Consider Akerlof's adverse selection model with three-types of workers,low, medium and high productivity. A worker of each type can produce a valueto firms equal to 1, 3 and 5, respectively. The fraction of workers of eachtype is the same (one third), and hence the average productivity of workersis 3. Workers differ not only in their productivity but also in their reservationwage. In particular, the reservation wages of low, medium, and high productivityworkers are 0, 2+a, and 2+b, respectively, where a, b are exogenous parametersthat satisfy b > a > 0. Think of these reservation wages as the disutilityassociated to being employed. Thus, the payoff of an employed worker is equalto the wage minus their reservation wage, and zero if unemployed. There aretwo identical firms that are unable to distinguish between different worker types.However, individual workers do know their own productivity. The distributionof worker types, productivities and associated reservation wages are commonknowledge.1. Consider the first best. That is, consider a social planner that is ableto distinguish between different worker types, can assign workers to either toemployment or unemployment, and aims at maximizing total surplus. Findconditions on parameter values, a, b, such that all types should be employed inthe first best.2. Now consider the following game. In the first stage, the two firms, 1 and2, simultaneously set wages w1, w2. In the second stage, workers choose whichoffer to accept if any. Find conditions on parameter values, a, b, such that inthe unique subgame perfect Nash equilibrium only low types are employed.

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Chapter1: Making Economics Decisions

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Question

Consider Akerlof's adverse selection model with three-types of workers,
low, medium and high productivity. A worker of each type can produce a value
to firms equal to 1, 3 and 5, respectively. The fraction of workers of each
type is the same (one third), and hence the average productivity of workers
is 3. Workers differ not only in their productivity but also in their reservation
wage. In particular, the reservation wages of low, medium, and high productivity
workers are 0, 2+a, and 2+b, respectively, where a, b are exogenous parameters
that satisfy b > a > 0. Think of these reservation wages as the disutility
associated to being employed. Thus, the payoff of an employed worker is equal
to the wage minus their reservation wage, and zero if unemployed. There are
two identical firms that are unable to distinguish between different worker types.
However, individual workers do know their own productivity. The distribution
of worker types, productivities and associated reservation wages are common
knowledge.
1. Consider the first best. That is, consider a social planner that is able
to distinguish between different worker types, can assign workers to either to
employment or unemployment, and aims at maximizing total surplus. Find
conditions on parameter values, a, b, such that all types should be employed in
the first best.
2. Now consider the following game. In the first stage, the two firms, 1 and
2, simultaneously set wages w1, w2. In the second stage, workers choose which
offer to accept if any. Find conditions on parameter values, a, b, such that in
the unique subgame perfect Nash equilibrium only low types are employed.

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Step 3: Explain the Subgame Perfect Nash Equilibrium with Only Low Types Employed

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