Privatization Neutrality Theorem When a Public Firm Maximizes Objectives Other than Social Welfare

Abstract

This chapter investigates the privatization neutrality theorem when a public firm has a different objective from social welfare maximization. The privatization neutrality theorem claims that when the government gives the optimal subsidy to both public and private firms, social welfare is exactly the same before and after privatization. We demonstrate that if the discriminatory subsidy scheme is adopted to public and private firms, the privatization neutrality theorem can be recovered in a variety of situations. Especially, we obtain a seemingly paradoxical result as follows: When a public firm incorrectly recognizes that a subsidy to firms by a government directly affects the welfare size, the privatization neutrality necessarily holds. In contrast, when a public firm correctly recognizes that a subsidy affects only income distribution but not social welfare itself, the situation in which the neutrality holds is limited.

Appendix

1.1 Derivation of the Optimal Subsidy Levels (s ^*₀ , s ^*₁ ) when ${\boldsymbol{\beta}}_0\ne \boldsymbol{\gamma}$

From the two equations, (7.30) and (7.31), $\left\{\left(\alpha -{\beta}_0-{\beta}_1\right)+\left(k+2\right)\left[{\beta}_0\left(k+2\right)-\alpha \right]\right\}\left[a-\left(k+1\right){q}_0-{q}_1\right]=0$ holds. Because generally $\left(\alpha -{\beta}_0-{\beta}_1\right)+\left(k+2\right)\left[{\beta}_0\left(k+2\right)-\alpha \right]$ is not equal to zero for any $k>0$, $a-\left(k+1\right){q}_0-{q}_1=0$ holds. Thus, it is confirmed that ${s}_1={q}_1$ and ${q}_0=\frac{a-{q}_1}{k+1}=\frac{a-{s}_1}{k+1}$. When arranging the latter equation, we obtain the following equations:

$$ \begin{array}{l} {q}_0=\frac{\left[\alpha +{\beta}_0\left(k+1\right)-{\beta}_1\right]a-\left(\gamma -{\beta}_0\right)\left(k+2\right){s}_0+\left(\alpha -{\beta}_0-{\beta}_1\right){s}_1}{\beta_0\left(k+1\right)\left(k+3\right)-{\beta}_1-\alpha \left(k+1\right)}=\frac{a-{s}_1}{k+1}\\ {} \iff \left(\gamma -{\beta}_0\right)\left(k+1\right)\left(k+2\right){s}_0+\left[{\beta}_1-{\beta}_0\left(k+1\right)\right]\left(k+2\right){s}_1\\ {} =\left[2\left(k+1\right)\alpha -2{\beta}_0\left(k+1\right)-{\beta}_1k\right]a,\end{array} $$

(A.7.1)

$$ \begin{array}{l} {q}_1=\frac{\left[{\beta}_0\left(k+1\right)-\alpha \right]a+\left(\gamma -{\beta}_0\right){s}_0+\left[{\beta}_0\left(k+2\right)-\alpha \right]{s}_1}{\beta_0\left(k+1\right)\left(k+3\right)-{\beta}_1-\alpha \left(k+1\right)}={s}_1\\ {} \iff \left(\gamma -{\beta}_0\right){s}_0+\left[\alpha k-{\beta}_0\left({k}^2+3k+1\right)+{\beta}_1\right]{s}_1\\ {} =\left[\alpha -{\beta}_0\left(k+1\right)\right]a.\end{array} $$

(A.7.2)

By solving the simultaneous equations, (A.7.1) and (A.7.2), we obtain the optimal subsidy levels as follows:

$$ {s}_0^{*}=\frac{\left(2\alpha -{\beta}_0-{\beta}_1\right)a}{\left(\gamma -{\beta}_0\right)\left(k+2\right)},\ {s}_1^{*}=\frac{a}{k+2}={q}_0={q}_1. $$

(A.7.3)