Critically discuss the trade-off between patent breadth and length.

Suppose that the innovation generates a notional maximum discounted social value W¯ that could be earned if it were available for free immediately but a deadweight loss, d, is incurred during each period of protection. Let the flow profits for each period of protection be π for the innovator. Profits fall to a baseline level of zero after protection expires. Suppose also that there are T periods of patent protection (for example, T = 20 years). This patent protection allows entrepreneur to gain a profit π, but also raises a penalty for social value, that equals the deadweight loss d. Now, let’s define the discounted rate ρ(T) we use to calculate discounted values for profits (π) and deadweight loss (d). Therefore, the net discounted benefits to society will be: N(T) = W¯ − dρ(T) where dρ(T) is the discounted value of deadweight losses accumulated over the years T of duration of the patent protection. dρ(T) = d + d*(1/(1+r )) + d*(1/(1+r )2) + d*(1/(1+r )3) + · · · + d*(1/(1+r )T)= d*SUM T, t=0 (1/(1+r )t) The expression N(T) = W¯ − dρ(T) is decreasing in T We can either think of maximising this expression with respect to T or minimising dρ(T) with respect to T... ...subject to the constraint that the discounted benefits generated from innovation, πρ(T), meet a value, c, required to induce innovation. Noting that πρ(T) is increasing in T, the solution to this problem is the minimum T that allows the constraint to be met. Formally, society’s problem is to maximise total discounted social welfare from innovating (or minimise the total discounted social costs), where welfare in each period decreases with the price premium over marginal cost and the patent expires at time T. If we take π to reflect the price cost margin, we have the following social planner’s problem: Max [W¯ − ρ(T)d] subject to the constraint that c ≤ πρ(T) What’s matter now is that in this maximisation program the social planner has two different tools to maximise social welfare by implementing patenting rules: 1) The statutory length T 2) The price-cost margin (or profit π) 3) Note that the deadweight loss function d is also a function of π (d(π)) MaxW¯ − [X(T)d(π)] subject to the constraint that c ≤ X(T)π