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Suppose the production of airframes is character- ized by a Cobb–Douglas production function: Q - LK. The marginal products for this production function are MPL - K and MPK - L. Suppose the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.?
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Suppose the production of airframes is character- ized by a Cobb–Dougl...
Introduction:
In this scenario, we are given a Cobb-Douglas production function for airframe production. The production function is represented by Q = L^αK^β, where Q is the quantity of airframes produced, L is the quantity of labor, K is the quantity of capital, and α and β are the output elasticities of labor and capital respectively. We are also given the prices of labor and capital, and we need to find the cost-minimizing combination of labor and capital to produce a given quantity of airframes.
Step 1: Calculate the Marginal Products:
The marginal product of labor (MPL) is the partial derivative of the production function with respect to labor (L), holding capital (K) constant. Similarly, the marginal product of capital (MPK) is the partial derivative of the production function with respect to capital (K), holding labor (L) constant.
MPL = ∂Q/∂L = αL^(α-1)K^β
MPK = ∂Q/∂K = βL^αK^(β-1)
Step 2: Set up the Cost-Minimization Problem:
To find the cost-minimizing combination of labor and capital, we need to minimize the cost of production subject to the constraint of producing a given quantity of airframes. The cost of production is given by the equation C = wL + rK, where w is the price of labor and r is the price of capital.
Step 3: Apply the Lagrange Multiplier Method:
To solve the cost-minimization problem, we can use the Lagrange multiplier method. We set up the Lagrangian function as follows:
L = C + λ(Q - 121,000)
Where λ is the Lagrange multiplier and Q is the given quantity of airframes to be produced.
Step 4: Take Partial Derivatives:
Next, we need to take partial derivatives of the Lagrangian function with respect to labor (L), capital (K), and the Lagrange multiplier (λ), and set them equal to zero to find the critical points.
∂L/∂L = w - αλL^(α-1)K^β = 0
∂L/∂K = r - βλL^αK^(β-1) = 0
∂L/∂λ = Q - 121,000 = 0
Step 5: Solve the System of Equations:
The system of equations can be solved simultaneously to find the values of labor (L) and capital (K) that minimize the cost of production.
From the first equation, we can solve for λ in terms of L and K:
λ = w / (αL^(α-1)K^β)
Substituting this expression for λ into the second equation, we can solve for r in terms of L and K:
r = βwL^αK^(β-1) / (αL^(α-1)K^β)
Next, we can substitute the expressions for λ and r into the third equation and solve for L:
Q - 121,000 = wL / (αL^(α-1)K^β)
Simplifying this equation, we get:
In this scenario, we are given a Cobb-Douglas production function for airframe production. The production function is represented by Q = L^αK^β, where Q is the quantity of airframes produced, L is the quantity of labor, K is the quantity of capital, and α and β are the output elasticities of labor and capital respectively. We are also given the prices of labor and capital, and we need to find the cost-minimizing combination of labor and capital to produce a given quantity of airframes.
Step 1: Calculate the Marginal Products:
The marginal product of labor (MPL) is the partial derivative of the production function with respect to labor (L), holding capital (K) constant. Similarly, the marginal product of capital (MPK) is the partial derivative of the production function with respect to capital (K), holding labor (L) constant.
MPL = ∂Q/∂L = αL^(α-1)K^β
MPK = ∂Q/∂K = βL^αK^(β-1)
Step 2: Set up the Cost-Minimization Problem:
To find the cost-minimizing combination of labor and capital, we need to minimize the cost of production subject to the constraint of producing a given quantity of airframes. The cost of production is given by the equation C = wL + rK, where w is the price of labor and r is the price of capital.
Step 3: Apply the Lagrange Multiplier Method:
To solve the cost-minimization problem, we can use the Lagrange multiplier method. We set up the Lagrangian function as follows:
L = C + λ(Q - 121,000)
Where λ is the Lagrange multiplier and Q is the given quantity of airframes to be produced.
Step 4: Take Partial Derivatives:
Next, we need to take partial derivatives of the Lagrangian function with respect to labor (L), capital (K), and the Lagrange multiplier (λ), and set them equal to zero to find the critical points.
∂L/∂L = w - αλL^(α-1)K^β = 0
∂L/∂K = r - βλL^αK^(β-1) = 0
∂L/∂λ = Q - 121,000 = 0
Step 5: Solve the System of Equations:
The system of equations can be solved simultaneously to find the values of labor (L) and capital (K) that minimize the cost of production.
From the first equation, we can solve for λ in terms of L and K:
λ = w / (αL^(α-1)K^β)
Substituting this expression for λ into the second equation, we can solve for r in terms of L and K:
r = βwL^αK^(β-1) / (αL^(α-1)K^β)
Next, we can substitute the expressions for λ and r into the third equation and solve for L:
Q - 121,000 = wL / (αL^(α-1)K^β)
Simplifying this equation, we get:
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Suppose the production of airframes is character- ized by a Cobb–Douglas production function: Q - LK. The marginal products for this production function are MPL - K and MPK - L. Suppose the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.?
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Suppose the production of airframes is character- ized by a Cobb–Douglas production function: Q - LK. The marginal products for this production function are MPL - K and MPK - L. Suppose the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.? for Commerce 2024 is part of Commerce preparation. The Question and answers have been prepared according to the Commerce exam syllabus. Information about Suppose the production of airframes is character- ized by a Cobb–Douglas production function: Q - LK. The marginal products for this production function are MPL - K and MPK - L. Suppose the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.? covers all topics & solutions for Commerce 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose the production of airframes is character- ized by a Cobb–Douglas production function: Q - LK. The marginal products for this production function are MPL - K and MPK - L. Suppose the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.?.
Suppose the production of airframes is character- ized by a Cobb–Douglas production function: Q - LK. The marginal products for this production function are MPL - K and MPK - L. Suppose the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.? for Commerce 2024 is part of Commerce preparation. The Question and answers have been prepared according to the Commerce exam syllabus. Information about Suppose the production of airframes is character- ized by a Cobb–Douglas production function: Q - LK. The marginal products for this production function are MPL - K and MPK - L. Suppose the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.? covers all topics & solutions for Commerce 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose the production of airframes is character- ized by a Cobb–Douglas production function: Q - LK. The marginal products for this production function are MPL - K and MPK - L. Suppose the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.?.
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Suppose the production of airframes is character- ized by a Cobb–Douglas production function: Q - LK. The marginal products for this production function are MPL - K and MPK - L. Suppose the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.?, a detailed solution for Suppose the production of airframes is character- ized by a Cobb–Douglas production function: Q - LK. The marginal products for this production function are MPL - K and MPK - L. Suppose the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.? has been provided alongside types of Suppose the production of airframes is character- ized by a Cobb–Douglas production function: Q - LK. The marginal products for this production function are MPL - K and MPK - L. Suppose the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.? theory, EduRev gives you an
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