Cross-Elasticity Matrix¶
Analytical matrix — instantiates Demand Curve Calibration and Response Design
Maps how demand for each item responds to price changes in every other item, exposing which goods are substitutes, which are complements, and where demand merely moves rather than disappears.
A Cross-Elasticity Matrix is a square table whose cell (i, j) records how much the quantity sought of item i changes when the generalized cost of item j moves. The diagonal holds own-price responses; everything off the diagonal is the story most single-good models miss — that raising the cost of one option pushes demand sideways into another. A positive cell means substitutes (dearer j sends buyers to i); a negative cell means complements (dearer j drags i down with it). Its defining purpose is not to size any one curve but to trace the connective tissue between them, so a reduction booked in one place can be recognized as a diversion to somewhere else rather than a genuine loss of demand.
Example¶
A metro transit agency is about to raise peak fares and wants to know where the riders go, not just how many leave. It assembles a cross-elasticity matrix across five "goods": peak rail, off-peak rail, the parallel bus, park-and-ride parking, and ride-hail. Using fare and usage histories, it estimates that a 10% peak-fare rise cuts peak rail trips by roughly 4% — but the matrix shows that most of that 4% reappears as positive entries elsewhere: off-peak rail (riders shift their trip an hour), the bus, and parking (they drive and park). Only a small slice shows up as trips not taken at all.
The off-diagonal cells change the decision. A revenue forecast built from the own-elasticity alone would have overstated the ridership loss and ignored that the agency's own parking lots would fill and its buses would crowd. One cell even comes back negative — evening rail falls slightly when peak fares rise, a complementarity, because some riders drop the whole round trip. The matrix turns "we will lose 4%" into "we will displace most of it onto assets we also run," which is a different capacity and pricing conversation.
How it works¶
- Fix the item set. Decide which goods form the rows and columns — your own options plus the substitutes and complements that actually absorb diverted demand.
- Estimate each cell. From price/usage variation, natural experiments, or a choice model, estimate the percentage change in i per percentage change in the cost of j.
- Read sign and magnitude. Positive off-diagonal = substitute; negative = complement; near-zero = independent. Diversion ratios fall straight out of the substitute cells.
- Reconcile the flows. Check that quantity leaving one good is accounted for by arrivals elsewhere versus true attrition, so displacement is not double-counted as disappearance.
Tuning parameters¶
- Item granularity — coarse rows (rail vs. road) are estimable but hide within-mode shifts; fine rows (each line, each hour) reveal the real diversion path but starve each cell of data.
- Symmetry assumption — imposing that cell (i, j) mirrors (j, i) stabilizes a sparse matrix but can hide genuine asymmetries in how two goods substitute.
- Cost dimension varied — the matrix can be built on money price, wait time, or any generalized cost; the substitution pattern can differ sharply by which lever moves.
- Estimation window — short windows catch immediate switching; long windows catch the slower migration (buying a car) that short ones miss.
When it helps, and when it misleads¶
Its strength is catching substitution leakage: it prevents a local demand reduction from being booked as a system-wide one, and it flags complementarities where cutting one good quietly kills another. For any decision where the alternatives are also yours to manage, the off-diagonal is where the money is.
Its danger is over-reading elasticities estimated at one set of prices as if they held everywhere — the cellophane fallacy is the canonical trap: at a high prevailing price, almost everything looks like a substitute, so a matrix read at that point can badly overstate how interchangeable goods really are at normal prices.[n1] Cells are also noisy and easily confounded (a promotion on j coincides with weather that moved i), and a symmetric-looking table can lend false confidence. The discipline is to estimate cells from real cost variation, annotate the price region each was measured in, and treat off-diagonal entries as directional evidence about where demand flows rather than precise coefficients.
How it implements the components¶
cross_elasticity_matrix— the mechanism is this component: the populated table of own- and cross-price responses.choice_set_and_substitution_map— deciding the rows and columns forces an explicit map of which alternatives compete, and the signs turn it from a list into a wiring diagram of substitutes and complements.demand_shift_classifier— the reconciliation step distinguishes demand that displaced to another good (off-diagonal arrival) from demand that genuinely left, which is the substitution-vs-attrition classification.
It does not size any single good's own-price slope or saturation curve (elasticity_and_threshold_profile) — that is the Price Sensitivity Experiment — nor does it value the goods in dollars (consumer_surplus_or_value_proxy), which belongs to the Conjoint or Discrete Choice Model; this matrix only measures how the goods pull on each other.
Related¶
- Instantiates: Demand Curve Calibration and Response Design — supplies the substitution and complementarity structure the schedule needs to avoid mistaking displacement for reduction.
- Consumes: Revealed Preference Choice Log supplies the observed switching behavior the cells are estimated from.
- Sibling mechanisms: Conjoint or Discrete Choice Model · Demand Curve Estimation Workbook · Demand Segmentation Dashboard · Equity Access Impact Review · Price Sensitivity Experiment · Revealed Preference Choice Log · Scenario Demand Stress Test · Shadow Price Probe · Waitlist and Stockout Analysis
Editorial Notes¶
Form Classification¶
Form family: Analysis, Modeling & Optimization
Rationale: Cross-Elasticity Matrix operates as a computation, comparison, model, or analytic representation used to infer, estimate, or choose because it maps how demand for each item responds to price changes in every other item, exposing which goods are substitutes, which are complements, and where demand merely moves rather than disappears.
Independent corroboration: The frozen evidence defines Cross-Elasticity Matrix as 'Maps how demand for each item responds to price changes in every other item, exposing which goods are substitutes, which are complements, and where demand merely moves rather than disappears', so its operative form is Analysis, Modeling & Optimization.
Review outcome: Independent reviewer agreement; high confidence.
Origin Attribution¶
Primary origin: Economics & Finance
Origin pattern: Single lineage
Present-day reach: Specialized
Rationale: Cross-elasticity is an economic measure of one quantity's response to another's price; statistical estimation operationalizes the matrix.
Related originating lineages:
- Statistics & Experimental Design — Regression and uncertainty estimation supply the empirical matrix of cross-elasticities.
Review resolution: Cross-elasticity is an economic measure of one quantity's response to another's price; statistical estimation operationalizes the matrix.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
[n1] The cellophane fallacy, named from a mid-century US antitrust case, is the error of inferring that goods are close substitutes because cross-elasticity is high at the prevailing price — when that price is already elevated, buyers are near the margin of switching, so almost everything looks substitutable. Cross-elasticities must be read together with the price level at which they were measured. ↩