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# Production Functions and Optimal Input

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Hello,

I have attached five HW problems that I need help on.

Thanks!

https://brainmass.com/economics/banking/production-functions-and-optimal-input-272742

#### Solution Preview

See the attached file. The chart may not print here. Thanks

1.

You are a manager at the Donnelly Corporation?a mirror and window supplier to the major automakers. Recently, you conducted a study of the production process for your DirectBond
single-side encapsulated window. The results from the study are summarized in the table below, and are based on the 5 units of capital currently available at your plant. Workers are paid \$50 per unit, per-unit capital costs are \$10, and you windows sell for \$5 each. Given this information, optimize your human resource and production decisions. Complete the table and answer to questions that follow:

Labor Capital Output Marginal Product Marginal Revenue Marginal Revenue
of Labor or Product
L K Q MP Price MRP
0 5 0
1 5 10 10 5 50
2 5 30 20 5 100
3 5 60 30 5 150
4 5 80 20 5 100
5 5 90 10 5 50
6 5 95 5 5 25
7 5 95 0 5 0
8 5 90 -5 5 -25
9 5 80 -10 5 -50
10 5 60 -20 5 -100
11 5 30 -30 5 -150

a. Identify the fixed and variable inputs.
Labor is a variable input as we vary the amount of labor available from 0 to 11. Capital is a fixed input in this case as the amount of capital used remains constant at 5.

b. What are the firm's fixed costs?
The fixed costs of the firm are the cost of capital. The firm deploys 5 units of cost @ \$10 per unit. So the fixed costs are \$50.

c. What is the variable cost of producing 60 units of output?
60 units are produced when we use 3 units of labor. The cost of labor is \$50 per unit. Hence, Variable cost of producing 60 units is 3*50=\$150.

d. How many units of the variable input should be used to maximize profits?
The profits are maximized where Marginal Revenue Product (MRP)is equal to the Marginal Cost. Since marginal cost of labor is \$50, this is equal to the MRP when we use 5 units of labor and produce 90 units of output. Thus, 5 units of variable input maximize the profit.

e. What are the maximum profits this firm can earn?
Maximum Profit is attained when we use 5 units of labor, hence
Profit = Total Revenue - Total Cost = Total Revenue - Total Fixed Cost - Total variable cost
=90*5 - 50 - 50*5
=\$150

f. Over what range of the variable input usage do increasing marginal returns exist?

Since MRP is increasing till we use 3 units of labor and then it starts decreasing, we get the increasing marginal returns in the range 0 to 3 units of labor.

2.

To better serve customers interested in buying cars over the Internet, Smart Motors, Inc., hired Nora Jones to respond to customer inquiries, offer price quotes, and write orders for leads generated by the company's Web site. During the last year, Jones averaged 1.5 vehicle sales per week. On average, these vehicles sold for a retail price of \$25,000 and brought the dealership a profit contribution of \$1,000 each.

A. Estimate Jones' annual (50 workweek) marginal revenue product.
MRP (per week) =1.5*1000=\$1500
For a 50 workweek year MRP for year =\$1500*50=\$75000

B. Jones earns a base salary of \$60,000 per year, and Smart ...

\$2.19