# Pricing Concept Card: [Product, market or course]

> A concept card on pricing with market power for the MBA managerial economics core: the profit-maximising price and
> the markup rule (Lerner index), price discrimination of the first, second and third degree, versioning, bundling,
> two-part tariffs, value-based pricing and dynamic pricing with capacity limits. It is for MBA students preparing
> for class, problem sets and exams, and for managers who set or review prices. It gives the idea in plain language,
> the formulas with a notation table, fully solved examples with Excel formulas, one half-solved example, one
> exercise, a chart, a decision diagram, common mistakes and a self-test with an answer key.
> Source basis: Cournot (1838), Pigou (1920), Lerner (1934), Oi (1971), Littlewood (1972), Adams and Yellen (1976),
> Mussa and Rosen (1978), written up in original words with fictional examples. Replace every [bracketed] value and
> every row marked *Example*. Competition and consumer law limits on pricing are named as dated facts only; this
> card is a learning aid, not legal advice.

<!--
Guidance: How to use this template
- Read "The idea in plain language" and the formulas, then rebuild Worked example A (parts A1 to A4) in a spreadsheet
  with the Excel formulas shown. Your numbers should match.
- Fill every [ ] in Worked example B (versioning) before you look at the answer key, then solve Exercise C on paper.
- Start your own pricing problem from the decision diagram: it tells you which tool fits which situation.
- Answer the self-test from memory; repeat after two days and after a week. Log misses in mba/mba-error-log.
- Every number was computed in Python (numpy, scipy) and checked against the Excel formula next to it.
- Related templates: mba/mba-demand-elasticity-card, mba/mba-microeconomics-guide, mba/mba-market-structure-card,
  mba/mba-game-theory-card, mba/mba-marketing-mix-card, mba/mba-customer-lifetime-value-card,
  mba/mba-stp-positioning-card, mba/pre-mba-economics-primer, mba/mba-cvp-breakeven-card, mba/mba-formula-sheet,
  finance/unit-economics-analysis, marketing/marketing-plan, marketing/go-to-market-plan.
-->

## Document control

<!-- Guidance: Record the product, the cost basis and the data behind every price. Pricing decisions are revisited often; a dated record shows what you assumed. -->

| Field | Value |
|---|---|
| Course or project | [Managerial economics, sessions 4 and 5 / pricing review for product X] |
| Learner or analyst | [Your name] |
| Product and market | [Corvane Robotics picking robot and software modules (fictional) / your product] |
| Marginal cost basis | [Source and date of the unit cost used] |
| Demand evidence | [Elasticity estimate, test, survey, conjoint, win-loss data] |
| Jurisdictions sold in | [Countries or regions, for the legal check] |
| Reviewed by | [Study partner, instructor, pricing lead or counsel] |
| Version | [0.1] |
| Last updated | YYYY-MM-DD |

## Contents

- [When to use it](#when-to-use-it)
- [The idea in plain language](#the-idea-in-plain-language)
- [Formulas and notation](#formulas-and-notation)
- [Assumptions, limits and legal checks](#assumptions-limits-and-legal-checks)
- [How managers use it](#how-managers-use-it)
- [Decision diagram](#decision-diagram)
- [Worked example A (fully solved): from one price to many](#worked-example-a-fully-solved-from-one-price-to-many)
- [Value-based and dynamic pricing](#value-based-and-dynamic-pricing)
- [Worked example B (half solved): versioning](#worked-example-b-half-solved-versioning)
- [Exercise C (unsolved)](#exercise-c-unsolved)
- [Connections to other cards](#connections-to-other-cards)
- [Common mistakes and misconceptions](#common-mistakes-and-misconceptions)
- [Self-test questions](#self-test-questions)
- [My notes and next steps](#my-notes-and-next-steps)
- [Sources and further reading](#sources-and-further-reading)
- [Answer key](#answer-key)
- [Revision history](#revision-history)

## When to use it

<!-- Guidance: Check that you have market power before you use any of these tools. A firm selling an undifferentiated product into a competitive market takes the market price. -->

Use this card when your firm **sets** its price rather than taking it from the market: the product is differentiated,
there are few rivals, switching costs are high, or the product is new. Typical decisions: launch price, price change,
list price and discount policy, price by segment or channel, plan or version design, bundles, subscriptions with a
fee plus usage charge, and prices that change with demand and remaining capacity.

| Situation | Tool in this card |
|---|---|
| One product, one price, demand elasticity known | Markup rule, $MR = MC$ |
| Identifiable segments with different elasticities that cannot resell to each other | Third-degree price discrimination |
| Buyers differ but you cannot tell who is who | Versioning, quantity discounts, menus (second degree) |
| Products whose values vary across buyers in opposite directions | Bundling (pure or mixed) |
| Buyers make repeated purchases of a service | Two-part tariff (access fee plus usage price) |
| B2B product with measurable savings for the buyer | Value-based pricing (economic value to the customer) |
| Fixed capacity, perishable inventory, demand that arrives over time | Dynamic pricing and booking limits |

## The idea in plain language

<!-- Guidance: Be able to state each idea in one sentence before moving to the formulas. -->

**One price.** A price setter faces a downward-sloping demand curve: to sell one more unit it must lower the price on
every unit. It should keep expanding output while the extra revenue from the last unit (marginal revenue) exceeds its
extra cost (marginal cost), and stop where the two are equal (Cournot 1838). Lerner (1934) turned this into a rule of
thumb: the percentage margin over price equals one over the absolute elasticity. Elastic demand means thin margins;
inelastic demand at the firm level means the price is too low.

**Many prices.** A single price leaves money on the table in two ways: some buyers would have paid more (consumer
surplus) and some buyers who value the product above its cost do not buy (deadweight loss). Pigou (1920) classified
the ways to charge different buyers different prices:

- **First degree** (perfect): each buyer pays their full willingness to pay. Rarely achievable, but individual
  negotiation, auctions and some personalised prices move toward it.
- **Second degree** (self-selection): the seller offers a menu (versions, quantity tiers, plans) and buyers choose;
  the menu must make each type prefer the option designed for it (Mussa and Rosen 1978 on quality menus).
- **Third degree** (segments): different prices for observable groups (students, regions, business size, channel),
  with a lower price for the more elastic group.

**Bundling** sells products together. When buyers' values for two products are negatively correlated, a bundle
evens out willingness to pay and captures more surplus; **mixed bundling**, which offers the bundle and the separate
items, usually does best when marginal costs are material (Adams and Yellen 1976). **Versioning** offers degraded or
enhanced versions of one product to make buyers sort themselves. A **two-part tariff** charges an access fee plus a
usage price; with identical buyers, setting the usage price at marginal cost and capturing consumer surplus in the
fee is optimal (Oi 1971).

**Value-based pricing** starts from the buyer, not from cost: what is the product worth compared with the next best
alternative? **Dynamic pricing** changes prices over time as demand information arrives and capacity is used up;
revenue management in airlines and hotels started with a simple rule for protecting seats for late, high-paying
customers (Littlewood 1972).

## Formulas and notation

<!-- Guidance: Learn the conditions under which each formula holds (constant elasticity, linear demand, identical buyers). Pitfall: applying the markup rule with an elasticity estimated far from the current price. -->

| Symbol | Meaning |
|---|---|
| $P$, $Q$ | Price and quantity |
| $MR$, $MC$ | Marginal revenue, marginal cost |
| $\varepsilon$ | Own-price elasticity of demand (negative) |
| $L$ | Lerner index, $(P - MC)/P$ |
| $i = 1, 2$ | Segments in third-degree price discrimination |
| $CS$ | Consumer surplus (area under demand above the price) |
| $F$, $p$ | Two-part tariff: fixed access fee and usage price |
| $r_j$, $m_j$ | Willingness to pay of buyer type $j$ for product R and product M |
| $EVC$ | Economic value to the customer |
| $p_H$, $p_L$ | High (late) fare and low (early) fare |
| $D_H$ | Random high-fare demand |
| $y$ | Protection level: units held back for high-fare demand |

**Single price.** Maximise $\pi = P(Q)\,Q - C(Q)$; the condition is $MR = MC$, with
$MR = P\left(1 + 1/\varepsilon\right)$. Rearranged, the **markup rule**:

$$L = \frac{P - MC}{P} = -\frac{1}{\varepsilon}, \qquad P^* = MC \cdot \frac{\varepsilon}{1 + \varepsilon} \quad (\varepsilon < -1)$$

For linear inverse demand $P = \alpha - \beta Q$: $MR = \alpha - 2\beta Q$.

**Third-degree discrimination.** Set output in each segment so that $MR_1 = MR_2 = MC$. Combined with the markup
rule:

$$\frac{P_1}{P_2} = \frac{1 + 1/\varepsilon_2}{1 + 1/\varepsilon_1}$$

so the segment with the more elastic demand gets the lower price.

**Two-part tariff with identical buyers** (Oi 1971): $p = MC$ and $F = CS(p)$. For individual demand
$q = a - bp$, $CS(p) = \tfrac{1}{2}\,(a - bp)\,(a/b - p)$.

**Bundling.** A buyer of type $j$ buys the bundle at price $P_B$ if $r_j + m_j \geq P_B$ and the bundle gives more
surplus than any separate purchase. Profit is computed type by type; with a handful of types, enumerate the
candidate prices (each type's value for each item and for the bundle) and pick the best.

**Economic value to the customer:**

$$EVC = \text{reference value} + \text{positive differentiation value} - \text{negative differentiation value}$$

The price should sit between your cost and the EVC; where in that range depends on competition and on how credibly
the value can be proven.

**Littlewood's rule** (two fare classes, low fare books first): protect $y$ units for high-fare demand, where

$$P(D_H > y) = \frac{p_L}{p_H}, \qquad y = F^{-1}_{D_H}\!\left(1 - \frac{p_L}{p_H}\right)$$

and the booking limit for the low fare is capacity minus $y$.

## Assumptions, limits and legal checks

<!-- Guidance: Read the legal rows with your counsel if you sell across borders. The facts below are dated; check the official pages before you rely on them. -->

| Assumption or limit | Why it matters | What to do |
|---|---|---|
| Market power exists | Without it, any price above the market price loses most sales | Check substitutes and switching costs first |
| Elasticity is known near the price you choose | The markup rule uses a local elasticity | Estimate it (mba/mba-demand-elasticity-card) and test |
| Segments cannot resell (arbitrage) | Low-price buyers would resell to high-price buyers | Use non-transferable features: identity, location, time, contract terms |
| Menus satisfy self-selection | A high-value buyer may take the cheap version | Check incentive compatibility for every type |
| Rivals do not react | A price cut may be matched | Model the reaction (mba/mba-game-theory-card) |
| Fairness perceptions | Buyers punish prices they see as unfair, especially on essentials and in emergencies | Use transparent rules, cap surge prices, explain the reason |
| Law: competing business buyers (US) | As of October 2026, the Robinson-Patman Act bars selling the same commodity to competing buyers at different prices where competition may be harmed, with cost-justification and meeting-competition defences; it covers goods, not services | Check the [FTC guidance page](https://www.ftc.gov/advice-guidance/competition-guidance/guide-antitrust-laws/price-discrimination-robinson-patman-violations) and counsel before you rely on it |
| Law: dominant firms (EU) | As of October 2026, Article 102(c) of the Treaty on the Functioning of the European Union lists applying dissimilar conditions to equivalent transactions with other trading parties, placing them at a competitive disadvantage, as an example of abuse of a dominant position | Check the [Article 102 text on EUR-Lex](https://data.europa.eu/eli/treaty/tfeu_2016/art_102/oj) and counsel before you rely on it |
| Law: personalised prices to consumers (EU) | As of October 2026, the Consumer Rights Directive as amended by Directive (EU) 2019/2161 (applying from 28 May 2022) requires traders to tell consumers, before a distance or off-premises contract, where applicable that the price was personalised on the basis of automated decision-making (new point (ea) of Article 6(1)) | Check [Directive (EU) 2019/2161 on EUR-Lex](https://data.europa.eu/eli/dir/2019/2161/oj), the consolidated Directive 2011/83/EU and counsel before you rely on it |

Other jurisdictions have their own rules on price discrimination, price display and surge pricing. Ask counsel for
each market before using segment, personalised or dynamic prices.

## How managers use it

<!-- Guidance: Read these uses and mark the ones relevant to your industry. -->

1. **Sanity check on a list price.** Compare the current margin $(P - MC)/P$ with $1/|\varepsilon|$. A large gap is a
   signal to investigate, not an instruction.
2. **Segment pricing.** Student, region, channel and company-size prices are third-degree discrimination; the
   question is always whether the segments differ in elasticity and whether arbitrage can be blocked.
3. **Plan design (good, better, best).** Software and service plans are second-degree discrimination; the entry plan
   is often deliberately limited so that high-value buyers do not take it.
4. **Bundles and suites.** Bundles reduce the spread of willingness to pay; mixed bundles also serve buyers who want
   only one item.
5. **Subscriptions and memberships.** A fee plus usage pricing captures surplus while keeping usage high.
6. **B2B pricing.** Quantify the customer's savings (value-based pricing) and share them; arm the sales team with the
   value calculation, not only the price list.
7. **Capacity businesses.** Airlines, hotels, events, rentals and logistics use booking limits and dynamic prices
   to fill capacity with the best mix of customers.
8. **Algorithmic pricing.** Software that sets prices automatically still needs the economics here, a legal review
   and limits on how far and how fast prices can move.

## Decision diagram

<!-- Guidance: Walk the diagram for your own product and write the tool you land on in My notes. -->

The diagram routes a pricing problem to the right tool. Several boxes can apply to one product; for example, a
software firm may use plans (second degree), a student discount (third degree) and a suite bundle at the same time.

```mermaid
flowchart TD
    A["Pricing decision"] --> B{"Do we have market power?"}
    B -- No --> C["Price at market, compete on cost and service"]
    B -- Yes --> D{"Do buyers differ in value?"}
    D -- "No, roughly identical" --> E{"Repeated usage?"}
    E -- Yes --> F["Two-part tariff: usage price = MC, fee = consumer surplus"]
    E -- No --> G["Single price: MR = MC, markup rule"]
    D -- Yes --> H{"Can we observe the segment and block resale?"}
    H -- Yes --> I["Third degree: lower price for the more elastic segment"]
    H -- No --> J{"Several products with mixed values?"}
    J -- Yes --> K["Bundling: pure or mixed"]
    J -- No --> L["Second degree: versions, tiers, quantity menus"]
    G --> M{"Fixed capacity and uncertain demand over time?"}
    I --> M
    K --> M
    L --> M
    F --> M
    M -- Yes --> N["Dynamic pricing and booking limits"]
    M -- No --> O["Legal and fairness check, then test"]
    N --> O
```

## Worked example A (fully solved): from one price to many

<!-- Guidance: Rebuild each part in its own spreadsheet tab. Money amounts in A1 and A2 are in USD thousands per unit, quantities in units per year. -->

Corvane Robotics *(fictional)* sells autonomous picking robots to warehouses, plus software modules and a robot
service plan. All numbers are invented.

### A1. Single price with the markup rule

The firm-level own-price elasticity for the picking robot is estimated at -2.5, and the marginal cost is USD 12k per
robot.

| Step | Calculation | Result | Excel |
|---|---|---|---|
| Profit-maximising price | $12 \times (-2.5)/(1 - 2.5)$ | USD 20k | `=12*-2.5/(1-2.5)` |
| Lerner index | $(20 - 12)/20$ | 0.400 | `=(20-12)/20` |
| Check | $1/2.5$ | 0.400 | `=1/2.5` |
| Markup over cost | $20/12 - 1$ | 66.7% | `=20/12-1` |

How the price responds to elasticity, at the same cost:

| Elasticity | Price (USD k) | Lerner index |
|---|---|---|
| -1.5 | 36.00 | 0.667 |
| -2.0 | 24.00 | 0.500 |
| -2.5 | 20.00 | 0.400 |
| -4.0 | 16.00 | 0.250 |
| -8.0 | 13.71 | 0.125 |

Excel: `=12*A2/(1+A2)` with the elasticity in column A. As demand becomes more elastic (more substitutes), the price
approaches marginal cost: the competitive limit.

### A2. Third-degree price discrimination

Corvane sells to two segments it can tell apart by contract: large third-party logistics operators (segment L) with
annual demand $Q_L = 120 - 2P$, and small e-commerce warehouses (segment S) with $Q_S = 80 - 2.5P$ (P in USD k). Robots
are installed and registered to a site, so resale is impractical. Marginal cost is USD 12k.

| Step | Segment L | Segment S | Excel (L; S) |
|---|---|---|---|
| Inverse demand | $P = 60 - Q/2$ | $P = 32 - 0.4Q$ | |
| MR = MC | $60 - Q = 12$ | $32 - 0.8Q = 12$ | |
| Quantity | 48 | 25 | `=(60-12)/1`; `=(32-12)/0.8` |
| Price (USD k) | 36 | 22 | `=60-48/2`; `=32-0.4*25` |
| Profit (USD k) | 1,152 | 250 | `=(36-12)*48`; `=(22-12)*25` |
| Elasticity at the price | -1.500 | -2.200 | `=-2*36/48`; `=-2.5*22/25` |
| Lerner index | 0.667 | 0.455 | `=(36-12)/36`; `=(22-12)/22` |

Total profit with discrimination: USD 1,402k on 73 robots. **Uniform price** for comparison: total demand
$Q = 200 - 4.5P$ while both segments buy ($P \leq 32$), profit $(P - 12)(200 - 4.5P)$, maximised at
$P = 254/9 = 28.22$ (`=(200+4.5*12)/(2*4.5)`), selling 63.56 to L and 9.44 to S, also 73 robots in total, for a
profit of USD 1,184.2k (`=(254/9-12)*(200-4.5*254/9)`). Discrimination raises profit by USD 217.8k (18.4%). With
linear demands and both segments served, total output is the same in both regimes; what changes is who gets the
robots and at what price. Consumer surplus falls from USD 1,027.7k to USD 701k.

The chart shows profit at each uniform price. There are two peaks: USD 28.22k (serve both segments) and USD 36k (serve
only segment L, profit USD 1,152k). When segments are very different, a uniform price may drop the low segment
entirely; discrimination lets the firm serve both.

```vega-lite
{
  "$schema": "https://vega.github.io/schema/vega-lite/v5.json",
  "title": "Corvane robots: profit at a uniform price (USD k, fictional)",
  "width": 420,
  "height": 260,
  "data": {"values": [
    {"price": 14, "profit": 274},
    {"price": 16, "profit": 512},
    {"price": 18, "profit": 714},
    {"price": 20, "profit": 880},
    {"price": 22, "profit": 1010},
    {"price": 24, "profit": 1104},
    {"price": 26, "profit": 1162},
    {"price": 28, "profit": 1184},
    {"price": 30, "profit": 1170},
    {"price": 32, "profit": 1120},
    {"price": 34, "profit": 1144},
    {"price": 36, "profit": 1152},
    {"price": 38, "profit": 1144},
    {"price": 40, "profit": 1120}
  ]},
  "layer": [
    {
      "mark": {"type": "line", "point": true},
      "encoding": {
        "x": {"field": "price", "type": "quantitative", "title": "Uniform price (USD k)"},
        "y": {"field": "profit", "type": "quantitative", "title": "Profit (USD k)", "scale": {"zero": false}}
      }
    },
    {
      "data": {"values": [{"label": "Discrimination: 1,402", "profit": 1402}]},
      "mark": {"type": "rule", "strokeDash": [4, 4]},
      "encoding": {"y": {"field": "profit", "type": "quantitative"}}
    }
  ]
}
```

### A3. Bundling (Adams and Yellen 1976)

Corvane sells two software modules for its robots, RouteOpt and PredictMaint, on monthly subscriptions per site.
Each module costs USD 20 a month per site to support (marginal cost). Four customer types, one site each, have these
monthly values (USD):

| Customer type | RouteOpt | PredictMaint | Sum |
|---|---|---|---|
| W1 | 95 | 10 | 105 |
| W2 | 75 | 40 | 115 |
| W3 | 40 | 75 | 115 |
| W4 | 10 | 95 | 105 |

| Strategy | Best prices (USD) | Who buys what | Profit (USD per month) |
|---|---|---|---|
| Separate prices | RouteOpt 75, PredictMaint 75 | W1, W2 buy RouteOpt; W3, W4 buy PredictMaint | 220 |
| Pure bundle | Bundle 105 | All four buy the bundle | 260 |
| Mixed bundle | RouteOpt 95, PredictMaint 95, bundle 115 | W1 RouteOpt, W2 and W3 bundle, W4 PredictMaint | 300 |

Excel: separate `=2*(75-20)*2`; pure bundle `=4*(105-40)`; mixed `=(95-20)+2*(115-40)+(95-20)`. The prices were
found by enumerating every candidate price (each type's value for each item and for the bundle) and keeping the best
combination.

Values are negatively correlated, so the bundle narrows the spread (105 to 115) and the pure bundle beats separate
selling. Mixed bundling does better still: it does not force W1 and W4 to take a module they value below its USD 20
cost (W1 values PredictMaint at 10), and it captures the full efficient surplus of USD 300 (every value above cost:
75 + 0 + 55 + 20 + 20 + 55 + 0 + 75). In practice set prices one dollar below each threshold (94, 94, 114) so that
indifferent buyers clearly buy: profit USD 296.

### A4. Two-part tariff (Oi 1971)

Corvane offers robot-as-a-service: a monthly access fee per warehouse plus a charge per thousand picks. Each
warehouse's monthly demand is $q = 500 - 20p$ (thousand picks, $p$ in USD per thousand picks); marginal cost is USD 5
per thousand picks. All warehouses are alike.

| Scheme | Calculation | Result per warehouse per month | Excel |
|---|---|---|---|
| Single usage price | $p = 25 - q/20$, $MR = 25 - q/10 = 5$ | $q = 200$, $p = 15$, profit USD 2,000 | `=(15-5)*200` |
| Two-part tariff: usage price | $p = MC$ | USD 5, $q = 400$ | `=500-20*5` |
| Two-part tariff: fee | $CS = \tfrac{1}{2} \times 400 \times (25 - 5)$ | USD 4,000 | `=0.5*400*(25-5)` |
| Two-part tariff: profit | fee + $(p - MC)q$ | USD 4,000 | `=4000+(5-5)*400` |

The two-part tariff doubles profit and doubles usage. The single price leaves USD 1,000 of consumer surplus and a
deadweight loss of USD 1,000 (`=0.5*(400-200)*(15-5)`); the two-part tariff captures both. With different buyer
types, the fee must be low enough for the smallest buyer you want to serve, and the usage price is then usually set
above marginal cost.

## Value-based and dynamic pricing

<!-- Guidance: Use EVC for B2B products where the buyer's savings can be measured; use the booking-limit rule wherever capacity is fixed and demand arrives in classes over time. -->

### Economic value to the customer

A warehouse operator's next best alternative to the Corvane robot is a rival robot at USD 60,000 *(Example)*. The
Corvane robot saves an extra USD 8,000 a year in labour for five years (discount rate 10%), but needs USD 6,000 of
extra integration work.

| Component | Calculation | USD | Excel |
|---|---|---|---|
| Reference value | Rival robot price | 60,000.00 | |
| Positive differentiation | PV of 8,000 a year for 5 years at 10% | 30,326.29 | `=PV(10%,5,-8000)` |
| Negative differentiation | Integration cost | 6,000.00 | |
| EVC | $60{,}000 + 30{,}326.29 - 6{,}000$ | 84,326.29 | `=60000+PV(10%,5,-8000)-6000` |

With a marginal cost of USD 12,000 (A1), the price range is USD 12,000 to about USD 84,300. A price of USD 72,000
*(Example)* leaves the buyer about USD 12,300 better off than with the rival, which a sales team can show in a
proposal. Value-based pricing needs evidence the buyer believes: pilots, references and guarantees. The A1 markup
price (USD 20,000) and the EVC price are different questions: the markup rule summarises how the whole market
responds, while EVC is the ceiling for one well-informed buyer; use EVC to segment and to negotiate.

### Dynamic pricing with fixed capacity (Littlewood 1972)

Northfield Outdoor Co. *(fictional)* runs a 60-place guided mountain course. Early-bird places sell at USD 250 and
book first; full-price places at USD 400 are bought later, by customers who decide close to the date. Full-price
demand is uncertain, roughly normal with mean 25 and standard deviation 8.

| Step | Calculation | Result | Excel |
|---|---|---|---|
| Fare ratio | $250/400$ | 0.625 | `=250/400` |
| Protection level | $y$ with $P(D_H > y) = 0.625$ | 22.45 places | `=NORM.INV(1-250/400,25,8)` |
| z value | | -0.3186 | `=NORM.S.INV(1-0.625)` |
| Early-bird booking limit | $60 - 22.45$ | 37.55, so sell at most 37 or 38 | `=60-NORM.INV(0.375,25,8)` |
| Check at 22 protected | $P(D_H > 22)$ | 0.6462 > 0.625: protect the 22nd place | `=1-NORM.DIST(22,25,8,TRUE)` |
| Check at 23 protected | $P(D_H > 23)$ | 0.5987 < 0.625: sell the 23rd place early | `=1-NORM.DIST(23,25,8,TRUE)` |

So Northfield should protect 22 places and cap early-bird sales at 38. The logic: hold back one more place only if
the chance of selling it at USD 400 is worth more than a certain USD 250. Modern dynamic pricing extends this with
many fare classes, demand forecasts updated daily and prices instead of fixed classes (Talluri and van Ryzin 2004).
Two cautions: customers who learn that prices fall late may wait (strategic buyers), and surge prices on essentials
attract regulatory and reputational risk.

## Worked example B (half solved): versioning

<!-- Guidance: Fill each [ ] and check every self-selection condition. Pitfall: pricing the premium version at the high type's full value, which makes the high type switch to the basic version. -->

Corvane sells its fleet-dashboard software in two versions. Standard limits reporting to daily data; Pro adds
real-time alerts. There are 40 high-type customers (large sites) and 60 low-type customers (small sites). Monthly
values and costs per customer (USD):

| | Standard | Pro |
|---|---|---|
| High type value | 30 | 50 |
| Low type value | 20 | 25 |
| Marginal cost | 2 | 5 |

Tasks:

1. Price Standard so that low types buy it: $P_S = $ [ ].
2. The high type must prefer Pro to Standard: $50 - P_{Pro} \geq 30 - P_S$, so the highest Pro price is
   $P_{Pro} = 50 - (30 - P_S) = $ [ ].
3. Profit with both versions: $60 \times (P_S - 2) + 40 \times (P_{Pro} - 5) = $ [ ] (`=60*(20-2)+40*(40-5)`).
4. Compare with three single-product alternatives: Pro only at 50 to high types: [ ]; Standard only at 20 to all:
   [ ]; Pro only at 25 to all: [ ].
5. The high type earns a surplus of [ ] even though the firm knows its value. Why can the firm not take it?
6. The firm considers making Standard even weaker, so that high types value it at 26 instead of 30 (low types still
   at 20). New Pro price: [ ]. New profit: [ ].

## Exercise C (unsolved)

<!-- Guidance: Solve on paper, then in a spreadsheet. Show the Excel formula for each answer. -->

Northfield's indoor climbing wall sells day passes to two groups it can identify by ID card: students with daily
demand $Q_S = 300 - 10P$ and other adults with $Q_A = 200 - 4P$ (P in USD). Marginal cost is USD 6 per visit.

1. Find the profit-maximising price and quantity for each group, and total profit.
2. Compute the elasticity at each price and verify the Lerner condition.
3. Find the best uniform price (check that both groups still buy), quantity and profit.
4. By how much does discrimination raise profit? Which group pays more, and why?
5. Name two ways the wall could stop adults from using student passes, and one fairness risk.

## Connections to other cards

<!-- Guidance: Follow at least one link after you finish this card. -->

- **Demand elasticity** (mba/mba-demand-elasticity-card): every tool here needs an elasticity or a demand curve.
- **Market structure** (mba/mba-market-structure-card): the amount of market power sets how far price can exceed
  marginal cost.
- **Game theory** (mba/mba-game-theory-card): price wars, matching, price leadership and signalling.
- **Marketing mix and segmentation** (mba/mba-marketing-mix-card, mba/mba-stp-positioning-card): price as part of the
  offer; segments as the basis for third-degree pricing.
- **Customer lifetime value** (mba/mba-customer-lifetime-value-card): subscription and two-part pricing affect
  retention and lifetime value.
- **Break-even** (mba/mba-cvp-breakeven-card): the volume needed to justify a price cut.
- **Course guide** (mba/mba-microeconomics-guide): the integrative Trailhead stove mini-case uses this card.

## Common mistakes and misconceptions

<!-- Guidance: Read before each problem set and add your own errors from mba/mba-error-log. -->

1. **Cost-plus by habit.** A fixed markup on cost ignores demand; it is right only by accident.
2. **Letting fixed costs set the price.** Fixed costs decide whether to sell at all, not the profit-maximising price.
3. **Using the market elasticity for one firm.** A firm faces a more elastic demand than its industry.
4. **Pricing in the inelastic range.** If $|\varepsilon| < 1$ at your price, raising it increases profit.
5. **Ignoring arbitrage.** Segment prices fail if low-price buyers can resell or if high-value buyers can pretend to
   be in the low segment.
6. **Forgetting self-selection in menus.** Price each premium version so the high type still prefers it.
7. **Assuming bundling always helps.** Pure bundling can lose to separate sales when values are positively correlated
   or marginal costs are high; check mixed bundling.
8. **Setting the usage price above cost in a two-part tariff with identical buyers.** It shrinks the surplus the fee
   can capture.
9. **Confusing EVC with price.** EVC is a ceiling; leave the buyer a reason to switch.
10. **Dynamic pricing without guardrails.** Strategic waiting, fairness backlash and legal disclosure duties are real
    costs.

## Self-test questions

<!-- Guidance: Answer in writing from memory before you open the answer key. -->

1. Elasticity is -3 and MC is USD 45. What is the profit-maximising price and the Lerner index?
2. A product has a 75% margin over price. What firm-level elasticity is consistent with profit maximisation?
3. In third-degree discrimination, which segment gets the lower price, and why?
4. State the two conditions needed for third-degree discrimination to work.
5. What makes a menu of versions a second-degree, not third-degree, scheme?
6. Why does mixed bundling beat pure bundling in Worked example A3?
7. In a two-part tariff with identical buyers, why is the usage price set at marginal cost?
8. A rival product costs USD 900; yours saves the buyer an extra USD 300 in present value and needs USD 100 of extra
   training. What is the EVC?
9. With $p_L = 120$, $p_H = 200$ and high-fare demand normal with mean 40 and standard deviation 10, how many units
   should be protected? Give the Excel formula.
10. Give two reasons why algorithmic dynamic pricing needs limits.

## My notes and next steps

<!-- Guidance: Write down one price you see every week and which tool in this card explains it. -->

| Item | Note |
|---|---|
| What I understood | [ ] |
| What is still unclear | [ ] |
| A price I observed and the tool behind it | [ ] |
| Next card | mba/mba-market-structure-card |
| *Example:* Error to avoid | Priced Pro at 50 in Example B and lost the high types to Standard |

## Sources and further reading

<!-- Guidance: Adams and Yellen and Oi are short and readable; textbooks are named only. -->

- Cournot, A. A. (1838). *Recherches sur les principes mathématiques de la théorie des richesses*. Hachette.
- Pigou, A. C. (1920). *The Economics of Welfare*. Macmillan.
- Lerner, A. P. (1934). The concept of monopoly and the measurement of monopoly power. *Review of Economic Studies*,
  1(3), 157–175.
- Oi, W. Y. (1971). A Disneyland dilemma: two-part tariffs for a Mickey Mouse monopoly. *Quarterly Journal of
  Economics*, 85(1), 77–96.
- Littlewood, K. (1972). Forecasting and control of passenger bookings. *AGIFORS Symposium Proceedings*, 12, 95–117.
  Reprinted (2005) in *Journal of Revenue and Pricing Management*, 4(2), 111–123.
- Adams, W. J., and Yellen, J. L. (1976). Commodity bundling and the burden of monopoly. *Quarterly Journal of
  Economics*, 90(3), 475–498.
- Mussa, M., and Rosen, S. (1978). Monopoly and product quality. *Journal of Economic Theory*, 18(2), 301–317.
- Talluri, K. T., and van Ryzin, G. J. (2004). *The Theory and Practice of Revenue Management*. Springer.
- US Federal Trade Commission, Price discrimination: Robinson-Patman violations (guidance page, checked 2026-10-28).
- Directive (EU) 2019/2161 of the European Parliament and of the Council of 27 November 2019, amending Directive
  2011/83/EU among others, *Official Journal of the European Union*, L 328, 18 December 2019,
  https://data.europa.eu/eli/dir/2019/2161/oj (checked 2026-10-28).
- Treaty on the Functioning of the European Union, Article 102, https://data.europa.eu/eli/treaty/tfeu_2016/art_102/oj
  (checked 2026-10-28).
- Textbooks (further reading, by name only): Nagle and Müller, *The Strategy and Tactics of Pricing*; Shapiro and
  Varian, *Information Rules*; Besanko, Dranove, Shanley and Schaefer, *Economics of Strategy*; Pindyck and
  Rubinfeld, *Microeconomics*.
- Open resource: OpenStax, *Principles of Economics* (3rd edition), chapter on monopoly, CC BY-NC-SA 4.0, openstax.org.

## Answer key

<!-- Guidance: Check only after writing your own answers. Note why any wrong answer was wrong. -->

**Worked example B.**

1. $P_S = 20$.
2. $P_{Pro} = 50 - (30 - 20) = 40$.
3. $60 \times 18 + 40 \times 35 = 1{,}080 + 1{,}400 = 2{,}480$ USD a month.
4. Pro only to high types at 50: $40 \times 45 = 1{,}800$. Standard to all at 20: $100 \times 18 = 1{,}800$. Pro to
   all at 25: $100 \times 20 = 2{,}000$. Versioning (2,480) beats all three.
5. USD 10 (50 - 40): the information rent. If Pro cost more than 40, high types would switch to Standard, which gives
   them 30 - 20 = 10. The firm cannot tell types apart, so it must leave this rent to make them choose Pro. (At
   exactly 40 the high type is indifferent; in practice price Pro at 39, profit USD 2,440.)
6. Weakening Standard reduces its appeal to high types: $P_{Pro} = 50 - (26 - 20) = 44$, profit
   $60 \times 18 + 40 \times 39 = 2{,}640$ (`=60*(20-2)+40*(44-5)`). Deliberately degrading the basic version is
   called damaging the good; it raises profit as long as low types still value it at 20.

**Exercise C.**

1. Students: $P = 30 - Q/10$, $MR = 30 - Q/5 = 6$, $Q = 120$, $P = 18$, profit $12 \times 120 = 1{,}440$. Adults:
   $P = 50 - Q/4$, $MR = 50 - Q/2 = 6$, $Q = 88$, $P = 28$, profit $22 \times 88 = 1{,}936$. Total USD 3,376 a day.
2. Students $\varepsilon = -10 \times 18/120 = -1.5$, Lerner $12/18 = 0.667$. Adults
   $\varepsilon = -4 \times 28/88 = -1.273$, Lerner $22/28 = 0.786 = 1/1.273$.
3. Total demand $Q = 500 - 14P$ (both buy below USD 30); profit $(P - 6)(500 - 14P)$ peaks at
   $P = 584/28 = 20.86$ (`=(500+14*6)/(2*14)`), $Q = 208$, profit USD 3,090.29. Both groups buy, since 20.86 < 30.
4. By USD 285.71 a day (3,376 - 3,090.29), about 9.2%. Adults pay more because their demand is less elastic at the
   relevant prices.
5. Check student ID at entry; issue named, non-transferable passes. Fairness risk: adults on low incomes may see the
   gap as unfair; an off-peak price open to all can soften it.

**Self-test.**

1. $P = 45 \times 3/2 = 67.50$; $L = 22.5/67.5 = 0.333$.
2. $L = 0.75 = 1/|\varepsilon|$, so $\varepsilon = -1.33$.
3. The more elastic segment, because the markup rule gives it a smaller margin over marginal cost.
4. The firm can identify the segments (with different elasticities), and buyers cannot resell or switch segments
   (no arbitrage). Market power is assumed.
5. The buyer chooses the option; the seller does not observe the buyer's type and cannot assign prices by segment.
6. Pure bundling forces W1 and W4 to take a module they value at USD 10, below its USD 20 cost, and it caps what W2
   and W3 pay at 105. Mixed bundling sells the single module to W1 and W4 at 95 and the bundle to W2 and W3 at 115.
7. Any usage price above cost reduces the quantity used and shrinks consumer surplus by more than the usage margin
   gains; with identical buyers the fee captures the whole surplus, so pricing usage at cost maximises the total.
8. $900 + 300 - 100 = 1{,}100$ USD.
9. $y = $ `=NORM.INV(1-120/200,40,10)` $= 40 + 10 \times (-0.2533) = 37.47$, so protect about 37 units.
10. Strategic customers learn to wait and erode prices; fairness and reputational backlash; legal duties such as
    disclosure of personalised prices; and pricing algorithms can amplify errors or move prices in ways that raise
    competition-law questions.

## Revision history

<!-- Guidance: Add a row when you change the cost basis, the demand evidence or the legal check. -->

| Version | Date | Author | Change summary |
|---|---|---|---|
| 0.1 | YYYY-MM-DD | [Your name] | First version from the template |
| 0.2 | YYYY-MM-DD | [Your name] | *Example:* Added segment prices for own product, legal check with counsel |

## Standard and licence note

Based on the standard treatment of pricing with market power in managerial economics (Cournot 1838, Pigou 1920,
Lerner 1934, Oi 1971, Littlewood 1972, Adams and Yellen 1976, Mussa and Rosen 1978) and on the official FTC and
EUR-Lex pages named in the text, checked in October 2026. The text, the examples and the questions are
original; no textbook text is copied or paraphrased, and Corvane Robotics and Northfield Outdoor Co. are fictional.
The template is independent and not affiliated with or endorsed by any business school, authority or software
vendor. Check the current official versions of every dated fact and legal rule before you rely on them.
Template licence: MIT.