3. First and Second Welfare TheoremsThere are two goods A and B and two inputs K and L. The production functions areAK L-aBK{L,Ya%3DY, =1-8The production functions display the standard properties, including constant returnsto scale.A representative household has utilityU(ca, Cb)where c; is the consumption of good i. The total supply of each factor is fixedK = Ka+K,L = La+Lo(a) A social planner allocates consumption of both goods and determines productionand the allocation of inputs. Derive the optimal allocation and demonstratethe marginal rate of substitution is equal to the marginal rate of transformationbetween the consumption goods. Draw a graph of the production possibilityfrontier.(b) Now assume a market system.i. Households own the labor and capital, which is rented to firms. The householdfaces a budget constraintwaLa + wyLb +raKa + rµK» 2 PaCa + PpCbwhere wi is the real wage paid in industry i and r is the rental rate inindustry i. The total supply of each factor is fixed, as described above. Solvethe household's maximization problem.ii. Firms maximize profits. A firm's profits in industry A areTha = PaFa(Ka, L.) – W.L – raKaand a firm's profits in industry B areTb = P,F(K, L1) – w,Ls – roK,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 problemand the competitive equilibrium - are equivalent. Provide a statement of the Firstand Second Welfare theorems using this example.

Answered: and Second Welfare Theorems There are…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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3. First and Second Welfare Theorems
There are two goods A and B and two inputs K and L. The production functions are
AK L-a
BK{L,
Ya
%3D
Y, =
1-8
The production functions display the standard properties, including constant returns
to scale.
A representative household has utility
U(ca, Cb)
where c; is the consumption of good i. The total supply of each factor is fixed
K = Ka+K,
L = La+Lo
(a) A social planner allocates consumption of both goods and determines production
and the 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 market system.
i. Households own the labor and capital, which is rented to firms. The household
faces a budget constraint
waLa + wyLb +raKa + rµK» 2 PaCa + PpCb
where wi is the real wage paid in industry i and r 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
Tha = PaFa(Ka, L.) – W.L – raKa
and a firm's profits in industry B are
Tb = P,F(K, L1) – w,Ls – roK,
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 statement of the First
and Second Welfare theorems using this example.

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Hi , as you have posted multiple questions , as per our guidelines we can only solve 1 at a time . Kindly repost the question to get remaining parts solved.

Marginal Rate of transformation refers to rate which one good is substituted for production , for the utilization of the limited resources available in the economy (So remain on same production possibility frontier ).

Marginal Rate of substitution refers to the rate at which a consumer substitutes one good for at constant utility level .

Now ,

We have Utility Function : U(ca ,cb)

Production functions :

Ya = A Ka^$α$La^{$1 - α$} (For good A )

Yb = B Kb^$β$ Lb^{$1 - β$} (For good B )