The Encyclopedia of Economic Principals

Chapter 37

Natural Monopoly, Regulation, and Public-Utility Pricing

Pricing capacity, testing for natural monopoly, spreading markups and capping prices.

Four of the chapter's worked examples, made interactive: peak-load pricing, a subadditivity test, Ramsey fares against a uniform markup, and a price ceiling with its shortage and queue. 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

Who pays for capacity?

Which users should pay for the capacity a utility builds?

Capacity is built for the period that fills it. When only the peak constraint binds, peak users carry the whole capacity cost and off-peak users pay operating cost alone.

Equation, written in LaTeX: p_O=20, p_P=20+40=60.

Equation, written in LaTeX: q_O=100-p_O

Equation, written in LaTeX: q_P=160-p_P

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Off-peak demand is q_O = 100 - p_O and peak demand q_P = 160 - p_P. Operating cost is 20 a unit and each unit of shared capacity K costs the capacity cost shown.

Predict first. If capacity cost rises from 40 to 60, does the efficient off-peak price change?

Your prediction

Choose an example

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Figure: Who pays for capacity? Off-peak and peak demand lines with prices 20 and 60; capacity is 100 and 20 units sit idle off-peak.
Capacity cost per unit: 40, Tariff design: Efficient peak-load
Constructed example: the chapter's hypothetical utility (demands 100 - p and 160 - p, cost 20, capacity cost 40, uniform price 40); capacity costs 20 and 60 are added.

Calculated values

Off-peak price p_O
20
Peak price p_P
60
Off-peak quantity q_O
80
Peak quantity q_P
100
Capacity K
100
Idle off-peak capacity
20

Only the peak constraint binds, so off-peak users pay operating cost, p_O = 20, and peak users also pay for capacity, p_P = 20 + 40 = 60. Then q_O = 100 - 20 = 80 and q_P = 160 - 60 = 100, so K = 100. Idle off-peak capacity is 100 - 80 = 20, so another off-peak unit triggers no investment.

Worked steps

  1. p_O = 20
  2. p_P = 20 + 40 = 60
  3. q_O = 100 - 20 = 80
  4. q_P = 160 - 60 = 100
  5. K = 100
  6. Idle = 100 - 80 = 20

Use the idea

When a network is congested at some hours and idle at others, charge the busy hours for the capacity they require rather than spreading it evenly.

Where the conclusion applies

Two periods of equal length, linear demands, constant costs and a firm peak. If the off-peak quantity exceeded peak capacity, both periods would share the capacity cost.

Check your understanding: With capacity cost 20, how much capacity does the efficient tariff require?
p_P = 20 + 20 = 40, q_P = 160 - 40 = 120, so K = 120; off-peak q_O = 80 is below it.

Chapter 37 source: section "Peak-load pricing".

Demonstration 2 of 4

Is one network cheaper than two?

Does one firm serve the whole market more cheaply than several?

With a large fixed cost and constant marginal cost, every extra firm duplicates the fixed cost, so one firm is cheapest. With congestion, a split can save more than it duplicates.

Equation, written in LaTeX: C(q)=100+2q, 0<q\leq100.

Equation, written in LaTeX: \sum_{j=1}^{m}(100+2q_j)=100m+2(100).

Equation, written in LaTeX: C(q)=40+q+0.02q^2.

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Demand is 100 units, split equally among the active firms. C(q) is each firm's cost: a fixed-cost technology or one with congestion. Subadditivity means one firm's cost is below any split's.

Predict first. Under congestion, does splitting into four firms keep lowering cost?

Your prediction

Choose an example

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Figure: Is one network cheaper than two? Bars of industry cost with C(q) = 100 + 2q and a dashed line at the one-firm cost: one firm 300.00, two firms 400.00, four firms 600.00.
Cost technology: Fixed cost, 100 + 2q, Number of active firms: 1
Constructed example: the chapter's hypothetical district network (demand 100, both cost functions, one and two firms); the four-firm split is added.

Calculated values

Output per firm
100.00
Cost per firm
300.00
Industry cost
300.00
One-firm cost
300.00
Difference from one firm
0.00

With C(q) = 100 + 2q and 1 firm each serving 100.00, each firm costs 100 + 2 x 100.00 = 300.00, and the industry pays 1 x 300.00 = 300.00. This is the single-firm benchmark.

Worked steps

  1. q = 100 / 1 = 100.00
  2. Cost per firm = 100 + 2 x 100.00 = 300.00
  3. Industry = 1 x 300.00 = 300.00
  4. One firm = 300.00; difference 0.00

Use the idea

Judge natural monopoly by comparing the cost of one supplier with the cost of realistic splits at the actual demand, not by counting firms.

Where the conclusion applies

Equal splits of a fixed demand and identical cost functions. The book also notes a 60 to 40 linear split costs 400, the same as an equal split.

Check your understanding: With linear cost, what does a three-firm split cost?
100 x 3 + 2 x 100 = 500, above 300, so subadditivity holds.

Chapter 37 source: section "Natural monopoly by subadditivity".

Demonstration 3 of 4

Ramsey fares versus a uniform markup

How should a transit authority spread the markups it needs to break even?

Inverse-elasticity pricing puts higher markups where demand responds least, which raises the money with fewer lost trips. When elasticities are equal there is nothing to exploit.

Equation, written in LaTeX: 1{,}000(m_C+m_L)=900, m_C+m_L=0.90.

Equation, written in LaTeX: |\Delta q_C|\approx0.5(\frac{0.72}{2})1{,}000=180

Equation, written in LaTeX: |\Delta q_L|\approx2(\frac{0.18}{2})1{,}000=180.

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Both trip types cost $2 and start at 1,000 trips. m_C and m_L are markups over cost; they must raise $900. |e_C| is the commuter elasticity and the leisure elasticity is 2. Lost trips use a small-change approximation and deadweight loss is one half of markup times lost trips.

Predict first. As commuter demand becomes as elastic as leisure demand, what happens to the gap between Ramsey and uniform deadweight loss?

Your prediction

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Figure: Ramsey fares versus a uniform markup. Markups 0.72 and 0.18 on commuter and leisure trips, with 180.0 and 180.0 trips lost; deadweight loss 81.00.
Commuter elasticity |e_C|: 0.5, Markup scheme: Ramsey inverse elasticity
Constructed example: the chapter's hypothetical transit fares (cost $2, 1,000 trips each, $900 target, elasticities 0.5 and 2, uniform 0.45); a commuter elasticity of 1 is added.

Calculated values

Commuter markup m_C
0.72
Leisure markup m_L
0.18
Commuter fare
$2.72
Leisure fare
$2.18
Commuter trips lost
180.0
Leisure trips lost
180.0
Deadweight loss
$81.00

Markups must sum to 900 / 1,000 = 0.90 and inverse elasticity sets m_C / m_L = 2 / 0.5 = 4, so m_C = 0.72 and m_L = 0.18. Lost trips are 0.5 x (0.72 / 2) x 1,000 = 180.0 and 2 x (0.18 / 2) x 1,000 = 180.0. Deadweight loss is 0.5 x 0.72 x 180.0 + 0.5 x 0.18 x 180.0 = 64.8000 + 16.2000 = 81.0000, about $81.00.

Worked steps

  1. m_C + m_L = 900 / 1,000 = 0.90
  2. m_C / m_L = 2 / 0.5 = 4: m_C = 0.72, m_L = 0.18
  3. Lost commuter trips = 0.5 x 0.360 x 1,000 = 180.0
  4. Lost leisure trips = 2 x 0.090 x 1,000 = 180.0
  5. DWL = 64.8000 + 16.2000 = 81.0000

Use the idea

When a regulated service must cover a deficit, compare the trips or sales lost under each markup plan, not just the revenue raised.

Where the conclusion applies

Small-change approximation, equal initial prices and quantities, and constant elasticities. The book's figures are approximate; large markups would need the full demand curves.

Check your understanding: At commuter elasticity 1, what is the Ramsey commuter markup?
m_C / m_L = 2 / 1 = 2 and m_C + m_L = 0.90, so m_C = 0.60 and m_L = 0.30.

Chapter 37 source: section "Ramsey-Boiteux pricing".

Demonstration 4 of 4

A price ceiling, the shortage, and the queue

What does a binding price ceiling cost once queuing is counted?

A ceiling below the clearing price cuts the quantity supplied, so trades worth more than their cost never happen; the hatched triangle is that loss. Queuing burns more surplus on top of it.

Equation, written in LaTeX: Q_D=120-2p, Q_S=3p.

Equation, written in LaTeX: \int_0^{36}(60-\frac{q}{2}-\frac{q}{3})dq =\$1{,}620.

Equation, written in LaTeX: \int_0^{72}(60-\frac{q}{2}-\frac{q}{3})dq =\$2{,}160.

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Demand Q_D = 120 - 2p and supply Q_S = 3p clear at p* = 24 and Q* = 72. Inverse demand is p_D(q) = 60 - q/2. Surplus assumes the supplied units go to the buyers who value them most; queue and search cost is time spent getting them.

Predict first. Does raising the ceiling from 12 to 18 halve the shortage?

Your prediction

Choose an example

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Figure: A price ceiling, the shortage, and the queue. Demand and supply lines crossing at 72 units and price 24, with a ceiling at 12. Supply falls to 36 against demand 96.
Price ceiling: 12, Queue and search cost when the ceiling binds: $300
Constructed example: the chapter's hypothetical controlled market (Q_D = 120 - 2p, Q_S = 3p, ceilings 12 and 30, queue cost $300); ceilings of 18 and 24 are added, with the same $300 queue cost at 18.

Calculated values

Quantity demanded
96
Quantity supplied
36
Shortage
60
Marginal willingness to pay
$42
Surplus with best allocation
$1,620
Lost to the ceiling
$540
Queue and search cost
$300
Realized surplus
$1,320

At the ceiling of 12, demand is 120 - 2 x 12 = 96 and supply 3 x 12 = 36, a shortage of 60. Buyers value the last unit at 60 - 36/2 = 42, so resale could carry a premium up to 42 - 12 = 30. Surplus on 36 units is 60 x 36 - (5/12) x 36^2 = 1,620, against 2,160 at Q* = 72, a loss of 540. Queuing and search worth $300 cut realized surplus to 1,620 - 300 = $1,320, $840 below the benchmark.

Worked steps

  1. Q_D = 120 - 2 x 12 = 96; Q_S = 3 x 12 = 36
  2. Shortage = 96 - 36 = 60
  3. p_D(36) = 60 - 36/2 = 42
  4. Surplus = 60 x 36 - (5/12) x 36^2 = 2,160 - 540 = 1,620
  5. Lost = 2,160 - 1,620 = 540
  6. Realized = 1,620 - 300 = 1,320

Use the idea

When judging a price cap, count the lost trades and the time buyers spend competing for the available units, not only the lower price paid.

Where the conclusion applies

Linear schedules, the best possible allocation of the units supplied, and a fixed queue cost when the ceiling binds. A nonbinding ceiling creates no queue.

Check your understanding: At a ceiling of 18 with frictionless allocation, how much surplus is lost relative to 2,160?
Q_S = 54; surplus = 60 x 54 - (5/12) x 54^2 = 3,240 - 1,215 = 2,025; lost 2,160 - 2,025 = 135.

Chapter 37 source: section "Price Controls, Rationing, and Black Markets".