First and second welfare theorems
There are two goods A and B and two inputs K and L. The production functions are
Ya=AKaαLa1-α
Yb=BKbβLbβ
The production functions display the standard properties including constatnt returns to scale
A representative household has utility
U(ca,cb)
where ci is the consumption of good i. The total supply of each factor is fixed.
K=Ka+Kb
L=La+Lb
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
b) Now assume a maret system
i. Households own the labor and capital which is rented to firms. The households face a budget constraint
waLa+wbLb+raKa+≥PaCa+PbCb
where wi is the real wage paid in industry i and ri 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
πa=PaFa(Ka,La)-waLa-raKa
and a firm's profit in industry B are
πb=PbFb(Kb,Lb)-wbLb-rbKb
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.
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