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Unformatted text preview: HOMEWORK 10 Sample Solution 1. Consider a single firm with cost function C(q) = 400 + 10q + q2 for q > 0 and c(0) = 0,
where q is the firm’s output. Note the “fixed cost” of 400 can be avoided by producing no output,
so it does enter the ﬁrm’s decision whether to shutdown. a. What is the firm’s marginal cost function?
. . . . . dc
Marglnal cost rs the derlvatlve of the cost functlon, MC(q) = d— = 10 + 2g for q > 0. Note
4 cost is not differentiable at q = 0. [It is not even continuous at q = 0.]
b. What is the firm’s average total cost function? Average total cost is the ratio of total cost to output, ATC(q) = c(q)/q = 400/9;r + 10 + q for
q > 0. c. For which output level is marginal cost equal to average total cost, and what is the
average cost at that output? Setting MC(q) = ATC(q), 10 + 26] = (400/q) + 10 + q, and solving for q, we obtain q2 =
400 or q = 20. The average cost at output 20 is ATC(20) = (400/20) + 10 + 20 = 50. d. For which output prices, p, is it optimal for the firm to produce nothing? Note that C(O) = 0, so no costs are sunk, and all costs are relevant for the decision whether
to shut down. For p < 50, the firm cannot cover its costs, and it is optimal to produce nothing.
For p = 50, the firm could just cover its costs by producing output 20, but output 0 is just as good. Thus the firm finds producing nothing optimal whenever p S 50. e. What is the firm’s supply function? From Part (d), for p < 50, the unique best output is 0, so 309) = 0 for p < 50. For p = 50,
outputs 0 and 20 tie for best, so 3(50) is 0 and 20. (It is not a function in the mathematical sense.)
For p > 50, we must solve p = MC(q) for q to find the optimal output. Setting p = 10 + 2g and
solving for q, we obtain 309) = (p12) —~ 5 for p > 50. f. Suppose the government introduces a subsidy scheme under which the firm will be paid
$300 if it stays in business (i .e., produces a strictly positive amount of output). What is its new
supply function? The effect of the subsidy is to reduce the net cost of any output q > 0 to c*(q) = (400 —
300) + We] + q2 = 100 + IOq + qz. Marginal cost is the same as without the subsidy, but average
net cost is now 100/q + 10 + q, and marginal cost equals average net cost at q = 10, with average
net cost of 30 at q = 10. For p < 30, the firm cannot cover its net costs, and it is optimai to
produce nothing, so s*(p) = 0 for p < 30. For p = 30, the firm could just cover its costs by
producing output 10, but output 0 is just as good. Thus 3*(30) = O and 10. For p > 30, we must solve p = MC(q) for q to find the optimal output. Setting p = 10 + Zq and solving for q, we obtain
3*(p) = (19/2) — 5 forp > 30. g. For each output price, p, how much extra output does the firm produce under the subsidy
scheme compared to the unsubsidized original situation? The additional output is s*(p) — s(p). For p < 30, both supply functions are 0, and the
difference is O. Forp > 50, both supply functions are (p12) — 5, and the difference is 0. For 30 <p
< 50, s*(p) = (p12) — 5 while s(p) = O, and the extra output is equal to (p12)— 5. Atp = 30, the
extra output could be either 0 or 10 (because 3*(30) has two values). At p = 50, the extra output
could be either 0 or 20 (because 3(50) has two values). 2. Suppose there are 10 firms identical to the firm in Problem 1, and aggregate demand is
D(p) = 1750 — 4p.
a. What are the competitive equilibrium price and quantity without the subsidy? The aggregate supply function is the sum of the supply functions for the individual firms.
At price p = 50, each firm was indifferent between two different outputs, so the sum must take
account of all the different possible combinations of optimal choices. Because the firms are
identical, for prices other than p = 50, S(p) = 103(p). Thus aggregate supply is 0 ifp < 50
S(p) = 0, or 20, or 40, or ..., or 180, or 200 ifp = 50. 5p- 50 ifp > 50 At p < 50, S(p) = 0 while D(p) > 0, so the equilibrium price must be at least 50. At p = 50,
S(50) is no more than 200 while D(50) = 1550 > 200. Thus the equilibrium price must exceed 50.
At p > 50, S(p) = 5p — 50 = 1750 —— 4p = D(p) at equilibrium price p* = 200 > 50. The
corresponding aggregate quantity is Q* = D(p*) = 1750 — 4(200) = 950. Each firm produces
5(50) = 95 units of output, and obtains profit p*s(p*) — c(s(p*)) = (200)(95) — [400 + (10)(95) +
(95)2] = 19000 — 10375 = 8625. b. What are the competitive equilibrium price and quantity with the subsidy? The analysis is similar to Part (a). The aggregate supply is
0 if p < 30 S*(p) = 0, or 10, or 20, or ..., or 90, or 100 ifp = 30
Sp—SO ifp>30 and the equilibrium price, aggregate quantity, and quantity per firm are the same as in Part (a). c. What difference does the subsidy make in terms of the properties of the equilibrium? The only change is that each firm has profit $8625 + $300 = $8925 instead of $8625.
(This is exactly counterbalanced by a government expenditure of $300 per firm.) The subsidy
was unnecessary because demand was already strong enough to keep all 10 firms in business. 3. Suppose there are 10 ﬁrms identical to the firm in Problem 1, and aggregate demand is
D(p) = 350 — 5p.
a. What are the competitive equilibrium price and quantity without the subsidy? Aggregate supply is the same as in 2a. At p < 50, S(p) = 0 while D(p) > 0, so the
equilibrium price must be at least 50. If we use the formula for supply that applies when p > 50,
and set S(p) = D(p), we obtain 5p — 50 = 350 —— 5p, with solution p a 40. This is not an
equilibrium price! The formula we used for supply was correct for p > 50, but 40 < 50. Thus the
equilibrium price must be 50. At p = 50, S(50) is any multiple of 20 between 0 and 200 while
D(50) = 100, which is a possible value for supply at price 50. The equilibrium price is p* = 50,
with corresponding aggregate quantity Q* «1 D(p*) = 350 — 5(50) = 100. Each firm produces an
output from s(50), but both 0 and 20 are allowed. To obtain the required aggregate output of 100,
five of the ten firms must produce output 20 while the other five firms produce output 0. Each
firm, whether producing output 20 or output 0, obtains profit 0. b. What are the competitive equilibrium price and quantity with the subsidy? Aggregate supply is the same as in 2b. Atp < 30, S(p) = 0 while D(p) > 0, so the
equilibrium price must be at least 30. At p = 30, 3*(30) is no more than 100 while D(30) = 200 >
100. Thus the equilibrium price must exceed 30. Atp > 30, S*(p) = 5p — 50 = 350 — Sp z D(p) at
equilibrium price p* = 40 > 30. The corresponding aggregate quantity is Q* = D(p*) = 350 —
5(40) = 150. Each firm produces 3*(40) = 15 units of output, and obtains proﬁt p*s*(p*) —
c(s*(p*)) = (40)(15) — [100 + (10)(15) + (15):] = 600 — 475 = 125. c. What difference does the subsidy make in terms of the properties of the equilibrium? The equilibrium price is lower ($40 instead of $50) and the aggregate quantity is larger
(150 instead of 100). The number of firms producing output is larger (10 instead of 5), the output
per active firm is smaller (1501' 10 = 15 instead of 100/5 2 20) and the per—firm net profit is larger
($125 instead of $0). The total subsidy paid by the government is (10)($300) = $3000. There is a
$500 reduction in total surplus ($1250 consumer surplus gain plus $1250 producer surplus gain
minus $3000 government expenditure). Note that having 10 ﬁrms produce 15 units each does not minimize the production
cost for 150 units! Using N ﬁrms each producing ISO/N [with the other firms producing 0]
would cost N[400 + 10(150/N) + (ISO/NY] = 400N + 1500 + 22500/N. It would be cheaper to
have 8 ﬁrms produce 18.75 each. The subsidy has distorted the firms’ incentives and led to
a productively inefficient outcome. ...
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- Spring '08