Equilibrium Analysis of Vertical Mergers

Abstract

Vertical mergers are known to potentially create an incentive for the merged firm to raise the price of inputs it supplies to its rivals (raising rivals’ cost [RRC]). At the same time, vertical mergers are known to create efficiencies in the form of elimination of double marginalization (EDM). Competitive effects of vertical mergers are evaluated as the net effect of RRC and EDM. Conventional antitrust techniques treat the two effects—RRC and EDM—as separable and analyze each in isolation before evaluating their net effect. We show that in an equilibrium treatment, RRC and EDM are not separable; instead, they are inseparably linked because the size of EDM is an important determinant of the strength of the RRC incentive. When the link between EDM and RRC is taken into account, predicted price effects of a vertical merger can turn out to be significantly different relative to those predicted by conventional techniques. Under certain commonly used assumptions, a vertical merger may even create an incentive for the merged firm to lower its rivals’ cost. The precise price effect depends on two things: the shape of demand and the bargaining power of the upstream input supplier in its price negotiations with downstream firms.

Keywords

antitrust vertical mergers

I. Introduction

Vertical mergers are known to potentially create both competitive harm and competitive benefits. Both are created by the change in the objective function of the merging firms from own profit maximization (before the merger) to joint profit maximization (after the merger). On the harm side, a vertical merger can create incentives for the merged firm to raise its wholesale price to downstream competitors that buy inputs from it to impair their competitiveness (raising rivals’ costs [RRC]). On the benefit side, a vertical merger can lead to the elimination of double marginalization (EDM) between the upstream and downstream merging firms which can lead to a reduction in retail price. Competitive analysis of a vertical merger involves a trade-off between these potential harms and benefits.^1,2

Conventional antitrust analysis of vertical mergers typically treats RRC and EDM as two separate effects. For example, in the Department of Justice (DOJ) recent challenge to AT&T/DirecTV’s acquisition of Time Warner, the DOJ acknowledged that the merger will likely create benefits through the EDM in DirecTV’s rates of Time Warner programming. At the same time, the DOJ failed to persuade the District Court that the merger would lead to an increase in the prices of Time Warner programming paid by rival TV distributors.³ In this article, we show that RRC and EDM are not two separate effects. Instead, they are inseparably linked because the size of EDM affects the strength of the RRC incentive, making EDM to be not just a stand-alone competitive benefit to be weighed against RRC.

The intuition for the link between EDM and RRC is as follows. Even if the merged firm were to leave its wholesale prices to rivals’ unchanged, EDM—and any resulting decrease in the downstream price of the merged firm—would serve to shift demand from rivals to the downstream division of the merged firm. The shift leaves rivals with a reduced demand relative to the premerger. The merged firm’s optimal postmerger wholesale prices depend on the characteristics of these reduced demands facing rivals. Thus, the extent of EDM—which determines the extent of demand shift—is a determinant of the merged firm’s RRC incentive.⁴

Two factors affect the link between RRC and EDM: the relative bargaining power of the upstream merging firm (in its price negotiations with downstream firms) and the shape of demand. The higher the bargaining power of the upstream firm, the bigger is its premerger margin and, thus, bigger is the size of the potential EDM. A large EDM reduces rivals’ demand more than a small EDM. Second, the extent to which the merged firm benefits by raising its wholesale price to rivals depends on the shape of the demand facing rivals. Depending on its shape, a reduction in demand may increase or decrease the gains from RRC.

Through a series of examples—that use varying assumptions about relative bargaining power and shape of demand—we show that the link between RRC and EDM can lead to very different price effects than those predicted when RRC and EDM are mistakenly taken to be two separable effects. In some cases, the wholesale price to rivals may even decrease when the link between EDM and RRC is taken into account. We also find that whether, and by how much, any increase in wholesale prices to rivals gets passed through to consumers also depends on the same two factors.

From the literature on horizontal mergers, it is already well known that pass-through of cost increases can depend on the shape of demand.⁵ We show that the weaker the bargaining power of the upstream firm, larger is the premerger margin earned by rival downstream firms and greater are their incentive and ability to absorb an input price increase when competing against a more efficient rival (the merged firm). Thus, whether a vertical merger’s competitive effects are measured solely by wholesale prices to rivals or by average retail prices paid by consumers (consumer welfare), relative bargaining power and shape of demand are important determinants of a merger’s net effect via the link between EDM and RRC. As a result, reliable predictions of the competitive effects of a vertical merger warrant an equilibrium analysis which incorporates the dependence of RRC on EDM.

The remainder of this article is organized as follows. Section II describes the structure of our examples followed by Section III that derives equilibrium expressions for determination of wholesale and retail prices before, and after, a vertical merger. Section III also describes conventional techniques that are used in antitrust analyses that treat RRC and EDM as separate effects. In Section IV, we present comparisons between equilibrium predictions and those obtained by using the conventional techniques. Section V contains a few concluding thoughts.

II. Structure of Examples

A. Model

We consider a simple model in which an upstream producer (denoted by U) supplies an input to two downstream distributors (denoted by D1 and D2). Modeling the upstream as a monopoly allows us to focus attention on vertical mergers involving inputs that do not have adequate substitutes and are, thus, more rather than less likely to be anticompetitive.⁶ We also assume that D1 and D2 offer products that are differentiated in nature.⁷

Our focus is on how a merger between U and one of the downstream distributors (D1) affects the market outcome relative to premerger. Premerger, D1, and D2 are independent firms that each maximize their own profits taking as given their wholesale costs. Wholesale prices charged by U are determined through separate bilateral bargains between U and D1 and between U and D2. Postmerger, U and D1 maximize their joint profits. The postmerger wholesale price charged to D2 is determined through bilateral bargaining between the merged firm and D2. D2 continues to maximize its own profit, given its wholesale cost.

We assume that the solution to each bilateral bargain is the equilibrium outcome of Nash bargaining (NBE).⁸ The NBE objective function is the geometric mean of the differences between the bargaining parties’ agreement payoffs and their respective disagreement payoffs, each weighted by a factor that reflects the bargaining parties’ relative bargaining power. The outcome of a Nash Bargain is the wholesale price that maximizes this objective function. Disagreement payoff of a firm (also known as its “bargaining leverage”) when bargaining with a counterparty is the amount of profit it earns if bargaining fails and it has to make without the product or service offered by the counterparty. Bargaining power of a firm can be thought of as a measure of its relative bargaining “strength” vis-à-vis the counterparty.⁹

1. Timing

Prices are determined through the following two-stage game:

Wholesale stage: In the first stage, U bargains one-on-one with D1 and D2 to determine wholesale prices, $ω_{1}$ and $ω_{2}$ , respectively.

Retail stage: In the second stage, D1 and D2 compete with each other, by each setting their own retail prices, p ₁ and $p_{2,}$ respectively, taking as given its wholesale cost as determined in the first stage, as well as observing the wholesale cost of its rival distributor.

2. Demand

Retail demand faced by D1 and D2 is of the form

q_{i} = D_{i} (p_{i}, p_{j}), \{i, j\} \in \{1, 2\}, i \neq j,

where q_i denotes the output of $D_{i} .$

We consider, in turn, the linear and logit demand functions. Linear demand takes the form

q_{1} = a_{1} - b_{1} \times p_{1} + d_{12} \times p_{2},

q_{2} = a_{2} - b_{2} \times p_{2} + d_{21} \times p_{1},

where the parameters $a_{i}, b_{i}, d_{i, j} > 0$ .

Logit demand takes the form

q_{1} = \frac{e^{d_{1} - a \times p_{1}}}{1 + e^{d_{1} - a \times p_{1}} + e^{d_{2} - a \times p_{2}}},

q_{2} = \frac{e^{d_{2} - a \times p_{2}}}{1 + e^{d_{1} - a \times p_{1}} + e^{d_{2} - a \times p_{2}}},

where the parameter $a > 0$ .

3. Costs

We assume that the upstream firm U produces the input at a constant marginal cost, which we denote by c. We also assume that to produce one unit of their retail product, D1 and D2 need one unit of U’s input and that they have identical additional retail costs per unit (other than the cost of the input). We normalize these additional retail costs to zero.

B. Analytical Steps for Solving the Model

We solve the model by backward induction. Premerger, we first derive the profit-maximizing retail prices, outputs, and profits of D1 and D2 for any given pair of wholesale prices and then solve for the NBE wholesale prices corresponding to the bilateral bargaining games between U and D1 and between U and D2, respectively. Postmerger, we follow the same analytical steps except this time we set the transfer price within the merged firm to be equal to c, that is, ( $ω_{1} = c$ ), and solve for the NBE wholesale price corresponding to the bilateral bargaining game between the vertically integrated firm ( $U + D 1$ ) and the rival distributor D2.¹⁰

III. Derivation of Objective Functions

With the pricing game as described above, we now turn to deriving the equilibrium NBE objective functions for the wholesale stage. Recall that the bargaining leverage of each firm is the amount of profit it can earn in the event of disagreement with its bargaining counterparty (assuming that all other negotiations with third parties result in agreement). Thus, in our model, disagreement in the bargaining game between U and, say, D2, makes D1 the monopolist at the retail stage. On the other hand, agreement in the bargaining game between U and D2 makes D1 and D2 duopolists at the retail stage. We will define the Nash Equilibrium retail prices, outputs, and profits of D1, D2, and U using superscripts to reflect whether the retail market structure is a monopoly (M) or duopoly (D) and subscripts 1, 2, and U to denote whether the player in question is D1, D2, or U. We start by deriving the premerger NBE objective functions and then go on to derive the postmerger NBE objective function.

A. Premerger Bargaining

Consider first premerger bargaining between U and D2.

Payoff for D ₂ from bargaining success is

π_{2}^{D} \equiv (p_{2}^{D} (ω_{1}, ω_{2}) - ω_{2}) \times q_{2}^{D} (ω_{1}, ω_{2}),

where $q_{2}^{D} (ω_{1}, ω_{2})$ and $p_{2}^{D} (ω_{1}, ω_{2})$ denote, respectively, the profit-maximizing duopoly retail output and the price of D2, given wholesale prices of $ω_{1}$ and $ω_{2}$ . Payoff for D ₂ from bargaining failure (D2’s bargaining leverage) is 0.

Payoff for U from bargaining success is the total wholesale profit it earns by selling its product to both $D_{1}$ and D ₂ which equals

π_{U}^{D} \equiv (ω_{2} - c) \times q_{2}^{D} (ω_{1}, ω_{2}) + (ω_{1} - c) \times q_{1}^{D} (ω_{1}, ω_{2}) .

Payoff for U from bargaining failure (U’s bargaining leverage) is the wholesale profit it earns from selling only to D ₁ who is then the monopolist in the retail stage.

π_{U}^{M} \equiv (ω_{1} - c) \times q_{1}^{M} (ω_{1}) .

Using µ and (1 − µ) to denote the relative bargaining powers of U and each downstream firm, respectively, the NBE objective function is then

\begin{array}{l} {[(ω_{2} - c) \times q_{2}^{D} (ω_{1}, ω_{2}) + (ω_{1} - c) \times q_{1}^{D} (ω_{1}, ω_{2}) - (ω_{1} - c) \times q_{1}^{M} (ω_{1})]}^{μ} \\ \times {[(p_{2}^{D} (ω_{1}, ω_{2}) - ω_{2}) \times q_{2}^{D} (ω_{1}, ω_{2})]}^{1 - μ} . \end{array}

The first square-bracketed term is the difference between U’s payoff in the event of bargaining success between U and D2 and its payoff in case of disagreement (i.e., its bargaining leverage). The second square-bracketed term is the difference between D2’s payoff in the event of bargaining success between U and D2 and its bargaining leverage (which, by the structure of the model, and as described above, is zero). Equilibrium NBE wholesale price to D2 is the value of $ω_{2}$ that maximizes this objective function for bargaining between U and $D 2,$ taking as given the value of $ω_{1}$ .

The objective function for premerger bargaining between U and D1 is analogous. Equilibrium NBE wholesale price is the value of $ω_{1}$ that maximizes the objective function for bargaining between U and $D 1,$ taking as given the value of $ω_{2}$ . Equilibrium NBE wholesale prices are the solutions of the simultaneous equations consisting of the first-order conditions that result from maximizing these objective functions. (All algebraic expressions that are involved in solving the model are derived in a self-contained Appendix which is available upon request).

B. Postmerger Bargaining Between (U + D1) and D2

We will use similar notation as before except to drop the subscript whenever we refer to the merged firm $U + D 1$ . Payoff for D ₂ from bargaining success, as in premerger, is $(p_{2}^{D} (ω_{1}, ω_{2}) - ω_{2}) \times q_{2}^{D} (ω_{1}, ω_{2})$ . Payoff for D ₂ from bargaining failure, as in premerger, is 0. Payoff for $U + D 1$ from bargaining success is $(ω_{2} - c) \times q_{2}^{D} (ω_{1}, ω_{2}) + π^{D} (ω_{1}, ω_{2})$ . Payoff for U from bargaining failure is $π^{M} (ω_{1})$ . Note that in these expressions, $π^{D} (., .)$ refers to the total profit of the merged firm $U + D 1$ when the retail stage is a duopoly (following bargaining agreement between $U + D 1$ and D2) excluding the profit from wholesale sales to D2. $π^{M} (.)$ is the total profit of the merged firm $U + D 1$ when the retail stage is a monopoly (following bargaining disagreement between (U + D1) and D2). $π^{M} (.)$ is thus the bargaining leverage of $U + D 1$ . $π^{D} (., .)$ and $π^{M} (.)$ both incorporate the efficiency due to EDM through appropriate choice of postmerger retail price by the merged firm.

Collecting together these expressions, the postmerger NBE objective function to be maximized with respect to $ω_{2}$ is

{[(ω_{2} - c) \times q_{2}^{D} (ω_{1}, ω_{2}) + π^{D} (ω_{1}, ω_{2}) - π^{M} (ω_{1})]}^{μ} \times {[(p_{2}^{D} (ω_{1}, ω_{2}) - ω_{2}) \times q_{2}^{D} (ω_{1}, ω_{2})]}^{1 - μ} .

C. Change in Surplus and Bargaining Leverage due to Merger

The merger induces two countervailing sources of change to the size of the bargaining surplus between U and D2 that affect the postmerger wholesale price $ω_{2}$ . First, gains to trade between them (evaluated at premerger prices) increase. This is reflected in the fact that in the event of agreement between them, U’s postmerger payoff is $[(ω_{2} - c) \times q_{2}^{D} (ω_{1}, ω_{2}) + π^{D} (ω_{1}, ω_{2})]$ , whereas its corresponding premerger payoff is $[(ω_{2} - c) \times q_{2}^{D} (ω_{1}, ω_{2}) + (ω_{1} - c) \times q_{1}^{D} (ω_{1}, ω_{2})]$ . Since $π^{D} (ω_{1}, ω_{2}) > (ω_{1} - c) \times q_{1}^{D} (ω_{1}, ω_{2})$ , the total gains to trade between $U (+ D 1)$ and D2 are higher after the merger than before. The second change reflects increase in U’s bargaining leverage. This can be noted by observing that $π^{M} (ω_{1}) > (ω_{1} - c) \times q_{1}^{M} (ω_{1})$ . Both changes are due to the fact that after the merger, when bargaining with D2, U takes into account the full profit of the integrated firm, ( $U + D 1$ ) compared to before the merger when it takes into account only its own wholesale profit.

D. Limiting Case of a Price-Setting Upstream Firm

Having derived the NBE expressions, it is helpful to note that when µ = 1 (i.e., all of the bargaining power rests with $U$ ), so long as $ω_{1}$ and $ω_{2}$ do not equal or exceed the “choke” price for the demand facing D1 and D2, respectively (i.e., so long as D1 and D2 make positive profits in equilibrium), the premerger NBE optimization problem simplifies to the standard profit maximization problem of an upstream firm that selects $ω_{1}$ and $ω_{2}$ to maximize its profit, and the postmerger NBE problem simplifies to the standard problem of a vertically integrated (merged) firm that selects $ω_{2}$ to maximize its overall profit. For example, in the postmerger NBE objective function, the term ${[(p_{2}^{D} - ω_{2}) \times q_{2}^{D} (c, ω_{2})]}^{1 - μ}$ in the NBE objective function above takes on the value 1 when µ = 1 and $ω_{2}$ does not equal or exceed the choke price of the demand facing D2. In that case, the postmerger NBE problem boils down to

max_{ω_{2}} [\{(ω_{2} - c) \times q_{2}^{D} (ω_{2}, ω_{2}) + π^{D} (ω_{2}, ω_{2})\} - π^{M} (ω_{1})] .

The expression inside the curly brackets is the total profit of an integrated firm, and the maximization problem is that of the integrated firm that sets its wholesale price to its rival in order to maximize its total profit. (The term outside the curly brackets, $π^{M} (ω_{1})$ , is independent of $ω_{2}$ .) In the same way, when µ = 1, the premerger NBE objective function simplifies to the objective function for profit maximization by an independent upstream firm.

E. Conventional Analysis of Vertical Mergers That Treats RRC and EDM as Separable Effects

In practice, equilibrium simulation of the competitive effects of vertical mergers—an exercise that would take into account the effect of EDM on RRC—is eschewed in favor of a separable approach that leaves out the demand shift effect. The separable approach is based on a calculation of the additional profit that the downstream merging firm can earn if the upstream firm were to (hypothetically) stop supplying the input to a downstream rival. The additional profit can be thought of as the merger-induced opportunity cost to the merged firm of supplying the input to the rival. This opportunity cost is then monetized into a wholesale price increase to rivals by multiplying it by the relative share of surplus that the rival gets to keep when it bargains with the upstream firm and dividing the product so obtained by the rival’s premerger output. The calculation assumes that the merging downstream firm’s retail price and the rival’s output remain at their premerger levels. Thus, the calculation ignores the demand shift effect that stems from EDM.¹¹

The opportunity cost is estimated as the product of three numbers: (i) the number of customers that rivals stand to lose if they are denied access to the input (customer loss rate), (ii) the fraction of those departing customers who would turn to the merged firm for their purchases (diversion ratio), and (iii) the premerger margin of the merging downstream firm.¹² All three numbers are typically positive. The technique is, thus, hardwired to predict a wholesale price increase. Although in theory this framework can be used to estimate the price effect when the upstream firm gets to set the price (as opposed to bargain)—by analyzing the case in which all of the bargaining power rests with the upstream firm—Moresi and Salop have proposed a framework that is specific to the price-setting case.¹³ The framework—which is called vertical GUPPI (vGUPPI)—resembles the opportunity cost calculations described above.¹⁴ Neither approach explicitly models the effect of any increase in wholesale prices on rivals’ retail prices, that is, prices that are paid by final consumers. Instead, it is commonly assumed that the pass-through of wholesale price increase by the rival is 50%.

IV. Results

We are now ready to use the NBE model described in the previous section to simulate the examples of vertical mergers and compare the price effects from NBE simulations with those predicted by the conventional approach. An important objective is to glean insights about how the predicted effect of a given merger may vary with the investigator’s assumption regarding the functional form of retail demand. Toward that end, we choose the linear and logit demand parameters in a way such that they generate the same premerger diversion ratios and retail margins.¹⁵ A second important objective is to understand how the predicted merger effect may vary depending on the investigator’s assumptions regarding relative bargaining powers of the upstream firm and the downstream rival. Toward that end, we consider, in turn, several different values of the NBE parameter µ.

For each set of parameter values, we compare the equilibrium merger effect with those predicted by the conventional approach that treats EDM and RRC as separable effects. (For the price-setting case corresponding to $μ = 1$ , we compare equilibrium changes to the predictions of vGUPPI in place of the Rogerson (2014) approach for reasons explained in footnote 13. In each case, our focus is on the merger’s effect on two variables in particular—the wholesale price paid by D2 ( $ω_{2}$ ) and the postmerger share weighted industry-wide average retail price.

The following model parameters generate the same premerger diversion ratios and retail margins when µ = 1:

– Linear demand: a ₁ = a ₂ = 454.30; d ₁₂ = d ₂₁ = 0.64; b ₁ = b ₂ = 1.29; U’s cost = 252.33.

– Logit demand: d ₁ = d ₂ = 11.0; a = 0.02; U’s cost = 290.0.

We have chosen the demand parameters to be symmetric between D1 and D2. In particular, for the linear demand, we have ensured that the cross-price coefficients, $d_{12}$ and $d_{21},$ that govern shift in demand between D1 and D2 as a result of changes in rival’s price, are identical.¹⁶ Finally—for reasons described earlier—we set the value of the postmerger transfer price between U and D1, $ω_{1}$ , to be equal to the upstream cost (c) reflecting complete EDM.¹⁷ (All calculations are contained in an Excel workbook, which is available from the authors on request.)

Table 1 is a snapshot of how equilibrium merger effects compare with those predicted by the standard antitrust technique. From left to right, Table 1 has three panels that summarize merger effects for three values of the bargaining parameter: µ = 1 (upstream firm is a price setter), µ = 0.75 (upstream firm earns 75% of bargaining surplus, i.e., 75/25 sharing rate), and µ = 0.5 (upstream firm earns 50% of bargaining surplus, i.e., 50/50 sharing rate). Within each panel, there are two rows; the top row is for linear demand and the bottom row for logit demand. The first column within each panel shows the equilibrium merger effect on the wholesale price paid by D2; the second column shows the effect predicted by the standard antitrust technique and the third column shows the equilibrium change in average share weighted retail price. While the summary table is restricted to these two prices, a larger table in Appendix A reports changes in all the variables including the extent of equilibrium demand shift from the rival to the merged firm and equilibrium retail prices of the merged firm and the rival. Several of our findings are worth noting.

Table 1.

Comparison of Equilibrium Merger Effects with Predictions from Conventional Analysis.

Shape of Demand	Price Setting (µ = 1)			Nash Bargaining (µ = 0.75)			Nash Bargaining (µ = 0.5)
	% Change in w ₂		Equilibrium Change in Industry-Wide Average Retail Price	% Change in w ₂		Equilibrium Change in Industry-Wide Average Retail Price	% Change in w ₂		Equilibrium Change in Industry-Wide Average Retail Price
	Equilibrium Simulation	Conventional Antitrust Analysis (vGUPPI)	Equilibrium Change in Industry-Wide Average Retail Price	Equilibrium Simulation	Conventional Antitrust Analysis	Equilibrium Change in Industry-Wide Average Retail Price	Equilibrium Simulation	Conventional Antitrust Analysis	Equilibrium Change in Industry-Wide Average Retail Price
Linear	−0.7	3.9	−8.5	8.8	2.9	−2.8	18.7	8.2	2.6
Logit	7.8	3.9	−6.0	11.5	2.8	−1.9	18.0	8.0	3.8

A. Limiting Case of a Price-Setting Upstream Firm

First, when all of the bargaining power rests with the upstream firm (i.e., µ = 1), and demand is linear, the merger leads to a reduction in the wholesale price of the upstream firm’s product to the rival downstream firm (−0.7%). This contrasts sharply with the prediction of the standard technique that a vertical merger can only lead to an increase in the wholesale price to rival downstream firms.^18
,19

The intuition for this result is as follows. Two important characteristics of the reduced demand of a rival that affect whether the merged firm finds it profitable to raise its wholesale price to the rival, and by how much, are its own-price elasticity and the diversion ratio from the rival to the merged firm, both evaluated at the premerger wholesale price charged to the rival.²⁰ The higher is the own-price elasticity of the reduced demand of a rival, the weaker is the merged firm’s incentive to raise its wholesale price. On the other hand, the higher is the diversion ratio (from the rival to the merged firm) of the reduced demand, the stronger is the merged firm’s incentive to raise its wholesale price. The net effect of these price pressures—and the standard RRC incentive—determines whether a merged firm finds it profitable to raise or lower the rival’s wholesale price.

For commonly used functional forms of demand-like linear and logit, own-price elasticity of reduced demand is higher than the own-price elasticity of premerger demand (evaluated at the premerger wholesale price charged to the rival). This creates a downward price pressure (incentive to lower rival’s cost [LRC]). On the other hand, demand shift from rival to merged firm leaves diversion ratio from the rival to the merged firm unchanged for a linear demand but raises it for a logit demand.²¹ As a result, for a linear demand, there is no incremental upward price pressure created by the EDM-driven demand shift whereas for a logit demand there is. We show through examples, that in the case of linear demand, LRC dominates RRC, that is, it turns out to be profitable for a merged firm to reduce the wholesale price to rivals so long as substitutability between the rival’s product and the merged firm’s product (as measured by the cross-price coefficients of retail demand) is sufficiently symmetric.²² In the case of logit demand, the combined upward price pressure from the higher diversion ratio of residual demand and the standard RRC incentive tends to outweigh LRC leading to a price increase to rivals.

B. Role of Bargaining Power

Second, regardless of the shape of demand, the predicted wholesale price effect increases as the bargaining power of the upstream firm (reflected by the value of the parameter µ) decreases. Starting from the limiting case of a price-setting upstream firm, as the relative bargaining power of the upstream firm decreases, the magnitude of price reduction to rivals in the case of linear demand shrinks and eventually turns into a price increase. Analogously, as the bargaining power of the upstream firm decreases, the magnitude of price increase in the case of logit demand increases. These comparative statics are intuitive. Since relative bargaining power is a determinant of the division of surplus prior to the merger (and the relative bargaining power is the same before and after the merger), the lower the bargaining power of the upstream firm, the smaller is its premerger margin. The smaller the premerger margin, the smaller is the size of the EDM from the merger and thus smaller is the demand shift effect due to EDM. With a sufficiently small demand shift, the standard RRC incentive prevails over LRC, leading to a price increase to rivals. This indicates that in an NBE analysis, the wholesale price effect may critically depend on the value of the relative bargaining power parameter.²³

C. Predicted Price Effects

Third, except for the linear demand and price-setting case, the equilibrium effect on wholesale prices to rivals is significantly larger than what is predicted by the standard technique. For example, when µ = 0.5, the equilibrium price effect with a logit demand is 18% while the price effect predicted by the standard technique is only 8%.

The reason for the difference is intuitive. For the portion of the NBE objective function that consists of the merged firm’s profit from the sale of the upstream product to rivals, demand shift engenders the same upward and downward pricing pressures as in the price-setting context. In addition, even if the wholesale price to rivals is left unchanged, in the event of disagreement the profit of the merged firm—once reoptimized to reflect EDM—is greater than the corresponding combined premerger profits of the merging firms. The latter increases the bargaining leverage of the merged firm and thus creates an additional upward price pressure.²⁴ This increase in the bargaining leverage of the merged firm due to EDM is not reflected in the standard technique. The net effect of these price pressures—together with the standard RRC incentive—determines the postmerger wholesale price.²⁵

D. Effect on Retail Prices

Fourth, when the overall competitive effects of a merger are determined by average retail prices that are paid by consumers, equilibrium analysis may show a decline in average retail prices notwithstanding an increase in wholesale price to rivals. For example, for both linear and logit demand, and µ = 0.75, the equilibrium wholesale price effect is significant (8.8% and 11.5%, respectively), yet the average retail price paid by consumers is lower after the merger (by 2.8% and 1.9%, respectively). Additionally, even though the wholesale price to the rival increases by 8.8% (linear demand), the rival’s retail price increases only by 2.2%.²⁶

Equilibrium analysis reveals that the incentives of a rival to pass through any increase in wholesale price is also affected by EDM. First, the EDM-induced efficiency of the merging downstream firm creates downward pressure on the rival’s retail prices as it seeks to compete against a more efficient competitor. Second, a reduction in rivals’ demand (due to EDM-induced demand shift) also puts downward pressure on the rival’s retail price. Whether, and by how much, a rival increases its retail prices when it faces a higher wholesale cost depends on the mitigating effects of these downward price pressures.²⁷

V. Concluding Thoughts

A key takeaway from our results is that the standard technique—that does not account for the link between EDM and RRC—can significantly miss the mark when it comes to predicting price effects. Equilibrium simulation can help to account for all sources of postmerger pricing incentives but, crucially, relies on the shape of the assumed demand function.²⁸ An important parameter that affects both the equilibrium approach as well as the standard technique is the relative bargaining power of the upstream firm. As demonstrated by the results contained in Table 1, it is of critical importance to choose the appropriate value of this parameter when investigating the competitive effects of a merger. Ad hoc assumptions regarding the value of the parameter µ can introduce a significant risk of getting the estimated price effect to be different from the equilibrium price effect.²⁹

In this article, we have focused attention on one of several ways in which a vertical merger may potentially reduce competition. Our focus is on whether a vertical merger necessarily creates an incentive for the merged company to raise rivals’ cost and how to estimate the magnitude of such an increase in rivals’ cost. Even if a vertical merger does not raise rivals’ cost, it might still need to be scrutinized for other potential anticompetitive effects. For example, the shift in demand from rival distributors to the merged firm may leave rivals with insufficient scale to compete effectively. This may be a concern, for example, in industries in which rivals need to make substantial investments in facilities—investments that are not profitable unless they can serve a sufficiently large share of customers. For example, a video distributor that needs to deploy or maintain video cable lines to households in order to distribute video may find such investment to be unprofitable unless it can serve sufficiently many customers in a local area. Although the shift in demand to the merged firm, which has a lower wholesale cost (due to EDM), may seem efficient in the short run, its longer run consequence may be to impair rivals’ competitiveness by reducing their investment incentives.

Footnotes

Appendix A—Detailed Results

Table A1.

Comparison of Equilibrium Merger Effects with Predictions from Incremental Approaches.

	Price Setting (µ = 1)						Nash Bargaining Setting (µ = 0.75)						Nash Bargaining Setting (µ = 0.5)
	Linear Demand			Logit Demand			Linear Demand			Logit Demand			Linear Demand			Logit Demand
	Premerger	Postmerger	% Change	Premerger	Postmerger	% Change	Premerger	Postmerger	% Change	Premerger	Postmerger	% Change	Premerger	Postmerger	% Change	Premerger	Postmerger	% Change
Quantities
q ₁	96.6	164.8	70.7%	96.6	176.1	82.3%	124.5	168.0	35.0%	110.4	169.4	53.4%	149.7	171.7	14.7%	124.4	162.5	30.7%
q ₂	96.6	79.1	−18.1%	96.6	45.0	−53.4%	124.5	92.0	−26.1%	110.4	54.9	−50.3%	149.7	106.9	−28.6%	124.4	66.1	−46.8%
Market total	193.2	243.9	26.3%	193.2	221.1	14.5%	249.0	260.1	4.4%	220.8	224.2	1.5%	299.3	278.6	−6.9%	248.7	228.6	−8.1%
Downstream prices
p ₁	550.7	489.4	−11.1%	550.7	503.8	−8.5%	507.7	479.5	−5.6%	527.3	503.5	−4.5%	468.9	468.1	−0.2%	496.1	502.2	1.2%
p ₂	550.7	533.8	−3.1%	550.7	572.1	3.9%	507.7	518.8	2.2%	527.3	559.9	6.2%	468.9	501.7	7.0%	496.1	547.1	10.3%
Market avg.	550.7	503.8	−8.5%	550.7	517.7	−6.0%	507.7	493.4	−2.8%	527.3	517.3	−1.9%	468.9	481.0	2.6%	496.1	515.2	3.8%
Wholesale prices
w ₁	475.8	252.3	−47.0%	475.8	290.0	−39.1%	411.2	252.3	−38.6%	446.8	290.0	−35.1%	353.0	252.3	−28.5%	408.8	290.0	−29.1%
w ₂	475.8	472.5	−0.7%	475.8	513.0	7.8%	411.2	447.5	8.8%	446.8	498.2	11.5%	353.0	418.9	18.7%	408.8	482.4	18.0%
Market avg.	475.8	323.7	−32.0%	475.8	335.3	−29.5%	411.2	321.4	−21.8%	446.8	341.0	−23.7%	353.0	316.2	−10.4%	408.8	345.7	−15.5%
Diversion ratios
D1->D2	49.7%	49.7%		49.7%	39.1%		49.7%	49.7%		61.1%	45.1%		49.7%	49.7%		74.6%	51.4%
D2->D1	49.7%	49.7%		49.7%	71.6%		49.7%	49.7%		61.1%	71.7%		49.7%	49.7%		74.6%	72.2%
Own price elasticity
D1	−7.4	−3.8		−7.4	−4.0		−5.3	−3.7		−6.5	−4.2		−4.0	−3.5		−5.7	−4.4
D2	−7.4	−8.7		−7.4	−9.7		−5.3	−7.3		−6.5	−9.1		−4.0	−6.1		−5.7	−8.5

Note. Demand parameters and upstream cost selected in order to generate the same equilibrium prices and diversion ratios premerger in the linear and logit price-setting scenarios:

– Linear demand: a ₁ = a ₂ = 454.30; d ₁₂ = d ₂₁ = 0.64; b ₁ = b ₂ = 1.29; U’s cost = 252.33.

– Logit demand: d ₁ = d ₂ = 11.0; a = 0.02; U’s cost = 290.0. Market size assumed to be such that, in the price setting case, total sales premerger are the same as in the linear demand premerger equilibrium.

Authors’ Note

Any ideas or opinions expressed in this article are solely those of the authors and do not necessarily represent the views of Charles River Associates or its clients. Any errors or omissions are our own.

Acknowledgments

The authors thank (without implicating) Luke Froeb, Mark Glick, Bill Rogerson, and Ralph Winter.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Equilibrium Analysis of Vertical Mergers

Abstract

Keywords

I. Introduction

II. Structure of Examples

A. Model

1. Timing

2. Demand

3. Costs

B. Analytical Steps for Solving the Model

III. Derivation of Objective Functions

A. Premerger Bargaining

B. Postmerger Bargaining Between (U + D1) and D2

C. Change in Surplus and Bargaining Leverage due to Merger

D. Limiting Case of a Price-Setting Upstream Firm

E. Conventional Analysis of Vertical Mergers That Treats RRC and EDM as Separable Effects

IV. Results

A. Limiting Case of a Price-Setting Upstream Firm

B. Role of Bargaining Power

C. Predicted Price Effects

D. Effect on Retail Prices

V. Concluding Thoughts

Footnotes

Appendix A—Detailed Results

Authors’ Note

Acknowledgments

Declaration of Conflicting Interests

Funding

Notes