antitrust (version 0.9)

cmcr.bertrand: Compensating Marginal Cost Reductions and Upwards Pricing Pressure (Bertrand)

Description

Calculate the marginal cost reductions necessary to restore premerger prices (CMCR), or the net Upwards Pricing Pressure (UPP) in a merger involving firms playing a differentiated products Bertrand pricing game.

Usage

cmcr.bertrand(prices, margins,  diversions, ownerPre,
              ownerPost=matrix(1,ncol=length(prices), nrow=length(prices)),
              labels=paste("Prod",1:length(prices),sep=""))


upp.bertrand(prices, margins, diversions, ownerPre,
             ownerPost=matrix(1,ncol=length(prices), nrow=length(prices)),
             mcDelta=rep(0,length(prices)),
             labels=paste("Prod",1:length(prices),sep=""))

Arguments

Let k denote the number of products produced by the merging parties.

prices

A length-k vector of product prices.

margins

A length-k vector of product margins.

diversions

A k x k matrix of diversion ratios with diagonal elements equal to -1.

ownerPre

EITHER a vector of length k whose values indicate which of the merging parties produced a product pre-merger OR a k x k matrix of pre-merger ownership shares.

ownerPost

A k x k matrix of post-merger ownership shares. Default is a k x k matrix of 1s.

mcDelta

A vector of length k where each element equals the proportional change in a product's marginal costs due to the merger. Default is 0, which assumes that the merger does not affect any products' marginal cost.

labels

A length-k vector of product labels.

Value

• cmcr.bertrand returns a length-k vector whose values equal the percentage change in each products' marginal costs that the merged firms must achieve in order to offset a price increase. upp.bertrand returns a length-k vector whose values equal upwards pricing pressure for each of the merging's parties' products , net any efficiency claims.

Details

All `prices' elements must be positive, all `margins' elements must be between 0 and 1, and all `diversions' elements must be between 0 and 1 in absolute value. In addition, off-diagonal elements (i,j) of `diversions' must equal an estimate of the diversion ratio from product i to product j (i.e. the estimated fraction of i's sales that go to j due to a small increase in i's price). Also, `diversions' elements are positive if i and j are substitutes and negative if i and j are complements. `ownerPre' will typically be a vector whose values equal 1 if a product is produced by firm 1 and 0 otherwise, though other values including firm name are acceptable. Optionally, `ownerPre' may be set equal to a matrix of the merging firms pre-merger ownership shares. These ownership shares must be between 0 and 1. `ownerPost' is an optional argument that should only be specified if one party to the acquisition is assuming partial control of the other party's assets. `ownerPost' elements must be between 0 and 1.

References

Farrell, Joseph and Shapiro, Carl (2010). ``Antitrust Evaluation of Horizontal Mergers: An Economic Alternative to Market Definition.'' The B.E. Journal of Theoretical Economics, 10(1), pp. 1-39. Jaffe, Sonia and Weyl Eric (2012). ``The First-Order Approach to Merger Analysis.'' SSRN eLibrary Werden, Gregory (1996). ``A Robust Test for Consumer Welfare Enhancing Mergers Among Sellers of Differentiated Products.'' The Journal of Industrial Economics, 44(4), pp. 409-413.

See Also

cmcr.cournot for a homogeneous products Cournot version of CMCR, and cmcr-methods for calculating CMCR and UPP after calibrating demand system parameters and simulating a merger.

Examples

## Let k_1 = 1 and and k_2 = 2 ##

    p1 = 50;      margin1 = .3
    p2 = c(45,70); margin2 = c(.4,.6)
    isOne=c(1,0,0)
    diversions = matrix(c(-1,.5,.01,.6,-1,.1,.02,.2,-1),ncol=3)

    cmcr.bertrand(c(p1,p2), c(margin1,margin2), diversions, isOne)
     upp.bertrand(c(p1,p2), c(margin1,margin2), diversions, isOne)


     ## Calculate the necessary percentage cost reductions for various margins and
     ## diversion ratios in a two-product merger where both products have
     ## equal prices and diversions (see Werden 1996, pg. 412, Table 1)


    margins = seq(.4,.7,.1)
    diversions = seq(.05,.25,.05)
    prices = rep(1,2) #assuming prices are equal, we can set product prices to 1
    isOne = c(1,0)
    result = matrix(ncol=length(margins),nrow=length(diversions),dimnames=list(diversions,margins))

    for(m in 1:length(margins)){
        for(d in 1:length(diversions)){

           dMatrix = -diag(2)
           dMatrix[2,1] <- dMatrix[1,2] <- diversions[d]

           firmMargins = rep(margins[m],2)

           result[d,m] = cmcr.bertrand(prices, firmMargins, dMatrix, isOne)[1]

    }}

    print(round(result,1))