# A define a competitive equilibrium hint do not forget

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(a) Define a competitive equilibrium. (Hint: Do not forget to include the real wage.) (b) Solve for a competitive equilibrium. Answer
w = k α (1 α ) n α 20. This question uses a model of two countries with two time periods to make some predictions about international correlations in income and consumption. Suppose that there is one good. Each country receives a non-storable endowment of it in each time period. The first country has endowments y 1 , y 2 in the two time periods, and a typical consumer there seeks to maximize U = ln( c 1 ) + β ln( c 2 ) , where c denotes consumption. The other country receives endowments x 1 , x 2 . A typical consumer there seeks to maximize W = ln( d 1 ) + β ln( d 2 ) , where d is consumption. We shall suppose that we observe a competitive equilibrium. (a) Find a competitive equilibrium by solving the following Pareto optimum problem: max λ · U + (1 λ ) · W subject to resource constraints. Solve for consumptions and relative prices in terms of endowments and λ . (b) Suppose y 1 = 2, y 2 = 1, x 1 = 1, and x 2 = 2. Let β = 0 . 9. Find the value of λ such that the Pareto optimum satisfies all the conditions of a competitive equilibrium. (c) With the same values, solve for the trade balance in both countries and time periods. (d) Show that there is intertemporal external balance. (e) Are any qualitative predictions of this model inconsistent with empirical evidence? Answer
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which holds with these values. (e) The predictions of a procyclical trade balance and perfect consumption correlations are not supported empirically. 21. One of the central predictions of financial economics is that assets with riskier payoffs should have higher returns. To study this idea simply, take a two-period, competitive exchange economy with a representative agent who has non-storable endowment y in period 1, and faces a distribution of risky income for period 2 given by: State 1: y + D with probability 0.5; State 2: y D with probability 0 . 5. Assume the agent has preferences u = ln c 1 + E 1 β ln c 2 .