Subscription & Membership Pricing

Most subscription plans—from streaming services to SaaS platforms—can be understood as instances of the two-part tariff: a fixed access fee $F$ plus a per-unit usage rate $p$ . The central design problem is choosing $F$ and $p$ to maximize profit when customers differ in how intensively they use the service. Set the access fee too high and light users leave; set it too low and the firm leaves money on the table from heavy users.

Two-Part Tariffs

A two-part tariff charges every subscriber a lump-sum access fee $F \ge 0$ for the right to use the service, plus a marginal price $p \ge 0$ for each unit consumed. The total bill for a customer who consumes $q$ units is $F + p \cdot q$ . This structure is ubiquitous: wireless plans charge a monthly fee plus per-gigabyte overage, warehouse clubs charge annual membership plus item prices, and many SaaS products charge a platform fee plus per-seat or per-API-call rates.

Definition — Two-Part Tariff

A pricing schedule $T(q) = F + p \cdot q$ consisting of a fixed access fee $F$ and a variable usage rate $p$ . The access fee is paid regardless of consumption; the usage rate applies to each additional unit. When $p = 0$ , the tariff reduces to a flat-rate subscription. When $F = 0$ , it reduces to simple per-unit pricing.

The power of the two-part tariff lies in the fact that it gives the firm two instruments rather than one. A single per-unit price faces a fundamental tension: raising it increases margin per unit but reduces volume. The access fee relaxes this constraint by extracting surplus through a lump sum that does not distort the customer’s usage decision at the margin. As Wilson (1993) demonstrates, two-part tariffs are the simplest member of the broader family of nonlinear pricing schedules.

Customer Demand

Following Phillips (2021), we model customers by a type parameter $\theta$ that indexes their usage intensity. A customer of type $\theta$ facing a per-unit price $p$ demands:

q(\theta, p) = \max(0, \; \theta - p)

(1)

Higher-type customers have greater willingness to use the service. When $p < \theta$ , the customer consumes $\theta - p$ units; when $p \ge \theta$ , the customer would consume nothing even if granted access. This linear demand specification is deliberately simple, but it captures the essential feature: usage decreases in price, and customers differ in how much they use at any given price.

Consumer Surplus and Participation

A customer of type $\theta$ who subscribes at tariff $(F, p)$ obtains consumer surplus equal to the area of the triangle between the demand curve and the price line, minus the access fee:

\mathrm{CS}(\theta, F, p) \;=\; \frac{(\theta - p)^2}{2} - F

(2)

The customer subscribes if and only if this surplus is non-negative. This yields the participation constraint (also called the individual rationality, or IR, constraint):

\frac{(\theta - p)^2}{2} \;\ge\; F

(3)

Equation 3 reveals the core tradeoff. Raising the access fee $F$ extracts more surplus from subscribers who remain, but causes the lowest-type subscribers to drop out because their surplus triangle is too small to cover the fee. The per-unit price $p$ affects the size of every customer’s surplus triangle: higher $p$ shrinks the triangle, making it harder for the access fee to be covered.

Self-Selection

When the firm serves a heterogeneous population, the participation constraint binds at different access-fee levels for different customer types. Consider a market with two segments: light users ( $\theta_L = 12$ ) and heavy users ( $\theta_H = 24$ ). Light users account for 60% of the population and heavy users for 40%.

At any given usage rate $p$ , the maximum access fee that light users will tolerate is:

F_{\max}^L = \frac{(\theta_L - p)^2}{2}

(4)

while heavy users can tolerate a much higher fee:

F_{\max}^H = \frac{(\theta_H - p)^2}{2}

(5)

If the firm sets $F$ between these two thresholds, only heavy users subscribe. If the firm sets $F \le F_{\max}^L$ , both segments participate. This is the mechanism of self-selection: the firm does not observe customer types directly, but the tariff structure induces customers to reveal their type through their subscription and usage decisions.

Participation Thresholds

With $\theta_L = 12$ , $\theta_H = 24$ , and a usage rate of $p = 4$ :

Light users: $q_L = 12 - 4 = 8$ units, maximum tolerable fee $F_{\max}^L = 8^2 / 2 = \$32$
Heavy users: $q_H = 24 - 4 = 20$ units, maximum tolerable fee $F_{\max}^H = 20^2 / 2 = \$200$

Any access fee above $32 excludes the light segment. The gap between $32 and $200 represents the informational rent that heavy users enjoy when the firm prices to retain both segments.

The firm faces a classic screening problem studied by Oren, Smith, and Wilson (1983). With a single tariff, it cannot simultaneously extract full surplus from both types. Binding the heavy-type IR constraint (setting $F = F_{\max}^H$ ) would extract all surplus from heavy users but exclude light users entirely. Binding the light-type IR constraint (setting $F = F_{\max}^L$ ) retains both segments but leaves heavy users with substantial informational rent.

Interactive Tier Designer

The visualization below models the two-segment market described above. The demand lines for light ( $\theta = 12$ ) and heavy ( $\theta = 24$ ) users are shown in the chart. Adjust the access fee and per-unit price to observe how each segment’s surplus, participation decision, and the firm’s profit respond. The optimal tariff (displayed in the bottom-right metric card) is computed by binding the light-type IR constraint and searching over usage prices. Marginal cost is fixed at $c = 2$ .

The shaded triangles represent consumer surplus for each segment. When the access fee exceeds a segment’s surplus, that segment drops out and the firm collects zero from those customers. The metric cards report participation status, average profit per customer, and the theoretical optimum.

The Optimal Single Tariff

For a single two-part tariff serving both segments, the profit-maximizing strategy binds the low-type IR constraint. That is, the firm sets the access fee just high enough to make the light user indifferent between subscribing and not subscribing:

F^* = \frac{(\theta_L - p)^2}{2}

(6)

Substituting this into the firm’s profit expression, the profit per customer of type $\theta$ who participates is:

\pi(\theta) = F^* + (p - c) \cdot q(\theta, p) = \frac{(\theta_L - p)^2}{2} + (p - c)(\theta - p)

(7)

The firm’s expected profit per customer, averaging over the population, is:

\bar{\pi}(p) = \lambda \cdot \pi(\theta_L) + (1 - \lambda) \cdot \pi(\theta_H)

(8)

where $\lambda$ is the share of light users. The firm optimizes over $p$ to maximize $\bar{\pi}(p)$ , subject to the constraint that both types participate.

Optimal Two-Part Tariff (Two Segments)

When the firm offers a single two-part tariff to a market with two customer types $\theta_L < \theta_H$ , shares $\lambda$ and $1 - \lambda$ , and marginal cost $c$ , the optimal tariff binds the low-type participation constraint: $F^* = (\theta_L - p^*)^2 / 2$ . The optimal usage rate $p^*$ satisfies a tradeoff: raising $p$ above marginal cost $c$ generates usage-rate profit from both segments but shrinks the access fee (because the low-type surplus triangle contracts). The solution balances these two effects.

Two limiting cases are instructive. When $\lambda \to 1$ (almost all light users), the optimal tariff approaches $p^* = c$ and $F^* = (\theta_L - c)^2 / 2$ —the firm sets usage price equal to cost and extracts all surplus through the access fee. This is the classic result for a homogeneous market. When $\lambda \to 0$ (almost all heavy users), the firm can raise $p$ above cost to profit from the high usage volume, since it no longer needs the light-type access fee to be large.

Optimal Tariff Computation

With $\theta_L = 12$ , $\theta_H = 24$ , $\lambda = 0.6$ , and $c = 2$ , the optimal tariff can be found by numerical search over $p$ . Using the interactive chart above, note that the optimal tariff balances the access fee and usage-rate revenue. Verify your answer against the “Optimal profit” card: the displayed $F^*$ and $p^*$ achieve the highest average profit per customer while retaining both segments.

From Theory to Practice

The two-part tariff model, while stylized, maps directly to subscription design decisions across industries.

SaaS and Cloud Platforms

Software-as-a-service products commonly offer tiered plans—Free, Pro, Enterprise—each of which is a distinct two-part tariff. The free tier sets $F = 0$ with strict usage limits, serving as a customer acquisition tool. Paid tiers charge a monthly access fee with per-unit charges for additional seats, storage, or API calls. The theory predicts that the firm should set per-unit prices close to marginal cost for the lowest paid tier (to maximize the access fee it can extract without losing price-sensitive users) and can afford higher per-unit margins on enterprise tiers where customer types $\theta$ are large.

Media and Streaming

Most streaming services operate as pure flat-rate subscriptions ( $p = 0$ ), with multiple tiers differentiated by features (ad-supported, HD, number of simultaneous screens) rather than by metered usage. In the two-part tariff framework, this corresponds to offering multiple access fees with zero usage rates, where the “usage” dimension is replaced by quality versioning. The participation constraint still governs tier design: the basic tier must be priced low enough to retain the light segment, while the premium tier captures surplus from heavy users who value the enhanced features.

Telecommunications

Mobile plans historically exemplified the two-part tariff in its purest form: a monthly fee plus per-minute or per-gigabyte charges. The industry’s shift toward “unlimited” plans (effectively $p \approx 0$ ) reflects a market in which marginal cost has fallen close to zero and competitive pressure forces firms to compete primarily on the access fee. The model predicts this outcome: when $c \to 0$ , the optimal usage rate approaches zero and the firm extracts value entirely through $F$ .

Warehouse Clubs and Membership Retailers

Costco and similar warehouse clubs charge an annual membership fee ( $F$ ) and price goods at low margins ( $p \approx c$ ). This is precisely the two-part tariff structure, and it is consistent with the theoretical prediction for a firm that wants to maximize participation: set the per-unit price near cost to make the surplus triangle as large as possible, then capture value through the membership fee. The participation constraint ensures that light shoppers (low $\theta$ ) find the membership fee worthwhile only if they shop frequently enough.

In each of these applications, the core tradeoff identified by the model persists: the access fee captures surplus without distorting usage, while the usage rate balances margin extraction against participation retention. The firm’s optimal tariff depends on the distribution of customer types, the marginal cost of service, and the competitive environment—factors that the interactive visualization above allows you to examine directly.

Capstone: Subscription Platform

This capstone lets you run a delivery platform (DoorDash/Prime-like) over 24 months. In Play mode, you configure 2-4 tiers (free/basic/premium), set monthly fees, per-order delivery fees, and minimum orders, then watch acquisition, churn, and unit economics play out month by month. In Design mode, you configure customer heterogeneity and compare your tier structure against single-tier, usage-only, and unlimited-delivery models.

References

Oren, S. S., Smith, S. A. & Wilson, R. B. (1983). “Competitive Nonlinear Tariffs.” Journal of Economic Theory, 29(1), 49–71.
Phillips, R. L. (2021). Pricing and Revenue Optimization, 2nd ed.. Stanford University Press.
Wilson, R. (1993). Nonlinear Pricing. Oxford University Press.