Prizes and Patents: Using Market Signals to Provide Incentives for Innovations

we then have the following sequence of inequalities leading to a contradiction:

0 ≥ τ (0J v (0₂) + T (0ι) - K > τ (0ι) v (0_i) + T (0J - K ≥ 0.

Here, the first inequality is the incentive compatibility constraint. The second inequality fol-
lows because V (0) is strictly increasing. The last inequality follows from the voluntary participation
constraint of the type 0ι. This argument establishes the critical threshold result.

Next we show that the incentive compatibility constraint implies that for the set of the
innovated goods, the patent length is nondecreasing in the quality of the good 0. Adding and
subtracting τ 0) V ^0) to the incentive compatibility constraint (4), we have that for any 0, 0

V (0, 0, δ (0)) ≥ V 0θ, 0, δ (⁰)) + τ (⁰) (π (0) - v (0)) .

A similar argument implies that:

V 00, 0, δ (0)) ≥ V (0, 0, δ (0)) + τ (0) (π (0) - v (0)) .

These two inequalities imply that if 0 > 0, then τ (0) ≥ τ (0).

Then, since social surplus is decreasing in the length of the patent, having a constant patent
length is optimal.

Next, we show that T(0) = 0, V0. Incentive compatibility for the type 0 = 0 implies that
T(0) ≤ 0. Voluntary participation by the threshold type 0* implies that

-v (0*) + T (0*) - K ≥ 0.

Suppose T (0*) < 0. Welfare maximization implies that the threshold type must satisfy

τS^m (0*) + (1 - -) S (0*) - K ≥ 0.

Because S^m (0*) ≥ V^m (0*) and S (0*) > 0 it follows that

τS^m (0*) + (1 - -) S (0*) - K > 0.

Then, it is optimal to reduce the threshold type and reduce the tax T(0*) the threshold type must

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