game theory techniques (see Chapter 18). However, other than the obvious prediction that transactions will occur at prices above the price ceiling, no general results have been obtained. The types of black-market transactions undertaken will depend on the speciﬁc institutional details of the situation. Tax Incidence Analysis The partial equilibrium model of competitive markets has also been widely used to study the impact of taxes. Although, as we will point out, these applications are necessarily limited by their inability to analyze tax effects that spread through many markets, they do provide important insights on a number of issues. Chapter 12: The Partial Equilibrium Competitive Model 443 A mathematical model of tax incidence The effect of a per-unit tax can be most easily studied using the mathematical model of supply and demand that was introduced previously. Now, however, we need to make a distinction between the price paid by demanders (PD) and the price received by suppliers (PS) because a per-unit tax (t) introduces a ‘‘wedge’’ between these two magnitudes: PS ¼ If we let the demand and supply functions for this taxed good be given by D(PD) and S(PS), respectively, then equilibrium requires that PD & (12:66) t. Differentiation with respect to the tax rate, t, yields: D PDÞ ¼ ð S PSÞ ¼ ð S PD & ð t : Þ DP dPD dt ¼ SP dPD dt & SP: Rearranging terms then produces the ﬁnal result that dPD dt ¼ SP SP & DP ¼ eS eS &, eD (12:67) (12:68) (12:69) where eS and eD represent the price elasticities of supply and demand and the ﬁnal equation is derived by multiplying both numerator and denominator by P/Q. A similar set of manipulations for the change in supply price gives Because eD + 0 and eS ( dPS dt ¼ eD eS & : eD 0, these calculations provide the obvious results dPD dt ( dPS dt + 0, 0: (12:70) (12:71) (12:72) If eD ¼ 1 and the per-unit tax is com0 (demand is perfectly inelastic), then dPD/dt 1 and the tax
ities of demand and supply in the market. A linear approximation to the size of this deadweight loss triangle for a small tax, t, is given by: DW 0:5t ¼ & dQ dt ’ t 0:5t2 dQ dt : ¼ & (12:74) Here the negative sign is needed because dQ/dt < 0, and we wish our deadweight loss ﬁgure to be positive. Now, by deﬁnition, the price elasticity of demand at the initial equilibrium (P0, Q0) is eD ¼ dQ dP ’ P0 Q0 ¼ dQ=dt dP=dt ’ P0 Q0 or dQ dt ¼ eD dP dt ’ Q0 P0 : (12:75) Chapter 12: The Partial Equilibrium Competitive Model 445 Thus, we can combine Equations 12.74, 12.75, and 12.69 to get a ﬁnal expression for the deadweight loss of this tax: DW 0:5t2 ¼ & eDeS eS & eD ’ Q0 P0 ¼ & 0:5 t P0! " 2 eDeS eS & eD P0Q0: (12:76) Clearly, deadweight losses are zero in cases in which either eD or eS is zero because then the tax does not alter the quantity of the good traded. More generally, deadweight losses are smaller in situations where eD or eS is small. In principle, Equation 12.76 can be used to evaluate the deadweight losses accompanying any complex tax system. This information might provide some insights on how a tax system could be designed to minimize the overall ‘‘excess burden’’ involved in collecting a needed amount of tax revenues (see Problems 12.9 and 12.10). Notice also that DW is proportional to the square of the tax rate—marginal excess burden increases with the tax rate. Transaction costs Although we have developed this discussion in terms of tax incidence theory, models incorporating a wedge between buyers’ and sellers’ prices have a number of other applications in economics. Perhaps the most important of these involve costs associated with making market transactions. In some cases these costs may be explicit. Most real estate transactions, for example, take place through a third-party broker, who charges a fee for the service of
bringing buyer and seller together. Similar explicit transaction fees occur in the trading of stocks and bonds, boats and airplanes, and practically everything that is sold at auction. In all these instances, buyers and sellers are willing to pay an explicit fee to an agent or broker who facilitates the transaction. In other cases, transaction costs may be largely implicit. Individuals trying to purchase a used car, for example, will spend considerable time and effort reading classiﬁed advertisements and examining vehicles, and these activities amount to an implicit cost of making the transaction. EXAMPLE 12.7 The Excess Burden of a Tax In Example 12.6 we examined the loss of consumer and producer surplus that would occur if automobile sales were cut from their equilibrium level of 12.8 (million) to 11 (million). An auto tax of $2,640 (i.e., 2.64 thousand dollars) would accomplish this reduction because it would introduce exactly the wedge between demand and supply price that was calculated previously. Because we have assumed eD ¼ & 1.0 in Example 12.6 and because initial spending on automobiles is approximately $126 (billion), Equation 12.76 predicts that the excess burden from the auto tax would be 1.2 and eS ¼ DW 0:5 ¼ 2:64 9:87 2 1:2 2:2! " ¼ 126 2:46: (12:77)! " This loss of 2.46 billion dollars is approximately the same as the loss from emissions control calculated in Example 12.6. It might be contrasted to total tax collections, which in this case amount to $29 billion ($2,640 per automobile times 11 million automobiles in the post-tax equilibrium). Here, the deadweight loss equals approximately 8 percent of total tax revenues collected. Marginal burden. An incremental increase in the auto tax would be relatively more costly in terms of excess burden. Suppose the government decided to round the auto tax upward to a ﬂat $3,000 per car. In this case, car sales would drop to approximately 10.7 (million). Tax collections would amount to $32.1 billion, an increase of $3.1 billion over what was computed previously. 446 Part 5: Competitive Markets Equation 12.76 can be used to show that deadweight losses now amount to $3.17 billion—an increase of $0.71 billion above the losses experienced with the lower tax. At the margin, additional deadweight losses amount to approximately
23 percent (0.72/3.1) of additional revenues collected. Hence marginal and average excess burden computations may differ signiﬁcantly. QUERY: Can you explain intuitively why the marginal burden of a tax exceeds its average burden? Under what conditions would the marginal excess burden of a tax exceed additional tax revenues collected? To the extent that transaction costs are on a per-unit basis (as they are in the real estate, securities, and auction examples), our previous taxation example applies exactly. From the point of view of the buyers and sellers, it makes little difference whether t represents a per-unit tax or a per-unit transaction fee because the analysis of the fee’s effect on the market will be the same. That is, the fee will be shared between buyers and sellers depending on the speciﬁc elasticities involved. Trading volume will be lower than in the absence of such fees.15 A somewhat different analysis would hold, however, if transaction costs were a lump-sum amount per transaction. In that case, individuals would seek to reduce the number of transactions made, but the existence of the charge would not affect the supply–demand equilibrium itself. For example, the cost of driving to the supermarket is mainly a lump-sum transaction cost on shopping for groceries. The existence of such a charge may not signiﬁcantly affect the price of food items or the amount of food consumed (unless it tempts people to grow their own), but the charge will cause individuals to shop less frequently, to buy larger quantities on each trip, and to hold larger inventories of food in their homes than would be the case in the absence of such a cost. Effects on the attributes of transactions More generally, taxes or transaction costs may affect some attributes of transactions more than others. In our formal model, we assumed that such costs were based only on the physical quantity of goods sold. Therefore, the desire of suppliers and demanders to minimize costs led them to reduce quantity traded. When transactions involve several dimensions (such as quality, risk, or timing), taxes or transaction costs may affect some or all of these dimensions—depending on the precise basis on which the costs are assessed. For example, a tax on quantity may cause ﬁrms to upgrade product quality, or informationbased transaction costs may encourage ﬁrms to produce less risky, standardized commodities. Similarly, a per-transaction cost (travel costs of getting to the store) may
cause individuals to make fewer but larger transactions (and to hold larger inventories). The possibilities for these various substitutions will obviously depend on the particular circumstances of the transaction. We will examine several examples of cost-induced changes in attributes of transactions in later chapters.16 15This analysis does not consider possible beneﬁts obtained from brokers. To the extent that these services are valuable to the parties in the transaction, demand and supply curves will shift outward to reﬂect this value. Hence trading volume may expand with the availability of services that facilitate transactions, although the costs of such services will continue to create a wedge between sellers’ and buyers’ prices. 16For the classic treatment of this topic, see Y. Barzel, ‘‘An Alternative Approach to the Analysis of Taxation,’’ Journal of Political Economy (December 1976): 1177–97. Chapter 12: The Partial Equilibrium Competitive Model 447 • If shifts in long-run equilibrium affect input prices, this will also affect the welfare of input suppliers. Such welfare changes can be measured by changes in long-run producer surplus. • The twin concepts of consumer and producer surplus provide useful ways of measuring the welfare impact on market participants of various economic changes. Changes in consumer surplus represent the monetary value of changes in consumer utility. Changes in producer in the monetary returns that inputs receive. represent changes surplus • The competitive model can be used to study the impact of various economic policies. For example, it can be used to illustrate the transfers and welfare losses associated with price controls. • The competitive model can also be applied to study taxation. The model illustrates both tax incidence (i.e., who bears the actual burden of a tax) and the welfare losses associated with taxation (the excess burden). Similar conclusions can be derived by using the competitive model to study transaction costs. SUMMARY In this chapter we developed a detailed model of how the equilibrium price is determined in a single competitive market. This model is basically the one ﬁrst fully articulated by Alfred Marshall in the latter part of the nineteenth century. It remains the single most important component of all of microeconomics. Some of the properties of this model we examined may be listed as follows. • Short-run equilibrium prices are determined by the interaction of what demanders are willing to pay (demand) and what existing ﬁrms are willing to produce (supply). Both demanders and suppliers act as price-takers in making their respective decisions. •
In the long run, the number of ﬁrms may vary in response to proﬁt opportunities. free entry is assumed, then ﬁrms will earn zero economic proﬁts over the long run. Therefore, because ﬁrms also maximize proﬁts, the long-run equilibrium condition is P MC AC. ¼ If ¼ • The shape of the long-run supply curve depends on how the entry of new ﬁrms affects input prices. If entry has no impact on input prices, the long-run supply curve will be horizontal (inﬁnitely elastic). If entry increases input prices, the long-run supply curve will have a positive slope. PROBLEMS 12.1 Suppose there are 100 identical ﬁrms in a perfectly competitive industry. Each ﬁrm has a short-run total cost function of the form C q ð Þ ¼ 1 300 q3 þ 0:2q2 4q þ þ 10: a. Calculate the ﬁrm’s short-run supply curve with q as a function of market price (P). b. On the assumption that there are no interaction effects among costs of the ﬁrms in the industry, calculate the short-run industry supply curve. c. Suppose market demand is given by Q 200P ¼ & þ 8,000. What will be the short-run equilibrium price–quantity combination? 12.2 Suppose there are 1,000 identical ﬁrms producing diamonds. Let the total cost function for each ﬁrm be given by where q is the ﬁrm’s output level and w is the wage rate of diamond cutters. C(q, w) q2 ¼ þ wq, a. If w 10, what will be the ﬁrm’s (short-run) supply curve? What is the industry’s supply curve? How many diamonds will ¼ be produced at a price of 20 each? How many more diamonds would be produced at a price of 21? b. Suppose the wages of diamond cutters depend on the total quantity of diamonds produced, and suppose the form of this relationship is given by here Q represents total industry output, which is 1,000 times the output of the typical ﬁrm. 0.002Q ; w ¼ 448 Part 5: Competitive Markets In this situation
, show that the ﬁrm’s marginal cost (and short-run supply) curve depends on Q. What is the industry supply curve? How much will be produced at a price of 20? How much more will be produced at a price of 21? What do you conclude about the shape of the short-run supply curve? 12.3 A perfectly competitive market has 1,000 ﬁrms. In the very short run, each of the ﬁrms has a ﬁxed supply of 100 units. The market demand is given by 160,000 Q ¼ & 10,000P. a. Calculate the equilibrium price in the very short run. b. Calculate the demand schedule facing any one ﬁrm in this industry. c. Calculate what the equilibrium price would be if one of the sellers decided to sell nothing or if one seller decided to sell 200 units. d. At the original equilibrium point, calculate the elasticity of the industry demand curve and the elasticity of the demand curve facing any one seller. Suppose now that, in the short run, each ﬁrm has a supply curve that shows the quantity the ﬁrm will supply (qi) as a function of market price. The speciﬁc form of this supply curve is given by Using this short-run supply response, supply revised answers to (a)–(d). qi ¼ & 200 þ 50P. 12.4 A perfectly competitive industry has a large number of potential entrants. Each ﬁrm has an identical cost structure such that long-run average cost is minimized at an output of 20 units (qi ¼ 20). The minimum average cost is $10 per unit. Total market demand is given by 1,500 Q ¼ & 50P. a. What is the industry’s long-run supply schedule? b. What is the long-run equilibrium price (P$)? The total industry output (Q$)? The output of each ﬁrm (q$)? The number of ﬁrms? The proﬁts of each ﬁrm? c. The short-run total cost function associated with each ﬁrm’s long-run equilibrium output is given by & Calculate the short-run average and marginal cost function. At what output level does short-run average cost reach a minimum? ¼ þ C(q) 0.5
q2 10q 200. d. Calculate the short-run supply function for each ﬁrm and the industry short-run supply function. e. Suppose now that the market demand function shifts upward to Q 2,000 ¼ part (b) for the very short run when ﬁrms cannot change their outputs. & 50P. Using this new demand curve, answer f. In the short run, use the industry short-run supply function to recalculate the answers to (b). g. What is the new long-run equilibrium for the industry? 12.5 Suppose that the demand for stilts is given by 1,500 Q ¼ & 50P and that the long-run total operating costs of each stilt-making ﬁrm in a competitive industry are given by C(q) ¼ 0.5q2 10q. & Entrepreneurial talent for stilt making is scarce. The supply curve for entrepreneurs is given by where w is the annual wage paid. QS ¼ 0.25w, Chapter 12: The Partial Equilibrium Competitive Model 449 Suppose also that each stilt-making ﬁrm requires one (and only one) entrepreneur (hence the quantity of entrepreneurs hired is equal to the number of ﬁrms). Long-run total costs for each ﬁrm are then given by C(q, w) 0.5q2 ¼ 10q w. þ & a. What is the long-run equilibrium quantity of stilts produced? How many stilts are produced by each ﬁrm? What is the long-run equilibrium price of stilts? How many ﬁrms will there be? How many entrepreneurs will be hired, and what is their wage? b. Suppose that the demand for stilts shifts outward to How would you now answer the questions posed in part (a)? c. Because stilt-making entrepreneurs are the cause of the upward-sloping long-run supply curve in this problem, they will receive all rents generated as industry output expands. Calculate the increase in rents between parts (a) and (b). Show that this value is identical to the change in long-run producer surplus as measured along the stilt supply curve. 2,428 Q ¼ & 50P. 12.6 The handmade snuffbox industry is composed of 100 identical ﬁrms, each having short-run total costs given by and short
-run marginal costs given by where q is the output of snuffboxes per day. STC ¼ 0.5q2 10q 5 þ þ SMC q þ ¼ 10, a. What is the short-run supply curve for each snuffbox maker? What is the short-run supply curve for the market as a whole? b. Suppose the demand for total snuffbox production is given by 1,100 Q ¼ & 50P. What will be the equilibrium in this marketplace? What will each ﬁrm’s total short-run proﬁts be? c. Graph the market equilibrium and compute total short-run producer surplus in this case. d. Show that the total producer surplus you calculated in part (c) is equal to total industry proﬁts plus industry short-run ﬁxed costs. e. Suppose the government imposed a $3 tax on snuffboxes. How would this tax change the market equilibrium? f. How would the burden of this tax be shared between snuffbox buyers and sellers? g. Calculate the total loss of producer surplus as a result of the taxation of snuffboxes. Show that this loss equals the change in total short-run proﬁts in the snuffbox industry. Why do ﬁxed costs not enter into this computation of the change in short-run producer surplus? 12.7 The perfectly competitive videotape-copying industry is composed of many ﬁrms that can copy ﬁve tapes per day at an average cost of $10 per tape. Each ﬁrm must also pay a royalty to ﬁlm studios, and the per-ﬁlm royalty rate (r) is an increasing function of total industry output (Q): Demand is given by 0.002Q. r ¼ 1,050 Q ¼ & 50P. a. Assuming the industry is in long-run equilibrium, what will be the equilibrium price and quantity of copied tapes? How many tape ﬁrms will there be? What will the per-ﬁlm royalty rate be? b. Suppose that demand for copied tapes increases to 1,600 Q ¼ & 50P. 450 Part 5: Competitive Markets In this case, what is the long-run equilibrium price and quantity for copied tapes? How many tape ﬁrms are there? What is the per-ﬁ
lm royalty rate? c. Graph these long-run equilibria in the tape market, and calculate the increase in producer surplus between the situations described in parts (a) and (b). d. Show that the increase in producer surplus is precisely equal to the increase in royalties paid as Q expands incrementally from its level in part (b) to its level in part (c). e. Suppose that the government institutes a $5.50 per-ﬁlm tax on the ﬁlm-copying industry. Assuming that the demand for copied ﬁlms is that given in part (a), how will this tax affect the market equilibrium? f. How will the burden of this tax be allocated between consumers and producers? What will be the loss of consumer and producer surplus? g. Show that the loss of producer surplus as a result of this tax is borne completely by the ﬁlm studios. Explain your result intuitively. 12.8 The domestic demand for portable radios is given by & where price (P) is measured in dollars and quantity (Q) is measured in thousands of radios per year. The domestic supply curve for radios is given by ¼ Q 5,000 100P, 150P. Q ¼ a. What is the domestic equilibrium in the portable radio market? b. Suppose portable radios can be imported at a world price of $10 per radio. If trade were unencumbered, what would the new market equilibrium be? How many portable radios would be imported? c. If domestic portable radio producers succeeded in having a $5 tariff implemented, how would this change the market equilibrium? How much would be collected in tariff revenues? How much consumer surplus would be transferred to domestic producers? What would the deadweight loss from the tariff be? d. How would your results from part (c) be changed if the government reached an agreement with foreign suppliers to ‘‘voluntarily’’ limit the portable radios they export to 1,250,000 per year? Explain how this differs from the case of a tariff. 12.9 Suppose that the market demand for a product is given by QD ¼ given by C(q) bq2. aq k ¼ þ þ BP. Suppose also that the typical ﬁrm’s cost function is A & a. Compute the long-run equilibrium output and price for the typical ﬁrm in this market. b. Calculate the equilibrium
number of ﬁrms in this market as a function of all the parameters in this problem. c. Describe how changes in the demand parameters A and B affect the equilibrium number of ﬁrms in this market. Explain your results intuitively. d. Describe how the parameters of the typical ﬁrm’s cost function affect the long-run equilibrium number of ﬁrms in this example. Explain your results intuitively. Analytical Problems 12.10 Ad valorem taxes Throughout this chapter’s analysis of taxes we have used per-unit taxes—that is, a tax of a ﬁxed amount for each unit traded in the market. A similar analysis would hold for ad valorem taxes—that is, taxes on the value of the transaction (or, what amounts to the same thing, proportional taxes on price). Given an ad valorem tax rate of t (t 0.05 for a 5 percent tax), the gap between the price demanders pay and what suppliers receive is given by PD ¼ t)PS. (1 ¼ þ Chapter 12: The Partial Equilibrium Competitive Model 451 a. Show that for an ad valorem tax d ln PD dt ¼ eS eS & eD and d ln PS dt ¼ eD eS & : eD b. Show that the excess burden of a small tax is c. Compare these results with those derived in this chapter for a unit tax. DW 0:5 ¼ & eDeS eS & eD t2P0Q0: 12.11 The Ramsey formula for optimal taxation The development of optimal tax policy has been a major topic in public ﬁnance for centuries.17 Probably the most famous result in the theory of optimal taxation is due to the English economist Frank Ramsey, who conceptualized the problem as how to structure a tax system that would collect a given amount of revenues with the minimal deadweight loss.18 Speciﬁcally, suppose there are n goods (xi with prices pi) to be taxed with a sequence of ad valorem taxes (see Problem 12.10) whose rates n are given by ti (i 1 tipixi. Ramsey’s problem is for a ﬁxed T to choose i ¼ tax rates that will minimize total deadweight loss DW a. Use the Lagrange multiplier method to show that the solution to Ramsey’s problem requires ti ¼ b

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