Una experiencia de diseño curricular en torno a la variación conjunta

Our next proposition gives a necessary and sufficient condition such that under As- sumption 2.1, equilibrium payoffs can be obtained using menu-contracts which are strictly higher than any separating equilibrium payoff using point contracts.

Proposition 2.2 If Assumption 1 holds, then P_SP and P_SM intersect only at the

RSW (or least-cost separating) equilibrium payoff and there exist a payoff v ∈ PM

S

such that v > λU(H) + (1−λ)U(L) if and only if

h ¯ U +a(L)− (a(L)−a1)πLs(a2) πLs(a2)−πLs(a1) > w. (2.5)

Proof The proposition is a result of the following lemma that proves that (2.5) is a necessary and sufficient condition for menu-contracts to deliver higher ex ante payoff to the principal than any separating point-contract.

Lemma 2.2 If Assumption 1 holds, there exists an equilibrium in menu-contracts that gives higher payoffs to both principal types than the least-cost separating equilib- rium in point-contracts if and only if (2.5) holds.

To see why this lemma holds, note that w_f∗(a∗(L);L) =h ¯ U+a2− (a2−a1)πLs(a2) πLs(a2)−πLs(a1)

and observe Figure 2.1 (note that a contract that implements effort a∗(L) must be below the curve wf,AIC and above the curve wf,IR). In panel (a), the least costly

contract that implements a∗(L) is strictly interior. As we have noted, in the least cost separating equilibrium the type-Hprincipal gives the agent utility strictly greater than her reservation utility. Moreover, separation requires the type-H principal to increase the cost of her contract to dissuade the type-L principal from mimicking her. Using menu-contracts and the inscrutability principle, we can transfer some of the rents ceded to the agent by the type-H principal to the type-L principal, effectively shifting her individual rationality constraint down, allowing the type-L principal to offer a less costly contract and earn more profits. This eases the incentive compatibility constraint between the principals, allowing the type-H principal to make her contract less costly and earn more profits herself.

On the other hand, if condition (2.5) fails, as in panel (b) of Figure 2.1, relaxing the type-Lprincipal’s individual rationality constraint does not generate a less costly contract for her to offer.

Proof Sufficiency. Suppose the a∗(L) = a2. The contract w∗(a2;L) is the unique

solution to

πLf(a2)U(wf) +πLs(a2)U(ws) = ¯U +a2 (2.6)

(a) Condition (2.5) is satisfied. (b) Condition (2.5) is not satisfied.

Figure 2.1: Examples of when condition (2.5) is and isn’t satisfied.

Solving (3.7) and (2.7) we get

w∗_f(a2;L) =h ¯ U+a2− (a2 −a1)πLs(a2) πLs(a2)−πLs(a1) > w by assumption. By hypothesis πHs(a2) πHf(a2) > πLs(a2) πLf(a2)

and it is clear thatw_s∗(a2;H)> w∗f(a2;H). Thus we can apply Lemma 1 of Silvers [10]

to conclude thatIR(a2;{1,0}) is satisfied with strict inequality: when the principal’s

type is her private information, in the least cost separating equilibrium the principal of type-H cedes rents to the agent ex ante. Thus,

∆ := λ 2(1−λ) " X n πHnU( ˆwn)−a2−U¯ # >0.

Since for menu-contracts individual rationality only needs to be satisfied in expecta- tion, we can transfer the half of rents ceded to the agent from the principal of type-H to the principal of type-L, essentially relaxing her individual rationality constraint by

∆. The type-Lprincipal’s contract can then be solved as the unique solution to (2.7) and

πLf(a∗(L))U(wf) +πLs(a∗(L)U(ws) = ¯U +a∗(L)−∆. (2.8)

I can solve (3.7), (2.7) and (2.8) for wf as a function of ws:

wf,AIR(ws) =h ¯ U+a2−πLs(a2)U(ws) πLf(a2) wf,AIC(ws) =h U(ws)− a2−a1 (πLs(a2)−πLs(a1)) wf,IR∆(ws) =h ¯ U+a2−∆−πLs(a2)U(ws) πLf(a2) .

Taking the derivative of wf,AIC(ws):

w_f,AIC0 (ws) =h0 U(ws)− a2−a1 (πLs(a2)−πLs(a1)) U0(ws)>0.

Note that w∗(a2;L) is the solution towf,IR(ws) =wf,AIC(ws); that is,

wf,IR∆(w

∗

s(a2;L)) = wf,AIC(w∗s(a2;L))

=w_f∗(a2;L).

Let w˜(a2;L) be the solution to wf,IR∆(ws) = wf,AIC(ws) if ˜wf(a2;L) ≥ w and

˜

w(a2;L) := (w, wf,IR∆(w)) otherwise. Sincewf,AIC(ws) is decreasing and wf,IR∆ < wf,IR,

we have w˜(a2;L) w∗(a2;L).6 Thus, w˜(a2;L) implements a2 at a lower cost than

w∗(a2;L).

6_{Ifaand bare two vectors of the same size,abindicates that each element ofais strictly} less than each element ofb.

Meanwhile, atw˜(a2;L), the constraintP IC(a∗(L);L) is relaxed (since the type-L

principal now implementsa∗(L) at a lower cost and therefore receives a higher payoff) and therefore the type-H principal can implement her effort at a lower cost. Thus, both types of the principal are strictly better off.

If a∗(L) = a1, then wf∗(a1;L) = ws∗(a1;L) =h( ¯U +a1). Let z = min{U¯ +a1, w}.

Then ˜wf(a1;L) = ˜ws(a1;L) = h(z) implements a1 in menu-contracts. As before

at w˜(a1;L), the constraint P IC(a∗(L);L) is relaxed (since the type-L principal now

implementsa∗(L) at a lower cost and therefore receives a higher payoff) and therefore the type-H principal can implement her effort at a lower cost. Thus, both types of the principal are strictly better off.

Necessity. To show necessity, suppose that w_f∗(a∗(L),2) =w. By definition,

wf,AIC(w∗s(a

∗

(L);L)) =w.

But then, since wf,AIC is strictly increasing, there is no ˜ws that implements a∗(L)

such that ˜ws < w∗s(a

∗_{(L);L), even for the relaxed individual rationality constraint.}

So the least cost contract that implements a∗(L) remains (w, w_s∗(a∗(L);L)).