Menu Pricing of Large Language Models

$$
\Pi(\theta, q(\theta))= \begin{cases}c_{F} q(\theta)^{\left(\sigma+\hat{\sigma}_{L}\right) / \sigma}-c_{L} q(\theta), & \text { if } q(\theta)<\hat{q}(\theta), \\ \varphi(\theta) q(\theta)^{\sigma+\hat{\sigma}_{L}}-c_{L} q(\theta)-\psi(\theta)+\frac{1-F(\theta)}{f(\theta)} \psi^{\prime}(\theta), & \text { if } q(\theta)>\hat{q}(\theta) .\end{cases}
$$

Consider the pointwise maximization of $\Pi(\theta, q(\theta))$. In the region $q(\theta) \in[0, \hat{q}(\theta)], \Pi(\theta, q(\theta))$ is convex in $q(\theta)$ and therefore attains its maximum at a corner, $q(\theta)=0$ or $q(\theta)=\hat{q}(\theta)$.

Direct calculation shows that $\Pi(\theta, 0)>\Pi(\theta, \hat{q}(\theta))$ if and only if $\theta<\theta_{1}$, where

$$
\theta_{1} \triangleq \frac{c_{F}}{\sigma}\left(\frac{c_{L}}{c_{F}}\right)^{(1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right) / \hat{\sigma}_{L}}
$$

In the region $q(\theta) \geq \hat{q}(\theta)$, if $\varphi(\theta) \leq 0$, then optimally $q(\theta)=\hat{q}(\theta)$. If $\varphi(\theta)>0$, then $\Pi(\theta, q(\theta))$ is concave in $q(\theta)$ and therefore the maximum is either at the corner, $q(\theta)=\hat{q}(\theta)$, or in the interior, in which case it equals the optimal monopoly quantity

$$
q^{\mathrm{m}}(\theta)=\left(\frac{\left(\sigma+\hat{\sigma}_{L}\right) \varphi(\theta)}{c_{L}}\right)^{1 /\left(1-\sigma-\hat{\sigma}_{L}\right)}
$$

Thus, the solution over the region $q(\theta) \geq \hat{q}(\theta)$ is interior if and only if $\Pi_{q}(\theta, \hat{q}(\theta))>0$, or

$$
\varphi(\theta)>\frac{c_{L}}{\sigma+\hat{\sigma}_{L}}\left(\frac{\theta \sigma}{c_{F}}\right)^{\frac{\sigma\left(1-\sigma-\sigma_{L}\right)}{(1-\sigma)\left(\sigma+\tilde{\sigma}_{L}\right)}} .
$$

Condition (48) can hold on disjoint intervals of $\theta$ even if $\varphi(\theta)$ is increasing, because the right-hand side is increasing in $\theta$. However, the power of $\theta$ on the right-hand side is strictly less than 1 . Thus, if $F$ satisfies monotone hazard rate, then $\varphi^{\prime}(\theta) \geq 1$, and condition (48) holds, if at all, for all $\theta>\theta_{2}$ where $\theta_{2}$ solves:

$$
\varphi\left(\theta_{2}\right)=\frac{c_{L}}{\sigma+\hat{\sigma}_{L}}\left(\frac{\theta_{2} \sigma}{c_{F}}\right)^{\frac{\sigma\left(1-\sigma-\sigma_{L}\right)}{(1-\sigma)\left(\sigma+\sigma_{L}\right)}}
$$

If (49) doesn't admit a solution, then the solution is never interior and thus for all $\theta<\theta_{1}$, $q(\theta)=0$ and for all $\theta>\theta_{1}, q(\theta)=\hat{q}(\theta)$. If (49) admits a solution at $\theta_{2}>\theta_{1}$, then: for all $\theta<\theta_{1}, q(\theta)=0$; for all $\theta \in\left(\theta_{1}, \theta_{2}\right), q(\theta)=\hat{q}(\theta)$; for all $\theta \geq \theta_{2}, q(\theta)=q^{\mathrm{m}}(\theta)$. Finally, if (49) admits a solution at $\theta_{2}<\theta_{1}$, then $\theta_{3} \in\left[\theta_{2}, \theta_{1}\right]$ exists such that for all $\theta<\theta_{3}, q(\theta)=0$ and for all $\theta \geq \theta_{3}, q(\theta)=q^{\mathrm{m}}(\theta)$. (The threshold $\theta_{3}>\theta_{2}$ is determined by the condition $\Pi\left(\theta_{3}, q^{\mathrm{m}}\left(\theta_{3}\right)\right)=0$.)

Since all these (relaxed) solutions are implementable, the result follows. $\square$


Menu Pricing of Large Language Models

Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.

Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.

## Leader-Fringe Competition with Uniform Types

Competitive Mechanism- With $\hat{\sigma}_{F}=0$ the buyer's first-order condition implies

$$
q_{L}^{\left(\sigma+\hat{\sigma}_{L}\right) / \sigma}+q_{F}=\left(\frac{\theta \sigma}{c_{F}}\right)^{1 /(1-\sigma)}
$$

On the fringe-indifference boundary $\left(q_{F}=0\right)$ the leader's quantity solves

$$
\hat{q}(\theta)=\left(\frac{\theta \sigma}{c_{F}}\right)^{\frac{\sigma}{(1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right)}} .
$$

When the buyer single-homes on the leader and the fringe constraint is slack,

$$
q^{\mathrm{m}}(\theta)=\left(\frac{\left(\sigma+\hat{\sigma}_{L}\right) \varphi(\theta)}{c_{L}}\right)^{\frac{1}{1-\sigma-\hat{\sigma}_{L}}}=\left(\frac{\left(\sigma+\hat{\sigma}_{L}\right)(2 \theta-1)}{c_{L}}\right)^{\frac{1}{1-\sigma-\hat{\sigma}_{L}}} .
$$

We want the optimal menu to be characterized by $0<\theta_{1}<\theta_{2}<1$ such that

$$
q^{L F}(\theta)=\left\{\begin{array}{ll}
0, & \theta \leq \theta_{1}, \\
\hat{q}(\theta), & \theta_{1}<\theta<\theta_{2}, \\
q^{\mathrm{m}}(\theta), & \theta \geq \theta_{2},
\end{array} \quad q_{F}^{L F}(\theta)= \begin{cases}\left(\frac{\theta \sigma}{c_{F}}\right)^{\frac{1}{1-\sigma}}, & \theta \leq \theta_{1}, \\
0, & \theta>\theta_{1} .\end{cases}\right.
$$

The entry cutoff is

$$
\theta_{1}=\frac{c_{F}}{\sigma}\left(\frac{c_{L}}{c_{F}}\right)^{\frac{(1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right)}{\hat{\sigma}_{L}}} .
$$

The boundary-interior switch $\theta_{2}$ is the unique solution to

$$
\varphi(\theta)=\frac{c_{L}}{\sigma+\hat{\sigma}_{L}}\left(\frac{\theta \sigma}{c_{F}}\right)^{\frac{\sigma\left(1-\sigma-\hat{\sigma}_{L}\right)}{(1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right)}},
$$

which exists in (1/2, 1) if and only if $\frac{c_{L}}{\sigma+\hat{\sigma}_{L}}\left(\frac{\sigma}{c_{F}}\right)^{\frac{\sigma\left(1-\sigma-\hat{\sigma}_{L}\right)}{(1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right)}}<1$. Furthermore, $\theta_{2}>\theta_{1}$ if and only if $\theta_{1}<\left(\sigma+\hat{\sigma}_{L}\right) /\left(\sigma+2 \hat{\sigma}_{L}\right)$. Therefore, for $0<\theta_{1}<\theta_{2}<1$ to take place, the necessary and sufficient conditions are:

$$
\frac{c_{L}}{\sigma+\hat{\sigma}_{L}}\left(\frac{\sigma}{c_{F}}\right)^{\frac{\sigma\left(1-\sigma-\hat{\sigma}_{L}\right)}{(1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right)}}<1, \quad \frac{c_{F}}{\sigma}\left(\frac{c_{L}}{c_{F}}\right)^{\frac{(1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right)}{\hat{\sigma}_{L}}}<\frac{\sigma+\hat{\sigma}_{L}}{\sigma+2 \hat{\sigma}_{L}} .
$$

Efficient Allocation - Efficiency features single-homing with a model-switch cutoff $\hat{\theta}$ :

$$
\left(q_{L}^{*}(\theta), q_{F}^{*}(\theta)\right)= \begin{cases}\left(0,\left(\frac{\theta \sigma}{c_{F}}\right)^{\frac{1}{1-\sigma}}\right), & \theta<\hat{\theta} \\ \left(\left(\frac{\theta\left(\sigma+\hat{\sigma}_{L}\right)}{c_{L}}\right)^{\frac{1}{1-\sigma-\hat{\sigma}_{L}}}, 0\right), & \theta \geq \hat{\theta}\end{cases}
$$

with

$$
\hat{\theta}=\left(\frac{1-\sigma}{1-\sigma-\hat{\sigma}_{L}}\right)^{\frac{\left(1-\sigma-\hat{\sigma}_{L}\right)(1-\sigma)}{\hat{\sigma}_{L}}}\left(\frac{\sigma}{c_{F}}\right)^{\frac{\sigma\left(1-\sigma-\hat{\sigma}_{L}\right)}{\hat{\sigma}_{L}}}\left(\frac{c_{L}}{\sigma+\hat{\sigma}_{L}}\right)^{\frac{\left(\sigma+\hat{\sigma}_{L}\right)(1-\sigma)}{\hat{\sigma}_{L}}} .
$$

Multi-Model Monopoly- Virtual type is $\varphi(\theta)=2 \theta-1$. Types $\theta<1 / 2$ are excluded. Among
served types the monopolist assigns a single model, switching at $\theta^{\mathrm{m}}=(1+\hat{\theta}) / 2$. Optimal quantities are

$$
\left(q_{L}^{\mathrm{m}}(\theta), q_{F}^{\mathrm{m}}(\theta)\right)= \begin{cases}(0,0), & \theta<\frac{1}{2}, \\ \left(0,\left(\frac{\sigma \varphi(\theta)}{c_{F}}\right)^{\frac{1}{1-\sigma}}\right), & \frac{1}{2} \leq \theta<\theta^{\mathrm{m}}, \\ \left(\left(\frac{\left(\sigma+\hat{\sigma}_{L}\right) \varphi(\theta)}{c_{L}}\right)^{\frac{1}{1-\sigma-\hat{\sigma}_{L}}}, 0\right), & \theta \geq \theta^{\mathrm{m}} .\end{cases}
$$


Menu Pricing of Large Language Models

## B Contractible Tasks

Throughout the manuscript, we focus on the more realistic case in which the LLM provider cannot contract on the allocation of tokens across the buyer's tasks. However, the fullcontracting setting provides a natural benchmark and we study it in this section. Thus, the seller designs a direct menu that specifies a token allocation across different tasks:

$$
\left\{\left(\left(x_{i 1}(w), \ldots, x_{i J}(w)\right)_{i \in[0,1]}, z_{1}(w), \ldots, z_{K}(w), t(w)\right)\right\}_{w} .
$$

The allocation being contractible means the buyer has no freedom to reallocate tokens across tasks. This naturally increases the scope for screening. This also makes the problem intractable in full generality, because the reduction to a one-dimensional aggregate type and quality as in Proposition 2 does not apply. Nevertheless, in this section we obtain complete characterizations in two special settings: the case of separable type distributions (Section B.1) and the case of two types (Section B.2). In the former case, the optimal solution can be obtained via a menu of token budgets. In the latter case, the optimal solution can be obtained by identifying the structure of binding incentive constraints.


Menu Pricing of Large Language Models

## B. 1 Separable Type Distribution and Cost-Based Pricing

In this section, we provide a sufficient condition on type distribution under which contracting on tasks is not profitable. In fact, under that condition an even less contractually restrictive class of mechanisms, cost-based tariffs, is optimal:

Definition 4 (Cost-Based Tariff). A cost-based tariff is a menu of monetary budgets and transfers $\left\{\left(B_{n}, T_{n}\right)\right\}_{n}$ such that, upon purchasing item $n$, the buyer pays $T_{n}$ for access to budget $B_{n}$, which he can freely spend on tokens priced at their marginal costs.

We build on the analysis of Armstrong (1996) and introduce the conditions that are jointly sufficient for the optimality of cost-based tariffs: demand separability and type separability. We show that the former is always satisfied in our setting, whereas the latter is equivalent to the aggregate type not being informative about the relative task weights.

To this end, consider a problem of optimal token allocation by type $w$ given a monetary budget $B$ and token prices equal to marginal costs:

$$
V(w, B)=\max _{\left\{\left(x_{i}\right)_{i \in[0,1], z \geq 0\}}\right.} \int_{0}^{1} w_{i} g\left(x_{i}, z\right) d i, \quad \text { s.t. } \int_{0}^{1} \sum_{j=1}^{J} c_{j} x_{i j} d i+\sum_{k=1}^{K} \hat{c}_{k} z_{k}=B,
$$

Definition 5 (Demand Separability). Demand is separable if there exist functions $V_{1}(w)$ and $V_{2}(B)$ such that

$$
V(w, B)=V_{1}(w) V_{2}(B) .
$$

If demand is separable, then faced with a cost-based tariff, all types $w$ with the same $V_{1}(w)$ purchase the same item.

Demand separability always holds in our setting. Indeed, by Proposition 2, if type $w$ purchases a total amount of inference tokens $X=\left(X_{1}, \ldots, X_{J}\right)$ and fine-tuning tokens $Z=$ $\left(Z_{1}, \ldots, Z_{K}\right)$, then his optimal payoff is $\theta(w) g(X, Z)$. Therefore, (51) holds with $V_{1}(w)=$ $\theta(w)$ and

$$
V_{2}(B)=\max _{X \geq 0, Z \geq 0} g(X, Z), \quad \text { s.t. } \sum_{j=1}^{J} c_{j} X_{j}+\sum_{k=1}^{K} \hat{c}_{k} Z_{k}=B .
$$

Definition 6 (Type Separability). Type distribution is separable if $f_{1}, f_{2}$ exist such that $f(w)=f_{1}(\theta(w)) f_{2}(w)$ for all $w$ and $f_{2}$ is homogeneous of degree zero.

A separable type distribution means that knowing $\theta(w)$ provides no information about which ray from the origin $w$ lies on, that is, about the relative weights the buyer assigns to different tasks. Equivalently, denoting by $\|\cdot\|_{p}$ a standard $L_{p}$ norm, $\theta(w)=\|w\|_{1 /(1-\sigma)}$ and the type separability is equivalent to $\|w\|_{1 /(1-\sigma)}$ and $w /\|w\|_{1 /(1-\sigma)}$ to be independent. Thus, any type distribution generated by a draw of a "total size" $\|w\|_{1 /(1-\sigma)}$ according to an arbitrary distribution together with an independent draw of "relative weights" $w /\|w\|_{1 /(1-\sigma)}$ according to any other distribution, is separable.

Proposition 8 (Cost-Based Optimality). If the type distribution is separable, then a cost-based tariff is optimal.

Proof. Since in our setting demand is always separable, the result follows from the arguments presented by Armstrong (1996). Specifically, we can relax the IC constraints between types located on different rays from the origin. Then, applying the Envelope Theorem ray by ray, we can find a solution to the relaxed problem as follows:

$$
\left(x^{*}, z^{*}\right) \in \arg \max _{\left\{\left(x_{i}\right)_{i \in[0,1],} z \geq 0\right\}}\left(1-\frac{\int_{1}^{\infty} r^{n-1} f(r w) d r}{f(w)}\right) u(w, x, z)-c(x, z) .
$$

Given the demand separability, this condition can be rewritten as:

$$
\left(x^{*}, z^{*}\right) \in \arg \max _{\left\{\left(x_{i}\right)_{i \in[0,1]}, z \geq 0\right\}} u(w, x, z), \quad \text { s.t. } c(x, z) \leq B^{*}(w),
$$

and

$$
B^{*}(w) \in \arg \max _{B \geq 0}\left(1-\frac{\int_{1}^{\infty} r^{n-1} f(r w) d r}{f(w)}\right) \theta(w) V_{2}(B)-B .
$$

At the same time, the optimal cost-based tariff implements an allocation

$$
\left(x^{*}, z^{*}\right) \in \arg \max _{\left\{\left(x_{i}\right)_{i \in[0,1]}, z \geq 0\right\}} u(w, x, z), \quad \text { s.t. } c(x, z) \leq B^{*}(w),
$$

where

$$
B^{*}(w) \in \arg \max _{B}\left(\theta(w)-\frac{1-F(\theta(w))}{f(\theta(w))}\right) V_{2}(B)-B .
$$

If the type distribution is separable, then

$$
\left(1-\frac{\int_{1}^{\infty} r^{n-1} f(r w) d r}{f(w)}\right) \theta(w)=\theta(w)-\frac{1-F(\theta(w))}{f(\theta(w))},
$$

and the cost-based tariff implements the solution to the relaxed problem. Therefore, it is (indirectly) optimal. $\square$


Menu Pricing of Large Language Models

## B. 2 Binary Types

Alternatively, let there be only two types, $w_{1}$ and $w_{2}$, which occur with prior strictly positive probabilities $f_{1}$ and $f_{2}$, respectively. In this case, the optimal mechanism depends on what happens if the seller attempts to extract the first-best level of surplus, that is, if she offers a menu containing the efficient amounts of tokens for each type with prices equal to their respective added values. If this menu is incentive compatible, then it is clearly optimal, and we call the type with the higher payment "high" and associate the label $H$ with it. If this menu is not incentive compatible, then we call "high" the type whose incentive constraint is violated. We call the other type "low" and associate the label $L$ with it.

To design an optimal menu, it is important to determine which type out of $w_{1}$ and $w_{2}$ is high and which one is low. If $w_{2}$ dominates $w_{1}$ for every task, then it is clearly high; alternatively, the types are in some sense horizontally differentiated and the distinction is less clear. However, it turns out that the ranking can be derived from the aggregate types, and the optimal menu admits a simple characterization.

Proposition 9 (Binary Types). The high type is the one with the higher aggregate type, i.e., $\theta\left(w_{H}\right) \geq \theta\left(w_{L}\right)$. In the optimal menu, the token allocation of $w_{H}$ is always efficient. If

$$
\int_{0}^{1}\left(w_{H i}-w_{L i}\right) w_{L i}^{\frac{\sigma}{1-\sigma}} d i \leq 0
$$

then the token allocation of $w_{L}$ is also efficient and the seller extracts full surplus. Otherwise, the token allocation of $w_{L}$ is efficient with respect to a virtual type $w_{L}-\left(w_{H}-w_{L}\right) f_{H} / f_{L}$.

Proof. With a small abuse of notation relative to previous sections, denote by $q=\left(q_{i}\right)_{i \in[0,1]}$ the profile of qualities delivered to each task, $q_{i} \triangleq g\left(x_{i}, z\right)$, and denote by $C(q)$ the minimal total cost of generating a given profile $q$ :

$$
C(q) \triangleq \min _{x_{i}, z \geq 0} \int_{0}^{1} \sum_{j=1}^{J} c_{j} x_{i j} d i+\sum_{k=1}^{K} \hat{c}_{k} z_{k}, \quad \text { s.t. } g\left(x_{i}, z\right)=q_{i}, \forall i \in[0,1] .
$$

Because the set of feasible profiles $q$ is convex, it follows from the analysis of Haghpanah and Siegel (2024) on general screening problems with two buyer types that in our setting either (i) the seller extracts full surplus; or (ii) the incentive constraint of type $w_{H}$ and the individual rationality constraint of type $w_{L}$ bind.

It follows that if the seller cannot extract full surplus, then the seller's problem can be written as

$$
\begin{aligned}
\max _{q_{L}, q_{H}, t_{L}, t_{H}} & f_{L}\left(t_{L}-C\left(q_{L}\right)\right)+f_{H}\left(t_{H}-C\left(q_{H}\right)\right) \\
\text { s.t. } & \int_{0}^{1} w_{H i} q_{H i} d i-t_{H}=\int_{0}^{1} w_{H i} q_{L i} d i-t_{L}, \int_{0}^{1} w_{L i} q_{L i} d i-t_{L}=0
\end{aligned}
$$

Solving for transfers from the constraints, the problem can be restated as:

$$
\max _{q_{L}, q_{H}} f_{L}\left(\int_{0}^{1}\left(w_{L i}-\frac{f_{H}}{f_{L}}\left(w_{H i}-w_{L i}\right)\right) q_{L i} d i-C\left(q_{L}\right)\right)+f_{H}\left(\int_{0}^{1} w_{H i} q_{H i} d i-C\left(q_{H}\right)\right),
$$

which is solved by the allocation efficient relative to the virtual types.
It is left to determine which type is high and provide conditions for full surplus extraction. Consider a mechanism that attempts full surplus extraction. By the efficiency analysis behind Proposition 1, under this mechanism type $w$, when reporting type $\tilde{w}$, obtains added value

$$
u(w, \tilde{w})=\int_{0}^{1} w_{i} \Psi\left(x_{i}(\tilde{w})\right) \Phi(z(\tilde{w})) d i=\int_{0}^{1} w_{i} \tilde{w}_{i}^{\frac{\sigma}{1-\sigma}} \Psi(d) \Phi(z(\tilde{w}))^{\frac{1}{1-\sigma}} d i
$$

Because under truth-telling each type obtains zero rents, the corresponding payment is
$t(w)=u(w, w)$, and the incentive constraint $u(w, w)-t(w) \geq u(w, \tilde{w})-t(\tilde{w})$ is violated if and only if:

$$
\Psi(d) \Phi(z(\tilde{w}))^{\frac{1}{1-\sigma}} \int_{0}^{1}\left(w_{i}-\tilde{w}_{i}\right) \tilde{w}_{i}^{\frac{\sigma}{1-\sigma}} d i>0
$$

By Jensen's inequality and the concavity of the logarithm:

$$
w_{i} \tilde{w}_{i}^{\frac{\sigma}{1-\sigma}}=w_{i}^{\frac{1-\sigma}{1-\sigma}} \tilde{w}_{i}^{\frac{\sigma}{1-\sigma}} \leq(1-\sigma) w_{i}^{\frac{1}{1-\sigma}}+\sigma \tilde{w}_{i}^{\frac{1}{1-\sigma}} .
$$

Therefore, inequality (53) implies

$$
\int_{0}^{1}\left(w_{i}^{\frac{1}{1-\sigma}}-\tilde{w}_{i}^{\frac{1}{1-\sigma}}\right) d i>0
$$

and the incentive violation is possible only if $\theta(w)>\theta(\tilde{w})$, i.e., from high to low aggregate type. Therefore, if full surplus extraction is not incentive compatible, then the high type is the one with the higher aggregate type. If full surplus extraction is incentive compatible, then the high type is the one with the higher aggregate type directly by the analysis behind Proposition 1. The result follows. $\square$

Note that even though the virtual types in Proposition 9 follow the standard formula of the single-dimensional case, each type there is infinite-dimensional.

One might wonder whether the correspondence between the incentive order and aggregate types extends beyond the binary-type case. This is not the case. In Section B.3, we study a special case of our model in which the buyer's type can be effectively parameterized by two variables: the number of ex ante homogeneous tasks to which the buyer attaches positive value, and the value he attaches to each of those tasks. We show that the incentive constraints that bind in the optimal mechanism do not admit a one-dimensional structure. Instead, they form an infinite collection of one-dimensional segments. In that setting the optimal mechanisms also admit a natural implementation via a menu of two-part tariffs. However, in contrast to the indirect implementations described in Section 4.2, each item in the menu must be accompanied by task caps, that is, restrictions on the number of tasks the buyer can process.


Menu Pricing of Large Language Models

## B. 3 Value-Scale Heterogeneity

In this section, we characterize an optimal menu with contractible token allocations across tasks in the case of value-scale heterogeneity. In this case, each multidimensional type $w$ is
characterized (with a small abuse of notation) by two parameters $(w, s)$ such that

$$
w_{i}=\left\{\begin{array}{l}
w, \text { if } i \leq s \\
0, \text { if } i>s
\end{array}\right.
$$

in which $w$ and $s$ are independently distributed according to CDFs $F_{w}$ and $F_{s}$, with $F_{w}$ featuring increasing virtual values. We will argue that in this case, under additional assumptions, the binding incentive constraints are those within each scale, and the seller is able to generate the same profit as if the scale were observable.

For this section, we drop the product structure and the homogeneity requirements of (1) and instead require that $g\left(x_{i}, z\right)$ is (i) positive and continuous on $\mathbb{R}_{+}^{J+K}$, (ii) strictly monotone, twice continuously differentiable, strictly concave with negative definite Hessian, and with all cross-partial derivatives strictly positive on $\mathbb{R}_{++}^{J+K}$, (iii) $g(0)=0$, and (iv) Inada at zero, i.e., $\lim _{y_{m} \downarrow 0} g_{y_{m}}\left(y_{m}, y_{-m}\right)=+\infty$ for all $y_{-m} \in \mathbb{R}_{+}^{J+K-1}$ such that $g\left(0, y_{-m}\right)>0 .{ }^{18}$

Consider the problem in which the scale is commonly known to be $s>0$. The buyer's payoff from any given item on the menu (50) is

$$
w \int_{i=0}^{s} g\left(x_{i}, z\right) d i-t
$$

For any reported $w$ the seller should optimize token allocation to deliver a promised level of (total) quality $q$,

$$
w q-t,
$$

with the minimal cost function $C(q, s)$ of delivering a given quality being:

$$
C(q, s)=\min _{\left(x_{i}\right)_{i \in[0,1], z \geq 0}} \int_{i=0}^{s} \sum_{j=1}^{J} c_{j} x_{i j} d i+\sum_{k=1}^{K} \hat{c}_{k} z_{k}, \quad \text { s.t. } \int_{i=0}^{s} g\left(x_{i}, z\right) d i=q \text {. }
$$

Since $g$ is strictly concave, the solution to this problem is achieved by allocating the inference tokens uniformly across the $s$ tasks. The problem can be equivalently stated as:

$$
C(q, s)=\min _{x, z \geq 0} s \sum_{j=1}^{J} c_{j} x_{j}+\sum_{k=1}^{K} \hat{c}_{k} z_{k}, \quad \text { s.t. } s g(x, z)=q .
$$

The resulting cost function $C(q, s)$ satisfies, over the domain of admissible $(q, s)$, the following properties:

[^16]


Menu Pricing of Large Language Models

## Lemma 5 (Cost Function).

1. $C(q, s)$ is strictly increasing and strictly convex in $q$ with $C_{q}(0, s)=0$.
2. $C(q, s)$ is strictly decreasing in $s$ for $q>0$.
3. $C(q, s)$ is submodular, i.e., $C_{q}(q, s)$ is decreasing in $s$ for all $q$.

Proof. The first two statements follow directly from our assumptions on $g$. To establish the third property, posit the Lagrangian for the cost minimization problem:

$$
L=s \sum_{j=1}^{J} c_{j} x_{j}+\sum_{k=1}^{K} \hat{c}_{k} z_{k}+\lambda(q-s g(x, z)) .
$$

By the Envelope Theorem, $C_{q}(q, s)=\lambda(q, s)$. Thus, it suffices to show that $\lambda(q, s)$ is decreasing in $s$ for a fixed $q$.

To this end, for any $q>0$, by the Inada condition, the solution must be interior, $(x, z) \gg$ 0 . The constraint binds, so $\lambda>0$. Denoting $(x, z)$ by $y$, the first-order conditions are:

$$
g_{x_{j}}(y)=c_{j} / \lambda, \quad g_{z_{k}}(y)=\hat{c}_{k} /(\lambda s) .
$$

Define the Hessian function $H(y) \triangleq \nabla^{2} g(y)$. Denote by $A, B \in \mathbb{R}^{J+K}$ vectors such that $A_{r}=c_{r} / \lambda^{2}$ if $r \leq J$ and $=\hat{c}_{r-J} /\left(\lambda^{2} s\right)$ if $r>J, B_{r}=0$ if $r \leq J$ and $=\hat{c}_{r-J} /\left(\lambda s^{2}\right)$ if $r>J$. Then, differentiating (57) with respect to $s$, we obtain:

$$
H \frac{d y}{d s}=-A \frac{d \lambda}{d s}-B .
$$

At the same time, differentiating the constraint $s g(y(s))=q$ with respect to $s$, we obtain:

$$
g(y)+s \nabla g(y)^{\top} \frac{d y}{d s}=0 .
$$

Solving for $d y / d s$ in (58) and substituting it into (59), we obtain, omitting the dependence on $y$ :

$$
\frac{d \lambda}{d s} s \nabla g^{\top}\left(-H^{-1} A\right)=-g-s \nabla g^{\top}\left(-H^{-1} B\right) .
$$

Now, observe that for all $y \in \mathbb{R}_{++}^{J+K}$, by assumption on $g, H(y)$ is symmetric negative definite; moreover, by the positivity of cross-derivative, all off-diagonal entries of $H(y)$ are strictly positive. Then, $-H(y)$ is a Stieltjes matrix and, consequently, all elements of $H(y)^{-1}$ are negative. It immediately follows from (60) that whenever $q>0$ and $s>0, d \lambda / d s<0$. □

As such, the seller's problem for any given $s$ is analogous to Mussa and Rosen (1978). Since $w$ and $s$ are independently distributed, we can drop the dependence on $s$ and define the virtual value as

$$
\varphi(w) \triangleq w-\frac{1-F_{w}(w)}{f_{w}(w)} .
$$

Since $\varphi(w)$ is increasing, all $w$ with $\varphi(w) \leq 0$ are excluded and all other $w$ receive the quality level $q(w, s)$ that solves:

$$
\varphi(w)=C_{q}(q(w, s), s) .
$$

The corresponding optimal transfers are

$$
t(w, s)=w q(w, s)-\int_{0}^{w} q(r, s) d r
$$

Lemma 5 shows that it is cheaper to generate an extra unit of (total) quality when you have more tasks. This property is intuitive given that the returns on each task are diminishing. Thus, the optimal quality, and hence the buyer's rent, increase in scale for any given value $w$.

Assumption 2 (Bounded Rent Increase). For all $w, s$, the function $q(w, s)$ defined in (62) satisfies $\int_{0}^{w} s q_{s}(r, s) d r \leq w q(w, s)$.

Assumption 2 requires that the buyer's rent does not grow too quickly and, specifically, that the marginal increase of buyer rent from having an additional task is smaller than the average equilibrium value generated by LLM across existing tasks.

Proposition 10 (Optimal Menu of Token Allocations). Under Assumption 2, an optimal menu is

$$
\left(\left(x_{i}(w, s)\right)_{i \in[0,1]}, z(w, s), t(w, s)\right)_{(w, s)},
$$

where for each $(w, s),\left(\left(x_{i}(w, s)\right)_{i \in[0,1]}, z(w, s)\right)$ are cost-minimizing tokens from (55) that deliver quality $q(w, s)$ as defined in (62), and $t(w, s)$ is as defined in (63).

Proof. If each type reports truthfully, then the menu attains the profits of the observablescale benchmark and is thus optimal.

If type $(w, s)$ deviates to $(w, \tilde{s})$ with $\tilde{s} \leq s$, then, under the proposed menu, he obtains exactly the same payoff as type $(w, \tilde{s})$, because he processes the same number of tasks with the same willingness to pay for quality. By Lemma $5, C_{q}(q, s)$ is decreasing in $s$ for all $q$, and thus $q(w, s)$ is increasing in $s$ for all $w$. Therefore, the rents accrued by type $(w, s)$ under truth-telling,

$$
U(w, s)=\int_{0}^{w} q(r, s) d r
$$

are increasing in $s$ for all $w$. Therefore, $(w, s)$ does not want to deviate to $(w, \tilde{s})$ with $\tilde{s} \leq s$. Furthermore, by incentive compatibility within a given $\tilde{s},(w, \tilde{s})$ does not want to deviate to $(\tilde{w}, \tilde{s})$. Therefore, $(w, s)$ does not want to deviate to any $(\tilde{w}, \tilde{s})$ with $\tilde{s} \leq s$.

If type $(w, s)$ deviates to $(\tilde{w}, \tilde{s})$ with $\tilde{s}>s$, then he obtains gross payoff $w q(\tilde{w}, \tilde{s}) s / \tilde{s}$ and pays the transfer $t(\tilde{w}, \tilde{s})$. Therefore, the optimal double deviation strategy for a misreporting type solves

$$
\max _{\tilde{w} \geq 0}\left[w q(\tilde{w}, \tilde{s}) \frac{s}{\tilde{s}}-\tilde{w} q(\tilde{w}, \tilde{s})+\int_{0}^{\tilde{w}} q(r, \tilde{s}) d r\right] .
$$

Because the mechanism incentivizes truthful reporting by any $\tilde{s}$-truthtelling type, including the type ( $w s / \tilde{s}, \tilde{s}$ ), it follows that

$$
\tilde{w}^{*}=\frac{w s}{\tilde{s}}, \quad U(w ; s, \tilde{s})=\int_{0}^{\frac{w s}{\tilde{s}}} q(r, \tilde{s}) d r
$$

The condition that discourages local deviations to $\tilde{s}>s$ is:

$$
s \int_{0}^{w} q_{s}(r, s) d r \leq w q(w, s)
$$

for all $(w, s)$, which is precisely Assumption 2. Furthermore, observe that for $\tilde{s}>s$ :

$$
\begin{aligned}
U_{\tilde{s}}(w ; s, \tilde{s}) & =\int_{0}^{\frac{w s}{\tilde{s}}} q_{s}(r, \tilde{s}) d r-\frac{w s}{\tilde{s}^{2}} q\left(\frac{w s}{\tilde{s}}, \tilde{s}\right) \\
& =\frac{1}{\tilde{s}}\left(\tilde{s} \int_{0}^{\frac{w s}{\tilde{s}}} q_{s}(r, \tilde{s}) d r-\frac{w s}{\tilde{s}} q\left(\frac{w s}{\tilde{s}}, \tilde{s}\right)\right) \leq 0
\end{aligned}
$$

where the inequality holds by Assumption 2 with $(w, s)$ replaced by $(s w / \tilde{s}, \tilde{s})$. Therefore, the upward global deviations in $\tilde{s}$ are suboptimal and the result follows. $\square$


Menu Pricing of Large Language Models

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[^0]:    *Bergemann: Department of Economics, Yale University, dirk.bergemann@yale.edu. Bonatti: MIT Sloan, bonatti@mit.edu. Smolin: Toulouse School of Economics, alexey.v.smolin@gmail.com. We thank Mark Armstrong, Mert Demirer, Scott Kominers, Antonio Russo, Ron Siegel, and Frank Yang for valuable comments and discussions, as well as audiences at the Triangle Conference, USC, Caltech, Virginia, the Chicago workshop, the Sciences Po workshop, the NBER Market Design workshop, VSET, Brown, LSE, the TSE Digital Economics Conference, the IESE Economics of AI Conference, and Cambridge. An extended abstract of an earlier version of this paper appeared as "The Economics of Large Language Models: Token Allocation, Fine-Tuning, and Optimal Pricing" in the proceedings of EC'25. Dirk Bergemann gratefully acknowledges financial support from NSF SES 2049754 and ONR MURI. Alessandro Bonatti gratefully acknowledges financial support from NSF SES 2519401. Alex Smolin gratefully acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future program (grant ANR-17-EURE-0010) and through the AI Interdisciplinary Institute ANITI (grant ANR-23-IACL-0002).

[^1]:    ${ }^{1}$ For the practical challenges associated with LLM pricing, see https://www.wsj.com/articles/no-one-knows-how-to-price-ai-tools-f346ea8a.

[^2]:    ${ }^{2}$ That is, for all $j$ and $x_{i,-j} \in \mathbb{R}_{+}^{J-1}$ such that $\Psi\left(0, x_{i,-j}\right)>0, \lim _{x_{i j} \downarrow 0} \Psi_{j}\left(x_{i j}, x_{i,-j}\right)=+\infty$.
    ${ }^{3}$ That is, there exists $z_{0} \geq 0, z_{0} \neq 0$, such that $\lim _{r \downarrow 0} \Phi\left(r z_{0}\right) / r^{1-\sigma}=+\infty$. Sufficient conditions for this property are (i) $\Phi(0)>0$ or (ii) $\Phi$ is homogeneous of degree $\hat{\sigma} \in(0,1)$ with $\sigma+\hat{\sigma}<1$.

[^3]:    ${ }^{4}$ For tractability, we will often study the case $g(x, 0)=0$. This can be viewed as a normalization that doesn't affect the economics of the problem.

[^4]:    ${ }^{5}$ The parameter counts of leading models are estimated to be in the several-billion range; see https: //codingscape.com/blog/most-powerful-llms-large-language-models.
    ${ }^{6}$ Recent progress has leveraged this channel; see, for instance, https://arcprize.org/blog/oai-o3-pub-breakthrough.

[^5]:    ${ }^{7}$ Note that fine-tuning typically does not change the model's architecture or size and therefore should not directly affect marginal inference costs.

[^6]:    ${ }^{8}$ In Section 5, we show that if $\Phi(Z)$ is also a homogeneous function, then $C(Q)$ is simply a power function.

[^7]:    ${ }^{9}$ Equivalently, we could assume that the buyer can combine tokens from two models on any task, with a performance given by the sum of individual gain functions.

[^8]:    ${ }^{10}$ See Calzolari and Denicolò (2013) for a related analysis of nonlinear competition between a dominant firm and a competitive fringe under private information about demand.


Menu Pricing of Large Language Models

[^9]:    ${ }^{11}$ Indeed, $\underline{\hat{\theta}}=\left[\left(\frac{1-\sigma}{1-\sigma-\hat{\sigma}_{L}}\right)^{1-\sigma-\hat{\sigma}_{L}}\left(\frac{\sigma}{\sigma+\hat{\sigma}_{L}}\right)^{\sigma+\hat{\sigma}_{L}}\right]^{(1-\sigma) / \hat{\sigma}_{L}}$, and thus, by the strict concavity of the logarithm and Jensen's inequality, $\ln \left(\frac{\hat{\theta}}{\underline{\theta}}\right)=\frac{1-\sigma}{\hat{\sigma}_{L}}\left[\left(1-\sigma-\hat{\sigma}_{L}\right) \ln \left(\frac{1-\sigma}{1-\sigma-\hat{\sigma}_{L}}\right)+\left(\sigma+\hat{\sigma}_{L}\right) \ln \left(\frac{\sigma}{\sigma+\hat{\sigma}_{L}}\right)\right]<0$.

[^10]:    ${ }^{12}$ Calzolari and Denicolò (2015) show that a dominant firm with a competitive advantage can profitably impose exclusive dealing on privately informed buyers; the mechanism here is analogous, though exclusivity arises from the optimal nonlinear tariff rather than from contractual restrictions.

[^11]:    ${ }^{13}$ For instance, a message to GPT-4o-mini costs 9 points, while a message to Claude Opus 4 costs 4,105 points, a ratio of roughly 450×.

[^12]:    ${ }^{14}$ As in the theory, the overage charge interacts with the multiplier system: a query to a 1 × model costs \$0.04 in the overage region, while a query to a $3 \times$ model (Claude Opus 4.5) costs \$0.12.

[^13]:    ${ }^{15}$ By homogeneity, for all $x \in \mathbb{R}_{+}^{J}$ and $r>0, \nabla \Psi(r x)=r^{\sigma-1} \nabla \Psi(x)$, so any token ray corresponds to a gradient ray. By strict concavity, if $x_{1} \neq x_{2}$, then $\nabla \Psi\left(x_{1}\right) \neq \nabla \Psi\left(x_{2}\right)$, so distinct token rays map to distinct gradient rays.

[^14]:    ${ }^{16}$ In the general case of heterogeneous $\sigma_{l}$, the optimal task split and the resulting buyer's payoff do not admit tractable closed-form solutions. In particular, the fine details of the profile $w$ could matter.

[^15]:    ${ }^{17}$ Non-monotone partitions may also be optimal, but $\theta_{l}\left(I_{l}^{*}\right)$ are uniquely determined.

[^16]:    ${ }^{18}$ The Inada condition is imposed only for simplicity of arguments in Lemma 5.


Menu Pricing of Large Language Models

## Citation and provenance

**Authors:** Dirk Bergemann; Alessandro Bonatti; Alex Smolin

**Canonical citation:** Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.

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