Physical Capital Accumulation and Partial Privatization

Abstract

This chapter investigates the effect of capital accumulation on partial privatization. We extend the dynamic oligopoly model proposed by Cellini and Lambertini in 1998 to a dynamic mixed oligopoly model. We show that (i) when a steady state is characterized by a demand-driven (i.e., static) equilibrium, partial privatization is adopted and the privatization ratio perfectly corresponds to the static model; (ii) when a public firm produces Ramsey output, the level of social welfare in a steady state does not depend on the privatization ratio; and (iii) when a private firm produces Ramsey output, the government adopts a full nationalization policy. The results of (ii) and (iii) are in contrast with the static result that partial privatization is optimal.

Appendix

1.1 A.4.1 Derivation of the Reaction Function of Private Firms

From (4.5), the optimal reaction function, ${q}_{1,t}^r$, becomes

$$ {q}_{1,t}^r=\frac{a-{q}_{0,t}^r-{c}_0-{\lambda}_{1,t}}{2}. $$

(A4.1)

Differentiating (A4.1) with respect to time, we obtain $\frac{d{q}_{1,t}^r}{dt}=-\frac{1}{2}\left(\frac{d{q}_{0,t}^r}{dt}+\frac{d{\lambda}_{1,t}}{dt}\right)$.

Substituting (4.6) into this, we obtain $\frac{d{q}_{1,t}^r}{dt}=-\frac{1}{2}\left\{\frac{d{q}_{0,t}^r}{dt}+{\lambda}_{1,t}\left[\delta +\rho -{f}^{\prime}\left({k}_{1,t}\right)\right]\right\}$. Using ${\lambda}_{1,t}=a-{q}_{0,t}^r-2{q}_{1,t}^r-{c}_0$, we obtain (4.7).

1.2 A.4.2 Derivation of the Reaction Function of the Public Firm

From (4.8), an optimal reaction function becomes

$$ {q}_{0,t}^r=\frac{a-{q}_{1,t}^r-{c}_0-{\lambda}_{1,t}}{1+\theta }. $$

(A4.2)

Differentiating (A4.2) with respect to time, we obtain $\frac{d{q}_{0,t}^r}{dt}=-\frac{1}{1+\theta}\left(\frac{d{q}_{1,t}^r}{dt}+\frac{d{\lambda}_{0,t}}{dt}\right)$. Substituting (4.9) into this, we obtain $\frac{d{q}_{0,t}^r}{dt}=-\frac{1}{1+\theta}\left\{\frac{d{q}_{1,t}^r}{dt}+{\lambda}_{0,t}\left[\delta +\rho -{f}^{\prime}\left({k}_{0,t}\right)\right]\right\}$. Using ${\lambda}_{1,t}=a-\left(1+\theta \right){q}_{0,t}^r-{q}_{1,t}^r-{c}_0$, we obtain (4.10).