Controlling Prices to Increase Social Welfare

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**Exceptional Cases**

We must carefully note that arbitrarily long patents are not always optimal in this model. Here, we design a demand curve specifically to break the assumptions of Gilbert and Shapiro. We select the curve

,

plotted below.(3)

As you can see from the graph below, the social cost to profit ratio does not increase monotonically as it did for the curves we selected previously. Hence, if the minimum feasible price falls within a particular range - say the minimal price is 3 on this graph - it will not be optimal. Rather, a lower social cost to profit ratio can be achieved at higher price. Hence, a higher price results in a lower total social cost over the length of the patent.

3. Written in MiniScheme, the demand curve is (+ (/ 1 (power p 3) ) 0.5). A subtle problem is that this curve never approaches zero. This can be corrected by subtracting away a small linear function, so that the demand becomes zero at some large value of price. This does not affect the results.

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