Vertical Mergers in a Model of Upstream Monopoly and Incomplete Information

Abstract

We examine the role of private information on the impact of vertical mergers. A vertical merger can improve the information that is available to an upstream monopolist because, after the merger, the monopolist can observe the cost of its downstream merger partner. In the pre-merger world, because the costs of the downstream firms are private information, the monopolist has incomplete information and cannot implement the monopoly outcome: The expected pre-merger equilibrium price of the downstream product is lower than the monopoly price. After a vertical merger, the equilibrium input price that is charged to the downstream rival can either increase or decrease—depending on whether the downstream merger partner’s cost is low or high, respectively. However, in all cases the equilibrium price of the downstream product increases to the monopoly price. Therefore, the merger leads to consumer harm even when it leads to a reduction in the input price. The merged firm, however, cannot extract all of the monopoly profit: The merger causes production inefficiency (when the downstream rival has a relatively small cost advantage) and the downstream rival still earns an information rent (when it has a relatively large cost advantage). These results also have implications for vertical merger policy.

Appendix

Proof of Proposition 2

We begin with the case when downstream production by UD1 is not viable: $c+{C}_{1}>V$. UD1 participates in the market only as a supplier to D2 and sets $W$ to maximize upstream expected profits. Those profits are given by $\left(W-c\right)F\left(V-W\right)$, are strictly increasing in $W$ at $W=c$, and are strictly decreasing in $W$ at $W=V-A$. Thus, the optimal input price $\widehat{W}\in (c,V-A)$. Since $F$ is log-concave, profits are strictly quasi-concave in $W$ for all $W\in (c,V-A)$. From the first-order condition, $\widehat{W}$ satisfies

$$\widehat{W}=c+\frac{F\left(V-\widehat{W}\right)}{f\left(V-\widehat{W}\right)}.$$

(A1)

When costs are uniformly distributed, $\widehat{W}=\frac{1}{2}\left(V-A+c\right)$.

To summarize, when ${C}_{1}>V-c$, downstream production by UD1 is not viable, UD1 sets the input price $\widehat{W}$ given by (A1), UD1 does not participate in the auction, and either D2 participates and wins (if ${C}_{2}\le V-\widehat{W}$), in which case the buyer pays $V$ to D2, or D2 does not participate (if ${C}_{2}>V-\widehat{W}$), in which case there is no sale.

We now turn to the case when downstream production by UD1 is viable: $c+{C}_{1}\le V$. It is helpful to partition the range of possible values for $W$ into three regions:

1.1 Region 1: ${\varvec{W}}>{\varvec{V}}-{\varvec{A}}$

If $W$ is in this region, even the most efficient type of D2 would not bid, since $W+A>V$. This is in effect an exclusionary strategy by UD1, where it is the only bidder in the auction. Without competition, UD1 always bids $V$. Therefore, expected profit from an exclusionary input price strategy is $V-{C}_{1}-c$.

1.2 Region 2: ${\varvec{V}}-{{\varvec{C}}}_{1}\le {\varvec{W}}\le {\varvec{V}}-{\varvec{A}}$

Since $V<{C}_{1}+W$ for $W$ in (the interior of) this region, if D2 participates in the auction, then UD1 is better off letting D2 win, since the gains from losing and only supplying the input, which are $W-c$, are larger than the gains from winning at price $V$, which are $V-{C}_{1}-c$.¹⁵ If instead D2 does not participate, then UD1 maximizes its profit by bidding $V$ and earning $V-{C}_{1}-c$. Thus, in Region 2 the optimal downstream bid of UD1 is $V$.

Given this auction strategy, it remains to determine what wholesale price within Region 2 is optimal and what is UD1’s (expected) profit at that price. Profits for UD1 in Region 2 are

$$\Pi \left(W\right)=F\left(V-W\right)\left(W-c\right)+\left[1-F\left(V-W\right)\right]\left(V-{C}_{1}-c\right).$$

The first-order condition is

$$\frac{d\Pi }{dW}=-\left(W+{C}_{1}-V\right)f\left(V-W\right)+F\left(V-W\right)=0.$$

(A2)

Note that when $W=V-{C}_{1}$, $\Pi =V-{C}_{1}-c$ and $\frac{d\Pi }{dW}=F\left({\mathrm{C}}_{1}\right)>0$; while at the other end of the region when $W=V-A$, $\Pi =V-{C}_{1}-c$ and $\frac{d\Pi }{dW}=-\left({C}_{1}-A\right)f\left(A\right)<0$.¹⁶ Thus, UD1 increases profits by picking $W$ in the interior of Region 2, and in so doing earns higher expected profits than $V-{C}_{1}-c$ and, hence, higher expected profits than it receives from the exclusionary strategy in Region 1.

From the first-order condition, $\widehat{W}$ satisfies

$$\widehat{W}=V-{C}_{1}+\frac{F\left(V-\widehat{W}\right)}{f\left(V-\widehat{W}\right)}.$$

(A3)

When costs are uniformly distributed, the solution to the first-order condition is halfway between the two endpoints of Region 2: $\widehat{W}=V-\frac{1}{2}\left(A+{C}_{1}\right)$. For later use, note that (A2) implies that $\widehat{W}$ is a strictly decreasing function of ${C}_{1}$ (for ${C}_{1}\le V-c$).

1.3 Region 3: ${\varvec{W}}<{\varvec{V}}-{{\varvec{C}}}_{1}$

In this region, UD1’s profit if D2 wins is equal to $W-c$ (which is smaller than $V-{C}_{1}-c$) and UD1’s profit if it wins is at most equal to $V-{C}_{1}-c$. Recall that $c+{C}_{1}\le V$, and so a sale will be made regardless of the chosen value of $W.$ Thus, if UD1 sets $W$ in this region, it earns lower expected profits than if it sets $W$ in Region 2.

To summarize, when ${C}_{1}\le V-c$, downstream production by UD1 is viable, UD1 sets the input price $\widehat{W}({C}_{1})$ given by (A3), UD1 bids $V$, and either D2 participates and wins (if ${C}_{2}<V-\widehat{W}({C}_{1})$), in which case the buyer pays $V$ to D2, or D2 does not participate and UD1 wins (if ${C}_{2}\ge V-\widehat{W}({C}_{1})$), in which case the buyer pays $V$ to UD1.

1.4 Proof that the main results in Proposition 2 also apply if the buyer uses a first-price auction

We want to show that the following strategies and beliefs constitute a perfect Bayesian equilibrium:

• If downstream production by UD1 is not viable (if ${C}_{1}>V-c$), UD1 sets the input price equal to $\widehat{W}$ (as given by (A1)), and does not bid. If instead downstream production by UD1 is viable (if ${C}_{1}\le V-c$), UD1 sets the input price equal to $\widehat{W}({C}_{1})$ (as given by (A3)), and bids $V$.

• If D2 observes $\widehat{W}$, D2 believes that UD1 does not bid, and D2 either bids $V$ or does not bid (depending on whether D2 is viable or not). If D2 observes $\widehat{W}\left({C}_{1}\right)>\widehat{W}$, D2 believes that UD1 bids $V$, and D2 either bids slightly below $V$ or does not bid. If D2 observes an input price either strictly lower than $\widehat{W}$ or strictly higher than $\widehat{W}(A)$, D2 believes that UD1 does not bid, and D2 either bids $V$ or does not bid.

One can check that (1) given the input price strategy of UD1, D2’s beliefs satisfy Bayes rule; (2) given the bid strategy of UD1 and the beliefs of D2, D2’s bid strategy is a best response; (3) given UD1’s input price strategy and D2’s bid strategy, UD1’s bid strategy is a best response. Finally, one can check that UD1 has no incentive to deviate and use a different input price strategy.

To see this last point, suppose first that downstream production by UD1 is not viable and, instead of setting the input price equal to $\widehat{W}$, UD1 were to set $W\in (\widehat{W},\widehat{W}\left(A\right)]$. Then, D2 would (wrongly) believe that downstream production by UD1 is viable and that UD1 will bid $V$ (instead of not bidding). Thus, D2 would either bid slightly below $V$ (instead of bidding $V$) or not bid if it is not viable to bid. Such deviation would reduce UD1’s expected profit because its effect on the probability of a sale is virtually the same as when the buyer uses a descending-price auction (and we know from Proposition 2 that the deviation is not profitable). If instead UD1 were to set $W\notin [\widehat{W},\widehat{W}\left(A\right)]$, D2 would still believe that downstream production by UD1 is not viable, and such deviation also would reduce UD1’s expected profit for the same reason.

Suppose now ${C}_{1}<V-c$ so that downstream production by UD1 is viable, but UD1 sets $W>\widehat{W}\left(A\right)$ or $W\le \widehat{W}$, instead of setting $W=\widehat{W}\left({C}_{1}\right)$. Then, D2 would (wrongly) believe that downstream production by UD1 is not viable and that UD1 will not bid (instead of bidding $V$). Thus, D2 would either bid $V$ (instead of bidding slightly below $V$) or not bid if it is not viable. Such deviation would reduce the expected profit of UD1, because (i) its effect on the probability of a sale would be the same as when the buyer uses a descending-price auction, and (ii) when D2 is viable and $W>V-{C}_{1}$, UD1 and D2 would tie but UD1 would prefer D2 to win. If instead UD1 were to deviate and set a different price $W\in (\widehat{W},\widehat{W}\left(A\right)]$, D2 would still believe that downstream production by UD1 is viable, and thus such deviation also would reduce UD1’s expected profit, as is the case when the buyer uses a descending-price auction.

Proof of Proposition 3

• (i) If downstream production by UD1 is not viable, then $\widehat{{\varvec{W}}}<{{\varvec{W}}}^{\boldsymbol{*}}$ .

From the main text, the first-order condition for ${W}^{*}$ is

$$\frac{d{\Pi }_{U}}{dW}=1-\left[1-F\left(V-{W}^{*}\right)\right]\left\{1-F\left(V-{W}^{*}\right)+2\left({W}^{*}-c\right)f\left(V-{W}^{*}\right)\right\}=0.$$

Rearranging terms, ${W}^{*}$ satisfies

$$F\left(V-{W}^{*}\right)-\left({W}^{*}-c\right)f\left(V-{W}^{*}\right)=\frac{{-F\left(V-{W}^{*}\right)}^{2}}{2\left[1-F\left(V-{W}^{*}\right)\right]}.$$

(A4)

The post-merger first-order condition for $\widehat{W}$ in the case when downstream production by UD1 is not viable ($c+{C}_{1}>V$) is

$$\frac{d\Pi }{d\widehat{W}}=F\left(V-\widehat{W}\right)-\left(\widehat{W}-c\right)f\left(V-\widehat{W}\right)=0.$$

Using (A4) to evaluate this first-order condition at $\widehat{W}={W}^{*}$,

$${\left.\frac{d\Pi }{d\widehat{W}}\right|}_{\widehat{W}={W}^{*}}=\frac{-{F\left(V-{W}^{*}\right)}^{2}}{2\left[1-F\left(V-{W}^{*}\right)\right]}<0 .$$

Therefore, in this case, the profit-maximizing value of $\widehat{W}$ satisfies $\widehat{W}<{W}^{*}$.

• (ii) If downstream production by UD1 is viable, then $\widehat{{\varvec{W}}}\left({{\varvec{C}}}_{1}\right)>{{\varvec{W}}}^{\boldsymbol{*}}$ if ${{\varvec{C}}}_{1}$ is sufficiently low.

Rearranging the first-order condition for ${W}^{*}$ once again, ${W}^{*}$ satisfies

$$f\left(V-{W}^{*}\right)=\frac{F\left(V-{W}^{*}\right)}{{W}^{*}-c}\left[1+\frac{F\left(V-{W}^{*}\right)}{2\left[1-F\left(V-{W}^{*}\right)\right]}\right].$$

(A5)

The first-order condition for $\widehat{W}\left({C}_{1}\right)$ when downstream production by UD1 is viable is given by (A2). When we evaluate (A2) at ${W}^{*}$ with the use of (A5),

$${\left.\frac{d\Pi }{d\widehat{W}}\right|}_{\widehat{W}={W}^{*}}=F\left(V-{W}^{*}\right)\left\{1-\left(\frac{{W}^{*}+{C}_{1}-V}{{W}^{*}-c}\right)\left[1+\frac{F\left(V-{W}^{*}\right)}{2\left[1-F\left(V-{W}^{*}\right)\right]}\right]\right\}.$$

As shown in Proposition 1, ${W}^{*}<V-A$, and so ${W}^{*}+{C}_{1}-V<0$ if ${C}_{1}$ is sufficiently low (sufficiently close to $A$). Thus, for sufficiently low ${C}_{1}$, ${\left.\frac{d\Pi }{d\widehat{W}}\right|}_{\widehat{W}={W}^{*}}>0$. Therefore, for sufficiently low ${C}_{1}$, $\widehat{W}\left({C}_{1}\right)>{W}^{*}$.