BargainingLogit-Functions {antitrust}

Nash Bargaining Model with Logit Demand

Description

Calibrates consumer demand using Logit and then simulates the price effect of a merger between two firms under the assumption that firms and customers in the market are playing a differentiated products Nash Bargaining game.

Let k denote the number of products produced by all firms playing the Nash Bargaining game below.

Usage

bargaining.logit(
  prices,
  shares,
  margins,
  ownerPre,
  ownerPost,
  bargpowerPre = rep(0.5, length(prices)),
  bargpowerPost = bargpowerPre,
  normIndex = ifelse(isTRUE(all.equal(sum(shares), 1, check.names = FALSE)), 1, NA),
  mcDelta = rep(0, length(prices)),
  subset = rep(TRUE, length(prices)),
  priceStart = prices,
  insideSize = NA_real_,
  priceOutside = 0,
  control.slopes,
  control.equ,
  labels = paste("Prod", 1:length(prices), sep = "")
)

bargaining2nd.logit(
  prices,
  shares,
  margins,
  ownerPre,
  ownerPost,
  bargpowerPre = rep(0.5, length(prices)),
  bargpowerPost = bargpowerPre,
  normIndex = ifelse(isTRUE(all.equal(sum(shares), 1, check.names = FALSE)), 1, NA),
  mcDelta = rep(0, length(prices)),
  subset = rep(TRUE, length(prices)),
  insideSize = NA_real_,
  mcDeltaOutside = 0,
  control.slopes,
  labels = paste("Prod", 1:length(prices), sep = "")
)

Arguments

prices

A length k vector of product prices.

shares

A length k vector of product (quantity) shares. Values must be between 0 and 1.

margins

A length k vector of product margins (in levels, not percents), some of which may equal NA.

ownerPre

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

ownerPost

EITHER a vector of length k whose values indicate which firm produced a product after the merger OR a k x k matrix of post-merger ownership shares.

bargpowerPre

A length k vector of pre-merger bargaining power parameters. Values must be between 0 (sellers have the power) and 1 (buyers the power). NA values are allowed, though must be calibrated from additional margin and share data. Default is 0.5.

bargpowerPost

A length k vector of post-merger bargaining power parameters. Values must be between 0 (sellers have the power) and 1 (buyers the power). NA values are allowed, though must be calibrated from additional margin and share data. Default is ‘bargpowerPre’.

normIndex

An integer equalling the index (position) of the inside product whose mean valuation will be normalized to 1. Default is 1, unless ‘shares’ sum to less than 1, in which case the default is NA and an outside good is assumed to exist.

mcDelta

A vector of length k where each element equals the (level) 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.

subset

A vector of length k where each element equals TRUE if the product indexed by that element should be included in the post-merger simulation and FALSE if it should be excluded.Default is a length k vector of TRUE.

priceStart

A vector of length k who elements equal to an initial guess of equilibrium prices. default is ‘prices’.

insideSize

An integer equal to total pre-merger units sold. If shares sum to one, this also equals the size of the market.

priceOutside

A positive real number equal to the price of the outside good. Default equals 0 for Logit demand.

control.slopes

A list of optim control parameters passed to the calibration routine optimizer (typically the calcSlopes method).

control.equ

A list of BBsolve control parameters passed to the non-linear equation solver (typically the calcPrices method).

labels

A k-length vector of labels. Default is "Prod#", where ‘#’ is a number between 1 and the length of ‘prices’.

mcDeltaOutside

A length 1 vector indicating the change in the marginal cost of the outside good. Default is 0.

Details

Using product prices, quantity shares and all of the product margins from at least one firm, auction2nd.logit is able to recover the price coefficient and product mean valuations in a Logit demand model. auction2nd.logit then uses these calibrated parameters to simulate a merger between two firms, under the assumption that firms are participating in a A Nash Bargaining Game (bargaining.logit) or splitting the full surplus (bargaining2nd.logit).

Value

bargaining.logit returns an instance of BargainingLogit, a child class of Logit. bargaining2nd.logit returns an instance of Bargaining2ndLogit, a child class of Auction2ndLogit.

Author(s)

Charles Taragin [e-mail]

References

Miller, Nathan (2014). “Modeling the effects of mergers in procurement” International Journal of Industrial Organization , 37, pp. 201-208.

See Also

logit for simulating mergers under a Nash-Bertrand pricing game with Logit demand, and ,auction2nd.logit for simulating mergers under a 2nd score auction with Logit demand.

Examples

## Calibration and simulation results from a merger between firms 2 and 3
## of a 4-firm market
## Source: Miller 2014 backup materials http://www.nathanhmiller.org/research

share = c(0.29,0.40,0.28,0.03)
bargpower <- rep(0.6,4) # buyer has advantage
price = c(35.53,  154, 84.08, 53.16)
cost  =  c(NA, 101, NA, NA)

ownerPre <- ownerPost <- diag(length(share))

#Suppose products 2 and 3 merge
ownerPost[2,3] <- ownerPost[3,2] <- 1

margin = (price - cost)/price

result.barg <- bargaining.logit(price,share,margin,bargpowerPre=bargpower,
                           ownerPre=ownerPre,ownerPost=ownerPost,normIndex=2)


print(result.barg)
summary(result.barg,revenue=FALSE)


## Get a detailed description of the 'BargainingLogit' class slots
showClass("BargainingLogit")

## Show all methods attached to the 'BargainingLogit' Class
showMethods(classes="BargainingLogit")