# Matching And Job Quality Consider The Following Bewley Model A Version Of Which Daro 1946875

Matching and job quality Consider the following Bewley model, a version of which Daron Acemoglu and Robert Shimer (2000) calibrate to deduce quantitative statements about the effects of government supplied unemployment insurance on equilibrium level of unemployment, output, and workers’ welfare. Time is discrete. Each of a continuum of ex ante identical workers can accumulate nonnegative amounts of a single risk-free asset bearing gross one-period rate of return R; R is exogenous and satisfies βR g and bad jobs with wage wb g. Both wages are exogenous. Unemployed workers must decide whether to search for good jobs or bad jobs. (They cannot search for both.) If an unemployment worker devotes h units of time to search for a good job, a good job arrives with probability mgh; h units of time devoted to searching for bad jobs makes a bad job arrive with probability mbh. Assume that mg b. Good jobs terminate exogenously each period with probability δg, bad jobs with probability δb. Exogenous terminations entitle an unemployed worker to unemployment compensation of b, which is independent of the worker’s lagged earnings. However, each period, an unemployed worker’s entitlement to unemployment insurance is exposed to an i.i.d. probability of φ of expiring. Workers who quit are not entitled to unemployment insurance. Workers choose {ct, ht}∞t=0 to maximize

Where β ∈ (0, 1), and θ is a coefficient of relative risk aversion, subject to the asset accumulation equation

And the no-borrowing condition at+1 ≥ 0; η governs the substitutability between consumption and leisure. Unemployed workers eligible for u.i. receive income yt = b,

while those not eligible receive 0. Employed workers with good jobs receive after tax income of yt = wgh(1 − τ), and those with bad jobs receive yt = wbh(1 − τ). In equilibrium, the flat rate tax is set so that the government budget for u.i. balances. Workers with bad jobs have the option of quitting to search for good jobs. Define a worker’s composite state as his asset level, together with one of four possible employment states: (1) employed in a good job, (2) employed in a bad job, (3) unemployed and eligible for u.i.; (4) unemployed and ineligible for u.i.

a. Formulate value functions for the four types of employment states, and describe Bellman equations that link them.

b. In the fashion of Bewley, define a stationary stochastic equilibrium, being careful to define all of the objects composing an equilibrium.

c. Adjust the Bellman equations to accommodate the following modification. Assume that every period that a worker finds himself in a bad job, there is a probability δupgrade that the following period, the bad job is upgraded to a good job, conditional on not having been fired.

d. Acemoglu and Shimer calibrate their model to U.S. high school graduates, then perform a ‘local’ analysis of the consequences of increasing the unemployment compensation rate b. For their calibration, they find that there are substantial benefits to raising the unemployment compensation rate and that this conclusion prevails despite the presence of a ‘moral hazard problem’ associated with providing u.i. benefits in their model. The reason is that too many workers choose to search for bad rather than good jobs. They calibrate β so that workers are sufficiently impatient that most workers with low assets search for bad jobs. If workers were more fully insured, more workers would search for better jobs. That would put a larger fraction of workers in good jobs and raise average productivity. In equilibrium, unemployed workers with high asset levels do search for good jobs, because their assets provide them with the ‘self-insurance’ needed to support their investment in search for good jobs. Do you think that the modification suggested in part in part (c) would affect the outcomes of increasing unemployment compensation b?

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