The project consists of multiple prompts related to what is known as the Cobb-DouglasProduction Model. Each prompt takes you closer to the complete solution to the followingoptimization problem. Suppose the quantity, q, of a product manufactures depends on the number of workers.W , and the amount of capital invested, K, and is represented by the Cobb-Douglas functionq = 64W^(3/4)K^(1/4). Suppose further that labor costs $18 per worker and capital costs $28 per unit, and thebudget is $4600.b. What is the optimum number of workers and the optimum number of units of capital?c. Show that at the optimum values of W and K, the ratio of the marginal productivityof labor( ∂q/∂W) to the marginal productivity of capital( ∂q/∂K) is the same as the ratio of the cost of a unit of labor to the cost of a unit of capital.d. Now assume that the product manufactured can be sold for $10 per unit. Give a function P (K, W ) for the profit associated with producing q units.e. What optimum combination of capital and labor corresponds to maximum profit if there are no budget constraints?f. What is the optimal profit when we apply the budget constraint of $4600 to the formulation?g. Recompute the optimum values of W , K and profit P (K, W ) when the budget is increased by one dollar.h. Let λ be the Lagrange multiplier. Does increasing the budget by $1 allow the production of λ extra units of the product? Explain why.i. Does the value of λ change if the budget changes from $4600 to $5600? j. What condition must a Cobb-Douglas production function q = cK^(α)W^(β) satisfy toensure that the marginal increase of production is not affected by the size of the budget?
The project consists of multiple prompts related to what is known as the Cobb-DouglasProduction Model. Each prompt takes you closer to the complete solution to the followingoptimization problem. Suppose the quantity, q, of a product manufactures depends on the number of workers.W , and the amount of capital invested, K, and is represented by the Cobb-Douglas functionq = 64W^(3/4)K^(1/4). Suppose further that labor costs $18 per worker and capital costs $28 per unit, and thebudget is $4600.b. What is the optimum number of workers and the optimum number of units of capital?c. Show that at the optimum values of W and K, the ratio of the marginal productivityof labor( ∂q/∂W) to the marginal productivity of capital( ∂q/∂K) is the same as the ratio of the cost of a unit of labor to the cost of a unit of capital.d. Now assume that the product manufactured can be sold for $10 per unit. Give a function P (K, W ) for the profit associated with producing q units.e. What optimum combination of capital and labor corresponds to maximum profit if there are no budget constraints?f. What is the optimal profit when we apply the budget constraint of $4600 to the formulation?g. Recompute the optimum values of W , K and profit P (K, W ) when the budget is increased by one dollar.h. Let λ be the Lagrange multiplier. Does increasing the budget by $1 allow the production of λ extra units of the product? Explain why.i. Does the value of λ change if the budget changes from $4600 to $5600? j. What condition must a Cobb-Douglas production function q = cK^(α)W^(β) satisfy toensure that the marginal increase of production is not affected by the size of the budget?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
Related questions
Question
The project consists of multiple prompts related to what is known as the Cobb-Douglas
Production Model. Each prompt takes you closer to the complete solution to the following
optimization problem. Suppose the quantity, q, of a product manufactures depends on the number of workers.
W , and the amount of capital invested, K, and is represented by the Cobb-Douglas function
q = 64W^(3/4)K^(1/4). Suppose further that labor costs $18 per worker and capital costs $28 per unit, and the
budget is $4600.
b. What is the optimum number of workers and the optimum number of units of capital?
c. Show that at the optimum values of W and K, the ratio of the marginal productivity
of labor( ∂q/∂W) to the marginal productivity of capital( ∂q/∂K) is the same as the ratio of the cost of a unit of labor to the cost of a unit of capital.
d. Now assume that the product manufactured can be sold for $10 per unit. Give a function P (K, W ) for the profit associated with producing q units.
e. What optimum combination of capital and labor corresponds to maximum profit if there are no budget constraints?
f. What is the optimal profit when we apply the budget constraint of $4600 to the formulation?
g. Recompute the optimum values of W , K and profit P (K, W ) when the budget is increased by one dollar.
h. Let λ be the Lagrange multiplier. Does increasing the budget by $1 allow the production of λ extra units of the product? Explain why.
i. Does the value of λ change if the budget changes from $4600 to $5600?
Production Model. Each prompt takes you closer to the complete solution to the following
optimization problem. Suppose the quantity, q, of a product manufactures depends on the number of workers.
W , and the amount of capital invested, K, and is represented by the Cobb-Douglas function
q = 64W^(3/4)K^(1/4). Suppose further that labor costs $18 per worker and capital costs $28 per unit, and the
budget is $4600.
b. What is the optimum number of workers and the optimum number of units of capital?
c. Show that at the optimum values of W and K, the ratio of the marginal productivity
of labor( ∂q/∂W) to the marginal productivity of capital( ∂q/∂K) is the same as the ratio of the cost of a unit of labor to the cost of a unit of capital.
d. Now assume that the product manufactured can be sold for $10 per unit. Give a function P (K, W ) for the profit associated with producing q units.
e. What optimum combination of capital and labor corresponds to maximum profit if there are no budget constraints?
f. What is the optimal profit when we apply the budget constraint of $4600 to the formulation?
g. Recompute the optimum values of W , K and profit P (K, W ) when the budget is increased by one dollar.
h. Let λ be the Lagrange multiplier. Does increasing the budget by $1 allow the production of λ extra units of the product? Explain why.
i. Does the value of λ change if the budget changes from $4600 to $5600?
j. What condition must a Cobb-Douglas production function q = cK^(α)W^(β) satisfy to
ensure that the marginal increase of production is not affected by the size of the budget?
ensure that the marginal increase of production is not affected by the size of the budget?
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