Consider the following possible new invention. It has not yet been invented. A single firm becomes aware of the possibility of the invention, and can immediately invent it by sinking a fixed cost of $20million in research and development.

a. Suppose that there are strong patent laws, and if the firm invents the product it can get a patent at no additional expense. With a patent the firm is a monopolist with a constant marginal cost of $6 per unit produced. Each consumer in the market has an inverse demand of for the product = $10 and the market has a million consumers annually. What is the optimal price for the patent holding firm to charge for its product?

b. Let the flow profitsm be the amount of profit the patent-holding firm makes annually with the above prices. Find the flow profits for this firm.

c. Now suppose that the flow of customers and profits comes continuously over the course of a year, and that the firm calculates the present value of such profits according to the continuous compounding method with an interest rate of . Suppose also that once awarded the patent lasts a length of time equal to T.

Find a simplified formula for the present value of the patent. (Hint: The starting point for present value formula in this case is:0Tmertdt

d. Solve for the smallest T necessary to incentivize the patent-holder to invent the product