Input price discrimination and non-controlling vertical shareholding

Abstract

We study how input price discrimination and non-controlling, vertical shareholding interact. We first discuss the implications of the invariance principle (Greenlee and Raskovich in Eur Econ Rev 50:1017–1041, 2006) for input price discrimination across independent downstream markets. We show that, in the short term, price discrimination based on non-controlling, backward shareholding is more likely to improve welfare than other forms of input price discrimination, because it results in a positive consumption reallocation effect. However, if the derived demand for the input is convex, and less so as output increases, input price discrimination is irrelevant in the long term, because the ownership structures that prevail under discriminatory and uniform input pricing yield the same market outcome. The main contribution of the paper is to show that input price discrimination matters, in the short and the long term, if downstream firms compete in prices. In particular, we find that input price discrimination improves welfare in the short term, for any given ownership structure, and in the long term, through its impact on shareholding.

Appendix

In this appendix, we detail the cases of downstream price competition (Sect. 7.1 in Appendix) and quantity competition (Sect. 7.2 in Appendix).We also present the study of forward shareholding (Sect. 7.3 in Appendix). We consider imperfect product substitution, as modeled by Shubik and Levitan (1980). In this model, a representative consumer has quasi-linear utility

$$\begin{aligned} U=q_{1}+q_{2}-\frac{1}{1+\mu }\left( q_{1}^{2}+q_{2}^{2}+\frac{\mu }{2}\left( q_{1}+q_{2}\right) ^{2}\right) +I, \end{aligned}$$

where I is the consumption of other goods than those provided by downstream firms 1 and 2. Parameter $\mu \in [0,\infty )$ describes the degree of product substitutability. If $\mu =0$, then downstream firms operate in independent markets. If $\mu \rightarrow \infty$, then they sell perfect substitutes.

1.1 Price competition

Maximization of the representative consumer’s utility function yields

$$\begin{aligned} q_{i}=\left( 1-p_{i}-\mu \left( p_{i}-\left( p_{i}+p_{-i}\right) /2\right) \right) /2,\,\forall i\in \{1,2\}. \end{aligned}$$

Equilibrium downstream prices are

$$\begin{aligned} p_{i}= & {} (8+8(1-\alpha _{i})w_{i}+\mu ^{2}((8-8\alpha _{i}+\alpha _{-i})w_{i}+2(1+2\alpha _{i}-\alpha _{-i})w_{-i})\\&+\,2\mu (5+8(1-\alpha _{i})w_{i}+(1+2\alpha _{i}-\alpha _{-i})w_{-i}))/(16+32\mu +15\mu ^{2}) \end{aligned}$$

and equilibrium input prices satisfy $q_{i}(w_{i},w_{-i})+w_{i}(\partial q_{i}/\partial w_{i})+w_{-i}(\partial q_{-i}/\partial w_{i})=0$.

If $\alpha _{1}\le \alpha _{2}$, then $w_{1}\le w_{2}$ at the unconstrained equilibrium. As in Sect. 3, we denote $r\equiv w_{2}-w_{1}$ the constrained degree of input price discrimination. By total differentiation of the upstream firm’s first-order condition, we obtain

$$\begin{aligned} \frac{dw_{1}(r)}{dr}=-\frac{dw_{2}(r)}{dr}=\frac{16+6(6+\alpha _{1})\mu +(20+7\alpha _{1})\mu ^{2}-\alpha _{2}(16+34\mu +17\mu ^{2})}{(2(4+5\mu )(-4(1+\mu )+\alpha _{1}(2+\mu )+\alpha _{2}(2+\mu )))}<0. \end{aligned}$$

If follows that $dq_{1}(r)/dr>0$ and $dq_{2}(r)/dr<0$.

For all $r\in [0,r^{*}]$, the total equilibrium output is

$$\begin{aligned} Q(r)=\frac{8(1+\mu )(4+5\mu )+(\alpha _{2}-\alpha _{1})\mu (2+\mu )(2+3\mu )r}{8(4+3\mu )(4+5\mu )}, \end{aligned}$$

which increases with r for all $\alpha _{1}\le \alpha _{2}$: As the degree of input price discrimination increases, $q_{1}(r)$ increases more than $q_{2}(r)$ decreases, and total output increases.

Proof of Proposition 1

i. For all given ownership structure, the relationship between welfare and the degree of input price discrimination is given by

$$\begin{aligned} \frac{dW(r)}{dr}=\underset{Output\,effect}{\underbrace{\frac{dU}{dq_{1}}\frac{dQ}{dr}}}\underset{Reallocation\,effect}{\underbrace{-\frac{dq_{2}}{dr}\left( \frac{\partial U}{\partial q_{1}}-\frac{\partial U}{\partial q_{2}}\right) }}, \end{aligned}$$

where the derivatives are evaluated at $q_{1}(r),q_{2}(r)$. This mirrors Eq. (1), which was introduced to study the case of independent markets.

The first term on the RHS is positive, because of non-satiation and because total output increases with input price discrimination. Unlike the case of independent markets, a marginal increase in r does not always yield a positive reallocation effect. Indeed, $\partial U/\partial q_{1}-\partial U/\partial q_{2}=2(q_{2}(r)-q_{2}(r))/(1+\mu )$ and $q_{2}(r)-q_{1}(r)\ge 0$ iff r is low enough. However, it is just a matter of computation to show that $\intop _{0}^{r^{*}}\left( \partial U/\partial q_{1}-\partial U/\partial q_{2}\right) dr>0$, which implies that the total reallocation effect induced by a shift from uniform to unconstrained discriminatory input pricing is positive. Therefore, $W(r^{*})-W(0)>0$: discriminatory input pricing improves welfare for all given ownership structure.

ii. We just need to prove the existence of cases where discriminatory input pricing reduces consumer surplus. This is the case if $\mu =0$, that is, for independent downstream markets: then, we find $CS(r^{*})-CS(0)=-((\alpha _{2}-\alpha _{1})^{2}/(32(2-\alpha _{1}-\alpha _{2})^{2}))<0$. This also occurs for strictly positive values of $\mu$, that is, if the downstream products are imperfect substitutes. For example, if $\mu =1/2$ and $\alpha _{1}=0$, we find

$$\begin{aligned}&CS(r^{*})-CS(0)\\&\quad = -\frac{15120\alpha _{2}^{2}(186316416-400473216\alpha _{2}+182031096\alpha _{2}^{2}+32045260\alpha _{2}^{3}-256725\alpha _{2}^{4})}{20449(12-5\alpha _{2})^{2}(20592-20592\alpha _{2}-35\alpha _{2}^{2})^{2}}\\&\quad <0, \end{aligned}$$

which is negative for all $\alpha _{2}<1/2$. $\square$

Proof of Proposition 2

We now study the impact of shareholding on total industry profit $\pi$. We find that the derivative of $\pi$ with respect to $\alpha _{i}$ is zero if and only if $\alpha _{1}=\alpha _{2}$, both under uniform and discriminatory input pricing. Besides, at $\alpha _{1}=\alpha _{2}$, we find

$$\begin{aligned} \frac{d^{2}\pi }{d\alpha _{i}^{2}} =-\frac{(1+\mu )^{2}(2+3\mu )}{(4+5\mu )^{2}(-2(1+\mu )+\alpha _{i}(2+\mu ))^{2}}<0 \end{aligned}$$

in the former case and

$$\begin{aligned}&\frac{d^{2}\pi }{d\alpha _{i}^{2}} =\frac{\mu (1+\mu )^{2}(2+3\mu )(\alpha _{i}(64+136\mu +78\mu ^{2}+11\mu ^{3})-2(32+84\mu +69\mu ^{2}+17\mu ^{3}))}{(4+3\mu )^{3}(4+5\mu )^{2}(-2(1+\mu )+\alpha _{i}(2+\mu ))^{2}(-2(1+\mu )+\alpha _{i}(2+3\mu ))}\\&\quad >0 \end{aligned}$$

in the latter. It follows that total industry profit is maximized at $\alpha _{1}=\alpha _{2}$ under uniform input pricing. By contrast, under discriminatory input pricing, it is minimized at $\alpha _{1}=\alpha _{2}$ and the more asymmetric the ownership structure, the higher the total industry profit. $\square$

Proof of Proposition 3

Like the derivatives of total industry profit, that of consumer surplus and welfare with respect to $\alpha _{i}$ are zero if and only if $\alpha _{1}=\alpha _{2}$. At such symmetric ownership structure, we find, under uniform input pricing,

$$\begin{aligned} \frac{d^{2}CS}{d\alpha _{i}^{2}}=-\frac{2+\mu }{2+3\mu }\frac{d^{2}W}{d\alpha _{i}^{2}}=\frac{(1+\mu )(2+\mu )(2+3\mu )}{4(4+5\mu )^{2}(-2(1+\mu )+\alpha _{i}(2+\mu ))^{2}}>0. \end{aligned}$$

Under discriminatory input pricing, we find

$$\begin{aligned} \frac{d^{2}CS}{d\alpha _{i}^{2}}= & {} \frac{\mu (4+12\mu +11\mu ^{2}+3\mu ^{3})}{4(4+3\mu )^{3}(4+5\mu )^{2}(-2(1+\mu )+\alpha _{i}(2+\mu ))^{2}(-2(1+\mu )+\alpha _{i}(2+3\mu ))}\\&\times (-2(32+172\mu +327\mu ^{2}+267\mu ^{3}+80\mu ^{4})\\&+\,\alpha _{i}(64+312\mu +522\mu ^{2}+353\mu ^{3}+80\mu ^{4})))>0. \end{aligned}$$

and

$$\begin{aligned} \frac{d^{2}W}{d\alpha _{i}^{2}}= & {} \frac{\mu (2+5\mu +3\mu ^{2})(-2(192+840\mu +1438\mu ^{2}+1205\mu ^{3}+495\mu ^{4}+80\mu ^{5})}{4(4+3\mu )^{3}(4+5\mu )^{2}(-2(1+\mu )+\alpha _{i}(2+\mu ))^{2}(-2(1+\mu )+\alpha _{i}(2+3\mu ))}\\&+\,\frac{\alpha _{i}(384+1488\mu +2212\mu ^{2}+1584\mu ^{3}+557\mu ^{4}+80\mu ^{5}))}{4(4+3\mu )^{3}(4+5\mu )^{2}(-2(1+\mu )+\alpha _{i}(2+\mu ))^{2}(-2(1+\mu )+\alpha _{i}(2+3\mu ))}>0. \end{aligned}$$

Therefore, the more asymmetric the ownership structure, the higher the consumer surplus, under both uniform and discriminatory input pricing. In the latter case, this is also true for welfare. By contrast, under uniform input pricing, the more asymmetric the ownership structure, the higher welfare. $\square$

1.2 Quantity competition

We now consider that the downstream firms compete in quantities. The inverse demand derived from the representative consumer’s utility function is $p_{i}=1-(2q_{i}+\mu Q)/(1+\mu )\forall i\in \{1,2\}$. Equilibrium quantities are

$$\begin{aligned} q_{i}(w_{i},w_{-i})=\frac{(\mu +1)(2(\alpha _{i}-1)(\mu +2)w_{i}-\alpha _{-i}\mu {{w_{-i}}}+\mu w_{-i}+\mu +4)}{(\mu +4)(3\mu +4)} \end{aligned}$$

and

$$\begin{aligned} w_{i}(r)=\frac{8-\mu \left( (2+\alpha _{i}-3\alpha _{-i})r-2\right) -8(1-\alpha _{-i})r}{2(2-\alpha _{1}-\alpha _{2})(4+\mu )}, \end{aligned}$$

for all $r\in [0,r^{*}]$. At the subgame perfect equilibrium, total output is

$$\begin{aligned} Q(r)=\frac{(1+\mu )(8+\mu (2+r\alpha _{1}-r\alpha _{2}))}{6\mu ^{2}+32\mu +32}. \end{aligned}$$

Proof of Lemma 1

The welfare effect of input price discrimination can be described as follows. On the one hand, total output decreases: $dQ/dr\le 0.$ On the other hand, the consumption reallocation effect is positive, because $q_{2}(r)-q_{1}(r)\ge 0\,\forall r\in [0,r^{*}]$. It is just a matter of computation to show that there exist a unique threshold ${\widetilde{\mu }}$ such that $W(0)\le W(r^{*})$ if and only if $\mu \le {\widetilde{\mu }}$. $\square$

Proof of Lemma 2

We now turn to the relationship between total industry profit and welfare on the one hand, and ownership structure on the other hand. As in case of price competition, we find that the derivatives of total industry profit and welfare with respect to the ownership share $\alpha _{i}$ are zero if and only if downstream firms own equal shares of the input provider. At $\alpha _{1}=\alpha _{2}$, we find

$$\begin{aligned} \frac{d^{2}\pi }{d\alpha _{i}^{2}}=2\frac{d^{2}W}{d\alpha _{i}^{2}}=-\frac{1+\mu }{2(1-\alpha _{i})^{2}(4+\mu )^{2}}<0 \end{aligned}$$

in case of uniform input pricing, and

$$\begin{aligned} \frac{d^{2}\Pi }{d\alpha _{i}^{2}}=-\frac{\mu (32+76\mu +63\mu ^{2}+21\mu ^{3}+2\mu ^{4})}{2(1-\alpha _{i})^{2}(4+\mu )^{2}(4+3\mu )^{3}}<0 \end{aligned}$$

and

$$\begin{aligned} \frac{d^{2}W}{d\alpha _{i}^{2}}=-\frac{\mu (96+236\mu +203\mu ^{2}+71\mu ^{3}+8\mu ^{4})}{2(1-\alpha _{i})^{2}(4+\mu )^{2}(4+3\mu )^{3}}<0 \end{aligned}$$

under discriminatory input pricing. If follows that total profit and welfare are maximized at $\alpha _{1}=\alpha _{2}$ in both input pricing regimes. $\square$

1.3 Forward shareholding

Consider a market where an upstream monopolist owned by a single shareholder sells an input to two downstream firms, each of which is controlled by a majority shareholder, but also partially owned by the shareholder of the upstream firm. Managers of the upstream firm maximize shareholder value $\Pi _{u}=\pi _{u}+\alpha _{1}\pi _{1}+\alpha _{2}\pi _{2}$. Those of downstream firm i maximize $\Pi _{i}=(1-\alpha _{i})\pi _{i}$.

Then, the equilibrium quantities are

$$\begin{aligned} q_{i}=\frac{(1+\mu )(8(1-w_{i})+2\mu (8w_{i}-w_{-i}-5)-\mu ^{2}(7w_{i}-2w_{-i}))}{32+64\mu +30\mu ^{2}} \end{aligned}$$

if downstream firms compete in prices and

$$\begin{aligned} q_{i}=\frac{(1+\mu )(4(1-w_{i})+g(1-2w_{i}+w_{-i}))}{16+16\mu +3\mu ^{2}} \end{aligned}$$

if they compete in quantities. We derive the equilibrium input prices, for all $w_{2}(r)=w_{1}(r)+r$ and $r\in [0,r^{*}]$. Under price competition,

$$\begin{aligned} w_{1}(r)=\frac{(16+16\mu +3\mu ^{2})(r-1)+\alpha _{1}(2+\mu )(4+\mu +\mu r)-\alpha _{2}(2+\mu )(4r+2\mu r-4-\mu )}{(4+\mu )(-8-6\mu +\alpha _{1}(2+\mu )+\alpha _{2}(2+\mu ))} \end{aligned}$$

and

$$\begin{aligned} r^{*}=\frac{(\alpha _{1}-\alpha _{2})(8+6\mu +\mu ^{2})}{32+32\mu +6\mu ^{2}+2\alpha _{1}(-2+\alpha _{2})(2+\mu )^{2}-4\alpha _{2}(2+\mu )^{2}}. \end{aligned}$$

Under quantity competition,

$$\begin{aligned} w_{1}(r)= & {} ((16+32\mu +15\mu ^{2})((2+\mu )r-2)+2\alpha _{1}(2+\mu )(4+\mu ^{2}r+\mu (5+r))\\&-\,\alpha _{2}(2+\mu )(8(r-1)+7\mu ^{2}r\\&+\,2\mu (8r-5)))/((2+\mu )(4+5\mu )((2+\mu )(\alpha _{1}+\alpha _{2})-8-6\mu )) \end{aligned}$$

and

$$\begin{aligned} r^{*}= & {} \frac{(\alpha _{1}-\alpha _{2})(4+5\mu )}{16+32\mu +15\mu ^{2}-\alpha _{2}(8+16\mu +7\mu ^{2})+\alpha _{1}(-8-16\mu -7\mu ^{2}+\alpha _{2}(4+8\mu +3\mu ^{2}))}. \end{aligned}$$

Based on these results, it is just a matter of computation to show that, under both price and quantity competition, $dQ(r)/dr>0$, $q_{2}(r)-q_{1}(r)>0$, $dq_{2}(r)/d(r)<0$, and, therefore, $dW(r)/dr>0$. Likewise, $d\pi /d\alpha _{i}>0$ and $dW/d\alpha _{i}>0$, under both uniform and discriminatory input pricing, regardless of whether firms compete in prices or in quantities.