Suppose that in addition to repairing your car, you can also spend money adverti

Question

Suppose that in addition to repairing your car, you can also spend money advertising. If you repair your car for H hours and spend A dollars advertising, your benefit (the amount you receive when you sell your car) is B(H, A) = 220H H2 +2A. Both H and A are finely divisible. What is your benefit of repair hours when H = 4 and A = 100? What is your marginal benefit of advertising dollars at that choice? Suppose that in Calculus Problem #2 repair work costs $100 per hour and you have $1500 to allocate between repair work and advertising. Write the constrained optimization problem you face. What is your best choice of repair hours and advertis- ing? Interpret the condition that characterizes your best choice in terms of marginal benefits and marginal (opportunity) costs of repair work?

Explanation / Answer

Benefit, B = 220H - H2 + 2A

(1)

When H = 4, A = 100,

B = (220 x 4) - 16 + (2 x 100) = 880 - 16 + 200

B = 1,064

(2)

To calculate marginal benefit from advertising, let us increase A by 1 unit, from 100 to 101.

B (when A = 101) =  (220 x 4) - 16 + (2 x 101) = 880 - 16 + 202 = 1,066

So, marginal benefit = 1,066 - 1,064 = 2

Mathematically, marginal benefit from advertising is found by differentiating total benefit with respect to A:

MB = dB / dA = 2

(3)

Total cost, C = 100H + A

So, the optimization prblem is

Maximize B = 220H - H2 + 2A

Subject to C = 100H + A

100H + A <= 1500

A >= 0, H >= 0

(4)

The best (optimum) condition requires that

Marginal benefit, MB = Marginal Cost, MC

Or:

dB / dH = dC / dH

220 - 2H = 100

2H = 120

H = 60

So, Total repair hours = 60, repair cost = 60 x $4 = $240, Advertising cost = $(1500 - 240) = $1260

(5)

This condition implies that the slopes of total benefit and total cost curves are equal.

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