Linear-Functions {antitrust}

Linear and Log-Linear Demand Calibration and Merger Simulation

Description

Calibrates consumer demand using either a linear or log-linear demand system and then simulates the prices effect of a merger between two firms under the assumption that all firms in the market are playing a differentiated products Bertrand game.

Let k denote the number of products produced by all firms.

Usage

linear(
  prices,
  quantities,
  margins,
  diversions,
  symmetry = TRUE,
  ownerPre,
  ownerPost,
  mcDelta = rep(0, length(prices)),
  subset = rep(TRUE, length(prices)),
  priceStart = prices,
  control.slopes,
  labels = paste("Prod", 1:length(prices), sep = ""),
  ...
)

loglinear(
  prices,
  quantities,
  margins,
  diversions,
  ownerPre,
  ownerPost,
  mcDelta = rep(0, length(prices)),
  subset = rep(TRUE, length(prices)),
  priceStart = prices,
  control.equ,
  labels = paste("Prod", 1:length(prices), sep = ""),
  ...
)

Arguments

prices

A length k vector product prices.

quantities

A length k vector of product quantities.

margins

A length k vector of product margins. All margins must be either be between 0 and 1, or NA.

diversions

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

symmetry

If TRUE, requires the matrix of demand slope coefficients to be symmetric and homogeneous of degree 0 in prices, both of which suffice to make demand consistent with utility maximization theory. Default is TRUE.

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.

mcDelta

A length k vector 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.

priceStart

A length k vector of prices used as the initial guess in the nonlinear equation solver. Default is ‘prices’.

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’.

...

Additional options to feed to the solver. See below.

control.equ

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

Details

Using price, quantity, and diversion information for all products in a market, as well as margin information for (at least) all the products of any firm, linear is able to recover the slopes and intercepts in a Linear demand system and then uses these demand parameters to simulate the price effects of a merger between two firms under the assumption that the firms are playing a differentiated Bertrand pricing game.

loglinear uses the same information as linear to uncover the slopes and intercepts in a Log-Linear demand system, and then uses these demand parameters to simulate the price effects of a merger of two firms under the assumption that the firms are playing a differentiated Bertrand pricing game.

‘diversions’ must be a square matrix whose off-diagonal elements [i,j] estimate 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). Off-diagonal elements are restricted to be non-negative (products are assumed to be substitutes), diagonal elements must equal -1, and rows must sum to 0 (negative if you wish to include an outside good) . If ‘diversions’ is missing, then diversion according to quantity share is assumed.

‘ownerPre’ and ‘ownerPost’ values will typically be equal to either 0 (element [i,j] is not commonly owned) or 1 (element [i,j] is commonly owned), though these matrices may take on any value between 0 and 1 to account for partial ownership.

Under linear demand, an analytic solution to the Bertrand pricing game exists. However, this solution can at times produce negative equilibrium quantities. To accommodate this issue, linear uses constrOptim to find equilibrium prices with non-negative quantities. ... may be used to change the default options for constrOptim.

loglinear uses the non-linear equation solver BBsolve to find equilibrium prices. ... may be used to change the default options for BBsolve.

Value

linear returns an instance of class Linear. loglinear returns an instance of LogLin, a child class of Linear.

Author(s)

Charles Taragin [e-mail]

References

von Haefen, Roger (2002). “A Complete Characterization Of The Linear, Log-Linear, And Semi-Log Incomplete Demand System Models.” Journal of Agricultural and Resource Economics, 27(02). doi: 10.22004/ag.econ.31118.

See Also

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

Examples

## Simulate a merger between two single-product firms in a
## three-firm market with linear demand with diversions
## that are proportional to shares.
## This example assumes that the merger is between
## the first two firms



n <- 3 #number of firms in market
price    <- c(2.9,3.4,2.2)
quantity <- c(650,998,1801)
margin <- c(.435,.417,.370)


#simulate merger between firms 1 and 2
owner.pre <- diag(n)
owner.post <- owner.pre
owner.post[1,2] <- owner.post[2,1] <- 1



result.linear <- linear(price,quantity,margin,ownerPre=owner.pre,ownerPost=owner.post)

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

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

diversion(result.linear,TRUE)  # returns premerger diversion ratios
diversion(result.linear,FALSE) # returns postmeger diversion ratios

cmcr(result.linear)            # returns the compensating marginal cost reduction

CV(result.linear)              # returns representative agent compensating variation


## Implement the Hypothetical Monopolist Test
## for products 1 and 2 using a 5\% SSNIP

#HypoMonTest(result.linear,prodIndex=1:2)


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

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

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

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