Peak-Load Pricing

Peak-Load Pricing is the microeconomic and regulatory theory of intertemporal pricing and capacity investment for non-storable services and infrastructure networks—such as electrical power grids, telecommunications and broadband networks, urban transit systems, toll highways, airports, and seasonal hospitality—that experience systematic cyclical fluctuations in demand across time periods while operating under a common physical capacity constraint. Formulated in its modern mathematical form by French economist Marcel Boiteux in his seminal 1949 paper, "La tarification des demandes en pointe: application de la théorie de la vente au coût marginal" ("Peak-Load Pricing"), and independently developed with geometric clarity by Peter O. Steiner (1957) in "Peak Loads and Efficient Pricing" and Jack Hirshleifer (1958), the theory resolves the fundamental question of how common capital capacity costs should be allocated between peak and off-peak consumers to achieve Pareto efficiency. Because installed plant capacity K is a joint input across all time periods of the cycle (exactly analogous to a joint public good in the Samuelson Condition), marginal social cost during off-peak periods with idle capacity equals short-run marginal operating cost b alone, whereas marginal social cost during peak periods that exhaust capacity equals short-run operating cost plus the marginal cost of capacity (b + β)—or the market-clearing rationing price that equates peak demand to installed capacity K*.

Consider a public utility or network operator producing a non-storable service (such as kilowatt-hours of electricity) across a repeating cycle partitioned into T equal-length sub-periods (t = 1, …, T; in the canonical two-period Steiner model, T = 2 with Period 1 denoting the peak and Period 2 denoting the off-peak). The baseline Boiteux–Steiner model rests on four structural conditions:

1. Non-Storability of Output: Output produced in period t cannot be stored in inventory to satisfy demand in period t′ ≠ t. Consequently, production in every period t must satisfy instantaneous consumption q_t ≤ K, where K denotes installed physical capacity.
2. Periodic Cyclical Demand: Each sub-period t is characterized by a distinct, known inverse demand function P_t(q_t) with ∂P_t / ∂q_t < 0. In the two-period model, peak inverse demand lies everywhere above off-peak inverse demand (P₁(q) > P₂(q) for all q > 0) and demands are independent across periods (∂P_t / ∂q_s = 0 for t ≠ s).
3. Leontief Fixed-Proportions Operating and Capacity Costs: Producing 1 unit of output in any single sub-period t incurs a constant short-run marginal operating cost b > 0 (such as fuel and variable maintenance). Installing 1 unit of physical plant capacity K that remains available across all T periods of the cycle incurs a constant per-cycle long-run marginal capacity cost β > 0 (representing annualized capital interest and depreciation).
4. Joint Availability of Capacity: Once a unit of capacity K is constructed at cost β, it can deliver up to 1 unit of output in Period 1 and up to 1 unit of output in Period 2 at no additional capital cost.

Total cost across the two-period cycle for output profile (q₁, q₂) and installed capacity K ≥ max{q₁, q₂} is therefore:

TC(q₁, q₂, K) = b ( q₁ + q₂ ) + β K

To determine the Pareto-efficient pricing vector (p₁*, p₂*) and optimal capacity K*, a social planner maximizes aggregate net social surplus W across the T = 2 periods subject to the physical capacity constraints q₁ ≤ K and q₂ ≤ K:

max_q1,q2,K W = ∫₀^q₁ P₁(u) du + ∫₀^q₂ P₂(u) du − b ( q₁ + q₂ ) − β K

Forming the Lagrangian with non-negative Kuhn–Tucker shadow multipliers μ₁ ≥ 0 and μ₂ ≥ 0 on the capacity constraints K − q_t ≥ 0 yields:

ℒ(q₁, q₂, K, μ₁, μ₂) = ∫₀^q₁ P₁(u) du + ∫₀^q₂ P₂(u) du − b(q₁ + q₂) − βK + μ₁(K − q₁) + μ₂(K − q₂)

The Kuhn–Tucker necessary and sufficient conditions for an interior optimum (q₁, q₂, K > 0) are:

1. Period Pricing Conditions (∂ℒ / ∂q_t = 0):

p₁* = P₁(q₁*) = b + μ₁ and p₂* = P₂(q₂*) = b + μ₂

1. Optimal Capacity Investment Condition (∂ℒ / ∂K = 0):

μ₁ + μ₂ = β ⇔ ( p₁* − b ) + ( p₂* − b ) = β

1. Complementary Slackness Conditions:

μ₁ ( K* − q₁* ) = 0, μ₂ ( K* − q₂* ) = 0, μ₁ ≥ 0, μ₂ ≥ 0

Observe the deep structural parallel between Condition (2) and the Samuelson Condition for pure public goods: because a unit of capacity K serves both Period 1 and Period 2 non-rivalrously across time, each period's net willingness to pay above operating cost, μ_t = P_t(K*) − b (known by Steiner as the "demand for capacity" in period t), is summed vertically across all binding periods and equated to the marginal cost of capacity β.

In the Firm-Peak Case (when peak demand P₁ is substantially higher than off-peak demand P₂), off-peak consumers do not fully utilize the capacity K* built for peak consumers even when off-peak price is set all the way down to short-run operating cost b (q₂* < K*). By complementary slackness, excess capacity in Period 2 (K* − q₂* > 0) implies μ₂ = 0, which directly forces μ₁ = β > 0 and q₁* = K*:

• Off-Peak Price (p₂*): Equals short-run marginal operating cost alone:

p₂* = b

• Peak Price (p₁*): Equals short-run operating cost plus 100% of long-run marginal capacity cost:

p₁* = b + β

• Optimal Capacity (K*): Determined by peak demand alone at price b + β: P₁(K*) = b + β.

In the Firm-Peak equilibrium, peak users bear 100% of the capital capacity cost β, while off-peak users pay zero capacity charge because an additional off-peak kilowatt-hour requires zero expansion of plant K*. Charging off-peak users any positive markup above b would create an unnecessary Harberger deadweight loss by idling already-built generating equipment.

What happens if off-peak demand P₂ is relatively close to peak demand P₁, so that setting p₂ = b and p₁ = b + β would cause off-peak quantity demanded D₂(b) to exceed the peak quantity demanded D₁(b + β)—reversing which period is the peak?

As Steiner (1957) proved, this Shifting-Peak Case implies that both periods bind against optimal capacity (q₁* = q₂* = K*) with positive shadow prices μ₁ > 0 and μ₂ > 0 in both periods:

• Optimal Capacity (K*): Determined where the vertical sum of the two periods' marginal willingness to pay for capacity equals β:

[ P₁(K*) − b ] + [ P₂(K*) − b ] = β

• Optimal Prices (p₁*, p₂*): Set to the market-clearing rationing prices that equate demand in each period to K*:

p₁* = P₁(K*) = b + μ₁ and p₂* = P₂(K*) = b + μ₂, with p₁* > p₂* > b

Even though physical plant is utilized at 100% capacity in both Period 1 and Period 2 (q₁* = q₂* = K*), the stronger-demand period still pays a higher price (p₁* > p₂*) and a larger share of capacity cost (μ₁ > μ₂) because its marginal valuation of capacity at K* is higher. Notice also that under constant returns to scale (TC = b(q₁ + q₂) + βK), total revenue in both the Firm-Peak and Shifting-Peak cases equals total cost (TR = p₁*q₁* + p₂*q₂* = b(q₁* + q₂*) + (μ₁ + μ₂)K* = TC): first-best peak-load pricing achieves full budget balance without needing Ramsey–Boiteux second-best markups unless there are additional fixed overhead costs F > 0.

Subsequent advances extended the Boiteux–Steiner model to real-world electrical power and network engineering:

1. Multi-Technology Generation Mix (Turvey, 1968; Crew & Kleindorfer, 1976): Real electricity grids do not rely on a single plant type; they combine baseload plants (nuclear, hydro, or combined-cycle gas with high capital cost β₁ and low fuel cost b₁) with peaker plants (open-cycle gas turbines with low capital cost β₂ < β₁ and high fuel cost b₂ > b₁). Crew and Kleindorfer (1976) proved that according to the screening curve / load-duration curve optimum, baseload plants run in both periods (K₁* = q₂*) while peaker turbines run only during the peak (K₂* = q₁* − q₂*). Consequently, even in a firm-peak regime, off-peak users pay a positive capacity charge p₂* = b₁ + (β₁ − β₂) > b₁ reflecting the capital premium paid to build fuel-efficient baseload capacity rather than cheap peakers, while peak users pay the marginal cost of a peaker turbine p₁* = b₂ + β₂.
2. Stochastic Demand and Loss-of-Load Probability (Brown & Johnson, 1969; Carlton, 1977): When demand fluctuates stochastically and real-time spot prices cannot adjust instantaneously, capacity shortages are rationed via brownouts or service curtailment with Value of Lost Load (VOLL). Optimal peak price incorporates the expected marginal outage cost p_t = b + LOLP_t · (VOLL − b), where LOLP_t is the Loss-of-Load Probability in period t.
3. Rate-of-Return Regulatory Distortion (Bailey & White, 1974): When a peak-load utility is regulated under a rate-of-return constraint (see Averch–Johnson Effect), the firm has an incentive to expand its capital rate base K by underpricing peak-period electricity (p₁ < b + β, since only peak expansion justifies new capital K) while charging a monopoly markup to off-peak users—inverting the socially optimal differential until independent system operators (ISOs) and real-time locational marginal pricing (LMP) replaced traditional rate-base tariffs.

Why is Peak-Load Pricing not Considered Price Discrimination by Economists?

Although peak and off-peak consumers pay different prices (p₁* > p₂*) for what appears to be the same physical commodity (e.g., 1 kWh of electricity), peak-load pricing is cost-based differential pricing, not price discrimination. Because electricity is non-storable, 1 kWh delivered at 5:00 PM on a hot summer afternoon (when the grid is at 100% capacity and requires b + β of marginal fuel plus turbine capacity) is a distinct economic good with a much higher marginal social cost than 1 kWh delivered at 3:00 AM (when existing turbines sit idle and marginal social cost is fuel b alone). In contrast, charging a flat uniform price across both periods is what actually discriminates against off-peak users relative to marginal cost.

What is the Difference Between a Firm Peak and a Shifting Peak?

In a Firm Peak, peak-period demand is so much larger than off-peak demand that even when off-peak price is set down to bare operating cost (p₂ = b) and peak price carries the full capacity charge (p₁ = b + β), off-peak quantity remains below peak quantity (q₂* < q₁* = K*). In a Shifting Peak, peak and off-peak demands are close enough that charging 100% of β to Period 1 and 0% to Period 2 would cause Period 2 quantity to exceed Period 1 quantity; efficiency therefore requires both periods to utilize 100% of capacity (q₁* = q₂* = K*) and share the capacity cost β = μ₁ + μ₂ according to their marginal valuations at K*.

Why do Off-Peak Electricity Users Pay More than Pure Operating Cost b in a Multi-Technology Power Grid?

In Boiteux and Steiner's single-technology model, off-peak firm-peak price equals b. However, Crew and Kleindorfer (1976) showed that in a grid with both capital-intensive baseload plants (b₁ low, β₁ high) and fuel-intensive peaker turbines (b₂ high, β₂ low), expanding off-peak demand by 1 kW requires replacing 1 kW of cheap peaker capacity (β₂) with 1 kW of expensive baseload capacity (β₁) so it can run across both periods. Off-peak price therefore equals baseload operating cost b₁ plus the net capital upgrade cost (β₁ − β₂).

1. Boiteux, M. (1949). La tarification des demandes en pointe: Application de la théorie de la vente au coût marginal. Revue Générale de l'Électricité, 58(8), 321–340 (English translation: Peak-load pricing, The Journal of Business, 33(2), 157–179, 1960). https://doi.org/10.1086/294331
2. Steiner, P. O. (1957). Peak loads and efficient pricing. The Quarterly Journal of Economics, 71(4), 585–610. https://doi.org/10.2307/1885712
3. Williamson, O. E. (1966). Peak-load pricing and optimal capacity under indivisibility constraints. The American Economic Review, 56(4), 810–827. https://www.jstor.org/stable/1813529
4. Crew, M. A., & Kleindorfer, P. R. (1976). Peak load pricing with a diverse technology. The Bell Journal of Economics, 7(1), 207–231. https://doi.org/10.2307/3003197
5. Bailey, E. E., & White, L. J. (1974). Reversals in peak and offpeak prices. The Bell Journal of Economics and Management Science, 5(1), 75–92. https://doi.org/10.2307/3003093