3. First and Second Welfare Theorems There are two goods A and B and two inputs K and L. The production functions are AKOL-a 1-3 Ya %3D Y, = The production functions display the standard properties, including constant returns to scale. A representative household has utility U(Ca, C6) where c is the consumption of good i. The total supply of each factor is fixed K = Ka+ K, L = La+L» (a) A social planner allocates consumption of both goods and deterımines 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
3. First and Second Welfare Theorems There are two goods A and B and two inputs K and L. The production functions are AKOL-a 1-3 Ya %3D Y, = The production functions display the standard properties, including constant returns to scale. A representative household has utility U(Ca, C6) where c is the consumption of good i. The total supply of each factor is fixed K = Ka+ K, L = La+L» (a) A social planner allocates consumption of both goods and deterımines 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
Chapter1: Making Economics Decisions
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Transcribed Image Text:3. First and Second Welfare Theorems
There are two goods A and B and two inputs K and L. The production functions are
Ya
AK L-a
%3D
Y,
BK{L
The production functions display the standard properties, including constant returns
to scale.
A representative household has utility
U(ca, Cs)
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.
Expert Solution
Step 1
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 La (For good A )
Yb = B Kb Lb (For good B )
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