Design of the contingent royalty rate as related to the type of investment

Financial Innovation

Table 1 Summary of the comparison of the optimal royalty rate across three investment types and the effect on the optimal royalty rate of increasing one variable when all others remain fixed

Variable	\(\upphi_{i}^{*}\)	\(\upphi_{i}^{\prime }\)	\(\upphi_{i}^{\prime \prime }\)
Capital productivity, \(\upalpha\)	+	?	?
Expected growth rate of output price, \(\upmu\)	+	\(-\)	\(-\)
Discount rate, \(\uprho\)	\(-\)	+	+
Output price volatility, \(\upsigma\)	+	\(-\)	\(-\)
Unit cost of capital, \(\uptheta_{i}\)	?	?	?
Fixed scale of operation, \(\overline{K}\)	N.A.	+	N.A.
Marginal cost of production, \(w\)	0	\(-\)	N.A.

\(\upphi_{i}^{*}\), \(\upphi_{i}^{\prime }\), and \(\upphi_{i}^{\prime \prime }\) are the optimal royalty rate for lumpy investment with variable intensity, lumpy investment with an exogenously given intensity, and incremental investment, respectively. If \(w = 0,\) then \(\upphi_{i}^{\prime } > \upphi_{i}^{\prime \prime }\).
+ indicates that an increase in the variable causes the optimal royalty rate to increase.
\(-\) indicates that an increase in the variable causes the optimal royalty rate to be reduced.
? indicates that the relationship is uncertain.
0 indicates no effect.
N.A. indicates that the result is not applicable.