Assume that a monopolist faces a demand curve for its product given by: p 80 2q

Question

Assume that a monopolist faces a demand curve for its product given by: p 80 2q Further assume that the firm's cost function is: TC 560 +13q Using calculus and formulas (but no tables or spreadsheets) to find a solution, what is the profit (rounded to the nearest integer) for the firm at the optimal price and quantity? Round the optimal quantity to the nearest hundredth before computing the optimal price, which you should then round to the nearest cent. Note: Non-integer quantities may make sense when each unit of q represents a bundle of many individual items. Hint: Define a formula for Total Revenue using the demand curve equation. Then take the derivative of the Total Revenue and Total Cost formulas. Use these derivative equations to perform a marginal analysis

Explanation / Answer

P = 80-2Q ------ (1)

Multiplying eq. 1 with Q to the both side, will give revenue function.

Revenue = 80Q-2Q^2 ------------ (2)

Differentiation of eq. 2 w.r.t. Q, will give marginal revenue (MR).

MR = 80-4Q

TC = 560 + 13Q ------------- (3)

Differentiation of eq. 3 w.r.t. Q, will give marginal cost (MC).

MC = 13

For profit maximizing output for a monopolist, MR = MC

80-4Q = 13

Q = (80-13)/4

Q = 16.75

P = 80-2*16.75

P = $46.5

Profit = P*Q – (560+13*Q)

Profit = 46.5*16.75 - (560+13*16.75)

Profit = $1.125 or $1

So, profit is $1 at the optimal output of 16.75 and optimal price of $46.5.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.