Optimal Technology Design

### 3.2 Optimal Project

Each binary project, $c$, can be described by specifying the cost of each probability of success through a function $C:[0,1] \rightarrow \mathbb{R}_{+}$. In particular, we set $C(\mu)=c\left(B_{\mu}\right)$ for all $\mu$, and we keep in mind that the cost of a non-binary distribution is infinity. Recall that, in binary projects, it is without loss of generality to restrict attention to bonus contracts, $w_{b}$, where the agent is paid $b$ if output is one. Finally, the agent's choice of distribution $B_{\mu}$ can be identified by its mean, $\mu$ (equivalently, by its "completion probability"). In what follows, we describe each binary outcome $\left(c, w_{b}, B_{\mu}\right)$ by a triple $(C, b, \mu)$.

We are ready to state the main result of this section.

Proposition 2. There is an optimal binary project, $C^{*}$, and an agent-optimal equilibrium in $C^{*}$, $\left(b^{*}, \mu^{*}\right)$, such that

(i) $C^{* \prime}(\mu)=1-1 /(e \mu)$ if $\mu \geq 1 / e$ and zero otherwise,
(ii) $b^{*}=1-1 / e$, and
(iii) $\mu^{*}=1$.

Proposition 2 describes an optimal binary project in terms of the marginal costs of the possible completion probabilities. Of course, adding a fixed cost has no impact on incentives, so $C^{*}(0)=0$.

We explain below that the functional form of the marginal cost in Part (i) is pinned down by the requirement that the principal be indifferent between implementing a large range of completion probabilities. To this end, we sketch an argument for Proposition 2 that restricts the agent to choose cost functions $C$ that are non-decreasing, differentiable, and such that equilibrium can be determined using the first-order approach. The formal proof in the Appendix uses an envelope type argument to determine the agent's payoff and does not rely on any such restrictions.

We view the agent's problem of finding an optimal binary project as a maximization problem subject to incentive compatibility constraints. Our argument can be understood in terms of backward induction. We first determine a condition relating the principal's choice of bonus $b$ to the agent's optimal choice of completion probability $\mu$. We can then determine a condition on the project $C$ for the principal to implement a given completion probability $\mu$. Finally, we consider optimizing the agent's payoff over the project $C$ and the completion probability $\mu$ to be implemented by the principal, subject to the constraints determined in the previous steps.

Let us then describe the agent's incentive constraint in a subgame $(C, b)$. Note that, at this stage, the agent's problem is to solve $\max _{\widehat{\mu} \in[0,1]}\{\widehat{\mu} b-C(\widehat{\mu})\}$. Then the requirement on the reward $b$ ensuring the agent chooses a given completion probability $\mu$ can be described by the first-order condition

$$
b=C^{\prime}(\mu) .
$$

We now turn to the incentive constraint of the principal. In project $C$, the principal's problem is

$$
\max _{\mu \in[0,1], b \in \mathbb{R}_{+}} \mu(1-b)
$$

subject to the constraint that $\mu$ and $b$ satisfy Equation (8). Plugging the constraint into the maximand, the principal's problem can be expressed solely in terms of the completion probability she wants to implement; that is, $\max _{\mu \in[0,1]} \mu\left(1-C^{\prime}(\mu)\right)$. So, in project $C$, the principal's choice of $\mu$ must satisfy

$$
\mu\left(1-C^{\prime}(\mu)\right) \geq \widetilde{\mu}\left(1-C^{\prime}(\widetilde{\mu})\right),
$$

for all $\widetilde{\mu} \in[0,1]$.
We are now ready to state the project selection problem. As anticipated above, we include the completion probability $\mu$ as a choice variable alongside the cost function $C$. This is important because the principal may be indifferent between implementing various completion probabilities, generating different payoffs for the agent. The interpretation of including $\mu$ as a choice is that,
after designing the project, the agent makes a recommendation to the principal regarding which $\mu$ to implement. The agent's first-stage problem can be written

$$
\begin{aligned}
& \max _{C, \mu, b} \mu b-C(\mu) \\
& \text { s.t. (8) and (9). }
\end{aligned}
$$

Alternatively, plugging the constraint (8) into the maximand, it can be written as

$$
\begin{aligned}
& \max _{C, \mu} \mu C^{\prime}(\mu)-C(\mu) \\
& \text { s.t. (9). }
\end{aligned}
$$

Now let us explain that, if $(\widehat{C}, \widehat{\mu})$ solves this problem, the constraint (9) must bind at each $\widetilde{\mu}<\widehat{\mu}$ such that $\widehat{C}^{\prime}(\widetilde{\mu})>0$. To understand this observation, notice first that the agent's cost of $\widehat{\mu}$ must be given by $\widehat{C}(\widehat{\mu})=\int_{0}^{\widehat{\mu}} \widehat{C}^{\prime}(\widetilde{\mu}) d \widetilde{\mu}$, since $\widehat{C}(0)=0 .{ }^{17}$ Hence, reducing the marginal cost $\widehat{C}^{\prime}(\widetilde{\mu})$ for any $\widetilde{\mu}<\widehat{\mu}$ implies a reduction in the cost $\widehat{C}(\widehat{\mu})$. However, the possibility to reduce the marginal costs $\widehat{C}^{\prime}(\widetilde{\mu})$ is restricted by the constraint (9) evaluated at $(C, \mu)=(\widehat{C}, \widehat{\mu})$. We can conclude that, for the optimum $(\widehat{C}, \widehat{\mu})$, the constraint (9) binds for all $\widetilde{\mu}<\widehat{\mu}$ as long as $\widehat{C}^{\prime}(\widetilde{\mu})>0$, implying that the principal is indifferent between implementing any completion probability below $\widehat{\mu}$ which has a strictly positive marginal cost.

Our next aim is to use this indifference condition to reduce the agent's problem in (10) to a twodimensional problem by replacing the domain of projects by the principal's possible equilibrium payoffs. To this end, note that if $(\widehat{C}, \widehat{\mu})$ solves (10), then $\widehat{C}^{\prime}$ can be expressed in terms of the principal's equilibrium payoff, $\widehat{\pi} \equiv \widehat{\mu}\left(1-\widehat{C}^{\prime}(\widehat{\mu})\right)$. Indeed, the binding constraint (9) evaluated at $(C, \mu)=(\widehat{C}, \widehat{\mu})$ can be written as

$$
\widehat{C}^{\prime}(\widetilde{\mu})=\left\{\begin{array}{cc}
0 & \text { if } \widetilde{\mu}<\widehat{\pi} \\
1-\frac{\widehat{\pi}}{\mu} & \text { if } \widetilde{\mu} \in[\widehat{\pi}, \widehat{\mu}] .
\end{array}\right.
$$

Consequently, the agent's problem in (10) can be rewritten as


Optimal Technology Design

Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.

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

$$
\max _{\widehat{\mu}, \widehat{\pi} \in[0,1]} \widehat{\mu}\left(1-\frac{\widehat{\pi}}{\widehat{\mu}}\right)-\int_{\widehat{\pi}}^{\widehat{\mu}}\left(1-\frac{\widehat{\pi}}{\widetilde{\mu}}\right) d \widetilde{\mu}
$$

[^9]To conclude Proposition 2, observe that

$$
\widehat{\mu}\left(1-\frac{\widehat{\pi}}{\widehat{\mu}}\right)-\int_{\widehat{\pi}}^{\widehat{\mu}}\left(1-\frac{\widehat{\pi}}{\widetilde{\mu}}\right) d \widetilde{\mu}=\int_{\widehat{\pi}}^{\widehat{\mu}} \frac{\widehat{\pi}}{\widetilde{\mu}} d \widetilde{\mu}=\widehat{\pi}[\log \widehat{\mu}-\log \widehat{\pi}],
$$

which is maximized at $\widehat{\mu}=1$ and $\widehat{\pi}=1 / e$. This explains how we obtain Parts (ii) and (iii) of the proposition. In particular, the principal's profit in an optimal outcome is $\pi^{*}=\mu^{*}\left(1-b^{*}\right)=1 / e$, and since $\mu^{*}=1$, we have $b^{*}=1-1 / e$. Finally, note that evaluating the right-hand side of (11) at $\widehat{\mu}=1$ and $\widehat{\pi}=1 / e$ yields Part (i) of the proposition.

Finally, we compute the payoffs of the agent and the principal in the outcome $\left(C^{*}, b^{*}, \mu^{*}\right)$. As mentioned above, the principal's equilibrium payoff is $\mu^{*}\left(1-b^{*}\right)=1 / e$. The agent's payoff is pinned down by evaluating (13) at $\left(\mu^{*}, \pi^{*}\right)=(1,1 / e)$, also yielding $1 / e$.


Optimal Technology Design

## 4 Discussion

Payoff Possibility Set.-We can use our results to characterize the set of possible payoff combinations which can arise in principal-agent models for some arbitrary production technology. To be more specific, we still consider an environment where the agent is risk neutral and has limited liability. However, the production technology is exogenously given and satisfies the constraint that the largest output cannot exceed one. Moreover, the agent has an outside option we take to be zero. We wish to characterize those payoff profiles which can arise in equilibrium for some production technology. We next show that there exists a production technology where the principal receives $\widehat{\pi}$ and the agent receives $\widehat{u}$ if and only if $\widehat{\pi} \in[0,1]$ and $\widehat{u} \in[0,-\widehat{\pi} \log \widehat{\pi}]$; see Figure 1 for illustration.

First note that the principal's payoff, $\widehat{\pi}$, must be between zero and one. The reason is that the contract $w_{0}$ guarantees at least zero profit. Since output is less than one, limited liability implies that the profit cannot exceed one. Next, we identify the frontier of the payoff possibility set. That is, we compute the largest payoff the agent can get if the principal's payoff is $\widehat{\pi}$. By Proposition 1, we know that this largest payoff is achieved in a binary project. Furthermore, recall that in Section 3.2 we rewrote the problem of designing the optimal binary project as a maximization problem with respect to the equilibrium probability of success, $\widehat{\mu}$, and the principal's equilibrium profit, $\widehat{\pi}$; see (12). So the problem of designing the agent-optimal binary technology which generates $\widehat{\pi}$ can be reduced to a similar maximization problem except $\widehat{\pi}$ is treated as a parameter instead of a choice variable. By Equation (13), the agent's maximal payoff is $-\widehat{\pi} \log \widehat{\pi}$.

Since the agent's outside option is zero, his equilibrium payoff cannot be negative. It remains to

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 1: The shaded area represents the set of payoffs for the principal and the agent that arise for some technology of the agent. The players' payoffs under the optimal project characterized in Proposition 2 are illustrated by the red dot.

argue that given the principal's payoff, $\widehat{\pi}$, for each $\widehat{u} \in[0,-\widehat{\pi} \log \widehat{\pi})$, there is a production technology so that the equilibrium payoff profile is $(\widehat{\pi}, \widehat{u})$. To do so, consider the production technology generating $(\widehat{\pi},-\widehat{\pi} \log \widehat{\pi})$ and add a fixed cost of $-\widehat{\pi} \log \widehat{\pi}-\widehat{u}$. That is, the agent's cost of each distribution is increased by this quantity. Adding this fixed cost does not change the agent's incentives, but it lowers his payoff to $\widehat{u}$.

Competition Among Agents.- As is standard in the literature, we maintained the assumption that the principal has full bargaining power and makes a take-it-or-leave-it offer to the agent. The principal's strong bargaining power is often motivated by fierce competition among the agents, which is usually left unmodeled. At first glance, such a justification might be questioned in our setting because competition could limit the rent an agent can earn from contracting with any principal. It is therefore of interest to examine which projects may arise in markets with imperfect competition among agents.

One way to understand competition among agents is to consider embedding our model into a standard search and matching framework in which one-to-one matches occur over time. Each
steady state equilibrium in such a model is associated with a continuation value of an unmatched principal. So, when an agent designs a project, he must bear in mind that any principal prefers to remain unmatched whenever her value from contracting with the agent is less than the continuation value from further search. As before, the agent's problem of designing his equilibrium project can be understood in terms of maximizing the expression in Equation (13) by choice of the completion probability and principal expected profits. The difference, however, is that the principal profit, $\widehat{\pi}$, is bounded from below by a constant which makes a principal indifferent between contracting with the agent and remaining unmatched. The solution to that problem has features similar to those of the optimal project described by Proposition 2. In particular, the equilibrium completion probability is one and the principal is indifferent between offering the equilibrium bonus and anything less than that.

Other Constraints on the Technology.- While we believe there are a range of situations where the agent can determine the technology in advance of contracting, in reality the agent's choice of projects would typically be more limited than in our setting where the only constraint is the bounds on output. Such constraints on technology design may ultimately be responsible for the key properties of the selected project. An example is where noise in the environment prevents distributions that put mass at the extremes of the possible output values, ruling out binary projects. Note also that imposing additional constraints on the set of available technologies is related to the possibility that some designs are more costly than others, e.g. binary technologies might be feasible but too costly to set up. We have treated in this paper the case where the agent is only subject to output bounds, and where all designs are costless to the agent. This was intended as a natural first step and helps shed light on the forces at play in the agent's design problem, while future work could examine the choice of projects given different constraints or costly technology choice.


Optimal Technology Design

In spite of these observations, the analysis in the paper does already allow determination of the agent's optimal project in some situations where he is more constrained. For instance, suppose that the choice of project $c$ is restricted to come from a set that contains $c^{*}$, a project determined to maximize the agent's payoff in our setting with only bounds on output. Then the same project $c^{*}$ remains optimal in the more constrained setting. An example is where there is, in addition to the output bounds, a lower bound $L(\mu)$ on the cost of generating mean output $\mu$. Recall that $C^{*}$ is the optimal binary project defined in Proposition 2. Provided $L(\mu) \leq C^{*}(\mu)$ for all $\mu \in[0,1]$, then $C^{*}$ remains an optimal project for the agent.

There are also cases where constraints on project selection bind, but the optimal project follows
closely from our analysis. Suppose that, in addition to output being bounded between zero and one, mean output is restricted to be no greater than some $m \in(0,1)$. This constraint does not affect any of the arguments needed to obtain Proposition 1, so there is an optimal project for the agent which is binary. ${ }^{18}$ Also, the agent's maximal payoff in an equilibrium where the probability of output one is $\widehat{\mu}$ and the principal earns payoff $\widehat{\pi}$ is still given by the expression in Equation (13), following the arguments in the proof of Proposition 2. An optimal project is then defined by Equation (11) with $\widehat{\mu}=m$ and $\widehat{\pi}=m / e$. There is an equilibrium given this binary project where the principal offers a bonus payment $1-1 / e$ for output one (and pays zero otherwise), and the agent chooses probability $m$ of output one. We have thus seen that, in the mean-constrained version of the problem, the agent's equilibrium probability of a high output is less than one.

Risk Aversion.- Our results can be generalized to the case where the agent is risk averse. To explain it more formally, let us modify our model so that, if the agent receives payment $w$ and chooses $F$ at cost $c(F)$, then his payoff is $v(w)-c(F)$, where $v$ is an increasing, concave and continuously differentiable function. It can be shown that, even in this case, binary projects remain optimal. More precisely, the following version of the statement of Proposition 1 remains valid. For each project $c^{*}$ and any equilibrium $\left(w^{*}, F^{*}\right)$ in $c^{*}$, there is a binary project $\widetilde{c}$ with an equilibrium $\left(\widetilde{w}, B_{1}\right)$ such that the outcome $\left(\widetilde{c}, \widetilde{w}, B_{1}\right)$ Pareto dominates $\left(c^{*}, w^{*}, F^{*}\right)$. The characterization of optimal binary projects follows the same steps as for the risk-neutral case. The marginal cost of any completion probability $\mu>\pi^{*}$ is $v\left(1-\left[\pi^{*} / \mu\right]\right)$, where $\pi^{*}$ is the equilibrium payoff of the principal. These claims are proved in the Online Appendix.

Relation to Another Hold-up Problem.- As mentioned in the Introduction, Condorelli and Szentes (2020) study a different hold-up problem in the context of bilateral trade. In their model, the buyer first chooses the distribution of her valuation for the seller's good. Then the seller, after observing the value distribution, makes a take-it-or-leave-it price offer. It turns out that, when the support of any value distribution must be in $[0,1]$, the buyer's equilibrium CDF, $F^{*}$, has the same functional form as $C^{* \prime}$ described in Proposition 2. More precisely, $F^{*}(v)=1-1 /(e v)$ on $(1 / e, 1)$. In what follows, we attempt to illuminate this similarity and the relationships between the two models by transforming the buyer's problem into a strategically equivalent one and show that the transformed problem is identical to the constrained maximization problem (10).

First note that by choosing a value distribution, $F$, the buyer determines his demand curve,

[^10]$1-F$. That is, the probability of trade at price $p$ is $1-F(p)$. The idea behind the transformation is that we replace the buyer's choice set by inverse demand curves and assume that the seller sets a quantity instead of a price, i.e., the probability of trade. Since there is a bijection between demand curves and inverse demand curves and the price is pinned down by inverse demand curve for each quantity, the transformed model is strategically equivalent to that of Condorelli and Szentes (2020). Since both the buyer's willingness-to-pay and the probability of trade are in [0, 1], the domain as well as the range of the inverse demand curve must also be in [0,1]. Let $\mathcal{P}$ denote the set of such inverse demand functions. So, if the buyer chooses $P \in \mathcal{P}$ and the seller sets $Q$ then the price is $P(Q)$. Next, we explain that the buyer's problem can be written as
$$
\begin{aligned}
& \max _{q \in[0,1], P \in \mathcal{P}} \int_{0}^{Q}(P(x)-P(Q)) d x \\
& \text { s.t. } Q P(Q) \geq \tilde{Q} P(\tilde{Q}) \text { for all } \tilde{Q} \in[0,1]
\end{aligned}
$$
Note that the objective function is the familiar expression for the buyer's payoff from Consumer Theory, for a given inverse demand curve $P$ and quantity $Q$. In addition, the constraint guarantees that when the buyer's inverse demand curve is $P$, the seller indeed prefers to set $Q$ to any smaller quantity. Now, observe that replacing $P$ by $1-C^{\prime}$ in this problem yields (10), that is, the two problems are equivalent. This means that the buyer-optimal inverse demand curve, $P^{*}$, is given by $P^{*}(Q)=1-C^{* \prime}(Q)$, where $C^{*}$ is characterized Proposition 2, that is, $P^{*}(Q)=1 /[e Q]$ on $[1 / e, 1]$. Finally, notice that for each inverse demand curve $P$, the buyer's value distribution is determined by the following equation: $P(1-F(v))=v$. Coincidently, applying this formula for $P^{*}$ yields $F^{*}(v)=1-1 /(e v)$ on $(1 / e, 1)$, confirming the main result in Condorelli and Szentes (2020).


Optimal Technology Design

Our discussion about the similarity of Proposition 2 to the main result in Condorelli and Szentes (2020) has followed the heuristic argument presented in Section 3.2. This assumes the applicability of the first-order approach to derive that the agent's bonus is equal to his marginal cost, i.e., to derive Equation (8). Importantly, the validity of this approach must be verified, and hence the proof of Proposition 2 in the appendix relies on a different argument, one which has no parallel in Condorelli and Szentes.


Optimal Technology Design

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Optimal Technology Design

## Appendix A: Omitted proofs


Optimal Technology Design

## Proofs of results in Section 3.1

Proof of Lemma 1. Consider a sequence $\left(\mu_{n}\right)$, with $\mu_{n} \in\left[0, \mu_{F^{*}}\right)$ for all $n$, such that

$$
u\left(\widehat{c}, w_{b}\right)=\lim _{n \rightarrow \infty} U\left(\widehat{c}, w_{b}, B_{\mu_{n}}\right) \text { and } \pi\left(\widehat{c}, w_{b}\right)=\lim _{n \rightarrow \infty} \Pi\left(w_{b}, B_{\mu_{n}}\right) .
$$

For each $k \in \mathbb{N}$, there exists $n_{k}$ such that, for all $\mu \in[0,1]$,

$$
\mu_{n_{k}} b-\hat{c}\left(B_{\mu_{n_{k}}}\right)+\frac{1}{k} \geq \mu b-\hat{c}\left(B_{\mu}\right) .
$$

Equivalently, for all $k$, and all $\mu \in\left[0, \mu_{F^{*}}\right)$,

$$
\mu_{n_{k}} b-\inf \left\{c^{*}(F): \mu_{F}=\mu_{n_{k}}\right\}+\frac{1}{k} \geq \mu b-\inf \left\{c^{*}(F): \mu_{F}=\mu\right\} .
$$

For each $k$, we can pick a distribution $F_{n_{k}}$ with mean $\mu_{n_{k}}$ such that

$$
c^{*}\left(F_{n_{k}}\right)<\inf \left\{c^{*}(F): \mu_{F}=\mu_{n_{k}}\right\}+\frac{1}{k} .
$$

Then, for all $k$ and all $F \in \mathcal{F}$ with mean $\mu_{F} \in\left[0, \mu_{F^{*}}\right)$,

$$
\mu_{n_{k}} b-c^{*}\left(F_{n_{k}}\right)+\frac{2}{k}>\mu_{F} b-c^{*}(F) .
$$

There are then two cases. In the first, the inequality (14) holds for all $k$ and all $F \in \mathcal{F}$ (not only those $F$ with $\mu_{F}<\mu_{F^{*}}$ ). Then

$$
u\left(c^{*}, w_{b}\right)=\lim _{k \rightarrow \infty} U\left(c^{*}, w_{b}, F_{n_{k}}\right)
$$

and hence

$$
\pi\left(c^{*}, w_{b}\right) \geq \lim _{k \rightarrow \infty} \Pi\left(w_{b}, F_{n_{k}}\right)=\lim _{k \rightarrow \infty} \Pi\left(w_{b}, B_{\mu_{n_{k}}}\right)=\pi\left(\widehat{c}, w_{b}\right)
$$

as desired. In the second, the inequality (14) fails to hold for some $k$ and some $F \in \mathcal{F}$ with $\mu_{F} \geq \mu_{F^{*}}$, which implies

$$
u\left(c^{*}, w_{b}\right)=\sup \left\{\mu_{F} b-c^{*}(F): F \in \mathcal{F}\right\}>\sup \left\{\mu_{F} b-c^{*}(F): F \in \mathcal{F}, \mu_{F}<\mu_{F^{*}}\right\} .
$$

This means that there is a sequence of distributions in $\mathcal{F}$ along which the agent's payoff converges to his value $u\left(c^{*}, w_{b}\right)$ and for which every distribution has mean at least $\mu_{F^{*}}$. By the definition of the principal's value, we have

$$
\pi\left(c^{*}, w_{b}\right) \geq \mu_{F^{*}}(1-b) \geq \pi\left(\widehat{c}, w_{b}\right),
$$

where the second inequality follows because any distribution with mean at least $\mu_{F^{*}}$ is assigned an infinite cost in the project $\widehat{c}$. $\square$

Proof of Lemma 2. Let us fix $b \in\left[0, b^{*}\right)$. We first show that $w_{b}$ does not implement $B_{\mu_{F^{*}}}$ in $\left(\widetilde{c}, w_{b}\right)$. Suppose for a contradiction that $B_{\mu_{F^{*}}}$ satisfies the agent's incentive constraint in $\left(\widetilde{c}, w_{b}\right)$, that is,

$$
\mu_{F^{*}} b-\bar{c} \geq \sup _{\mu<\mu_{F^{*}}}\left\{\mu b-\widetilde{c}\left(B_{\mu}\right)\right\} .
$$

Therefore,

$$
\bar{c} \leq-\sup _{\mu<\mu_{F^{*}}}\left\{\left(\mu-\mu_{F^{*}}\right) b-\widetilde{c}\left(B_{\mu}\right)\right\} \leq-\sup _{\mu<\mu_{F^{*}}}\left\{\left(\mu-\mu_{F^{*}}\right) b^{*}-\widetilde{c}\left(B_{\mu}\right)\right\}=\bar{c},
$$

where the first inequality is just the previous displayed inequality rearranged, the second inequality follows from $b<b^{*}$, and the equality is the definition of $\bar{c}$. Since the farthest left term and the farthest right term are equal in the previous chain, all inequalities must be equalities. Note that the second inequality is an equality only if the supremum in Equation (15) is approached along a sequence of $\mu^{\prime}$ s converging to $\mu_{F^{*}}$. Since $\widetilde{c}\left(B_{\mu}\right)=\widehat{c}\left(B_{\mu}\right)$ whenever $\mu \neq \mu_{F^{*}}$, and since $\widehat{c}\left(B_{\mu}\right)=\infty$ for $\mu \geq \mu_{F^{*}}$, it follows that the supremum of $\mu b-\hat{c}\left(B_{\mu}\right)$ is approached by the same sequence. Hence, $\pi\left(\widehat{c}, w_{b}\right)=\mu_{F^{*}}(1-b)$. We can conclude that

$$
\pi\left(c^{*}, w_{b}\right) \geq \pi\left(\widehat{c}, w_{b}\right)=\mu_{F^{*}}(1-b)>\mu_{F^{*}}\left(1-b^{*}\right)=\Pi\left(w^{*}, F^{*}\right),
$$

where the first inequality follows from Lemma 1, the strict inequality is implied by $b<b^{*}$ and the second equality follows from the definition of $b^{*}$. This inequality implies that $w^{*}$ is not incentive compatible in project $c^{*}$, a contradiction.

Since $w_{b}$ does not implement $B_{\mu_{F^{*}}}$ in $\widetilde{c}, U\left(\widetilde{c}, w_{b}, B_{\mu_{F^{*}}}\right)<u\left(\widetilde{c}, w_{b}\right)$. We next show that $\mathbf{F}^{\widetilde{c}, w_{b}}=$ $\mathbf{F}^{\widehat{c}, w_{b}} .{ }^{19}$ Note that, for each $\left(F_{n}\right) \in \mathbf{F}^{\widetilde{c}, w_{b}} \cup \mathbf{F}^{\widehat{c}, w_{b}}$, there exists $K \in \mathbb{N}$ such that $F_{k} \neq B_{\mu_{F^{*}}}$

[^11]if $k>K$. If $\left(F_{n}\right) \in \mathbf{F}^{\widehat{c}, w_{b}}$, it follows from $\widehat{c}\left(B_{\mu_{F^{*}}}\right)=\infty$. If $\left(F_{n}\right) \in \mathbf{F}^{\widetilde{c}, w_{b}}$, it is implied by $U\left(\widetilde{c}, w_{b}, B_{\mu_{F^{*}}}\right)<u\left(\widetilde{c}, w_{b}\right)$. Since $\widetilde{c}(F)=\widehat{c}(F)$ whenever $F \neq B_{\mu_{F^{*}}}$, this means that, for each $\left(F_{n}\right) \in \mathbf{F}^{\widetilde{c}, w_{b}} \cup \mathbf{F}^{\widehat{c}, w_{b}}$,
$$
\lim _{n \rightarrow \infty} U\left(\widetilde{c}, w_{b}, F_{n}\right)=\lim _{n \rightarrow \infty} U\left(\widehat{c}, w_{b}, F_{n}\right),
$$
implying that $\mathbf{F}^{\widetilde{c}, w_{b}}=\mathbf{F}^{\widehat{c}, w_{b}}$. Consequently,
$$
\begin{aligned}
\pi\left(\widetilde{c}, w_{b}\right) & \equiv \sup \left\{\limsup _{n \rightarrow \infty} \Pi\left(w_{b}, F_{n}\right):\left(F_{n}\right) \in \mathbf{F}^{\widetilde{c}, w_{b}}\right\} \\
& =\sup \left\{\limsup _{n \rightarrow \infty} \Pi\left(w_{b}, F_{n}\right):\left(F_{n}\right) \in \mathbf{F}^{\widehat{c}, w_{b}}\right\}=\pi\left(\widehat{c}, w_{b}\right) .
\end{aligned}
$$

QED


Optimal Technology Design

## Proof of Proposition 2

This section proves Proposition 2. We begin by considering an arbitrary binary project $C$. Let us drop the dependence on $C$ and write the value for the agent when the bonus is $b$ as $u(b)$. The value is given by

$$
u(b)=\sup _{\mu \in[0,1]}\{b \mu-C(\mu)\}
$$

Note that $u$ is non-decreasing. Moreover, as the upper envelope of linear functions, it is convex and hence continuous.

Let $\Gamma(b)$ be the set of values $\mu$ such that there is a sequence $\left(\mu_{n}\right)$ with $\mu_{n} \rightarrow \mu$ and $b \mu_{n}-C\left(\mu_{n}\right) \rightarrow$ $u(b)$. Take $\bar{\mu}(b)=\max \Gamma(b)$, and note that the maximum is attained. Similarly, let $\underline{\mu}(b)$ be the minimum of $\Gamma(b)$ (also attained). Note that, if the principal offers a bonus $b \in[0,1]$ in project $C$, then she obtains value $\bar{\mu}(b)(1-b)$.

For any $b \geq 0$, let $u_{+}^{\prime}(b)$ be the right derivative of $u$ at $b$. For any $b>0$, let $u_{-}^{\prime}(b)$ be the left derivative of $u$ at $b$. We next show a result that is analogous to Theorem 1 of Milgrom and Segal (2002), but adjusted for the possibility that the agent's payoff $u(b)$ is not attained by values $\mu \in \Gamma(b)$.

Lemma 3. For all $b \geq 0, u_{+}^{\prime}(b) \geq \bar{\mu}(b)$. For all $b>0, u_{-}^{\prime}(b) \leq \underline{\mu}(b)$.
Proof. Consider a sequence $\left(\mu_{n}\right)$ with $\mu_{n} \rightarrow \bar{\mu}(b)$ and $b \mu_{n}-C\left(\mu_{n}\right) \rightarrow u(b)$. Then, for any $b^{\prime}>b$, we have

$$
\left(b^{\prime}-b\right) \bar{\mu}(b)=\lim _{n \rightarrow \infty}\left\{b^{\prime} \mu_{n}-C\left(\mu_{n}\right)\right\}-\lim _{n \rightarrow \infty}\left\{b \mu_{n}-C\left(\mu_{n}\right)\right\} \leq u\left(b^{\prime}\right)-u(b) .
$$

Dividing by $b^{\prime}-b$ and taking limits as $b^{\prime}$ approaches $b$ from above yields $u_{+}^{\prime}(b) \geq \bar{\mu}(b)$.
Let $b>0$ and consider a sequence $\left(\mu_{n}\right)$ with $\mu_{n} \rightarrow \underline{\mu}(b)$ and $b \mu_{n}-C\left(\mu_{n}\right) \rightarrow u(b)$. For any $b^{\prime}<b$, we have

$$
\left(b-b^{\prime}\right) \underline{\mu}(b)=\lim _{n \rightarrow \infty}\left\{b \mu_{n}-C\left(\mu_{n}\right)\right\}-\lim _{n \rightarrow \infty}\left\{b^{\prime} \mu_{n}-C\left(\mu_{n}\right)\right\} \geq u(b)-u\left(b^{\prime}\right) .
$$

Dividing by $b-b^{\prime}$ and taking limits as $b^{\prime}$ approaches $b$ from below yields $u_{-}^{\prime}(b) \leq \underline{\mu}(b)$.
QED
We can further use the convexity of $u$ to determine its right derivative in terms of the completion probability attainable with a given bonus.

Lemma 4. For all $b \geq 0, u_{+}^{\prime}(b)=\bar{\mu}(b)$.
Proof. Fix $b \geq 0$ and suppose for a contradiction that $u_{+}^{\prime}(b)>\bar{\mu}(b)$. By convexity of $u$ and the previous lemma

$$
\bar{\mu}(b)<u_{+}^{\prime}(b) \leq u_{-}^{\prime}\left(b^{\prime}\right) \leq \underline{\mu}\left(b^{\prime}\right)
$$

for all $b^{\prime}>b$. For each $n \in \mathbb{N}$, let

$$
b_{n} \in\left(b, b+\frac{1}{n}\right)
$$

and let $\mu_{n} \in\left[\frac{\bar{\mu}(b)+u_{+}^{\prime}(b)}{2}, 1\right]$ and such that

$$
b_{n} \mu_{n}-C\left(\mu_{n}\right)>u\left(b_{n}\right)-\frac{1}{n}
$$

(that such a choice is possible follows because $\frac{\bar{\mu}(b)+u_{+}^{\prime}(b)}{2}<u_{-}^{\prime}\left(b_{n}\right) \leq \underline{\mu}\left(b_{n}\right)$ for all $n$ ). Consider a subsequence ( $b_{n_{k}}$ ) such that $\mu_{n_{k}} \rightarrow \mu^{*} \geq \frac{\bar{\mu}(b)+u_{+}^{\prime}(b)}{2}$ for some $\mu^{*}$. We have

$$
\lim \left\{b \mu_{n_{k}}-C\left(\mu_{n_{k}}\right)\right\}=\lim \left\{b_{n_{k}} \mu_{n_{k}}-C\left(\mu_{n_{k}}\right)\right\}=\lim u\left(b_{n_{k}}\right)=u(b)
$$

where the final equality follows by continuity of $u$. The fact that $\mu^{*}>\bar{\mu}(b)$ contradicts the definition of $\bar{\mu}(b)$.

QED

Note now that, because $u$ is convex, it is absolutely continuous and hence differentiable almost everywhere. This means that

$$
u(b)=u(0)+\int_{0}^{b} \bar{\mu}(s) d s
$$

It is immediate from the agent's problem that we must have $u(0) \leq 0$; i.e., the agent cannot obtain a strictly positive payoff if the bonus is set to zero.

Consider now a project $C$ with an equilibrium in which the principal offers bonus $\widehat{b}$ for project completion, the agent chooses completion probability $\widehat{\mu}$, and therefore the principal's payoff is given by $\widehat{\pi}=\widehat{\mu}(1-\widehat{b})$. Note that, if $(C, \widehat{b}, \widehat{\mu})$ is an optimal outcome for the agent, then we must have $\widehat{\mu}>0$. Incentive compatibility of the principal offering bonus $\widehat{b}$ requires that, for all $b$,

$$
\begin{aligned}
\widehat{\pi} & \geq \bar{\mu}(b)(1-b) \\
& =u_{+}^{\prime}(b)(1-b) .
\end{aligned}
$$

Hence, if the agent is to get positive rent in outcome $(C, \widehat{b}, \widehat{\mu})$, we must have also $\widehat{\pi}>0$. Assume from now on that $\widehat{\mu}, \widehat{\pi}>0$.

Now let us determine the highest agent value, across projects $C$, that can occur for an equilibrium in which the principal offers bonus $\widehat{b}$ for completion and the agent chooses completion probability $\widehat{\mu}$. Consider then the problem of maximizing the agent's equilibrium payoff

$$
u(\widehat{b})=u(0)+\int_{0}^{\widehat{b}} u_{+}^{\prime}(b) d b
$$

by choice of convex function $u: \mathbb{R}_{+} \rightarrow \mathbb{R}$ satisfying (i) $u(0) \leq 0$, and (ii) $\widehat{\pi} \geq u_{+}^{\prime}(b)(1-b)$ for all $b$. The first requirement reflects the above observation that the agent cannot obtain a positive payoff if the bonus is zero. The second condition is a re-statement of Condition (17). Any solution to this problem involves $u(0)=0$ and

$$
u_{+}^{\prime}(b)=\frac{\widehat{\pi}}{1-b}
$$

for all $b \in[0, \widehat{b}]$. In other words, the constraint (ii), or equivalently (17), holds with equality over $b \in[0, \widehat{b}]$ (in which case, the principal must obtain payoff $\widehat{\pi}$ from all such bonuses $b$ ). The agent's value function is therefore given on $[0, \widehat{b}]$ by

$$
u(b)=\int_{0}^{b} \frac{\widehat{\pi}}{1-z} d z .
$$

Now, recall that $\widehat{\pi}=\widehat{\mu}(1-\widehat{b})$, or $\widehat{b}=1-\frac{\widehat{\pi}}{\mu}$. The agent's equilibrium payoff can then be written as


Optimal Technology Design

$$
\int_{0}^{1-\frac{\widehat{\pi}}{\mu}} \frac{\widehat{\pi}}{1-z} d z=[-\widehat{\pi} \log (1-z)]_{0}^{1-\frac{\widehat{\pi}}{\mu}}=\widehat{\pi}(\log (\widehat{\mu})-\log (\widehat{\pi}))
$$

which is the expression given in Equation (13) (hence establishing also the one in Equation (12)). As explained in the main text, this payoff is maximized across feasible equilibrium values of $\widehat{\mu}$ and $\widehat{\pi}$ by $\widehat{\mu}=1$ and $\widehat{\pi}=\frac{1}{e}$. The corresponding equilibrium bonus must be $\widehat{b}=1-1 / e$.

Note then that, if the project is $C^{*}$ as given in the proposition (with $C^{*}(0)=0$, as explained in the main text), and the principal offers any $b \in[0,1-1 / e]$, the agent best responds by choosing $\mu$ such that

$$
b=1-\frac{1}{e \mu},
$$

i.e. $\mu=\frac{1}{e(1-b)}$. All such bonuses therefore generate profit $1 / e$ for the principal. Hence, it is indeed an equilibrium of project $C^{*}$ for the principal to offer bonus $b^{*}=1-1 / e$, and the agent to choose completion probability equal to $\mu^{*}=1$. This completes the proof of the proposition. $\square$


Optimal Technology Design

## Appendix B: Discussion of uniqueness of optimal projects

The optimal project, $C^{*}$, described in Proposition 2 is not unique. To see this, note that $C^{*}$ can be arbitrarily modified at completion probabilities strictly below $1 / e$. Since $C^{*}(1 / e)=0$, any of these probabilities are weakly dominated by $1 / e$, so the modified project is still optimal. The goal of this appendix is to argue that optimal projects differ only in non-essential ways.

Uniqueness of the Optimal Binary Project.- Recall that the optimal project $C^{*}$ in Proposition 2 is determined via the principal's incentive constraint, (9), evaluated now at $(C, \mu)=\left(C^{*}, \mu^{*}\right)$ and taken to hold with equality for completion probabilities $\widetilde{\mu}$ above $\pi^{*}$. This means that the principal can also generate her equilibrium payoff by setting any bonus smaller than $b^{*}$. Appendix A showed that these conclusions are valid for each optimal binary project. We also showed that, in any optimal binary project, the agent's optimal payoff is determined by an equilibrium which satisfies Parts (ii) and (iii) of the proposition. We now state formally these observations. We abuse notation and write the players' values, $u$ and $\pi$, as functions of a binary outcome.

Remark 1. In any optimal binary project $C$,

(i) $\left(b^{*}, \mu^{*}\right)=(1-1 / e, 1)$ is an agent-optimal equilibrium,
(ii) for all $b \in[0,1-1 / e], u(C, b)=\int_{0}^{b} 1 /[e(1-z)] d z$, and
(iii) for all $b \in[0,1-1 / e], \pi(C, b)=1 / e$.

Proof of Remark 1. The proof of Remark 1 recalls the proof of Proposition 2 in Appendix A.

Part (i) of the remark follows because the expression in Equation (19) represents the agent's highest possible expected payoff in any project in which the completion probability is $\widehat{\mu}$ and the principal's expected payoff is $\widehat{\pi}$. Moreover, it is uniquely maximized by $\widehat{\mu}=\mu^{*}=1$ and $\widehat{\pi}=\pi^{*}=1 / e$. Part (ii) of Remark 1 then follows from Equation (18). Part (iii) of Remark 1 follows because the constraint (17) holds with equality in an optimal binary project over the relevant range of bonus payments. That is, because

$$
\pi^{*}=\bar{\mu}(b)(1-b)
$$

for all $b \in[0,1-1 / e]$, where recall $\bar{\mu}(b)(1-b)$ coincides with the principal's value $\pi(C, b)$ in any subgame $(C, b)$. $\square$

QED

We now argue that Remark 1 implies optimal binary projects are "close" to uniquely determined. We begin with a remark which compares any optimal binary project to the project $C^{*}$ of Proposition 2.

Remark 2. In any optimal binary project $C$,

(i) for all $\mu \in[0,1], C(\mu) \geq C^{*}(\mu)$, and
(ii) for all $\mu \in[1 / e, 1]$, there is a sequence $\left(\mu_{n}\right)$ with $\mu_{n} \rightarrow \mu$ and $C\left(\mu_{n}\right) \rightarrow C^{*}(\mu)$.

Proof of Remark 2. To show Part (i) of Remark 2, consider the optimal project $C^{*}$ in Proposition 2. Notice that, as we reduce the bonus $b$ from $b^{*}$ to zero, the agent's best response in the subgame $\left(C^{*}, b\right)$ decreases continuously from one to zero. Suppose now that $C$ is a project with $C\left(\mu^{\prime}\right)<C^{*}\left(\mu^{\prime}\right)$ for some $\mu^{\prime}$, and let $b^{\prime} \in\left(0, b^{*}\right]$ be the bonus that implements $\mu^{\prime}$ in project $C^{*}$. Then we have

$$
u\left(C, b^{\prime}\right) \geq b^{\prime} \mu^{\prime}-C\left(\mu^{\prime}\right)>b^{\prime} \mu^{\prime}-C^{*}\left(\mu^{\prime}\right)=u\left(C^{*}, b^{\prime}\right)=\int_{0}^{b^{\prime}} 1 /[e(1-z)] d z
$$

The first equality follows because $\mu^{\prime}$ is a best response in the subgame ( $C^{*}, b^{\prime}$ ). The second equality follows because $C^{*}$ is an optimal project, and by Part (ii) of Remark 1. Hence, by Part (ii) of Remark 1, $C$ is not an optimal project.

To show Part (ii) of Remark 2, fix an optimal binary project $C$ and consider a completion probability $\mu \in[1 / e, 1]$. By Part (iii) of Remark 1, it can be attained by a bonus $b=1-1 /(\mu e) \in$ $[0,1-1 / e]$. Formally, we mean that there exists a sequence $\left(\mu_{n}\right)$ convergent to $\mu$ with $\mu_{n} b-C\left(\mu_{n}\right) \rightarrow$ $u(C, b)$. Note that the bonus $b$ also implements $\mu$ in project $C^{*}$. By Part (ii) of Remark 1, the
agent's value in subgame ( $C, b$ ) is the same as in subgame ( $C^{*}, b$ ); that is, $u(C, b)=u\left(C^{*}, b\right)$. In particular, considering the aforementioned sequence $\left(\mu_{n}\right)$, we have

$$
\mu_{n} b-C\left(\mu_{n}\right) \rightarrow u\left(C^{*}, b\right)=\mu b-C^{*}(\mu),
$$

implying $\lim _{n \rightarrow \infty} C\left(\mu_{n}\right)=C^{*}(\mu)$, which is what we wanted to show. $\square$

We argue that, in spite of possible differences between any optimal binary project $C$ and the project $C^{*}$ of Proposition 2, such projects can be viewed almost equivalently from the agent's perspective. First, note that Remark 2 admits that some completion probabilities in $[1 / e, 1]$ may be more costly under $C$ than $C^{*}$. In this case, however, there are arbitrarily close probabilities which are as affordable to the agent as in $C^{*}$, or for which the difference in costs is negligible (this follows by Part (ii) of Remark 2). In addition, it turns out that, for any optimal binary project $C$, the specification of costs for probabilities below $1 / e$ is irrelevant. The reason is that, using Part (ii) of Remark 2, the agent can generate a completion probability at least $1 / e$ at negligible cost. ${ }^{20}$ Hence, the agent's value can be approached by completion probabilities at least $1 / e$ irrespective of the bonus.

We now formalize further the equivalence of optimal binary projects $C$ on $[1 / e, 1]$. Note first that if $C$ is restricted to be continuous, then $C$ is unique on $[1 / e, 1]$ and equal to $C^{*}$ on this interval by Part (ii) of Remark 2. For cost functions $C$ that are not continuous, we now demonstrate formally that the completion probabilities $\mu$ that are relevant for the agent's problem are only those for which the cost is close to the one given by $C^{*}$.

To achieve our goal we let, for any $\varepsilon>0$,

$$
P(\varepsilon) \equiv\left\{(\mu, C(\mu)): \mu \in[1 / e, 1], C(\mu)-C^{*}(\mu) \leq \varepsilon\right\} .
$$


Optimal Technology Design

Suppose the agent can choose only probabilities $\mu$ with $(\mu, C(\mu)) \in P(\varepsilon)$, and suppose the associated costs are $C(\mu)$. Then $P(\varepsilon)$ describes the agent's technology in project $C$ but after removing his ability to choose probabilities $\mu$ that we anticipate being redundant, either because they are less than $1 / e$ or because their costs exceed $C^{*}(\mu)$ by more than $\varepsilon$. By Parts (i) and (ii) of Remark 2, for a fixed bonus payment $b$, the agent's value is the same for the restricted technology $P(\varepsilon)$ as if

[^12]the agent could choose any completion probability with a cost specified by $C$. In fact, the feature of $P(\varepsilon)$ that is relevant in determining the agent's value is the lower boundary of its closure. For any $\varepsilon>0$, any $\mu \in[1 / e, 1]$, we have
$$
\min \{y:(\mu, y) \in \operatorname{cl}(P(\varepsilon))\}=C^{*}(\mu),
$$
which again can be seen directly from Parts (i) and (ii) of Remark 2. ${ }^{21}$ This demonstrates a further sense of equivalence between $C$ and $C^{*}$.

Uniqueness beyond binary projects. - The above discussion describes a qualified sense in which optimal binary projects are uniquely determined. Still, the possibility of optimal but non-binary projects may also be a source of non-uniqueness. Nonetheless, we show that properties analogous to those described above continue to hold, even among non-binary projects.

We first explain that output is one in any agent-optimal equilibrium of any optimal project. To this end, let $c^{*}$ be an optimal project and $\left(w^{*}, F^{*}\right)$ an agent-optimal equilibrium in $c^{*}$. Proposition 1 applied to the optimal outcome $\left(c^{*}, w^{*}, F^{*}\right)$ implies that the corresponding binary outcome $\left(\widetilde{c}, w_{b^{*}}, B_{\mu_{F^{*}}}\right)$ is also optimal. Then, Part (i) of Remark 1 implies that we must have $\mu_{F^{*}}=1$, and hence $F^{*}=B_{1}$.

Next we argue that output realizations in $(0,1)$ are redundant in the sense that replacing these realizations by output zero has no impact on equilibrium behavior. To this end, we first show that it can be assumed that $w^{*}(x)=0$ for all $x<1$. The intuition is that if the principal wants to implement output one, she should not reward the agent for any other output realization by offering a positive payment. To state it formally, let us define the payments scheme, $\rho_{b}$, for each $b$ such that $\rho_{b}(1)=b$ and $\rho_{b}(x)=0$ for $x \neq 1$. Observe that replacing $w^{*}$ by $\rho_{w^{*}(1)}$ makes choosing $B_{1}$ no less attractive to the agent, so $\left(\rho_{w^{*}(1)}, B_{1}\right)$ is also an agent-optimal equilibrium in $c^{*}$.

Provided that the compensation scheme is $\rho_{w^{*}(1)}$, when the agent is contemplating choosing a distribution, all that matters is the probability that output is one. So, moving all the probability mass from $(0,1)$ to zero has no impact on the agent's choice of a distribution. Moreover, these new distributions generate smaller expected outputs, hence the principal still prefers to implement $B_{1}$. To make these claims precise, let us define a binary project $\widetilde{C}$ such that $\widetilde{C}(\mu)=\inf \left\{c^{*}(F): \Delta(F)=\mu\right\}$, where $\Delta(F)$ denotes the atom at one specified by $F .{ }^{22}$ Given the payment schedule $\rho_{w^{*}(1)}$, the change in cost function from $c^{*}$ to $\widetilde{C}$ does not affect the agent's

[^13]willingness to choose completion probability one. We conclude that the binary project $\widetilde{C}$ is optimal and $\left(w^{*}(1), 1\right)$ is an agent-optimal equilibrium in $\widetilde{C}$.

To give a further sense in which the results in the previous section are robust to the considerations of non-binary projects, we state the following.

Remark 3. If $c^{*}$ is an optimal project and $\left(w^{*}, F^{*}\right)$ is an agent-optimal equilibrium in $c^{*}$, then

(i) $F^{*}=B_{1}$,
(ii) $\Pi\left(w^{*}, F^{*}\right)=U\left(c^{*}, w^{*}, F^{*}\right)=1 / e$,
(iii) $u\left(c^{*}, \rho_{b}\right)=\int_{0}^{b} 1 /[e(1-z)] d z$ for all $b \in[0,1-1 / e]$, and
(iv) $\pi\left(c^{*}, \rho_{b}\right)=1 / e$ for all $b \in[0,1-1 / e]$.

We have already established Part (i) and that the binary outcome $\left(\widetilde{C}, w^{*}(1), 1\right)$ is agent-optimal. So, by Part (iii) of Remark 1, $w^{*}(1)=1-1 / e$, and the principal's payoff in project $c^{*}$ must be $1 / e$, establishing Part (ii). By the construction of $\widetilde{C}, u\left(c^{*}, \rho_{b}\right)$ is equal to $u(\widetilde{C}, b)$ for all $b$. Optimality of $\widetilde{C}$ and Part (ii) of Remark 1 then imply Part (iii) of Remark 3. Part (iii) of Remark 1 implies $\pi(\widetilde{C}, b)=1 / e$ for any $b \in[0,1-1 / e]$ and hence $\pi\left(c^{*}, \rho_{b}\right) \geq 1 / e$. Part (ii) of Remark 3 then implies $\pi\left(c^{*}, \rho_{b}\right)=1 / e$ for all $b \in[0,1-1 / e]$ establishing Part (iv).

An interpretation of Remark 3 is that the conclusions reached in Remark 1 remain valid even when projects are not restricted to be binary. For instance, consider any optimal project $c^{*}$ and an optimal binary project $C$. The players' values are the same across both projects for any payment schedule $\rho_{b}, b \in[0,1-1 / e]$, as follows from Parts (iii) and (iv) of Remark 3. This can be compared to Parts (ii) and (iii) of Remark 1.


Optimal Technology Design

[^0]:    *We are grateful to Alessandro Bonatti, Florian Ederer, Uli Hege, Harry Di Pei, Luis Rayo, Phil Reny, Dan Spulber, Jean Tirole, as well as to participants at several seminars and conferences for helpful comments. In the initial stages of this project, D. Garrett was based at Toulouse School of Economics, University of Toulouse Capitole. He is grateful for funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 714147). D. Garrett and A. Smolin are grateful for funding from the French National Research Agency (ANR) under the Investments for the Future (Investissements d'Avenir) program (grant ANR-17-EURE-0010).
    ${ }^{\dagger}$ D. Garrett: University of Essex, Wivenhoe Park, Colchester CO4 3SQ, U.K., mailto:d.garrett@essex.ac.uk; G. Georgiadis: Kellogg School of Management, Northwestern University, Evanston, IL 60208, U.S.A., mailto:ggeorgiadis@kellogg.northwestern.edu; A. Smolin: Toulouse School of Economics, University of Toulouse Capitole, 31080 Toulouse Cedex 06, France, mailto:alexey.v.smolin@gmail.com; B. Szentes: Department of Economics, London School of Economics, London, WC2A 2AE, U.K., mailto:b.szentes@lse.ac.uk.

[^1]:    ${ }^{1}$ A sizable literature on entrepreneurship examines the decisions of entrepreneurs when developing a business. See Gans et al. (2019) for a recent review.
    ${ }^{2}$ To be precise, we characterize this set as a function of the largest possible output realization.
    ${ }^{3}$ See also Garrett (2021) for a related exercise in a setting with moral hazard and adverse selection.

[^2]:    ${ }^{4}$ We extend our analysis to environments where the agent is risk averse in the Online Appendix.
    ${ }^{5}$ Gans et al. (2019) argue that many of the decisions an entrepreneur takes during business development cannot be revisited or renegotiated.
    ${ }^{6}$ Another consideration that may rule out secret deviations by the agent is monitoring by the principal. The principal may not permit the agent to adjust the project initially proposed, especially if she finds it costly to evaluate all the implications of the adjustments.
    ${ }^{7}$ While there is some multiplicity of optimal projects, we show in Appendix B that all optimal projects share the same essential attributes, so we write informally of "the" optimal project.

[^3]:    ${ }^{8}$ For example, Averch and Johnson (1962) observe that a regulated firm has incentives to inflate capital costs.
    ${ }^{9}$ See the Online Appendix.

[^4]:    ${ }^{10}$ Also related are Carroll (2015) who studies a principal facing ambiguity with respect to the agent's technology, Laux (2001) who considers agents with multiple projects, and Jewitt et al. (2008) who explore general bounds on payments beyond limited liability.

[^5]:    ${ }^{11}$ Such an indifference argument has appeared in the contexts of incentivizing monitoring (Ortner and Chassang (2018)), optimal testing (Perez-Richet and Skreta (2018)) and monopoly pricing in the presence of ambiguity (Bergemann and Schlag (2011)).
    ${ }^{12}$ Also related, Roesler and Szentes (2017) consider a setting where signals inform an otherwise uninformed buyer of his value, and asks which signal structure yields the highest information rent.

[^6]:    ${ }^{13}$ Non-negativity of payments encodes the limited-liability constraint.
    ${ }^{14}$ That the agent can choose any distribution on [0, 1] departs from much of the moral hazard literature, where the output distribution is parameterized by a one-dimensional variable called "effort". We view the agent's chosen distribution as synonymous with his action, an approach also taken for instance by Carroll (2015).

[^7]:    ${ }^{15}$ Recall that $w_{b^{*}}(x)=b^{*} x$ for all $x$.

[^8]:    ${ }^{16}$ We take the infimum in case no cheapest distribution exists.

[^9]:    ${ }^{17}$ Note that $\widehat{C}(0)=0$ follows because, if $\widehat{C}(0)>0$, we could reduce all costs by this amount, keeping the players' incentives unchanged, but increasing the agent's equilibrium payoff.

[^10]:    ${ }^{18}$ Given a possibly non-binary equilibrium distribution $F^{*}$, we showed that agent upward deviations to distributions with means above $\mu_{F^{*}}$ could be ignored in the analysis (recall the construction of the binary project $\widetilde{c}$ ).

[^11]:    ${ }^{19}$ Recall that $\left(F_{n}\right) \in \mathbf{F}^{c, w}$ if and only if $\lim _{n \rightarrow \infty} U\left(c, w, F_{n}\right)=u(c, w)$.

[^12]:    ${ }^{20}$ Formally, there is a sequence $\left(\mu_{n}\right)$ that approaches $1 / e$ from above, and for which $C\left(\mu_{n}\right) \rightarrow 0$. This follows from Part (ii) of Remark 2 and the continuity of $C^{*}$.

[^13]:    ${ }^{21}$ Here, " $\operatorname{cl}(\cdot)$ " refers to the closure of the set.
    ${ }^{22}$ That is, $\Delta(F)=F(1)-\lim _{x \nearrow 1} F(x)$.


Optimal Technology Design

## Citation and provenance

**Authors:** Daniel F. Garrett; George Georgiadis; Alex Smolin; Balázs Szentes

**Canonical citation:** Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.

**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/optimal-technology-design.md

**Source record:** https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf

**Published record:** https://doi.org/10.1016/j.jet.2023.105621

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

Provenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt


Disclosure and Pricing of Attributes

> Machine-readable author manuscript.
> Authors: Alex Smolin.
> Canonical citation: Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.
> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt


Disclosure and Pricing of Attributes

# Disclosure and Pricing of Attributes

**Authors:** Alex Smolin

**Manuscript date:** 2022-08-08

#### Abstract

A monopolist sells an object characterized by multiple attributes. A buyer can be one of many types, differing in their willingness to pay for each attribute. The seller can provide arbitrary attribute information in the form of a statistical experiment. To screen different types, the seller offers a menu of options that specify information prices, experiments, and object prices.

I characterize revenue-maximizing menus. All experiments belong to a class of linear disclosure rules. An optimal menu may be nondiscriminatory and qualitatively depends on the structure of buyer heterogeneity. The analysis highlights the importance of demand microstructure and the benefits of information control in trade settings.

Keywords: advertising, attributes, bilateral trade, demand transformation, information design, intermediaries, mechanism design, multidimensional disclosure, persuasion

JEL Codes: D11, D42, D82, D83, L15

[^0]


Disclosure and Pricing of Attributes

## 1 Introduction

In many important markets, sellers have considerable control over the information available to their buyers. Business brokers can control the extent of the firm investigation and documentation that they supply, recruiting platforms can decide what parts of a job candidate's profile to reveal to employers, and producers can decide what features of their consumer goods to advertise. In all of these markets, the products (i.e., a business, meeting with a job candidate, and consumer good) are characterized by multiple attributes that appeal to different types of buyers. To maximize revenue, sellers need to understand what attribute information to provide, how to price their products, and whether and how to price the information provided. These questions require unification of information and mechanism design paradigms to allow for joint control over information and monetary incentives.

As a concrete example, consider the operation of Ziprecruiter.com, a major online recruiting platform. The platform facilitates matching job seekers with employers: employers subscribe to the platform to advertise their open vacancies and obtain access to a large database of resumes. ${ }^{1}$ The recruitment market features substantial heterogeneity on both sides. Candidate profiles vary in many attributes, including work experience, education levels, and standardized test scores. The employers belong to distinct types, such as tech start-ups, chain stores, and government agencies. Naturally, different types of employers are looking for different attributes in their candidates.

Ziprecruiter.com has access to a large amount of data about prospective candidates and, by programming its algorithms, it can commit to coarsening the data or denying access to some attributes. Moreover, the platform can price both information, through upfront fees, and the decision to contact a job candidate, through contact fees. Indeed, the platform currently employs a nonlinear pricing scheme for subscriptions, which varies in the breadth of information provided and the ability to contact preferred candidates (Dubé and Misra (2019)). My goal is to study the trade-offs that the platform faces and to evaluate the allocation distortions introduced by its information control.

In this paper, I develop a framework to study information disclosure and the pricing of multiattribute products. I consider a monopolist seller who has an indivisible object for sale to a single buyer and aims to maximize her revenue. The object has several attributes, and the buyer is uncertain about their values. The buyer's valuation for the object is linear in attributes. The strengths of the preferences are the buyer's private information and constitute the buyer's type. The seller controls the pricing and attribute information available to the buyer.

[^1]Both the object and the information about its attributes are valuable for a buyer, and I allow the seller to price them separately. In particular, the seller offers a menu of options that differ in their informativeness. Each option consists of an information price, paid upfront, attribute information, and a price for the object. The attribute information is modeled as an arbitrary statistical experiment informative about attributes. Information control enables price discrimination. By varying the information price, the experiment, and the object price, the seller can screen buyer types. I illustrate the qualitative features of multiattribute disclosure and pricing in Section 3.

I emphasize that the model formulation implies the seller is not privately informed and cannot condition the price on the outcome of the experiment. In other words, the seller can provide the buyer access to attribute information about the product but cannot condition the price on the outcome of this information. This kind of information provision is referred to in the economic literature as "private disclosure" and has been widely used. ${ }^{2}$ One common motivation for the privacy of disclosure is that it may be hard for the seller to assess the impact of information on the buyer. As such, the analysis is directly applicable to the settings of intermediaries who organize sales of goods and services and in which the buyer has more expertise in assessing the product information than the intermediary. However, the analysis also provides insights into the more traditional buyer-seller settings, such as advertising, as long as the seller does not condition the price on the outcome of the information provided.

The general revenue-maximization problem features information design and multidimensional screening. As such, it entails two methodological challenges. First, the class of stochastic experiments is large since the underlying uncertainty covers a continuum of possible states, each having multiple dimensions. To understand the distortions driven by the information design, it is important to determine the structure of optimal experiments. Second, in the absence of a single-dimensional structure, it is not clear which incentive constraints are relevant for optimal design. This difficulty is further exacerbated by the distinct feature of information according to which different buyer types can respond differently to the same signal. I progress in both directions in turn.

In Section 4, I study the design of disclosure rules. ${ }^{3}$ Providing disclosure serves two functions. First, it swings the buyer's expectations and may persuade him to purchase the object at a higher price. Second, providing several disclosure options may facilitate screening because different buyer types prefer learning about different aspects of the object. Theorem 1 shows that an optimal way to combine these two functions is through a specific class

[^2]of experiments-linear disclosure rule. A linear disclosure rule informs whether a linear combination of attributes is above or below a specified threshold. This disclosure guides the allocation and can be seen as informing the buyer about the valuation of a virtual type that can differ from the demanding type to account for incentive constraints. ${ }^{4}$ Notably, this result requires no assumptions about the distributions of types or attributes and, as such, can be generalized to arbitrary valuation functions.


Disclosure and Pricing of Attributes

In addition, I establish an analog of the "no distortion at the top" property, which is common in mechanism design settings: If gains from trade are commonly known to be positive, then some type is provided with no information and always purchases the object.

In Section 5, I study optimal pricing mechanisms. In Theorem 2, I establish that if all buyer types value the same, always positive, attribute, then no information is optimally provided and the seller posts a single price for the object. The intuition behind this result lies in the product structure of the buyer's valuation. When all types value the same attribute, any disclosure realization simply scales their valuations and the corresponding demand curve. Even if the seller could condition the price on this realization, she would charge scaled prices, serve the same types, and obtain scaled revenue. By the martingale property of Bayesian expectations, the seller can obtain the same revenue by providing no information. ${ }^{5}$

The case of several attributes is qualitatively different because types can be differentiated not only vertically but also horizontally. Therefore, an optimal allocation may depend on attribute realizations: the seller should aim to allocate the object to types who value the realization the most. To guide the allocation, she should provide some attribute information.

I formalize this intuition in the setting in which each type values one of many independent attributes (Section 5.2) but the seller does not know which attribute or the strength of the preference. I show that an optimal menu features free-of-charge partial disclosure and no price discrimination. Information is not priced since the payment can be backloaded into the object price. Price dispersion is not profitable, because it implies that the seller could extract more surplus on some items by simultaneously lowering their object prices and changing their informational content. The optimal menu admits nondiscriminatory implementation-posting a single price for the object and informing the buyer whether the object is sufficiently good along each attribute.

In the limit case, in which there is only one type per attribute, in the optimal menu the seller effectively persuades each type to purchase the object at a fixed price separately by

[^3]informing him whether his valuation is sufficiently high. In Section 5.3, I show that similar mechanisms are optimal in a broad range of settings, as long as the types can be seen as belonging to distinct cohorts-with low valuation correlation across them.

I conclude the analysis with a discussion in Section 6. First, I discuss what product information must be priced and when. Second, I emphasize that attribute information can rotate the demand curve locally and can justify attribute shrouding. Finally, I indicate how a multiattribute framework complements the existing disclosure frameworks, and I emphasize the importance of explicitly modeling demand microstructure, i.e., how the consumer valuation is formed, in the settings with information control.

Related Literature This paper is about information provision and pricing. One strand of the related literature focuses on nondiscriminatory mechanisms-in which a seller commits to a single disclosure rule. Lewis and Sappington (1994) introduce these mechanisms in a setting where a buyer has no prior information. They show that within a simple parameterized class, an optimal disclosure rule is extreme: either full or no disclosure. Bergemann and Pesendorfer (2007) further observe that if there is common knowledge of positive trade gains, then no disclosure dominates any other possible disclosure rule because it allows the seller to extract the full expected surplus. ${ }^{6}$ Johnson and Myatt (2006) extend the analysis to settings in which the buyer has prior information. They focus on disclosure rules that correspond to the global rotation of a demand curve and show, once again, that extreme disclosure rules are optimal. My paper contributes to this literature by showing that if the product has several attributes, then a partial disclosure can dominate both full and no disclosure, even if there is common knowledge of positive trade gains (Sections 4.4, 6.3).

At the same time, when the buyer has private information, it is natural to study discriminatory mechanisms and how they can be used to screen buyer types. In an influential paper, Eső and Szentes (2007) study settings in which the attribute and the buyer's type enter the valuation "additively." In such settings, under certain distributional assumptions, the seller optimally provides full disclosure. However, Li and Shi (2017) show that the seller should withhold some information if the types represent private information about the object. ${ }^{7}$ Section 6.3 contains a detailed discussion related to these two papers.

All of the works described above operate in single-dimensional settings. Under complete object information, when comparing any two objects, all buyer types agree on the ranking.

[^4]However, in practice, many products are multidimensional, with different attributes that appeal to different buyers. In this paper, I demonstrate that these settings can be successfully studied and lead to qualitatively different results. Despite the richness of the attribute space, optimal experiments belong to a tractable class of linear disclosure rules (Section 4.4). Optimal mechanisms feature partial disclosure but can be remarkably simple (Section 5.2).

This paper builds on several existing frameworks. The multiattribute buyer's valuation follows the characteristic model of Lancaster (1966). An unrestricted search for an optimal disclosure rule is a defining feature of the Bayesian persuasion literature (Rayo and Segal (2010), Kamenica and Gentzkow (2011)). ${ }^{8}$ The screening analysis builds on the mechanism design machinery of Myerson $(1981,1982)$ and Kolotilin, Mylovanov, Zapechelnyuk, and Li (2017). Finally, information design with screening and monetary transfers has already appeared in my previous work (Bergemann, Bonatti, and Smolin (2018)). There, the seller can price only information-the buyer's action is not contractable. In contrast, in this paper, the seller can price both services and as a result, in many settings, provides the information free of charge.


Disclosure and Pricing of Attributes

## 2 Model

A buyer (he) decides whether to buy a single indivisible object from a seller (she). The object has a finite number $J$ of characteristics or attributes. The attribute values constitute an attribute vector $x=\left(x_{1}, \ldots, x_{J}\right) \in X=\mathbb{R}^{J}$. The buyer's preferences toward each attribute constitute the buyer's type $\theta=\left(\theta_{1}, \ldots, \theta_{J}\right) \in \Theta \subseteq \mathbb{R}^{J}$. The ex post buyer's valuation for the object is: ${ }^{9}$

$$
v(\theta, x)=\theta \cdot x=\sum_{j=1}^{J} \theta_{j} x_{j} .
$$

The buyer's utility is quasilinear in transfers. The seller maximizes her revenue.
Prior Information Attributes are distributed over $X$ according to a cumulative distribution function $G$. The buyer and seller are symmetrically informed about the attributes. The type space $\Theta$ can be finite or infinite. The buyer's type is his privately known preferences, which are uncorrelated with attributes. From the seller's perspective, the types are distributed according to a cumulative distribution function $F$. Until Section 5, I do not impose any structural assumptions on the attribute and type distributions. The only technical

[^5]requirement is that the ex ante expectations of all attributes are finite.
Information Disclosure The seller can disclose attribute information to the buyer. This information is modeled as a statistical experiment $E=(S, \pi)$ that consists of a signal set $S$ and a likelihood function: ${ }^{10}$
$$
\pi: X \rightarrow \Delta(S) .
$$

The experiment can be arbitrarily informative about the attributes. That is, the experiment can provide no information, or no disclosure, $\underline{E} \triangleq(\underline{S}, \underline{\pi})$, with $\underline{S}$ being a singleton; it can fully reveal attributes, or provide full disclosure, $\bar{E} \triangleq(\bar{S}, \bar{\pi})$, with $\bar{S}=X$ and $\bar{\pi}(x)$ placing a probability of 1 on $s=x$; or it can provide partial information.
Selling Mechanism For the environments in which the designer can control players' private information, it is not generally known what class of mechanisms one can look at without loss of generality. To make progress while avoiding trivialities, I follow Eső and Szentes (2007) and Li and Shi (2017) and focus on a class of menu mechanisms, so that the seller designs a menu of items, $i \in \mathcal{I}$ :

$$
M=(r(i), E(i), p(i))_{i \in \mathcal{I}} .
$$

Each item consists of an experiment $E(i)$ and two tariffs $r(i) \geq 0$ and $p(i) \geq 0$. The first tariff captures the price of information-the upfront payment made to observe the signal of experiment $E(i)$, irrespective of whether the buyer decides to purchase the object later. The second tariff captures the price of the object, paid only if the trade occurs. Effectively, the menu is a collection of call options that differ in monetary terms and information disclosure, designed to screen different buyer types. ${ }^{11}$ This class of mechanisms provides a natural and rich framework to study how information disclosure and pricing interact in design problems.

The timing is analogous to that of Courty and Li (2000) and is as follows. The seller posts a menu $M$. The attribute vector $x$ and the buyer's type $\theta$ are realized. If the buyer refuses to participate, then the players obtain zero payoffs. Otherwise, the buyer chooses an item $i \in \mathcal{I}$ and pays the corresponding price $r(i)$. Next, the buyer observes a signal $s$ from the experiment $E(i)$ and decides whether to buy the object at the price $p(i)$. Finally, the payoffs are realized. The timing is illustrated in Figure 1.

The timing implies that the seller commits to a menu before the realization of the attributes $x$ and the type $\theta$. The attributes $x$ and signals $s$ are not contractible, corresponding to the setting of "private disclosure" (Li and Shi (2017)). ${ }^{12}$ Sales are deterministic-a pay-

[^6]

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 1: Timeline of the selling mechanism.

ment guarantees a transaction. Sequential interactions between players are excluded, so belief-elicitation schemes and scoring rules are not available. ${ }^{13}$

My goal is to characterize a revenue-maximizing menu, i.e., a menu upon which the seller cannot strictly improve by offering another menu.


Disclosure and Pricing of Attributes

## 3 Illustrative Example

I begin by illustrating the workings of disclosure and pricing through a simple example. There are two attributes that are uniformly and independently distributed, $J=2, x_{1} \sim U[0,1]$, $x_{2} \sim U[0,2]$. There is a continuum of buyer types split into two cohorts. Types $\theta_{1} \in \Theta_{1}$ value only the first attribute, i.e., each $\theta_{1}$ is of the form $\left(\theta_{11}, 0\right)$, whereas types $\theta_{2} \in \Theta_{2}$ value only the second attribute, i.e., each $\theta_{2}$ is of the form $\left(0, \theta_{22}\right)$. Each cohort is equally likely, and the marginal type distributions are uniform over [0, 2]. The sets of attributes and types are illustrated in Figure 2.

Perhaps the simplest way to think about the impact of attribute disclosure is in terms of the demand curves that the seller faces and how these curves are affected by the release of information. To this end, consider a simple class of mechanisms in which the seller provides some information free of charge and follows it with posting a single object price. These mechanisms can be viewed as marketing strategies that combine pricing with informative, persuasive advertising.

If the seller provides no disclosure, then the expectation of the first and second attributes stay at their prior values of 1/2 and 1, respectively. As such, expected valuations of types in the first cohort are distributed uniformly over the interval [0, 1] and those in the second cohort over the interval [0, 2]. These valuations translate into a piecewise-linear demand curve with a kink at price $p=1$, as illustrated in Figure 3. Given this demand, the optimal no-disclosure price is $p_{n}=2 / 3$ and it generates revenue of 1/3.

[^7]

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 2: Illustrative example: Attributes are distributed uniformly over the gray rectangle (left); types are distributed uniformly over the L-shaped segment (right).

In contrast, if the seller provides full disclosure, then the valuation of each type stochastically changes according to the realization of a relevant attribute. As a result, a valuation of a first-cohort type with intensity $\theta_{11}$ is distributed uniformly over $\left[0, \theta_{11}\right]$, and a valuation of a second-cohort type with intensity $\theta_{22}$ is distributed uniformly over $\left[0,2 \theta_{22}\right]$. The demand curve aggregates these valuations across all types and is presented in Figure 3. Compared to the benchmark of no disclosure, the demand decreases at lower prices and increases at higher prices. The change is determined by the microstructure of the consumer types and the spread of their valuations. Roughly, one can think of this change as being driven by the types with intermediate ex ante valuations who, instead of remaining uninformed, learn that the attributes are too low, decreasing the demand at lower prices, or too high, increasing the demand at higher prices. The overall effect for the seller is negative: The optimal full-disclosure price is $p_{f} \simeq 0.82$ and it leads to revenue $0.28<1 / 3$.

To increase revenue, the seller should provide partial disclosure. Consider an experiment that reveals whether the first attribute is above some threshold $\alpha_{0}$ but provides no information about the second attribute. This disclosure affects only valuations of first-cohort types $\theta_{1}$ and can be viewed as transforming this cohort into two others. One new cohort captures valuations of types $\theta_{1}$ who observed that the first attribute is below the threshold; these valuations are uniformly distributed over an interval of $\left[0,2 \mathbb{E}\left[x_{1} \mid x_{1}<\alpha_{0}\right]\right]$. Another new cohort captures valuations of types $\theta_{1}$ who observed that the first attribute is above the threshold; these valuations are uniformly distributed over an interval of $\left[0,2 \mathbb{E}\left[x_{1} \mid x_{1} \geq \alpha_{0}\right]\right]$. As a result, the demand curve is piece-wise linear with two kinks. Relative to no disclosure,

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 3: Attribute disclosure and demand transformation. Left: demand curves under no disclosure and full disclosure. Right: demand curves under no disclosure and optimal disclosure. Vertical lines indicate revenue-maximizing prices.

demand decreases at low prices, increases at medium prices, and remains the same at high prices. If the disclosure threshold is chosen optimally, $\alpha_{0}^{*} \simeq 0.27$, then this demand transformation benefits the seller. The optimal price is $p_{\text {opt }} \simeq 0.80$, resulting in an approximate revenue of $0.35>1 / 3$ (Figure 3). Intuitively, this disclosure targets types with lower ex ante valuations, i.e., those in the first cohort, and persuades them to buy at a higher price.

The effectiveness of partial disclosure raises the question of whether the seller can increase revenue even further if she employs a discriminatory menu with upfront payments, offering a range of experiments at different prices. After all, the cohort structure presents a clear opportunity to screen types from different cohorts by offering information about different attributes. Perhaps surprisingly, the answer is negative, as I show in Section 5.2. The simple partial-disclosure mechanism presented here is optimal.

In the example that we have just studied, the attributes were independently distributed and each type cared about a single attribute. As a result, an optimal disclosure rule provided information only about one attribute. In a more general case, one can expect an optimal disclosure rule to be richer-providing information about several attributes simultaneously. In the next section, I show how such a disclosure rule should optimally be constructed.


Disclosure and Pricing of Attributes

## 4 Design of Disclosure Rules

I proceed with studying the optimal menu design in general settings. I begin by discussing the buyer's incentives and formalizing his choice in an arbitrary menu. I use this formalization to approach the design problem in two consecutive steps. First, I identify the class of optimal disclosure rules without explicitly characterizing a pricing scheme; this step is presented in the current section. Second, I derive optimal pricing details and complete the characterization of optimal menus in several leading settings; that step is presented in Section 5.


Disclosure and Pricing of Attributes

### 4.1 Buyer's Problem

Consider the buyer's incentives when he chooses an item from a given menu. Let his type be $\theta$. If he chooses option $i$, then he pays the upfront price $r(i)$. Then, a signal $s$ is realized according to the likelihood function $\pi(E(i))$, leading to the interim valuation:

$$
V(i, s, \theta) \triangleq \mathbb{E}[v(\theta, x) \mid E(i), s] .
$$

Finally, the buyer decides whether to buy the object and does so optimally if and only if $V(i, s, \theta)-p(i)$ is greater than 0. By integrating over signal realizations, I can define the resulting (total) trade probability as:

$$
Q(i, \theta) \triangleq \operatorname{Pr}(V(i, s, \theta)-p(i) \geq 0 \mid E(i)) .
$$

The corresponding indirect utility of choosing option $i$ can be written as:

$$
U(i, \theta)=-r(i)+\mathbb{E}[\max \{0, V(i, s, \theta)-p(i)\} \mid E(i)],
$$

Type $\theta$ chooses an option with the largest indirect utility. Naturally, types seek information that most fits their interests. This feature gives the seller the opportunity to discriminate among the types by carefully designing the menu.


Disclosure and Pricing of Attributes

### 4.2 Responsive Menus

The seller's problem lies at the intersection of the mechanism and information design because the seller can both control the information available to the buyer and charge monetary transfers. In principle, she can offer complex experiments in an attempt to better discriminate among types; however, I show that an optimal class of experiments is simple and tractable.

I begin approaching the seller's problem by binding the size of the optimal menus and signal sets. First, there is no need to have more items than there are types, so I can focus on direct menus:

$$
M=(r(\theta), E(\theta), p(\theta)),
$$

which effectively ask the buyer his type and assign the experiment and the tariffs as functions of his report. Second, I can bound the size of the signal sets. For a given direct mechanism $M$, I call an experiment $E(\theta)$ responsive if $S(\theta)=\left\{s^{+}, s^{-}\right\}$, and type $\theta$, when choosing this experiment, purchases the object if and only if $s=s^{+}$. I call the menu responsive if all of its experiments are responsive.

Proposition 1. (Responsive Menus)
The outcome of every menu can be replicated by a direct and responsive menu.
Proof. Detailed proofs of all formal statements can be found in the Appendix. $\square$

Proposition 1 imposes an elementary structure on the exchange of information between the seller and the buyer. The buyer should inform the seller about his preferences, and the seller should provide a recommendation about whether to buy the object. The proof is analogous to the argument of the revelation principle of Myerson (1982). If the menu contains nonresponsive experiments, then the seller can replace them with responsive experiments that replicate the behavior of truth-telling types. After this modification, truth telling delivers the same payoff as before. Dishonesty, however, becomes weakly less appealing (Blackwell (1953)).

In a responsive menu, every experiment $E$ is characterized by its trade function:

$$
q(x) \triangleq \operatorname{Pr}\left(s^{+} \mid E, x\right) .
$$

The function defines a probability of the trade recommendation for each attribute realization. The probability of the no-trade recommendation is then the complimentary $1-q(x)$. A responsive menu features a collection of experiments, one per each buyer's type, that corresponds to a collection of trade functions. With slight abuse of notation, I refer to the trade function of type $\theta$ as $q(\theta, x)$.


Disclosure and Pricing of Attributes

### 4.3 Seller's Problem

Proposition 1 enables the association of each experiment with its trade function (8). Vice versa, any trade function $q: X \rightarrow[0,1]$ determines a responsive experiment that recommends trading with probability $q(x)$. As such, the seller's problem can be written in a standard
mechanism design form as a maximization of the expected revenue over the tariff and trade functions:

$$
\max _{(r(\theta), q(\theta, x), p(\theta))} \int_{\theta \in \Theta}\left(r(\theta)+p(\theta) \int_{x \in X} q(\theta, x) \mathrm{d} G(x)\right) \mathrm{d} F(\theta),
$$

subject to the incentive-compatibility constraints and individual rationality constraints. The seller's revenue obtained from a particular type consists of the upfront payment $r(\theta)$ and, if the buyer decides to purchase the object, the object price $p(\theta)$. The incentive-compatibility constraints require that, for all $\theta, \theta^{\prime} \in \Theta$ :

$$
\int_{x \in X}(\theta \cdot x-p(\theta)) q(\theta, x) \mathrm{d} G(x)-r(\theta) \geq \int_{x \in X}\left(\theta \cdot x-p\left(\theta^{\prime}\right)\right) \sigma\left(q\left(\theta^{\prime}, x\right), k\right) \mathrm{d} G(x)-r\left(\theta^{\prime}\right),
$$

where $\sigma\left(q\left(\theta^{\prime}, x\right), k\right)$ is a deviation function equal to $q\left(\theta^{\prime}, x\right), 1-q\left(\theta^{\prime}, x\right)$, 1, and 0 for $k=1, \ldots, 4$. These constraints ensure that each type prefers truth telling over all doubledeviating strategies: misreporting and following the recommendations, "swapping" the buying decisions, always buying, or never buying. Deviations from $\theta$ to $\theta$ are included and ensure that the types are obedient on-path after truth telling.

The individual-rationality constraints require that, for all $\theta \in \Theta$ :

$$
\int_{x \in X}(\theta \cdot x-p(\theta)) q(\theta, x) \mathrm{d} G(x)-r(\theta) \geq 0,
$$

so that the seller cannot force the buyer to purchase an item from the menu.
Several challenges are involved in this problem. First, the seller maximizes over a large class of all functions from a multidimensional space $X$. Second, it is a priori not clear what kinds of deviations are binding and, hence, relevant for the design problem: the buyer's type has no single-dimensional structure, and there is an additional multiplicity of constraints caused by double deviations. ${ }^{14}$

The following observation is crucial to address the experimental complexity: only two coarse statistics, not the entire trade function, matter for the revenue-maximizing problem. Namely, for a given responsive experiment $E$, the associated trade function $q$ achieves the

[^8]attribute surplus and the (total) trade probability: ${ }^{15}$
$$
\begin{aligned}
& \mathcal{X}(q) \triangleq \int_{x \in X} x q(x) d G(x) \in \mathbb{R}^{J} \\
& \mathcal{Q}(q) \triangleq \int_{x \in X} q(x) d G(x) \in[0,1]
\end{aligned}
$$

The formulations (9), (10), and (11) reveal that, due to the linearity of integration, these statistics are the only economically relevant parameters of the problem. A change in the trade function $q(\theta, \cdot)$ that does not affect the attribute surplus and the trade probability affects neither the buyer's incentives nor the seller's revenue. Accordingly, the seller can maximize directly over attribute surpluses and trade probabilities. In what follows, I refer to $\mathcal{X}(q(\theta, \cdot))$ and $\mathcal{Q}(q(\theta, \cdot))$ as $\mathcal{X}(\theta)$ and $\mathcal{Q}(\theta)$, respectively.

Not all attribute surpluses and trade probabilities can be achieved by some trade function. At one extreme, if the trade probability is nil, then the trade never occurs, $q(\cdot) \equiv 0$, so the attribute surpluses must also be nil. At the other extreme, if the trade probability is 1, then the trade always occurs, $q(\cdot) \equiv 1$, so the attribute surplus is equal to its ex ante expectation $\mathbb{E}[x]$. Intermediate values of trade probability provide more freedom to choose attribute surpluses because the seller can select the regions in which the trade recommendations are sent. The corresponding feasibility set $\mathcal{F} \subseteq \mathbb{R}^{J+1}$, which is generated by all measurable trade functions, is:

$$
\mathcal{F} \triangleq\{(\mathcal{X}(q), \mathcal{Q}(q)) \mid q: X \rightarrow[0,1]\} .
$$

The set $\mathcal{F}$ is convex, and its shape is determined by the attribute distribution $G$.
These observations reduce the search to the following problem:

$$
\max _{\{r(\theta), \mathcal{X}(\theta), \mathcal{Q}(\theta), p(\theta)\}} \int_{\theta \in \Theta}(r(\theta)+\mathcal{Q}(\theta) p(\theta)) \mathrm{d} F(\theta)
$$

subject to incentive-compatibility constraints: $\forall \theta, \theta^{\prime} \in \Theta$,

$$
\begin{aligned}
& \theta \cdot \mathcal{X}(\theta)-\mathcal{Q}(\theta) p(\theta)-r(\theta) \geq \theta \cdot \mathcal{X}\left(\theta^{\prime}\right)-\mathcal{Q}\left(\theta^{\prime}\right) p\left(\theta^{\prime}\right)-r\left(\theta^{\prime}\right), \\
& \theta \cdot \mathcal{X}(\theta)-\mathcal{Q}(\theta) p(\theta)-r(\theta) \geq \theta \cdot\left(\mathbb{E}[x]-\mathcal{X}\left(\theta^{\prime}\right)\right)-\left(1-\mathcal{Q}\left(\theta^{\prime}\right)\right) p\left(\theta^{\prime}\right)-r\left(\theta^{\prime}\right), \\
& \theta \cdot \mathcal{X}(\theta)-\mathcal{Q}(\theta) p(\theta)-r(\theta) \geq \theta \cdot \mathbb{E}[x]-p\left(\theta^{\prime}\right)-r\left(\theta^{\prime}\right), \\
& \theta \cdot \mathcal{X}(\theta)-\mathcal{Q}(\theta) p(\theta)-r(\theta) \geq-r\left(\theta^{\prime}\right),
\end{aligned}
$$

[^9]the individual-rationality constraints: $\forall \theta \in \Theta$,
$$
\theta \cdot \mathcal{X}(\theta)-\mathcal{Q}(\theta) p(\theta)-r(\theta) \geq 0,
$$
and the feasibility constraints: $\forall \theta \in \Theta$,
$$
(\mathcal{X}(\theta), \mathcal{Q}(\theta)) \in \mathcal{F} .
$$

Even though the seller sells a single object, information disclosure allows her to control the multidimensional attribute surpluses $\mathcal{X}(\theta)$ at the time of a purchase. Moreover, the surpluses directly affect the feasible trade probability in a nonlinear fashion.


Disclosure and Pricing of Attributes

### 4.4 Optimal Disclosure

I begin by observing a special feature of a responsive experiment that always recommends the buyer to buy and, as such, provides no information about attributes. If all attributes are strictly positive, $X \subseteq \mathbb{R}_{++}^{J}$, then this experiment is a unique maximizer of the attribute surplus along all dimensions. If all types are strictly positive, $\Theta \subseteq \mathbb{R}_{++}^{J}$, then this experiment is also a unique maximizer of the trade surplus.

Proposition 2. (No Disclosure)
If all attributes and types are strictly positive, $X \subseteq \mathbb{R}_{++}^{J}$ and $\Theta \subseteq \mathbb{R}_{++}^{J}$, and the number of types is finite, then in any optimal menu, some type buys the object with probability one. That is, no disclosure, $\underline{E}$, is part of any optimal responsive menu.

Proposition 2 is consistent with the "no distortion at the top" property, common in mechanism design problems: there is a type that is optimally served an efficient allocation. However, recall that a responsive experiment only recommends allocation, and the buyer always has an option to disobey. Hence, it is important that no disclosure also provides minimal information to the buyer and, thus, maximally limits the scope of deviation. No disclosure arises in an optimal mechanism because it maximizes efficiency and minimizes incentive costs simultaneously.

Furthermore, Proposition 2 emphasizes the distinctive feature of the seller's problem, which combines information and mechanism design. In a typical information design problem, the payoff structure is exogenously fixed and, unless the receiver's indirect utility is concave everywhere, disclosure appears in the optimal mechanism for some prior distributions. Indeed, if the prices were fixed and the buyer's preferences had sufficiently low intensity, then the seller would have to provide some information to persuade the buyer to buy the object.

In contrast, when the seller has control over monetary incentives, she can compensate for the lack of information with lower prices and does find it optimal to do so.

To provide a further understanding of optimal experiments, it is useful to understand the general properties of the feasibility set $\mathcal{F}$. To this end, I define a key class of experiments.

Definition 1. (Linear Disclosure)
A responsive experiment $E$ is a linear disclosure if, for some coefficients $\alpha=\left(\alpha_{1}, \ldots, \alpha_{J}\right) \in$ $\mathbb{R}^{J}$ and $\alpha_{0} \in \mathbb{R}$ not all being equal to zero, its trade function is:

$$
q(x)= \begin{cases}1, & \text { if } \alpha \cdot x>\alpha_{0}, \\ 0, & \text { if } \alpha \cdot x<\alpha_{0} .\end{cases}
$$

A linear disclosure informs the buyer whether a linear combination of attributes is above or below a specified threshold. A linear disclosure assigns probability one to some signal everywhere outside of the defining hyperplane, $\left\{x \mid \alpha \cdot x=\alpha_{0}\right\}$, on which is can possibly randomize. In the case of a single attribute, a linear disclosure corresponds to a binary monotone partition of the attribute space.

A linear disclosure can be viewed as a "reference" disclosure that informs the buyer whether some virtual type $\hat{\theta}=\alpha$ would like to buy the object at price $p=\alpha_{0}$. If attributes are always positive and independently distributed, a linear disclosure admits additional interpretations. If elements of the coefficient vector $\alpha$ are positive, this disclosure can be viewed as a "level" disclosure. Observing a "trade" recommendation uniformly increases attribute expectation, whereas observing a "no-trade" recommendation uniformly decreases it. In contrast, if the elements of a coefficient vector $\alpha$ have different signs, then a linear disclosure can be viewed as a "comparative" disclosure between the attribute groups of different signs. A "trade" recommendation increases the attribute expectations in one group and decreases them in the other group.

Note that the likelihood function of a linear disclosure is not restricted on the defining hyperplane. Furthermore, the hyperplane does not exist for $\alpha \equiv 0$ and $\alpha_{0}$ being strictly positive or negative. Those disclosure rules correspond to never-trade and always-trade uninformative experiments.

Lemma 1. (Feasibility)
The feasibility set $\mathcal{F}$ is compact and convex. Any linear disclosure achieves some boundary point of $\mathcal{F}$. Any boundary point of $\mathcal{F}$ is achieved by some linear disclosure.

To prove this central result, I first show that $\mathcal{F}$ is compact as a continuous image of a compact set. Second, I show that $\mathcal{F}$ is convex because a convex combination of trade
functions achieves a convex combination of attribute surpluses and trade probabilities. Then, I appeal to the supporting hyperplane theorem to show that a given trade function achieves a boundary point if and only if it maximizes a linear combination of attribute surpluses and trade functions. Any such trade function corresponds to a linear disclosure.

In general multidimensional screening problems, one cannot be sure that all optimal bundles can be found at the boundary of a feasibility set. However, the current problem is an exception. To this end, say that an allocation $(\mathcal{X}(\theta), \mathcal{Q}(\theta))_{\theta \in \Theta}$ is implementable if there exist tariff functions $r(\theta), p(\theta)$ such that each buyer's type $\theta \in \Theta$ reports his type truthfully.

Lemma 2. (Implementability)
For any implementable allocation $(\mathcal{X}(\theta), \mathcal{Q}(\theta))_{\theta \in \Theta}$ there exists an allocation $\left(\mathcal{X}(\theta), \mathcal{Q}^{\prime}(\theta)\right)_{\theta \in \Theta}$ such that (1) it can be implemented with the same revenue and the same payoffs for all types and (2) for all $\theta \in \Theta,\left(\mathcal{X}(\theta), \mathcal{Q}^{\prime}(\theta)\right)$ is on the boundary of $\mathcal{F}$ and $\mathcal{Q}^{\prime}(\theta) \leq \mathcal{Q}(\theta)$.