production functions

First and second welfare theorems

There are two goods A and B and two inputs K and L. The production functions are

Y_a=AK_a^αL_a^1-α

Y_b=BK_b^βL_b^β

The production functions display the standard properties including constatnt returns to scale

A representative household has utility

U(c_a,c_b)

where c_i is the consumption of good i. The total supply of each factor is fixed.

K=K_a+K_b

L=L_a+L_b

a) A social planner allocates consumption of both goods and determines production and allocation of inputs. Derive the optimal allocation and demonstrate the marginal rate of substitution is equal to the marginal rate of transformation between the consumption goods. Draw a graph of the production possibility frontier.

b) Now assume a maret system

i. Households own the labor and capital which is rented to firms. The households face a budget constraint

w_aL_a+w_bL_b+r_aK_a+≥P_aC_a+P_bC_b

where w_i is the real wage paid in industry i and r_i is the rental rate in industry i. The total supply of each factor is fixed, as described above. Solve the household's maximization problem.

ii. Firms maximize profits. A firm's profits in industry A are 

π_a=P_aF_a(K_a,L_a)-w_aL_a-r_aK_a 

and a firm's profit in industry B are

π_b=P_bF_b(K_b,L_b)-w_bL_b-r_bK_b 

solve the maximization problem for each firm

iii. state the equilibrium conditions assuming households and firms are price takers.

c) Demonstrate how the two allocations -the solution to the social planning problem and the competitive equilibrium - are equivalent. Provide a statemnt of the first and second welfare theorem using this example.