Chapter 34: Monopoly Pricing, Product Design, Bundling, and Versioning. The Encyclopedia of Economic Principals, interactive companion

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Two-part tariff with two member types

With a high-demand and a low-demand member, which membership fee and visit price earn the most?

A fee extracts surplus without distorting use, but one fee must suit both types. Keeping the low type caps the fee at its surplus, so the seller raises the usage price to tax the high type's extra visits instead.

$Equation, written in LaTeX: CS=\frac{1}{2}(20-4)(16)=\$128.$

$Equation, written in LaTeX: F=\frac{1}{2}(12-p)^2.$

Equation, written in LaTeX: 2F+(p-4)[(20-p)+(12-p)]=16+16p-p^2.

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The high type's inverse demand is 20 - q and the low type's 12 - q, in dollars per visit. Each visit costs 4. F is the monthly fee and p the usage price; F is set so the low type just joins.

Predict first. Does any inclusive tariff beat serving only the high type?

Your prediction Yes No

Usage price ($ per visit)

Figure: Two-part tariff with two member types. Profit from a tariff that keeps both types, 16 + 16p - p^2, peaks at 80 when p = 8 and stays below the 128 line from serving only the high type. At p = 8 it is 80.00. — Usage price ($ per visit): 8
Constructed example: the chapter's hypothetical gym (demands 20 - q and 12 - q, cost 4, usage prices 4, 8 and 12); a usage price of 6 is added for comparison.

Membership fee F: $8.00
High-type visits: 12
Low-type visits: 4
Profit serving both: $80.00
Profit serving only the high type: $128.00

Keeping the low type requires F = (12 - 8)^2 / 2 = 8.00. Fees bring 16.00 and the usage margin (8 - 4) x 16 brings 64.00, for 80.00. This is the best inclusive tariff. Serving only the high type at F = 128 and p = 4 earns 128, more than any inclusive tariff.

Worked steps

F = (12 - 8)^2 / 2 = 8.00
Visits: 20 - 8 = 12 and 12 - 8 = 4
Fees 2 x 8.00 = 16.00; usage margin (8 - 4) x 16 = 64.00
Profit = 16.00 + 64.00 = 80.00
Compare with 128 from the high type alone

Use the idea

With heterogeneous customers, compare the best tariff that keeps everyone with simply serving the high-value segment at full extraction.

Where the conclusion applies

Two members with known linear demands, one common tariff and no outside option beyond not joining.

Check your understanding: At p = 6, what fee and profit result?

F = (12 - 6)^2 / 2 = 18; profit = 16 + 96 - 36 = 76.

Chapter 34 source: section "Two-part tariff".

Demonstration 2 of 4

Damaged-good versioning

How much should a seller degrade its basic version, and how does that depend on the gap between buyers?

Every unit of low-version quality lets the high type gain theta_H - theta_L by imitating, so the seller must leave that rent. A wide gap makes the rent so costly that the low version is degraded to nothing.

$Equation, written in LaTeX: (4q_L-\frac{q_L^2}{2})+(100-6q_L-50)=50-2q_L-\frac{q_L^2}{2}.$

$Equation, written in LaTeX: 18+2q_L-\frac{q_L^2}{2},$

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Two equally weighted buyers value quality q at theta q; theta_L = 4 and theta_H is the control. Quality costs q^2/2 per buyer and the premium version keeps quality theta_H. t_L and t_H are payments.

Predict first. As the type gap shrinks (theta_H falls toward 6), does the low version get better or worse?

Your prediction Better Worse

High type valuation theta_H

Figure: Damaged-good versioning. Seller profit against low-version quality for high-type valuation 10; the maximum is at q_L = 0 with profit 50.00. — High type valuation theta_H: 10
Constructed example: the chapter's hypothetical software menu (theta_L = 4, theta_H 10 and 6); theta_H of 7 and 8 are added for comparison.

Low-version quality q_L: 0
Low payment t_L: 0.00
Premium payment t_H: 100.00
High type's information rent: 0.00
Seller profit: 50.00

With theta_H = 10, profit is 50.00 + (-2) q_L - q_L^2/2, maximized at q_L = 0. Payments are t_L = 0.00 and t_H = 100.00, and profit is 50.00. Each unit of low quality saves at least as much high-type rent as it earns from the low type, so no useful low version is offered.

Worked steps

Profit = 10^2 / 2 + (8 - 10) q_L - q_L^2 / 2
Slope at zero is -2, so q_L = max(0, -2) = 0
t_L = 4 x 0 = 0.00; t_H = 10^2 - 6 x 0 = 100.00
Profit = 50.00 + (-2) x 0 - 0.00 = 50.00

Use the idea

Read a stripped-down basic tier as a screen: its quality falls as premium buyers value quality far more than basic buyers.

Where the conclusion applies

One-dimensional private information, single crossing, equal type weights and quadratic cost.

Check your understanding: At theta_H = 7, what low quality is offered?

Profit = 24.5 + q_L - q_L^2/2, maximized at q_L = 1, giving profit 25.

Chapter 34 source: section "Versioning".

Demonstration 3 of 4

Bundling and correlated values

When does selling two tools only as a package earn more than pricing them separately?

Bundling helps when buyers disagree about the parts but agree about the whole: adding offsetting values makes willingness to pay less dispersed. Aligned values stay dispersed after adding.

$Equation, written in LaTeX: 90+90=\$180.$

$Equation, written in LaTeX: 30(4)=\$120.$

$Equation, written in LaTeX: 120(2)=\$240.$

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Ada and Ben each value tools A and B; marginal cost is zero. A buyer takes the bundle when the sum of the two values is at least the bundle price.

Predict first. Does bundling still help when values are positively aligned?

Your prediction Yes No

Valuation pattern

Bundle price ($)

Figure: Bundling and correlated values. Buyers plotted by their values of the two tools with the line where the package value equals 120. Bundle revenue 240 against best separate revenue 180. — Valuation pattern: Offsetting (90, 30) and (30, 90), Bundle price ($): 120
Constructed example: the chapter's hypothetical Ada and Ben (offsetting and aligned values, bundle prices 60, 120 and 180); the partial pattern and a bundle price of 150 are added.

Best separate revenue: $180
Bundle buyers: 2
Bundle revenue: $240

Ada values the pair at 90 + 30 = 120 and Ben at 30 + 90 = 120. At a bundle price of 120, 2 of 2 buy and revenue is 120 x 2 = 240, more than the best separate revenue 180.

Worked steps

Package values: 90 + 30 = 120 and 30 + 90 = 120
Buyers at 120: 2, revenue 120 x 2 = 240
Best separate prices earn 180

Use the idea

Before bundling, check whether customers' values for the components are negatively correlated.

Where the conclusion applies

Two buyers, zero marginal cost, pure bundling only and buyers who purchase when indifferent.

Check your understanding: In the partial pattern, what does a 150 bundle earn?

Each buyer values the pair at 150, both buy, revenue 300 against best separate revenue 240 (60 x 4).

Chapter 34 source: section "Bundling".

Demonstration 4 of 4

Lerner rule and residual demand

How does the markup over cost respond to the elasticity of the demand a firm faces?

At the profit-maximizing price the markup ratio equals one over the absolute elasticity. A substitute that lowers residual demand makes demand more elastic at the chosen price and squeezes the markup.

Equation, written in LaTeX: P=100-0.5Q,

$Equation, written in LaTeX: L=\frac{60-20}{60}=\frac{2}{3}.$

$Equation, written in LaTeX: \varepsilon=\frac{dQ}{dP}\frac{P}{Q}=(-2)\frac{60}{80}=-1.5,$

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P is dollars per dose and Q thousands of doses per month; the demand intercept and marginal cost are the controls and the slope stays 0.5. L = (P - MC)/P is the Lerner index and epsilon the price elasticity of demand.

Predict first. Does a lower demand intercept raise or lower the markup ratio?

Your prediction Raises it Lowers it

Demand intercept

Marginal cost

Quantity Q (thousand): 80
Price P: $60.00
Lerner index L: 0.667
Elasticity: -1.500
1 / |elasticity|: 0.667
Operating profit ($ million): 3.20

Setting MR = MC gives Q = 100 - 20 = 80 thousand and P = 60. The Lerner index is (60 - 20) / 60 = 0.667, equal to 1/|epsilon| with epsilon = -1.500. Operating profit is 3.20 million.

Worked steps

MR = MC: 100 - Q = 20, so Q = 80
P = 100 - 0.5 x 80 = 60
L = (60 - 20) / 60 = 0.667
epsilon = (-2) x 60 / 80 = -1.500, so 1/|epsilon| = 0.667
Profit = (60 - 20) x 80,000 = 3.20 million

Use the idea

Use the Lerner index to read how much pricing power a firm has, remembering it reflects the residual demand the firm faces.

Where the conclusion applies

Linear residual demand, constant marginal cost and an interior, uniform, unconstrained price.

Check your understanding: At a = 120 and c = 40, what are P and L?

Q = 120 - 40 = 80, P = 120 - 40 = 80, L = 40/80 = 0.5 and epsilon = -2 x 80/80 = -2.

Chapter 34 source: section "Lerner inverse-elasticity pricing rule".

Two-part tariff with two member types

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Damaged-good versioning

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Calculated values

Worked steps

Use the idea

Where the conclusion applies

Bundling and correlated values

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Lerner rule and residual demand

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies