Investment-driven mixed firms: partial privatization by local governments

Abstract

We analyze partial privatization by local governments, driven by investment and credit constraints, and provide a theory of monopolistic mixed firms based on strategic interaction between local politicians and private shareholders. Minority participation by private investors—as empirically observed—arises endogenously in the model to prevent investment expropriation. We consider the example of water supply with perfectly inelastic demand and fixed-price regulation, coupled with price discretion at a local level. Welfare-maximizing local governments face a trade-off between the increase in consumers’ surplus and the reduction in costly public funds. Therefore, private shareholders choose investment to keep the government share at a threshold such that the politician always sticks to the price-cap and dividends are then maximized. To consider normative issues, we compare investment by mixed firms and by a social planner.

Appendices

Appendix 1

In this appendix, we study the monotonicity of the function V with respect to K.

$$\begin{aligned} V=\left\{ \begin{array} [c]{l} (1-p)\left[ \beta (i-\underline{i})K\left[ 2-\frac{(i+\underline{i})K}{L_{0} }\right] +(\rho -rc)K+\rho S^{\circ }\right] -(1-c)\alpha \rho K,\\ \mathrm {if}\ K\le S^{\circ }\frac{\lambda }{1-c}\\ 0,\mathrm {if}\ K>S^{\circ }\frac{\lambda }{1-c} \end{array} \right. \end{aligned}$$

When $K\le S^{\circ }\frac{\lambda }{1-c}$, the complete form of V is

$$\begin{aligned} V= & {} \frac{(1-c)K}{S^{\circ }+(1-c)K}\left[ \beta (i-\underline{i})K\left[ 2-\frac{(i+\underline{i})K}{L_{0}}\right] +(\rho -rc)K+\rho S^{\circ }\right] \nonumber \\&-(1-c)\alpha \rho K \end{aligned}$$

which can be written as

$$\begin{aligned} V=\frac{(1-c)K}{S^{\circ }+(1-c)K}\left[ \beta (i-\underline{i})K\left[ 2-\frac{(i+\underline{i})K}{L_{0}}\right] +(\rho -rc-(1-c)\alpha \rho )K+\rho (1-\alpha )S^{\circ }\right] \end{aligned}$$

where the two factors are positive:

$$\begin{aligned}&\frac{(1-c)K}{S^{\circ }+(1-c)K}=1-s>0,\\&\beta (i-\underline{i})K\left[ 2-\frac{(i+\underline{i})K}{L_{0}}\right]>0\ \mathrm {if}\ K<2\frac{L_{0}}{i+\underline{i}},\ \mathrm {where} \ 2\frac{L_{0}}{i+\underline{i}}>2\frac{L_{0}}{i+i}=\frac{L_{0}}{i},\\&(\rho -rc-(1-c)\alpha \rho )=\rho [1-(1-c)\alpha ]-rc\underbrace{>} _{\rho>r}r[1-(1-c)\alpha -c]=r(1-\alpha )(1-c)>0,\\&\rho (1-\alpha )S^{\circ }>0 \end{aligned}$$

The first factor $\frac{(1-c)K}{S^{\circ }+(1-c)K}$ has a positive derivative $\frac{(1-c)S^{\circ }}{[S^{\circ }+(1-c)K]^{2}}$, whereas the second factor has the derivative

$$\begin{aligned} 2\beta (i-\underline{i})+\rho -rc-(1-c)\alpha \rho -2\beta \frac{i^{2} -\underline{i}^{2}}{L_{0}}K, \end{aligned}$$

which is positive when

$$\begin{aligned} K<\frac{2\beta (i-\underline{i})+\rho -rc-(1-c)\alpha \rho }{2\beta \frac{i^{2}-\underline{i}^{2}}{L_{0}}}=\frac{L_{0}}{i+\underline{i}}+\frac{\rho -rc-(1-c)\alpha \rho }{2\beta (i^{2}-\underline{i}^{2})}. \end{aligned}$$

This shows that an analytic conclusion about the monotonicity of V cannot be easily found. A sufficient condition is that for $K<\frac{L_{0}}{i+\underline{i}}$, the function V is strictly increasing in K. However, we obtained an interesting result by simulation as follows:

• We randomly generate 100,000 parameter sets in the ranges, $\underline{i} = 0.074925$;

  

• We computed the $K^{\circ }$ that maximizes V in the range $K\in \left( 0,2\frac{L_{0}}{i}\right)$.

• We computed the number of times when $K^{\circ }< \frac{S^{\circ }\, \lambda }{1-c}$ and $K^{\circ }< \frac{L_{0}}{i}$. We found that this number is 0.

We are therefore reasonably sure that the function V is strictly increasing for $K\in \left( 0,\frac{S^{\circ }\,\lambda }{1-c}\right)$.

Appendix 2

In this appendix, we formally prove that local welfare is strictly increasing in c for any P.

• 1. $P=\widehat{P}$. By substituting and $U_{s}(\widehat{P})=\alpha \rho S^{\circ }$ [see eq. (10)] in (8) we get the maximization problem

  $$\begin{aligned} \max _{c}W&=\max _{c}\left\{ P^{\max }-\widehat{P}-d\left[ L_{0} -2iK+\frac{(iK)^{2}}{L_{0}}\right] +(1+\lambda )\alpha \rho S^{\circ }\right\} \\ \mathrm {s.t.}&\widehat{P}=\left( \frac{1-c}{1-s}\,\alpha \rho +1+cr\right) K+\beta \left( 1+L_{0}-2iK+\frac{(iK)^{2}}{L_{0}}\right) \\&c\le \overline{c}. \end{aligned}$$

  By substitution of $(1-s)=\frac{(1-c)K}{S^{\circ }+(1-c)K}$ in $\widehat{P}$, we obtain

  $$\begin{aligned} \widehat{P}=S^{\circ }\alpha \rho +(1-c)K\alpha \rho +K+crK+\beta \left( 1+L_{0}-2iK+\frac{(iK)^{2}}{L_{0}}\right) , \end{aligned}$$

  and differentiating W w.r.t. $\widehat{P}$, we get

  $$\begin{aligned} \frac{\partial W}{\partial c}=-\frac{\partial \widehat{P}}{\partial c} =(\alpha \rho -r)K>0\ \mathrm {for}\ \alpha \rho >r. \end{aligned}$$

• 2. $P=\overline{P}$, then the welfare maximization problem is

  $$\begin{aligned} \max _{c}W&=\max _{c}\left\{ P^{\max }-\overline{P}-d\left[ L_{0} -2iK+\frac{(iK)^{2}}{L_{0}}\right] +(1+\lambda )s(c,K)\varPi \left( \overline{P},K\right) \right\} \\ \mathrm {s.t.}&\overline{P}=\beta \left( 1+L_{0}-2\underline{i} K+\frac{(\underline{i}K)^{2}}{L_{0}}\right) +(1+\rho )K+\rho S^{\circ }\\&\varPi (\overline{P})=\beta (i-\underline{i})K\left[ 2-\frac{(i+\underline{i})K}{L_{0}}\right] +(\rho -cr)K+\rho S^{\circ }\\&s(c,K)=\frac{S^{\circ }}{S^{\circ }+(1-c)K}. \end{aligned}$$

  By differentiating w.r.t. to c, we get

  $$\begin{aligned} \frac{\partial W}{\partial c}&=\frac{\partial s}{\partial c}(1+\lambda )\varPi (\overline{P},K)+s(c,K)(1+\lambda )\frac{\partial \varPi (\overline{P} ,K)}{\partial c}=\\&=\frac{Ks(c,K)}{S^{\circ }+(1-c)K}(1+\lambda )\varPi (\overline{P} ,K)-s(c,K)(1+\lambda )rK. \end{aligned}$$

  By giving prominence to $(1+\lambda )Ks(c,K)$ and by substitution of $\varPi (\overline{P},K)$, we get

  $$\begin{aligned} \frac{\partial W}{\partial c}=\frac{\beta K(i-\underline{i})}{S^{\circ }+(1-c)K}\left[ 2-\frac{(i+\underline{i})K}{L_{0}}\right] +\frac{(\rho -cr)K+\rho S^{\circ }}{S^{\circ }+(1-c)K}-r \end{aligned}$$

Notice that

$$\begin{aligned} \frac{(\rho -cr)K+\rho S^{\circ }}{S^{\circ }+(1-c)K}-r>0\Longrightarrow \frac{\partial W}{\partial c}>0\Longrightarrow c=\overline{c}. \end{aligned}$$

By solving we obtain

$$\begin{aligned} \frac{(\rho -cr)K+\rho S^{\circ }}{S^{\circ }+(1-c)K}-r=\frac{(\rho -r)(S^{\circ }+K)}{S^{\circ }+(1-c)K}>0. \end{aligned}$$

Appendix 3: Calibration exercise and simulations

Given the expression of optimal investment chosen by the social planner and those arising in equilibrium with partial privatization, it is difficult to consider the issue of investment distortions only in theoretical terms. We expect, however, that in most real situations, with credible parameter values, this distortion can be well defined. For this reason, in this section we calibrate the model to a plausible real case. Due to the difficulty to retrieve real data, we use a couple of studies to find the parameter values and to cross-check the consistence of some common parameter estimates. Calibration data are obtained from South-West France (Garcia and Thomas 2003) and Norway (Venkatesh 2012). Total demand is normalized to 1, so every water quantity is rescaled accordingly; we assume that, without any investment, water leaks would amount to 40 % of the total demand, i.e., $L_{0}=0.4$. Monetary values in millions of Euros, M€.

• France Table 1 presents the main figures from the French case. We use the average values for calibration. The variable cost is equal to $\beta (1+L)$, therefore $\beta \simeq 0.18$.

• Norway The study presents many data from which we can infer technical and economic values. We use the extrapolated data concerning: total demand, leak volumes, rehabilitation cost, avoided leaks, cost savings. Monetary values are converted to Euros and considered in current terms. We obtain the following values: $i\simeq 0.15$, variable costs can be estimated 0.568 M€/Mm³, therefore, in our normalized example, $\beta \simeq 0.227$.

Table 1 Extrapolated data from Garcia and Thomas (2003), approximated figures

Summing up, we obtain reasonably close values for $\beta$. Moreover, we can set $i\simeq 0.15$ and $d\simeq 0.04$,¹⁸ corresponding to 0.1 €/m³. Therefore, we consider a “current” setting with parameters values

Starting from these reference settings, we explore a wide set of scenarios, for i ranging from 0.075 to 1.8 and $\beta$ from 0.016 to 3.178. All cases are evaluated for $\lambda$ from 0.01 to 0.3 (Snow and Warren 1996). This means that, for robustness check, we extend the parameter values well beyond a reasonably realistic range. For the sake of brevity, we present what we consider to be the most interesting and illustrative results. The full results are available upon request. Figure 2 shows the optimal investment chosen by a social planner, Fig. 3 shows the optimal investment with mixed firms, and Fig. 4 includes both types of investments to show when overinvestment or underinvestment is likely to arise. The values of optimal investments are plotted with respect to the efficiency parameter i in the cases of low/current/high variable cost and low/regular/high marginal cost of public funds.