Question

Monopoly:

TC = 10,000 + 100Q + 0. 20Q^2

QD = 20,000 − 10P

Find Total Revenue, Total Cost, Price, Quantity, Profit and Elasticity

Answer 1

Demand function is as follows -

Q = 20,000 - 10P

Inverse demand function is as follows -

10P = 20,000 - Q

P = 2000 - 0.10Q

Calculate the Total Revenue -

TR = P * Q = (2,000 - 0.10Q) * Q = 2,000Q - 0.10Q²

Calculate Marginal Revenue -

MR = dTR/dQ = d(2,000Q - 0.10Q²)/dQ = 2,000 - 0.20Q

TC = 10,000 + 100Q + 0.20Q²

Calculate the marginal cost

MC = dTC/dQ = d(10,000 + 100Q + 0.20Q²)/dQ = 100 + 0.40Q

A monopolist maximizes profit when it produce that level of output corresponding to which marginal revenue equals marginal cost.

Equating MR and MC

2,000 - 0.20Q = 100 + 0.40Q

0.60Q = 1900

Q = 3,166.67 or 3,167

P = 2,000 - 0.10Q = 2,000 - (0.10 * 3,167) = 1,683.30 or 1,683

Calculate the Total Revenue -

TR = 2,000Q - 0.10Q² = (2,000 * 3167) - [0.10(3167)²] = $6,334,000 - $1,002,988.9 = $5,331,011.1 or $5,331,011

Calculate the Total Cost -

TC = 10,000 + 100Q + 0.20Q²

TC = 10,000 + (100*3167) + 0.20(3167)²

TC = $10,000 + $316,700 + $2,005,977.8

TC = $2,332,677.8 or $2,332,678

Calculate Profit

Profit = TR - TC = $5,331,011 - $2,332,678 = $2,998,333

Calculate Elasticity -

Elasticity = (dQ/dP) * (P/Q) = [d(20,000 - 10P)/dP] * (1683/3167) = -10 * 0.53 = -5.3

So,

Total revenue = $5,331,011

Total Cost = $$2,332,678

Price = $1,683

Quantity = 3,167

Profit = $2,998,333

Elasticity = -5.3