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Berkeley ECON 101A - ECON 101A Midterm

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Econ101A: Fall 1996Goldman/Chan/WangAnswers to Midterm 21. (a) The short-run cost function is the minimum expenditure level needed to produce a givenlevel of output when at least one of the factor inputs is held constant.(b) Let A and B be lotteries. Now for any lottery C, de¯ne D as the lottery where you playA with probability p and C with probability 1-p and de¯ne B as the lottery where youplay B with probability p and C with probability 1-p. The independence axiom statesthat if an individual prefers A to B then for any lottery C, the individual prefers thelottery D to E.(c) A nash equilibrium is a set of strategies where no individual can improve her payo®s bydeviating from her strategy when everyone else's strategy is ¯xed.(d) A Cournot equilibrium is the nash equilibrium of a game where players choose outputwithout observing the output levels of the other players.2. (a) FalseAlthough it is true that all individuals who prefer A to B are risk averse, it is not truethat all risk-averse individuals prefer A to B. B has greater expected value than A butin some sense A is riskier than B. Consequently, very risk-averse individuals prefer Ato B (e.g. consider u(w) =pw). However, individuals who are not very risk aversemay prefer B because its higher expected payo®s more than compensate for the greaterriskiness (e.g. u(w) = ln w).(b) FalseSuppose ® > ¯ and consider any t > 1 thenF(tK; tL) = ((tK)®+ (tL)®)1¯= t®¯(K®+ L®)1¯< t (K®+ L®)1¯for ® > ¯= tF(K; L)Therefore, when ® > ¯, the production function exhibits increasing returns to scale.(c) FalseCost minimization implies that the marginal product of labor and hence the marginalcosts will be equal across plants. Consequently, if the plants have identical productionfunctions then the statement will be true for production function exhibiting constantreturns to scale.However, the statement will not be true in general. For instance, for all Cobb-Douglasproduction functions that exhibit deminishing marginal productivilty of labor and in-creasing returns to scale, the plant with twice the capital will produce more than twiceof the output of the other plant.1(d) FalseA monopoly chooses the pro¯t-maximizing output level q¤which occurs when MC(q¤) =MR(q¤). This output level can occur when AC is decreasing, increasing or at its mini-mum as illustrated by the following graphs:(e) TrueBy de¯nition of nash equilibrium, no player has incentive to deviate from her strategytaking the other player's strategy is ¯xed.First, consider Player 1. We can describe any possible strategy that Player 1 playsby (p U, (1-p) D) where p = probability Player 1 chooses U and 1-p= probability Player1 chooses D.Given Player 2's strategy of playing L with probability14and playing R with proba-bility34, Player 1's payo® from playing an arbitrary strategy isEU1(pU; (1 ¡ p)D) = p ¢µ14¢3 +34¢ 0¶+ (1 ¡ p) ¢µ14¢0 +34¢ 1¶=34:Therefore, Player 1 has expected payo®s of34for any strategy she plays. Consequently,given Player 2's strategy, Player 1 can do no better than play U with probability14andto choose D with probability34.Similarly, we can show that Player 2 will have no incentive to deviate if Player 1 playsU with probability14and to choose D with probability34.Therefore, these strategies constitute a nash equilibrium.3. (a) Since the market is perfectly competitive, each ¯rm will choose an output level suchthat P = MC(q) = 2q. There are 10 ¯rms in the market, therefore the market supplyis equal toQ = 10q = 5P: (1)Equating the demand and supply gives: P = 200 and q = 100.The pro¯t of each ¯rm is equal to:¦ = Pq ¡q2= 20000 ¡10000 = 10000:2(b) Let assume that each overseas ¯rm is going to sell 72 units at the new equilibrium. Thedomestic ¯rms are still going to choose q such that P = 2q. Thus, the market supply isequal toQ = 5q + 360 =52P + 360Equating demand and supply gives: P = 240 and qd= 120.Since P = 240 > MC(72) = 144, the overseas ¯rms will use up their quota as weassume.(c) The pro¯t of a foreign ¯rm is equal to¦o= P 72 ¡722= 12096:The pro¯t of the overseas ¯rms are actually higher under the quota system. When theoverseas ¯rms are forced to sell less, the domestic ¯rms are going to sell more. However,since the marginal cost of production is increasing, the increase in output by the domestic¯rms will not be big enough to o®et the reduction in imports. As a result, the totalmarket supply decreases and the market price goes up. The increase in price is bigenough that the pro¯t of the overseas ¯rm go up despite that they are forced to sell less.4. (a) The pro¯t function of the monopolist is given by:¦m= (1 ¡Q)Q ¡ FThe ¯rst order condition implies that@¦m@Q= 0 ) Q = 1=2. The consumer surplus is equal to 1=2:1=2 = 1=8, the monpoly pro¯t is equal to 1=4¡F,and the total surplus is equal to 3=8 ¡F .(b) Let q1and q2be the outputs of ¯rm 1 and 2 respectively. The pro¯t of ¯rm i, i=1,2 isequal to ¦i= (1 ¡ q1¡ q2)qi¡ F . In cournot equilibrium, each ¯rm chooses its outputto maximize pro¯t based on its expectation about the other ¯rm's output. The reactionfunctions of the ¯rms are:q1=1 ¡ ¹q22q2=1 ¡ ¹q12where ¹qiis the ¯rm j's (j 6= i) expectation of ¯rm i's output.In equilibrium, the expectations are correct. Solving the equations gives q1= q2= 1=3:The pro¯t of each ¯rm is equal to 1=9 ¡ F, and the consumer surplus is equal to(1 ¡ P )Q=2 = 2=9. The total surplus is equal to CS + PS = 4=9 ¡ 2F .(c) The monopolist will make a postive pro¯t when F < 1=4 and the duopolists will su®erlosses when F > 1=9. Thus the range of ¯xed cost in which only one ¯rm can producepro¯tably is 1=9 > F > 1=4.(d) The total surplus under monopoly is bigger then that under duopoly and that whenthere is no production when 3=8 ¡ F > 4=9 ¡ 2F and 3=8 ¡F > 0. Thus the range of¯xed cost in which it is socially desirable to have only one ¯rm is 5=72 < F < 3=8.3(e) When 1=4 < F < 3=8, the total surplus is the highest under monopoly but there will beno production because not even a monopolist can make a pro¯t. There are too few ¯rmsin the market. Even under monopoly pricing, the ¯rm cannot extract all the surplus.As a result, it may not have enough incentive to produce.When 5=72 < F < 1=9, the total surplus is again the highest under monopoly but sinceduopoly is pro¯table, two ¯rms will produce. There are too many ¯rms in the market.There is a trade o® between pricing e±ciency and production e±ciency


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