Optimal Pricing by Season: Does Countercyclical Pricing Make Sense?

The Lerner rule derived in the previous article gives a direct way to answer this. It does not care about intuitions or seasonal folklore — it only cares about one number, own-price elasticity, and it returns a markup. Split that number by season, run the formula twice, and the question stops being a matter of instinct.

The Executive Question

Progresso raises price in the off-season, when the category is at its seasonal trough. Campbell's does not. Is Progresso's behavior a pricing error, or does the elasticity data justify it?

Answering this requires exactly two ingredients: an elasticity estimate for winter, an elasticity estimate for everything else, and one formula applied twice.

Splitting Demand by Season

The same log-log specification from Chapter 8.1 — $\ln Q = \beta_0 + \beta_1 \ln P + \cdots$ — is re-estimated separately on winter weeks and non-winter weeks, both with store fixed effects. The two samples are large and the two estimates are precise:

Both estimates sit comfortably in the elastic zone ($\epsilon < -1$), so the pricing formula from Chapter 8.2 applies cleanly to each. The confidence intervals do not overlap: winter demand for Progresso is measurably, not just numerically, more elastic than non-winter demand. That is the opposite of what a "discount when demand is weak" instinct would assume — it is not that off-season shoppers are indifferent to price and winter shoppers are price-sensitive; the data says the reverse.

A quick intuition for why: in winter, soup is a category everyone is buying, private label and competing brands included, so a shopper who finds Progresso pricier than usual has the easiest possible substitution sitting right next to it on the shelf. In the off-season, soup buyers are a smaller, more specific group — closer to a habitual purchase than an impulse comparison — and that selection makes remaining demand less sensitive to price.

Applying the Lerner Rule, Twice

Recall the constant-elasticity optimal pricing rule derived last chapter:

Holding marginal cost fixed at $MC = \$1.00$ — the same assumption used throughout this chapter — plug in each season's elasticity separately:

The formula recommends a higher price in the non-winter months and a lower price in winter — roughly a 12-cent swing on a dollar of marginal cost, or about an 8% seasonal price differential.

Reading the Result: Does the Real Strategy Hold Up?

The mechanism is the Lerner index working exactly as designed: less elastic demand means a smaller $1/|\epsilon|$, which means a bigger markup survives before volume erodes it away. Non-winter demand is the less elastic of the two seasons, so the formula assigns it the bigger markup — a higher optimal price — and winter, the more elastic season, gets the smaller markup and the lower optimal price.

That is precisely the shape of Progresso's real pricing behavior: raise price in the off-season, hold it lower when the category is at its winter peak. Run through the logic plainly and it stops looking backwards:

• Winter demand is more elastic, not less — shoppers stocking up on soup in peak season have the most substitutes conveniently in view, and Progresso's volume is more exposed to a price increase precisely when everyone is buying.
• Non-winter demand is less elastic — the shoppers still buying soup in July are a more captive, less price-comparing group, and Progresso can carry a higher markup without losing them.

On this evidence, Progresso's countercyclical habit is not an error to be corrected — it is what the inverse-elasticity rule would recommend if you ran it by season instead of once a year. Campbell's flat, un-seasonalized pricing is the more suspect strategy by comparison: if its own elasticity swings similarly by season (this analysis does not establish that — it would take the identical split re-run on Campbell's data), a flat price is leaving exactly the seasonal margin on the table that Progresso is capturing.

From a Quarterly Recalculation to Continuous Repricing

Nothing about this exercise is conceptually different from what a ride-hailing app does when it raises fares during a Friday-night surge, or what an e-commerce repricer does dozens of times a day on a single SKU. The economics are the same Lerner logic: elasticity is not a single number belonging to a product, it is a number that shifts with context — time of day, season, local competition, even the specific customer being priced. Surge pricing is, structurally, the seasonal split in this article carried to its limit: instead of re-estimating elasticity once a quarter and setting a price for the season, an algorithmic pricing engine re-estimates (or approximates) elasticity continuously and recomputes $P^*$ in real time.

What changes moving from this chapter's hand calculation to an automated system is not the formula — it is the frequency of re-estimation, the granularity of the segment the elasticity is computed on, and the speed of the feedback loop that catches a bad estimate. The economic logic taught here is the same logic running, largely invisibly, behind every dynamic price a customer sees today.