Econ 101A Problem Set 6 Solutions Due on Monday Dec. 9. No late Problem Sets accepted, sorry!

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1 Econ 0A Problem Set 6 Solutions Due on Monday Dec. 9. No late Problem Sets accepted, sry! This Problem set tests the knowledge that you accumulated mainly in lectures 2 to 26. The problem set is focused on dynamic games and general equilibrium. General rules f problem sets: show your wk, write down the steps that you use to get a solution (no credit f right solutions without explanation), write legibly. If you cannot solve a problem fully, write down a partial solution. We give partial credit f partial solutions that are crect. Do not fget to write your name on the problem set! Problem. Dynamic Games (8 points) Two companies produce the same good. In the first period, firm sells its product as a monopolist on the West Coast. In the second period, firm competes with firm 2 on the East Coast as a Cournot duopolist. There is no discounting between the two periods. Firm produces quantity x W c/α at the West at cost cx W. On the East Coast, and that s what makes this problem interesting, firm produces quantity x E at cost (c αx W ) x E, where 0 <α<c</2. The parameter α captures a fm of learning by doing. The me firm produces on the West Coast, the lower the marginal costs are going to be on the East Coast. As f firm 2, it produces in the East market with cost cx 2. The inverse demand functions are p W (x W )= x W and p E (x E,x 2 )= x E x 2. Each firm maximizes profit. In particular, firm maximizes the total profits from its West and East coast operations.. Consider first the case of simultaneous choice. Assume that firm 2 does not observe x W befe making its production decision. This means that, although fmally firm chooses output x W first, that you should analyze the game as a simultaneous game between firm and firm 2. Use Nash Equilibrium. Write down the profit functionthatfirm maximizes (careful here) and the profit function that firm 2 maximizes (5 points) 2. Write down the first der conditions of firm with respect to x W and x E, and the first der condition of firm 2 with respect to x 2. Solve f x W, x E, and x 2. ( points). Check the second der conditions f firm and f firm 2. ( points). What is the comparative statics of x W and x E with respect to α? Does it make sense? How about the comparative statics of x 2 with respect to α? ( points) 5. Compute the profits of firm 2 in equilibrium. How do they vary as α varies? (compute the comparative statics) Why are firm 2 s profits affected by α even though the parameter α does not directly affect the costs of firm 2? (5 points) 6. Now consider the case of sequential choice. Assume that firm 2 observes x W befe making its production decision x 2. This means that you should analyze the game as a dynamic game between firm and firm 2, and use the concept of subgame-perfect equilibrium. Remember, we start from the last period. Write down the profit functions that firm and firm 2 maximize on the East Coast ( points) 7. Write down the first der conditions of firm with respect to x E, and the first der condition of firm 2withrespecttox 2. Solve f x E and x 2 as a function of x W ( points) 8. Compute the comparative statics of x E and x 2 with respect to x W. Do these results make sense? ( points) 9. Compute the profits of firm on the East Coast as a function of x W. (2 points) 0. Using the answer to point 9, write down the maximization problem of firm in the first period, that it, when it decides the production on the West Coast. ( points). Write down the first der conditions of firm with respect to x W. Solve f x W solution f x W, find the solution f x E and x 2. (5 points) and then, using the

2 2. Compare the solutions f x W under simultaneous and under sequential choice. What can you conclude? Under which conditions the firm does me preemption, that is, produces me on the West Coast in der to reduce the production in equilibrium of firm 2? (6 points) Solution to Problem.. Firm maximizes the sum of the profits on the East and West coast, that is, the maximization program is max x W ( x W ) cx W + x E ( x E x 2 ) (c αx W ) x E. x W,x E Firm 2 simply maximizes the profits on the East Coast: max x 2 ( x E x 2 ) cx The first der conditions of firm with respect to x W and x E are 2x W c + αx E =0 () and 2x E x 2 (c αx W )=0. (2) The first der condition of firm 2 with respect to x 2 is x E 2x 2 c =0 () From equation () we obtain x W =( c + αx E ) /2 which we can substitute into (2) to obtain 2x E x 2 c + α ( c + αx E ) /2 =0 2 α 2 /2 x E = x 2 c + α ( c) /2. () From () we obtain x E = 2x 2 c which we can substitute into () to get 2 α 2 /2 ( 2x 2 c) = x 2 c + α ( c) /2 We can use () to obtain x 2 α 2 = +α 2 /2+α/2 ( c) x 2 = 2 α 2 α ( c) 6 2α 2. (5) x E =( c) 2x 2 = Finally, from () we get x W =( c) /2+αx E /2=. The Hessian matrix f firm is 6 2α α 2 α ( c) (2 + 2α)( c) 6 2α 2 = 6 2α 2. (6) α 2 + α ( + α) ( c) ( + α)( c) 6 2α 2 = 6 2α 2 (7) 2 H = α α 2 where the first min is - and the determinant is α 2 which is positive since α<2. As f firm 2, the second derivative of the profit function with respect to x 2 is 2 < 0. The second der conditions are satisfied. 2

3 . From expressions (6) and (7) it is clear that both x E and x W are increasing in α: the higher the degree of learning by doing, the higher the production by firm on the East and West market. The learning by doing induces the firm to produce me on the West coast, since this will reduce the costs of production on the East Coast. Once this high production has taken place, the firm has effectively reduced the costs of producing on the East coast and therefe produces me. As f firm 2, we can differentiate expression (5) to get x 2 α = ( c) ( 2α ) 6 2α 2 2 α 2 α ( α) c (6 2α 2 ) 2 = 2α 6+α (6 2α 2 ) 2 +2α 2 +8α α α 2 = c = α 6 2α 2 (6 2α 2 ) 2 < 0. Therefe, as the degree of learning-by-doing of firm increases, the production of firm 2 decreases. Essentially, by learning by doing, firm decreases the own production costs and pushes firm 2 to lower and lower levels of production. 5. The profits of firm 2 are 2 α 2 α Ã ( c) (2 + 2α)( c) 2 α 2 α! ( c) 2 α 2 α ( c) π 2 = 6 2α 2 6 2α 2 6 2α 2 c 6 2α 2 = 2 α 2 α ( c) = Ã( 6 2α 2 c) ( c) +α α 2! " 2 α 2 α # 2 ( c) 6 2α 2 = 6 2α 2 Since the profits of firm 2 π 2 equal (x 2) 2, it is clear that π 2 / α has the same sign as x 2/ α, which is negative. As the learning by doing of firm increases, the profits of firm 2 decrease. This is since the increased production of firm induces firm 2 to produce less and reduces its profits. Interestingly, this occurs despite firm 2 not observing the decisions of firm befe it takes its own decisions. It all happens through equilibrium arguments. 6. In the sequential version, the firms produce on the East Coast after observing production x W West Coast. The profit functionoffirm in period 2 is on the x E ( x E x 2 ) (c αx W ) x E and the profit function of firm 2 is x 2 ( x E x 2 ) cx The first der conditions are 2x E x 2 c + αx W =0 and x E 2x 2 c =0. Solving f x 2 in the first equation we get x 2 = 2x E c + αx W which we can substitute in the second expression to get x E 2( 2x E c + αx W ) c =0 x E (x W )= c +2αx W. Wecanthenobtain x 2 (x W )= c c +2αx W = ( c) αx W Notice that x E is increasing in x W since higher past production decreases current marginal costs f firm. As f firm 2, x 2 is decreasing in x W since higher α implies that firm will produce me on the East coast.

4 9. The profits f firm are π (x W ) = x E ( x E x 2 (c αx W )) = c +2αx W = c µ +2αx W c + αx W 2( c)+αx W 0. The profit functioninperiodffirm is. The first der condition with respect to x W is x W ( x w ) cx W + ( c +2αx W ) 2 2x W c ( c +2αx W )2α =0 µ c + αx W c +2αx W 9 = ( c +2αx W )2 9 ( c) αx W = x W µ2 89 α2 =( c) µ+ 9 α x W =( c)(9+α) / 8 8α 2 Wecanthenplugx W into x E (x W ) to obtain x E (x W )= c +2α ( c)(9+α) / 8 8α 2. Similarly, we get x 2 (x W )= ( c) α ( c)(9+α) / 8 8α 2 2. The production on the West coast under simultaneous production is x W = whereas in the sequential version we obtain ( + α)( c) 6 2α 2 x W =( c)(9+α) / 8 8α 2. The production in the sequential game is higher than in the simultaneous game if ( c)(9+α) / 8 8α 2 ( + α)( c) 6 2α 2 (9 + α) 6 2α 2 ( + α) 8 8α 2 5 8α 2 +2α 8α 5 2α 2 +8α 8α 6α 2 +6α =6α ( + α) 0 α 0. That is, as long as there is learning by doing (α >0), firm produces me on the West coast if the game is sequential than if it is simultaneous. In other wds, if firm 2 can observe x W then, as in a standard Stackelberg duopoly, the leader gets to produce me and earn me profits. I realize that computing the profits here would be very cumbersome (sry), but i am ready to bet 0: that firm overall makes me profits in the sequential than in the simultaneous case. :)

5 Problem 2. General Equilibrium (2 points) Consider the case of pure exchange with two consumers. Both consumers have Cobb-Douglas preferences, but with different parameters. Consumer has utility function u(x,x 2)=(x ) α (x 2) α. Consumer 2 has utility function u(x 2,x 2 2)=(x 2 ) β (x 2 2) β. The endowment of good j owned by consumer i is ω i j. The price of good is p, while the price of good 2 is nmalized to without loss of generality.. Only f point, assume ω =,ω 2 =,ω 2 =,ω 2 2 =. (that is, total endowment of each good is ). Assume further α =/2, β=/2. Draw the Pareto set and the contract curve f this economy in an Edgewth box. (you do not need to give the exact solutions, only a graphical representation) What is the set of points that could be the outcome under barter in this economy? (5 points) 2. F each consumer, compute the utility maximization problem. Solve f x i j f j =, 2 and i =, 2 as a function of the price p and of the endowments. [This problem to be solved with closed eyes!] (5 points). Assume again ω =,ω 2 =,ω 2 =,ω 2 2 =and α =/2, β=/2. Do a qualitative plot of the offer curve f consumer. [Trick to do this is to compute the values of x and x 2 as you increase p from 0] What happens to the consumption of good and good 2 as the price p increases? Plot also the offer curve of consumer 2. Graphically, find the intersection, the general equilibrium point. (7 points). We now solve analytically f the general equilibrium. Require that the total sum of the demands f good equals the total sum of the endowments, that is, that x + x 2 = ω + ω 2. Solve f the general equilibrium price p. (6 points) 5. What is the comparative statics of p with respect to the endowment of good, that is, with respect to ω i f i =, 2? What about with respect to the endowment of the other good? Does this make sense? What is the comparative statics of p with respect to the taste f good, that is, with respect to α and β? Does this make sense? ( points) 6. Now require the same general equilibrium condition in market 2. Solve f p again, and check that this solution is the same as the one you found in the point above. In other wds, you found a property that is called Walras Law.In an economy with n markets, if n markets are in equilibrium, the nth market will be in equilibrium as well. (5 points) Solution to Problem 2.. See Figure. 2. Instead of solving the Lagrangean problem (make sure you know how to do this), i will use a general feature of Cobb-Douglas utility functions. The consumer consumes share α of income in the first commodity. This implies x p ω = α + 2 ω and Fconsumer2 and. See Figure. p x 2 =( α) p ω + ω 2. x 2 p ω 2 + ω 2 = β 2 p x 2 2 =( β) p ω 2 + ω

6 . We have already imposed the firstpartofthedefinition of Walrasian Equilibrium, that is, that each consumer chooses the optimal allocation given the price p. We now impose the second part of the definition, that is, that each market be in equilibrium. We first impose the condition f the first market, that is, x + x 2 = ω + ω 2. This implies p α ω + ω2 p + β ω 2 + ω2 2 = ω + ω 2 p p αω αω + βω βω2 2 = ω + ω 2 αω 2 + βω 2 2 = p ( α) ω +( β) ω 2 p = αω 2 + βω 2 2 / ( α) ω +( β) ω Notice that, as the endowment of good ω ω 2 increases, the price of good decreases. this makes sense. If the quantity available of good increases, while holding constant the quantity of good 2, good 2 becomes relatively scarcer and this induces the relative price of good to decrease. This is like an increase in supply that decreases the price. Notice that p is not the absolute price of good, but rather the relative price of good relative to the price of good 2, which we nmalized to. Symmetrically, if the endowment of good 2 increases, good becomes scarcer and the price p increases. As f the other comparative statics, as the taste f good increases (increase in α β) the equilibrium price of good goes up. Since increase in taste f good means a positive shift in demand, we are not surprised that this induces an increase in the equilibrium price of good. Notice a difference, though, between the general equilibrium increase in the demand and the partial equilibrium. If both α and β increase, that is, if both consumers like good me, the price of good will go up, but the quantity consumed of good will go down ( at least not go up) f one of the consumers. In this setting, the total quantity available of each good is constant. 6. We now impose the condition f the second market, that is, x 2 + x 2 2 = ω 2 + ω 2 2. This implies ( α) p ω + ω 2 +( β) p ω 2 + ω 2 2 = ω 2 + ω 2 2 ( α) ω +( β) ω2 p = αω 2 + βω2 2 p = αω 2 + βω 2 2 / ( α) ω +( β) ω 2 which is the same solution that we found befe. If there are n markets, it s enough to impose equilibrium conditions in n of them, and the n-th market will automatically be in equilibrium. p 6

7 Problem. Mal Hazard (6 points, due to Botond Koszegi). We analyze here a principal-agent problem with hidden action (mal hazard). The principal is hiring an agent. The agent can put high efft e H low efft e L. If the agent puts high efft e H, the output is y H =8with probability / and y L = with probability /. If the agent puts low efft e L, the output is y H =8with probability / and y L = with probability /. The principal decides the pay of the agent w. The utility of the agent is w c (e), where c (e H )=. and c (e L )=0. The reservation utility of the agent is.. The principal maximizes expected profits.. Assume first that the principal can observe the efft of the agent (that is, there is no hidden action). We want to determine the optimal contract. In this case, the principal pays a wage w (e) that can depend on the efft. The way you solve a problem like this with a discrete number of efft levels is to study first the case in which the principal wants to implement high efft e H. In this case, the wage will be w if the agent chooses e H and 0 otherwise. Solve the problem s.t. w c (e H ). max w 8 + w Argue that the constraint is satisfied with equality and solve f w. Compute the expected profit in this case Eπ H. (5 points) 2. We are still in the case of no hidden action. Assume that the principal wants to implement low efft e L. Similarly to above, set up the maximization problem and solve f w. Compute the expected profit in this case Eπ L and compare with Eπ H. Which profit level is higher? The higher one is the contract chosen by the principal and hence the action implemented. (5 points). Now consider the case with hidden action. The wages can only be a function of the outcomes: w H when y = y H and w L when y = y L. We study this case in two steps. Assume first that the principle wants to implement e H. We study the optimal behavi of the agent after signing the contract. Write the inequality that indicates under what condition the agent prefers action e H to action e L. (This is a function of w H and w L and goes under the name of incentive compatibility constraint). (5 points). Now consider the condition under which the agent prefers the contract offered to the reservation utility.. (This is the individual rationality constraint) ( points) 5. Argue that the two inequalities you just derived will be satisfied with equality. Solve the two equations to derive wl and w H. Compute the profits f the principal from implementing the high action under hidden action: Eπ HA. (5 points) H 6. Now, assume that the principal wants the agent to take the low action e L. In this case, you do not need to wry that the agent will deviate to the high action, since that will take me efft. The principal will pay a flat wage w. Write the individual rationality constraint f the agent if he takes the action e L and the pay is w. Set this constraint to equality (Why?) and derive w. Compute the implied profits f the principal from implementing the low action under hidden action: Eπ HA.(5 points) L 7. Compare the profits in point 6 (profits from low efft) and in point 7 (profits from low efft). Under hidden action, what contract does the principal choose to implement, the one that guarantees high efft the one that guarantees low efft? ( points) 8. Compare the profits f the principal under perfect observability and under hidden action. Why are they different despite the fact that the action chosen by the agent in equilibrium will be the same? (6 points) 9. There is a moniting system that allows the principal to perfectly observe the actions (and hence to implement the perfect observability contract). How much would the principal be willing to pay f it? ( points) 0. Compare the utility of the agent under perfect observability and under hidden action. ( points) 7

8 Solution to Problem.. In the problem s.t. w c (e H ). max w 8 + w the principal would be foolish to pay a salary w higher than the one that satisfies the constraint with equality, since that would lover expected profits. The solution f w hence is w. =., which implies w = /00. This implies an expected profit of Eπ = 55/ / In this case, we follow the same logic, but the individual rationality constraint is now w c (e L )=., w =., which implies w =/00. This implies an expected profitofeπ = 8+ w =2/ /00.. In the case of hidden action, the principal cannot enfce a particular action. S/he can only pay as a function of the observe output y L y H. The agent prefers the action e H to the action e L if wh + wl c (e H ) wh + wl c (e L ). If this condition holds, the expected utility of putting high efft e H is higher than the expected utility of putting low efft e L. Hence, the agent will choose to put high efft.. The agent prefers to put high efft to the reservation utility if wh + wl c (e H ).. 5. The individual rationality inequality will be satisfied with equality because otherwise the principal could lower the pay under w L and increase profits. (Notice that this would make the first constraint me likely to be satisfied, so the argument is kosher) As f the first constraint, the incentive compatibility one, if the constraint were not satisfied with equality, the principal could increase the profits by lowering w H and increasing w L so as to keep the individual rationality constraint satisfied. This would increase profits because of the risk-aversion. Given that the agents are risk-averse, they value me the increase in w L than the decrease in w H (since in the contract it will be the case that w H >w L ). This is not a full proof, but it tries to convey the intuition. If this is hard to comprehend, do not wry too much. The proof of this part is beyond the level required in this class. Moving on, now that we know that the constraints are satisfied with equality, we can solve f wl and w H. The system is wh + wl c (e H ) = wh + wl c (e L ) wh + wl c (e H ) =. The equations reduce to wh = 2 wl +. 2 wh + wl =.2 Substituting w H into the second equation gives wl wl =.2 w L =/20, wl =/00. It follows that w H =25/00. Hence, the expected profits are Eπ = =

9 6. If the principal wants to implement the low action, she will pay a constant wage f the high and low output given the risk-aversion of the agents. The individual rationality constraint is w + w c (el ). The inequality is satisfied with equality because otherwise the principal could lower the wage w and increase the profits. It follows that w =., w =/00. The implied profits are Eπ = = The profits from implementing the high efft ( ) are substantially higher than the profits from implementing the low efft ( 2 00 ). Hence, the principal chooses to implement the contract with high efft. 8. It is interesting that both under perfect observability and under hidden action the principal implements the high action e H. That is, in equilibrium, the agent will choose e H in both cases. However, the profits are higher in the case of perfect observability. Compare and The difference is due to the fact that under hidden action the principal has to introduce different wages in case of high outcome and low outcome, paying the high outcome me, in der to make sure that the agent puts high efft. Since the agent is risk-averse, she does not like this, and is less willing to take the contract, which in turn reduces the profits of the firm. 9. The principal would be willing to pay µ 55 9 = Notice that in all cases the agent is offered a contract that makes her indifferent relative to the outside option. Therefe, her utility is always.. This is because the principal here is a monopolist which can extract all the surplus. 9

EC476 Contracts and Organizations, Part III: Lecture 2

EC476 Contracts and Organizations, Part III: Lecture 2 Leonardo Felli 32L.G.06 19 January 2015 Moral Hazard: Consider the contractual relationship between two agents (a principal and an agent) The principal