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The Lagrangian method

Solve this economic application must use the Lagrangian method. Show all work and answer has to be all
correct.
Check the image of the problem first, however if the image does not show up I have copied and pasted it as well.
A producer employs labor (L paid a wage w > 0 per unit) and capital (K paid a rental rate r > 0
per unit) to produce output according to the Cobb-Douglas production function Q = F(L, K) =
AL'Kß, where A > 0, a > 0, and B > 0. The producer chooses its input demands to minimize
its total cost of employing L and K, TC (L, K, r, w) = wL + rK, subject to producing output
quantity Co > 0.
a. Prove that this producer's long run costs exhibit economies of scale for all O if and only if the producer's long
run production function exhibits increasing returns to scale for all L and K.
b. Suppose that capital is fixed at K = Ko in the short run. Prove that this producer's short
run costs exhibit increasing marginal cost of production for all for all Q if and only if the producer's short run
production function exhibits diminishing marginal product of labor

Full Answer Section

2. Finding the Minimum Cost:

We take partial derivatives of L with respect to L, K, and λ and set them equal to zero:

∂L/∂L = w + λaAL^(a-1)K^β = 0
∂L/∂K = r + λβAL^aK^(β-1) = 0
∂L/∂λ = AL^aK^β - Q = 0

3. Analyzing Economies of Scale:

Economies of scale exist if doubling all inputs (L and K) leads to more than double the output (Q). Mathematically, this translates to:

If L' = 2L and K' = 2K, then Q' > 2Q

From the constraint equation:

Q' = A(L')^a(K')^β = A(2L)^a(2K)^β = 2^a 2^β AL^aK^β = 2^(a+β) Q

For economies of scale to hold: 2^(a+β) > 2

This simplifies to: a + β > 1

4. Analyzing Increasing Returns to Scale:

Increasing returns to scale exist if doubling all inputs leads to more than double the output. Mathematically, this translates to:

If L' = 2L and K' = 2K, then Q' > 2^(a+β) Q

From the constraint equation:

Q' = A(L')^a(K')^β = A(2L)^a(2K)^β = 2^a 2^β AL^aK^β = 2^(a+β) Q

For increasing returns to scale to hold: 2^(a+β) > 2^(a+β) (This is always true)

Conclusion:

We can see that economies of scale (a + β > 1) is a necessary and sufficient condition for increasing returns to scale (always true). Therefore, the producer's long-run costs exhibit economies of scale for all outputs if and only if the long-run production function exhibits increasing returns to scale for all L and K.

b) Short-Run Marginal Cost and Diminishing Marginal Product of Labor

1. Short-Run Scenario:

In the short run, capital (K) is fixed at Ko. We need to minimize the cost function subject to this constraint.

2. Setting Up the Lagrangian:

Similar to part (a), we define the Lagrangian function (L) with an additional constraint for fixed capital:

L(L, λ) = wL + rKo + λ (AL^a(Ko)^β - Q)

3. Finding the Minimum Cost:

We take partial derivatives of L with respect to L and λ and set them equal to zero:

∂L/∂L = w + λaAL^(a-1)(Ko)^β = 0
∂L/∂λ = AL^a(Ko)^β - Q = 0

4. Analyzing Marginal Cost:

Marginal cost (MC) is the change in total cost with respect to a change in output (Q). We can derive it using the relationship between the Lagrangian multiplier (λ) and the marginal cost:

MC = ∂TC/∂Q = ∂L/∂λ = AL^a(Ko)^β

5. Analyzing Diminishing Marginal Product of Labor:

Diminishing marginal product of labor occurs when increasing the amount of labor employed (L) leads to a smaller increase in output (Q). Mathematically:

∂Q/∂L < 0 (for a given K)

From the production function:

∂Q/∂L = AaL^(a-1)(Ko)^β

For diminishing marginal product to hold:

AaL^(a-1)(Ko)^β < 0 (since A, a, and Ko are positive)

This simplifies to: a < 1 (assuming L is positive)

6. Relating Marginal Cost and Diminishing Marginal Product:

From the

Sample Answer

Solving the Economic Application with Lagrangian Method

a) Economies of Scale and Increasing Returns to Scale

We can use the Lagrangian method to analyze the relationship between economies of scale and increasing returns to scale.

1. Setting Up the Lagrangian:

Objective Function (Minimize Total Cost): TC(L, K) = wL + rK
Constraint (Production Function): Q = AL^aK^β
Lagrange Multiplier: λ

The Lagrangian function (L) is:

L(L, K, λ) = wL + rK + λ (AL^aK^β - Q)