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:

[1]

While total profit is similarly,

[2]

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 [1] and [2] above, we get

[3]

So minimizing total cost to society, S, is equivalent to minimizing the ratio s / π. In this, we are constrained by equation [2], 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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