Hidden Information and Self-Selection
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OPTIMAL CONTRACTING
Principal \(P\) / Agent \(A\) problem
\(x\) = output
\(t\) = payment
\(s\) = type
Agent maximises: \[ U(x,t,s) = t - C(x,s) \]
Principal maximises:
\[ V(x,t) = R(x) - t \]
Timing
- \(A\) observes \(s\)
- \(P\) believes \(s \sim D(\cdot)\)
- \(P\) offers a contract
- \(A\) accepts or rejects
- \(A\) chooses output, then \(P\) pays \(A\)
Full information benchmark
Let the reservation utility be zero. Then, if \(P\) observes \(s\), the optimal contract solves:
\[ \max_{x,t} \; R(x) - t \]
s.t. no rent for \(A\):
\[ t - C(x,s) = 0 \]
\[ \Rightarrow R'(x^*) = C_x(x^*, s), \quad t^* = C(x^*, s) \]
Delegation approach
\(P\) commits to a contract \(t = T(x)\)
If \(s \in \{L,H\}\), then \(T(x)\) is defined at two values.
Revelation approach
\(P\) offers a DRM:
\[ \{x(\hat{s}), t(\hat{s})\} \]
Agent solves:
\[ \max_{\hat{s}} \; t(\hat{s}) - C(x(\hat{s}), s) \]
Truthful DRM:
\[ \hat{s} = s \]
Revelation principle
Given any contract \(t = T(x)\) or DRM, there exists a truthful DRM yielding the same \((x,t)\).
P-A model
\(s \in \{L,H\}\) where s enters the cost of production, with \(L < H\).
\[ P(s=H)=q, \quad P(s=L)=1-q \]
Principal solves:
\[ \max_{x_H,x_L,t_H,t_L} \; q(R(x_H)-t_H) + (1-q)(R(x_L)-t_L) \]
s.t.
\[ t_H - C(x_H,H) = 0 \qquad\text{(P for $H$)} \]
\[ t_L - C(x_L,L) \ge 0 \qquad\text{(P for $L$ - always satisfied)} \]
\[ t_H - C(x_H,H) \ge t_L - C(x_L,H) \qquad\text{(IC for $H$ - never bites)} \]
\[ t_L - C(x_L,L) = t_H - C(x_H,L) \qquad\text{(IC for $L$)} \]
Full information benchmark
\[ R'(x_s^*) = C_x(x_s^*, s), \quad t_s^* = C(x_s^*, s) \]
SIGNALLING
One informed agent, multiple uninformed principals. The agent always moves first. Differences with respect to optimal contracting (OC):
- S: the type enters the P’s utility function - OC: one principal, one agent; S: multiple principals, one agent - OC: principal moves first; S: agent moves first - OC: unique equilibrium; S: multiple equilibria - OC: good type envies bad; S: bad type envies good
Agent:
\[ U(w,e,n) = w - C(e,n) \]
Principal:
\[ \pi = y(n,e) - w \]
where
\(w\) = wage
\(n\) = ability (type)
\(e\) = education
\(y\) = output
Timing
- \(A\) observes \(n\)
- \(P\) has prior \(P(n=H)=q\)
- \(A\) chooses \(e\)
- \(P\) offers \(w\)
- \(A\) accepts or rejects
Strategies
\[ A: e(n), \quad P: w(e) \]
Beliefs:
\[ \mu(H|e) \]
Full information benchmark
As as consequence of perfect competition, \(\pi=0\) and \(w(e) = \mu(H|e) y(H,e) + (1-\mu(H|e)) y(L,e)\). Follows:
\[ w(e) = y(n,e) \]
The agent solves:
\[ \max_e \; y(n,e) - C(e,n) \]
Define envy as the situation in which the agent of type \(L\) prefers the full-information contract designed for type \(H\): \[ y(H,e^*(H)) - C(e^*(H),L) > y(L,e^*(L)) - C(e^*(L),L) \]
Pooling
By definition:
\[ e(H) = e(L) = e_p \]
\[ \mu(H|e_p) = q \]
The it must be:
\[ w(e_p) = q y(H,e_p) + (1-q) y(L,e_p) \]
usually pooling equilibria are enforced with strong off-patch beliefs.
Intuitive criterion
Set beliefs:
\[ \mu(H|e') = 0 \]
for deviations that are never profitable for \(H\)
Separating equilibria:
No envy: \(\mu(H|e) = 1\) for \(e \ge e^*(H)\), \(\mu(H|e) = 0\) for \(e < e^*(H)\)
\[ e(L) = e^*(L), \quad e(H) = e^*(H) \]
Envy: \(\mu(H|e) = 1\) for \(e \ge e^s\), \(\mu(H|e) = 0\) for \(e < e^s\): H overinvest in education to separate from L
\[ e(L) = e^*(L), \quad e(H) = e^s > e^*(H) \]
where \(e^s\) is such that L is indifferent between its full-information contract and the one designed for H:
\[ y(H,e^s) - C(e^s,L) = y(L,e^*(L)) - C(e^*(L),L) \]
SCREENING
Multiple competing uninformed principals move first, then one informed agent moves.
Timing
- \(A\) observes \(n \in \{L,H\}\)
- \(P\) has prior \(P(n=H)=q\)
- \(P\) offers contract \((w,e)\)
- \(A\) accepts or rejects
- \(A\) acquires e and receives w
Strategies
\[ P_i: (w_i,e_i), \quad A: (w,e) \in \{(w_i,e_i)\} \]
Results
- Any contract yields zero profits due to competition
- No pooling is possible: let \(P_i\) offer a pooling contract \((w_p,e_p)\), then any \(P_j\) can offer a slightly better contract \((w_p+\epsilon,e_p)\) to attract only the high type and make positive profits (cream skimming).
Uniqueness
L always chooses \((w(L)^*, e(L)^*)\) since every contract of the type \((w(L)^*-\varepsilon, e(L)^*)\) could generate a profit for some \(P_i\).
No envy: H chooses \((w(H)^*, e(H)^*)\) for a similar reason.
Envy: H chooses \((y(H,e^s), e^s)\) since any contract of the type \((y(H,e^s)+\varepsilon, e^s)\) could generate a profit for some \(P_i\) while only attracting the high type.
Existence
No envy: PBE always exists
Envy: $ = $ such that
\[ q \le \hat{q} \Rightarrow \text{unique PBE} \]
\[ q > \hat{q} \Rightarrow \text{no PBE} \]
Social planner
The social planner solves:
\[ \max_{x_H,x_L} \; q\big(R(x_H)-C(x_H,H)\big) + (1-q)\big(R(x_L)-C(x_L,L)\big) \]
s.t. participation and IC
\[ R'(x_s^*) = C_x(x_s^*, s) \]