Screening Menus

A seller lays out a menu, and buyers pick for themselves.

The seller cannot tell big spenders from small ones. So the menu offers two sizes. The big option is made in full. The small option is shrunk on purpose. Shrinking it stops big spenders from lying. In return, big spenders pocket a bonus called information rent. The formulas below set the two sizes.

For example, a 🏪 shop sells 🍬 candy in a big 📦 box and a small box. Big spenders choose the big box. The small box is made tiny on purpose. Pretending to be a small buyer would only hurt. So big spenders honestly pick the big box.

qL=θL−λ1−λ(θH−θL),qH=θHq_L = \theta_L - \frac{\lambda}{1-\lambda}(\theta_H - \theta_L), \qquad q_H = \theta_H
UH=(θH−θL) qLU_H = (\theta_H - \theta_L)\, q_L

Part of these learning paths

Examples & Exercises

In the Mussa-Rosen model with θH=2\theta_H = 2, θL=1\theta_L = 1, P(θH)=λ=0.3P(\theta_H) = \lambda = 0.3, buyer utility θq−t\theta q - t and seller cost q2/2q^2/2, find the optimal menu (qL,tL)(q_L, t_L), (qH,tH)(q_H, t_H);

verify the high type's information rent and that IRLIR_L and ICHIC_H bind.

qL=θL−λ1−λ(θH−θL)=1−0.30.7=47q_L = \theta_L - \frac{\lambda}{1-\lambda}(\theta_H - \theta_L) = 1 - \frac{0.3}{0.7} = \frac{4}{7}
qH=θH=2q_H = \theta_H = 2
λ=0.3\lambda = 0.3 — the bottom is distorted to qL=4/7q_L = 4/7; no distortion at the top.
Show proof

  1. The downward-distortion formula gives qL=1−(0.3/0.7)⋅1=4/7≈0.5714q_L = 1 - (0.3/0.7)\cdot 1 = 4/7 \approx 0.5714, and no distortion at the top gives qH=2q_H = 2.

  2. IRLIR_L binds, so tL=θLqL=4/7≈0.5714t_L = \theta_L q_L = 4/7 \approx 0.5714 and the low type's utility is UL=4/7−4/7=0U_L = 4/7 - 4/7 = 0. ICHIC_H binds, so tH=θHqH−(θH−θL)qL=4−4/7=24/7≈3.4286t_H = \theta_H q_H - (\theta_H - \theta_L)q_L = 4 - 4/7 = 24/7 \approx 3.4286.

  3. The high type's utility is UH=2⋅2−24/7=4/7≈0.5714U_H = 2\cdot 2 - 24/7 = 4/7 \approx 0.5714.

    Picking the low contract would also give 2⋅(4/7)−4/7=4/72\cdot(4/7) - 4/7 = 4/7, so the slack of ICHIC_H is 00.

    The information rent is (θH−θL)qL=4/7(\theta_H-\theta_L)q_L = 4/7 — what the high type extracts merely for holding private information.

    UH=47,tL=47,tH=247,rent=47U_H = \frac{4}{7}, \qquad t_L = \frac{4}{7}, \qquad t_H = \frac{24}{7}, \qquad \text{rent} = \frac{4}{7}

  4. The welfare loss comes only from the low type's distortion: (1−λ) [S(1,1)−S(1,4/7)]=0.7 (0.5−20/49)≈0.0643(1-\lambda)\,[S(1,1) - S(1,4/7)] = 0.7\,(0.5 - 20/49) \approx 0.0643.

In the same model with λ=0.05\lambda = 0.05, find the optimal menu

and verify that as the high type gets rarer the distortion vanishes but the information rent grows.

qL=1−0.050.95=1819≈0.9474q_L = 1 - \frac{0.05}{0.95} = \frac{18}{19} \approx 0.9474
λ=0.05\lambda = 0.05 — qL=18/19q_L = 18/19, almost at the first-best.
Show proof

  1. qL=1−(0.05/0.95)⋅1=18/19≈0.9474q_L = 1 - (0.05/0.95)\cdot 1 = 18/19 \approx 0.9474, nearly the first-best qLFB=1q^{FB}_L = 1. When the high type is rare there is little reason to distort the low type's contract.

  2. The information rent is (θH−θL)qL=18/19≈0.9474(\theta_H-\theta_L)q_L = 18/19 \approx 0.9474 — larger, because mimicking is easier when the low contract's quality rises while qH=2q_H = 2 stays put. The welfare loss is 0.95 (0.5−180/361)≈0.00130.95\,(0.5 - 180/361) \approx 0.0013, nearly gone.

19. Adverse Selection and Signaling