Auction2ndLogit-Functions {antitrust}

2nd Score Procurement Auction Model with (Nested) Logit Demand

Description

Calibrates consumer demand using (Nested) Logit and then simulates the price effect of a merger between two firms under the assumption that all firms in the market are playing a differentiated products 2nd score auction game.

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

Usage

auction2nd.logit(
  prices,
  shares,
  margins,
  ownerPre,
  ownerPost,
  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 = "")
)

auction2nd.logit.nests(
  prices,
  shares,
  margins,
  nests,
  diversions,
  ownerPre,
  ownerPost,
  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,
  parmsStart,
  constraint = TRUE,
  control.slopes,
  labels = paste("Prod", 1:length(prices), sep = "")
)

auction2nd.logit.alm(
  prices,
  shares,
  margins,
  ownerPre,
  ownerPost,
  mktElast = NA_real_,
  insideSize = NA_real_,
  mcDelta = rep(0, length(prices)),
  subset = rep(TRUE, length(prices)),
  mcDeltaOutside = 0,
  parmsStart,
  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.

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.

insideSize

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

mcDeltaOutside

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

control.slopes

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

labels

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

nests

A length k factor of product nests.

diversions

A k x k matrix of diversion ratios with diagonal elements equal to -1. Default is missing, in which case diversion according to share is assumed.

parmsStart

For auction2nd.logit.alm, a length 2 vector of starting values used to solve for price coefficient and the share of the outside good. The first element should always be the price coefficient and the second should be the outside good. For auction2nd.logit.nests, a length n+1 vector of starting values used to solve for price coefficient and the nesting parameters. The first element should always be the price coefficient and the remaining elements should be the nesting parameters.

constraint

if TRUE, then the nesting parameters for all non-singleton nests are assumed equal. If FALSE, then each non-singleton nest is permitted to have its own value. Default is TRUE.

mktElast

a negative value indicating market elasticity. Default is NA.

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 2nd score procurement auction.

auction2nd.logit.nests is identical to auction2nd.logit except that it assumes that products can be grouped into nests. Additional margin information is needed to identify th nesting parameters.

auction2nd.logit.alm is identical to auction2nd.logit except that it assumes that an outside product exists and uses additional margin information to estimate the share of the outside good.

Value

auction2nd.logit returns an instance of Auction2ndLogit, a child class of Logit. auction2nd.logit.nests returns an instance of Auction2ndLogitNests. auction2nd.logit returns an instance of Auction2ndLogitALM.

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,logit.nests for simulating mergers under a Nash-Bertrand pricing game 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)

price = c(35.53,  154, 84.08, 53.16)*1e3
cost  =  c(NA, 101, NA, NA)*1e3

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

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

margin = price - cost

result.2nd <- auction2nd.logit(price,share,margin,
                           ownerPre=ownerPre,ownerPost=ownerPost,normIndex=2)


print(result.2nd)
summary(result.2nd,revenue=FALSE)

##re-run without any price information except Firm 2

price <- rep(NA_real_, length(price))

result.noprice <- auction2nd.logit(price,share,margin,
                                   ownerPre=ownerPre,ownerPost=ownerPost,normIndex=2)

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

##changing the units of prices and margins can yield dramatically different results 

price = c(35.53,  154, 84.08, 53.16)
cost  =  c(NA, 101, NA, NA)
margin <- price - cost

result.units <- auction2nd.logit(price,share,margin,
                                   ownerPre=ownerPre,ownerPost=ownerPost,normIndex=2)

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

## Get a detailed description of the 'Auction2ndLogit' class slots
showClass("Auction2ndLogit")

## Show all methods attached to the 'Auction2ndLogit' Class
showMethods(classes="Auction2ndLogit")