CMCRBertrand-Functions {antitrust}

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.

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

Usage

cmcr.bertrand(
  prices,
  margins,
  diversions,
  ownerPre,
  ownerPost = matrix(1, ncol = length(prices), nrow = length(prices)),
  rel = c("cost", "price"),
  labels = names(prices)
)

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

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.

rel

A length 1 character vector indicating whether CMCR should be calculated relative to pre-merger cost (“cost”) or pre-merger price (“price”), Default is “cost”.

labels

A length-k vector of product labels.

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.

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.

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 the generalized pricing pressure (GePP) for each of the merging's parties' products, net any efficiency claims. GePP is a generalization of Upwards Pricing Pressure (UPP) that accomodates multi-product firms.

Author(s)

Charles Taragin [e-mail]

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))