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Chapter 11: Q17E (page 413)

Consider a firm with monopoly power that faces the demand curve

P= 100 - 3Q+ 4A1/2

and has the total cost function

C= 4Q2 + 10Q+ A

where Ais the level of advertising expenditures, and Pand Qare price and output.

a.Find the values of A, Q, and Pthat maximize the firm’s profit.

b.Calculate the Lerner index, L = (P - MC)/P, for this firm at its profit-maximizing levels of A, Q, and P.

Short Answer

Expert verified

(a) The profit is maximized when the value of A = 900, the value of Q = 15, and the value of P = 175.

(b) The Lerner index is 0.2571.

Step by step solution

01

Computing the values of A, Q, and P

\(\begin{aligned}TR &= P \times Q\\ &= (100 - 3Q + 4{A^{\frac{1}{2}}})Q\\ &= 100Q - 3{Q^2} + 4{A^{\frac{1}{2}}}Q\end{aligned}\)

Profit refers to the difference between the total revenue and cost incurred on fixed and variable factors.

Profit = TR – C

\({\rm{Pr = (100Q - 3}}{{\rm{Q}}^{\rm{2}}}{\rm{ + 4}}{{\rm{A}}^{\frac{{\rm{1}}}{{\rm{2}}}}}{\rm{Q) - (4}}{{\rm{Q}}^{\rm{2}}}{\rm{ + 10Q + A)}}\)

First-order condition:

\(\begin{aligned}\frac{{\partial \Pr }}{{\partial Q}} &= 0\\100 - 6Q + 4{A^{\frac{1}{2}}} - 8Q - 10 &= 0\\90 - 14Q + 4{A^{\frac{1}{2}}} &= 0{\rm{ - - - - - - - - - - - - - - - - - - equation (i)}}\end{aligned}\)

\(\begin{aligned}\frac{{\partial \Pr }}{{\partial A}} &= 0\\\left( {\left( {\frac{1}{2}} \right)\frac{4}{{{A^{\frac{1}{2}}}}}} \right)Q - 1 &= 0\\{A^{\frac{1}{2}}} &= 2Q{\rm{ - - - - - - - - - - - - - - - - - equation (ii)}}\end{aligned}\)

From equation (i) and (ii)

\(\begin{aligned}90 - 14Q + 4{A^{\frac{1}{2}}} &= 0\\90 - 14Q + 4\left( {2Q} \right) &= 0\\6Q &= 90\\Q &= 15\end{aligned}\)

Putting the value of Q in equation (ii)

\(\begin{aligned}{A^{\frac{1}{2}}} &= 2(15)\\A &= {\left( {2 \times 15} \right)^2}\\A &= \left( {4 \times 225} \right)\\A &= 900\end{aligned}\)

Putting the value of A and Q in the demand curve, the value of P is calculated:

\(\begin{aligned}P &= 100 - 3\left( {15} \right) + 4{\left( {900} \right)^{\frac{1}{2}}}\\ &= 100 - 45 + {\rm{120}}\\ &= 220 - 45\\ &= 175\\\end{aligned}\)

The value of A is 900, the value of Q is 15, and the value of P is 175.

02

Calculate Lerner’s index for the firm at profit-maximizing levels

The firm's marginal cost is calculated by differentiating the total cost of a firm. It is calculated below:

\(\begin{aligned}MC &= \frac{{d\left( {4{Q^2} + 10Q + A} \right)}}{{dQ}}\\ &= 8Q + 10\\ &= 8 \times 15 + 10\\ &= 130\\\end{aligned}\)

The formula for Lerner’s index is:

\(\begin{aligned}L &= \frac{{\left( {P - MC} \right)}}{P}\\ &= \frac{{\left( {175 - 130} \right)}}{{175}}\\ &= 0.2571\end{aligned}\)

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Most popular questions from this chapter

Some years ago, an article appeared in the New York Times about IBM’s pricing policy. The previous day,IBM had announced major price cuts on most of itssmall and medium-sized computers. The article said:

IBM probably has no choice but to cut prices periodicallyto get its customers to purchase moreand lease less. If they succeed, this could makelife more difficult for IBM’s major competitors.Outright purchases of computers are needed for ever larger IBM revenues and profits, says Morgan Stanley’s Ulric Weil in his new book, InformationSystems in the 80’s. Mr. Weil declares that IBM cannot revert to an emphasis on leasing.

a. Provide a brief but clear argument in support of the claim that IBM should try “to get its customers to purchase more and lease less.”

b. Provide a brief but clear argument against this claim.

c. What factors determine whether leasing or selling is preferable for a company like IBM? Explain briefly.

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  2. Does prohibiting the use of coupons make German producers better off or worse off?

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  1. Suppose that to maintain a “professional” atmosphere, you want to limit membership to serious players. How should you set the annual membership dues and court fees (assume 52 weeks per year) to maximize profits, keeping in mind the constraint that only serious players choose to join? What would profits be (per week)?
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  3. Suppose that over the years, young, upwardly mobile professionals move to your community, all of whom are serious players. You believe there are now 3000 serious players and 1000 occasional players. Would it still be profitable to cater to the occasional player? What would be the profit-maximizing annual dues and court fees? What would profits be per week?
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