What is the right price?
The sales team wants a lower price. They are closest to the customer, and they can see the obvious effect: when price drops, units move.
The finance team wants a higher margin. Looking at the same tradeoff from the other side, they can see that revenue can go up while profit disappears if the extra volume is bought too cheaply.
Marketing sits between these views, but not by averaging them. Its job is to connect customer response, brand position, and commercial objectives: when to chase units or revenue, when to protect margin, and when price itself sends a positioning signal.
Both sides are partly right. Price affects volume, but volume is not always the main goal. Sometimes the business wants revenue, sometimes margin, sometimes traffic or share, or even just to clear inventory or defend a price position. The real missing piece is not the objective. It is understanding how much volume actually changes when price moves.
That is what price elasticity gives us. It does not decide the business goal. It gives the response curve that lets us compare pricing decisions against the goal, whether that goal is units, revenue, margin, or positioning.
Imagine you are deciding whether to raise the price of a top-selling cereal by 10%. Will you lose 5% of units, or 20%? That difference decides whether the price change improves sales, revenue, or margin, or whether it simply pushes shoppers away.
Price elasticity of demand measures the sensitivity of quantity sold to a change in price:
\[ \varepsilon = \frac{\% \Delta \text{Quantity}}{\% \Delta \text{Price}} \]
For normal goods, elasticity is negative: raise the price, sell fewer units. For simplicity, we usually talk about the absolute value.
The same elasticity estimate can support different pricing questions:
To estimate price elasticity, you need sales data with real price variation. Typical sources are POS scanner data, panel data, or internal online retail records. Use the effective price—revenue divided by units sold—instead of the shelf price:
\[ \text{Price}_{\text{effective}} = \frac{\text{Revenue}}{\text{Units}} \]
Effective price captures discounts, multi-buy offers, and midweek price changes more reliably than the listed price.
Two design choices set the foundation: the time grain and the product grain. Weekly is usually the best time grain because daily data is often noisy, while monthly data can hide price changes inside the month. On the product side, start at the SKU level. A single SKU has one package size, one baseline price, and one demand response, so the price signal is easier to read.
Three things to check before fitting anything:
The workhorse model is the constant-elasticity, or power-law, model:
\[ Q = b_0 \times P^{b_1} \]
where \(Q\) is quantity, \(P\) is price, \(b_0\) is a scale parameter, and \(b_1\) is the elasticity. For normal goods, \(b_1\) is negative. Taking logs gives a linear relationship, \(\log Q = \log b_0 + b_1 \log P\), so \(b_1\) is the slope in log-log space. The model assumes elasticity is constant across the price range. That is a strong assumption, but it is a reasonable starting point for many consumer goods.
Fit the curve in the original, non-log, space with scipy.optimize.curve_fit to avoid the retransformation bias that comes from fitting in log-space and converting back.
import numpy as np
from scipy.optimize import curve_fit
def power_model(price, b0, b1):
"""Q = b0 * Price^b1"""
return b0 * np.power(price, b1)
# Initial parameter guesses
p0 = [df["units"].mean(), -1.5]
# Fit the model
popt, pcov = curve_fit(
power_model,
df["price"].values,
df["units"].values,
p0=p0,
maxfev=10000,
)
b0, b1 = popt
print(f"Scale (b0): {b0:.1f}")
print(f"Elasticity (b1): {b1:.2f}")The companion notebook also runs a two-pass estimation: fit once on all data, flag observations where actual divided by predicted units falls outside a sensible band, such as 0.3 to 3.0, and refit without them.
# Pass 1: fit on all data
popt_1, _ = curve_fit(power_model, prices, units, p0=p0, maxfev=10000)
# Identify outliers
predicted = power_model(prices, *popt_1)
ratio = units / predicted
mask = (ratio > 0.3) & (ratio < 3.0)
# Pass 2: refit on clean data
popt_2, pcov_2 = curve_fit(
power_model, prices[mask], units[mask], p0=popt_1, maxfev=10000
)Promotional spikes are the usual culprit. Include them blindly, and the model treats promotional volume as normal price response, biasing elasticity away from its true value.
Across the eight top-selling Fluid Milk items, elasticities range from \(-1.2\) for gallon jugs, a staple format with lower elasticity, to \(-3.9\) for smaller formats, which are easier to substitute. Figure 2 shows the fitted curves side by side.
In practice, an elasticity in the range \(-0.5\) to \(-5\) with \(R^2 > 0.4\) and several distinct price points is usable for pricing decisions. Estimates outside that range are often data problems: too few prices, a single promotional discount dominating the fit, or too short a window. If the elasticity is not believable, neither is any pricing recommendation built on it.
Once we have a demand curve, we can ask different pricing questions.
The first is which price maximizes revenue:
\[ \text{Revenue}(P) = P \times Q(P) \]
This is often the most intuitive starting point because it requires only price and units. If the business does not have reliable variable cost by product or category, revenue may be the practical metric for a first-pass pricing analysis.
But revenue is not profit. If variable cost is available, we can ask the stronger question: which price maximizes contribution margin?
\[ \text{Contribution Margin}(P) = (P - C) \times Q(P) \]
where \(C\) is variable cost per unit and \(Q(P) = b_0 \times P^{b_1}\).
The two answers can differ. A lower price can sell more units and increase revenue while reducing total margin. A higher price can reduce revenue but improve margin if the remaining units are profitable enough. The point is not that margin is always the only objective. The point is that the objective must be explicit before choosing the price.
Take a private-label cereal, the same category as the part-introduction promotion story, with these parameters:
| Price | Units | Revenue | Margin per Unit | Total Margin |
|---|---|---|---|---|
| $3.59 (10% lower) | 1,776 | $6,376 | $1.39 | $2,469 |
| $3.99 (current) | 1,500 | $5,985 | $1.79 | $2,685 |
| $4.39 (10% higher) | 1,288 | $5,654 | $2.19 | $2,821 |
| $4.79 (20% higher) | 1,121 | $5,369 | $2.59 | $2,903 |
The table shows the core tradeoff. Revenue is maximized by cutting the price: the 10% lower price has the highest revenue at $6,376. Margin is maximized by raising the price: the 20% higher price has the highest total margin at $2,903. At 10% higher, revenue falls 5.5% while margin rises 5.1%. Revenue and margin point in opposite directions. This is the discount trap from the part introduction, now with numbers attached.
Figure 3 visualizes the same logic for five real Dunnhumby products. In every case, the revenue curve and contribution margin curve peak at different prices. For these highly elastic private-label dairy items, the margin-maximizing price is consistently lower than the current price because the extra volume from a price cut is large enough to lift total margin.
The optimization itself is a one-dimensional search. scipy.optimize.minimize_scalar over the negative margin function does the job in a few lines, and the companion notebook walks through it.
from scipy.optimize import minimize_scalar
def neg_margin(price, b0, b1, variable_cost):
"""Negative contribution margin (for minimization)."""
quantity = b0 * price ** b1
return -(price - variable_cost) * quantity
result = minimize_scalar(
neg_margin,
bounds=(2.50, 6.00),
method="bounded",
args=(b0_fitted, b1_fitted, 2.20),
)
optimal_price = result.x
optimal_margin = -result.fun
print(f"Optimal price: ${optimal_price:.2f}")
print(f"Max weekly margin: ${optimal_margin:,.0f}")The full pipeline covers data loading, quality checks, two-pass estimation, and the sensitivity table.
price_elasticity_scanner_data.ipynb — Open in Colab · nbviewer
Committed with outputs; Colab is best-effort. Full list on the Notebooks & Code page.
The same category can contain products with very different shopper roles. In Fluid Milk Products, a gallon of white milk, a single-serve chocolate milk, and a quart of coffee creamer are not interchangeable pricing decisions. They may sit in the same broad commodity, but shoppers buy them for different occasions and compare them against different alternatives.
That is why SKU-level estimates should be aggregated only into comparable SKU groups, such as the same sub-commodity and similar package size. The goal is not to create as many segments as possible. The goal is to avoid averaging together products whose price response should be different.
Before applying one pricing rule across a category, check whether major product roles respond differently. If they do, the average elasticity is not a pricing rule. It is a warning that the category needs to be split more carefully.
Endogeneity. Prices are not set at random. If a retailer raises prices during high-demand periods like back-to-school, the observed elasticity is biased toward zero because high prices and high demand co-occur for reasons unrelated to pricing. This is the same selection bias problem from Part 3. Address it with instrumental variables, randomized price experiments, or quasi-experimental designs.
Reference price effects. Customers remember what they paid before. Raising the price usually causes a bigger drop in volume than lowering the price brings in new buyers. The JCPenney “Fair and Square” case is the famous example: switching from promotions to everyday low prices at the same average level led to a sales collapse because customers lost their price anchor.
Competitor reaction. A price cut only works if competitors hold. In concentrated categories, competitors typically match. Everyone sells about the same volume, but at lower margins. Game theory matters more than demand curves there.