An auto-service establishment has estimated its monthly cost function as follows

Question

An auto-service establishment has estimated its monthly cost function as follows:
TC = 6000 + 10 Q
where Q is the number of cars it services each months and TC represents its total cost. The firm is targeting 35,000 net monthly profit servicing 2000 cars.
a. What price should the firm charge to realize the targeted profit?
b. What would be its (cost-based) markup ratio?
b. Now suppose the demand curve the firm faces is:Q = 3000 - 50 P. Is the firm going to achieve its profit goal? Explain.
c. If your to answer to (b) is "no", what would be the optimal markup ratio for this firm?

Explanation / Answer

(a)

Let targeted price be P.

Revenue = Cost + Profit

P x Q = 6000 + 10Q + 35,000

2000P = 6000 + (10 x 2000) + 35,000

2000P = 6000 + 20,000 + 35,000 = 61,000

P = 61,000 / 2,000 = 30.50

(b)

Marginal cost, MC = dTC / dQ = 10

Mark-up = (P - MC) / MC x 100

= (30.50 - 10) / 10 x 100 = 205% **

(c)

Q = 3000 - 50P

P = (3000 - Q) / 50

Assuming target number of cars is unchanged at 2000,

P = (3000 - 2000) / 50 = 1000 / 50 = 20

Revenue = P x Q = 20 x 2000 = 40,000

TC = 6000 + 10Q = 6000 + (10 x 2000) = 6000 + 20,000 = 26,000

Profit = Revenue - TC = 40,000 - 26,000 = 14,000

So actual profit < Target profit.

(d)

Clarification required: "Optimal Mark-up Ratio" for what? For achieving target profit of 35,000 or for maintaining a price at 30.5?

** Mark-up is based on Marginal Cost mark-up.

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