Optimal Privatization in a Vertical Chain: A Delivered Pricing Model

Abstract

The chapter considers the introduction of a mixed ownership firm into a classic model in which downstream firms locate strategically so as to achieve accommodating upstream price reductions. These reductions happen endogenously but the strategic locations harm welfare. It shows that a mixed ownership firm downstream can limit such inefficiency but that its ability to do so depends on the extent to which its costs match those of a private firm. Thus, reconfirming in this spatial setting that the optimal share of privatization set by a government depends positively upon the cost disadvantage of the public firm.

In addition, Yes would like to add the following for his participation: “Financial support from the National Natural Science Foundation of China (No.71773129) and National Social Science Foundation of China (No. 19ZDA110) is gratefully acknowledged.”

Appendices

A.1 Appendix 1

The possible highest costs in the spatial market for firms are reached when x takes the values of 0, 1, or $\frac{L_1+{L}_2-\left(1-\lambda \right)c}{2}$, which is the sale bound between firms 1 and 2. See Fig. 9.1. When the two firms’ cost at any x exceeds r, there is no sale existing; therefore, the following conditions must hold:

$$ x=0:w+\left(1-\lambda \right)c+{L}_1\le r $$

$$ x=\frac{L_1+{L}_2-\left(1-\lambda \right)c}{2}:w+{L}_2-\frac{L_1+{L}_2-\left(1-\lambda \right)c}{2}\le r $$

$$ x=1:w+1-{L}_2\le r $$

Thus the wholesale price must satisfy

$$ w\le r-\max \left\{{L}_1+\left(1-\lambda \right)c,{L}_2-\frac{L_1+{L}_2-\left(1-\lambda \right)c}{2},1-{L}_2\right\} $$

As the upstream firm wants to maximize his profit, namely, the wholesale price, we have that

$$ w=r-\max \left\{{L}_1+\left(1-\lambda \right)c,{L}_2-\frac{L_1+{L}_2-\left(1-\lambda \right)c}{2},1-{L}_2\right\} $$

B.1 Appendix 2

As $w=r-\max \left\{{L}_1+\left(1-\lambda \right)c,{L}_2-\frac{L_1+{L}_2-\left(1-\lambda \right)c}{2},1-{L}_2\right\}$, the wholesale price in stage 2 can take three forms:w = r − (L ₁ + (1 − λ)c), or $w=r-\left({L}_2-\frac{L_1+{L}_2-\left(1-\lambda \right)c}{2}\right)$, orw = r − (1 − L ₂).

First we will prove that $w=r-\left({L}_2-\frac{L_1+{L}_2-\left(1-\lambda \right)c}{2}\right)$ is not equilibrium. When $w=r-\left({L}_2-\frac{L_1+{L}_2-\left(1-\lambda \right)c}{2}\right)$, we denote the public firm’s objective function as G ₁. FOC of G ₁ with respect to L ₁ yields public firm’s optimal location of L ₁(L ₂) as the function of L ₂. Then we obtain G ₁(L ₂) as the function of L ₂ and the associated wholesale price as ${w}_1=r-\left({L}_2-\frac{L_1\left({L}_2\right)+{L}_2-\left(1-\lambda \right)c}{2}\right)$.

If the public firm chooses a large location of ${L}_1^{\hbox{'}}$ that satisfies ${L}_1^{\hbox{'}}=\left\{{L}_1:{L}_1+\left(1-\lambda \right)c={L}_2-\frac{L_1\left({L}_2\right)+{L}_2-\left(1-\lambda \right)c}{2}\right\}$ and meanwhile keep the wholesale price remain at the same level as w ₁ (so that the upstream firm is indifferent), we can obtain the public firm’s associated objective as the function of L ₂, and we denote it as ${G}_2\left({L}_2\right)=G\left({L}_1^{\hbox{'}}\right)$.

We find that ${G}_2\left({L}_2\right)-{G}_1\left({L}_2\right)=\frac{\lambda^2{\left( c\lambda +{L}_2-c\right)}^2}{{\left(6+\lambda \right)}^2}\ge 0$, and the equality holds only when λ = 0. Therefore we have that the wholesale price of $w=r-\left({L}_2-\frac{L_1+{L}_2-\left(1-\lambda \right)c}{2}\right)$ is not equilibrium.

Now we derive downstream locations associated with the other two expressions.

Take w = r − (L ₁ + (1 − λ)c) as an example. The FOCs of $\left\{\frac{\partial G}{\partial {L}_1}=0,\frac{\partial {\pi}_2}{\partial {L}_2}=0\right\}$ yield the optimal downstream location as

${L}_1^a=\frac{4c{\lambda}^2-2 c\lambda -2c+2\lambda +1}{2\left(2+\lambda \right)}$,${L}_2^a=\frac{2c{\lambda}^2-2c+2\lambda +3}{2\left(2+\lambda \right)}$

Then the associated wholesale price is ${w}^a=r-\left({L}_1^a+\left(1-\lambda \right)c\right)$.

When the private firm is critical, FOCs yield the downstream locations of other forms as ${\overline{L}}_1^b=\frac{1}{6}-\frac{2\left(1-\lambda \right)c}{3}$, ${\overline{L}}_2^b=\frac{1}{2}-\left(1-\lambda \right)c\le \frac{1}{2}$, we will prove that this is not equilibrium.

When the private firm chooses ${\overline{L}}_2^b$ and the mixed firm chooses to jump to the right of the private competitor, then mixed firm is on the right side while the private firm is in the left side, and the equilibrium price of mixed firm becomes

$$ {p}_1(x)=\left\{\begin{array}{l}w+x-{L}_2^b,\frac{L_1+{\overline{L}}_2^b+\left(1-\lambda \right)c}{2}\le x\le r+{\overline{L}}_2^b-w\\ {}r,r+{\overline{L}}_2^b-w\le x\le 1\end{array}\right. $$

The equilibrium price of private firm becomes

$$ { \begin{array}{l}{p}_2(x)=\left\{\begin{array}{l}r,0\le x\le {L}_1+w+\left(1-\lambda \right)c-r\\ {}w+{L}_1-x+\left(1-\lambda \right)c,{L}_1+w+\left(1-\lambda \right)c-r\le x\le \frac{L_1+{\overline{L}}_2^b+\left(1-\lambda \right)c}{2}\end{array}\right.\end{array}} $$

Denote the new profit functions and social welfare as ${\pi}_1^{\hbox{'}}$, ${\pi}_2^{\hbox{'}}$, and W ^'. The objective of the mixed firm is ${G}^{\hbox{'}}={\lambda \pi}_1^{\hbox{'}}+\left(1-\lambda \right){W}^{\hbox{'}}$.

The wholesale price may take three forms: $w=r-{\overline{L}}_2^b$, or $w=r-\left(\frac{L_1+{\overline{L}}_2^b+\left(1-\lambda \right)c}{2}-{\overline{L}}_2^b\right)$, or w = r − (1 − L ₁ + (1 − λ)c).

Take $w=r-{\overline{L}}_2^b$ as an example. The FOC of ${\pi}_1^{\hbox{'}}$ with respect to L ₁ yields the optimal location of mixed firm as ${L}_1^{\hbox{'}}=\frac{5}{6}$. Denote the associated maximized objective of mixed firm as G ^'∗ and the one under the original location of $\left\{{\overline{L}}_1^b,{\overline{L}}_2^b\right\}$ asG ^∗. Then we have

$$ {G}^{\ast }-{G}^{\hbox{'}\ast }=\frac{2c}{3}\left(\lambda -1\right)\left(\left(3\lambda +2\right)\left(\lambda -1\right)c+1\right)\le 0 $$

“ = ” holds only when λ = 1. (Notice that in this case ${\overline{L}}_2^b$ is the largest among$\left\{{\overline{L}}_2^b,\frac{L_1+{\overline{L}}_2^b+\left(1-\lambda \right)c}{2}-{\overline{L}}_2^b,1-{L}_1+\left(1-\lambda \right)c\right\}$, which indicates that (1 − λ)c ≤ 1/6.)

The above condition indicates that the mixed firm can achieve higher value of its objective function if he jumps to the right side of the private firm who chooses ${\overline{L}}_2^b$.

The cases of $w=r-\left(\frac{L_1+{\overline{L}}_2^b+\left(1-\lambda \right)c}{2}-{\overline{L}}_2^b\right)$ and w = r − (1 − L ₁  + (1 − λ)c) are similar; therefore, the mixed firm would jump to the right side when the private firm chooses ${\overline{L}}_2^b$.

Therefore, the private firm cannot locate anywhere left of $\frac{1}{2}$. The optimal location for the private firm becomes ${L}_2^b=\frac{1}{2}$. In this case, FOC yields the mixed firm’s optimal location as ${L}_1^b=\frac{1}{6}-\frac{\left(1-\lambda \right)c}{3}$, which is right to ${\overline{L}}_1^b=\frac{1}{6}-\frac{2\left(1-\lambda \right)c}{3}$.

Notice that when λ = 1, $\left\{{L}_1^a,{L}_2^a\right\}=\left\{\frac{1}{2},\frac{5}{6}\right\}$ and $\left\{{L}_1^b,{L}_2^b\right\}=\left\{\frac{1}{6},\frac{1}{2}\right\}$. This is precisely as in Gupta et al. (1994).

C.1 Appendix 3

When the mixed firm is critical, the derivative of the associated social welfare W ^∗ with respect to λ is $\frac{\partial {W}^{\ast }}{\partial \lambda }=- AS$, where $A=\frac{1-2c+2\lambda c}{{\left(2+\lambda \right)}^2}>0$ and S = 2cλ ³ + 6cλ ² − 18cλ − 8c + 3λ.

When $0<c<\frac{1}{6}$, $\frac{\partial S}{\partial \lambda }=6c{\lambda}^2+12 c\lambda +3\left(1-6c\right)>0$; therefore, S increases as λincreases. As S(λ = 0) =  − 8c < 0, S(λ = 1) = 3(1 − 6c) > 0, there exists only a threshold value of $\overline{\lambda}$ so that $S\left(\overline{\lambda}\right)=0$ and $0<\overline{\lambda}<1$. Then we have that for $0\le \lambda <\overline{\lambda}$, S(λ) < 0, $\frac{\partial {W}^{\ast }}{\partial \lambda}\left(\lambda \right)=-A\cdotp S\left(\lambda \right)>0$, for $\overline{\lambda}<\lambda \le 1$, S(λ) > 0, $\frac{\partial {W}^{\ast }}{\partial \lambda}\left(\lambda \right)=-A\cdotp S\left(\lambda \right)<0$, and $\frac{\partial {W}^{\ast }}{\partial \lambda}\left(\lambda =\overline{\lambda}\right)=0$. Thus, it can be concluded that an interior solution of $\overline{\lambda}$ for optimal privatization is reached when $0<c<\frac{1}{6}$.

When $\frac{1}{6}<c<\frac{1}{2}$, let the two solutions of equation $\left\{\lambda :\frac{\partial S}{\partial \lambda }=0\right\}$ be λ ₁ and λ ₂. As ${\lambda}_1+{\lambda}_2=-\frac{12c}{2\cdotp 6c}=-1<0$ and ${\lambda}_1\cdotp {\lambda}_2=\frac{3\left(1-6c\right)}{6c}=\frac{1-6c}{2c}<0$, only one of λ ₁ and λ ₂ can be in λ ∈ [0, 1]. As ${\left.\frac{\partial S}{\partial \lambda}\right|}_{\lambda =0}=3\left(1-6c\right)<0$, ${\left.\frac{\partial S}{\partial \lambda}\right|}_{\lambda =1}=3>0$; there is one threshold value satisfying $\left\{\lambda :\frac{\partial S}{\partial \lambda }=0\right\}$, so S first decreases and then increases with λ ∈ [0, 1], and then S ≤  max {S(λ = 0), S(λ = 1)} =  max {−8c, 3(1 − 6c)} < 0; therefore, $\frac{\partial {W}^{\ast }}{\partial \lambda }=-A\cdotp S>0$, and it can be concluded that the optimal privatization is λ = 1 when $\frac{1}{6}<c<\frac{1}{2}$.