itive markets S (0), the surplus function under the monopoly Sm (0), and the function for monopoly
profits of the form π (0).
3. Benchmark with full information
In this section, we set up a benchmark example of the environment in which the quality of
an idea is known to the planner.
The classic analysis of the optimal patent length problem is the work of Nordhaus (1969).
The planner seeks to maximize the discounted value of the social surplus. The only instrument
available to the planner is a patent if length rT. The problem of the planner is to determine the
length of time rT that a patent will be valid, which solves the following problem:
s.t.
max
^
T
l{l t e"'tsm (0)dt+∕τ"
e~rtS (0) dtldF (0)
^
T
Jo
e^rtπ (0) dt > K.
(1)
In the objective function, the social surplus is equal to Sm (0) for the time period between O and T
as the good is produced by the monopoly under the patent granted. Afterward, the social surplus
is equal to S (0) as the good is produced under the competitive markets. The equation (1) is a
participation constraint that guarantees that the innovator granted a patent of length rT at least
breaks even.
Letting τ = r ∕q e rtdt, this problem reduces to
max ʃ [τSm (0) + (1 - τ) S (0)] dF (0)
(2)
s.t.
τπ (0) > K.
Suppose now that prizes are available, and prizes can be a function of the quality of the good.
Then the problem of the social planner becomes that of maximizing (2) subject to
τπ (0) + T (0) > K,