Nurturing an Infant Industry by Markovian Subsidy Schemes

Abstract

We model a small open economy with an infant industry facing competition from imports. We derive the welfare maximizing output path and knowledge path for an infant industry under the central planner who can dictate the industry output. We next show how the social planner’s optimal path can be achieved when the infant industry is in private hand, focusing on two cases: the case where the infant industry consists of a monopoly and the case where it is a duopoly. In the case of a monopoly we show that free trade can induce the monopoly to choose the socially optimal production path. Contrary to conventional wisdom, we show that the volume of imports is large when the stock of knowledge is small, and gradually declines as this stock grows. In the case of a duopoly with knowledge spillovers we derive a family of subsidy rules capable of inducing a Markov perfect Nash equilibrium that replicates the social optimum. When the subsidy rule is linear affine in the state variable, we show that the subsidy rate per unit of output must be an increasing function of the stock. The underlying intuition is that the government should put domestic firms under a tough competition in their infancy with a promise to make their life easier as their knowledge grows.

Appendix

Proof of Corollary 2.2

From (24) we have:

$$\begin{aligned} \sigma \left( \frac{\phi ^{so}}{2}-\delta K\right) -\left( \rho +\delta - \frac{\phi ^{so\prime }}{2}\right) (\sigma K+\alpha )=\psi (K)\left( \frac{1 }{2Z}-\delta \right) . \end{aligned}$$

For this to hold for all K we must have

$$\begin{aligned} \alpha =\frac{\sigma \frac{\phi ^{so}\left( 0\right) }{2}-\psi (0)\left( \frac{1}{2Z}-\delta \right) }{\left( \rho +\delta -\frac{\phi ^{so\prime }}{2 }\right) } \end{aligned}$$

or

$$\begin{aligned} \alpha =\frac{\sigma \frac{\phi ^{so}\left( 0\right) }{2}-\psi (0)\left( \frac{1}{2Z}-\delta \right) }{\left( \rho +\delta -\frac{1}{2Z}\right) } \end{aligned}$$

From the proof of Corollary 2.1, we have established (34) that is $\frac{1}{2Z}<\delta$ and therefore $\rho +\delta -\frac{1}{2Z}>0$.

Since $\sigma >0$ and we are focusing on the case where the regulator wishes to build a domestic industry, we have $\phi ^{so}\left( 0\right) >0.$ We establish below that $\psi (0)>0$. Using (29), we have

$$\begin{aligned} \psi \left( 0\right) =-\left( p^{*}-c-\gamma \bar{K}-\beta \phi ^{so}\left( 0\right) \right) \end{aligned}$$

(A.1)

and we obtain after simplification

$$\begin{aligned} \psi (0)=\left( p^{*}-c-\gamma \overline{K}\right) \left( \frac{\frac{ 2\delta +\rho }{\delta +\rho }\gamma -\frac{\beta }{Z}}{\delta \beta -\frac{ 2\delta +\rho }{\delta +\rho }\gamma }\right) . \end{aligned}$$

Now, given assumptions A1–A5, and given that we have assumed $\left( p^{*}-c-\gamma \overline{K}\right) >0$, we argue that

$$\begin{aligned} \frac{2\delta +\rho }{\delta +\rho }\gamma -\frac{\beta }{Z}>0. \end{aligned}$$

(A.2)

Indeed, after simple substitution of Z from (30) inequality (A.2) becomes

$$\begin{aligned} \frac{2\delta +\rho }{\delta +\rho }\gamma -\frac{2\gamma }{\left( 1+\sqrt{1- \frac{4\gamma }{\left( 2\delta +\rho \right) \beta }}\right) }>0 \end{aligned}$$

which holds iff

$$\begin{aligned} \frac{2\delta +\rho }{\delta +\rho }-\frac{2}{1+\sqrt{1-\frac{4\gamma }{ \left( 2\delta +\rho \right) \beta }}}>0 \end{aligned}$$

or after simplification

$$\begin{aligned} \sqrt{1-\frac{4\gamma }{\left( 2\delta +\rho \right) \beta }}>\frac{\rho }{ (2\delta +\rho )}. \end{aligned}$$

This last inequality is in turn satisfied iff

$$\begin{aligned} 1-\frac{4\gamma }{\left( 2\delta +\rho \right) \beta }>\left( \frac{\rho }{ 2\delta +\rho }\right) ^{2} \end{aligned}$$

or after simplification

$$\begin{aligned} \frac{\beta }{\gamma }>\frac{4(2\delta +\rho )}{4\delta (\delta +\rho )}= \frac{(2\delta +\rho )}{\delta (\delta +\rho )} \end{aligned}$$

which holds from Assumption A5. Therefore

$$\begin{aligned} \psi (0)=\left( p^{*}-c-\gamma \overline{K}\right) \left( \frac{\frac{ 2\delta +\rho }{\delta +\rho }\gamma -\frac{\beta }{Z}}{\delta \beta -\frac{ 2\delta +\rho }{\delta +\rho }\gamma }\right) >0 \end{aligned}$$

and

$$\begin{aligned} \alpha =\frac{\sigma \frac{\phi ^{so}\left( 0\right) }{2}-\psi (0)\left( \frac{1}{2Z}-\delta \right) }{\left( \rho +\delta -\frac{1}{2Z}\right) }>0. \end{aligned}$$

This complete the proof of Corollary 2.2. $\square$