The Encyclopedia of Economic Principals

Chapter 36

Vertical Control, Foreclosure, Integration, and Hold-Up

Markups that stack, contracts that exclude, and the volume that justifies making.

Four of the chapter's worked examples, made interactive: double marginalization, exclusive dealing as a coordination failure, raising rivals' costs, and the make-or-buy choice by volume. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

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

Double marginalization

Why do two independent markups give a higher price and lower joint profit than one?

Each firm adds its own markup without counting the profit the other loses when quantity falls. Integration, or a two-part tariff with w equal to cost, removes the second markup.

Equation, written in LaTeX: p=\frac{100+w}{2}, Q=\frac{100-w}{2}.

Equation, written in LaTeX: \max_p (p-10)(100-p).

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Household demand is Q = 100 - p subscriptions. The channel owner has marginal cost c and charges the platform a wholesale fee w; the platform sets the retail price p. An integrated firm transfers at cost (w = c), so its whole profit appears as the retail margin.

Predict first. At upstream cost 10, how much lower is the coordinated price than the separate-firm price?

Your prediction

Choose an example

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Figure: Double marginalization. Demand p = 100 - Q with the separate-firm price 77.50 and the integrated price 55.00 at upstream cost 10; bars compare profits across the three structures.
Upstream marginal cost: 10, Structure: Separate firms
Constructed example: the chapter's hypothetical sports package (demand 100 - p, cost 10, three structures); upstream costs 0, 20 and 40 are added.

Calculated values

Wholesale fee w
55.00
Retail price p
77.50
Quantity Q
22.50
Upstream profit
$1,012.50
Downstream profit
$506.25
Joint profit
$1,518.75
Price gap, separate minus integrated
$22.50

The channel owner sets w = (100 + 10) / 2 = 55.00; the platform charges p = (100 + 55.00) / 2 = 77.50 and sells Q = 100 - 77.50 = 22.50. Upstream earns 45.00 x 22.50 = $1,012.50, downstream 22.50 x 22.50 = $506.25, joint $1,518.75, below the integrated $2,025.00.

Worked steps

  1. w = (100 + 10) / 2 = 55.00
  2. p = (100 + 55.00) / 2 = 77.50
  3. Q = 100 - 77.50 = 22.50
  4. Upstream = 45.00 x 22.50 = $1,012.50
  5. Downstream = 22.50 x 22.50 = $506.25
  6. Joint = $1,012.50 + $506.25 = $1,518.75

Use the idea

When a supplier and a retailer both have pricing power, look for contracts that price the input at cost and move profit through fixed fees.

Where the conclusion applies

Linear demand, one supplier and one retailer, no retail cost and full information. With competitive retail there is no second markup to remove.

Check your understanding: At c = 20 with separate firms, what are w, p and joint profit?
w = 60, p = 80, Q = 20; upstream 40 x 20 = 800, downstream 400, joint 1,200 against integrated 40^2 = 1,600.

Chapter 36 source: section "Double marginalization".

Demonstration 2 of 4

Exclusive dealing and coordination failure

Can cheap exclusivity payments keep out a lower-cost entrant?

Each distributor alone is too small to make entry viable, so it takes the payment; together the rejecting group would gain far more than the payments. Exclusion rests on that coordination failure.

Equation, written in LaTeX: 6(\$500)=\$3{,}000,

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Ten distributors each represent ten purchases. The entrant needs the threshold number of open units to enter; competition saves buyers $50 a unit. The incumbent pays $100 for each exclusive.

Predict first. With a 40-unit threshold, do five exclusives still foreclose entry?

Your prediction

Choose an example

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Figure: Exclusive dealing and coordination failure. Ten blocks of ten purchases: 5 open and 5 signed, with the entry threshold at 60 units; entry is blocked.
Entry threshold (units): 60, Distributors who sign: 5
Constructed example: the chapter's hypothetical machine distribution (10 distributors of 10 units, thresholds 60 and 40, payment $100, saving $50, five signers); threshold 80 and 3 or 7 signers are added.

Calculated values

Open units
50
Entry
blocked
Smallest rejecting coalition
6 distributors
Coalition buyer gain
$3,000
Coalition payments given up
$600

With 5 of 10 distributors signed, 5 x 10 = 50 units stay open; 50 is below 60, so entry is blocked. A single distributor that expects the others to sign cannot reach 60 units alone, so taking the $100 is individually rational. Yet 6 distributors rejecting together would open 60 units and gain 6 x $500 = $3,000, against 6 x $100 = $600 in payments.

Worked steps

  1. Open units = (10 - 5) x 10 = 50
  2. 50 is below 60: entry is blocked
  3. Coalition size = 60 / 10 = 6
  4. Coalition gain = 6 x 10 x $50 = $3,000
  5. Payments given up = 6 x $100 = $600

Use the idea

When judging an exclusive contract, compare the share it locks up with the volume an entrant needs, and ask whether buyers can coordinate.

Where the conclusion applies

Identical distributors, a fixed entry threshold and a single offer round. A large buyer able to reach the threshold alone breaks the scheme.

Check your understanding: Threshold 80 with 3 signers: does entry occur?
Open units = 70, below 80, so no: even 3 signers block entry.

Chapter 36 source: section "Exclusive-dealing foreclosure".

Demonstration 3 of 4

Raising rivals' costs

When does paying to raise a rival's cost beat competing on equal terms?

Raising the rival's cost shifts its best response inward, so D sells more at a higher price. The strategy pays only if the asymmetric cost gap is large enough to cover the contract and any rise in D's own cost.

Equation, written in LaTeX: Q_D=Q_R=\frac{100-2(20)+20}{3}=26.67.

Equation, written in LaTeX: Q_D=31.67, Q_R=16.67,

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Inverse demand is P = 100 - Q_D - Q_R. Firm D pays $100 for an upstream contract that sets the rival's cost c_R; D's own cost after the contract is c_D. Without the contract both costs are 20 and each firm earns $711.11.

Predict first. If D's own cost rises to 30 while the rival's is 35, does the strategy still pay?

Your prediction

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Figure: Raising rivals' costs. Cournot best-response lines for D with cost 20.00 and R with cost 35.00; they cross at Q_D = 31.67 and Q_R = 16.67, compared with 26.67 each without the contract.
Rival cost c_R: 35, D's own cost after the contract c_D: 20
Constructed example: the chapter's hypothetical duopoly (P = 100 - Q, costs 20 and 35, D's cost 30, contract $100); rival costs 27.5 and 45 and D's cost 25 are added.

Calculated values

Q_D
31.67
Q_R
16.67
Price P
51.67
D's profit before contract
$1,002.78
D's net profit
$902.78
Gain over 711.11 baseline
$191.67

Q_D = (100 - 2 x 20 + 35) / 3 = 95 / 3 = 31.67 and Q_R = (100 - 2 x 35 + 20) / 3 = 50 / 3 = 16.67, so P = (100 + 20 + 35) / 3 = 155 / 3 = 51.67. Because P - c_D = Q_D, D earns Q_D^2 = 9,025.00 / 9 = $1,002.78, or $902.78 after the $100 contract. The contract pays: the baseline with both costs at 20 is $711.11, so the change is 191.67.

Worked steps

  1. Q_D = (100 - 2 x 20 + 35) / 3 = 95 / 3 = 31.67
  2. Q_R = (100 - 2 x 35 + 20) / 3 = 50 / 3 = 16.67
  3. P = (100 + 20 + 35) / 3 = 155 / 3 = 51.67
  4. D's profit = Q_D^2 = 9,025.00 / 9 = 1,002.78
  5. Net = 1,002.78 - 100 = 902.78
  6. Change from 711.11 = 902.78 - 711.11 = 191.67

Use the idea

When a firm locks up an input, compare the cost it imposes on rivals with the cost it bears itself, and look at what happens to price.

Where the conclusion applies

Cournot competition with linear demand and constant costs, and both firms producing. Results are rounded to cents; the calculation uses unrounded quantities.

Check your understanding: Rival cost 45, D's cost 25: what is D's net profit?
Q_D = (100 - 50 + 45) / 3 = 31.67; profit 31.6667^2 - 100 = 1,002.78 - 100 = 902.78.

Chapter 36 source: section "Raising rivals' costs".

Demonstration 4 of 4

Make or buy by volume

At what volume does making a specialized component beat buying it?

Integration trades a higher fixed cost for a lower unit cost and less hold-up exposure. The answer depends on volume because the setup must be spread over enough output.

Equation, written in LaTeX: C_M=100{,}000(\$40)+\$400{,}000+\$600{,}000=\$5.0\text{ million}.

Equation, written in LaTeX: C_I=\$1.8\text{ million}+100{,}000(\$28)+\$300{,}000=\$4.9\text{ million}.

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V is annual volume. Buying costs $40 a unit plus $400,000 of contracting and $600,000 of expected hold-up cost. Making costs $1.8 million of annualized setup, $28 a unit and $300,000 of administration.

Predict first. Roughly at what annual volume does making start to beat buying?

Your prediction

Choose an example

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Figure: Make or buy by volume. Expected cost of buying and of making against volume; the lines cross at 91,667 units. At 100,000 units buying costs $5,000,000 and making $4,900,000.
Annual volume (units): 100,000
Constructed example: the chapter's hypothetical component (100,000 and 50,000 units, $40 price, $28 internal cost, fixed terms as stated); volumes of 75,000 and 150,000 are added.

Calculated values

Market cost C_M
$5,000,000
Internal cost C_I
$4,900,000
Saving from integration
$100,000
Break-even volume
91,667

In dollars, C_M = 100,000 x 40 + 400,000 + 600,000 = 5,000,000 and C_I = 1,800,000 + 100,000 x 28 + 300,000 = 4,900,000. The difference is $100,000, so making in house is cheaper. Making wins above 91,667 units, where the $12 unit saving covers the $1,100,000 net fixed cost.

Worked steps

  1. C_M = 100,000 x 40 + 400,000 + 600,000 = $5,000,000
  2. C_I = 1,800,000 + 100,000 x 28 + 300,000 = $4,900,000
  3. C_M - C_I = 100,000
  4. Break-even: 1,100,000 / 12 = 91,667 units

Use the idea

Write both options as fixed cost plus unit cost times volume, include expected hold-up costs, and find the break-even volume before deciding.

Where the conclusion applies

Fixed and hazard terms do not change with volume, and expected costs are known. Different hazards or setup costs move the break-even point.

Check your understanding: At 150,000 units, how much does integration save?
C_M = 6.0 + 1.0 = 7.0 million; C_I = 1.8 + 4.2 + 0.3 = 6.3 million; saving 700,000.

Chapter 36 source: section "Vertical Integration".