# Analyzing the given demand functions

1. Using the following equation, what is the demand equation as a function of Ps if the price of other pastas (Po) is $2, the individual's income (Y) in thousands is $25, and tastes (Z) are represented by 20? What happens if the individual's income increases to $30?

Qd= 500 - 10Ps + 5Po + 20Y +40Z

2. Given the regression estimate of the demand equation of

Qx = 1,000 - 3.3Px + 0.001Y

where Y is income, what is the change in demand if price rises by $1, holding income constant? What is the percentage change in demand if price rises by $1 from an initial price of Px = $200 given Y = 10,000? What is the effect on demand of a $1 increase in income, holding price constant?

3. Consider the estimate demand equation of

Qx = 1,000 -3.3Px -0.2Pz + 0.001Y

(3.5) (2.1) (0.5)

with t values in parenthesis, where Pz is the price of another good Z, and Y is income. Is good Z a substitute or a complement? Can we say confidently whether good X is a normal good or an inferior good?

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#### Solution Preview

1. Using the following equation, what is the demand equation as a function of Ps if the price of other pastas (Po) is $2, the individual's income (Y) in thousands is $25, and tastes (Z) are represented by 20? What happens if the individual's income increases to $30?

Qd= 500 - 10Ps + 5Po + 20Y +40Z

Put Po=$2, Y=$25, Z=20

Qd=500-10Ps+5*2+20*25+40*20=1810-10Ps

In the first case, demand equation will be given by

Qd=1810-10Ps

Now let us assume that individual income rises to $30 thousands. Then

Qd=500-10Ps+5*2+20*30+40*20=1910-10Ps

Demand equation will change to

Qd=1910-10Ps

Change in demand=(1910-10Ps)-(1810-10Ps)=100 ...

#### Solution Summary

Solutions study the variations in demand due to changes in income.

Price and Output decisions.

Problem 1: Apex, Inc. is a monopolist. The demand function for its product is estimated to be

Q = 60 - 0.4P + 6Y +2A

Where Q=quantity of units sold

P= price per unit

Y = per capita disposable personal income (thousands of dollars)

A = hundreds of dollars of advertising expenditures

The firm's average variable cost function is

AVC = Q2 - 10Q + 60 Y is equal to 3 (thousand) for the period being analyzed.

a. If fixed costs are equal to $1,000, derive the firm's total cost function and marginal cost function.

b. Derive a total revenue function and marginal revenue function for the firm.

c. Calculate the profit-maximizing level of price and output for Apex, Inc.

d. What profit or loss will Apex earn?

e. If fixed costs were $1,200, how would your answers change for parts a through d?

Problem 2: XYZ Mining, Inc. is a leading manufacturer of magnesium, which is used in many products, estimates the following demand schedule for its product:

Price ($/pound) Quantity (Pounds /Period)

$25 0

18 1000

16 2000

14 3000

12 4000

10 5000

8 6000

6 7000

4 8000

2 9000

Fixed costs of manufacturing chromium are $14,000 per period. The firm's variable cost schedule is as follows:

Output (Pounds /period) Variable Cost (per Pound)

0 1000

1,000 10.00

2,000 8.50

3,000 7.33

4,000 6.25

5,000 5.40

6,000 5.00

7,000 5.14

8,000 5.88

9,000 7.00

a. Find the total revenue and marginal revenue schedules for the firm.

b. Determine the average total cost and marginal cost schedules for the firm.

c. What are the XYZ Mining's profit-maximizing price and output levels for the production and sale of magnesium?

d. What is XYZ Mining's profit (or loss) at the solution determined in part c?