Personalized Pricing Tutorial#

This tutorial demonstrates how to use deep-inference for personalized pricing analysis with heterogeneous demand, following Dube & Misra (2022, Journal of Political Economy).

Motivation#

Problem: Firms set prices, but consumers respond differently based on their characteristics. How do we:

Estimate heterogeneous price elasticities \(\eta(X)\)?
Compute personalized optimal prices?
Measure welfare implications of price discrimination?

Approach: Use structural deep learning to estimate heterogeneous demand parameters \(\theta(X) = (\alpha(X), \beta(X))\) from a logit demand model, then compute economic targets with valid inference.

When to Use#

Use this approach when:

You have binary purchase data (buy/no-buy)
Treatment is price or a continuous instrument
You want heterogeneous price elasticities, not just average effects
You need valid confidence intervals on economic quantities

Mathematical Setup#

Logit Demand Model#

\[P(\text{buy} \mid X, P) = \sigma(\alpha(X) + \beta(X) \cdot P)\]

where \(\beta(X) < 0\) is the heterogeneous price sensitivity.

Three Economic Targets#

Target	Formula	Interpretation
Elasticity	\(\eta = (1-p) \cdot \beta \cdot P\)	% change in demand per % change in price
WTP	\(-\beta_{\text{attr}} / \beta_{\text{price}}\)	Dollar value of a one-unit attribute improvement
Consumer Surplus	\(\log(1 + e^V) / \lvert\beta_{\text{price}}\rvert\)	Expected utility in monetary units (Small & Rosen, 1981)

Example: Heterogeneous Price Sensitivity#

Step 1: Generate Data#

import numpy as np
from scipy.special import expit

np.random.seed(42)
n = 2000

# Consumer characteristics
X = np.random.randn(n, 3)

# Heterogeneous parameters
alpha_true = 1.0 + 0.3 * X[:, 0]       # Base utility
beta_true = -0.5 - 0.2 * X[:, 0]       # Price sensitivity (always negative)

# Price (treatment)
T = np.random.uniform(0.5, 3.0, n)

# Purchase probability and outcome
p_true = expit(alpha_true + beta_true * T)
Y = np.random.binomial(1, p_true).astype(float)

Step 2: Estimate Price Elasticity#

from deep_inference import inference

# Elasticity at price point P=2.0
result_elast = inference(
    Y, T, X,
    model='logit',
    target='elasticity',
    t_tilde=2.0,
    epochs=100,
    n_folds=50,
)

print(f"Average elasticity at P=2.0: {result_elast.mu_hat:.4f}")
print(f"SE: {result_elast.se:.4f}")
print(f"95% CI: [{result_elast.ci_lower:.4f}, {result_elast.ci_upper:.4f}]")

Step 3: Compute Consumer Welfare#

# Consumer surplus at current price P=2.0
result_welfare = inference(
    Y, T, X,
    model='logit',
    target='welfare',
    t_tilde=2.0,
    epochs=100,
    n_folds=50,
)

print(f"Consumer surplus: {result_welfare.mu_hat:.4f}")
print(f"95% CI: [{result_welfare.ci_lower:.4f}, {result_welfare.ci_upper:.4f}]")

Step 4: Dose-Response Curve#

Evaluate elasticity at multiple price points to trace out the demand curve:

prices = np.linspace(0.5, 3.0, 6)
elasticities = []

for p in prices:
    r = inference(Y, T, X, model='logit', target='elasticity', t_tilde=float(p),
                  epochs=100, n_folds=50)
    elasticities.append((p, r.mu_hat, r.se))
    print(f"P={p:.1f}: elasticity={r.mu_hat:.4f} +/- {r.se:.4f}")

Optimal Pricing#

With estimated elasticity \(\hat{\eta}\), the Lerner markup rule gives:

\[\frac{P - MC}{P} = -\frac{1}{\hat{\eta}}\]

For a firm with marginal cost \(MC\):

MC = 1.0  # marginal cost
eta_hat = result_elast.mu_hat  # estimated elasticity

# Optimal markup: P* = MC * eta / (1 + eta)
P_optimal = MC * eta_hat / (1 + eta_hat)
print(f"Optimal price: {P_optimal:.4f}")

Uniform vs. Personalized Pricing#

The key insight from Dube & Misra (2022): personalized pricing based on heterogeneous \(\beta(X)\) can increase both profits and consumer welfare relative to uniform pricing, when firms can identify consumers with high willingness to pay.

References#

Dube, J.-P. & Misra, S. (2022). Personalized Pricing and Consumer Welfare. Journal of Political Economy, 131(1).
Farrell, M. H., Liang, T., & Misra, S. (2021). Deep Neural Networks for Estimation and Inference. Econometrica, 89(1).
Small, K. A. & Rosen, H. S. (1981). Applied Welfare Economics with Discrete Choice Models. Econometrica, 49(1).