Suggested solutions: Final Exam Winter 2004, by M-O Moisan-Plante

Section A.

#1. The monopolist will maximize its profits: PQ – C. So: Max (20000-100Q)Q – 5000Q-50Q^2. The FOC is: 20000-200Q-5000-100Q=0. Q*=50, P*=15000.

#2. The production function is a case of perfect substitutes. The firm will use the input with the highest marginal product per $ spent. Here MPK/r and MPL/w are equal. Any combination such that 10K+4L=100 will

do.

#3. The minimum efficiency scale is the smallest quantity at which the long-run average cost curve attains its minimum. Here we are given the short-run total cost function. The short-run average cost function is declining in Q: SAV = 2/Q + 4. A function like Q=k^0.5L, will have a short-run and a long-run average cost function declining in Q.

#4. The production functions for which the firm wouldn’t hire more labor if the wage rate decreases are the perfect complements (as the firm always use the inputs in the same proportions) and the perfect substitutes if the price change is not large enough to induce the firm to use only labor (assuming it was using only capital before the price change). It is also possible to observe this with a quasi-linear production fuction. This was for the long-run. In the short-run, if capital is fixed, the firm will not hire more labor.

#5. If X=1000, the firm will choose K* and L* such that: L*=1000 and 10K*=1000. Total costs are: rK*+wL*.

Section B.

#1. a. I got P* = 150 and Q* = 500.

b. P^d = 1.1P^s. Solving I got P^d = 150.65, Q*=370. The producers receive P’=0.90909 * 150.65 = 136.96. Consumer surplus: 342.25, producer surplus: 6838 and tax revenues: $5065 (approximatively due to rounding).

c. The market price will be: P* = 150.65. Consumers are indifferent – they pay the same price - but producers are better off since they get the surplus that was paid in taxes before.

#2. a. I found 2 Nash equilibria: A1-A2 with outcome (10,10) and C1-B2 with outcome (8,12).

b. There is only one Nash equilibrium here (hence subgame perfect), A1-B2 played and A2 if B1.

c. There is only one Nash equilibrium here (hence subgame perfect), B2-B1 played and A1 if A2.

#3. a. The reaction function of player 1 is found by solving: Max (2000-50y1-50y2)y1-10000-20y1. Taking the FOC, I got: y1 = (1980-50y2)/100. Similarly: y2=(1995-50y1)/100.

b. Solving the system of equations found in part a., I got y1*=13.1, y2*=13.4, P*=675, profits for 1 = -1419.5 and profits for 2 = -1022. (actually one firm should leave the market).

c. Costs are linear, so only firm 2 will produce as it has the lowest marginal cost. If we set y1=0 in the reaction function of firm 2, we get y2*=19.95, P*=1002.5 and profits = 9900.

#4. a. At the optimum: MPK/r = MPL/w. Substituting the expressions for the marginal products we get: wL=rK. For K*: we isolate K in Q=(rK/w)^0.5K^0.5 and K*=(w/r)^0.5Q. For L*: we isolate L in Q=(wL/r)^0.5L^0.5 and L*=(r/w)^0.5Q. Long-run costs are: rK*+wL* = 2(rw)^0.5Q.

b. The long-run cost function will shift by a factor of (w’/w)^0.5.

c. L* = 10000/256 = 39.0625. SR = 256r+39w = 17360. LR = 2(rw)^0.5*100 = 8000.

#5. a. Q* can be found by minimizing the average long-run cost: Min 4000-50Q+0.2Q^0.2. FOC: -50+0.4Q=0. Q*=125. We can find P* by equating: MC(Q*)=P. P*=875. Finally nQ* = 50000-40P*. N* = 120.

b. nQ* = 60000-40P*. n* = 200.

c. I think there is a typo in the new entrants cost function. But here is the logic: new entrants have higher costs. Find their min AV(Q’) and the new price on the market will be P’=MC(Q’), which will be higher than P*. Find the production done by the 120 old firms, Q’’, by solving: P’=MC(Q’’), the MC function of the old firms. Finally equate (120Q’’ + n’Q’) = 60000-40P’ and solve for n’. The old firm will make positive economic profits, but not the new firms.

I would like to thank the students from the Fall term 2005 who found typos in the previous version of this answer key.

Good luck for the final,