Microeconomics Problem Set 3 Solutions PDF

Micro II Solutions to exercises Problem Set 3: Solutions Chapter 4: Time and uncertainty Exercise 4.1: Economy with Storage for one Agent Theorem 15: Economy with Storage for one Agent Consider an economy with...

Micro II Solutions to exercises Problem Set 3: Solutions Chapter 4: Time and uncertainty Exercise 4.1: Economy with Storage for one Agent Theorem 15: Economy with Storage for one Agent Consider an economy with two consumers, a single good x and two time periods. Intertemporal utility functions for the consumers are u(x1 , x2 ) = x1 x2 , and endowments are ω1 = (19, 1) and ω2 = (1, 9). Consumer 1 can store the good. That is, consumer one can transform one unit of the good in period one into one unit of the good in period 2. 1. Suppose the consumers cannot trade with each other. How much does each consumer consume? How well off are the consumers? How much storage takes place. 2. Now suppose the consumers can trade. Treat the storage capacity of consumer one as a firm in a production economy which is fully owned by the consumer 1. What are the Walrasian equilibrium prices? How much storage takes place? How well off are the consumers? 3. Now suppose that the storage is costly. That is, for each unit stored, only δ are recovered in the second period. What is the optimal choice of consumers when they cannot trade. How well off is consumer 1? 4. Continue assuming that storage is costly, but suppose that the consumers can trade. What are the equilibrium prices? For what δ does storage take place? For what δ is consumer 1 better off than without trade? Part 1 First, observe that the consumer 2 has no choice to make. He simply consumes his en- dowment in each period and receives utility: u2 (ω21 , ω22 ) = u2 (1, 9) = 9. Agent 1, on the other hand, can store the good. His maximization problem is max x1 x2 x1 ,x2 s.t. x1 + x2 = 20 and x2 ≥ 1 ⇒ (ignore and verify) 18 Micro II Solutions to exercises Form the Lagrangian, take the FOC with respect to each good and combine L = x1 x2 − λ(x1 + x2 − 20) ⎫ x2 − λ = 0⎬ 1 ! FOCs : x = x2 ⇒ x1 = x2 = 10. ! x1 − λ = 0⎭ Consumer 1 receives utility of 100 and 9 units are stored. Part 2 First, consider the problem of the firm to find the possible equilibrium prices. In equilib- rium, we need that the firm makes zero profits. If p1 < p2 , the firm could make infinite profits. If p1 = p2 , the firm makes 0 profits for any production. If p1 > p2 , the firm optimally chooses y = (0, 0) and makes makes 0 profits. Hence, we have two possible cases. Now consider the problem of consumers: max x1i x2i x1i ,x2i s.t. p1 x1i + p2 x2i = mi where m1 = p1 · 19 + p2 · 1 and m2 = p1 · 1 + p2 · 9. L = x1i x2i − λ(p1 x1i + p2 x2i − mi ) ⎫ x2i − λp1 = 0⎬ x2i ! FOCs : x1i = x1 − λp2 = 0⎭ p ! 1 p2 i ⇒ p2 x2i = p1 x1i. From BC: 2p1 x1i = 2p2 x2i = mi mi mi so that x1i = 1 , x2i = 2. 2p 2p We have to check if the equilibrium allocation without production (p1 > p2 ) is feasible. If y = (0, 0), then feasibility implies m1 m2 x1 = + = 20 = ω 1 2p1 2p1 m1 m2 x2 = 2 + 2 = 10 = ω 2 2p 2p m1 + m2 = 40p1 m1 + m2 = 20p2 2p1 = p2. 19 Micro II Solutions to exercises Which is a contradiction with the requirement that p1 > p2. Thus, it must be the case p1 = p2. Let p1 = p2 = 1. Then m1 x11 = x21 = = 10 2 m2 x12 = x22 = = 5. 2 Feasibility requires that x11 + x12 = ω11 + ω21 + y 1 15 = 20 + y 1 y 1 = −5 ⇒ y 2 = 5 ⇒ storage x21 + x22 = ω12 + ω22 + y 2 10 + 5 = 10 + 5. So there is less storage then in autarky. Consumer 1 does not benefit, but consumer 2 does. Part 3 Consumer 1 now solves max x11 x21 x11 ,x21 s.t. x21 = ω12 + δ(ω11 − x11 ) ⇔ δx11 + x21 = ω12 + δω11 and x11 ≤ ω11 (ignore and verify) L = x11 x21 − λ(δx11 + x21 −ω12 − δω11 )  FOCs: x21 − λδ = 0 x21 = x11 x11 − λ = 0 δ Substitute into BC: 2x21 = 2δx11 = ω12 + δω11 ω12 + δω11 x11 = , 2δ ω 2 + δω11 x21 = 1 2 1 1 Valid only if: x1 ≤ ω1 ω12 + δω11 ≤ ω11 2δ ω12 ≤ 2δω11 − δω11 ω12 ≤ δω11 1 δ≥. 19 20 Micro II Solutions to exercises 1 1 If δ < 19 , the solution is x1 = (19, 1); and the utility of the consumer 1 is 19. If δ ≥ 19 , the optimal consumption is the derived above and the utility is (1 + δ19)2 U1 (x11 , x21 ) =. 4δ Part 4 As in part 2, if p1 = δp2 , the firm produces any output and makes zero profits. If p1 > δp2 , the firm shuts down. Suppose the firm shuts down. As calculated in Part 2, this can be an equilibrium only if p1 = 12 p2. Thus, this equilibrium will exist whenever δ < 12. Again from mi mi Part 2, the demands will be x1i = 2p 2 1 2 1 1 , xi = 2p2 , where m1 = 19p + p and m2 = p + 9p. 2 Let p1 = 1 so that p2 = 2. We get m1 = 21 and m2 = 19: 21 21 19 19 x11 = x21 = x12 = x22 = 2 4 2 4 212 The utility of consumer 1 is 8. Compare it to the utilities derived in part 3:   212 (1 + δ · 19)2 1 1 > ∀δ ∈ , 8 4δ 19 2 212 1 > 19 when δ <. 8 19 Where the first expression can deducted by noticing that it needs to be increasing in δ and is equal δ = 12. Now suppose the firm does not shut down. Then, it must be p1 = δp2. Let p1 = δ and p2 = 1. The demands are then 1 + 19δ 1 + 19δ x11 = x21 = 2δ 2 δ + 9 δ + 9 x12 = x22 = 2δ 2 Take good 2, feasibility requires x21 + x22 = ω12 + ω22 + y 2 10 + 20δ = 1 + 9 + y2 2 5 + 10δ = 10 + y 2 y 2 = 10δ − 5. Since y 2 ≥ 0, this equilibrium only exists if δ ≥ 12. Note that the utility of consumer 1 is (1+19δ)2 4δ as in autarky! Thus, consumer 1 benefits only in those instances when the firm shuts down! 21 Micro II Solutions to exercises Exercise 4.2: Argentinia and Brazil Theorem 16: Argentinia and Brazil Consider two agents A (an Argentinian) and B (a Brazilian). Suppose that Ar- gentina and Brazil are playing the World Cup finals. If Argentina wins, A will receive happiness of 1 and if Brazil wins A receives happiness of 0. Agent B has exactly the opposite preferences. Agents can trade happiness (for example by doing laundry for the other agent). Let πA be the belief of agent A that Argentina will win, and πB the belief of agent B that Argentina will win. The agents are going to bet, if Argentina wins A will do a amount of laundry for B, and if Brazil wins, √ B will do b amount of laundry for A. The utility of agent from happiness h is h. 1. Represent the situation above as an exchange economy with uncertainty. 2. Suppose that πA = πB = 1/4 (both agents agree that Brazil is more likely to win). What is the equilibrium allocation? Represent the situation in an Edgeworth box. 3. Suppose now that πA = 3/4 while πB = 1/4 (the Argentinian is overopti- mistic). What is the equilibrium allocation? Represent the situation in an Edgeworth box. Part 1 We can think of two states of the world, A (Argentina wins) and B (Brazil wins). Then, there are two contingent commodities xA and xB , x = (xA , xB ). The initial endowment is ωA = (1, 0) and ωB = (0, 1). Notice that aggregate endowment is   1 ω = ωA + ωB = 1 Which corresponds to the total endowment in different states of the world. Since total endowment is the same, we say there is⎛no⎞aggregate uncertainty, i.e. we have constant 1 endowment across states. If, say, ω = ⎝0⎠, there would be aggregate uncertainty. Our 3 model can also capture this! To finish, we need utility, which is given by uA (xA B A , xA ) = ΠA xA A + (1 − ΠA ) xB A uB (xA B B , xB ) = ΠB xA B + (1 − ΠB ) xB B 22 Micro II Solutions to exercises Part 2 Denote the prices of commoditites as pA and pB. Agent i solves max Πi xA i + (1 − Πi ) xB i xA B i ,xi s.t. pA xA B B i + p x i = mi. Let pB = 1. A A B L = Πi xA i + (1 − Πi ) xB i − λ(p xi + xi − mi ) ⎫ FOCs: Πi 12 √1 A − λpA = 0 ⎬ xi ∗ (1 − Πi ) 12 √1 B − λ = 0⎭ xi Πi (1 − Πi ) ∗: = 2 xA i p A 2 xB i A Π xB i = (1 − Πi )p xAi  2 1 − Πi xB i = (pA )2 xA i. Πi From BC: pA x A B i + x i = mi   1 − Πi 2 A 2 A pA x A i + (p ) xi = mi Πi mi xA i = 2 (∗) pA + 1−Π Πi i (p A )2   B 1 − Πi 2 A 2 mi xi = (p )  2 Πi pA + 1−Π Πi i (pA )2  2 1−Πi Πi pA mi B xi =  2 (∗∗) 1 + 1−Π Πi i p A  2 1 1− 14 In this part, ΠA = ΠB = 4 and mA = pA , mB = 1. Then 1 = 9, and: 4 pA 1 xA A = ; xA B = pA + 9(pA )2 pA + 9(pA )2 9(pA )2 9pA xB A = ; xB B = 1 + 9pA 1 + 9pA 23 Micro II Solutions to exercises Market clearing requires pA 1 A A 2 + A =1 p + 9(p ) p + 9(pA )2 pA + 1 = pA + 9(pA )2 1 pA =. 3 We can use this to calculate the equilibrium demand 1 1 1 3 xA A = 1= ;3 xA B = = ; 1 +13 4 +1 4 3 1 1 3 3 xB A = = ; xB B = = ; 1+3 4 1+3 4 The consumption of both agents is constant across states. Thus, the agents are fully insured! Note that the agents are fully insured on the 45◦ line. Part 3 1 Since ΠB = 4 is not changed, the demand of agent B is as before, 1 9pA xA B = , xB B =. pA + 9(pA )2 1 + 9pA 3 But ΠA = 4 and the demand of agent A changes. Note that  2  2 1 − ΠA ( 14 ) 1 = = ΠA ( 34 ) 9 24 Micro II Solutions to exercises Using (∗) we get pA xA A = pA + 19 (pA )2 1 A 2 9 (p ) xB A =. 1 + 19 pA Market clearing requires that xA A A + xB = 1 1 1 1 A + pA + 9(pA )2 = 1. 1 + 9p This equation is a bit tedious, but it is easy to verify that pA = 1 is a solution. This gives demands 9 1 xA A = ; xA B = ; 10 10 1 9 xB A = ; xB B = ; 10 10 As you can see, differing beliefs lead the agents to not insure fully. 25 Micro II Solutions to exercises Chapter 5: The core of an economy Exercise 5.1 Theorem 17 Consider an exchange economy with two goods and three agents with the identical utility functions ui (x1i , x2i ) = x1i x2i. The endowments are ω1 = (2, 0), ω2 = (4, 0) and ω3 = (0, 4). 1. Consider an allocation x1 = (1, 1), x2 = (3, 1) and x3 = (2, 2). Are all agents better off than with their initial allocations? Is this allocation blocked? If so, find the blocking coalition and their preferred allocation. 2. Consider an allocation x1 = (2, 0), x2 = (2, 2) and x3 = (2, 2). Is this alloca- tion blocked? If so, find the blocking coalition and their preferred allocation. 1. The utility of all agents at their initial endowment is 0. Utility at allocations x1 , x2 , x3 is strictly positive. Thus, everyone is better off. Let S = {2, 3} and consider allocation x2 = (2, 2) = x3 , u2 (x2 ) < u2 (x2 ) and u3 (x3 ) = u3 (x3 ). We need that both agents strictly improve. Take a bit away from consumer 2 and give it to consumer 3 u2 (1.9, 1.9) > 3 and u3 (2.1, 2.1) > 4. Hence this is a blocking allocation. 2. Let S = {1, 3}. Consider an allocation x1 = (0.5, 1) and x3 = (1.5, 3). Both agents are clearly better off so this is a blocking coalition. Exercise 5.2 Theorem 18 Suppose that a feasible allocation x is not Pareto efficient. Then x is not in the core. Find a blocking coalition and a blocking allocation. Since x not Pareto efficient is, there exists a feasible allocation x , such that ui (xi ) > ui (xi ) ∀ i. But then the grand coalition, s = N , blocks the allocation with x. 26 Micro II Solutions to exercises Exercise 5.3 Theorem 19 Consider the exchange economy as in Exercise 2.1. The agent 1 is endowed with ω11 = 3 of good 1 and ω12 = 1 of good 2. The agent 2 is endowed with ω21 = 1 of good 1 and ω22 = 2 of good 2. The utility function of both agents is ui (x1i , x2i ) = x1i x2i. We found the equilibrium allocation to be x1 = (13/6, 13/8) and x2 = (11/6, 11/8). 1. In an Edgeworth box, depict the core of this endowment. Explain what char- acterizes the allocations in the core. 2. Depict all allocations that are in the core for an arbitrary reallocation of initial endowments. In the core, the agents cannot profitably trade. That is, their indifference curves are tangent to each other! The contract curve is the set of all tangent points of an Edgeworth box. The core of a specific endowment is the part of the contract curve that is inside of the indifference curves of this endowment. 27 Micro II Solutions to exercises Exercise 5.4 Theorem 20 Consider again the exchange economy as in Exercise 2.1. We found the equilibrium allocation to be x∗1 = (13/6, 13/8) and x∗2 = (11/6, 11/8). It can be shown (for example by demonstrating that the indifference curves are tangent) that the allo- cation x1 = (14/6, 14/8) and x2 = (10/6, 10/8) is in the core. Now suppose that there are two identical agents 1 and two identical agents 2. 1. Argue that the Walrasian equilibrium prices are the same as before and that the equilibrium allocation is the same, so that both agents of type 1 get x∗1 and both agents of type 2 get x∗2. 2. Show that the allocation where both agents of type one get x1 and both agents of type two get x2 is no longer in the core. Part 1 Note that x∗i was a solution to the problem with two agents max ui (x1i , x2i ) x1i ,x2i s.t.p1 x1i + p2 x2i = mi. Now we double the economy. Let agents 1 and 3 be identical and agents 2 and 4 be identical. Observe that identical agents have identical maximization problems, so the solutions must be identical as well. That means the equilibrium prices are unchanged and the demands for consumer 1 and 3 are identical, x∗1 = x∗3. The same for consumer 2 and 4, x∗2 = x∗4. The only remaining thing to check is if the allocation is feasible. Since x∗1 , x∗2 is an equilibrium allocation with two agents, then x∗1 + x∗2 = ω1 + ω2 and then x∗1 + x∗2 + x∗3 + x∗4 = ω1 + ω2 + ω3 + ω4 2x∗1 + 2x∗2 = 2ω1 + 2ω2. x∗1 + x∗2 = ω1 + ω2 Hence the allocation is also feasible with four agents. Part 2 Consider a coalition S = {2, 3, 4}. Remember ω1 = (3, 1) and ω2 = (1, 2), x1 = ( 14 14 6 , 8 ), x2 = ( 10 10 6 , 8 ). For consumer 2 and 4, consider an allocation 1 1 x2 = x4 = x2 + ω2 2 2    1 10 10 1 6 16 = , + , 2 6 8 2 6 8   8 13 = , 6 8 28 Micro II Solutions to exercises For consumer 3 suppose x3 = x1. Is this allocation feasible? ω2 + ω3 + ω4 = (1, 2) + (3, 1) + (1, 2)   30 40 = (5, 5) = , 6 8          8 13 14 14 8 13 x2 + x3 + x4 = , + , + , 6 8 6 8 6 8   30 40 = , 6 8 Agent 3 is just as well off as before since we did not change his allocation. However, agent 2 and 4 are better off 10 10 100 u2 (x2 ) = · = 6 8 48  8 13 104 u2 (x2 ) = · =. 6 8 48 It is clear that agents 2 and 4 can give up a small amount of their consumption and give it to agent 3 to create an allocation in which all agents are strictly better off. Hence this is a blocking coalition. 29