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marginal cost function

(See attached file for full problem description)

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1. Consider the following short-run production function (where L = variable input, Q = output):

Q = 6L2 - .4L3

a. Determine the marginal product function (MPL).
b. Determine the average product function (APL).
c. Find the value of L that maximizes Q.
d. Find the value of L at which the marginal product function takes on its maximum value.
e. Find the value of L at which the average product function takes on its maximum value.
f. Plot the (i) total, (ii) marginal, and (iii) average product functions for values of L = 0, 1, 2, 3, . . ., 12.

2. Suppose that a firm's production function is given by the following relationship:

Q = 2.5 LK (square root) (i.e., Q -= 2.5L.5K.5)

Where Q = output
L = labor input
K = capital input

a. Determine the percentage increase in output if labor input is increased by 10 percent (assuming that capital input is held constant).

b. Determine the percentage increase in output if capital input is increased by 25 percent (assuming that labor input is held constant).

c. Determine the percentage increase in output if both labor and capital are increased by 20 percent.

3. Consider the following variable cost function (Q = output):

VC = 200Q - 9Q2 + .25Q3

Fixed costs are equal to $150.
Output (Units Per Week) TFC TVC TC AFC AVC ATC MC

0 _______ _______ _______ _x______ _x______ _x______ _x______
1,000 _______ _______ _______ _______ _______ _______ _______
3,000 _______ _______ _______ _______ _______ _______ _______
4,000 _______ _______ _______ _______ _______ _______ _______
5,000 _______ _______ _______ _______ _______ _______ _______
6,000 _______ _______ _______ _______ _______ _______ _______
7,000 _______ _______ _______ _______ _______ _______ _______
8,000 _______ _______ _______ _______ _______ _______ _______
9,000 _______ _______ _______ _______ _______ _______ _______

a. Determine the total cost function

b. Determine the (i) average fixed, (ii) average variable, (iii) average total, and (iv) marginal cost functions.

c. Determine the value of Q at which point the average variable cost function takes on its minimum value. Hint: Take the first derivative of the AVC function, set the derivative equal to 0, and solve for Q. Also use the second derivative to check for a maximum or minimum.

d. Determine the value of Q at which point the marginal cost function takes on its minimum value.

4. Sisneros has just completed his MBA degree and is considering pursuing doctoral (Ph.D.) studies in economics. If Sisneros takes a job immediately after his MBA, he could earn $60,000 during the first year, with an anticipated raise of $5,000 per year over the next four years. If Sisneros pursues the doctorate, four more years of school are required. Sisneros has been offered an assistantship paying $14,000 per year plus tuition. Books and computer purchases needed for his study will cost an average of $2,000 per year. These costs will not be incurred if Sisneros takes a job immediately. Upon graduation, Sisneros expects an annual income level of $75,000 during his first year of teaching. The growth rate in Sisneros' teaching salary is expected to equal the growth rate of the income he would make if he did not pursue the Ph.D. How should Sisneros evaluate his decision to pursue a Ph.D.? What other information do you need? What factors other than salary should be considered?
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Solution Summary

Determine the value of Q at which point the average variable cost function takes on its minimum value. Hint: Take the first derivative of the AVC function, set the derivative equal to 0, and solve for Q. Also use the second derivative to check for a maximum or minimum.

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