Suppose that we are trying to determine the optimal patent length for an inventi

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 ¡ exp (¡ 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.

Explanation / Answer

an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is indifferent. That is, the consumer has no preference for one combination or bundle of goods over a different combination on the same curve. One can also refer to each point on the indifference curve as rendering the same level of utility (satisfaction) for the consumer. In other words, an indifference curve is the locus of various points showing different combinations of two goods providing equal utility to the consumer. Utility is then a device to represent preferences rather than something from which preferences come.[1] The main use of indifference curves is in the representation of potentially observable demand patterns for individual consumers over commodity bundles.[2]

There are infinitely many indifference curves: one passes through each combination. A collection of (selected) indifference curves, illustrated graphically, is referred to as an indifference map.

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