Private provision of public good and immiserizing growth

Abstract

This paper develops a simple R&D driven endogenous growth model with a public good financed by private contributions. I show that a larger endowment of population or resources can be immiserizing for the economy as a whole. With a larger population, the economy grows at a higher rate but per-capita income from asset holdings falls unambiguously. Since people voluntarily contribute a part of their instantaneous income toward the public good, reduced asset income may lead to lower level of provision of the public good. This brings in the possibility of immiserizing growth where higher rate of growth is associated with lower level of welfare. I also show that the socially optimum level of public good provision may well fall below the equilibrium level and that the problem of underprovision of the public good need not be aggravated in larger economies.

Appendix

Proof of Lemma 1

To solve the optimal expenditure path, we write the current value Hamiltonian of the dynamic utility maximization problem as

$$\begin{aligned} H=ln(u)+\delta [w+rA-e], \end{aligned}$$

where $ln(u)$ is the indirect utility function given in Eq. (11) and $\delta$ is the co-state variable associated with the intertemporal budget constraint in Eq. (5). The first order conditions of this maximization problem are

$$\begin{aligned} \frac{\partial {H}}{\partial {e}}&= \frac{1}{e}-\delta =0;\\ \dot{\delta }&= \rho \delta - \frac{\partial {H}}{\partial {A}}=\delta (\rho -r). \end{aligned}$$

From the first equation we have $\frac{\dot{\delta }}{\delta }=-\frac{\dot{e}}{{e}}$. Using this in the second equation we get

$$\begin{aligned} \frac{\dot{e}}{e}=r-\rho . \end{aligned}$$

$\square$

Proof of Proposition 1

First we show the derivation of Eq. (26). Using Eq. (10) and (23), the indirect utility in (11) takes the following form,

$$\begin{aligned} ln(u)=ln(L+a\rho )-ln\left( 1+L\frac{\sigma }{1-\sigma }\right) +\sigma ln\left( \frac{\sigma \theta }{(1-\sigma )\beta } \right) +\frac{\sigma (1-\theta )}{\theta }ln(n). \end{aligned}$$

Since $n$ grows at a constant rate rate $g$ in the steady state, we must have $n(t)=n(0)e^{gt}$, where $n(0)$ is the initial number of product variety. So,

$$\begin{aligned} \int _0^\infty e^{-\rho t}ln (n(t))=\frac{1}{\rho }\left( ln(n(0))+\frac{g}{\rho } \right) . \end{aligned}$$

Using Eq. (1), the expression of welfare under the steady state can be given by Eq. (26). Next, differentiating $g$ in Eq. (25) with respect to $L$ and plugging back this expression into Eq. (27) gives us

$$\begin{aligned} \frac{dW}{dL} = -\; \frac{1}{\rho }\frac{a\rho \frac{\sigma }{1-\sigma }-1}{(L+a\rho )(1+L\frac{\sigma }{1-\sigma }) } + \frac{\sigma (1-\theta )^2}{a\theta \rho ^2}\left[ \frac{ \left( \frac{\sigma }{1-\sigma }L\right) ^2+2\frac{\sigma }{1-\sigma }L+ a\rho \frac{\sigma }{1-\sigma } }{\left( \frac{\sigma }{1-\sigma }L+1 \right) ^2 } \right] . \end{aligned}$$

This expression is negative iff condition (28) in text is satisfied. $\square$

Proof of Proposition 2

Let $S^o$ denote the socially optimal level of $S$ and let $S^*$ denote its market equilibrium given by Eq. (24). Define the gap between $S^o$ and $S^*$ as

$$\begin{aligned} S^o-S^*=\frac{1-\sigma }{\sigma }\;\left[ \frac{a\rho \theta }{1-\theta }- \frac{L+a\rho }{L+\frac{1-\sigma }{\sigma }} \right] . \end{aligned}$$

Note that in the special case when $a\rho =\frac{1-\sigma }{\sigma }$ and $\sigma =\theta$, one obtains $S^o=S^*$. There are no distortions in this case. Part (i) of proposition 2 takes place under the following parametric restriction:

$$\begin{aligned} \theta < min\left\{ \frac{L+a\rho }{L+a\rho +a\rho (L+\frac{1-\sigma }{\sigma })}, \; \frac{L-\frac{a\rho (1-\sigma )}{\sigma L}}{L+a\rho }\right\} . \end{aligned}$$

The first part of the above inequality guarantees that $S^o < S^*$. This implies that the public good is overprovided in the laissez-faire economy. The second part guarantees that the rate of growth is positive [see Eq. (25)]. Also we have already seen (in Sect. 2) that $S^*$ can be an increasing function of $L$. If we start from a situation where public good is underprovided (i.e., $S^o>S^*$), the extent of underprovision would decrease as $L$ grows larger. This proves part (ii) of Proposition 2. $\square$