Model Answer
0 min readIntroduction
The Cobb-Douglas production function is a widely used economic model representing the relationship between inputs and output. It assumes that production is determined by combining labor and capital, with constant returns to scale. Understanding how a firm optimally chooses its input mix to maximize output for a given cost is crucial for analyzing firm behavior and industry dynamics. The expansion path illustrates this optimal input combination as output levels change. This answer will derive the expansion path for a firm operating with a Cobb-Douglas production function, demonstrating the firm’s cost-minimizing input choices.
Derivation of the Expansion Path
Let the Cobb-Douglas production function be represented as:
Q = A * Kα * Lβ
Where:
- Q = Output
- A = Total factor productivity
- K = Capital
- L = Labor
- α and β = Output elasticities of capital and labor, respectively (0 < α, β < 1)
1. Cost Minimization Problem
The firm aims to minimize cost (C) subject to a given output level (Q). The cost function is:
C = rK + wL
Where:
- r = Rental rate of capital
- w = Wage rate of labor
To minimize cost, we use the Lagrangian method:
L = rK + wL + λ(A * Kα * Lβ - Q)
2. First-Order Conditions (FOCs)
Taking partial derivatives with respect to K, L, and λ and setting them to zero, we get:
- ∂L/∂K = r + λAαK(α-1)Lβ = 0 => r = -λAαK(α-1)Lβ
- ∂L/∂L = w + λAβKαL(β-1) = 0 => w = -λAβKαL(β-1)
- ∂L/∂λ = A * Kα * Lβ - Q = 0 => A * Kα * Lβ = Q
3. Deriving the Optimal Input Ratio (K/L)
Dividing the first FOC by the second FOC, we get:
r/w = (λAαK(α-1)Lβ) / (λAβKαL(β-1))
Simplifying, we obtain:
r/w = (α/β) * (L/K)
Rearranging to find the optimal capital-labor ratio (K/L):
K/L = (β/α) * (r/w)
4. The Expansion Path
The expansion path shows the optimal combination of K and L for different levels of output (Q). Substituting the optimal K/L ratio into the production function (A * Kα * Lβ = Q):
A * ( (β/α) * (r/w) * L )α * Lβ = Q
Simplifying, we get:
A * (β/α)α * (r/w)α * Lα + β = Q
Since α + β = 1 (constant returns to scale in the Cobb-Douglas function), we have:
L = [Q / (A * (β/α)α * (r/w)α)]1/(α+β) = [Q / (A * (β/α)α * (r/w)α)]1
And, substituting this value of L back into K/L = (β/α) * (r/w), we get the optimal capital level (K):
K = (β/α) * (r/w) * L
Therefore, the expansion path is defined by the relationship between K and L, given the output level (Q), input prices (r and w), and the parameters of the production function (A, α, and β). It shows how the firm scales up its inputs in a cost-minimizing manner as output increases.
Conclusion
The derivation of the expansion path for a Cobb-Douglas production function demonstrates the firm’s optimal input choices in response to changes in output levels, given input prices. The expansion path is a crucial concept in understanding firm behavior and cost minimization. The ratio of capital to labor is determined by the relative prices of capital and labor, and the output elasticities. This analysis provides a foundational understanding of production theory and its application in economic modeling.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.