9
PROFIT CHAPTER MAXIMIZATION
Problems in this chapter consist mainly of applications of the P = MC rule for profit maximization by a price-taking firm. A few of the problems (9.29.5) ask students to derive marginal revenue concepts, but this concept is not really used in the monopoly context until Chapter 13. The problems are also concerned only with the construction of supply curves and related concepts since the details of price determination have not yet been developed in the text. Comments on Problems 9.1 9.2 A very simple application of the P = MC rule. Results in a linear supply curve. Easy problem that shows that a tax on profits will not affect the profit-maximization output choice unless it affects the relationship between marginal revenue and marginal cost. Practice with calculating the marginal revenue curve for a variety of demand curves. Uses the MR-MC condition to illustrate third degree price discrimination. Instructors might point out the general result here (which is discussed more fully in Chapter 13) that, assuming marginal costs are the same in the two markets, marginal revenues should also be equal and that implies price will be higher in the market in which demand is less elastic. An algebraic example of the supply function concept. This is a good illustration of why supply curves are in reality only two-dimensional representations of multi-variable functions. An introduction to the theory of supply under uncertainty. This example shows that setting expected price equal to marginal cost does indeed maximize expected revenues, but that, for risk-averse firms, this may not maximize expected utility. Part (d) asks students to calculate the value of better information. A simple use of the profit function with fixed proportions technology. This is a conceptual examination of the effect of changes in output price on input demand. A very brief introduction to the CES profit function. This problem describes some additional mathematical relationships that can be derived from the profit function.
9.3 9.4
9.5
9.6
9.7 9.8 9.9 9.10
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Solutions
9.1 a. MC = C q = 0.2 q + 10 set MC = P = 20, yields q* = 50 b. = Pq C = 1000 800 = 200
c. 9.2
( q) = R ( q) C ( q) With a lump sum tax T ( q) = R( q) C ( q) T R C = 0 = 0 MR = MC , no change q q q
Proportional tax
( q) = (1 t )[ R( q) C ( q)]
MR = MC , no change
= (1 t )( MR MC ) = 0, q
Tax per unit ( q) = R ( q) C ( q) tq = MR MC t = 0 , so MR = MC + t, q is changed: a per unit tax does affect output. q dP q a 2q a = + q (1/ b) = dq b b
9.3
a. q = a + bP, MR = P + q Hence, q = (a + bMR)/2.
Because the distance between the vertical axis and the demand curve is q = a + bP, it is obvious that the marginal revenue curve bisects this distance for any line parallel to the horizontal axis. b. If q = a + bP ; b < 0; P = MR = 2q a b qa b 1 q b
P MR =
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Chapter 9/Profit Maximization y 40
c. Constant elasticity demand curve: q = aPb, where b is the price elasticity of demand. MR = P + q P q = q a
1/ b
( q / a )1 / b + b
Thus, vertical distance = P MR = b < 0)
( q / a )1 / b P (which is positive because = b b
d. If eq,P < 0 (downward-sloping demand curve), then marginal revenue will be less than price. Hence, vertical distance will be given by P MR. dP dP dq , vertical distance is q , and since = b is the slope of dq dP dq 1 the tangent linear demand curve, the distance becomes q as in Part (b). b Since MR = P + q
e. 9.4 Total cost = C = .25q2 = .25(qA + qL)2 qA = 100 2PA PA = 50 qA/2 qL = 100 4PL PL = 25 qL/4
2 2 R A = P A q A = 50q A q A / 2 RL = PLqL = 25q L q L / 4
MRA = 50 qA MCA = .5(qA + qL)
MRL = 25 qL/2 MCL = .5(qA + qL)
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Set MRA = MCA and 50 qA = .5qA + .5qL
MRL = MCL 25 qL = .5qA + .5qL these 2
Solving simultaneously gives qA = 30 qL = 10 9.5 PA = 35 PL = 22.5 = 1050 + 225 400 = 875 C = wl = wq 2 / 4 .
a. Since q = 2 l , q 2 = 4l
Profit maximization requires P = MC = 2wq/4. Solving for q yields q = 2P/w. b. Doubling P and w does not change profit-maximizing output level. = Pq TC = 2P2/w P2/w = P2/w, which is homogeneous of degree one in P and w. c. It is algebraically obvious that increases in w reduce quantity supplied at each given P. 9.6 a. Expected profits = E() = .5[30q C(q)] + .5[20q C(q)] = 25q C(q). Notice 25 = E(P) determines expected profits. For profit maximum set E(P) = MC = q + 5 so q = 20 E() = E(P)q C(q) = 500 400 = 100. b. In the two states of the world profits are P = 30 = 600 400 = 200 P = 20 = 400 400 = 0 and expected utility is given by E (U ) = .5 200 + .5 0 = 7.1 c. Output levels between 13 and 19 all yield greater utility than does q = 20. Reductions in profits from producing less when P is high are compensated for (in utility terms) by increases in profits when P is low. Calculating true maximum expected utility is difficultit is approximately q = 17. d. If can predict P, set P = MC in each state of the world. When P = 30 q = 25 = 212.5, P = 20 q = 15 = 12.5 E() = 112.5 E (U ) = .5 212.5 + .5 12.5 = 9.06 a substantial improvement. 9.7 a. In order for the second order condition for profit maximization to be satisfied, marginal cost must be decreasing which, in this case, requires diminishing returns to scale.
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Chapter 9/Profit Maximization y 42
b. q = 10k 0.5 = 10l 0.5 so k = l = q 2 100 C = vk + wl = q 2 ( v + w) 100 Profit maximization requires P = MC = q( v + w) 50 or q = 50 P (v + w) . ( v, w, P ) = Pq C = 50 P 2 ( v + w) [50 P ( v + w)]2 ( v + w) 100 = 25P 2 ( v + w) . c. If v = 1000, w = 500, P = 600 then q = 20, = 6000 . d. If v = 1000, w = 500, P = 900 then q = 30, = 13500 .
e. 9.8 a. With marginal cost increasing, an increase in P will be met by an increase in q. To produce this extra output, more of each input will be hired (unless an input is inferior). b. The Cobb-Douglas case is best illustrated in two of the examples in Chapter 9. In Example 9.4, the short-run profit function exhibits a positive effect of P on labor demand. A similar result holds in Example 9.5 where holding a third input constant leads to increasing marginal cost. c. l P = [ w] P = 2 Pw = q w . The sign of the final derivative may be negative if l is an inferior input. 9.9 b. Diminishing returns is required if MC is to be increasingthe required second order condition for profit maximization. c. determines how easily firms can adapt to differing input prices and thereby shows the profitability obtainable from a given set of exogenous prices. d. q = P = K (1 ) 1 P 1 ( v1 + w1 ) (1 )( 1) . This supply function shows that does not affect the supply elasticity directly, but it does affect the shift term that involves input prices. Larger values for imply smaller shifts in the supply relationship for given changes in input prices. e. See the results provided in Sydsaeter, Strom, and Berck.
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9.10
a. l v = 2 vw = 2 wv = k w . This shows that cross price effects in input demand are equal. The result is similar to the equality of compensated cross-price effects in demand theory. b. The direction of effect depends on whether capital and labor are substitutable or complementary inputs. c. q w = 2 wP = 2 Pw = l P . This shows that increases in wages have the same effect on reducing output that a fall in the product price has on reducing labor demand. This is, the effects of wages and prices are in some ways symmetrical. d. Because it seems likely that l P > 0 (see Problem 9.8), we can conclude that q w < 0 that is, a tax on labor input should reduce output.
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