Question

Consider a monopolist who has a total cost function C(q) = 1000 + 10q The demand function for the market is D(p) = 600 - 12.5p Answer each part below:(a) Use the inverse demand equation to derive the monopolist's marginal revenue equation, MR(q). (1 points) (b) What is the marginal cost, MC(q)? (0.5 points) (c) Solve for the profit-maximizing quantity and price. Note: Explain or show your work! (2 points) (d) Compute the price elasticity of demand at the profit-maximizing price and quantity (round to 3 decimal places). Then, interpret the elasticity and state whether demand is elastic, inelastic, or unit elastic. (1.5 points)

Answer 1

(a)

Demand function D(P): q = 600 - 12.5p

=> 12.5p = 600 - q

=> p = (600 -q) / 12.5

=> p = 48 - 0.08q

The inverse demand function: p = 40 - 0.08q

Total revenue: TR = pq

=> TR = (48 - 0.08q)*q

=> TR = (48q - 0.08q2)

Marginal revenue: MR = ΔTR / Δq

=> MR = 48 - 0.08(2) q^2-1

=> MR = 48 - 0.16q

---------------------------------------------------------------------

(b) C = 1000 + 10q

=> MC = ΔC/Δq

=> MC = 10

-------------------------------------------------------------

(c) A monopolist maximizes profit at MR = MC

=> 48 - 0.16q = 10

=> 48 -10 = 0.16q

=> 38 = 0.16q

=> q = (38 / 0.16)

=> q = 237.5

Profit maximizing quantity is 237.5

and

p = 48 - 0.08q

=> p = 48 - 0.08(237.5)

=> p = 29

Profit maximizing price is 29

----------------------------------------------------------

(d) Price elasticity of demand (ed) = (Δq/Δp ) * (p/q)

q = 600 - 12.5p

=> Δq /Δp = -12.5

and

q = 237.5 , p = 29 (profit maximizing price and quantity)

--

ed = (-12.5) (29 / 237.5)

=> ed = (-362.5 / 237.5)

=> ed = -1.526

The ed is-1.526. It means change in price by 1 unit leads to change in quantity by 1.526 units in opposite direction.

The absolute value of ed is 1.526 (i.e., ignore negative sign), it is greater than one. It means demand is elastic.