A Firm with a Patent Facing Consumer Demand
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Effect of the Length of a Patent
Define π as the flow rate of profit to the inventor, and s as the flow rate of cost to the society. s is equal to the difference between the societal value generated by the invention under conditions of perfect competition and that generated with the patent in place.
Now, given s and π, suppose that we can design a patent to last for a certain period of time, L. Then, total social cost is found by discounting the flow rate of cost over time:
While total profit is similarly,
Our problem is to minimize S, subject to some fixed reward, V.
Klemperer observes that increasing L multiplies both the total cost to society and the total profit to the inventor by the same factor. If π is large enough, we can achieve any desired reward, V, by assigning an appropriate value for L; that is, by choosing how long to make the patent.
By combining equations  and  above, we get
So minimizing total cost to society, S, is equivalent to minimizing the ratio s / π. In this, we are constrained by equation , which sets a minimum possible value for π.
In the rest of this tutorial, we turn our attention to the task of minimizing the ratio of flow rate of social cost to flow rate of profit, s / π.
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