Solve the problem.
-The Cobb-Douglas function for a new product is given by where x is the number of units of labor and y is the number of units of capital required to produce N(x, y) units of the product. Each unit of
Labor costs $40, and each unit of capital costs $80. If $400,000 has been budgeted for the production of this
Product, determine how this amount should be allocated in order to maximize production, and find the
Maximum production.

Question

Solve the problem.
-The Cobb-Douglas function for a new product is given by  where x is the number of units of labor and y is the number of units of capital required to produce N(x, y) units of the product. Each unit of
Labor costs $40, and each unit of capital costs $80. If $400,000 has been budgeted for the production of this
Product, determine how this amount should be allocated in order to maximize production, and find the
Maximum production.

Quizplus · Accepted Answer

The budget constraint is given by:
40x+80y=400,000
Simplifying the constraint, we get:
x+2y=10,000
Now, we need to maximize the production function N(x,y) subject to the above constraint.
We can use Lagrange Multipliers for this:
L(x,y,λ)= x^1/2 * y^1/2 - λ(x+2y-10,000)

Taking partial derivatives and solving the system of equations, we get:
x=4,000 and y=3,000

Therefore, maximum production is:
N(4,000,3,000) = (4,000)^1/2 * (3,000)^1/2 = 2,000 * (3,000)^1/2

Which is approximately equal to 34,641 units of the product.

Allocating the budget of $400,000 accordingly, we get:
40x+80y=400,000
40(4,000)+80(3,000)=400,000

Therefore, the correct answer is B) 11ebb6d7_4898_cfa3_b437_6d5ad5bb933b_TB8593_11

Solve the Problem. -The Cobb-Douglas Function for a New Product Is Given by Is