EconS 301 Homework #8 Answer key

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1 EconS 0 Homework #8 Answer key Exercise # Monopoly Consider a monopolist facing inverse demand function pp(qq) 5 qq, where qq denotes units of output. Assume that the total cost of this firm is TTTT(qq) 5 + qq. a) Find the monopolist s marginal revenue, and its marginal cost. Answer: MMMM dd(pp(qq) qq) MMMM ddtttt(qq) dd(5qq qq ) dd(5 + qq) 5 6qq and b) Set marginal revenue equal to marginal cost, to find the optimal output for this monopolist qq MM. Answer: Let MRMC, we obtain the optimal output for monopolist: 5 6qq qq MM 6 c) Find the monopoly price, and profits. Find the consumer surplus, and total welfare. Answer: Given the optimal output above, we plug into demand function to obtain the optimal price, that is pp MM 5 qq MM Therefore, the consumer surplus is the triangular area between pp 0 when qq 0 and pp MM, that is CCCC (pp0 pp MM ) qq MM (5.5) And finally, because MC curve is constant, meaning the producer surplus is rectangular the area between pp ss when pp ss MMMM and pp MM,so PPPP (pp MM pp ss ) qq MM (.5 ) Therefore, the total welfare is CS + PPPP See Figure below.

2 d) Assume now that the market operated under perfect competition. Find the equilibrium output. Answer: If market operated under perfect competition, we must have optimal price equals to marginal cost. That is, pp MM MMMM. Then we plug it back into demand function and obtain the optimal output, pp MM 5 qq qq MM e) Find equilibrium price, profits, consumer surplus, and total welfare, under the perfectly competitive market you analyzed in part (e). Answer: The consumer surplus is, CCCC (pp0 pp MM ) qq MM (5 ) And finally, because MC curve is a constant, meaning producer surplus is zero. So W CS f) Compare your results under a monopoly (part c) and those under a perfectly competitive market (part e). Answer: Compared with the results in part c and part e, we can see that the monopoly captured more consumer surplus than in the perfectly competitive market, causing a deadweight loss. DDDDDD WW PPPP WW MMMMMMMMMMMMMMMM (CCCC PPPP + PPPP PPPP ) (CCCC MMMMMMMMMMMMMMMM + PPPP MMMMMMMMMMMMMMMM ) Exercise # Monopoly with constant elasticity demand curve Consider a monopolist facing a constant elasticity demand curve qq(pp) pp. a) Assume that the total cost function is TTTT(qq) 5 + qq. Use the inverse elasticity pricing rule (IEPR) to obtain the profit-maximizing price that this monopolist should charge. Answer: Since given above is the constant elasticity demand curve, then we know that εε QQ,PP. Using IEPR rule, we obtain: PP MMMM PP And,

3 MMMM ddtttt(qq) dd(5 + qq) Therefore, PP PP pp MM 6 b) How would your result in part (a) change if the demand curve changes to qq(pp) pp 5, but still assuming the same cost function as in part (a)? Interpret. Answer: Since given above is the constant elasticity demand curve, then we know that εε QQ,PP 5. Using IEPR rule, we obtain: PP MMMM PP 5 5 And, MMMM ddtttt(qq) dd(5 + qq) Therefore, PP PP 5 ppmm 5 The results above can be interpreted as εε QQ,PP gets bigger the price that monopoly charge will become smaller. In other words, εε QQ,PP and pp MM has negative relationship. c) Assume that the total cost function is TTTT(qq) 5 + (qq). Use the inverse elasticity pricing rule (IEPR) to obtain the profit-maximizing price that this monopolist should charge. Answer: Since given above is the constant elasticity demand curve, then we know that εε QQ,PP. Using IEPR rule, we obtain: PP MMMM PP And, MMMM ddtttt(qq) dd(5 + (qq) ) qq Therefore, PP qq PP 0qq 0 PP Solving for price PP, we plug into demand function to obtain PP 7 0 Solving it, pp MM.0 d) How would your result in part (c) change if the demand curve changes to qq(pp) pp 5, but still assuming the same cost function as in part (c)? Interpret. Answer: Since given above is the constant elasticity demand curve, then we know that εε QQ,PP 5. Using IEPR rule, we obtain:

4 And, PP MMMM PP 5 5 MMMM ddtttt(qq) dd(5 + (qq) ) qq Therefore, PP qq PP 5 PP 0qq 0 We plug demand function into above function and finally have, PP 0(pp 5 ) 0 PP Solving it, pp MM.786 The results above can be interpreted as εε QQ,PP gets bigger the price that monopoly charge will become smaller. In other words, εε QQ,PP and pp MM has negative relationship. Exercise # - Multiplant monopoly Consider a monopoly facing inverse demand function pp(qq) 0 QQ, where QQ qq + qq denotes the monopolist s production across two plants, and. Assume that total cost in plant is given by TTCC (qq ) ( + qq )qq, while that of plant is TTCC (qq ) [ + ( + dd)qq ]qq, where parameter dd 0 represents plant s inefficiency to plant. When dd 0, the total (and marginal) cost of both plants coincide; but when dd > 0, plant has a higher total and marginal cost than plant. a) Write down the monopolist s joint profit maximization problem ππ ππ + ππ. Answer: First, we find our profit for plant. In this setting, our profit can be defined as Similarly, for plant, ππ pp(qq)qq TTCC (qq ) ππ (0 qq qq )qq ( + qq )qq ππ pp(qq)qq TTCC (qq ) ππ (0 qq qq )qq ( + ( + dd)qq )qq To find the joint profit, we add our profit functions together ππ ππ + ππ ππ (0 qq qq )qq ( + qq )qq + (0 qq qq )qq ( + ( + dd)qq )qq ππ (0 qq qq )(qq + qq ) ( + qq )qq ( + ( + dd)qq )qq b) Differentiate with respect to the output in plant, qq, and then in plant, qq. Set each expression equal to zero and use your results to find the optimal production in each plant.

5 Answer: Differentiating our profit function with respect to qq gives us Solving for qq, we obtain qq 0 qq qq qq qq 0 qq + qq 0 qq qq 6qq 7 qq qq 7 qq 6 Next, we differentiate our profit function with respect to qq, Solving for qq, we find qq qq + 0 qq qq qq ddqq 0 qq + qq + ddqq 0 qq qq 6qq + ddqq 7 qq qq (6 + dd) 7 qq qq 7 qq 6 + dd To find our optimal consumption of qq and qq, we will substitute qq 7 qq 6 qq function. Thus, qq qq 6 + dd that we found into our We seek to solve for qq. In order to do that, we first multiply both sides of the above expression by (8 + 6dd) to get rid of the fraction, as follows, qq (8 + 6dd) 7 6 (8 + dd) (7 qq ) 8qq + 6ddqq + 7dd 7 + qq qq (6 + 6dd) + 7dd qq 5 + 7dd 6 + 6dd Next, we plug the optimal output in plant, qq +7dd, back into our qq 6+6dd 7 qq expression to find 6+dd the optimal amount of production in plant. qq + 7dd dd 6 + dd + 7dd 7 (8 + dd) 6 + dd

6 7(8 + dd) ( + 7dd) 8 + dd 6 + dd + dd (8 + dd)(6 + dd) dd c) How does your result in part () change in the inefficiency of plant, dd. What is the optimal production in each plant if dd 0? Answer: Since dd is in the denominator for qq, we know that as dd increases, or our inefficiency in plant increases, the quantity produced in plant decreases. We also see that as dd increases, our optimal production in qq is increasing. In summary, the monopolist produces more in plant and less in plant as the inefficiency of plant increases. For illustration purposes, the next figure plots output qq (increasing red line) and qq (decreasing yellow line). When dd 0, our optimal production in plant, qq, becomes qq which coincides with that in plant, since + 7dd + (7 0) 6 + 6dd 6 + (6 0) qq dd ( 0) 7 8 Hence, when dd 0, both plants have the same total cost function, leading the monopolist to produce the same amount in plant and. This is depicted in the vertical intercept of the figure where dd 0. However, when dd > 0, production in plant becomes more expensive than in plant, leading the monopolist to reduce its output on plant but increase its output on plant As a curiosity, note that the aggregate production in both plants is QQ qq + qq 7( + dd) 6 + 6dd which is decreasing in the inefficiency of firm, since its derivative with respect to dd is 6

7 (8 + dd) which is negative for all values of dd. In words, aggregate production decreases in the inefficiency of firm. That is, while an increase in dd decreases qq and increases qq, the decrease in qq dominates, producing a reduction in the total amount that this monopolist produces. Exercise # - Monopsony with linear supply curve Consider a firm who is the only employer in a small town (a labor monopsony). It faces an international price pp > 0 for each unit of output that it produces, its production function is qq LL αα where αα > 0, and the labor supply is given by ww(ll) LL where > 0. Intuitively, the firm s production function is concave (when αα < ), linear (when αα ), or convex (when αα > ). a) Find the monopsonist marginal revenue product, and its marginal expenditure on labor. Answer: MMMMMM LL (pp LLαα ) ααααll αα And, MMMM LL (ww(ll)) ww(ll) + LL + b) Set the marginal revenue product and marginal expenditure on labor equal to each other to find the optimal number of workers that the firm hires. Answer: Let MMMMMM LL MMMM LL, we can find: ααααll αα αα LL αααα c) Find the equilibrium wage in this monopsony. Answer: Given the optimal number of workers that firm hires, we can find: αα αα αα ww(ll ) LL αααα αα αααα d) Assume now that the labor market was perfectly competitive. Find the optimal number of workers the firm hires, and the equilibrium wage in this setting. Answer: If the labor market was perfectly competitive, then we have MMMMMM LL ww(ll), so ααααll αα LL LL αααα ww(ll ) αααα αα αα αα αα(αααα) e) Find the deadweight loss of the monopsony. Answer: The DWL is the area below the marginal revenue product and above the supply curve, between LL and LL PPPP workers ( ααααll αα LL PPPP αααα αα ), so αα 7

8 LL DDDDDD PPPP LL MMMMMM LL ww(ll) LL PPPP ααααll αα LL LL PPPP ppll αα LL LL αα αα αα αααα pp αα αα αααα αα f) Evaluate your results in parts (a)-(e) at parameter values, and pp $. Then, evaluate your results at αα /, at αα, and at αα. How are your results affected as αα increases (that is, as the firm s production function becomes more convex ) Answer:. αα /,, and pp $ MMMMMM LL ααααll αα ppll LL and MMMM LL LL; When MMMMMM LL MMMM LL, LL αααα αα pp 0.66 and optimum wage is ww(ll ) αα αα αααα αα pp 0.66; When MMMMMM LL ww(ll), LL αααα αα pp and optimum wage is ww(ll ) αα(αααα) αα αα pp ; And finally, DDDDDD αα αα pp αααα αα αα αα αααα αα pp pp pp 0.06pp pp pp αα,, and pp $ MMMMMM LL ααααll αα pp and MMMM LL LL; When MMMMMM LL MMMM LL, LL αααα αα pp and optimum wage is ww(ll ) αα αα αααα αα pp ; When MMMMMM LL ww(ll), LL αααα αα pp and optimum wage is ww(ll ) αα(αααα) αα αα pp ; And finally, DDDDDD αα αα pp αααα αα αα αα αααα αα pp pp pp 8

9 0.5 pp 0.75 pp 0.5 pp 0.5. αα,, and pp $ MMMMMM LL ααααll αα pppp LL and MMMM LL LL; When MMMMMM LL MMMM LL, LL αααα αα pp.587 and optimum wage is ww(ll ) αα αα αααα αα pp 0.66; When MMMMMM LL ww(ll), LL αααα αα pp and optimum wage is ww(ll ) αα(αααα) αα αα pp ; And finally, DDDDDD αα αα pp αααα αα αα αα αααα αα pp pp pp pp.5 Therefore, as αα increases, a) MMMMMM LL increases and MMMM LL is unchanged. b) When MMMMMM LL MMMM LL, LL, and wage ww(ll ) both increases if / αα ; but decrease if αα. When MMMMMM LL ww(ll), LL and wage ww(ll ) increase if / αα but decrease if αα. c) DDDDDD increases. Exercise #5 - Cournot vs. Stackelberg models of quantity competition Consider a market with two firms selling a homogeneous good, and competing in quantities. The inverse demand function is given by pp(qq, qq ) (qq + qq ), and both firms face a common marginal cost > cc > 0. a) Cournot model. Set up the profit-maximization problem of firm. b) Differentiate with respect to qq. Solve for qq to obtain firm s best response function qq (qq ). c) Depict firm s best response function qq (qq ). Interpret. d) Repeat steps (a)-(b) for firm, so you obtain firm s best response function qq (qq ). e) Find the point where the above two best response functions cross each other. f) What is the equilibrium price? And the equilibrium profits for each firm? g) Stackelberg model. Consider now that firm acts as a leader and firm as the follower. Find the optimal production for firm and in this sequential competition (Stackelberg model). Then find the equilibrium price, and the profits that each firm makes. h) Comparison. Compare the individual output for each firm, price, and profits in the Cournot model (where firms simultaneously select their output levels) and in the Stackelberg model (where firms sequentially choose their output).

10 Answer: a) With Cournot, our profit function is ππ pp(qq)qq ccqq ( qq qq )qq ccqq b) Taking the derivative with respect to qq gives us ππ qq : qq qq cc 0 Solving for qq we find firm s best response function qq (qq ) cc qq This best response function is often rearranged as follows qq (qq ) cc qq The first term does not depend on qq, and represents the vertical intercept of the best response function, while the second term represents the negative slope of the best response function,. In words, an increase in firm s production qq by one unit induces firm to reduce its output qq by half a unit. c) Firm s best response function can be depicted as: In this setting, the best response function originates at cc with a slope of. (when qq 0), and decreases in qq d) Firm has a symmetric profit function as firm (only the subscripts change). Hence, firm s best response function is symmetric to that of firm as follows: 0

11 qq cc qq e) Since firms are symmetric, we can say that, in equilibrium they must produce the same amount of output qq qq. Plugging this information in firm s best response function, we obtain qq cc qq Rearranging, yields qq cc qq Solving for qq gives us the equilibrium output for firm qq cc Because qq qq, we can say that firm s equilibrium output coincides with that of firm qq cc f) To obtain the equilibrium price, we insert the values of qq and qq that we found in part (e) into our price function to find the optimal price pp(qq) qq qq cc cc + cc + cc We then plug this back into our profit function. In this setting: ππ pp(qq)qq ccqq + cc cc cc cc ( + cc)( cc) (cc)( cc) Simplifying: ππ + cc cc cc + cc ( cc) By symmetry, the profit of firm is also: ( cc) ππ g) Using our best response function above from firm, qq cc qq function to obtain Simplifying, yields ππ qq cc qq qq ccqq ππ + cc qq qq ccqq + cc, we plug it into firm s profit Differentiating with respect to firm s output, qq, we obtain

12 ππ qq : + cc qq cc 0 Rearranging, and solving for qq, we find the equilibrium output of firm qq cc We then plug this back into firm s best response function: Simplifying qq cc qq cc qq cc cc cc cc + cc cc We insert this back into our price function to find the equilibrium price. ( cc) ( cc) + cc + cc pp + cc Finally, we plug these back into firm s profit function, to obtain firm s equilibrium profit ππ + cc cc cc cc + cc cc cc 8 + cc cc cc + cc 8 Similarly, we find firm s profit in equilibrium ππ + cc ( cc) 8 cc cc cc + cc + cc cc 6 + cc + cc cc + cc 6 ( cc) 6 h) First, we start with our quantity. From above we have the following: CCCCCCCCCCCCCC: qq qq cc (cc cc ) (cc cc ) SSSSSSSSSSSSSSSSSSSSSS: qq cc aaaaaa qq cc Thus, we can see that the leader produces more in Stackelberg than in Cournot, whereas the follower produces more in Cournot than in Stackelberg. Next, we compare our profits. From above:

13 ( cc) CCCCCCrrnnnnnn: ππ ππ ( cc) SSSSSSSSSSSSSSSSSSSSSS: ππ 8 and ππ ( cc) 6 Again these profits follow the same pattern as above. The leader obtains a higher profit under Stackelberg than under Cournot, whereas the follower s profit is larger under Cournot than Stackelberg. Finally, we compare price CCCCCCCCCCCCCC: pp + cc SSSSSSSSSSSSSSSSSSSSSS: pp + cc To compare which is higher, we set up an inequality. pp SS < pp cc Rearranging, + cc < + cc + cc < + 8cc cc < Thus, as long as cc <, the Stackelberg price is lower than the Cournot price. This condition must hold, as otherwise firms would be producing negative output levels.