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Chapter 16: Patents and Technological Change |
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(n). The larger the number of firms, the higher the probability, '(n) > 0; but the increase in probability occurs at a decreasing rate, ''(n) < 0. If the value to society of the discovery is B, the expected value of the discovery, prior to the research, is B(n).
Research takes place in period t. If a discovery is made, society benefits from it for the rest of time. For simplicity, we do not consider research in later periods if the discovery is not made in t.
Society wants to choose the number of competing research projects (firms), n, to maximize net social benefit, S(n), which equals expected benefit minus industry costs:
Equation 1
max n | S(n)  B(n) - C(n) = B(n) - nm. |
S is maximized at n*, where
Equation 2
S'(n*) = B'(n*) - C'(n*) = B'(n*) - m = 0.
That is, marginal benefit, B'(n*), equals marginal cost, m.
By assumption, (n) = 1 - e-
n (as in Table 16.2 and Figure 16.1 in your textbook), so we can rewrite Equation 2 as
Equation 2'
S'(n*) = Be-n* - m = 0.
We can solve Equation 2' for n*:
Equation 3
n* = -(1/)ln(m/(B)).
In Table 16.2, m = 1, B = 25, and = .2031, so n* = 8.
(n). In the case of ties, either the prize is shared or awarded randomly to one of the successful firms, so the probability that any one of the n firms wins the prize is (n)/n. In this case, a profit-maximizing firm undertakes a research project only if the expected benefit is greater than or equal to the expected cost:
Equation 4
| B | (n) ------ n |  m. |
In the example in Table 16.3, the equality holds in Equation 4 when n = 24.84.
A prize equal to Bn*'(n*)/(n*) leads to the optimal number of research projects. Each firm's expected payoff equals B'(n*), and from Equation 2, that payoff leads to the optimal number of firms = n* = 8. Because ''(n) < 0, the average probability of making the discovery is greater than the marginal probability: (n)/n > '(n*). Thus, Bn*'(n)/(n*) < B, so this alterative prize is smaller than B; thus it leads to fewer research projects.
In the example, demand is linear and the costs of producing are constant. The demand for the product is
Equation 5
p = a - bQ,
where a = 6 and b = 5. The marginal cost of producing the product is m = 1. Thus, if the product is sold at a competitive price (= m), then p = 1. The corresponding consumer surplus (benefit of the discovery), CS, in any period is the area under the demand curve above m = 1: CS = (a - m)2/(2b) = $2.50. The present value of consumer surplus, given competitive pricing, is CS/r, where r is the interest rate. If r = 10 percent, then the present value of consumer surplus = $25 = B, the social benefit from the discovery.
A monopoly maximizes profits by setting marginal revenue equal to marginal cost. Here, the revenues = aq - bq2, so marginal revenue = a - 2bq, and marginal cost is m = 1. Thus, a monopoly charges a price of pm = (a + m)/2 = $3.50, sells qm = (a - m)/(2b) = 0.5 units, and makes profits of 
m = $1.25. The consumer surplus, given monopoly pricing, is CSm = $0.625 per period, or one-fourth the consumer surplus of a competitive industry.
If the patent lasts forever, given that the discovery is made, present value of the monopoly profits is m/r = $12.50. If each firm has an equal chance of obtaining the patent, then the expected return to a firm undertaking research is $12.50(n)/n. Firms undertake projects if m = 1 
12.50(n)/n. With a permanent patent, a firm's net expected return, 12.50(n)/n - 1, equals zero when there are 11.22 firms racing to make the discovery. If a fractional number of firms is impossible, it does not pay for a firm to compete if there are more than 11 firms. A permanent patent leads to excessive research (11.22 > 8).
How long should the patent last to induce n* = 8 firms to enter the patent race? The present value of a patent, PVP(t), that lasts t years is
Equation 6
PVP(t) = m | t 0 | e-rsds = | m ----- r | (1 - e-rt). |
The expected return to any given firm in the patent race is PVP(t)(n)/n. A firm enters the race if the expected return is greater than or equal to the marginal cost, or PVP(t)(n)/n m. In the example, the length of a patent, t*, such that n = 8 firms is determined by solving PVP(t*)(8)/8 = 1. Thus, t* = -10 ln(1 - 8/[12.50(8)]) = 15.94.
Unlike prizes, however, patents cause a distortion after the discovery: The price, set at the monopoly level, is too high. Thus, the government is faced with a trade-off. The longer the duration of the patent, the greater the chance of success, but the larger the cost due to more research projects and the monopoly loss.
Given that the government uses patents, it should choose the length of a patent, t, to maximize expected net social benefit, taking into account monopoly pricing. Expected net social benefit for a patent of length t is
Equation 7
| NSB(t) = [(CSm + m) | t 0 | e-rsds + (CS/r)e-rt](nm) - nmm, |
where the first term on the right side is the present value of the sum of consumer surplus and profits for the t years that the patent grants exclusive rights, the second term is the present value of the consumer surplus for the rest of time after the patent expires, (nm) is the probability of success if nm firms race, and the last term is the total cost of research for nm firms. Firms enter until marginal cost equals their expected marginal return:
Equation 8
| m = 1 = | m(nm) ------------- rnm | (1 - e-rt). |
For a given t, the value for nm that solves this expression is used for the calculations in Table 16.4.
To find the optimal solution, the government should maximize Equation 7 with respect to t, subject to Equation 8. The resulting t = 11.4475 leads to a nm = 6.004, which is smaller than n* = 8. See Table 16.4 in the text.
Wright, Brian D. 1983. "The Economics of Invention Incentives: Patents, Prizes, and Research Contracts." American Economic Review 73:691-707.
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