Question

Monopoly problem: Find TR (total revenue), TC (total cost), P (price), Q (quantity), Profit & elasticity

Given: TC=10,000+100Q+0.20Q^2

Qd (demand)=20,000-10P

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

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

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

TC = 10,000 + 100Q + 0.20Q²

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

Equating MR and MC

2,000 - 0.20Q = 100 + 0.40Q

0.60Q = 1900

Q = 3,166.67

P = 2,000 - 0.10Q = 2,000 - (0.10 * 3,166.67) = 1,683.33

TR = 2,000Q - 0.10Q² = (2,000 * 3166.67) - 0.10(3166.67)² = $5,330,560.11

TC = 10,000 + 100Q + 0.20Q² = 10,000 + (100*3166.67) + 0.20(3166.67)² = $2,332,226.78

Profit = TR - TC = $5,330,560.11 - $2,332,226.78 = $2,998,333.33

Elasticity = (dQ/dP) * (P/Q) = [d(20,000 - 10P)/dP] * (1683.33/3166.67) = -10 * 0.53 = -5.3

So,

TR = $5,330,560.11

TC = $2,332,226.78

P = $1,683.33

Q = 3,166.67

Profit = $2,998,333.33

Elasticity = -5.3