Post-Estimation and Merger Simulation • rblp

Introduction

After estimating a demand model, the structural parameters can be used to compute policy-relevant quantities: elasticities, diversion ratios, consumer surplus, markups, and merger simulations. This vignette demonstrates these post-estimation tools.

We start with a logit model for fast, evaluable examples, then discuss how results extend to the random coefficients case. For estimation details, see vignette("logit-nested-logit") and vignette("random-coefficients").

Setup: Estimate a Logit Model

library(rblp)

products <- load_nevo_products()

# Logit with product fixed effects
f1 <- blp_formulation(~ prices, absorb = ~ product_ids)
problem <- blp_problem(list(f1), products)
results <- problem$solve(method = "1s")

cat(sprintf("Price coefficient: %.4f\n", results$beta[1]))
#> Price coefficient: -30.4205

Elasticities

Own-Price and Cross-Price Elasticities

The price elasticity matrix $E$ for market $t$ has entries:

$E_{jk} = \frac{\partial s_j}{\partial p_k} \cdot \frac{p_k}{s_j}$

Diagonal entries ( $E_{jj}$ ) are own-price elasticities: the percent change in $j$ ’s share for a 1% increase in $j$ ’s price. These should be negative.
Off-diagonal entries ( $E_{jk}$ , $j \neq k$ ) are cross-price elasticities: the percent change in $j$ ’s share for a 1% increase in $k$ ’s price. In a substitutes market, these are positive.

# Compute elasticities for the first market
first_market <- results$problem$unique_market_ids[1]
E <- results$compute_elasticities(first_market)

cat("Elasticity matrix dimensions:", dim(E), "\n")
#> Elasticity matrix dimensions: 24 24

# Own-price elasticities (diagonal)
own_elast <- diag(E)
cat("\nOwn-price elasticities summary:\n")
#> 
#> Own-price elasticities summary:
cat(sprintf("  Mean: %.3f\n", mean(own_elast)))
#>   Mean: -3.935
cat(sprintf("  Min:  %.3f\n", min(own_elast)))
#>   Min:  -5.286
cat(sprintf("  Max:  %.3f\n", max(own_elast)))
#>   Max:  -2.166

# Cross-price elasticities (first row, first 5 columns)
cat("\nCross-price elasticities (product 1 w.r.t. products 1-5):\n")
#> 
#> Cross-price elasticities (product 1 w.r.t. products 1-5):
print(round(E[1, 1:min(5, ncol(E))], 4))
#> [1] -2.1657  0.0271  0.0523  0.0229  0.0845

Logit vs. RC Logit Elasticities

In the plain logit, cross-price elasticities follow the IIA pattern: if product $k$ ’s price increases, all other products gain share proportionally to their current share. This means a luxury cereal and a budget cereal respond identically to a third product’s price change, which is unrealistic.

In the RC logit, cross-price elasticities depend on how similar products are in the random coefficient characteristic space. Products that appeal to similar consumer segments have higher cross-price elasticities, generating realistic substitution patterns.

Diversion Ratios

The diversion ratio $D_{jk}$ measures: if product $j$ is removed (or its price increases), what fraction of $j$ ’s lost demand is captured by product $k$ ?

$D_{jk} = -\frac{\partial s_k / \partial p_j}{\partial s_j / \partial p_j}$

Diversion ratios are central to merger analysis. The upward pricing pressure (UPP) test for a merger between firms owning products $j$ and $k$ is proportional to $D_{jk} \times m_k$ , where $m_k$ is product $k$ ’s margin.

# Compute diversion ratios for the first market
D <- results$compute_diversion_ratios(first_market)

cat("Diversion ratio matrix dimensions:", dim(D), "\n")
#> Diversion ratio matrix dimensions: 24 24

# Where does demand from product 1 go?
cat("\nDiversion from product 1 (first 5 products):\n")
#> 
#> Diversion from product 1 (first 5 products):
print(round(D[1, 1:min(5, ncol(D))], 4))
#> [1] 0.0000 0.0079 0.0132 0.0058 0.0182

# What fraction goes to the outside good?
cat(sprintf("\nDiversion to outside good from product 1: %.4f\n",
            1 - sum(D[1, ])))
#> 
#> Diversion to outside good from product 1: 0.5622

In the logit model, diversion is proportional to market shares (another manifestation of IIA). In the RC logit, diversion reflects the actual similarity of products to heterogeneous consumers.

Consumer Surplus

Consumer surplus in the logit family follows the Small-Rosen (1981) log-sum formula:

$CS_t = \frac{1}{\alpha} \cdot E_i\left[ \log\left(1 + \sum_{j=1}^{J_t} \exp(\delta_{jt} + \mu_{ijt})\right) \right]$

where $\alpha$ is the (negative) price coefficient (marginal utility of income). In the plain logit, $\mu_{ijt} = 0$ and the expectation is trivial. In the RC logit, the expectation is taken over the simulated consumer population.

cs <- results$compute_consumer_surplus()

cat("Consumer surplus per market (first 10 markets):\n")
#> Consumer surplus per market (first 10 markets):
print(round(cs[1:10], 4))
#>  C01Q1  C03Q1  C04Q1  C05Q1  C07Q1  C08Q1  C11Q1  C12Q1  C13Q1  C14Q1 
#> 0.0193 0.0176 0.0363 0.0182 0.0297 0.0180 0.0213 0.0204 0.0197 0.0249

cat(sprintf("\nMean CS across markets: %.4f\n", mean(cs)))
#> 
#> Mean CS across markets: 0.0220
cat(sprintf("Std. dev. of CS: %.4f\n", sd(cs)))
#> Std. dev. of CS: 0.0071

Changes in consumer surplus across policy scenarios (e.g., before and after a merger) provide a monetary measure of consumer welfare impact.

HHI (Herfindahl-Hirschman Index)

The HHI measures market concentration:

$HHI_t = \sum_{f} \left(\sum_{j \in f} s_{jt}\right)^2 \times 10000$

where the inner sum aggregates shares within each firm $f$ . The HHI ranges from near zero (perfect competition) to 10,000 (monopoly). The U.S. DOJ considers markets with HHI above 2,500 to be highly concentrated.

hhi <- results$compute_hhi()

cat("HHI per market (first 10 markets):\n")
#> HHI per market (first 10 markets):
print(round(hhi[1:10], 0))
#> C01Q1 C03Q1 C04Q1 C05Q1 C07Q1 C08Q1 C11Q1 C12Q1 C13Q1 C14Q1 
#>   711   643  1435   566  1048   540   679   712   793   817

cat(sprintf("\nMean HHI: %.0f\n", mean(hhi)))
#> 
#> Mean HHI: 834

Costs and Markups

Recovering Marginal Costs

Given estimated demand, marginal costs can be backed out from the Nash-Bertrand first-order conditions. In equilibrium, each firm sets prices to satisfy:

$p_j - mc_j = -\left(\Omega^{-1} s\right)_j$

where $\Omega$ is the ownership-weighted matrix of share derivatives. Rearranging:

$mc_j = p_j + \left(\Omega^{-1} s\right)_j$

costs <- results$compute_costs()

cat("Marginal costs (first 10 products):\n")
#> Marginal costs (first 10 products):
print(round(costs[1:10], 4))
#>  [1] 0.0348 0.0769 0.0951 0.0930 0.1175 0.0997 0.1069 0.0909 0.1123 0.0657

cat(sprintf("\nPrice range:  [%.4f, %.4f]\n",
            min(products$prices), max(products$prices)))
#> 
#> Price range:  [0.0455, 0.2257]
cat(sprintf("Cost range:   [%.4f, %.4f]\n", min(costs), max(costs)))
#> Cost range:   [-0.0001, 0.1878]

Markups (Lerner Index)

The Lerner index measures the fraction of price that is markup:

$L_j = \frac{p_j - mc_j}{p_j}$

markups <- results$compute_markups()

cat("Markups (first 10 products):\n")
#> Markups (first 10 products):
print(round(markups[1:10], 4))
#>  [1] 0.5176 0.3268 0.2818 0.2862 0.2410 0.2722 0.2587 0.2910 0.2494 0.3941

cat(sprintf("\nMean markup: %.4f\n", mean(markups)))
#> 
#> Mean markup: 0.3292
cat(sprintf("Median markup: %.4f\n", median(markups)))
#> Median markup: 0.3116

Merger Simulation

Merger simulation is the central counterfactual exercise in structural IO. The idea: when two firms merge, the merged entity internalizes the cross-price effects between the previously competing products, leading to higher prices.

Steps

Estimate demand and recover marginal costs $mc$ under the pre-merger ownership structure
Update the ownership matrix to reflect the merger (products of the acquired firm now belong to the acquirer)
Solve for new equilibrium prices under the post-merger ownership, holding costs fixed
Compare pre- and post-merger prices, shares, and consumer surplus

Simulation Example

We use the simulation framework for a clean merger example:

# Create a balanced panel: 50 markets, 20 products, 4 firms
id_data <- build_id_data(T = 50, J = 20, F = 4)
set.seed(42)
id_data$x <- runif(nrow(id_data), 0, 1)

# Simulate with known parameters
sim <- blp_simulation(
  product_formulations = list(blp_formulation(~ prices + x)),
  product_data = id_data,
  beta = c(0.5, -2, 0.8),
  xi_variance = 0.3,
  seed = 42
)

sim_data <- sim$replace_endogenous(
  iteration = blp_iteration("simple", list(atol = 1e-12, max_evaluations = 5000))
)

# Estimate
sim_problem <- sim_data$to_problem()
sim_results <- sim_problem$solve(method = "1s")

Now simulate a merger between firms 2 and 3:

# Create post-merger firm IDs: merge firm 3 into firm 2
new_firm_ids <- sim_data$product_data$firm_ids
new_firm_ids[new_firm_ids == 3] <- 2

cat(sprintf("Pre-merger firms:  %s\n", paste(sort(unique(sim_data$product_data$firm_ids)), collapse = ", ")))
#> Pre-merger firms:  1, 2, 3, 4
cat(sprintf("Post-merger firms: %s\n", paste(sort(unique(new_firm_ids)), collapse = ", ")))
#> Post-merger firms: 1, 2, 4

# Run the merger simulation
merger <- sim_results$compute_merger(
  new_firm_ids = new_firm_ids,
  iteration = blp_iteration("simple", list(atol = 1e-12, max_evaluations = 5000))
)

cat(sprintf("\nMean price change:   %.2f%%\n", mean(merger$price_change_pct)))
#> 
#> Mean price change:   7.61%
cat(sprintf("Max price change:    %.2f%%\n", max(merger$price_change_pct)))
#> Max price change:    25.48%
cat(sprintf("Mean CS change:      %.4f\n", mean(merger$delta_cs)))
#> Mean CS change:      -0.0786

Interpreting Merger Results

# Which products see the biggest price increases?
# Products of the merging firms should see larger increases
merging_products <- sim_data$product_data$firm_ids %in% c(2, 3)
non_merging <- !merging_products

cat("Price changes by group:\n")
#> Price changes by group:
cat(sprintf("  Merging firms (2 & 3):     mean = %.2f%%\n",
            mean(merger$price_change_pct[merging_products])))
#>   Merging firms (2 & 3):     mean = 14.22%
cat(sprintf("  Non-merging firms (1 & 4): mean = %.2f%%\n",
            mean(merger$price_change_pct[non_merging])))
#>   Non-merging firms (1 & 4): mean = 1.00%

# Consumer surplus change
cat(sprintf("\nTotal CS change: %.4f\n", sum(merger$delta_cs)))
#> 
#> Total CS change: -3.9287
cat(sprintf("CS change is %s\n",
            ifelse(sum(merger$delta_cs) < 0, "negative (consumers harmed)",
                   "positive (consumers benefit)")))
#> CS change is negative (consumers harmed)

Pre-Merger vs. Post-Merger HHI

# Compute pre-merger and post-merger HHI
hhi_pre <- sim_results$compute_hhi()

cat(sprintf("Pre-merger mean HHI:  %.0f\n", mean(hhi_pre)))
#> Pre-merger mean HHI:  1250
cat(sprintf("Delta HHI estimate: merging two of four equal firms\n"))
#> Delta HHI estimate: merging two of four equal firms

Specification Tests

Hansen J-Test

The Hansen J-test checks the validity of the overidentifying restrictions. Under the null that all instruments are valid, the J-statistic is asymptotically chi-squared with degrees of freedom equal to the number of moment conditions minus the number of parameters.

# Only meaningful for two-step GMM
results_2s <- problem$solve(method = "2s")
j_test <- results_2s$run_hansen_test()

cat(sprintf("J-statistic: %.4f\n", j_test$statistic))
cat(sprintf("Degrees of freedom: %d\n", j_test$df))
cat(sprintf("p-value: %.4f\n", j_test$p_value))

A small p-value suggests that at least some instruments may be invalid (correlated with the structural error $\xi$ ).

Wald Test

The Wald test can be used to test restrictions on the nonlinear parameters (sigma, pi). For example, to test whether all random coefficient variances are jointly zero (i.e., the logit is adequate):

# Test H0: all sigma parameters = 0
n_theta <- length(results$se)
R <- diag(n_theta)
r <- rep(0, n_theta)

wald <- results$run_wald_test(R, r)
cat(sprintf("Wald statistic: %.4f, p-value: %.4f\n",
            wald$statistic, wald$p_value))

Summary of Post-Estimation Methods

Method	What It Computes	Output
`compute_elasticities(market_id)`	$J \times J$ price elasticity matrix	Matrix
`compute_diversion_ratios(market_id)`	$J \times J$ diversion ratio matrix	Matrix
`compute_consumer_surplus()`	Small-Rosen CS per market	Named vector
`compute_hhi()`	HHI per market	Named vector
`compute_costs()`	Marginal costs from Bertrand FOC	Vector (length $N$ )
`compute_markups()`	Lerner index $(p - mc)/p$	Vector (length $N$ )
`compute_merger(new_firm_ids)`	Post-merger equilibrium	List
`run_hansen_test()`	Overidentification J-test	List
`run_wald_test(R, r)`	Wald test on parameters	List
`summary_table()`	All parameter estimates with SEs	Data frame

References

Small, K. & Rosen, H. (1981). Applied Welfare Economics with Discrete Choice Models. Econometrica, 49(1), 105-130.
Werden, G. (1996). A Robust Test for Consumer Welfare Enhancing Mergers Among Sellers of Differentiated Products. Journal of Industrial Economics, 44(4), 409-413.
Nevo, A. (2000). Mergers with Differentiated Products: The Case of the Ready-to-Eat Cereal Industry. RAND Journal of Economics, 31(3), 395-421.
Conlon, C. & Gortmaker, J. (2020). Best Practices for Differentiated Products Demand Estimation with pyblp. RAND Journal of Economics, 51(4), 1108-1161.