topic 4.1

Topic 4.1

A MODEL OF PRODUCTION


Consider a closed economy with a production function

Y = F(K,L) = A.K1/3.L2/3

A = productivity parameter
K = capital
L = labor

this production function uses a Cobb-Douglas form (Y = Ka.Y1-a)

this function generates constant returns to scale (since a + (1-a) = 1)

F(2K,2L) = 2.F(K,L)

if a + (1-a) <> 1 we have increasing returns to scale

how should capital and labor be allocated to maximize profit?

assuming this is the production function for a single firm acting competitively, and setting thr price of the good to 1

max(over K and L)

profit = A.F(k,L) - r.K - w.L

we get results for L and K


these results make sense:

as w increases the firm decreases L
as Y increases the firm increases L

What the firm is doing is to adjust L and K to match marginal productivities with the values of r and w. r and w are taken as given by the individual firm.

The macro level determination of r and w can be illustrated by two diagrams that show equilibrium in the markets for L and K.

As a consequence of the Cobb-Douglas formulation the total output is absorbed by the returns to labor and capital.

w*L* + r*K* = Y*