Consider an economy with one consumption good, 100 identical consumers and 100 identical firms. Each consumer is endowed with one unit of time and one unit of capital. The agent can spend time either working or enjoying leisure. A represen- tative consumerís utility function is u (x; l) = ln x + 2 ln l, where x is consumption of goods and l is leisure. Each firm hires consumers to work and rents capital from consumers to produce goods: y = f (K; L) = K^1/2L^1/2, where L is the amount of labor hired and K is the amount of capital rented. Both consumers and firms take goods price p, wage rate w and rental rate r as given. Normalize p = 1. 1. Set up a representative consumerís utility-maximization problem. Derive the Marshallian demands for consumption and leisure as functions of wage w and rental rate r 2. Set up a representative firm's profit-maximization problem. Write down the first-order conditions regarding the choices of K and L 3. Set up all the market-clearing conditions.
Consider an economy with one consumption good, 100 identical consumers and 100
identical firms. Each consumer is endowed with one unit of time and one unit of
capital. The agent can spend time either working or enjoying leisure. A represen-
tative consumerís utility function is u (x; l) = ln x + 2 ln l, where x is consumption
of goods and l is leisure. Each firm hires consumers to work and rents capital from
consumers to produce goods: y = f (K; L) = K^1/2L^1/2, where L is the amount of
labor hired and K is the amount of capital rented. Both consumers and firms take
goods price p, wage rate w and rental rate r as given. Normalize p = 1.
1. Set up a representative consumerís utility-maximization problem.
Derive the Marshallian
wage w and rental rate r
2. Set up a representative firm's profit-maximization problem. Write
down the first-order conditions regarding the choices of K and L
3. Set up all the market-clearing conditions.
4. Use the results from 1-3, to calculate equilibrium wage rate w*,
rent rate r*, and the consumer optimal choices of x* and l*
5.Imagine a social planner who cares about all agents equally. The
planner seeks to maximize a representative agentís utility subject to the feasi-
bility constraint:
max
(x;l) ln x + 2 ln l s:t: 100x <= 100(1-l)^1/2
Solve for the plannerís solutions and compare them with the results in (4). Do
the First and Second Welfare Theorems hold in this economy?
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