Screening menu derivation (2nd-degree PD)

When consumer types are unobservable, the optimal menu of two-part tariffs is derived by binding the low type's participation constraint and the high type's mimicking constraint, then optimising each contract independently.

1. Identify the two types. Label the one with the higher surplus at any given price the "high" type; the other is the "low" type.
2. Bind the low type's individual-rationality (IR) constraint: set $T_2 = CS_2(p_2)$ so the low type is exactly indifferent between buying and walking away.
3. Bind the high type's IC constraint: the high type must (weakly) prefer their own contract to the low-type bundle. This gives $T_1 = CS_1(p_1) - \big[CS_1(p_2) - CS_2(p_2)\big]$. The bracket is the Information Rent left to the high type.
4. Write firm profit as the sum of fees plus per-unit margin: $\pi = T_1 + T_2 + (p_1 - MC) q_1(p_1) + (p_2 - MC) q_2(p_2)$. Substitute the bound constraints and the problem separates into two independent maximisations over $p_1$ and $p_2$.
5. Take FOCs. The high-type's optimum gives $p_1^* = MC$ (no distortion at the top). The low-type's optimum gives $p_2^* > MC$ (downward distortion).
6. Compute the final fees, quantities, profit. Verify IC for both directions and IR for both types.

type: second-degree-pd

The geometric a-vs-b trade-off: shrinking the poor type's bundle by $\Delta q$ loses revenue $b$ (red, under the low-type demand) but gains $a$ (green, between the two demands) by relaxing the high-type IC. The monopolist keeps cutting while $a > b$ and stops at $a = b$ — the low-type quantity is distorted downward, the high-type quantity is not ("no distortion at the top").

Common pitfalls

• Treating both contracts as independent and setting both $p_i = MC$. This violates IC: the high type would prefer the cheap low-type contract and pocket the surplus difference.
• Mis-identifying the binding constraints. Always: IR binds for the low type, IC binds for the high type. The other two constraints (high's IR, low's IC) are slack at the optimum.
• Forgetting to leave the information rent. The high type's profit margin $T_1 < CS_1(p_1)$ by exactly $CS_1(p_2) - CS_2(p_2)$ — try to extract that and the high type defects.
• "Distortion at the bottom, no distortion at the top" — memorise the direction. The low type is the one whose quantity is reduced, not the high type.

Worked example

Two consumers $q_1 = 40 - 2P$, $q_2 = 20 - P$, $MC = 6$ (see EX-5 - Micro 3 Q3e). Useful identity: $CS_1(p) = 2 \cdot CS_2(p) = (20 - p)^2$.

Step 5 — pick $p_1$ to maximise $CS_1(p_1) + (p_1 - 6)(40 - 2p_1)$: $\to p_1^* = 6 = MC$. Pick $p_2$ to maximise $(p_2 - 6)(20 - p_2)$: $\to p_2^* = 13$. Quantities $q_1^* = 28$, $q_2^* = 7$.

Step 6 — $T_2 = CS_2(13) = \tfrac{1}{2}(7)^2 = \$24.50$. $T_1 = CS_1(6) - CS_2(13) = 196 - 24.5 = \$171.50$ — the high type keeps $24.50 of information rent. Profit $= 171.5 + 24.5 + 0 + 7 \times 7 = \$245$. Ranking across regimes: uniform ($220.50) < screening ($245) < perfect PD ($294).