Question

A producer estimated the dependence of sales volume on advertising expenditures and priced as follow: Q = 35,000 - 5,000 P + 0.8 A - 0.000025 A2 where Q is the monthly quantity sold, P the price and A the monthly level of advertising. The average cost is constant at $2.65 (and hence equal to marginal cost) between 5,000 and 20,000 units produced per month and there is no fixed cost other than the  where Q = output, P = the market price, and S = the amount of advertising, where the unit cost of advertising is constant and equal to 1. 2 advertising expenditures. Right now, the price is fixed by a 60% mark-up over average cost or P = 1.60 * 2.65 = $4.24 and the advertising expenditures amount to $9,000 per month.

a) At the actual level of advertising of $ 9,000 is the $ 4.24 price profit-maximizing? If not, determine the profit-maximizing price and the maximum amount of profit.

b) At the actual price of $4.24 what would be the profit-maximizing level of advertising expenditures and what would be the maximum amount of profit?

c) Verify if the Dorfman-Steiner condition is satisfied for a level of advertising of $ 8.524 and a price of $ 5.33.

Answer 1

Question 1.

For a given advertising level of $9,000,

the demand function condenses to

Q = 35,000 – 5,000 P + 0.8 * 9,000 – 0.000025 * 9,0002 = 40,175 – 5,000 P or

P = 8.035 – 0.0002 Q and MR = 8.035 – 0.0004 Q

Profit-maximising quantity equalises marginal cost to marginal revenue

2.65 = 8.035 – 0.0004 Q* or Q* = (8.035-2.65)/0.0004 = 13,462.5 units/month.

P* = 8.035 – 0.0002 * 13,462.5 = 5.3425 > 4.24

The maximum amount of profit is

13,462.5 * (5.3425 – 2.65) – 9,000 = 27,247.781

The actual quantity is Q' = 40,175 – 5,000 * 4.24 = 18,975

and the actual profit is 18,975 * (4.24 – 2.65) – 9,000 = 21,170.25 only.

So for that level of advertising the price is too low

Question 2.

For a price fixed at $ 4.24, the optimality condition is

∂ π /∂ A = 1 or (∂ π /∂Q)(∂Q/∂A) = 1

π = TR – TC = Q(4.24 – 2.65) = 1.59 Q is the contribution to fixed cost and profit hence

∂ π /∂Q = 1.59

and ∂Q/∂A = ∂(35,000 – 5,000 P + 0.8 A - 0.000025 A2 )/∂A = 0.8 – 0.00005 A

For P = 4.24 : ∂ π /∂ A or (∂ π /∂Q)(∂Q/∂A) = 1.59 * (0.8 – 0.00005 A) = 1

or 1,272 – 1 = 0.0000795 A or A* = 0.272/0.0000795 = $ 3,421.38

It is easy to see that at A = 9,000 :

∂ π /∂ A or (∂ π /∂Q)(∂Q/∂A) = 1.59(0.8 - 0.00005 * 9,000) = 0.5565 < 1 hence for a price

of $4.24 the advertising budget should be reduced since the last dollar spent on advertising brings in only $0.5565.

With A = 3,421.38 and P = 4.24, the quantity sold is Q = 16,244.46 units/month and profit π* = 16,244.46 (4.24-2.65) – 3.421.38 = 22,407.31 better than 21,170.25 actually.

Question 3.

The profit maximising price P and promotional budget A are such that the ratio of the advertising elasticity to minus the own-price elasticity is equal to the ratio of the advertising budget to total revenue or - EAP/EQP = A/PQ.

Now Q = 35,000 – 5,000 * 5.33 + 0.8 * 8,524 – 0.000025 * 8.5242 = 13,352.74

The own-price elasticity : EQP = (∂Q/∂P)(P/Q) = -5,000 * 5.33/13,352.74 = -1.9958

The elasticity with respect to advertising : EQA = (∂Q/∂A)(A/Q)

∂Q/∂A = (0.8 – 0.00005 A) = 0.8 – 0.4262 = 0.3738

EQA = 0.3738 * 8,524/13,352.74 = 0.2386

-EQA/QQP = 0.2386 / -1.9958 = + 0.1196

While A/PQ = 8.524/(13,352.74 * 5.33) = 0.1197

Yes, the firm indeed charges a profit-maximising price of $5.33 and spends a profitmaximising promotional budget of $8,524 simultaneously.

Profit is then : π* = 13,352.74 (5.33-2.65) – 8,524 = 27,259.36 much better