Question

 Consider an economy where an agent produces a consumption good with labor.

 The agent has the following preferences:

 u(c, l) = √c+ α√l

 where c is consumption, I is leisure, and a is a parameter. An agent splits her time between labor and leisure such that n +l= 1 where n is how much she works. Prices for the consumption good is p and labor is w. The agent works at a competitive firm (i.e. zero-profit) which transforms labor using the technology

 y =F(N) using labor with wage w.

 1. Define a competitive equilibrium. You need to include:

 .The household problem 

 .The firm problem 

 .Market clearing 

 2. Suppose F(N)= θN. Solve for the optimal c, n, and l.

Answer 1

According to the question:

1. The household problem:

Households maximize the consumption c and labor n

Firm problem:

Firms try to maximize their profit. The production function is given by:

y = F(N)

Market clearing condition :

The firm is able to sell all its produce.

Y = F(N) =c

2. Firm's problem

Household problem:

Using equation (1) and (2):

Using this equation (3):

l = 1-n