algo-price-elasticity

# Price Elasticity of Demand

## Overview

Price elasticity measures the percentage change in quantity demanded for a 1% change in price. Ed = %ΔQ / %ΔP. |Ed| > 1 = elastic (price-sensitive), |Ed| < 1 = inelastic (price-insensitive). Critical for pricing decisions and revenue optimization.

## When to Use

**Trigger conditions:**
- Estimating how a price change will affect unit sales and revenue
- Determining if demand is elastic or inelastic for a product
- Optimizing price for maximum revenue or profit

**When NOT to use:**
- When you need consumer willingness-to-pay distribution (use Van Westendorp or conjoint)
- When pricing multiple products together (use bundle pricing)

## Algorithm

```
IRON LAW: Elasticity Is NOT Constant Along a Linear Demand Curve
It varies at every price point. At high prices, demand is elastic
(small price increase → big volume drop). At low prices, demand is
inelastic. Always calculate at the SPECIFIC price point of interest.
Revenue-maximizing price is where Ed = -1 (unit elastic).
```

### Phase 1: Input Validation
Collect: price-quantity pairs over time (or across markets). Control for: seasonality, promotions, competitor actions, other confounders.
**Gate:** Minimum 10 price-quantity observations, confounders identified.

### Phase 2: Core Algorithm
**Point elasticity:** Ed = (dQ/dP) × (P/Q) at a specific price point
**Arc elasticity:** Ed = ((Q₂-Q₁)/((Q₂+Q₁)/2)) / ((P₂-P₁)/((P₂+P₁)/2)) between two points
**Regression method:** log(Q) = α + β×log(P) + controls → β is the elasticity (constant elasticity model)

### Phase 3: Verification
Check: sign should be negative (price up → quantity down). Cross-validate with holdout periods.
**Gate:** Elasticity is negative, confidence interval is reasonable.

### Phase 4: Output
Return elasticity estimate with revenue impact projection.

## Output Format

```json
{
  "elasticity": -1.5,
  "interpretation": "elastic — 1% price increase → 1.5% quantity decrease",
  "revenue_impact": {"price_change_pct": 10, "quantity_change_pct": -15, "revenue_change_pct": -6.5},
  "metadata": {"method": "log-log regression", "r_squared": 0.82, "observations": 52}
}
```

## Examples

### Sample I/O
**Input:** Price increased 10% from $100 to $110, quantity dropped from 1000 to 850
**Expected:** Arc elasticity = ((-150/925) / (10/105)) = -1.70 (elastic)

### Edge Cases
| Input | Expected | Why |
|-------|----------|-----|
| Luxury good | May be positive (Veblen) | Higher price → higher perceived value |
| Necessity (insulin) | Near zero | Demand barely responds to price |
| Perfect substitute available | Very elastic (< -3) | Customers switch immediately |

## Gotchas

- **Omitted variable bias**: Without controlling for advertising, seasonality, and competitor prices, elasticity estimates are biased.
- **Short-run vs long-run**: Short-run elasticity is typically lower (customers are locked in). Long-run gives them time to find substitutes.
- **Cross-price elasticity**: Demand for product A may depend on product B's price. Ignoring this in a portfolio context leads to suboptimal pricing.
- **Asymmetric elasticity**: Consumers may react differently to price increases vs decreases. Don't assume symmetry.
- **Small sample noise**: With few observations, elasticity estimates have wide confidence intervals. Report intervals, not just point estimates.

## Scripts

| Script | Description | Usage |
|--------|-------------|-------|
| `scripts/arc_elasticity.py` | Compute arc elasticity and revenue impact | `python scripts/arc_elasticity.py --help` |

Run `python scripts/arc_elasticity.py --verify` to execute built-in sanity tests.

## References

- For regression-based elasticity estimation, see `references/regression-estimation.md`
- For cross-price elasticity analysis, see `references/cross-price.md`