2. [25 marks] In an economy there are many identical price-taking firms, using a constant returns to scale technology, in which labor is the only input. The price of every firm’s output is 1 such that the profit per worker is θ − w, where θ is the worker’s productivity (i.e. the number of units of output this worker can produce) and w is the wage.
There are two types of workers: type-a workers have productivity θa = 3 and type-b workers have productivity θb = 1. Workers’ productivities are unobservable by firms but they know that the proportion of type-b workers is π. Both types of workers have a reservation wage of zero.
(a) What is the equilibrium wage set by the firms? What types of workers are going to
accept an employment offer? [5 marks]
Suppose workers can spend their own resources to acquire educational certificates in order to signal their productivity. It is common knowledge that the cost of acquiring an education level z equals z2 for type-b workers and z 2 2
for type-a workers.
(b) Characterize the set of separating Perfect Bayesian equilibria. For what values of π will the type-a and type-b workers be better off under the no-signalling outcome in part (a) than under the least-cost separating equilibrium? [10 marks]
Suppose now that workers don’t have access to educational certificates, but firms create jobs with different task levels t ≥ 0. Let the proportion of type-b workers be π =34
. The output produced by a worker of type i is θi (i = a, b) regardless of the worker’s task level, i.e. t is used as a screening mechanism and does not affect profit directly. Task levels affect workers’ utility: ua = w −14t2 and ub = w −12t2
. In stage 1, firms simultaneously announce a menu of contracts (w, t). In stage 2, workers decide whether they want to sign a contract and which one to sign:

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