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?

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

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?

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?

Expert Solution