Screening Contracts for Hidden Types

Why Offer a Menu?

A firm that cannot observe customers’ preferences may be unable to charge each person a price tailored to their type. It can instead offer a menu of contracts and let customers select among them. If the contracts differ in quality and price, customers with different preferences may choose different options. The firm is then using its offer to screen for information it does not possess.

This is a problem of hidden information: the customer knows their type when choosing, while the firm does not. It is distinct from a hidden-action problem, in which an action taken after a contract is signed cannot be observed. The simple model below isolates the screening choice and is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: A screening menu is a set of contracts offered by an uninformed firm to customers who privately know their types. A customer’s type determines their benefit from each contract. Self-selection occurs when customers choose from the menu in a way that reveals their types to the firm.

A Simple Screening Model

Suppose there are two customer types, low and high, with benefit parameters $\theta_L$ and $\theta_H$, where $0<\theta_L<\theta_H$. A contract specifies quality $q\geq 0$ and price $P$. Type $\theta$ receives utility $\theta q-P$ from that contract and utility zero from not buying. This is a quasi-linear utility model: quality contributes $\theta q$ to utility, while price reduces utility dollar for dollar.

A proposed separating menu has a low-type contract $(q_L,P_L)$ and a high-type contract $(q_H,P_H)$, with $q_H>q_L$. The model treats quality as observable and contract performance as certain; it does not model customers changing their actions after signing. Assume the firm can identify the contracts customers select, but cannot observe their types before they choose.

Definition: Incentive compatibility (IC) means each type prefers the contract intended for it to the other type’s contract. Individual rationality (IR), or the participation constraint, means a type’s chosen contract gives utility at least as high as the outside option, which here is zero.

For a separating menu, both types must have a reason to participate as well as a reason not to imitate the other type. A contract can satisfy one requirement and fail the other: for example, a type may prefer its intended contract to the other menu option but still prefer not to buy.

Incentive Compatibility and Participation

Write $\Delta q=q_H-q_L>0$ and $\Delta P=P_H-P_L$. The low type prefers its intended contract when $\theta_Lq_L-P_L\geq\theta_Lq_H-P_H$, which is equivalent to $\Delta P\geq\theta_L\Delta q$. The high type prefers its intended contract when $\theta_Hq_H-P_H\geq\theta_Hq_L-P_L$, or $\Delta P\leq\theta_H\Delta q$. Together, the incentive-compatible price difference must lie in an interval.

\theta_L\Delta q \leq \Delta P \leq \theta_H\Delta q

The lower bound is what makes the high-quality contract unattractive enough to the low type. The upper bound is what keeps the high type from preferring the low-quality contract. Because $\theta_L<\theta_H$, the interval is nonempty. These bounds establish IC only; participation must still be checked separately:

\theta_Lq_L-P_L\geq 0 \qquad\text{and}\qquad \theta_Hq_H-P_H\geq 0

Worked Example: Checking a Two-Contract Menu

A firm offers two service plans. Customers have $\theta_L=2$ or $\theta_H=5$, and utility is $U_\theta(q,P)=\theta q-P$. The low-type plan is $(q_L,P_L)=(1,1.50)$, and the high-type plan is $(q_H,P_H)=(3,10.50)$. Determine whether the menu separates the types and whether each type participates.

Method. Check the IC price-difference bounds, then calculate each type’s utility from both plans and from the outside option. This avoids assuming that a type chooses a plan merely because it is labelled for that type.

First, $\Delta q=3-1=2$ and $\Delta P=10.50-1.50=9$. The IC interval is $\theta_L\Delta q=2(2)=4$ through $\theta_H\Delta q=5(2)=10$. Since $4\leq 9\leq 10$, both IC conditions hold.

The low type receives $2(1)-1.50=0.50$ from the low plan and $2(3)-10.50=-4.50$ from the high plan. The outside option gives zero, so the low type strictly prefers the low plan to either alternative.

The high type receives $5(1)-1.50=3.50$ from the low plan and $5(3)-10.50=4.50$ from the high plan. The high type strictly prefers the high plan and receives positive utility, so participation is individually rational.

Check. Each type has a strictly preferred intended plan, and each intended plan gives positive utility. The menu therefore separates the types without relying on a tie-breaking assumption.

Why the Price Bounds Hold

The IC result follows directly from comparing the two contracts. If the low type is to choose the low contract, its utility there must be at least its utility from the high contract:

\theta_Lq_L-P_L\geq\theta_Lq_H-P_H \quad\Longleftrightarrow\quad P_H-P_L\geq\theta_L(q_H-q_L)

The same comparison for the high type reverses the restriction on the price difference:

\theta_Hq_H-P_H\geq\theta_Hq_L-P_L \quad\Longleftrightarrow\quad P_H-P_L\leq\theta_H(q_H-q_L)

Thus a larger quality increment requires a larger price increment to deter the low type, but the price increment cannot exceed what the high type is willing to pay for that additional quality. These comparisons assume the two types value the same quality increment according to their respective $\theta$ parameters and that price enters utility linearly. If $q_H=q_L$, the menu does not use quality to distinguish types; if $q_H<q_L$, the inequalities must be interpreted with care and the displayed interval for $\Delta q>0$ does not apply as written.

Common exam trap: It is wrong to say that the price difference must be at most $\theta_L\Delta q$. For the low type not to imitate the high type, the correct condition is $\Delta P\geq\theta_L\Delta q$. The high type’s no-imitation condition supplies the upper bound, $\Delta P\leq\theta_H\Delta q$.

Participation, Indifference, and Tie-Breaking

A weak inequality in an IC or IR condition means that a customer is indifferent; it does not guarantee a particular choice. If the low type’s intended contract gives utility zero, that type is indifferent between buying and taking the outside option. If the high type is exactly on its IC bound, it is indifferent between the two contracts. A claim that the menu separates the types must therefore state how these ties are resolved.

For instance, if $P_L=\theta_Lq_L$, the low type gets zero from the low contract. It may choose not to buy. In that case, the menu does not achieve separation in actual purchases, even if the low type weakly prefers its contract to the other menu option. Separation at such a boundary requires a tie-breaking rule that assigns the low type to the low contract. Similarly, if the high type is indifferent between contracts at its IC bound, assign it to the high contract to obtain the intended sorting.

A firm that needs strict choices can make small adjustments to prices so the intended contracts are strictly preferred and participation is strict. Such adjustments generally reduce the firm’s profit relative to a solution that extracts all available surplus, so boundary solutions are often described with their tie-breaking assumptions made explicit.

A Screening Menu and the Firm’s Profit

The IC constraints also explain why a high-type customer may receive positive utility in a profit-maximizing separating menu. Suppose the firm sets the low price at the maximum compatible with low-type participation, $P_L=\theta_Lq_L$, and sets the high price at the high type’s IC bound, $P_H=P_L+\theta_H(q_H-q_L)$. The low type then gets zero utility, while the high type gets:

\theta_Hq_H-P_H =\theta_Hq_H-\bigl(\theta_Lq_L+\theta_H(q_H-q_L)\bigr) =(\theta_H-\theta_L)q_L

This positive amount is the high type’s information rent: the utility it receives because the firm must leave the high contract attractive enough that it does not choose the low contract. The calculation uses binding constraints; it is not a claim that every menu gives the high type exactly this utility.

Worked Example: Finding a Profit-Maximizing Separating Menu

A firm serves a low type with probability $0.80$ and a high type with probability $0.20$. Their benefit parameters are $\theta_L=2$ and $\theta_H=5$. Quality is $q\geq 0$, the firm’s cost is $C(q)=q^2/2$, and profit on a contract is its price less its cost. Find the separating menu that maximizes expected profit under the binding-constraint method, and check its choices.

Method. Use the low-type participation constraint and the high-type IC constraint as equalities, then maximize expected profit over qualities subject to $q_H\geq q_L\geq0$. This avoids pricing the two types as if the firm observed their types directly.

Set $P_L=2q_L$. The high-type IC constraint binds at $P_H=P_L+5(q_H-q_L)=5q_H-3q_L$. The resulting high-type utility is $5q_H-P_H=3q_L$, its information rent. Expected profit is therefore

\begin{aligned} \Pi &=0.80\left(P_L-\frac{q_L^2}{2}\right) +0.20\left(P_H-\frac{q_H^2}{2}\right)\\ &=0.80\left(2q_L-\frac{q_L^2}{2}\right) +0.20\left(5q_H-3q_L-\frac{q_H^2}{2}\right)\\ &=q_L-0.40q_L^2+q_H-0.10q_H^2 \end{aligned}

Differentiate with respect to each quality: $\partial\Pi/\partial q_L=1-0.80q_L$, giving $q_L=1.25$; and $\partial\Pi/\partial q_H=1-0.20q_H$, giving $q_H=5$. Both second derivatives are negative, $-0.80$ and $-0.20$, and the solution satisfies $5\geq1.25\geq0$. Thus it is the maximum within the stated separating allocation and binding-constraint method.

The prices are $P_L=2(1.25)=2.50$ and $P_H=5(5)-3(1.25)=25-3.75=21.25$. The low type gets $2(1.25)-2.50=0$ from its plan and $2(5)-21.25=-11.25$ from the high plan. The high type gets $5(1.25)-2.50=3.75$ from the low plan and $5(5)-21.25=3.75$ from the high plan.

Expected profit is $0.80(2.50-1.25^2/2)+0.20(21.25-5^2/2)$. The low-type margin is $2.50-0.78125=1.71875$, contributing $1.375$; the high-type margin is $21.25-12.50=8.75$, contributing $1.75$. Total expected profit is $1.375+1.75=\$3.125$ per potential customer.

Check and tie-break. The high type is indifferent between the two contracts, and the low type is indifferent between its contract and not buying. The separating outcome and the stated expected profit therefore require tie-breaking that assigns the high type to the high contract and the low type to the low contract. Without that assumption, the low type may opt out, so the calculated profit is not assured. A small discount on the low contract, with a compatible adjustment to the high price, can make low-type participation strict but changes the profit.

Pooling as a Useful Contrast

A pooling offer gives every type the same contract rather than using a menu to sort them. It can be a useful comparison, but acceptance must still be checked type by type. In particular, a price that gives one type exactly zero utility does not guarantee that type buys. The firm’s expected profit depends on who actually accepts, not merely on who weakly prefers buying to the outside option.

Worked Example: A Pooling Price and an Indifferent Type

A firm offers one service level, $q=1$, at a single price. There are low and high types with $\theta_L=2$ and $\theta_H=5$, occurring with probabilities $0.60$ and $0.40$. The firm’s cost per customer served is $C(1)=\$1.25$. Calculate expected profit at prices $P=\$2$ and $P=\$4$, accounting for the outside option.

Method. Calculate each type’s utility first, identify actual purchasers, and then weight the margin by the probability of purchase. This avoids treating indifference as guaranteed acceptance.

At $P=\$2$, the low type’s utility is $2(1)-2=0$, while the high type’s utility is $5(1)-2=3$. The high type strictly buys. If the low type buys when indifferent, both types buy and the margin per customer is $2-1.25=\$0.75$, so expected profit per potential customer is $\$0.75$. If the low type opts out, only the high type buys, and expected profit is $0.40(0.75)=\$0.30$.

At $P=\$4$, the low type’s utility is $2-4=-2$, so it does not buy; the high type’s utility is $5-4=1$, so it strictly buys. The margin per high-type customer is $4-1.25=\$2.75$, giving expected profit $0.40(2.75)=\$1.10$.

Check. The price of $\$2$ does not ensure that all customers accept: its profit calculation depends on the low type’s tie-breaking behavior. At $\$4$, only the high type purchases, so the expected profit correctly weights the margin by $0.40$.

When Screening Analysis Is Useful

Screening analysis separates three questions that are easy to conflate: what each type values, which contract each type prefers, and whether each type chooses to participate. In hypothetical insurance settings, an insurer might offer different coverage and premium combinations to people whose risk characteristics are privately known. The details of such an application require an appropriate model of risk and utility, but the general screening logic remains: a menu is useful only if its options satisfy both the relevant incentive and participation conditions.

When analyzing a proposed menu, proceed in a fixed order: identify the types and outside option; calculate each type’s utility from every contract; check IC; check IR; then calculate profit using the types who actually purchase. If any condition holds only as an equality, state the tie-breaking assumption needed for the claimed outcome.

Key takeaway: With $q_H>q_L$, a separating menu must set the price difference between $\theta_L(q_H-q_L)$ and $\theta_H(q_H-q_L)$. Those IC bounds do not guarantee participation or resolve indifference. Verify both separately before claiming that a menu screens customers.

Check Your Understanding

For each problem, check incentive compatibility and participation separately, and state any tie-breaking assumption that affects the result.

Types have $\theta_L=1$ and $\theta_H=4$. A menu has $q_L=2$ and $q_H=5$. What interval of price differences $\Delta P$ satisfies both IC conditions? (Answer: $3\leq\Delta P\leq12$.)
Let $\theta_L=3$, $\theta_H=7$, and offer $(q_L,P_L)=(1,2)$ and $(q_H,P_H)=(4,20)$. Do the types strictly prefer their intended contracts, and do both participate? (Answer: Yes; the low type gets $1$ from its plan and $-8$ from the high plan, while the high type gets $5$ from the low plan and $8$ from the high plan.)
Let $\theta_L=2$, $\theta_H=6$, $q_L=1$, $q_H=3$, and $P_L=2$. What interval of $P_H$ satisfies IC, and does that interval by itself guarantee participation? (Answer: $6\leq P_H\leq14$; no, participation must be checked separately.)
A single contract has $q=1$, price $P=2$, and cost $C(1)=\$1$. Low and high types have $\theta_L=2$ and $\theta_H=5$, with probabilities $0.70$ and $0.30$. What is expected profit if the indifferent low type buys, and if it opts out? (Answer: $\$1.00$ if the low type buys; $\$0.30$ if it opts out.)
A firm has $\theta_L=2$, $\theta_H=6$, probabilities $0.80$ and $0.20$, utility $\theta q-P$, and cost $C(q)=q^2/2$. Under binding low-type IR and high-type IC, the expected-profit objective is $0.8(2q_L-q_L^2/2)+0.2(6q_H-4q_L-q_H^2/2)$. Find the maximizing qualities and prices. (Answer: $q_L=1$, $q_H=6$, $P_L=2$, and $P_H=32$; separation requires tie-breaking that assigns the low type to its zero-utility plan and the high type to the high plan.)

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