Nonlinear Pricing

Linear pricing charges the same per-unit price regardless of quantity. Nonlinear pricing—two-part tariffs and quantity discounts—allows the firm to extract more surplus by varying the effective per-unit price with the customer’s purchase volume.

This topic covers two-part tariffs and quantity discounts. For bundling strategies, see Bundling & Product Line Design. For quality-tier discrimination, see Versioning & Quality Tiers.

The Problem

A telecom company wants to price a data plan. A flat per-GB rate leaves money on the table: light users pay too little to be profitable, heavy users pay too much and churn. Can a fixed monthly fee plus a per-GB rate do better?

This is second-degree price discrimination—the firm offers a menu of options and lets customers self-select. Unlike third-degree discrimination (where the firm observes which segment a customer belongs to), the firm here cannot identify individual willingness to pay directly. Instead, it designs a tariff schedule that induces customers to reveal their type through their quantity choices.

Two-Part Tariffs

Definition — Two-Part Tariff

A two-part tariff charges the customer a total payment that is a linear function of quantity consumed:

T(q) = F + p \cdot q

where $F \ge 0$ is the fixed fee (access charge, subscription fee, or entry fee) and $p \ge 0$ is the per-unit price (usage charge).

A customer $i$ subscribes (pays the fixed fee and begins consuming) only if the surplus from doing so is non-negative. Let $v_i(x)$ denote the marginal valuation of customer $i$ for the $x$ -th unit, and let $q_i(p)$ be the quantity demanded at per-unit price $p$ . The consumer surplus from subscribing is:

CS_i(p) = \int_0^{q_i(p)} \bigl(v_i(x) - p\bigr)\, dx

(1)

Customer $i$ subscribes if and only if $CS_i(p) \ge F$ . The firm therefore faces a fundamental tradeoff: a high fixed fee extracts more surplus from heavy users who subscribe, but drives away light users whose surplus falls below $F$ . A low per-unit price encourages high usage and generates large consumer surpluses from which the fixed fee can then extract, but it reduces the margin on each unit sold.

Optimal Two-Part Tariff (Homogeneous Customers)

When all customers are identical, the optimal two-part tariff sets the per-unit price equal to marginal cost, $p^* = c$ , and the fixed fee equal to the entire consumer surplus at that price, $F^* = CS(c)$ . This extracts all surplus from the customer and converts it to profit.

With heterogeneous customers, the result is more nuanced. Setting $p = c$ maximizes total surplus per subscriber, but the fixed fee required to capture it may exclude too many low-value customers. The optimal tariff therefore typically sets $p$ somewhat above marginal cost and $F$ below the surplus of the heaviest user—a compromise that trades some surplus extraction per subscriber for a larger subscriber base.

Numerical Example

Suppose 100 customers have linear demand $q_i(p) = \alpha_i - \beta p$ where $\alpha_i \sim \text{Uniform}(5, 30)$ and $\beta = 1$ . Marginal cost is $c = 1$ .

At $F = 20$ and $p = 2$ , customer $i$ subscribes if $(\alpha_i - 2)^2 / 2 \ge 20$ , i.e. $\alpha_i \ge 2 + \sqrt{40} \approx 8.3$ .
About 87 of 100 customers subscribe. The firm earns $20 \times 87 = \$1{,}740$ in fixed fees plus usage margin on each subscriber’s quantity.
Compare with the best linear price (no fixed fee): the optimal single per-unit price of roughly $9 earns far less because it cannot extract surplus from heavy users.

Try these values in the interactive explorer below.

Quantity Discounts

Definition — Quantity Discount (Declining Block Pricing)

A quantity discount schedule sets a total payment $T(q)$ that is concave in quantity: the marginal price per unit decreases as the customer buys more. A common implementation is declining block pricing, where the first $q_1$ units cost $p_1$ each, the next $q_2 - q_1$ units cost $p_2 < p_1$ each, and so on.

Quantity discounts are widely used by utilities (electricity, natural gas) and in B2B procurement. They serve a dual purpose: they encourage larger purchases from each customer, and they function as a form of second-degree price discrimination. High-value customers who buy in large volumes enjoy a lower average price, while low-volume customers face a higher effective per-unit cost.

Formally, the connection to second-degree price discrimination is direct. Any concave tariff $T(q)$ can be interpreted as a menu of two-part tariffs: the customer selects the block whose marginal price is closest to their marginal valuation, and the inframarginal blocks act as implicit fixed fees. Wilson (1993) shows that the optimal nonlinear tariff under a continuous type distribution satisfies:

T'(q) = c + \frac{1 - F(\theta(q))}{f(\theta(q))} \cdot \frac{\partial v}{\partial \theta}

(2)

where $\theta$ indexes customer type, $F(\theta)$ and $f(\theta)$ are the CDF and PDF of the type distribution, and $v(\theta, q)$ is the marginal valuation. The second term is the familiar “information rent” markup from mechanism design.

Choosing the Right Price Metric

Before designing the tariff schedule, the firm must choose the price metric — the unit of consumption on which the price is based. A per-seat license, a per-GB charge, a per-API-call fee, and a per-successful-outcome fee are all different metrics for potentially the same underlying service. The metric choice determines how tightly the payment tracks value received.

Metric Design Criteria

Nagle and Müller (2018) argue that a good price metric satisfies three criteria:

Value alignment — the metric scales with the value the customer receives. A per-user metric works for collaboration tools (more users = more value) but fails for analytics platforms (value depends on query complexity, not headcount).
Difficulty of gaming — the metric should be hard for customers to manipulate. Per-page-view pricing invites bot traffic; per-transaction pricing on genuine business events is harder to inflate.
Ease of measurement — the metric must be observable and auditable by both parties. Per-outcome pricing (e.g., per successful hire) requires agreement on what counts as success.

Common Metric Types

Metric Type	Example	Value Alignment	Gaming Risk
Per-unit (flat)	$X per item	Low — ignores usage intensity	Low
Per-user (seat)	$X/seat/month	Medium — proxy for org size	Medium — shared logins
Per-usage	$X/GB, $X/API call	High — tracks consumption	Medium — batching
Per-outcome	$X/conversion	Very high — pays for value	High — attribution disputes
Hybrid	Base + per-use	Configurable	Low

Tie-Ins as Price Metrics

The classic razor-and-blade model (Gillette) and printer-and-ink model (HP) are examples where the consumable IS the price metric. The durable good (razor handle, printer) is sold at or below cost, and the firm monetizes through the ongoing consumable purchases. The consumable quantity correlates with usage intensity, making it a natural value-aligned metric.

The Math of Metric Choice

Formally, the choice of price metric $m$ determines how much residual WTP variation remains after conditioning on the metric. If we observe metric value $m$ for a customer, the firm’s pricing problem is constrained by the remaining uncertainty:

\text{Extractable Surplus} \propto \text{Var}(V) - \text{Var}(V \mid m) = \rho_{V,m}^2 \cdot \text{Var}(V)

where $\rho_{V,m}$ is the correlation between customer WTP $V$ and the observed metric $m$ . A perfect metric ( $\rho = 1$ ) eliminates all WTP uncertainty, enabling first-degree price discrimination. A poor metric ( $\rho \approx 0$ ) provides no segmentation value, leaving the firm stuck with uniform pricing. This formalizes the intuition that metric choice is fundamentally about how well the metric predicts willingness to pay (Nagle and Müller (2018); Sundararajan (2004)).

Interactive Explorer

Use the sliders below to design two-part tariff pricing schemes and compare their performance against linear pricing. The chart updates in real time.

The left panel shows 100 heterogeneous customers colored by subscription status. The right panel compares total profit under the two-part tariff (stacked bar: fixed fees in blue, usage margin in green) against the best linear price (dashed outline). Adjust the fixed fee $F$ and per-unit price $p$ to find the profit-maximizing combination.

Key Insights

1. Two-Part Tariffs Outperform Linear Pricing With Heterogeneous Customers

By separating the payment into a fixed fee and a usage charge, the firm gains a second instrument for extracting surplus. Even a modest fixed fee combined with a price near marginal cost typically outperforms the best single per-unit price, because the low usage price encourages high consumption and the fixed fee captures part of the resulting surplus.

2. The Fixed-Fee / Price Tradeoff Is Non-Trivial

Setting $p = c$ and $F = CS_{\max}$ works perfectly with identical customers. With heterogeneous customers, the optimal $p$ is above marginal cost and the optimal $F$ is below the maximum surplus. Use the two-part tariff explorer to see how moving $F$ too high drives away the light users whose fixed fees were contributing to profit.

3. Nonlinear Pricing Is Second-Degree Price Discrimination

Both mechanisms—two-part tariffs and quantity discounts—are forms of second-degree price discrimination. The firm cannot directly observe customer type, but designs a tariff schedule that induces customers to self-select through their quantity choices. The optimal tariff balances surplus extraction from high-value types against participation from low-value types.

Extensions

The models presented here are the building blocks for a wide range of practical pricing structures:

Multi-tier pricing—Instead of a single two-part tariff, the firm offers a menu of plans (e.g., Basic, Pro, Enterprise), each with a different fixed fee and usage rate. This is the dominant model in SaaS and telecommunications, and it directly extends the two-part tariff framework to multiple customer segments.
Versioning & quality tiers—The firm sells the same underlying product at different quality levels (e.g., standard vs. premium features). Versioning satisfies the same incentive-compatibility constraints as the optimal nonlinear tariff, but applied to quality rather than quantity. See Versioning & Quality Tiers for the full Mussa-Rosen treatment.
Freemium models—A special case of multi-tier pricing where the lowest tier has $F = 0$ and limited functionality. The goal is to maximize the conversion rate to paid tiers while maintaining a large free user base for network effects and word-of-mouth acquisition.
Product bundling—When a firm sells multiple distinct products, it can offer them as a bundle at a single price. Bundling reduces effective WTP dispersion and can capture more surplus than separate selling, especially when valuations are negatively correlated. See Bundling & Product Line Design for a full treatment.

References

Nagle, T. T. & Müller, G. (2018). The Strategy and Tactics of Pricing: A Guide to Growing More Profitably, 6th ed.. Routledge.
Sundararajan, A. (2004). “Nonlinear Pricing of Information Goods.” Management Science, 50(12), 1660–1673.
Wilson, R. (1993). Nonlinear Pricing. Oxford University Press.

Bundles, Tiers & Volume Discounts