Write down the maximization problem faced by the researcher

Reference no: EM132196908

Question: Suppose that we are trying to determine the optimal patent length for an invention that will generate the following each year if the owner is granted a monopoly: $50 million in Consumer Surplus, $35 in prots. If however, the invention has no patent protection, economic prots will be zero and consumer surplus $100 million. There is only one rm capable of performing the research necessary for this invention. Unfortunately, research devoted to this invention is uncer- tain and requires an investment on the side of the rm. Specically, the probability of a successful research endeavor is given by P(x) = 1 Â¡ exp (Â¡ x 50 ); where x is the number of dollars (in millions) devoted to research. Let the length of the patent granted be given by T and let = 1 1 + r , where r is the known and constant interest rate in the economy.

A. Write down the value of an invention to the researching rm (call this V (T)) and the value of the invention to all of society (call this D(T )) as functions of T and . Evaluate these values if the patent length is 20 years and the interest rate is 4%.

B. Write down the maximization problem faced by the researcher.

C. Solve the researcher's maximization problem. Label the resulting function x (T ). How does x change when T gets longer?

D. What is the probability of a successful innovation as a function of T , i.e. what is P(x (T))? Evaluate this value if T is 20 and the interest rate is 4%.

E. Write down the maximization problem faced by a benevelent dictator that is trying to maximize total social surplus by choosing T.

Reference no: EM132196908

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