AIDS-Functions: (Nested) AIDS Calibration and Merger Simulation

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(Nested) AIDS Calibration and Merger Simulation

Description

Calibrates consumer demand using (nested) AIDS 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 Bertrand game.

Below let k denote the number of products produced by all firms.

Usage

aids(
  shares,
  margins,
  prices,
  diversions,
  ownerPre,
  ownerPost,
  mktElast = NA_real_,
  insideSize = NA_real_,
  mcDelta = rep(0, length(shares)),
  subset = rep(TRUE, length(shares)),
  parmStart = rep(NA_real_, 2),
  priceStart = runif(length(shares)),
  isMax = FALSE,
  control.slopes,
  control.equ,
  labels = paste("Prod", 1:length(shares), sep = ""),
  ...
)

pcaids(
  shares,
  knownElast,
  mktElast = -1,
  prices,
  diversions,
  ownerPre,
  ownerPost,
  knownElastIndex = 1,
  insideSize = NA_real_,
  mcDelta = rep(0, length(shares)),
  subset = rep(TRUE, length(shares)),
  priceStart = runif(length(shares)),
  isMax = FALSE,
  control.slopes,
  control.equ,
  labels = paste("Prod", 1:length(shares), sep = ""),
  ...
)

pcaids.nests(
  shares,
  margins,
  knownElast,
  mktElast = -1,
  prices,
  ownerPre,
  ownerPost,
  nests = rep(1, length(shares)),
  knownElastIndex = 1,
  insideSize = NA_real_,
  mcDelta = rep(0, length(shares)),
  subset = rep(TRUE, length(shares)),
  priceStart = runif(length(shares)),
  isMax = FALSE,
  nestsParmStart,
  control.slopes,
  control.equ,
  labels = paste("Prod", 1:length(shares), sep = ""),
  ...
)

Arguments

shares	A length k vector of product revenue shares. All shares must be between 0 and 1.
margins	A length k vector of product margins. All margins must be either be between 0 and 1, or NA.
prices	A length k vector product prices. Default is missing, in which case demand intercepts are not calibrated.
diversions	A k x k matrix of diversion ratios with diagonal elements equal to -1. Default is missing, in which case diversion according to revenue share is assumed.
ownerPre	EITHER a vector of length k whose values indicate which firm produced a product before the 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.
mktElast	A negative number equal to the industry pre-merger price elasticity. Default is NA for aids and -1 for pcaids.
insideSize	Total expenditure (revenues) on products included in the simulation.
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.
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.
parmStart	aids only. A vector of length 2 who elements equal to an initial guess for "known" element of the diagonal of the demand matrix and the market elasticity.
priceStart	A vector of length k who elements equal to an initial guess of the proportional change in price caused by the merger. The default is to draw k random elements from a [0,1] uniform distribution.
isMax	If TRUE, checks to see whether computed price equilibrium locally maximizes firm profits and returns a warning if not. Default is FALSE.
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.
...	Additional options to feed to the BBsolve optimizer used to solve for equilibrium prices.
knownElast	A negative number equal to the pre-merger own-price elasticity for any of the k products.
knownElastIndex	An integer equal to the position of the ‘knownElast’ product in the ‘shares’ vector. Default is 1, which assumes that the own-price elasticity of the first product is known.
nests	A length k vector identifying which nest a product belongs to. Default is that all products belong to a single nest.
nestsParmStart	A vector of starting values used to solve for price coefficient and nest parameters. If missing then the random draws with the appropriate restrictions are employed.

Details

Using product market revenue shares and all of the product product margins from at least two firms, aids is able to recover the slopes in a proportionally calibrated Almost Ideal Demand System (AIDS) without income effects. aids then uses these slopes to simulate the price effects of a merger between two firms under the assumption that all firms in the market are playing a differentiated Bertrand pricing game.

If prices are also supplied, aids is able to recover the intercepts from the AIDS demand system. Intercepts are helpful because they can be used to simulate pre- and post-merger price levels as well as price changes. Whatsmore, the intercepts are necessary in order to calculate compensating variation.

aids assumes that diversion between the products in the market occurs according to revenue share. This assumption may be relaxed by setting ‘diversions’ equal to a k x k matrix of diversion ratios. The diagonal of this matrix must equal -1, the off-diagonal elements must be between 0 and 1, and the rows must sum to 1.

pcaids is almost identical to aids, but instead of assuming that at least two margins are known, pcaids assumes that the own-price elasticity of any single product, and the industry-wide own-price elasticity, are known. Demand intercepts cannot be recovered using pcaids.

pcaids.nests extends pcaids by allowing products to be grouped into nests. Although products within the same nest still have the independence of irrelevant alternatives (IIA) property, products in different nests do not. Note that the ‘diversions’ argument is absent from pcaids.nests.

pcaids.nests assumes that the share diversion between nests is symmetric (i.e for 2 nests A and B, the diversion from A to B is the same as B to A). Therefore, if there are w nests, 2 <= w <=k, then the model must estimate w(w-1)/2 distinct nesting parameters. To accomplish this, pcaids.nests uses margin information to produce estimates of the nesting parameters. It is important to note that the number of supplied margins must be at least as great as the number of nesting parameters in order for PCAIDS to work.

The nesting parameters are constrained to be between 0 and 1. Therefore, one way to test the validity of the nesting structure is to check whether the nesting parameters are between 0 and 1. The value of the nesting parameters may be obtained from calling either the ‘summary’ or ‘getNestsParms’ functions.

Value

aids returns an instance of class AIDS, a child class of Linear. pcaids returns an instance of class PCAIDS, while pcaids.nests returns an instance of PCAIDSNests. Both are children of the AIDS class.

Author(s)

Charles Taragin [e-mail]

References

Epstein, Roy and Rubinfeld, Daniel (2004). “Merger Simulation with Brand-Level Margin Data: Extending PCAIDS with Nests.” The B.E. Journal of Economic Analysis and Policy, advances.4(1), pp. 2.

Epstein, Roy and Rubinfeld, Daniel (2004). “Effects of Mergers Involving Differentiated Products.”

See Also

linear for a demand system based on quantities rather than revenue shares.

Examples

## Simulate a merger between two single-product firms A and B in a
## three-firm market (A, B, C). This example assumes that the merger is between
## the firms A and B and that A's own-price elasticity is
## known.
## Source: Epstein and Rubinfeld (2004), pg 9, Table 2.


prices    <- c(2.9,3.4,2.2) ## optional for aids, unnecessary for pcaids
shares    <- c(.2,.3,.5)


## The following are used by aids but not pcaids
## only two of the margins are required to calibrate the demand parameters
margins <- c(0.33, 0.36, 0.44)

## The following are used by pcaids, but not aids
knownElast<- -3
mktElast  <- -1


## Define ownership using a vector of firm identities
ownerPre <- c("A","B","C")
ownerPost <- c("A","A","C")

## Alternatively, ownership could be defined using matrices
#ownerPre=diag(1,length(shares))
#ownerPost=ownerPre
#ownerPost[1,2] <- ownerPost[2,1] <- 1


## AIDS: the following assumes both prices and margins are known.
##       Prices are not needed to estimate price changes



result.aids <- aids(shares,margins,prices,ownerPre=ownerPre,ownerPost=ownerPost,labels=ownerPre)



print(result.aids)           # return predicted price change
summary(result.aids)         # summarize merger simulation

elast(result.aids,TRUE)      # returns premerger elasticities
elast(result.aids,FALSE)     # returns postmerger elasticities

diversion(result.aids,TRUE)  # return premerger diversion ratios
diversion(result.aids,FALSE) # return postmerger diversion ratios


cmcr(result.aids)            #calculate compensating marginal cost reduction
upp(result.aids)             #calculate Upwards Pricing Pressure Index


## Implement the Hypothetical Monopolist Test
## for products A and B using a 5\% SSNIP

HypoMonTest(result.aids,prodIndex=1:2)


CV(result.aids)        #calculate compensating variation as a percent of
#total expenditure
#CV can only be calculated if prices are supplied


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

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

## Show which class have their own 'elast' method
showMethods("elast")

## Show the method definition for 'elast' and Class 'AIDS'
getMethod("elast","AIDS")



## PCAIDS: the following assumes that only one product's elasticity is
##       known as well as the market elasticity.



result.pcaids <- pcaids(shares,knownElast,mktElast,
                        ownerPre=ownerPre,ownerPost=ownerPost,
                        labels=ownerPre)

print(result.pcaids)           # return predicted price change
summary(result.pcaids)         # summarize merger simulation

elast(result.pcaids,TRUE)      # returns premerger elasticities
elast(result.pcaids,FALSE)     # returns postmerger elasticities

diversion(result.pcaids,TRUE)  # return premerger diversion ratios
diversion(result.pcaids,FALSE) # return postmerger diversion ratios


cmcr(result.pcaids)            #calculate compensating marginal cost reduction


## Implement the Hypothetical Monopolist Test
## for products A and B using a 5\% SSNIP

HypoMonTest(result.pcaids,prodIndex=1:2)



## Nested PCAIDS: in addition to the PCAIDS information requirements,
##                users must supply the nesting structure as well as margin infromation.

nests <- c('H','L','L') # product A assigned to nest H, products B and C assigned to nest L


result.pcaids.nests <- pcaids.nests(shares,knownElast,mktElast,margins=margins,
                                    nests=nests,ownerPre=ownerPre,
                                    ownerPost=ownerPost,labels=ownerPre)

antitrust documentation built on Aug. 24, 2022, 9:05 a.m.