Question

Q = 400 – 3P + 4T + .6A

Where:

Q = quantity demanded in units

P = price in dollars

T = tastes and preferences

A = Advertising expenditures in dollars

We are currently operating at the following values:

A = 10,000

T = 8000

P = 2800

In addition, suppose MC is 2800.

Given all this, please answer the following questions:

a. Derive the firm’s current demand curve:

b. Calculate and interpret the firm’s current price elasticity of

demand.

c. Given you answer in b, should the firm increase or decrease output

in order to maximize profits? Explain and show your work on a

well labeled graph.

d. What is the profit maximizing output and price? Show and explain

all your work and match up your answer to your work in part c.

e. Calculate a “advertising elasticity of demand.” Based on this,

by what percent should advertising expenditures increase in order

to increase sales by 10 percent? Note: work from the original

price and output levels, NOT the profit maximizing levels.

Answer 1

(a)

Q = 400 - 3P + 4 x 8,000 + 0.6 x 10,000

Q = 400 - 3P + 32,000 + 6,000

Q = 38,400 - 3P

(b)

When P = 2,800, Q = 38,400 - 3 x 2,800 = 38,400 - 8,400 = 30,000

Elasticity of demand (Ed) = (dQ/dP) x (P/Q) = - 3 x (2,800 / 30,000) = - 0.28

(c)

Since absolute value of Ed is less than 1, demand is inelastic. A firm facing a downward sloping demand curve operates only on the elastic portion of its demand curve. Since demand is more elastic the higher the price and the lower the output, the firm has to increase price and decrease output, to maximize profit.

In following graph, currently the firm is at point E of demand curve, with price P0 (= 2,800) and output Q0 (= 30,000). Since point E lies to the right of the mid-point of demand curve, demand is inelastic at point A. To maximize profit, firm moves to point F (which lies to the left of the mid-point of demand curve, so demand is elastic) with higher price P1 and lower output Q1.

(d)

P = (38,400 - Q)/3

TR = P x Q = (38,400Q - Q²)/3

MR = dTR/dQ = (38,400 - 2Q)/3

Profit is maximized when MR = MC.

(38,400 - 2Q)/3 = 2,800

(38,400 - 2Q) = 8,400

2Q = 30,000

Q = 15,000

P = (38,400 - 15,000)/3 = 23,400/3 = 7,800

In following graph, profit is maximized at point A where MR intersects MC with price Pm (= 7,800) and output Qm (= 15,000).

From demand function, when Q = 0, P = 38,400/3 = 12,800 (vertical intercept of demand curve)

Mid-point of demand curve = 12,800/2 = 6,400

Since Pm > Mid-point of demand curve, profit is maximized at the elastic segment of demand curve.

(e)

Advertising elasticity = (Q/A) x (A/Q) = 0.6 x (10,000/30,000) = 0.2

Advertising elasticity = % Change in sales / % Change in advertising

Therefore, to increase sales by 10%, Advertising will increase by (10%/0.2) = 50%.