Tiered Demand Curve Estimator

Pricing Analytics Demand Modeling

Estimate per-tier demand curves and substitution patterns when you sell multiple vertically-differentiated products (e.g., Basic / Premium / Elite) to one market and observe how each tier's sales respond to its own price changes — and to the prices of its sibling tiers.

OVERVIEW & WHAT THIS TOOL SOLVES

When you sell a vertically differentiated product line—such as Basic, Standard, and Premium subscription tiers—raising the price of your Standard plan doesn't just lose customers to competitors; it pushes some price-sensitive buyers down to Basic and nudges others up to Premium. Fitting separate single-product regressions for each tier ignores cannibalization and breaks down because sibling prices move together. Tiered Demand Curve Estimator applies Berry (1994) Aggregate Discrete-Choice Logit Inversion (with optional 2SLS instrumental variables) to estimate all tier demand curves jointly and recover the full own- and cross-price elasticity matrix from market-level sales snapshots.

🎯 Questions It Answers

When we raise the price of one product tier, how many lost buyers switch to our sibling tiers vs. abandoning us for the outside good?
What is each tier's intrinsic baseline quality appeal (α_j) after stripping out price differences?
Is historical price endogeneity biasing our price sensitivity estimate, and how much does 2SLS cost-instrumenting correct it?

📂 What Data You Need

Period-Level Market Snapshots: Observed price (p_jt) and unit sales or share (s_jt) for each tier across T time periods/markets
Total Market Potential (M_t): Addressable market size per period so the outside no-purchase share (s_0t) can be computed
Optional Cost Shifters / Instruments: A cost index or other shifter that moves price but not demand, to run Two-Stage Least Squares (2SLS)

💡 What You'll Discover

Multi-Tier Demand & Substitution Curves: How varying one tier's price bends both its own volume and every sibling tier's volume
Full J × J Elasticity Matrix: Own-price elasticities on the diagonal and cross-price substitution elasticities off-diagonal
Price Simulator & Customer Flow: How each tier's share and the no-purchase share move when you reprice one or more tiers

🔍 Key Nuances & Common Traps to Watch For

The Outside Good (s_0t) Is Essential: In Berry's inversion, shares are normalized against the fraction of the addressable market that bought nothing (s_0t = 1 − ∑ s_jt). Without an outside option, raising all tier prices by $10 simultaneously would falsely predict zero total customer loss!
Price Endogeneity & When to Use 2SLS: If managers raised prices during high-demand periods (unobserved demand shock ξ_jt > 0), OLS will underestimate how much customers dislike high prices (β will be biased toward zero or even positive). Supplying a cost instrument via 2SLS isolates clean price variation, but only when the instrument is strong: if the first-stage F is below 10, the 2SLS estimate is not reliable. The streaming case shows the correction working, the gym case shows prices that needed no correction, and the software case shows a weak instrument.
IIA Proportional Substitution Property: Standard aggregate logit implies that when Tier j raises its price, departing customers switch to Alternative k in proportion to k's market share (producing identical cross-elasticities across rows in column k).

📐 Under the Hood: Model Equations & Variable Definitions

1. Random-Utility Model & Multinomial Logit Market Shares

\[ U_{ijt} = \alpha_j - \beta p_{jt} + \xi_{jt} + \varepsilon_{ijt}, \qquad s_{jt} = \frac{\exp(\alpha_j - \beta p_{jt} + \xi_{jt})}{1 + \sum_{k=1}^{J} \exp(\alpha_k - \beta p_{kt} + \xi_{kt})} \]

💬 Read it in plain English: Each consumer chooses the tier (or outside no-purchase option, whose utility is normalized to 0) that gives them the highest net utility—balancing tier quality α_j against price penalty β·p_jt—which aggregates into the multinomial logit market share formula.

U_ijt

Consumer Utility for Tier jOUTPUT ⓘ

Net utility customer i derives from tier j in period/market t (outside option utility U_i0t = ε_i0t).

α_j

Tier Fixed Effect (Intrinsic Quality)ESTIMATED ⓘ

Baseline vertical quality/brand appeal of tier j relative to not buying at all.

Price Sensitivity ParameterESTIMATED ⓘ

Marginal disutility of paying $1 more (expected β > 0 since entered with a minus sign −β·p_jt). The results table reports the price coefficient itself, which is −β, so it should be negative there.

p_jt, s_jt

Tier Price & Market ShareYOUR DATA ⓘ

Observed price p_jt and share of total addressable market s_jt = q_jt / M_t for tier j in period t.

ξ_jt, ε_ijt

Unobserved Demand Shock & Idiosyncratic TasteINFERENCE ⓘ

Market-level unobserved demand shock ξ_jt and consumer i’s Type-I Extreme Value taste draw ε_ijt.

2. Berry (1994) Log-Share Inversion & Own/Cross-Price Elasticities

\[ \ln\!\left(\frac{s_{jt}}{s_{0t}}\right) = \alpha_j - \beta p_{jt} + \xi_{jt}, \qquad \eta_{jj} = -\beta p_j (1 - s_j), \qquad \eta_{jk} = \beta p_k s_k \; (j \neq k) \]

💬 Read it in plain English: Dividing each tier's share by the no-purchase share and taking the natural log transforms a nonlinear choice system into a clean linear regression we can fit with OLS or 2SLS, directly yielding every tier's own-price elasticity (η_jj) and cross-price substitution elasticity (η_jk).

s_0t

Outside Good (No-Purchase) ShareYOUR DATA ⓘ

Share of addressable market M_t that did not buy any tier in period t: s_0t = 1 − ∑_{k=1}^J s_kt.

ln(s_jt / s_0t)

Berry Linearized Log-Share RatioOUTPUT ⓘ

Dependent variable of the Berry inversion equal to mean tier utility δ_jt = α_j − β·p_jt + ξ_jt.

η_jj

Own-Price Elasticity of Tier jOUTPUT ⓘ

Percentage change in tier j’s share per +1% increase in its own price p_j (negative when β > 0).

η_jk (j ≠ k)

Cross-Price Substitution ElasticityOUTPUT ⓘ

Percentage increase in tier j’s share when sibling tier k raises its price p_k by +1%.

DATA SOURCE

Load a tiered-pricing case study:

📊 Tiered-Pricing Case Studies

Pre-built datasets where one firm sells multiple vertically-differentiated tiers. Each scenario varies tier prices over time so you can recover per-tier demand curves and substitution patterns. Loading a case shows what one period is, how many potential customers each period holds and what its instrument column measures.

Required columns: period, product, price, units_sold.
Optional: instrument (a numeric cost index or other price shifter that does not move demand; enables 2SLS), outside_share (per-period no-purchase share, enables that outside-option mode).
One row per (period × product). Tool supports 2 to 5 tiers.

Drag & drop CSV (.csv, .tsv, .txt, .xls, .xlsx)

Long format: period, product, price, units_sold (+ optional instrument, outside_share)

No file uploaded.

RESULTS

📖 How to read these results:

Coefficient table — sanity check that the price coefficient is negative and significant. The tier dummies tell you the implied quality ordering (higher α = more desirable tier at equal prices).
Demand curves — one per tier, showing predicted share at different prices for that tier (with sibling tiers held at their typical prices). Use the tabs to switch between tiers.
Elasticity heatmap — the marketer's payoff. Diagonal cells = own-price sensitivity (negative numbers). Off-diagonal cells = substitution (positive = those tiers steal from each other when prices move).
Interpretation block — we translate the numbers into plain-English takeaways for each tier. Read this first if you're in a hurry.

Coefficient estimates

Per-tier demand curves

Each curve sweeps one tier's price across ±20% of its observed range, holding the other tiers' prices at their observed medians. Dots = observed (price, share) pairs.

Price elasticity matrix

Cell (row j, col k) = % change in tier j's share per 1% change in tier k's price, evaluated at observed median prices. Diagonal = own elasticity (negative); off-diagonal = cross elasticity (positive for substitutes).

Interpretation

🎛️ Price simulator: what if you re-price the line?

Drag each tier's price and watch demand flow between products in real time. The model reallocates customers across tiers and the "No purchase" outside option — so you can see who steals from whom and who walks away entirely. Starts from each tier's typical (median) observed price.

Shares always sum to 100% (including no-purchase). This is a model extrapolation held at the estimated coefficients — treat large price moves (far outside your observed range) as directional, not precise.

Diagnostics

What am I looking at?

Residuals vs. fitted: Should be a structureless cloud. Funnels or curves indicate misspecification (e.g., the price coefficient is not constant across tiers — try the per-tier slopes toggle).

Per-tier price CV: The coefficient of variation of each tier's price. Below 5% = the price coefficient may be poorly identified for that tier.

Tiered Demand Curve Estimator

👨‍🏫 Professor Mode: Guided Learning Experience