Package 'mergersim'

Nash bargaining calibration

Description

Nash bargaining calibration

Usage

bargain_calibrate(
  param,
  own,
  price,
  shares,
  cost,
  weight,
  lambda,
  includeMUI = TRUE
)

Arguments

param	Price coefficient alpha and mean values delta, parameters to calibrate
own	Ownership matrix
price	Price
shares	Observed market shares
cost	Marginal costs for each product
weight	Weighting matrix
lambda	Bargaining power of the buyer
includeMUI	logical; whether to include marginal utility of income in buyer's payoff, thereby translating dollars to utility. Default is True, interpreted as buyer maximizing utility. Setting equal to False would have interpretation that buyer maximizes profts.

Details

This function calculate the first-order conditions from a Bertrand price-setting model of competition

Value

The first-order conditions

Examples

alpha  <- -0.9
delta <- c(.81,.93,.82)
c_j <- c(.05,.31,.30)
own_pre = diag(3)
p0 <- c_j*1.1
share1 <- (exp(delta + alpha*p0))/(1+sum(exp(delta + alpha*p0)))
wt_matrix <- diag(c(1,1,1,1000,1000,1000))

bargain_calibrate(param = c(alpha,delta),own = own_pre,price = p0,
shares = share1,cost = c_j, weight = wt_matrix,
lambda = 0.5)

---

Nash bargaining first-order conditions

Description

Nash bargaining first-order conditions

Usage

bargain_foc(price, own, alpha, delta, cost, lambda, includeMUI = TRUE)

Arguments

price	Price
own	Ownership matrix
alpha	Price coefficient
delta	Mean values
cost	Marginal costs for each product
lambda	Bargaining power of the buyer
includeMUI	logical; whether to include marginal utility of income in buyer's payoff, thereby translating dollars to utility. Default is True, interpreted as buyer maximizing utility. Setting equal to False would have interpretation that buyer maximizes profts.

Details

This function calculate the first-order conditions from a Bertrand price-setting model of competition

Value

The first-order conditions

Examples

alpha  <- -0.9
delta <- c(.81,.93,.82)
c_j <- c(.05,.31,.30)
own_pre = diag(3)
p0 <- c_j*1.1

bertrand_foc(price = p0,
own = own_pre, alpha= alpha,
delta = delta, cost = c_j)

---

Nash Product FOCs for use in vertical model with simultaneous timing

Description

Nash Product FOCs for use in vertical model with simultaneous timing

Usage

bargain_foc_vert_sim(
  price_w,
  own_down,
  own_up,
  alpha,
  delta,
  cost_w,
  cost_r,
  lambda,
  price_r,
  sumFOC = TRUE
)

Arguments

price_w	Upstream or wholesale prices
own_down	Ownership matrix for downstream firms
own_up	Ownership matrix for upstream firms
alpha	Price coefficient
delta	Mean values
cost_w	Marginal costs for upstream firm for each product
cost_r	Marginal costs for downstream firm for each product
lambda	Bargaining power of the buyer/retailer
price_r	Retail prices starting values
sumFOC	logical; if true, returns the sum of first-order conditions, if false, returns vector of FOC's for each product

Details

This function calculate the first-order conditions from a Nash bargaining model. For use in a vertical supply chain with simultaneous timing

Value

The first-order conditions

Examples

bargain_foc_vert_sim(price_w = c(1.6, 1.6, 1.6, 1.6, 1.6, 1.6),
own_down = paste0("R",rep(c(1,2,3),each=2)),
own_up = paste0("W",rep(c(1,2),3)),
alpha = -0.9, delta = c(0.2, 0.3, 0.9, 1.0, 0.8, 0.9),
cost_w = rep(.2, times = 6),
cost_r = rep(.1, times = 6),
lambda = 0.5,
price_r = c(2.9, 2.9, 3.0, 3.0, 3.0, 3.0))

---

Nash Product for use in vertical model with sequential timing

Description

Nash Product for use in vertical model with sequential timing

Usage

bargain_NP_vert_seq(
  w_start,
  product_max,
  price_w,
  own_down,
  own_up,
  alpha,
  delta,
  cost_w,
  cost_r,
  lambda,
  p_R0,
  sigma,
  showAll = FALSE,
  maxJointProfits = FALSE
)

Arguments

w_start	Initial wholesale price for product for which to maximize Nash Product
product_max	Index of product for which to maximize Nash Product
price_w	Upstream or wholesale prices
own_down	Ownership matrix for downstream firms
own_up	Ownership matrix for upstream firms
alpha	Price coefficient
delta	Mean values
cost_w	Marginal costs for upstream firm for each product
cost_r	Marginal costs for downstream firm for each product
lambda	Bargaining power of the buyer/retailer
p_R0	Retail prices starting values
sigma	Contract type; value between 0 and 1 where 0 is linear price, 1 is two-part tariff
showAll	logical; if true, returns gains for trade for every product
maxJointProfits	Experimental option; if true, will maximize joint profits rather than gains from trade

Details

This function calculate the Nash Product from a Nash bargaining model

Value

The first-order conditions

Examples

bargain_NP_vert_seq(w_start = 1.5, product_max = 1,
price_w = c(1.6, 1.6, 1.6, 1.6, 1.6, 1.6),
own_down = paste0("R",rep(c(1,2,3),each=2)),
own_up = paste0("W",rep(c(1,2),3)),
alpha = -0.9, delta = c(0.2, 0.3, 0.9, 1.0, 0.8, 0.9),
cost_w = rep(.2, times = 6),
cost_r = rep(.1, times = 6),
lambda = 0.5, sigma = 0,
p_R0 = c(2.9, 2.9, 3.0, 3.0, 3.0, 3.0))

---

Nash Product for use in vertical model with sequential timing

Description

Nash Product for use in vertical model with sequential timing

Usage

bargain_NP_vert_seq_gnl(
  w_start,
  product_max,
  price_w,
  own_down,
  own_up,
  alpha,
  delta,
  cost_w,
  cost_r,
  lambda,
  p_R0,
  sigma,
  nest_allocation,
  mu,
  showAll = FALSE
)

Arguments

w_start	Initial wholesale price for product for which to maximize Nash Product
product_max	Index of product for which to maximize Nash Product
price_w	Upstream or wholesale prices
own_down	Ownership matrix for downstream firms
own_up	Ownership matrix for upstream firms
alpha	Price coefficient
delta	Mean values
cost_w	Marginal costs for upstream firm for each product
cost_r	Marginal costs for downstream firm for each product
lambda	Bargaining power of the buyer/retailer
p_R0	Retail prices starting values
sigma	Contract type; value between 0 and 1 where 0 is linear price, 1 is two-part tariff
nest_allocation	For generalized nested logit demand, a J-by-K matrix where each element (j,k) designates the membership of good j in nest k. Rows should sum to 1.
mu	Nesting parameters for each nest
showAll	logical; if true, returns gains for trade for every product

Details

This function calculate the Nash Product from a Nash bargaining model. This version allows for nested logit with overlapping nests.

Value

The first-order conditions

Examples

own_down <- paste0("R",rep(c(1,2,3),each=2))
own_up <- paste0("W",rep(c(1,2),3))

B1 <- 1 * matrix( c(own_down == "R1",
                    own_down == "R2",
                    own_down == "R3",
                    own_up == "W1",
                    own_up == "W2"),
                  ncol = 5, nrow = 6)
a1 <- B1 * 0.5
mu1 <- rep(1.0,5)

bargain_NP_vert_seq_gnl(w_start = 1.5, product_max = 1,
                        price_w = c(1.6, 1.6, 1.6, 1.6, 1.6, 1.6),
                        own_down = own_down,
                        own_up = own_up,
                        alpha = -0.9, delta = c(0.2, 0.3, 0.9, 1.0, 0.8, 0.9),
                        cost_w = rep(.2, times = 6),
                        cost_r = rep(.1, times = 6),
                        lambda = 0.5, sigma = 0,
                        p_R0 = c(2.9, 2.9, 3.0, 3.0, 3.0, 3.0),
                        nest_allocation = a1, mu = mu1)

---

Calibrate costs for Nash bargain in vertical model with sequential timing

Description

Calibrate costs for Nash bargain in vertical model with sequential timing

Usage

bargain_vert_seq_calibrate(
  c_w_val,
  price_w,
  own_down,
  own_up,
  alpha,
  delta,
  lambda,
  cost_r,
  price_r,
  sigma,
  setTol = 0.01,
  setMaxIter = 500,
  showAll = FALSE,
  symmetricCosts = FALSE
)

Arguments

c_w_val	Wholesale/upstream costs
price_w	Upstream or wholesale prices
own_down	Ownership matrix for downstream firms
own_up	Ownership matrix for upstream firms
alpha	Price coefficient
delta	Mean values
lambda	Bargaining power of the buyer/retailer
cost_r	Marginal costs for downstream firm for each product
price_r	Retail prices
sigma	Contract type; value between 0 and 1 where 0 is linear price, 1 is two-part tariff
setTol	tolerance for convergence
setMaxIter	Maximum iterations for convergence loop
showAll	logical; if true, returns gains for trade for every product
symmetricCosts	Constrain costs to be symmetric

Details

This function can be used to calibrate the costs in a Nash bargain which is the upstream market of a vertical supply chain. Assumes logit demand.

Value

The value of objective function

Examples

bargain_vert_seq_calibrate(c_w_val = rep(.2, times = 6),
price_w = c(1.6, 1.6, 1.6, 1.6, 1.6, 1.6),
own_down = paste0("R",rep(c(1,2,3),each=2)),
own_up = paste0("W",rep(c(1,2),3)),
alpha = -0.9, delta = c(0.2, 0.3, 0.9, 1.0, 0.8, 0.9),
lambda = 0.5, cost_r = rep(.1, times = 6),
price_r = c(2.9, 2.9, 3.0, 3.0, 3.0, 3.0),
sigma = 0, setTol = 1.5)

---

Calibrate lambda for Nash bargain in vertical model with simultaneous timing

Description

Calibrate lambda for Nash bargain in vertical model with simultaneous timing

Usage

bargain_vert_sim_calibrate(
  lambda,
  price_w,
  own_down,
  own_up,
  alpha,
  delta,
  cost_w,
  cost_r,
  price_r
)

Arguments

lambda	Bargaining power of the buyer/retailer
price_w	Upstream or wholesale prices
own_down	Ownership matrix for downstream firms
own_up	Ownership matrix for upstream firms
alpha	Price coefficient
delta	Mean values
cost_w	Marginal costs for upstream firm for each product
cost_r	Marginal costs for downstream firm for each product
price_r	Retail prices

Details

This function can be used to calibrate the bargaining parameter in a Nash bargain which is the upstream market of a vertical supply chain. Assumes logit demand.

Value

The value of objective function

Examples

bargain_vert_sim_calibrate(lambda = 0.4,
price_w = c(1.6, 1.6, 1.6, 1.6, 1.6, 1.6),
own_down = paste0("R",rep(c(1,2,3),each=2)),
own_up = paste0("W",rep(c(1,2),3)),
alpha = -0.9,
delta = c(0.2, 0.3, 0.9, 1.0, 0.8, 0.9),
cost_w = rep(.2, times = 6),
cost_r = rep(.1, times = 6),
price_r = c(2.9, 2.9, 3.0, 3.0, 3.0, 3.0))

---

Calibrate upstream Nash bargain in vertical model with simultaneous timing

Description

Calibrate upstream Nash bargain in vertical model with simultaneous timing

Usage

bargain_vert_sim_calibrate_gnl(
  param,
  lambda = NA,
  cost_w = NA,
  own_down,
  own_up,
  alpha,
  delta,
  cost_r,
  price_r,
  price_w,
  nest_allocation = NA,
  mu = NA,
  sumFOC = FALSE
)

Arguments

param	vector of parameters to calibrate, will be either bargaining parameter or demand parameters, depending on what other information is supplied
lambda	Bargaining power of the buyer/retailer
cost_w	Marginal costs for upstream firm for each product
own_down	Ownership matrix for downstream firms
own_up	Ownership matrix for upstream firms
alpha	Price coefficient
delta	Mean values
cost_r	Marginal costs for downstream firm for each product
price_r	Retail prices
price_w	Upstream or wholesale prices
nest_allocation	For generalized nested logit demand, a J-by-K matrix where each element (j,k) designates the membership of good j in nest k. Rows should sum to 1.
mu	Nesting parameters for each nest
sumFOC	logical; whether to return the sum of squares of the first-order conditions. Defaults to FALSE, in which case it returns each product first-order condition as a vector.

Details

This function can be used to calibrate the bargaining parameter in a Nash bargain which is the upstream market of a vertical supply chain. Assumes logit demand.

Value

The value of objective function

Examples

bargain_vert_sim_calibrate_gnl(param = 0.4, lambda=NA,
cost_w = rep(.2, times = 6),
own_down = paste0("R",rep(c(1,2,3),each=2)),
own_up = paste0("W",rep(c(1,2),3)),
alpha = -0.9,
delta = c(0.2, 0.3, 0.9, 1.0, 0.8, 0.9),
cost_r = rep(.1, times = 6),
price_r = c(2.9, 2.9, 3.0, 3.0, 3.0, 3.0),
price_w = c(1.6, 1.6, 1.6, 1.6, 1.6, 1.6),
nest_allocation=NA, mu=NA, sumFOC = TRUE)

---

Bertrand model calibration

Description

Bertrand model calibration

Usage

bertrand_calibrate(
  param,
  own,
  price,
  shares,
  cost,
  weight,
  returnOutcomes = FALSE
)

Arguments

param	Price coefficient alpha parameter to calibrate
own	Ownership matrix
price	Price
shares	Observed market shares
cost	Marginal costs for each product
weight	Weighting matrix
returnOutcomes	logical; should equilibrium objects be returned (mean value parameter, prices, shares, costs) as a list.

Details

This function calculate the first-order conditions from a Bertrand price-setting model of competition. This function is only for standard logit demand. For nested logit or generalized nested logit, see bertrand_calibrate_gnl().

Value

Distance between observed values and model predicted values for prices and shares

Examples

alpha  <- -0.9
delta <- c(.81,.93,.82)
c_j <- c(.05,.31,.30)

own_pre = diag(3)

p0 <- c_j*1.1

share1 <- (exp(delta + alpha*p0))/(1+sum(exp(delta + alpha*p0)))
x00 <- c(-1)
wt_matrix <- diag(c(1,1,1,1000,1000,1000))

bertrand_calibrate( param = x00,
                    own = own_pre, price = p0,
                    shares = share1, cost  = c_j,
                    weight = wt_matrix)

---

Calibrate Bertrand model with GNL demand

Description

Calibrate Bertrand model with GNL demand

Usage

bertrand_calibrate_gnl(
  param,
  own,
  price,
  shares,
  cost,
  weight = c(1, 1, 1, 1),
  nest_allocation,
  div_matrix,
  mu_constraint_matrix = NA,
  div_calc_marginal = TRUE,
  optimizer = "BBoptim",
  optimizer_c = "optim",
  returnOutcomes = FALSE,
  maxitval = 1500,
  maxitval_c = 1500,
  fast_version = FALSE
)

Arguments

param	Vector of demand parameters (alpha,mu)
own	Ownership matrix
price	Observed prices
shares	Observed market shares
cost	Marginal costs for each product
weight	Vector of length four with weights given to prices, shares, diversions, and costs, respectively. Default is c(1,1,1,1).
nest_allocation	For generalized nested logit demand, a J-by-K matrix where each element (j,k) designates the membership of good j in nest k. Rows should sum to 1.
div_matrix	A matrix of observed diversions from product in row j to product in column k.
mu_constraint_matrix	is a (K-by-K') matrix indicating which nesting parameters are constrained to be equal to each other, where K is the number of nests and K' is the number of freely varying nesting parameters. mu_full = mu_constraint_matrix %*% mu_prime. Where mu_full is a vector of length K of the nesting parameter value for each nest, and mu_prime is a vector of length K' of parameters to be calculated. It must be the case that K is greater than K'.
div_calc_marginal	is a logical if function should match to marginal diversions (if TRUE) or second choice diversions (if FALSE). Default to TRUE.
optimizer	Which optimization routine should be used to find equilibrium prices, either BBoptim or multiroot
optimizer_c	Which optimization routine should be used to calibrate costs, either BBoptim or optim
returnOutcomes	logical; should equilibrium objects be returned (mean value parameter, prices, shares, costs) as a list.
maxitval	Max iterations during price optimization from foc
maxitval_c	Max iterations during cost optimization from foc
fast_version	logical; if True, uses only first-order conditions for products that have non-missing costs. If False, uses the model to predict costs for all products, and then uses all first-order conditions, which takes longer to calibrate. Default is False.

Details

This function calibrates a Bertrand model with generalized nested logit (GNL) demand

Value

Difference between model predicted and observed values of prices, shares, and diversions.

Examples

nest1 <- matrix( c(1, 0, 0, 0, 1, 1), ncol = 2, nrow = 3)
divmat <- matrix( c(0, .4, .4, .4, 0, .4, .4, .4, 0), ncol = 3, nrow = 3)

bertrand_calibrate_gnl(param = c(-0.9, 1, 1),
                       own = diag(3),
                       price = c(.05, .34, .33),
                       shares = c( 0.31, 0.27, 0.25),
                       cost = c(.05,.31,.30),
                       weight = c(1,1,1,1),
                       nest_allocation = nest1, div_matrix = divmat)

---

Bertrand first-order conditions

Description

Bertrand first-order conditions

Usage

bertrand_foc(
  price,
  own,
  alpha,
  delta,
  cost,
  nest_allocation = NA,
  mu = NA,
  sumFOC = FALSE
)

bertrand_foc_c(
  cost,
  price,
  own,
  alpha,
  delta,
  nest_allocation = NA,
  mu = NA,
  sumFOC = FALSE
)

Arguments

price	Price
own	Ownership matrix
alpha	Price coefficient
delta	Mean values
cost	Marginal costs for each product
nest_allocation	For generalized nested logit demand, a J-by-K matrix where each element (j,k) designates the membership of good j in nest k. Rows should sum to 1.
mu	Nesting parameters for each nest
sumFOC	logical; whether to return the sum of squares of the first-order conditions. Defaults to FALSE, in which case it returns each product first-order condition as a vector.

Details

This function calculate the first-order conditions from a Bertrand price-setting model of competition

Value

The first-order conditions

Examples

alpha  <- -0.9
delta <- c(.81,.93,.82)
c_j <- c(.05,.31,.30)

own_pre = diag(3)

p0 <- c_j*1.1

bertrand_foc(price = p0,
             own = own_pre, alpha= alpha,
             delta = delta, cost = c_j)

---

Downstream Bertrand first-order conditions

Description

Downstream Bertrand first-order conditions

Usage

bertrand_foc_vert(
  price_r,
  own_down,
  own_up,
  alpha,
  delta,
  cost_r,
  price_w,
  cost_w,
  sumFOC = FALSE
)

Arguments

price_r	Downstream or retail prices
own_down	Ownership matrix for downstream firms
own_up	Ownership matrix for upstream firms
alpha	Price coefficient
delta	Mean values
cost_r	Marginal costs for downstream firm for each product
price_w	Upstream or wholesale prices, treated as costs by downstream firms
cost_w	Marginal costs for upstream firm for each product
sumFOC	logical; whether to return the sum of squares of the first-order conditions. Defaults to FALSE, in which case it returns each product first-order condition as a vector.

Details

This function calculates the first-order conditions from a Bertrand price-setting model of competition as the downstream market of a vertical supply chain. Assumes logit demand.

Value

The first-order conditions

---

Downstream Bertrand first-order conditions

Description

Downstream Bertrand first-order conditions

Usage

bertrand_foc_vert_gnl(
  price_r,
  own_down,
  own_up,
  alpha,
  delta,
  cost_r,
  price_w,
  cost_w,
  nest_allocation = NA,
  mu = NA,
  sumFOC = FALSE
)

Arguments

price_r	Downstream or retail prices
own_down	Ownership matrix for downstream firms
own_up	Ownership matrix for upstream firms
alpha	Price coefficient
delta	Mean values
cost_r	Marginal costs for downstream firm for each product
price_w	Upstream or wholesale prices, treated as costs by downstream firms
cost_w	Marginal costs for upstream firm for each product
nest_allocation	For generalized nested logit demand, a J-by-K matrix where each element (j,k) designates the membership of good j in nest k. Rows should sum to 1.
mu	Nesting parameters for each nest
sumFOC	logical; whether to return the sum of squares of the first-order conditions. Defaults to FALSE, in which case it returns each product first-order condition as a vector.

Details

This function calculates the first-order conditions from a Bertrand price-setting model of competition as the downstream market of a vertical supply chain. This version allows for generalized nested logit demand.

Value

The first-order conditions

---

Calibrate Bertrand model demand parameters

Description

Calibrate Bertrand model demand parameters

Usage

bertrand_vert_calibrate(param, own_down, price_r, shares, cost, price_w)

Arguments

param	Price coefficient, alpha
own_down	Ownership matrix for downstream firms
price_r	Downstream prices
shares	Market shares
cost	Marginal costs of downstream firm
price_w	Upstream or wholesale prices, treated as costs by downstream firms

Details

This function can be used to calibrate the demand parameters of a Bertrand model of competition that is the downstream market of a vertical supply chain. Wholesale prices are treated as costs by the downstream firms. Assumes logit demand.

Value

Objective function value

---

Market share calculations

Description

Calculates choice probabilities based on logit or generalized nested logit demand parameters

Usage

diversion_calc(
  price,
  alpha,
  delta,
  nest_allocation = NA,
  mu = NA,
  marginal = FALSE,
  outsideOption = TRUE
)

Arguments

price	Price
alpha	Price coefficient
delta	Mean values
nest_allocation	For generalized nested logit demand, a J-by-K matrix where each element (j,k) designates the membership of good j in nest k. Rows should sum to 1.
mu	Nesting parameters for each nest
marginal	logical; if True, diversions are calculated as diversions from a small price increase. If False, diversions are calculated as second choice diversions.
outsideOption	logical; whether to include an outside option in choice set. Default is TRUE.

Details

This function calculates choice probabilities based on demand parameters for a logit or generalized nested logit demand system

Value

Returns vector of choice probabilities for each good

Examples

diversion_calc(price=c(2.1,2.4,2.1),alpha=-0.9,delta=c(.81,.93,.82))

---

Market share calculations

Description

Calculates choice probabilities based on logit or generalized nested logit demand parameters

Usage

match_share(price, delta, alpha, nest_allocation = NA, mu = NA, shares_obs)

Arguments

price	Price
delta	Mean values
alpha	Price coefficient
nest_allocation	For generalized nested logit demand, a J-by-K matrix where each element (j,k) designates the membership of good j in nest k. Rows should sum to 1.
mu	Nesting parameters for each nest
shares_obs	Observed market shares

Details

This function calculates the difference between model predicted choice probabilities and observed market shares.

Value

Returns vector of difference between predicted shares and observed shares

Examples

match_share(price=c(2.1,2.4,2.1), delta=c(.81,.93,.82), alpha=-0.9,
shares_obs = c(.2, .2, .2))

---

Market share calculations

Description

Calculates choice probabilities based on logit or generalized nested logit demand parameters

Usage

share_calc(
  price,
  delta,
  alpha,
  nest_allocation = NA,
  mu = NA,
  returnLogsum = FALSE,
  outsideOption = TRUE
)

Arguments

price	Price
delta	Mean values
alpha	Price coefficient
nest_allocation	For generalized nested logit demand, a J-by-K matrix where each element (j,k) designates the membership of good j in nest k. Rows should sum to 1.
mu	Nesting parameters for each nest
returnLogsum	logical; whether to return the denominator of the choice probabilities (also known as the log-sum term). Defaults to FALSE, in which case the function returns a vector with each product's choice probability.
outsideOption	logical; whether to include an outside option in choice set. Default is TRUE.

Details

This function calculates choice probabilities based on demand parameters for a logit or generalized nested logit demand system

Value

Returns vector of choice probabilities for each good

Examples

share_calc(price=c(2.1,2.4,2.1),delta=c(.81,.93,.82),alpha=-0.9)

---

Second score auction calibration

Description

Second score auction calibration

Usage

ssa_calibrate(param, own, price, share, cost, weight)

Arguments

param	The price coefficient alpha to be calibrated
own	Ownership matrix
price	Price
share	Share
cost	Marginal costs for each product
weight	Weighting matrix

Details

This function calculate the first-order conditions from a second score auction model of competition

Value

The first-order conditions

Examples

own_pre = diag(3)
p0 <- c(.05, .34, .33)
share1 <- c( 0.31, 0.27, 0.25)
c_j <- c(.05,.31,.30)
wt_matrix <- diag(c(1,1,1))

ssa_calibrate(param = -1,own = own_pre,price=p0,share=share1,cost=c_j,
              weight = wt_matrix)

---

Calibrate second score auction model with GNL demand

Description

Calibrate second score auction model with GNL demand

Usage

ssa_calibrate_gnl(
  param,
  own,
  price,
  shares,
  cost,
  weight,
  nest_allocation,
  mu_constraint_matrix = NA
)

Arguments

param	Vector of demand parameters (alpha,mu)
own	Ownership matrix
price	Observed prices
shares	Observed market shares
cost	Marginal costs for each product
weight	Weighting matrix of dimensions J-by-J
nest_allocation	For generalized nested logit demand, a J-by-K matrix where each element (j,k) designates the membership of good j in nest k. Rows should sum to 1.
mu_constraint_matrix	is a (K-by-K') matrix indicating which nesting parameters are constrained to be equal to each other, where K is the number of nests and K' is the number of freely varying nesting parameters. mu_full = mu_constraint_matrix %*% mu_prime. Where mu_full is a vector of length K of the nesting parameter value for each nest, and mu_prime is a vector of length K' of parameters to be calculated. It must be the case that K is greater than K'.

Details

This function calibrates a second score auction model with generalized nested logit (GNL) demand

Value

Difference between model predicted and observed values of prices, shares, and diversions.

Examples

nest1 <- matrix( c(1, 0, 0, 0, 1, 1), ncol = 2, nrow = 3)

ssa_calibrate_gnl(param = c(-0.9, 1, 1),
own = diag(3),
price = c(.05, .34, .33),
shares = c( 0.31, 0.27, 0.25),
cost = c(.05,.31,.30),
weight = diag(c(1,1,1)),
nest_allocation = nest1,
mu_constraint_matrix = NA)

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Second score bargaining first-order conditions

Description

Second score bargaining first-order conditions

Usage

ssbargain_calibrate(
  param,
  own,
  price,
  shares,
  cost,
  weight,
  lambda,
  includeMUI = TRUE
)

Arguments

param	Price coefficient alpha and mean values delta, parameters to calibrate
own	Ownership matrix
price	Price
shares	Observed market shares
cost	Marginal costs for each product
weight	Weighting matrix
lambda	Bargaining power of the buyer
includeMUI	logical; whether to include marginal utility of income in buyer's payoff, thereby translating dollars to utility. Default is True, interpreted as buyer maximizing utility. Setting equal to False would have interpretation that buyer maximizes profits.

Details

This function calculate the first-order conditions from a bargaining model that nests the second score auction.

Value

The first-order conditions

Examples

alpha  <- -0.9
delta <- c(.81,.93,.82)
own_pre = diag(3)
p0 <- c(.05, .34, .33)
c_j <- c(.05,.31,.30)
wt_matrix <- diag(c(1,1,1,1000,1000,1000))
share1 <- c( 0.31, 0.27, 0.25)

ssbargain_calibrate(param = c(alpha,delta),own = own_pre,price = p0,
                    shares = share1, cost = c_j, weight = wt_matrix,
                    lambda = 0.5)

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Second score bargaining first-order conditions

Description

Second score bargaining first-order conditions

Usage

ssbargain_foc(price, own, alpha, delta, cost, lambda, includeMUI = TRUE)

Arguments

price	Price
own	Ownership matrix
alpha	Price coefficient
delta	Mean values
cost	Marginal costs for each product
lambda	Bargaining power of the buyer
includeMUI	logical; TO BE ADDED

Details

This function calculate the first-order conditions from a bargaining model that nests the second score auction.

Value

The first-order conditions

Examples

alpha  <- -0.9
delta <- c(.81,.93,.82)
own_pre = diag(3)
p0 <- c(.05, .34, .33)
c_j <- c(.05,.31,.30)

ssbargain_foc(price = p0,own = own_pre,alpha=alpha,delta=delta,cost=c_j,
              lambda = 0.5)

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