Micro II Problem Set 2 Solutions PDF

Micro II Solutions to exercises Problem Set 2: Solutions Exercise 2.4 Theorem 7: 2.4 Suppose we have two agents with identical utility functions u(x1 , x2 ) = max{x1 , x2 }. The total endowment is 1 unit...

Micro II Solutions to exercises Problem Set 2: Solutions Exercise 2.4 Theorem 7: 2.4 Suppose we have two agents with identical utility functions u(x1 , x2 ) = max{x1 , x2 }. The total endowment is 1 unit of good 1 and 2 units of good 2. In an Edgeworth box, illustrate the strongly and weakly Pareto efficient allocations. Why does Proposition 2.13 not hold? Step 1: draw the 45◦ lines and split into two squares Step 2: argue that the interior of the squares is pareto dominated by the purple points Step 3: argue that the lines A and B are dominated by the purple points Step 4: start from the line separating the two squares and argue it is WPE, as are lines C and D Step 5: argue E and F are also WPE and purple dots are SPE The proposition does not hold because the utility functions are not strongly increasing. 11 Micro II Solutions to exercises Exercise 2.5: Revealed Preference Theorem 8: Revealed Preference Consider the revealed preference argument for the Second Welfare Theorem. Show that if utility functions are strictly quasiconcave, then xi = x∗i for all i. Suppose not. Then there exists some agent i such that xi = x∗i. Consider a consumption bundle xi = αxi + (1 − α)x∗i for some α ∈ (0, 1). Observe that since p xi = p ωi and p x∗i = p ωi then  p (αxi + (1 − α)x∗i ) = αp xi + (1 − α)p x∗i = αp ωi + (1 − α)p ωi = p ωi Thus the bundle xi is in the budget set of consumer i. By the argument in the proof of the Second Welfare Theorem, ui (xi ) = ui (x∗i ). But since ui is strictly quasiconcave, ui (xi ) = ui (αxi + (1 − α)x∗i ) > ui (xi ) = ui (x∗i ). Hence, x could not have formed the Walrasian equilibrium. A contradiction. 12 Micro II Solutions to exercises Exercise 2.6 Theorem 9: 2.6 Suppose that agents have identical utility functions u(x1 , x2 ) = min{x1 , x2 } and identical endowments (1, 1). In an Edgeworth box, argue that there is an infinite number of prices that are Walrasian equilibria. there is an infinity of price vectors for which the indifference curves are tangent at the endowment point this is natural as, for both agents, utility only increases if consumption of both goods increases. As such, they don’t want to trade! Exercise 2.7: Equal Divison is Pareto Efficient Theorem 10: Equal Divison is Pareto Efficient We have n agents with identical strictly concave utility functions. There is some initial bundle of goods ω. Show that equal division is a Pareto efficient allocation. Let us start with an intermediate result: Lemma: In this exchange economy, the sum of utilities is maximized by the equal division of goods. 13 Micro II Solutions to exercises Proof: Take any allocation x. If there exists two agents i and l such that xi = xl , then we can construct a new allocation, where we only make a change for agents i and l, such that they both get the average bundle: 1 1 xi = xl = xi + xl. 2 2 Since u is strictly concave then   1 1 1 1 u xi + xl > u(xi ) + u(xl ). 2 2 2 2 But then:   1 1 2u xi + xl > u(xi ) + u(xl ). 2 2 Hence, the new allocation increases the sum of utilities. As we did this for an arbitrary allocation, then the sum of utilities is maximized for the equal split. QED Now, suppose that ω n allocation was  not Pareto efficient. Then, there exists  another allocation x such that ui (xi ) ≥ ui ω n for all i, and for some l u l (x l ) > u l ω n. But then, this allocation x would produce a higher sum of utilities than the equal split, which is not possible by the lemma above. Exercise 2.8: Competitive Prices Theorem 11: Competitive Prices Consider an economy with 15 consumers and 2 goods. Consumer 3 has a utility function u3 (x13 , x23 ) = ln x13 + ln x23. At a certain Pareto efficient allocation x∗ , con- sumer 3 holds (10, 5). What are the competitive prices that support the allocation x∗ ? The consumer is maximizing her utility subject to the budget constraint. Suppose the prices are p1 and p2. Then the consumer solves max ln x1 + ln x2 x1 ,x2 s.t. p 1 x 1 + p 2 x 2 = m3 The FOCs are 1 ! 1 1 − λp1 = 0 ⇔ 1 1 = λ x x p 1 ! 1 2 − λp2 = 0 ⇔ 2 2 = λ x x p 1 1 2 2 x p =x p x1 p2 = x2 p1 p 2 10 1 = = 2 ⇔ p2 = 2p1. p 5 14 Micro II Solutions to exercises Exercise 2.9 Theorem 12 Suppose that agent 1 has the utility function u1 (x11 , x21 ) = x11 +x21 and the agent 2 has the utility function u2 (x12 , x22 ) = max{x12 , x22 }. Agents have identical endowments (1/2, 1/2). Illustrate this situation in an Edgeworth box. What is the equilibrium relationship between p1 and p2 ? What is the equilibrium allocation? Suppose that p1 = p2. For example p1 > p2. Then agent 1 wants to sell all units of good 1 and buy as many units of good 2 as possible. The agent 2 wants to do the same. This cannot be an equilibrium. We run into the same problem with p1 < p2. Thus, the equilibrium has to satisfy p1 = p2. With these prices, the agent 1 is indifferent between all allocations on the budget line. Agent 2 still wants to hold only one good, but it does not matter which. There are two possible equilibrium allocations. Either – x11 = 1, x21 = 0, x12 = 0, x22 = 1 or – x11 = 0, x21 = 1, x12 = 1, x22 = 0. 15 Micro II Solutions to exercises Chapter 3: Production Economy Exercise 3.1 Theorem 13 Consider three firms that can transform good 1 into good 2. Each firm has access to two units of good 1. Firm 1 can transform x units of good one into x units of √ good two, firm 2 can produce x with the same input, while firm 3 produces x2 units of good two. Let y (x) be the production plan of firm  when it uses x units of good 1 in production. 1. Represent Y1 , Y2 and Y3 graphically. 2. Mark the points y (0) and y (2) for  = 1, 2, 3. 3. Mark the set of points ȳ which satisfy the condition 1 1 ȳ ≥ y (0) + y (2) 2 2 (and equality does not hold) for  = 1, 2, 3. Which firm satisfies the point 3 from Assumption 3.2? 16 Micro II Solutions to exercises Exercise 3.2 Theorem 14 Consider an economy with two firms and two consumers. Firm 1 is entirely owned by consumer 1. It produces carrots from land via the production function c() = 2. Firm 2 is entirely owned by consumer 2. It produces potatoes from land via the production function p() = 3. Each consumer owns 10 units of land. Consumer 2 3 1’s utility function is u1 (c, p) = c 5 p 5 and consumer 2’s utility function is u2 (c, p) = 10 + 12 ln c + 12 ln p. 1. Find the market clearing prices for carrots, potatoes and land. 2. How many potatoes and carrots does each consumer consume. 3. How much land does each firm use? We know that only relative prices matter. So let us normalize pl = 1. Suppose pc > 12. Then, the profit of firm 1 would be Π1 (l) = pc (2l) − pl (l) = (2pc − 1)l > 0 and firm 1 would want to produce an infinite number of carrots. This cannot be an equilibrium. For any pc < 12 , the firm would produce 0. Hence, if an equilibrium with positive production exists, it must be pc = 12. Similar argument yields pp = 13. Observe that since pl = 1, the budget of each consumer is 10 (firm profits are zero). Observe that both utility functuons are Cobb-Douglas (why is the second one?) and you should know by now that consumer 1 will spend 25 of total income on c and 35 of total income on p. Consumer 2 spends half of income on each good. This yields 2 pc · c 1 = · 10 5 1 c1 = 4 2 c1 = 8 3 pp p1 = · 10 5 p1 = 18 1 pc · c2 = · 10 2 c2 = 10 1 pp · p2 = · 10 2 p2 = 15 Thus 9 units of land will be used for carrots and 11 will be used for potatoes. 17