Valuation of public goods: an intertemporal mixed demand approach

Abstract

This paper presents a new mixed demand model for measuring the value of public goods in an intertemporal optimization framework. From the specification of an indirect utility function allowing for public goods, direct demand functions for private goods are derived and estimated jointly with the Euler equation for intertemporal consumption behavior, using US data. This allows us to identify the marginal utility of private consumption and to obtain the inverse demand or shadow price of a public good, which is then related to its observed price to assess whether the public good is efficiently provided. There is evidence, though suggestive, that the public good as measured by national defense in the USA has been inefficiently provided over the past decades.

Appendices

Appendix A: Scaling of z and/or C for the MPIGLOG indirect utility function in Sect. 3.1

Estimation of the budget share system and Euler equation involves different units of measurement for the variables under consideration. In our analysis, the public good in particular has large units relative to private goods, which causes convergence problems. Specifically, in line with the representative agent framework, private consumption is measured in per capita value by dividing it by population. We use national defense as a public good in empirical analysis. While it is common to express national defense in per capita term (see, e.g., Barro and Redlick 2011; Ramey 2011), this procedure is not valid because of its nature of a public good, especially joint consumption. If not expressed in per capita term, national defense has a large unit. A common procedure is to scale down those variables that have large units of measurement. It can be shown that, with appropriate scaling of some parameters, parameter estimates are invariant to this scaling.

To do this, we introduce scaling factors for the public good and consumption, k₁ and k₂, and rewrite (14) as

$$\nu (\hat{C}_{t}^{{}} ,\,\,{\mathbf{p}}_{t}^{{}} ,{\tilde{\mathbf{z}}}_{t}^{{}} ) = \ln \left[ {\frac{{\hat{C}_{t}^{{}} \tilde{z}_{t}^{\omega } }}{{\bar{\kappa }P_{1} ({\mathbf{p}}_{t}^{{}} )}}} \right]\left[ {\frac{{\hat{C}_{t}^{{}} }}{{P_{2} ({\mathbf{p}}_{t}^{{}} )}}} \right]^{\eta } \left[ {\tilde{z}_{t}^{{}} - \tilde{\gamma }} \right]^{\phi } ,$$

(A1)

where $\tilde{z}_{t}^{{}} = k_{1} z_{t}^{{}} ,\hat{C}_{t}^{{}} = k_{2} C_{t} ,\bar{\kappa } = (k_{1} )_{{}}^{\omega } k_{2} \kappa ,$ and $\tilde{\gamma } = k_{1} \gamma .$ Re-expressing (A1),

$$\begin{aligned} \nu (\hat{C}_{t}^{{}} ,\,\,{\mathbf{p}}_{t}^{{}} ,{\tilde{\mathbf{z}}}_{t}^{{}} ) & = \ln \left[ {\frac{{\hat{C}_{t}^{{}} \tilde{z}_{t}^{\omega } }}{{\bar{\kappa }P_{1} ({\mathbf{p}}_{t}^{{}} )}}} \right]\left[ {\frac{{\hat{C}_{t}^{{}} }}{{P_{2} ({\mathbf{p}}_{t}^{{}} )}}} \right]^{\eta } \left[ {\tilde{z}_{t}^{{}} - \tilde{\gamma }} \right]^{\phi } \\ & = \ln \left[ {\frac{{k_{2} C_{t}^{{}} (k_{1} z_{t} )_{{}}^{\omega } }}{{(k_{1} )_{{}}^{\omega } k_{2} \kappa P_{1} ({\mathbf{p}}_{t}^{{}} )}}} \right]\left\{ {\frac{{(k_{2} C_{t} )^{\eta } }}{{\left[ {P_{2} ({\mathbf{p}}_{t}^{{}} )} \right]^{\eta } }}} \right\}\left[ {k_{1} z - k_{1} \gamma } \right]^{\phi } \\ & = \ln \left[ {\frac{{C_{t}^{{}} z_{t}^{\omega } }}{{\kappa P_{1} ({\mathbf{p}}_{t}^{{}} )}}} \right]\left\{ {\frac{{k_{2}^{\eta } C_{t}^{\eta } }}{{\left[ {P_{2} ({\mathbf{p}}_{t}^{{}} )} \right]^{\eta } }}} \right\}k_{1}^{\phi } \left[ {z_{t} - \gamma } \right]^{\phi } \\ & = k_{2}^{\eta } k_{1}^{\phi } \ln \left[ {\frac{{C_{t}^{{}} z_{t}^{\omega } }}{{\kappa P_{1} ({\mathbf{p}}_{t}^{{}} )}}} \right]\left[ {\frac{{C_{t}^{{}} }}{{P_{2} ({\mathbf{p}}_{t}^{{}} )}}} \right]^{\eta } \left[ {z_{t} - \gamma } \right]^{\phi } \\ & = k_{2}^{\eta } k_{1}^{\phi } \nu (C_{t}^{{}} ,\,\,{\mathbf{p}}_{t}^{{}} ,z_{t}^{{}} ). \\ \end{aligned}$$

(A2)

This is invariant to scaling because of the constant factors right out of the intertemporal criterion function. Thus scaling of either or both of C and z does not affect the optimal solution, provided the parameters $\kappa$ and $\gamma$ are scaled accordingly. In our estimation, we only allow z to be scaled.

Appendix B: Regularity conditions for the MPIGLOG indirect utility function in Sect. 3.1

Provided that the conditions on parameters noted in Sect. 3.1 are satisfied, the MPIGLOG indirect utility function (14) is regular with respect to the within-period properties over the region $C_{t}^{{}} z_{t}^{\omega } > \kappa P_{1} \left( {{\mathbf{p}}_{t} } \right)$ and $z_{t}^{{}} > \gamma .$ Intertemporal properties of MPIGLOG with respect to the concavity of $C_{t}^{{}}$ can be examined from (18). If $\eta \ge 1,$ the MPIGLOG indirect utility function (14) is convex in $C_{t}^{{}} .$ If $0 < \eta \le 0.5,$ on the other hand, it is concave in $C_{t}^{{}}$. If $0.5 < \eta < 1,$ the MPIGLOG indirect utility function is likely to be concave in $C_{t}^{{}}$ for certain combinations of data and parameters. Intertemporal optimization, however, requires the concavity of the intertemporal utility function (4). To this end, we can use the relevant expressions in (14), (15), and (16) to evaluate the concavity condition in (6). For $\zeta > 0$ and $0 < \eta < 0.5,$ with $C_{t}^{{}} z_{t}^{\omega } > \kappa P_{1} \left( {{\mathbf{p}}_{t} } \right)$ and $z_{t}^{{}} > \gamma$, the concavity of the indirect utility function (14) will preserve the concavity of the intertemporal utility function. However, if $\zeta > 0$ and $0 < \eta < 1,$ an additional sufficient condition is $\zeta > 1 + (0.5/\eta ).$ If $\eta$ is in the range $0.5 < \eta < 1,$ then the Box–Cox transformation has to be sufficiently concave to ensure overall concavity.