Intermediate Micro Problem set 4: Exchange Economies and Competitive Equilibrium June 18, 2023 1 Pareto-efficient allocations Three agents meet to trade three goods. Each agent has utility function u j ( x ) = √ x j 1 + √ x j 2 + √ x j 3 Agent 1 enters the market with 1 unit of good 1 (i.e., e 1 = (1 , 0 , 0)), agent 2 brings 4 units of good 2 (i.e., e 2 = (0 , 2 , 0)), and agent 3 brings 25 units of good 3 (i.e., e 3 = (0 , 0 , 25)). 1. Suppose the agents do not trade. Is the resulting allocation Pareto-optimal? If yes, calculate the pareto weights λ 1 , λ 2 , λ 3 that induce this allocation via the planner’s problem. If not, find a Pareto improvement (i.e., a new allocation which all agents prefer, one strictly). 2. Suppose that the available goods are divided equally between the three agents. (a) Is this allocation Pareto-optimal? If yes, calculate the associated set of Pareto weights. If not, find a Pareto improvement. (b) Does this allocation satisfy voluntary participation? 2 Competitive equilibrium Two agents meet to trade two goods. Agent 1 has utility function u 1 ( x 1 ) = ( x 1 1 ) 1 / 4 ( x 1 2 ) 3 / 4 and endowment e 1 = (1 , 0) , while agent 2 has utility function u 2 ( x 2 ) = ( x 2 1 ) 3 / 4 ( x 2 2 ) 1 / 4 and 1 endowment e 2 = (0 , 2) 1. Find a competitive equilibrium of this economy. 2. A well-meaning social planner re-allocates endowments in an attempt to achieve a fair outcome. The new endowments he comes up with are ̃ e 1 = ̃ e 2 = (1 / 2 , 1) Is either agent better off under this allocation (i.e., consuming ̃ e 1 and ̃ e 2 , respectively) than the equilibrium allocation from the previous part? 3. Now suppose markets are allowed to re-open after the social planner reallocates en- dowments as in the previous part. Find a competitive equilibrium of the new economy. 3 The second welfare theorem Two agents meet to trade two goods. Agent 1 has utility function u 1 ( x 1 ) = ln x 1 1 + ln x 1 2 and endowment e 1 = (3 , 4) , while agent 2 has utility function u 2 ( x 2 ) = 2 √ x 2 1 + √ x 2 2 and endowment e 2 = (0 , 8) 1. Find a competitive equilibrium of this economy. 2. A social planner wishes to redistribute wealth so that both agents consume an equal amount of good 2. Find a competitive equilibrium with transfers which achieves this outcome. 3. Calculate a set of Pareto weights associated with each of the allocations you found in the previous two parts, with the normalization that each set of weights sums to 1. 4 A market for lending We have seen that we can treat the problem of choosing consumption over time as just a version of a regular consumer problem. Let’s see now that we can as well use the notion of competitive equilibrium to think about the determination of interest rates. With this in mind, consider two agents trade intertemporal consumption in an exchange economy. Each lives for two periods and has utility function u j ( c j ) = √ c j 1 + δ √ c j 2 , 2 where δ = 1 / √ 3 Endowments are e 1 = (2 , 24) and e 2 = (10 , 12) Both agents can borrow or save at interest rate r, subject to the usual budget constraint of paying back all first-period borrowing in the second period. 1. Suppose r is determined as part of a competitive equilibrium. Calculate r. 2. Which agent acts as the creditor in period 1, and how much do they lend? How much does the borrower pay back in the second period? 3