Question

An independent inventor argues that the smartphone manufacturer in Q4 above is violating one of her patents on smartphone technology. Her legal team estimates that without the patented technology, the smartphone manufacturer’s demand function would be p = 460 – 2Q. Use this demand function, and the demand and cost information from Q4 (p = 500 – 2Q, where p is the price of the smartphone, and Q is the quantity (thousands) of smartphones demanded by consumers. The manufacturer’s total cost function is C = 200Q, where Q is the quantity of smartphones produced by the manufacturer.) to estimate the (upper-bound) economic damages to the inventor from the smartphone manufacturer’s illegal use of the patent.

Answer 1

Current demand function => P = 500 -2Q ----- (1) ; C = 200Q

Profit function = Revenue - cost

Revenue = P*Q = 500Q - 2Q^2

Cost = 200Q

Profit = 500Q - 2Q^2 - 200Q = 300Q - 2Q^2 ----- (2)

For profit to be maximum dP/dQ = 0 => 300 - 4Q = 0 => Q = 75

If Q = 75, P = 350 (from 1)

Therefore, profit = 11,250 (from 2)

Now let us consider the situation without the patent

demand function P = 460 - 2Q^2

C = 200Q

Profit = revenue - cost = 460Q - 2Q^2 - 200Q = 260Q - 2Q^2

Revenue = P*Q = 460Q - 2Q^2 ----- (3)

Again, for profit to be maximum dP/dQ = 0 => 260 - 4Q = 0 => Q = 65

Again putting value of Q in 3, we get profit = 260*65 - 2*65*65 = 8450

Therefore, the incremental profit being generated because of the patent is 11250 - 8450 = 2800, which is the maximum amount of damages which is possible due to the manufacturers illegal use of the patent.