Consider the signaling problem discussed in class. Assume that θL =
1, θH = 2, c (e, θL) = e2 and c (e, θH) = e2
k , where k > 1.
(i) Find the separating equilibrium with the lowest education level and
compute the utilities of both types in this equilibrium.
(ii) Consider a social planner who does not know θ but can offer workers
a menu of education and wages that allows for cross subsidies across the
two types. The constraints that the social planner faces are: (a) Incentive
compatibility: if (wL, 0) 6= (wH, eH) is the pair of wages and education
suggested to the two workers, the low type must be better off choosing (wL, 0)
and the high type must be better off choosing (wH, eH); (b) Budget Balance:
λwH + (1 − λ)wL = λθH + (1 − λ) θL = 1+λ. Assume that the utility that
you computed in part (i) for the high type is higher than 1 + λ (the utility
corresponding to the prohibition of signaling). Show that there is no menu
that the social planner can choose that satisfies (a) and (b), and that leads
to a Pareto improvement relative to the outcome computed in part (i).
(iii) Provide an example where in fact the social planner can find a menu
that satisfies (a) and (b), and that leads to a Pareto improvement relative to
the outcome computed in part (i).
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