Partial Privatization Upstream with Spatial Price Discrimination Downstream

Abstract

We consider a mixed duopoly selling to downstream retailers that are engaged in spatial price discrimination. We show that the optimal degree of privatization falls below—often far below—the level that is implied in the absence of the vertical chain. The size of this reduction in privatization reflects the extent to which increasing transport cost (differentiation) increases double marginalization in contested market regions—as opposed to simply reducing demand in uncontested market regions. Moreover, we show that despite higher costs of production, a fully public monopoly upstream can be welfare-superior to the optimal mixed duopoly. This would not be the case in the absence of downstream differentiation.

Appendix

1.1 Proof of Proposition 1 for the M-shaped equilibrium

We know that $\left. {\frac{{\partial W^{m} (\omega^{m*} )}}{\partial \lambda }} \right|_{{\lambda { = }0}}$ reaches its minimum when $c_{1} = 0.2$ and that the value at the minimum is positive: $\left. {\frac{{\partial W^{m} (\omega^{m*} )}}{\partial \lambda }} \right|_{{\lambda { = }0}} (c_{1} = 0.2) = 0.1977$. This implies that $\left. {\frac{{\partial W^{m} (\omega^{m*} )}}{\partial \lambda }} \right|_{{\lambda { = }0}} > 0$ and, therefore, that $\lambda { = }0$ is never optimal for the M-shaped equilibrium, and that $\lambda^{m*}$ must be positive.

1.2 Proof of Proposition 2

For the small t of the V-shaped equilibrium ($0 < t \le t^{s*}$), we know that $\omega^{ *} = 2c_{1} - \frac{t}{2}$, so it follows naturally that $\omega^{ *}$ decreases with t. When $0 < t \le 2c_{1}$, $\omega^{*} \ge c_{1}$ and when $2c_{1} < t \le t^{s*}$, $\omega^{*} < c_{1}$.

For the moderate t of the M-shaped equilibrium ($t^{s*} < t \le t^{T}$), we have $\omega^{*} (t = t_{T} ) < c_{1} ,\omega^{*} (t = t^{s*} ) < c_{1}$. Further, we show $\left. {\frac{{\partial \omega^{*} }}{\partial t}} \right|_{{t = t^{s*} }} < 0$, $\left. {\frac{{\partial \omega^{*} }}{\partial t}} \right|_{{t = t_{T} }} > 0$ and the second derivative of $\omega^{*}$ with respect to t, $\frac{{\partial^{2} \omega^{*} }}{{\partial t^{2} }} > 0$ as $t^{s*} \le t \le t^{T}$. Thus, $\omega^{*}$ first decreases with t and then increases with t for the M-shaped equilibrium (see "Appendix" Fig. 5).

Fig. 5

The optimal wholelsale price under optimal privatization level for the M-shaped equilibrium

Therefore, we have $\omega^{*} < \omega^{*} (t = t^{s*} ){ = }\omega^{*} (t = t_{T} ) < c_{1}$. In consequence,$\omega^{ *}$ first decreases with t and then increases with t, and when $0 < t < 2c_{1}$, $\omega^{*} > c_{1}$; when $2c_{1} < t < t_{T}$, $0 < \omega^{*} < c_{1}$.

1.3 Proof of Proposition 3

When t is small so that only the V-shaped equilibrium exists, the optimal privatization level is $\lambda^{s*} = \frac{{c_{1} }}{{1 - 4c_{1} + \frac{3t}{4}}}$, and $\frac{{\partial \lambda^{s*} }}{\partial t} = - \frac{{12c_{1} }}{{(4 + 3t - 16c_{1} )^{2} }} < 0$, therefore $\lambda^{s*}$ decreases as t increases.

When t becomes larger so that the M-shaped equilibrium exists, the optimal wholesale price becomes:

$$\omega^{m*} = \frac{{16\lambda t + 6c_{1} + 13\lambda + 2t + 5 - 3\sqrt {A_{1} } }}{11 + 19\lambda },$$

${\text{where }}A_{1} = 39\lambda^{2} t^{2} - 4c_{1} \lambda t + 4\lambda^{2} t + 9\lambda t^{2} + 4c_{1}^{2} - 8c_{1} \lambda - 12c_{1} t + 4\lambda^{2} + 16\lambda t - 2t^{2} - 8c_{1} + 8\lambda + 12t + 4$ The FOC of the associated social welfare $W^{m} (\omega^{m*} )$ with respect to $\lambda$ yields the optimal privatization level.

While the expression for the optimal privatization level of the M-shaped equilibrium (intermediate values of t) remains very complicated, we can graph it for any $c_{1}$ (see Fig. 3). Notice that when t is large enough that each firm is uncontested, privatization does not change with t.

We can also show analytically that the optimal privatization level of the M-shaped equilibrium first decreases with t and then increases with t until the upper bound, which is exactly what Proposition 3 indicates.

1.4 Proof of Proposition 4

We denote the optimal privatization level as $\lambda^{*}$. Since Proposition 3 shows that the optimal privatization level first decreases with t and then increases with t, we have the following inequality:

$$\mathop {\hbox{max} }\limits_{t} \lambda^{*} \le \{ \lambda^{s*} (t = 0),\lambda^{l*} (t = t_{T} )\}.$$

We have proven in the text that:

$$\lambda^{s*} (t = 0) > \lambda^{l*} (t = t_{T}).$$

So we have: $\mathop {\hbox{max} }\limits_{t} \lambda^{*} \le \lambda^{s*} (t = 0) = \frac{{c_{1} }}{{1 - 4c_{1} }}$: The optimal privatization level with no double marginalization is the highest, which is what Proposition 4 states.

1.5 Proof of Proposition 5

For small t such that only the contested regime exists downstream the comparison of welfare between a mixed upstream duopoly and a fully public upstream monopoly is: $W^{s*} (\lambda^{s*} ) - W(\omega^{*} ) = \frac{{c_{1} (3c_{1} - t)}}{2}$.

Recall that for t above $t^{s*}$ uncontested regimes begin to emerge downstream with mixed upstream duopoly. It is easy to prove that $t^{s*} < \frac{{2(1 - c_{1} )}}{3}$, the threshold value for t such uncontested regimes begin to appear with the fully public upstream monopolist. Then we have, for $0 \le t \le \hbox{min} \{ 3c_{1} ,t^{s*} \}$, $W^{s*} (\lambda^{s*} ) > W(\omega^{*} )$; for $\hbox{min} \{ 3c_{1} ,t^{s*} \} \le t \le t^{s*}$, $W^{s*} (\lambda^{s*} ) < W(\omega^{*} )$. Solving $3c_{1} - t^{s*} = 0$, we have $c_{1} = 0.154$. Therefore, 1) when $0.146 < c_{1} < 0.154$, $W^{s*} (\lambda^{s*} ) > W(\omega^{*} )$ as $0 \le t \le 3c_{1}$, $W^{s*} (\lambda^{s*} ) < W(\omega^{*} )$ as $3c_{1} \le t \le t^{s*}$; 2) when $0.154 < c_{1} < 0.2$, $W^{s*} (\lambda^{s*} ) > W(\omega^{*} )$ always holds. So for sufficiently small t, a mixed upstream duopoly yields higher social welfare.

For the large t of $t_{T}$ such that only uncontested regimes exist with a mixed upstream duopoly, solving $W^{l*} (\lambda^{l*} ) - W(\omega^{*} ) = 0$ we have $c_{1} = 0.184$. It is easy to verify that for $0.146 < c_{1} \le 0.184$, $W^{l*} (\lambda^{l*} ) < W(\omega^{*} )$; for $0.184 < c_{1} \le 0.2$, $W^{l*} (\lambda^{l*} ) > W(\omega^{*} )$. So, for a large t such that uncontested regimes exist, the result of the welfare comparison is ambiguous: It depends on the upstream production cost.

For moderate t such that the M-shaped equilibrium exists, analytic welfare comparisons are intractable. The comparison can be graphed:

"Appendix" Fig. 6 indicates that when $c_{1}$ is small, the upstream duopoly yields even less social welfare than does monopoly (area I); while when $c_{1}$ is large, the range of t such that the upstream duopoly yields higher social welfare emerges (area II). Then we know that when $c_{1}$ is large, the effect of the upstream duopoly’s increasing social welfare is stronger.

Fig. 6

Welfare comparison between an upstream duopoly and an upstream monopoly for the M-shaped equilibrium

The discussion above proves Proposition 5.