On the Private Provision of Intertemporal Public Goods with Stock Effects

Abstract

The provision of public goods is often hindered by a lack of powerful institutions that can sanction free riders or otherwise enforce private contributions to the public good. The simple deposit based solution introduced by Gerber and Wichardt (J Public Econ 93:429–439, 2009) solves this problem, but may require prohibitively large deposits, in particular in the context of intertemporal public goods. In this paper, we propose a modification of the deposit solution that relies only on comparably small deposits. The proposed modification improves the applicability of the procedure, most notably as it also allows to reduce deposits in static public goods problem by transforming them into dynamic ones with small per period contributions.

Appendix

1.1 Proof of Proposition 1

In order to see that zero contributions in all periods, i.e. $\xi _i^t=0$ for all $i$ and all $t$ is the unique subgame perfect Nash equilibrium of PG consider period 2 first.

For any given initial stock $z^0$ of the public good and any given contributions $c^1$ in period 1, the utility of player $i$ obtained in period 2 is given by

$$\begin{aligned} U_i(e_i^2-c_i^2, z^2)=e_i^2-c_i^2+u_i(z^2)=e_i^2-c_i^2+u_i(F(z^1,(c_i^2,c_{-i}^2))) \end{aligned}$$

where $z^1=F(z^0,c^1)$ denotes the stock retained from period 1 and $c_{-i}^2$ denotes the contributions of all players except player $i$ in period 2. From DOM it follows that

$$\begin{aligned} \frac{\partial U_i(e_i^2-c_i^2, z^2)}{\partial c^2_i}=-1+u_i^{\prime }(z^2)\frac{\partial F}{\partial c_i^2}(z^1,c^2)<0, \end{aligned}$$

and hence, it is strictly dominant for all players to contribute zero in period 2. Accordingly, each player’s utility at the end of period 2 is given by

$$\begin{aligned} U_i(e_i^2-c_i^2, z^2)=e_i^2+u_i(F(z^1,0)). \end{aligned}$$

Once the equilibrium contribution level in period 2 is fixed, the game can be reduced and the focus can be put on period 1. Taking period 2 behaviour as given, the utility of any player $i$ obtained in period 1 is given by

$$\begin{aligned} U_i(e_i^1-c_i^1, z^1)+\delta _i [e_i^2+u_i(F(F(z^0,c^1),0))]. \end{aligned}$$

(16)

Taking the partial derivative of (16) with respect to $c^1_i$ yields

$$\begin{aligned} -1+\frac{\partial F(z^0,c^1)}{\partial c^1_i}\left(u_i^{\prime }(z^1)+\delta _iu_i^{\prime }(z^2)\frac{\partial F(z^1,0)}{\partial z^1}\right)<0 \end{aligned}$$

by the second condition in DOM. Thus, again, contributing zero in period 1 is optimal for each player independent of the contributions of the other players.