The Constant Elasticity of Substitution (CES) production function is a flexible way to de- scribe how a firm combines capital and labor to produce output, allowing for different levels of substitutability between the two inputs. The elasticity of substitution, denoted by σ, measures how easily the firm can substitute capital for labor (or vice versa) while maintaining the same output level. The parameter p is related to the elasticity of substi- tution by the formula σ = 1/(1 - p). Now, let's consider a firm that operates for two periods (t and t + 1) and produces output according to the CES production function: F(K₁, N₁) = [aK? + (1 - a) No] 1, 0
The Constant Elasticity of Substitution (CES) production function is a flexible way to de- scribe how a firm combines capital and labor to produce output, allowing for different levels of substitutability between the two inputs. The elasticity of substitution, denoted by σ, measures how easily the firm can substitute capital for labor (or vice versa) while maintaining the same output level. The parameter p is related to the elasticity of substi- tution by the formula σ = 1/(1 - p). Now, let's consider a firm that operates for two periods (t and t + 1) and produces output according to the CES production function: F(K₁, N₁) = [aK? + (1 - a) No] 1, 0
Chapter9: Production Functions
Section: Chapter Questions
Problem 9.10P
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![The Constant Elasticity of Substitution (CES) production function is a flexible way to de-
scribe how a firm combines capital and labor to produce output, allowing for different
levels of substitutability between the two inputs. The elasticity of substitution, denoted
by σ, measures how easily the firm can substitute capital for labor (or vice versa) while
maintaining the same output level. The parameter p is related to the elasticity of substi-
tution by the formula σ = 1/(1 - p).
Now, let's consider a firm that operates for two periods (t and t + 1) and produces
output according to the CES production function:
F(K₁, N₁) = [aK? + (1 - a) No] 1, 0<a<1, p<1
The parameter a determines the relative importance of capital and labor in the pro-
duction process, with a higher a indicating that capital is more important and a lower a
indicating that labor is more important.
The CES production function is particularly useful for analyzing how changes in the
prices of capital and labor affect the firm's optimal choice of inputs, and how the firm's
ability to substitute between capital and labor influences its production decisions.
The firm starts with an initial capital stock K, in period t and can invest I, to increase
its capital stock in period t + 1. The capital accumulation equation is:
=
K+1 (18)Kt + It
where is the depreciation rate (0 < 5 < 1).
The firm borrows an amount BI, from a financial intermediary to finance its invest-
ment It at the risk-free interest rate rt.
The firm's dividends in each period are:
-
D₁ = A₁F(K₁, N₁) — w₁Nt
-
D+1=A+F(K₁+1, N+1) + (18) K₁+1 - W₁+1N++1 − (1 + r₁) BI₁
The firm's objective is to maximize its value, which is the present discounted value of
dividends:
1
V₁ = Dt +
·D₁+1
1+rt
(a) Writing Down the Firm's Optimization Problem
Write down the firm's optimization problem in terms of choosing N₁ and I, to maximize
V₁, subject to the capital accumulation equation and the constraint that BI₁ = I₁.
(b) Deriving First-Order Conditions
Derive the first-order conditions for N₁ and It.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e210e3b-7b89-45ac-81ce-3e543ee87969%2F4ad354e1-a9df-4abd-984f-f30f32c344d9%2F5xgw5zn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The Constant Elasticity of Substitution (CES) production function is a flexible way to de-
scribe how a firm combines capital and labor to produce output, allowing for different
levels of substitutability between the two inputs. The elasticity of substitution, denoted
by σ, measures how easily the firm can substitute capital for labor (or vice versa) while
maintaining the same output level. The parameter p is related to the elasticity of substi-
tution by the formula σ = 1/(1 - p).
Now, let's consider a firm that operates for two periods (t and t + 1) and produces
output according to the CES production function:
F(K₁, N₁) = [aK? + (1 - a) No] 1, 0<a<1, p<1
The parameter a determines the relative importance of capital and labor in the pro-
duction process, with a higher a indicating that capital is more important and a lower a
indicating that labor is more important.
The CES production function is particularly useful for analyzing how changes in the
prices of capital and labor affect the firm's optimal choice of inputs, and how the firm's
ability to substitute between capital and labor influences its production decisions.
The firm starts with an initial capital stock K, in period t and can invest I, to increase
its capital stock in period t + 1. The capital accumulation equation is:
=
K+1 (18)Kt + It
where is the depreciation rate (0 < 5 < 1).
The firm borrows an amount BI, from a financial intermediary to finance its invest-
ment It at the risk-free interest rate rt.
The firm's dividends in each period are:
-
D₁ = A₁F(K₁, N₁) — w₁Nt
-
D+1=A+F(K₁+1, N+1) + (18) K₁+1 - W₁+1N++1 − (1 + r₁) BI₁
The firm's objective is to maximize its value, which is the present discounted value of
dividends:
1
V₁ = Dt +
·D₁+1
1+rt
(a) Writing Down the Firm's Optimization Problem
Write down the firm's optimization problem in terms of choosing N₁ and I, to maximize
V₁, subject to the capital accumulation equation and the constraint that BI₁ = I₁.
(b) Deriving First-Order Conditions
Derive the first-order conditions for N₁ and It.
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