The Mathematical Model

Firm k ∈ K chooses the prices {p_i}_i = 1^n_k of its products so as to maximize profits. Mathematically, firm k solves:

where ω_ik is the share of product i’s profits earned by firm k, so that $\sum\limits_{k\in K} \omega_{ik}\le 1$. q_i, the quantity sold of product i, is assumed to be a twice differentiable function of all product prices.

Differentiating profits with respect to each p_i yields the following first order conditions (FOCs):

which may be rewritten as

where $r_i\equiv\frac{p_iq_i}{\sum\limits_{j=1}^np_jq_j}$ is product i’s revenue share, $m_i\equiv\frac{p_i-c_i}{p_i}$ is product i’s gross margin, and $\epsilon_{ij}\equiv\frac{\partial q_i}{\partial p_j}\frac{p_j}{q_i}$ is the elasticity of product i with respect to the price of product j.

The FOCs for all products may be stacked and then represented using the following matrix notation:

rearranging yields

where r and m are n-length vectors of revenue shares and margins, $E = \left(\begin{smallmatrix} \epsilon_{11}&\ldots&\epsilon_{1n}\\\vdots &\ddots&\vdots\\\epsilon_{n1}&\ldots&\epsilon_{nn} \end{smallmatrix}\right)$ is a n × n matrix of own- and cross-price elasticities, and $\Omega=\left(\begin{smallmatrix} \omega_{11}&\ldots&\omega_{1n}\\\vdots &\ddots&\vdots\\\omega_{n1}&\ldots&\omega_{nn} \end{smallmatrix}\right)$ is an n × n matrix whose i, jth element equals the share of product j’s profits owned by the firm setting product i’s price.³ In many cases, product i and j are wholly owned by a single firm, in which cases the i, jth element of Ω equals 1 if i and j are owned by the same firm and 0 otherwise. Under partial ownership, the columns of the matrix formed from the unique rows of Ω must sum to 1. ‘diag’ returns the diagonal of a square matrix and ‘∘’ is the Hadamard (entry-wise) product operator.

The solution to system @ref(eq:FOC1) yields equilibrium prices conditional on the ownership structure Ω. A (partial) merger is modeled as the solution to system @ref(eq:FOC1) where Ω is changed to reflect the change in ownership.

Adding Exogenous Capacity Constraints

The Bertrand model described above assumes that products are produced with constant marginal costs and no capacity constraints. Here, we extend this model to allow for exogenous capacity constraints.⁴

Firm k ∈ K chooses the prices {p_i}_i = 1^n_k of its products so as to maximize profits, subject to capacity constraints {t_i}_i = 1^n_k. Mathematically, firm k solves:

subject to

In general, either the capacity constraint for product i will bind and the firm will be forced to produce less of i than it would find optimal, or the capacity constraint will not bind, and the firm will produce the optimal amount implied by the FOCs. In the former, it can be shown that ∂p_i ≤ 0 and q_i − t_i = 0, while in the latter ∂p_i = 0 and q_i − t_i ≤ 0. Mathematically, these cases can be written as

Linear Demand

The Bertrand model with linear demand may be implemented using the linear function.

The linear demand system assumes that the demand for each product i ∈ n in the market is given by

which may be written in matrix notation as

where q ≥ 0, p > 0 are vectors of product quantities and prices, α is a vector of product specific demand intercepts and B is a matrix of slopes. This demand system yields the following own- and cross-price elasticities:

Without additional restrictions and/or data, there are 2n equations (n FOCs and n demand equations) but n(n + 1) unknown parameters, which means that there are more unknowns than equations and the demand parameters α, B cannot be recovered. To remedy this, we assume that the quantity diversion is observed.⁵ With the linear model, this assumption increases the number of equations to n(n + 1), allowing estimates of α and B to be recovered if prices, quantities and margins are observed for all products.

One known issue with the linear demand system is that, while analytically tractable, it is not rooted in consumer choice theory. Indeed, it has been shown that the linear demand system without income effects is consistent with the axioms of consumer choice if B is a symmetric matrix and satisfies homogeneity of degree 0 in prices.⁶ Imposing these additional assumptions reduces the number of unknown parameters to $\frac{n(n+3)}{2}$ ($\frac{n(n+1)}{2}$ elements of B and n intercepts), which means that the system is over-identified.⁷

The linear demand system can be modified to allow for substitution to an outside (numeraire) good. To accomplish this, assume that the price of the outside product is not set strategically (i.e. not set as part of the Nash-Bertrand game played by the manufacturers of the n products included in the simulation). It can be shown that if B is symmetric and satisfies homogeneity of degree 0 in prices when the outside good is included, then ∑_j ∈ nd_ij < 0.⁸ In other words, under symmetry and homogeneity of degree 0, users may include a non-strategically priced outside good by simply allowing the rows of the diversion matrix to sum to less than zero. The extent to which consumers substitute from product i to the outside good may be controlled by increasing the magnitude of ∑_j ∈ nd_ij.

By default, the calcSlopes, called by the linear function to calibrate the linear demand parameters, assumes that the above assumptions hold and uses a minimum distance algorithm to find the elements of B that best satisfy i) all the FOCs for which there is sufficient information and ii) the diversion equations $d_{ij}=-\frac{\beta_{ji}}{\beta_{ii}}$, subject to the constraints that β_ij ≥ 0 for all i ≠ j and ∑_jb_ij ≤ 0 for all i. The model intercepts can then be recovered from the linear demand equations.

For completeness, linear includes the ‘symmetry’ argument that, when set equal to FALSE, instructs calcSlopes to calibrate demand parameters without imposing symmetry and homogeneity of degree zero in prices on B. Note that when ‘symmetry’ is FALSE, the system of equations is just-identified, which means that prices, quantities, and margins must be observed for all products. Also, note that when ‘symmetry’ is FALSE, linear demand is unlikely to be consistent with consumer choice theory, and welfare measures such as compensating variation cannot be calculated.⁹

Log-Linear Demand

The Bertrand model with log-linear demand may be implemented using the loglinear function.

The log-linear demand system assumes that the demand for each product i ∈ n in the market is given by

which may be written in matrix notation as

where q, p are vectors of product quantities and prices, α is a vector of product specific demand intercepts and B is a matrix of slopes. This demand system yields the following own- and cross-price elasticities:

As with linear demand, there are 2n equations but n(n + 1) unknown parameters, which means the demand parameters α, B cannot be recovered without additional assumptions. As before, we will assume that quantity diversion is known and by default occurs according to quantity share. However, it turns out that the parameter restrictions needed to make log-linear demand consistent with consumer choice theory are likely to be inconsistent with the Bertrand model.¹⁰ As such, loglinear employs only the first assumption. Consequently, the demand parameters are just-identified, which means that users must supply loglinear with prices, margins, and quantities for all products in the market.

LA-AIDS Demand

The Bertrand model with the linear approximate Almost Ideal Demand System (LA-AIDS) may be implemented using the aids function.

The LA-AIDS without income effects assumes that the demand for each product i ∈ n in the market is given by

which may be written in matrix notation as

where r, p are vectors of product revenue shares and prices, α is a vector of product-specific demand intercepts and B is a matrix of slopes.¹¹

The LA-AIDS model yields the following own- and cross-price elasticities:

where ϵ is the market elasticity of demand.

As with the linear demand system, the LA-AIDS model assumes that B is symmetric, satisfies homogeneity of degree zero in prices, and that diversion is known. The LA-AIDS model, however, assumes that revenue diversion, rather than quantity diversion is observed.¹² Under these two assumptions, there are $\frac{n(n+3)}{2}$ unknown demand parameters ($\frac{n(n-1)}{2}$ diagonal elements in B, n diagonal elements, and n intercepts) and up to n(n + 1) equations (n(n − 1) diversion equations, n FOCs and n demand equations), in which case the system is over-identified.¹³

One interesting feature of the LA-AIDS that distinguishes it from the linear and log-linear demand systems included in antitrust is that LA-AIDS elasticities incorporate ϵ, the market elasticity. Roughly speaking, ϵ controls the extent to which consumers substitute to products outside the n products included in the simulation given a small change in market-wide product prices. Here, we assume that ϵ is a parameter whose value is not a function of product prices.¹⁴ While in some cases ϵ can be readily observed, in others it cannot. For the latter, the calcSlopes method (called by aids) exploits the fact that there are more equations than unknowns to identify both the unknown demand parameters described above as well as ϵ.

The calcSlopes method, called by the aids function to calibrate the AIDS parameters uses a minimum distance algorithm to find the ϵ and a single diagonal element of B that best satisfy i) all the FOCs for which there is sufficient information and ii) the diversion equations $d_{ij}=-\frac{\beta_{ji}}{\beta_{ii}}$, and iii) the market elasticity (if supplied using the ‘mktElast’ argument). The β_iis are recovered from the fact that ∑_jd_ij = 0; customers must switch to a product included in the model.

If only a single product’s own-price elasticity and ϵ is observed, then pcaids may be used in lieu of aids to calibrate the LA-AIDS parameters.¹⁵

Another distinguishing feature of the LA-AIDS model is that it does not require any information on product prices in order to simulate merger price effects. The LA-AIDS accomplishes this by using the supplied margin and revenue information to estimate B, but not α. There are however, a few drawbacks to not using pricing information. First, while merger-specific price changes may be calculated, pre- and post-merger price levels cannot. Second, welfare measures like compensating variation cannot be calculated. Prices are an optional input to aids and pcaids, and when they are supplied both price levels and welfare measures may be calculated.

Logit Demand

The Bertrand model with Logit demand may be implemented using the logit function.

Logit demand is based on a discrete choice model that assumes that each consumer is willing to purchase at most a single unit of one product from the n products available in the market. The assumptions underlying Logit demand imply that the probability that a consumer purchases product i ∈ n is given by

where s_i is product i’s quantity share and V_i is the (average) indirect utility that a consumer receives from purchasing product i. We assume that V_i takes on the following form

The Logit demand system yields the following own- and cross-price elasticities:

Logit demand has n + 1 parameters to estimate (n δs and α) and up to 2n equations with which to estimate them (up to n complete FOCs and n choice probabilities). calcSlopes exploits this over-identification by employing a minimum distance algorithm to find the value for α that best satisfies all the FOCs for which there are data. The δs are then recovered from the choice probabilities.

One feature of the logit function is that the function allows users to specify whether or not consumers must purchase one of the n products sold in the market or whether consumers can choose to purchase an “outside” good. logit determines whether users wish to include an outside option by determining if the user-supplied quantity shares s_i sum to 1. If the shares sum to 1, then no outside good is included and by default δ₁ is normalized to 0.¹⁹ Otherwise, an outside good is included whose δ is normalized to 0, price is set equal to ‘priceOutside’ (default 0), and whose share equals $s_0=1-\sum\limits_{i\in n}s_i$.²⁰

CES Demand

The Bertrand model with Constant Elasticity of Substitution (CES) demand may be implemented using the ces function.

Like the Logit, CES demand is based on a discrete choice model. However, CES differs from the Logit model in that under CES consumers do not purchase a single unit of a product but instead spend a fixed proportion of their budget on one of the n products available in the market.²³

The assumptions underlying CES demand imply that the probability that a consumer purchases product i ∈ n is given by where r_i is product i’s revenue share and V_i is the (average) indirect utility that a consumer receives from purchasing product i. We assume that V_i takes on the following form

The CES demand system yields the following own- and cross-price elasticities:

Functional form differences aside, one important difference between the CES and Logit demand systems is that the Logit model’s choice probabilities are based on quantity shares, while the CES model’s choice probabilities are based on revenue shares.

Like Logit demand, CES demand has n + 1 parameters to estimate (n, δs and γ) and up to 2n equations with which to estimate them (up to n complete FOCs and n choice probabilities). ces exploits this over-identification by employing a minimum distance algorithm to find the value for γ that best satisfies all the FOCs for which there are data. The δs are then recovered from the choice probabilities.

ces also allows users to specify whether or not consumers must purchase one of the n products sold in the market or whether consumers can choose to purchase an “outside” good. ces determines whether users wish to include an outside option by determining if the user-supplied revenue shares r_i sum to 1. If the shares sum to 1, then no outside good is included and by default δ₁ is normalized to 1.²⁴ Otherwise, an outside good is included whose price and δ are normalized to 1, and whose share equals $r_0=1-\sum\limits_{i\in n}r_i$.

In addition to specifying an outside option, ces has the ‘shareInside’ argument that may be used to specify the proportion of the representative consumer’s budget that the consumer is willing to spend on the n + 1 products that are within the market.²⁵ By default, ‘shareInside’ equals 1, which indicates that the customer spends her entire budget on the n + 1 products within the market.

Marginal Costs

If all n product margins are observed, estimating marginal costs can be accomplished by noting that $m_i\equiv\frac{p_i-c_i}{p_i}$ and using observed prices to calculate pre-merger marginal costs.

Rather than using observed margins to compute marginal costs, antitrust instead relies on the margins predicted by the Bertrand model. Rearranging the FOCs yields an expression for margins as a function of the demand parameters, product ownership, and revenue shares:

where E_pre, r_pre are elasticities and revenues calculated from the assumed demand model, evaluated at observed prices.

The main advantage of using m̂_pre over m is that not all of the product margins must be observed in order to estimate marginal costs.²⁸ Once m̂_pre has been calculated, observed prices and the margin definition may be used to estimate pre-merger marginal costs.

Because antitrust‘s Bertrand model assumes that marginal costs are constant, product i’s post-merger marginal costs are equal to its pre-merger marginal costs, multiplied by (1 + Δmc_i), the change in marginal costs due to any merger-specific efficiencies. All of the functions described above have a ’mcDelta’ argument that allows users to specify a length-n vector of marginal cost changes.²⁹ By default, ‘mcDelta’ is equal to a length-n vector of zeros, indicating that the merger will not yield any efficiencies.

For the Bertrand model with capacity constraints, product margins for capacity-constrained products cannot be recovered from the first-order conditions.³⁰ Therefore, marginal costs for capacity-constrained products must be recovered from user-supplied margins and prices.

The calcMC method may be used to calculate pre- and post-merger marginal costs.

Summarizing Results

The summary method may be used to summarize the results of a merger between two firms for a given demand model. By default, the summary method reports pre- and post-merger equilibrium prices, revenue shares, weighted average compensating marginal cost reduction, and compensating variation.³³ Quantity shares rather than revenue shares may be reported by setting summary‘s ’revenue’ argument equal to FALSE. Likewise, levels, either in units or in revenues, rather than shares, may be reported by setting summary‘s ’shares’ argument equal to FALSE. Calibrated demand parameters may be reported by setting summary‘s ’parameters’ argument equal to TRUE.The number of significant digits can be altered using the ‘digits’ argument.

In addition to printing the equilibrium price and output information to the screen, the summary method invisibly returns a matrix containing this information. Users can save this matrix to a new object for later use.

Plotting Results (experimental)

The plot method employs ggplot to plot pre- and post-merger product residual demand, marginal costs and equilibria. This method has a ‘scale’ argument, a number between 0 and 1 which controls how much of the demand curve above the equilibrium price and below marginal cost is plotted. The default for ‘scale’ is .1.

plot returns a ggplot object.

Simulating Price Effects with Efficiencies

Absent efficiencies, the Bertrand model with the demand systems described here will almost always produce a (possibly negligible) post-merger price increase among substitutes. These price increases, however, can be offset by merger-specific efficiencies that decrease the incremental costs of some of the merging firms’ products.³⁴

All the functions discussed above allow users to evaluate these efficiencies in two different ways. First, all of these functions contain the ‘mcDelta’ argument, which allows users to specify the proportional change in a product’s marginal costs that may result from a merger. These cost changes are factored into the post-merger price equilibrium calculation made by the calcPrices method.

Second, users can call the cmcr method on the output of any of the functions described above. This method computes the compensating marginal cost reduction (CMCR) on the merging parties’ products. CMCR is the percentage decrease in the marginal costs of the merging parties’ products necessary to prevent a post-merger price increase. See the cmcr help page for further details.

Excluding Products From the Market (experimental)

By default, the Bertrand model calculates a merger’s effects under the assumption that the acquisition does not change the set of products available to consumers. A merger’s effects, however, may differ if the merger induces either the merging parties or another market participant to eliminate some products from their portfolio.

To accommodate the possibility that some products may be removed from the market following an acquisition, all the constructor functions described above have a ‘subset’ argument that allows users to specify a length-n logical vector that equals TRUE if a product should be included in the post-merger simulation and FALSE otherwise. By default, ‘subset’ is equal to a length-n vector of TRUEs.

Measuring Changes In Consumer Welfare

All of the demand models included in antitrust’s Bertrand model have a CV method which may be used to calculate compensating variation. Compensating variation is the amount of money needed to make a consumer as well off as they were before the merger increased prices. Table @ref(tab:cvFormula) lists the formula for calculating compensating variation for the demand models included in antitrust. The last column in this table indicates whether the formula for compensating variation returns compensating variation in levels (e.g. dollars) or as a percent of the representative consumer’s total income.³⁵

Compensating variation can be calculated only if i) the demand system is consistent with both consumer choice theory as well as the Bertrand model described above and ii) all the demand parameters can be estimated. As discussed earlier, the parameter restrictions necessary for the Log-linear demand system to satisfy consumer choice theory will typically not satisfy the parameter restrictions implied by the Bertrand model. Consequently, there is no CV method defined for the Log-linear demand system. Similarly, the CV method returns an error if Linear demand is calibrated without imposing symmetry and homogeneity of degree 0 in prices on the matrix of slope coefficients B (i.e. setting ‘symmetry’ equal to FALSE). Lastly, the CV method for LA-AIDS demand will return an error if LA-AIDS demand is calibrated without prices. This occurs because prices are needed to uncover estimates of the LA-AIDS demand intercepts, which are needed to compute compensating variation.

Finally, it is worth noting that since none of the demand models included in antitrust contain income effects, it can be shown that compensating variation equals two other measures of consumer welfare: equivalent variation and consumer surplus.³⁶

Defining Antitrust Markets

According to the 2010 Horizontal Merger Guidelines issued by the U.S Department of Justice (DOJ) and the Federal Trade Commission (FTC), the purpose of market definition is twofold:

First, market definition helps specify the line of commerce and section of the country in which the competitive concern arises. In any merger enforcement action, the Agencies will normally identify one or more relevant markets in which the merger may substantially lessen competition. Second, market definition allows the Agencies to identify market participants and measure market shares and market concentration. (U.S Department of Justice and the Federal Trade Commission 2010, 7)

To assist users in identifying antitrust product and geographic markets, antitrust includes the HypoMonTest method. HypoMonTest assumes that i) firms are playing the differentiated Bertrand pricing game described earlier and ii) consumer demand is characterized by one of the demand systems described earlier, and then performs an implementation of the Hypothetical Monopolist Test described in the Guidelines for a set of products specified in HypoMonTest‘s ’prodIndex’ argument ³⁷

Specifically, HypoMonTest first determines if ‘prodIndex’ contains at least one of the merging parties’ products. If so, then by default HypoMonTest calls the calcPriceDeltaHypoMon method to find the profit-maximizing prices that the Hypothetical Monopolist would set on the products in ‘prodIndex’, holding the prices of all other products fixed at (predicted) pre-merger levels. HypoMonTest then compares the largest price change across the merging parties’ products indexed in ‘prodIndex’ to the specified ‘ssnip’. If this price change is greater than the specified ‘ssnip’, HypoMonTest returns TRUE. Otherwise, HypoMonTest returns FALSE.

The Guidelines state that

… if the market includes a second product, the Agencies will normally also include a third product if that third product is a closer substitute for the first product than is the second product. The third product is a closer substitute if, in response to a SSNIP on the first product, greater revenues are diverted to the third product than the second product (U.S Department of Justice and the Federal Trade Commission 2010, 9).

To facilitate such comparisons, antitrust‘s Bertrand model includes the diversionHypoMon method, which, for a set of products specified using the ’prodIndex’ argument, returns the revenue diversion (as defined by equation @ref(eq:divrev))³⁸ matrix for all products included in the merger simulation (i.e. all products placed under the Hypothetical Monopolist’s control as well as those outside of its control).

Simulating Merger Effects With Known Demand Parameters

Until now, most of the discussion has focused on how to recover demand parameters when users have information on shares, margins, and in most cases, prices. To accommodate known demand parameters (e.g. there is sufficient data to employ econometric methods to estimate demand parameters), antitrust contains the sim function. The sim function allows users to simulate price effects (or the output from any method listed in Table @ref(tab:selectMethods)) from a merger under the assumption that firms are playing a Bertrand differentiated pricing game. sim requires users to specify a vector of market prices, demand form (either “Linear”, “AIDS”, “LogLin”, “Logit”, “CES”, “LogitNests”, “LogitCap”, or “CESNests”), a list containing the known demand parameters, and pre- and post-merger ownership information. See the sim help page for further details.

Market Definition

As discussed above, HypoMon method is a post-simulation command and therefore is run only after the user has assumed i) which firms (and products) are playing a differentiated Bertrand pricing game, and ii) the demand system. As a result, HypoMon and diversionHypoMon can never be applied to a set of products that includes any product excluded from the merger simulation.

Log-Linear Demand

loglinear always predicts no price effects for single-product firms in the market who are not party to the acquisition. This occurs because the FOC for a single-product firm producing i is $m_i=\frac{1}{\epsilon_{ii}}=\frac{1}{\beta_{ii}}$. Since the Bertrand model described earlier assumes constant marginal costs, a constant ϵ_ii implies that prices are constant as well.

Although a single-product non-merging party’s prices will not change, its output will increase. This occurs because the acquisition will increase the price of the merging parties’ products as well as the prices of multi-product non-merging parties. These price increases will entice some customers to switch towards the single-product firms’ products, increasing their output.

LA-AIDS Demand

As discussed earlier, aids attempts to calibrate ϵ, the market elasticity parameter, by using the FOCs, LA-AIDS demand, and additional margins. For some combinations of margins and shares, however, this procedure can yield a very large market elasticity estimate, which in turn will yield small price effects from the merger. This issue appears to occur because the set of FOCs that are being used to calibrate this parameter are nearly colinear. Issues arising from colinearity can be often be remedied by supplying additional margin information, supplying margin information for products with disparate market shares, or supplying the market elasticity parameter directly.

The Mathematical Model

Firm k ∈ K chooses product output at each plant $\{q_j^r\}_{\substack{j\in j_k,\\ r\in n_k}}$ so as to maximize profits. Mathematically, firm k solves:

subject to

where p_j, the price sold of product i, is assumed to be a twice differentiable function of all firm quantities with $\frac{\partial p_j}{\partial q_j^r}< 0$ for all plants and firms. Likewise, additively separable plant variable costs, c^r, assumed to be twice differentiable with $\frac{\partial c^r}{\partial q_j^r}> 0$ .

Differentiating profits with respect to each q_j^r yields the following first order conditions (FOCs):

Summarizing Results

The summary method may be used to summarize the results of a merger between two firms. By default, the summary method reports pre- and post-merger equilibrium prices and quantities for each product included in the simulation. Also reported is the compensating marginal cost reduction (CMCR) as well as the change in consumer surplus from the merger. Setting the ‘market’ argument equal to FALSE returns plant-level equilibrium price effects.

Simulating Price Effects With Marginal Cost Efficiencies

Absent efficiencies, the Cournot model with the demand systems described here will almost always produce a (possibly negligible) post-merger price increase among substitutes. These price increases, however, can be offset by merger-specific efficiencies that decrease the incremental costs of some of the merging firms’ products.⁴⁰

All the functions discussed above allow users to evaluate these efficiencies in two different ways. First, all of these functions contain the ‘mcDelta’ argument, which allows users to specify the proportional change in a product’s marginal costs that may result from a merger. These cost changes are factored into the post-merger price equilibrium calculation made by the calcPrices method. If changes in marginal costs are not expected to be proportional, then users can instead set the ‘mcfunPre’ and ‘mcfunPost’ arguments equal to lists containing functions that return each plant’s pre- and post-merger marginal costs.⁴¹

Second, users can call the cmcr method on the output of any of the functions described above. This method computes the compensating marginal cost reduction (CMCR) on the merging parties’ products. CMCR is the percentage decrease in the marginal costs of the merging parties’ products necessary to prevent a post-merger price increase. See the cmcr help page for further details.

Simulating Price Effects With Capacity Constraints

The Cournot model allows users to impose capacity constraints that may potentially change as a result of the merger. The “capacitiesPre” argument allows users to specify a vector of pre-merger plant-specific capacity constraints (default is Inf, or no constraint). The “capacitiesPost” argument allows users to specify a vector of post-merger constraints (default is “capacitiesPre”).

Excluding Products or Plants

By default, the Cournot model calculates a merger’s effects under the assumption that the acquisition does not change the set of products available to consumers or the set of plants that are in production. A merger’s effects, however, may differ if the merger induces either the merging parties or another market participant to eliminate some products from their portfolio or to alter the mix of plants used in producing their products.

To accommodate the possibility that some products or plants may be removed from the market following an acquisition, all the constructor functions described above have a ‘productsPre’ and a ‘productsPost’ arguments that allows users to specify an n by J logical matrix which equals TRUE if a product produced at a particular plant should be included in the merger simulation and FALSE otherwise. By default, ‘productsPre’ equals TRUE if the ‘quantities’ is not NA, and FALSE otherwise, while ‘productsPost’ equals ‘productsPre’.

Measuring Changes In Consumer Welfare

The Cournot model has a CV method which may be used to calculate the change in consumer surplus from a merger.

Allowing For First-Mover Advantage

The package contains a stackelberg function that is similar to cournot, but also allows for one or more firms to be designated as output leaders for certain products. Output leaders are assumed to set their output levels first, and conditional on the leaders’ output decisions, output followers (defined as everyone who is not a leader) choose their levels of output. The stackelberg function contains four additional arguments: ‘isLeaderPre’ and ‘isLeaderPost’ for setting which firms are leaders and which are followers, and ‘dmcfunPre’ and ‘dmcfunPost’ for specifying the derivative of a plant’s marginal cost (only necessary when marginal and variable costs are also specified).

The Game

Suppose that a buyer is interested in either purchasing a single unit of a homogeneous product from one of the K firms who supply the product, or supplying the product herself. Although the product is homogeneous, the cost of producing it is not: firm k ∈ K uses constant marginal cost technology c_k ≥ 0 to produce the product.⁴⁴ Likewise, suppose that the cost to the buyer of self-supply is c₀. Moreover, marginal costs are assumed to be private information, with the buyer and each firm believing that firm costs are independently drawn from the distribution F(c) with support $(\underline{c},\overline{c})$. Although marginal costs are constant, firms are assumed to be capacity constrained in the sense that all buyers and the bidder believe that a firm endowed with capacity t_k ≥ 0 has a marginal cost c_k equal to the minimum of t_k cost draws from F(c). Define t = (t₁, …, t_K) as the vector of firm capacities and $\bar{t}=\sum\limits_{k\in K}t_k$ as total industry capacity.

The buyer is assumed to employ a 2nd price auction with a reserve price to determine who will supply the product. In this auction format, each firm submits a bid and the firm with the lowest bid that is also less than the buyer’s reserve price wins the auction but is paid the 2nd lowest bid.⁴⁵ The buyer’s reserve reflects the fact that the buyer has the ability to self-supply at cost c₀, and so is only willing to purchase the product from a bidder whose bid is less than c₀.

It can be shown that in a 2nd price auction, each firm has a weakly dominant strategy to bid its marginal cost. Waehrer and Perry (2003) also show that

• the probability that firm k’s cost draw is less than c, equals

• the probability that firm k wins the auction for a given reserve price r, is

• the probability that firm k wins conditional on I, the event that some firm wins, is

• firm k’s expected producer surplus is

• firm k’s expected cost is

• firm k’s expected price is

The buyer sets her reserve price r so as to minimize her expected costs (EC):

where p(r, t) is the expected price paid by the buyer conditional on the buyer purchasing from some firm, i.e. which is minimized at

Note that this last equation implies that the buyer has an incentive to set its optimal reserve below its self-supply cost.

Calibrating Model Parameters

In this model, we will assume that the user observes firm capacities, some price and margin information, and possibly the auction reserve price or the proportion of times a buyer opts to self-supply. The unknowns are the seller cost distribution F(c), the buyer’s cost of self-supply c₀, and if not observed, the auction’s reserve price.

Simulating Merger Effects (experimental)

The Game

Suppose that a buyer is interested in either purchasing a single unit of a differentiated product from one of K firms, each of whom manufacture J_k, k ∈ K variants of the product, or in supplying the product herself. Let $J=\bigcup_k^KJ_k$ denote the set of all products produced by any of the K firms. Not only do each of the variants have different characteristics, the cost of producing these products may also differ. In particular, suppose that variant J_k, k ∈ K uses constant marginal cost technology c_Jk ≥ 0 to produce the product.⁴⁶

Let the utility that a buyer receives from product j ∈ J with offer p_j equal to

where $\overline{V}_j$ represents an index of “vertical” quality differentiation that is decomposed into a non-offer component δ_j and an offer component αp_j. In addition, ϵ_j represents a buyer-specific idiosyncratic shock to utility (i.e. “horizontal” quality differentiation). Without loss of generality, we will assume that these shocks are independently drawn from the distribution F, with mean 0 and variance 1.⁴⁷ We will also assume that at the start of the auction, ϵ_j is known to j’s manufacturer but is not known to any other sellers.

The buyer is assumed to employ a 2nd score auction to determine which product she will purchase. In this auction format, each firm submits an offer and the firm with the highest score (which may be the buyer if the buyer self-supplies) wins the auction but pays the difference between the value of the highest scoring product and the value of the next-highest scoring product, less that seller’s offer on that product. It can be shown that it is a dominant strategy for each firm to offer the product with the highest surplus in its portfolio to the buyer at cost.

Let z_A = max_k ∈ AV_k(c_k) denote the the maximum surplus available from any product k ∈ A ⊆ J. It can be shown that the following must hold when firm k ∈ K wins with variant j ∈ J_k

• the probability that j wins is

• the expected price conditional on j winning is

• subtracting c_j from both sides of @ref(eq:auc22) yields the expected profit margin conditional on j winning:

Calibrating Model Demand and Cost Parameters

For this model, we will assume that users observe firm shares, prices, and some margin information. From the user’s perspective, the unknowns are the buyers’ distribution of valuations (F), the constant marginal costs (c_j) and the components of a buyer’s mean valuation (i.e. the δ_js and α).

In order to calibrate this model with the limited information available, we will need to assume a functional form for F. Currently, antitrust assumes that F follows a Gumbel distribution. Under this distributional assumption,

• the probability that j wins is

• the expected value of the maximum for products in A ⊆ J is

First, note that for product h ≠ j, equation @ref(eq:auc24) indicates

Equation @ref(eq:auc24) implies that at least one of the V_js is not separately identified. Therefore, one product must be selected as the numeraire so that all the other valuations are relative to the normalized product.⁴⁹

Next, substituting equation @ref(eq:auc25) into equation @ref(eq:auc23) yields the following closed form expression for margins:

Equations @ref(eq:auc26) and @ref(eq:auc27) form the basis of the calibration strategy. All the model parameters may be calibrated with just a single margin and market shares. Product prices only need to to be supplied for products whose marginal costs are assumed to change post-merger (i.e. products for which ‘mcDelta’ is not equal to 0).

For example, suppose that the margin for product j ∈ J_k is known. Equation @ref(eq:auc27) implies that α may be recovered using only j’s margin as well as the sum of the shares for all of firm j’s products. This estimate of α, in conjunction with equation @ref(eq:auc27) may then be used to estimate margins for all the other products in the market. Absent post-merger cost changes, these margin estimates, along with equation @ref(eq:auc26), may then be used to estimate each products’ non-offer mean valuations (δs).

When post-merger cost changes are included in the model, it becomes necessary to estimate marginal costs, but only for the products whose costs are changing due to the merger.⁵⁰ This is accomplished by including price information for these products. Products whose costs are not assumed to change do not require pricing data (i.e. ‘prices’ may be NA for products where ‘mcDelta’ is 0).

In some instances, there may be more than one margin available. In these cases a minimum distance algorithm is used to find the value of α that best satisfies all the margin equations @ref(eq:auc27) for which there is data.

Simulating Merger Effects

The Incentive to Collude Under Grim Trigger

In this model, Firm z has an incentive to collude with the other firms in the Z coalition only if the discounted value of the surplus it gains from colluding is greater than the discounted value of the surplus it receives from defecting from coalition prices for a single period, followed by the discounted value of surplus it receives from all firms earning surplus from the Bertrand game for all subsequent periods. Mathematically, this condition can be written as:

Since we are assuming that outside of the merger firms cannot enter or exit, or add or remove products, equilibrium prices and quantities are the same across all periods, and the above simplifies to