Consider an economy where the representative consumer has the utility function u(X, R)=x + R where...

 

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Consider an economy where the representative consumer has the utility function u(X, R)=√x + √R where X is consumption and R is leisure. The consumer is endowed with T = 24 hours of time. The representative firm has production technology f(L) = L. (a) Solve for the unconditional labor demand function Ld (w), the supply function X³ (w), and the profit function (w). (b) Formulate Min's utility maximization problem. (c) Solving Min's utility maximization problem, calculate the demand functions for consumption, Xd (w), and leisure, Rd (w). (You may want to divide your cases into when w ≤ 1, and when w >1 and think about the problem separately.) (d) Impose market clearing conditions and find the equilibrium wage w*. (e) Suppose the market is abolished and there is a central planner who knows both the utility function and the production function. Formulate the central planner's problem. (f) Solve the central planner's problem. Verify that the allocations X* and R* are the same as the equilibrium quantities from the market. Consider an economy where the representative consumer has the utility function u(X, R)=√x + √R where X is consumption and R is leisure. The consumer is endowed with T = 24 hours of time. The representative firm has production technology f(L) = L. (a) Solve for the unconditional labor demand function Ld (w), the supply function X³ (w), and the profit function (w). (b) Formulate Min's utility maximization problem. (c) Solving Min's utility maximization problem, calculate the demand functions for consumption, Xd (w), and leisure, Rd (w). (You may want to divide your cases into when w ≤ 1, and when w >1 and think about the problem separately.) (d) Impose market clearing conditions and find the equilibrium wage w*. (e) Suppose the market is abolished and there is a central planner who knows both the utility function and the production function. Formulate the central planner's problem. (f) Solve the central planner's problem. Verify that the allocations X* and R* are the same as the equilibrium quantities from the market.

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Solutiog Representative Consumer U 2Rx R 0 LR 24 t seisure labor Supplied a Fir... View the full answer