# The optimal length of a patent

Now consider the problem of designing an appropriate intellectual property rights regime. The critical policy variable in this set-up is the choice of the patent length, T. A moment's reflection reveals that in this set-up, a patent length of *T =* 0 is not efficient: if *T =* 0, then the inventor will be unable to recover his fixed costs K and will not innovate. On the other hand, a patent length of *T = * is also clearly not optimal, since that would involve consumers receiving no additional consumption benefits, with J

*ABe~*0 as

^{rs}ds ^*T ^<*>,*and so the innovation would be pointless from a benefit point of view.

Therefore, as a general rule, the optimal patent length is positive and finite. The optimal length is found by solving:

The first term in (7.13) is the expression for total welfare in (7.11). The constraint in (7.13) is the break-even constraint for the producer - for the invention to be produced, the producer must be compensated an amount that is at least equal to his sunk costs, so that he at least breaks even.

Since the objective function is decreasing in T and since the constraint is increasing in T, the optimal patent length here involves setting T so that the producer just breaks even. Therefore the optimal patent length T* satisfies:

or:

so that the optimal patent length is:

The determination of the optimal patent length in this model is illustrated in Figure 7.5.1. The curve ^{B} + ^Be is the present value of the gains

*r*

to consumers from innovation. It is maximised at *T =* 0, which is the first- best optimum. However, at *T =* 0, the innovator would make a loss and would therefore not innovate, and there would be no welfare gain. The

*P*

curve (1 - *e*~^{rT}) - *K* is the second-best constraint, and is the net present *r*

value of the innovator's profit stream. Setting *T* so that profits are zero maximises welfare subject to the second-best constraint holding, and gives

the optimal patent length *T* * = - ^log ^1 - -K j.

Given a patent length of *T**, the level of welfare is:

and the incremental welfare gain from innovation is:

It is straightforward to show both analytically and diagrammatically (using Figure 7.5.1 and 7.5.2 below) that in this model, all else being equal:

• The optimal patent length is increasing in K. If it is more costly for the firm to invent, it will need to be compensated with higher

*Figure 7.5.1* The optimal patent length

*Figure 7.5.2* The welfare gain from the optimally designed patent length revenues in present value terms for a fixed *P,* the only way that this can happen is if *T* rises. *A* rise in *K* also reduces the welfare gain from innovation, even if the patent length is adjusted optimally.

- • The optimal patent length is decreasing in P. If the original price of the good is higher, then for a given cost of inventing, the firm does not need as long a patent length in order to cover its costs. Thus it is efficient to have a lower
*T*in this case. A higher*P*also increases the welfare gain from innovation when the optimal patent length is chosen. - • The optimal patent length is increasing in r. If
*r*rises, then for any initial price*P*and any given patent length*T*, the present value of the revenue earned by the innovator falls, and research and development costs will not be fully recovered. The only way that revenues can be restored to previous levels (in present value terms) is if the patent length rises. A rise in*r*also reduces the welfare gain from innovation, even when the patent length is adjusted optimally.