Download Module 4: Moral Hazard and more Summaries Economics in PDF only on Docsity!
Module 4: Moral Hazard - Linear Contracts
Information Economics (Ec 515) · George Georgiadis
A principal employs an agent.
Timing:
- The principal o↵ers a linear contract of the form w (q) = ↵ + q.
- ↵ is the salary, is the bonus rate.
- The agent chooses whether the accept or reject the contract.
- If the agent accepts it, then goto t = 3.
- If the agent rejects it, then he receives his outside option U , the principal receives profit 0, and the game ends.
- The agent chooses action / e↵ort a 2 A ⌘ [0, 1 ].
- Output q = a + " is realized, where " ⇠ N (0, 2 )
- The principal pays the agent, and the parties’ payo↵s are realized.
The principal is risk neutral. His profit function is
E [q w (q)]
The agent is risk averse. His utility function is
U (w, a) = E
e r(w(q) c(a))^
with c (a) = ca^
2 2
Rationality assumptions:
- Upon observing the contract w (·), the agent chooses his action to maximize his expected utility.
- The principal, anticipating (1), chooses the contract w (·) to maximize his expected profit.
First Best
Benchmark: Suppose the principal could choose the action a.
- We call this benchmark the first best or the e cient outcome.
- Equivalent to say that the agent’s action is verifiable or contractible.
Principal solves:
a , wmax(q) E^ [a^ +^ ✏^ ^ w^ (q)] s.t. E
e r(w(q) c(a))^
U Individual Rationality (IR)
Solution approach:
- Jensen’s inequality =) E (^) x [ e rx^ ] e rE^ x^ [x]
- Because the principal chooses the action, optimal wage must be independent of q; i.e., w (q) = ↵
- Because a higher w (q) decreases the principal’s profit and increases the agent’s payo↵, (IR) must bind. So:
e r(↵ c(a))^ = U =) ↵ = c (a) ln ( rU^ )
- The last equation pins down the wage ↵ as a function of the action a.
- We now substitute into the objective function. We have:
max a
a ca^
2 2 ^
ln ( U ) r
- First order condition: 1 c a = 0
Optimal solution:
a ⇤^ =^1 c and hence w (q) = ln ( r U^ )+ 21 c
Therefore, the agent’s problem reduces to
max a
↵ + a c a 2 2 ^
2 r ^
The first-order condition for the agent’s optimal e↵ort choice is:
a ( ) = c
Unless 1, in equilibrium, e↵ort is less than first best.
The principal will then maximize
max a,↵, E [a + ✏ ↵ (a + ✏)] = (1 ) a ↵
s.t. a =
c ↵ + ^
2 2
c ^ r ^
2
ur
First equation is the incentive compatibility constraint (IC) and the second is the individual rationality (IR) with u = ln
U
The principal will choose ↵ = ur 2 2
c ^ r ^2
(s.t. IR binds).
Substituting into the principal’s objective function:
max
c +^
c ^ r ^
2
ur
Solution: ⇤^ = (^) 1 + 1 rc 2 (1) and ↵ ⇤^ = ur 1 ^ rc ^
2 2 c 2 (1 + rc 2 ) 2
Because negative salaries are allowed, the IR constraint is binding.
The equilibrium level of e↵ort is
a ⇤^ = (^) c (1 +^1 rc 2 )
which is always lower than the first-best level of e↵ort, a f b^ = (^1) c.
Comparative Statics ⇤^ = 1 1 + rc 2 Incentives are lower powered ; i.e., ⇤^ is lower when:
- the agent is more risk-averse; i.e., if r is larger
- e↵ort is more costly; i.e., if c is larger
- there is greater uncertainty; i.e., if 2 is larger.
Is a linear contract optimal (among all possible contracts)?
- NO!
- Mirrlees’s “shoot-the-agent” contract is optimal here:
q ⇤^ (x) =
w (^) H if x q (^0) w (^) L otherwise
where w (^) H > w (^) L.
- By choosing w (^) H , w (^) L and q 0 appropriately, it is possible to implement first best (approximately). ⇤ Agent receives w (^) H almost surely, yet has incentives from fear of w (^) L.
- But this result depends crucially on the assumption ✏ ⇠ N (0, 2 ).
What to make of linear contracts
- Even if linear contracts are not optimal here, they are attractive for their simplicity and for being easy to characterize and interpret.
- Nonlinear models are often very sensitive to the particular assumptions of the model (e.g., the distribution function of ✏).
Nonlinear contracts are also prone to “gaming”.
- Consider Mirrlees’ “shoot-the-agent” contract in a dynamic world.
- After output has reached q 0 , the agent has no incentive to exert e↵ort.
