Lab 4: Pricing on a Whiteboard
Textbook reading: Kreps, Chapters 6 and 7. This lab assumes you have had both classes 6 and 7.
Key Ideas
From data to a price
- Classes 6 and 7: know demand and marginal cost, set MR = MC
- No real firm knows its demand curve; it has data: a few prices tried, and what it sold at each
- Today: start from data and work all the way to a price, with nothing but a marker
Problem of the Day
Board notes
- Brightside Coffee: monthly subscription box
- Cost $12 per box; fixed cost $20,000 per month
- Current price $25 everywhere
- Data: 5 test cities, price and subscribers (below)
| Price | Subscribers |
|---|---|
| $20 | 3,650 |
| $25 | 2,950 |
| $30 | 2,450 |
| $35 | 1,750 |
| $40 | 1,150 |
Part 1. Find the demand line. Four analysts each propose a straight-line demand function:
- q = 5,000 − 90p
- q = 6,000 − 120p
- q = 7,000 − 150p
- q = 6,500 − 120p
Which of these four fit the data most closely?
Part 2. Elasticity at $25. Using the line from Part 1, compute predicted subscribers at $25 and at $26. Compute the elasticity. What does that elasticity tell you about whether Brightside coffee may be profit maximizing at a price of $25?
Part 3. Profit Grid. Brightside’s CEO wants a recommended price by the end of the meeting, and your group has time to compute profit at only a handful of prices. Using the line, the $12 cost per box, and the $20,000 fixed cost, compute monthly profit at $25 and then at prices of your choosing. What price do you recommend to the CEO?
Part 4. Check your profit maximization. Find the elasticity at the profit maximizing price you recommended to the CEO. In class 7, we learned that \(MR = P(1 + \frac{1}{\varepsilon})\). At the best price, \(MR\) should equal \(MC\). Is this what you find?
Part 5. What if. The box now costs $18 to fill. Which way must the profit-maximizing price move?
Summarize. List everything a firm needs to know to be able to set its price.
Extensions
Use the line from Part 1 to predict subscribers at $10 and at $60. Would you trust either prediction? How far from the tested prices should Brightside trust the model, and what does that say about the $31 recommendation?
In student markets Brightside believes the elasticity at $25 is −3. Using the markup rule and the $12 cost, what should the student price be?
Solutions
Published after the last section has met.
Part 1. Line (B). Its predictions are 3,600, 3,000, 2,400, 1,800, 1,200, within 50 of the data at every price. With price on the vertical axis, (A) is too steep: it loses only 90 subscribers per dollar, so it predicts too few subscribers at $20 and too many at $40. (C) is too flat: 150 per dollar, so too many at $20 and too few at $40. (D) has the right slope but sits about 500 subscribers to the right of the data at every price. Each dollar of price costs 120 subscribers; the line meets the price axis at $50.
Part 2. Predicted subscribers at $25 = 3,000; elasticity ≈ −1.0. Demand is about unit-elastic, so marginal revenue is roughly zero: Brightside is near the revenue-maximizing price, well below the profit-maximizing one given its $12 marginal cost.
Part 3. Profit peaks at $31: 2,280 subscribers and $23,320 per month, against $19,000 at $25, a gain of $4,320 a month. A typical search goes up from $25 (unit-elastic, so MR is zero and well below MC), overshoots to $35 or $40, and closes in from both sides. Profit is symmetric around the peak: $28 and $34 both earn $22,240, $25 and $37 both earn $19,000. With a straight-line demand curve, profit as a function of price is a symmetric hump, so once you have two prices with equal profit, the best price is halfway between them.
Part 4. At $31 the line predicts 2,280 subscribers, and at $32 it predicts 2,160, a fall of 5.3% for a 3.2% price rise, so \(\varepsilon\) ≈ −1.63. Then MR = 31 × (1 − 1/1.63) ≈ $12.0, equal to marginal cost, so $31 passes the check. The markup is (31 − 12)/12 ≈ 158%, and −100/(1 + \(\varepsilon\)) = −100/(−0.63) ≈ 158%. They must agree because the markup rule is MR = MC rearranged: setting P(1 + 1/\(\varepsilon\)) = MC and solving for P/MC gives the same equation. For comparison, at $25 the elasticity is −1, so MR = 0, far below $12: the price should rise, which is what Part 3 found.
Part 5. At $31, MR is still $12 but MC is now $18, so the price should rise. At $34 the line predicts 1,920 subscribers, \(\varepsilon\) ≈ −2.13, and MR = 34 × (1 − 1/2.13) ≈ $18.0, so $34 is the new best price, a rise of $3, half the cost increase. With straight-line demand the best price is halfway between marginal cost and the price at which demand hits zero, so a $6 cost rise moves it $3. That rule also explains the symmetry in Part 3.
The board. The firm needs its demand near the current price (from test markets), its marginal cost (counting only what changes with volume), and fixed costs only to decide whether to be in business. The data cannot tell you how competitors will respond to $31, whether the five cities are representative, or how current subscribers will react to a price increase.