MANAGERIAL ECONOMICS

Homework #1 - Optimization and Demand Analysis

The homework should be written on 8 1/2 by 11 paper. If you use paper torn out of a notebook, trim the ragged edges before you turn in your assignment. When calculations are required, you must show all work. If no work is shown, no credit will be given. The homework is due at the beginning of class on Thursday, February 12.  No late homework will be accepted.

1.  Suppose the average cost of production for a firm is given below:

AC = 200 - 24 Q + Q²

The point of minimum efficient scale is the level of output where the firm first produces at the lowest possible average cost.  What is the minimum efficient scale in this case?  Show that costs are minimized in this case, not maximized.

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2.  Consider a firm that produces two products, X and Y.  The profit that the firm makes depends on the quantities of the two products sold, and is given by the following equation:

π = 64 X - X² + 56 Y - 2 Y²

a.  If the firm had no constraints, how much X and how much Y would it produce to maximize profits?

b.  Suppose each unit of X cost $1 to produce and each unit of Y costs $2 to produce.  If the firm is limited to spending a total of $36 on producing the two products, how much X and how much Y would it produce to maximize profits?

c.  Calculate and interpret the Lagrangian multiplier from (b).

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3. Consider the case of a steel producer that faces the following demand for steel:

Q_S = 3,000 - 2,000 P_S + 0.12 I - 2,500 P_C

where

Q_S = steel demand in thousands of tons per year

P_S = price of steel in dollars per pound

I = income per capita

P_C = price of concrete in dollars per pound

Suppose the price of steel is $1 per pound, the income per capita is $50,000, and the price of concrete is $0.80 per pound.

a.  How much steel will be demanded at the current prices and income?

b.  What is the point price elasticity in this case?  What should the firm do with the price of steel if it wanted to increase its income?

c.  What is the point income elasticity in this case?  What type of product is involved in this case?

d.  What is the point cross-price elasticity in this case?  What does the elasticity imply about the relationship between steel and concrete?

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