Interactive Pricing Theory
Educational tools for understanding pricing and revenue optimization. Explore canonical models with interactive visualizations, full mathematical formulations, and real-world industry context.
What is Pricing & Revenue Optimization?
Pricing and Revenue Optimization (PRO) is the application of analytics—mathematical modeling, optimization, and statistical estimation—to the problem of setting prices. It sits at the intersection of microeconomics, operations research, and data science.
The illustrations on this site demonstrate how analytical pricing methods can improve outcomes relative to heuristic approaches. Each model isolates a core tradeoff—margin versus volume, early revenue versus residual inventory, protection versus spoilage—and lets the reader explore it interactively.
Based on Phillips, Pricing and Revenue Optimization (2nd ed., 2021) and Talluri & van Ryzin, The Theory and Practice of Revenue Management (2004).
Tracks
Eighteen thematic tracks — each a learning path from foundational concepts to a hands-on capstone simulator — with interactive visualizations and complete mathematical formulations.
First principles of pricing: the history of pricing as a discipline, demand curves, willingness to pay, cost pass-through, and the margin-versus-volume tradeoff.
How Pricing Became a Science
From ancient bazaars to algorithmic markets
Trace the evolution of pricing from antiquity through airline deregulation to modern algorithmic pricing, and discover why a 1% price improvement is the most powerful profit lever.
The Building Blocks of Pricing
Price waterfalls, approaches, and the analytics case
Explore the price waterfall — how twelve independent discounts compound to a 29% reduction — and compare cost-plus, market-based, and value-based pricing approaches.
What Customers Will Pay
Quantifying reference value and differentiation
Decompose a product's total economic value into the reference price plus differentiation value, and see how shifting individual value drivers changes the optimal price.
How Demand Responds to Price
Price-response functions and willingness to pay
Compare linear, logit, exponential, and constant-elasticity demand functions, and see how willingness-to-pay distributions generate demand curves.
Finding the Best Price
The margin-versus-volume tradeoff
Explore the fundamental tradeoff between margin and volume. Adjust cost structure, demand parameters, and observe the optimal price on the contribution curve.
Cost Pass-Through
How cost shocks reach consumers
Analyze how much of a cost increase reaches consumers, why pass-through depends on demand curvature rather than elasticity alone, and when firms over-shift costs beyond 100%.
Extracting more surplus through price discrimination, two-part tariffs, bundling, menu design, and pricing policy waterfalls.
Charging Different Prices
Segmenting customers by willingness to pay
Drag segment boundaries on a willingness-to-pay distribution and compare revenue under uniform pricing versus multi-segment strategies.
Bundles, Tiers & Volume Discounts
Two-part tariffs and quantity-based pricing
Design two-part tariffs that balance access fees against per-unit prices, and discover when bundling products extracts more surplus than selling separately.
Bundling & Product Line Design
When selling together beats selling apart
Explore Adams-Yellen theory of commodity bundling, compare pure and mixed bundling strategies, and see how negative WTP correlation determines when bundling dominates.
Restaurant Menu Pricing
RevPASH, item classification, and menu psychology
Classify menu items by profitability and popularity using the Kasavana-Smith matrix, optimize Revenue Per Available Seat Hour, and see how menu formatting shifts order patterns.
From List Price to Pocket Margin
How discounts compound into margin erosion
Build an interactive price waterfall from list price to pocket margin, discover how small discounts compound into massive margin erosion, and learn to evaluate volume discount traps.
Designing product tiers and quality ladders: conjoint analysis, hedonic pricing, the Mussa-Rosen versioning model, and menu optimization.
Versioning & Quality Tiers
Damaged goods and the information good ladder
Analyze optimal product line design through the Mussa-Rosen quality discrimination model, and discover why firms sometimes deliberately degrade products to create tiers.
What Customers Value Most
Measuring willingness to pay for each feature
Estimate willingness to pay for individual product attributes using choice-based conjoint, and predict market shares from part-worth utilities.
Hedonic Pricing
Decomposing prices into attribute values
Run hedonic regressions on synthetic market data to recover implicit prices for individual product attributes, and compare revealed-preference hedonic estimates to survey-based conjoint.
How Many Items to Offer
Coming SoonVariety vs decision overload
Explore how the number of items on a menu or product line affects choice quality and revenue, and discover the tradeoff between variety and decision complexity.
How psychology shapes pricing: reference dependence, loss aversion, framing effects, fairness norms, and the economics of price regulation.
The Psychology of Pricing
Fairness, loss aversion, and reference prices
Explore how reference prices, loss aversion, and framing effects shape customer responses, and see why the same price change can be accepted or rejected depending on context.
How You Present the Price
Anchoring, decoys, and charm pricing
Explore how anchoring, charm pricing, decoy effects, and partitioned pricing shift willingness-to-pay, and run side-by-side frame comparisons.
Price Fairness & Regulation
Anti-gouging laws, price controls, and welfare
Impose price ceilings and anti-gouging constraints on markets with demand shocks, measure welfare effects and shortages, and explore when fairness norms align with economic efficiency.
Discrete choice models, structural demand estimation, multi-product optimization, and optimal assortment design with mixed logit.
How Customers Choose Among Products
Logit models and the substitution problem
Model how customers choose among competing products using discrete choice frameworks, and discover why the Independence of Irrelevant Alternatives property matters for pricing.
Estimating Demand from Market Data
Recovering preferences with the BLP method
Recover consumer preferences from aggregate market-share data using the Berry-Levinsohn-Pakes contraction mapping, and compare simple logit vs random-coefficients elasticity patterns.
Multi-Product Pricing
Optimizing across substitutes and complements
Optimize prices across a product portfolio where cross-elasticities link demand. Explore the joint profit surface and see why optimizing products independently leaves money on the table.
Product Line Architect
Assortment + pricing with mixed logit
Design a product line of 3-5 SKUs, set features and prices, and see how heterogeneous customers choose using Mixed Logit. Compare your manual assortment against algorithmic optimization.
Capacity-constrained pricing for perishable inventory: Littlewood's rule, EMSR booking limits, network RM, bid-price controls, overbooking, and competitive simulation.
Pricing Scarce Inventory
Why capacity constraints push prices higher
See how capacity constraints push optimal prices above the unconstrained optimum, and learn why the shadow price of capacity is the bridge from static pricing to revenue management.
Filling Seats at the Right Price
Allocating capacity across fare classes
Observe an animated booking timeline filling fare classes, step through Littlewood's Rule, and see how nested allocation outperforms partitioned strategies.
Setting Booking Limits
EMSR heuristics and the spoilage-dilution tradeoff
Step through the EMSR-b algorithm, compare EMSR-a vs EMSR-b accuracy, and explore the spoilage-vs-dilution tradeoff at each booking limit.
How Many Extra Seats to Sell
Balancing empty seats against denied boardings
Run Monte Carlo simulations of show-up rates, and find the overbooking limit that maximizes expected net revenue after denied-boarding costs.
Managing a Route Network
Multi-leg itineraries and bid-price controls
Manage a small airline network: accept or reject booking requests across connecting itineraries, and compare manual decisions against LP-based bid prices.
Revenue Management Simulator
Comparing RM policies head-to-head
Pit FCFS, EMSR-b, bid-price, and dynamic pricing against each other in an airline market simulator. Watch bookings unfold in real time and compare revenue outcomes across hundreds of Monte Carlo replications.
Pricing under uncertainty and time pressure: the newsvendor problem, Gallego-van Ryzin dynamic pricing, advertising allocation, and seasonal markdown optimization.
Pricing Under Demand Uncertainty
The price-setting newsvendor problem
Jointly optimize price and order quantity when demand is uncertain, explore how the critical fractile shifts under price-dependent demand, and compare stochastic and deterministic solutions.
Real-Time Dynamic Pricing
Pricing under stochastic demand and finite inventory
Explore the Gallego and van Ryzin model of dynamic pricing. Observe the characteristic sawtooth pattern as prices rise with inventory depletion and drop on replenishment.
Advertising vs Price Cuts
The Dorfman-Steiner condition
Analyze the optimal allocation between advertising spend and price reductions using the Dorfman-Steiner condition, and explore when promotions dominate brand investment.
Fashion Season Manager
Optimizing price reductions as inventory dwindles
Simulate multi-period markdown schedules, compare optimal versus naive discounting, and observe how inventory depletion interacts with price trajectories.
Learning optimal prices from data: multi-armed bandits, structural estimation, strategic buyer behavior, and measuring price sensitivity from experiments.
Learning the Best Price
The explore-exploit tradeoff
Explore the explore-exploit tradeoff in pricing. Compare regret rates under different structural assumptions and watch bandit algorithms learn optimal prices in real time.
Pricing with Economic Models
Revealed preference and partial identification
See how revealed-preference theory and partial identification let pricing algorithms eliminate dominated prices and converge logarithmically fast.
When Buyers Game the Algorithm
Strategic behavior and corrected pricing
Discover why naive pricing algorithms fail when buyers strategically distort their features, and how corrected policies restore sublinear regret.
Why Identical Products Have Different Prices
Consumer search and price dispersion
Explore the Stigler search model, the Diamond paradox, and the Burdett-Judd resolution to understand why price dispersion persists even in competitive markets.
Pricing products that last: the Coase conjecture, leasing vs selling, planned obsolescence, launch strategies, and switching cost dynamics.
Durable Goods & the Coase Conjecture
Why durable good monopolists lose all pricing power
Prove why a patient monopolist selling a durable good must eventually price at marginal cost, and explore how the discount factor determines the rate of price decline.
Leasing vs Selling
How leasing restores monopoly pricing power
Analyze why Xerox leased and did not sell photocopiers, compare profit under three regimes (lease, commitment sell, no-commitment sell), and see how subscription models solve the durable goods problem for software.
Planned Obsolescence
Strategic durability reduction under self-competition
Explore why a monopolist may deliberately under-provide durability relative to the social optimum, and see how the durability gap varies with discount factor and production costs.
Penetration vs Skimming
Pricing strategy for new product launches
Simulate new-product launch strategies — skim high then lower price versus penetrate low to build share — and see how learning curves, network effects, and competitive entry timing determine which strategy wins.
Switching Costs & Lock-in
Bargains then ripoffs in two-period markets
Model how switching costs create invest-then-harvest dynamics: firms price below cost to build a customer base in period one, then raise prices to capture locked-in customers in period two.
Location-based pricing and real-time market clearing: Hotelling spatial competition, congestion pricing, and surge multiplier design.
Hotelling Spatial Competition
Location, transport costs, and the minimum differentiation paradox
Model how competing firms locate along a linear city and set prices in Nash equilibrium, and discover why transport costs create local market power even for identical products.
Congestion & Pigouvian Pricing
Correcting the traffic externality with optimal tolls
Derive the Pigouvian congestion toll from the BPR delay function, compare private and social marginal costs, and see how London and Singapore implemented real-world congestion charges.
City Surge Operator
How dynamic multipliers balance supply and demand
Model how surge multipliers balance rider demand against driver supply in real time, and evaluate the welfare effects of dynamic pricing versus flat-rate alternatives.
Designing subscription tiers, freemium funnels, and recurring revenue models: access fees, usage rates, churn, and lifetime value.
Subscription Platform
Access fees, usage rates, and self-selection
Design optimal subscription tiers using two-part tariff theory, explore the tradeoff between access fees and usage rates, and model how tier structure affects customer self-selection.
Where to Place the Paywall
Coming SoonConversion funnels and tier structure
Analyze the economics of freemium models — where to place the paywall, how conversion rates interact with lifetime value, and why Netflix, Spotify, and Prime structure their tiers differently.
Churn, Retention & Lifetime Value
Coming SoonModeling subscriber economics over time
Model the relationship between churn rate, retention curves, and customer lifetime value. Explore how pricing changes affect long-term subscriber economics and when to invest in retention vs acquisition.
Pricing across the value chain: double marginalization, channel coordination contracts, and transfer pricing between divisions.
Supply Chain Pricing
Double marginalization and channel coordination
Discover why independent markup decisions by manufacturers and retailers inflate the final price above the integrated monopoly optimum, and compare coordination contracts that restore channel efficiency.
Transfer Pricing
Internal prices between divisions
Set internal transfer prices between upstream and downstream divisions, compare decentralized outcomes to the integrated optimum, and discover when Hirshleifer's marginal-cost rule breaks down.
Two-sided market design: balancing buyers and sellers with asymmetric fees, network effects, adoption dynamics, and competitive platform strategies.
Network Effects & Adoption Pricing
Critical mass and same-side externalities
Simulate adoption dynamics under same-side network effects, identify critical mass thresholds, and discover how installed-base pricing differs from platform pricing.
Platform & Marketplace Pricing
Balancing buyers and sellers with asymmetric fees
Model how platforms set asymmetric prices to balance participation on both sides of the market, and discover when price structure dominates price level.
Pricing Against Competitors
Nash equilibrium and the race to the bottom
Simulate a two-player pricing game, find Nash equilibria on best-response curves, and discover why undifferentiated price competition drives margins to zero.
Welfare-maximizing pricing under regulatory constraints: peak-load pricing, the Ramsey rule, entry deterrence, and dynamic oligopoly games.
Power Grid Operator
Peak-load pricing and the Ramsey rule
Explore the Boiteux-Steiner model of peak-load pricing, where off-peak users pay only operating costs while peak users bear capacity charges, and the Ramsey rule for welfare-maximizing markups under a break-even constraint.
Competing Over Time
Entry, limit pricing, and Markov equilibrium
Model how firms make joint pricing and investment decisions in dynamic oligopolies, and explore entry deterrence, limit pricing, and Markov Perfect Equilibrium.
Testing and targeting prices: A/B test design, personalized pricing with Bayesian shrinkage, and individual deal optimization.
A/B Test Lifecycle
A/B tests, statistical power, and endogeneity
Design pricing experiments with proper statistical power, understand endogeneity threats to causal inference, and explore regression discontinuity and instrumental variable approaches.
Personalized Pricing
How finer data changes profits and consumer surplus
Design randomized pricing experiments, estimate heterogeneous demand from experimental data, construct personalized prices with Bayesian shrinkage, and analyze welfare redistribution under price targeting.
Pricing Individual Deals
Bid-response curves and quote optimization
Optimize individual price quotes using bid-response functions, and discover why the profit-maximizing win rate is well below 50%.
Measuring the causal impact of pricing changes: difference-in-differences, synthetic controls, causal forests, and staggered rollout evaluation.
Measuring Price Impact
Diff-in-diff, synthetic controls, and causal forests
Apply modern causal inference methods to pricing problems, including difference-in-differences, synthetic controls, and causal forests for heterogeneous treatment effects.
City Rollout Simulator
Staggered DiD and synthetic controls
Roll out surge pricing across 20 cities in waves. Design the rollout order, observe revenue data, and estimate the causal impact using difference-in-differences and synthetic control methods.
Price discovery through competition: auction formats, optimal reserves, position auctions for advertising, the winner's curse, and Nash bargaining.
Auctions: Private & Common Values
Formats, reserves, revenue equivalence, and the winner's curse
Compare auction formats, compute optimal reserve prices from the virtual value condition, understand the winner's curse in common-value settings, and learn rational bid shading.
Position Auctions & Ad Markets
GSP vs VCG and the keyword bidding game
Analyze the Generalized Second-Price auction used by Google and other search engines, compare GSP equilibria to VCG outcomes, and explore how click-through rates determine bid strategies.
Negotiating a Deal
Nash bargaining and alternating offers
Solve for Nash bargaining outcomes, simulate Rubinstein alternating-offers games, and discover why private information makes efficient trade impossible.
Mathematical foundations for pricing models: optimization methods, gradient ascent, convexity, probability distributions, and Bayes' rule.
Optimization for Pricing
Gradient ascent, convexity, KKT conditions
Visualize gradient ascent on pricing objective functions, explore why convexity guarantees a global optimum, and see KKT conditions in action.
Probability for Pricing
Distributions, expectations, Bayes' rule
Explore the probability distributions that underpin pricing models — Normal, Poisson, Binomial — and see how they connect to demand forecasting.
About This Project
Interactive Pricing Theory is an open-source educational project that brings canonical pricing and revenue optimization models to life through interactive web-based tools. Inspired by interactive-or.net, it is designed for graduate students in Operations Research, Economics, and Business Analytics.
The primary source material is Robert L. Phillips' Pricing and Revenue Optimization (2nd edition, Stanford University Press, 2021), supplemented by Talluri and van Ryzin's The Theory and Practice of Revenue Management (2004) and the Oxford Handbook of Pricing Management.
Key References
- Bulow, J. I. (1982). “Durable-Goods Monopolists.” Journal of Political Economy, 90(2), 314–332.
- Coase, R. H. (1972). “Durability and Monopoly.” Journal of Law and Economics, 15(1), 143–149.
- Gallego, G. & van Ryzin, G. (1994). “Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons.” Management Science, 40(8), 999–1020.
- McFadden, D. (1974). Conditional logit analysis of qualitative choice behavior. In Frontiers in Econometrics, 105–142.
- Nagle, T. T. & Müller, G. (2018). The Strategy and Tactics of Pricing: A Guide to Growing More Profitably, 6th ed.. Routledge.
- Phillips, R. L. (2021). Pricing and Revenue Optimization, 2nd ed.. Stanford University Press.
- Talluri, K. T. & van Ryzin, G. J. (2004). The Theory and Practice of Revenue Management. Springer.
- Tirole, J. (1988). The Theory of Industrial Organization. MIT Press.
- Train, K. E. (2009). Discrete Choice Methods with Simulation, 2nd ed.. Cambridge University Press.
- Wilson, R. (1993). Nonlinear Pricing. Oxford University Press.