Industry: Retail Product: ModelRisk Application: Pricing strategy
A retail category team had one SKU under price review: a private-label staple currently shelved at $27.99, costing $14.20 to land, moving about 9,000 units a week chain-wide. The pricing worksheet did what pricing worksheets do — it took the planners' best-guess own-price elasticity, plotted revenue against a few candidate prices, and pointed at $28.49 as the profit-maximizing point. Raise the price a little, it said, and weekly profit ticks up to about $124k.
The Monte Carlo rebuild in ModelRisk pointed the other way. Once the elasticity is treated as uncertain rather than a single number, and once the competitor's tendency to react to price moves is priced in, the profit surface peaks at $23.49 — four and a half dollars below the current shelf price — at a mean weekly profit of about $131,580, roughly $5,900/week (4.7%) above the status quo. The deterministic answer was not just imprecise; it was pointing uphill when the value was downhill.
Constant-elasticity demand, q(P) = q0 * (P/P0)^(-eta), has a clean closed-form profit optimum at P* = c * eta/(eta-1). That formula is exquisitely sensitive to eta: at an elasticity of 2.0 the optimal price is $28.40, but at 2.5 it is $23.67 and at 3.0 it is $21.30. A pricing committee that picks one elasticity is, without realising it, picking one of those prices and discarding the rest. Worse, the deterministic worksheet ignores the competitor entirely — and a rival that partially matches your moves makes the SKU behave more elastically than its own-price coefficient suggests, pulling the true optimum lower still. The honest object is not a price; it is a distribution of prices.
q(P) = q0 * (P/P0)^(-eta)
P* = c * eta/(eta-1)
eta
The whole decision lives in one chart — the candidate price swept across the grid, with mean weekly profit, its P10-P90 band, and the deterministic curve drawn alongside:
The simulated mean profit curve peaks at $23.49 at about $131,580/week and falls away on either side; the P10-P90 band runs from roughly $80k to $166k at the low end of the grid and narrows as price rises. The deterministic curve (mean elasticity 2.0, no competitor amplification) peaks far to the right at $28.49 (about $124,110) — almost exactly at the current shelf price, which is why the worksheet kept blessing the status quo. The current $27.99 price sits on the downhill side of the real optimum: holding it leaves an estimated $5,900/week on the table. The two curves crossing near the incumbent price is the visual signature of a deterministic model that has mistaken a slope for a peak.
Because elasticity, the market multiplier, and the competitor's intensity all vary trial to trial, each trial has its own profit-maximizing price. Collecting them gives a distribution:
The profit-maximizing price has a median of $23.99, a P10 of $20.49, and a P90 of $31.99 — a $11 spread. The single best expected-value price is $23.49, marked against the median. The body of the distribution clusters in the low-$20s, but a meaningful minority of trials (the low-elasticity, low-competition draws) want a price up near the top of the grid. This is the picture a point estimate cannot draw: the recommendation is "price around $23.50," but the confidence in that recommendation is exactly the width of this distribution, and that width is what tells the committee how much elasticity research is worth.
Setting the price at the EV-optimal $23.49 versus holding at $27.99 shifts the entire weekly-profit distribution:
At the current $27.99, weekly profit has a mean of $125,655 and a P90 of $152,248. At the EV-optimal $23.49, the mean rises to $131,580 and the P90 to $164,604 — the whole curve shifts right, not just the average. The probability that the optimal price beats the current price's profit is 61.6%. That is the honest headline: the recommended move is favourable, but it is a 62/38 bet, not a certainty — because in the high-elasticity-but-low-volume trials the cut does not pay off. A point estimate would have reported "+$5,900" with false confidence; the simulation reports the same expected gain and the one-in-three chance it disappoints.
A tornado around the EV-optimal $23.49 isolates what the profit at that price is most sensitive to:
Own-price elasticity dominates, with a +/-$9,612 half-spread on weekly profit — confirming that the single most valuable piece of pricing intelligence is a tighter elasticity estimate, worth far more than any other input. Competitor response intensity is second at +/-$6,207, which is why competitor monitoring earns its keep. The market demand multiplier (+/-$1,725) and the elasticity spread itself (+/-$1,098) are minor by comparison. The ranking tells the team exactly where to spend its next research dollar: on the elasticity coefficient, then on the competitor's reaction function.
Pricing is not "what number maximises the revenue line" — it is "where does the profit surface actually peak once elasticity and competitor reaction are both uncertain." The deterministic worksheet here recommended a price above the true optimum because it confused a single elasticity guess for a fact and left the competitor out of the model. ModelRisk put both back in, moved the recommended price down by four dollars, and attached a probability to the move so the committee knew it was making a 62/38 bet rather than a sure thing.