Shrink Ray vs. 20% Free Model
A simple story + transparent equations + an optional simulation check

A guided story

You will see why changing Q (quantity in the box) can mimic a price change, and when a firm prefers "Shrink Ray" (smaller quantity) versus "20% Free" (larger quantity).

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1) What changes when the sticker price stays the same?

Imagine a product that usually costs P dollars. One day the sticker price is unchanged, but the amount inside the package is different.

Key idea

Holding P fixed and changing Q changes the unit price P / Q. Some shoppers notice unit price changes; others mostly do not.

Try it (watch unit price move)
Unit price
Why not just change the sticker price?

In many retail settings, changing shelf prices is costly (menu costs), and consumers can be more sensitive to sticker price changes than quantity changes. That makes quantity an attractive "quiet" lever for firms.

2) Two kinds of shoppers

The model is built around a simple behavioral split:

Informed vs uninformed
  • Informed shoppers look at unit price and compare P / Q to their value per gram.
  • Uninformed shoppers mostly respond to the sticker price P, unless the quantity drop is obvious.

In the interactive model, values are "willingness to pay": how much someone personally thinks the product is worth.

Try it (one shopper of each type)
Informed decision
Uninformed decision
Math details (optional) 
Informed buys if:   V_i >= P/Q
Uninformed buys if: V_u >= P   and (optionally) Q > Q*

3) From people to demand

A single person is unpredictable, but a crowd becomes predictable if you assume a distribution of values. The paper uses a uniform distribution to get clean closed-form equations.

Assumptions you can see

Here V_I and V_U are the maximum possible values in each group. Bigger maxima mean more people are willing to buy.

Informed demand D_i(Q)
Uninformed demand D_u(P)
Total demand
Math details (optional) 
Assume: V_i ~ Uniform[0, V_I] and V_u ~ Uniform[0, V_U]

D_i(Q) = max(0, 1 - P/(Q * V_I))
D_u(P) = max(0, 1 - P/V_U)

Total demand = alpha * D_i(Q) + (1 - alpha) * D_u(P)
(In strict mode, D_u is multiplied by 1{Q > Q*}.)

4) Profit and regimes

The firm's margin per sold box is P - C*Q. Lowering Q reduces cost (good) but raises unit price (bad for informed demand).

Try it (see the tradeoff)
Margin per box
Expected profit (per customer)
Regime boundary
Two typical outcomes
  • Shrink Ray regime: Q is low enough that informed demand collapses, so profit comes mostly from uninformed buyers.
  • Normal regime: Q is high enough that informed buyers still participate, which pushes the firm toward a larger quantity.
Math details (optional) 
Margin per box: (P - C*Q)

Expected profit per customer:
pi(Q) = (P - C*Q) * [ alpha*D_i(Q) + (1-alpha)*D_u(P) ]

The simulation tab computes analytic expectations from these equations, then runs a Monte Carlo simulation to show how close the sampled outcome is.

5) Try Shrink Ray vs 20% Free

Both tactics can use the same packaging: the firm changes how much product is inside while keeping the sticker price the same.

  • Shrink Ray: decrease Q -> unit price goes up -> informed buyers are the first to leave.
  • 20% Free: increase Q -> unit price goes down -> informed buyers are more likely to buy.
What to do next
  1. Open the simulation.
  2. Pick a scenario (Shrink-ray likely vs "20% Free").
  3. Click Run, then compare simulated profit to analytic profit.
  4. Use "Set Q to argmax" to see what quantity the model prefers for your settings.

Transparency note: the simulation uses the same buy rules as the tutorial, sampled from uniform distributions.

Regime
Unit price
Margin/box
Analytic profit
Sim profit
Error
processed - sold () - analytic sold

Analytics

Total demand is the mixture alpha*D_i(Q) + (1-alpha)*D_u(P). In Strict mode, D_u is multiplied by 1{Q > Q*}.