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 piece of land. The agent can spend time either working or enjoying leisure. A representative consumer's utility function is u (x; r) = ln x + ln r, where x is consumption of goods and r is leisure. Each firrm hires consumers to work and rents land from consumers to produce goods: y = f (L; D) = (LD)^(1/2), where L is the amount of labor hired and D is the amount of land rented. Both consumers and firms take goods price p, wage rate w and rent rate q as given. Normalize p* = 1.
(a) Set up a representative consumer's utility-maximization problem. Derive the Marshallian demands x (w; q) and r (w; q). [Hint: A consumer's income is the sum of labor income and rent income: w (1 ? r) + q.]
(b) Set up a representative firm's profit-maximization problem. Write down the first-order conditions regarding the choices of L and D.
(c) In the Walrasian equilibrium, markets of goods, labor and land all clear. Set up all the market-clearing conditions for each of the 3 markets.
(d) Use the results from (a)-(c) to calculate equilibrium wage rate w*, rent rate q*, and the consumer's optimal choices of x* and r*.
(e) Imagine a social planner who cares about all agents equally. The planner seeks to maximize a representative agents utility subject to the feasibility constraint:
max ln x + ln r s:t: 100x
Solve for the planner's solutions and compare them with the results in (d). Do the First and Second Welfare Theorems hold in this economy?