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5. Consider the following economy. The representative consumer's preference is d...

Mar 25, 2024

5. Consider the following economy. The representative consumer's preference is described by

U(C, l)=\log (C)+a \log (l) .

The representative firm produces output by combining capital and labor input:

Y=z F\left(K, N^{d}\right)=z K N^{d} .

The government expenditure is 0 . For simplicity, assume that

z=1, a=2, K=1, G=0

h=24

. Solve for the social planner's problem. Specifically, solve for optimal employment, leisure, consumption, output, and the implied wage and profit. 5

Solution by Steps

step 1

Given the utility function

U(C, l) = \log(C) + a \log(l)

with

a = 2

, the production function

Y = z K N^d

with

z = 1

K = 1

, and the constraints

C + l + N^d = 24

Y = C

G = 0

, and

h = 24

, we need to solve for the optimal values of employment

N^d

, leisure

l

, consumption

C

, output

Y

, and the implied wage and profit

step 2

Since

G = 0

and

Y = C

, the constraint simplifies to

C + l + N^d = h

. With

h = 24

, we have

C + l + N^d = 24

step 3

To maximize the utility function subject to the constraint, we can set up the Lagrangian:

\mathcal{L} = \log(C) + 2\log(l) + \lambda(24 - C - l - N^d)

step 4

Taking the partial derivatives of

\mathcal{L}

with respect to

C

l

, and

N^d

and setting them to zero gives us the system of equations:

\frac{1}{C} - \lambda = 0

\frac{2}{l} - \lambda = 0

, and

-\lambda = 0

step 5

Solving the system of equations, we find that

\lambda = \frac{1}{C} = \frac{2}{l}

, which implies

l = 2C

step 6

Substituting

l = 2C

into the constraint

C + l + N^d = 24

, we get

C + 2C + N^d = 24

, which simplifies to

3C + N^d = 24

step 7

Since

Y = C

and

Y = K N^d

, we have

C = N^d

. Substituting

C = N^d

into

3C + N^d = 24

, we get

4C = 24

, which gives

C = 6

step 8

With

C = 6

, we find

l = 2C = 12

and

N^d = C = 6

step 9

The output

Y

is then

Y = K N^d = 1 \cdot 6 = 6

step 10

The implied wage can be found from the marginal product of labor, which is

\frac{\partial Y}{\partial N^d} = K = 1

. Since the entire output goes to consumption, there are no profits

Answer

Optimal employment

N^d = 6

, leisure

l = 12

, consumption

C = 6

, output

Y = 6

, wage

w = 1

, and profit

\pi = 0

Key Concept

Lagrangian optimization in economics

Explanation

The Lagrangian method is used to find the maximum of a function subject to constraints. In this case, it is used to maximize the utility function given the constraints of the economy.