Labor search model with acceptance decision

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I was wondering if I can use the toolkit to solve a lifecycle model with labor search and an acceptance decision.

Let me illustrate the model, which is a stripped-down version of Kitao (JME, 2014).

Individuals can be either employed or unemployed. The state variables of an employed/unemployed individual are (a,g,j) where a is assets, g is productivity and j is age. The choice variables are savings a' and search effort s.

Employed individuals choose a' to solve

V^E(a,g,j) = \max_{a'} u(c) + \beta E[ (1-\sigma) \max [V^E(a',g',j+1),V^U(a',g',j+1)]+ \sigma V^U(a',g',j+1) ]
subject to
c+a' = g\times w +(1+r)a, \quad a' \geq 0

where \sigma \in (0,1) denotes the job separation rate. Note that if the individual is lucky and does not lose the job, he/she can still decide to quit.
Unemployed individuals choose a' and s to solve

V^U(a,g,j) = \max_{s,a'} u(c)-v(s) + \beta E[ \pi^s(s) \max [V^E(a',g',j+1),V^U(a',g',j+1)]+ (1-\pi^s(s)) V^U(a',g',j+1) ]
subject to
c+a' = b_U +(1+r)a, \quad a' \geq 0

where \pi^s(s) denotes the job finding probability (increasing in effort s). The expectation E on the right-hand-side of the Bellman equations is with respect to the realization g' that follows the Markov chain \Gamma_j(g,g').

Now, if there wasn’t the max inside of the two Bellman equations, the problem would be easy: I can define a semi-exogenous state call it \ell \in \{E,U \} whose transition probabilities are affected by the decision variable s, in addition to the standard exogenous Markov g. Let the transition probability of \ell be defined as \pi^{\ell}(\ell,\ell'|s).

However, in this formulation we also have a stay/quit decision in the equation for employed and an accept/reject decision in the equation for unemployed. Do I need another decision variable? How can I set the problem in the toolkit?

Thanks!

2

I feel like there is some needed info missing here that will matter for whether toolkit can do it. Namely why reject the job? Is the idea just that there is some threshold gstar, so if g’>=gstar you want to accept the job, and g’<gstar you want to reject the job? Or is there something else on the firm side that is why you want to reject a job?

If it is about g’, I think you can sneak it into the toolkit through clever choice of setting up (e,g) as a joint-semi-exogenous state, but if the decision is from something else I suspect it won’t be possible in toolkit.

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You are right, I might have oversimplified the model in Kitao (2014). I think the model of Krusell et al (2010) is clearer

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The setting illustrated in the screenshot above is quite common in these Bewley models with search frictions (but no matching function: the job finding rate lambda_w is exogenous).

Let’s consider the unemployed individual with value N(k,s). With probability \lambda_w he gets a job offer and his value becomes V, which is the max between the value of staying unemployed and the value of accepting the offer and being employed in the next period. Without this accept/reject decision, the problem would become trivial: we don’t even need a semi-exogenous shock since there is no search effort affecting \lambda_w. I’m however struggling in mapping this Krusell model to the toolkit :frowning:

Question: why would unemployed worker ever reject an offer? I guess it depends on s'. Note that it is written as E[max(accept,reject)], so first you see s', then you decide whether to accept or reject: if s' is very low, it might be optimal to reject, get unemployed benefits next period and search again.

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Looking at Krusell, Mukoyama, Rogerson & Sahin (2010), it is going to be a threshold rule for s’ that determines whether you accept/reject (the threshold will depend on k). Call the threshold s^*(k), so you will accept for s'>s^*(k) and reject for s'<s^*(k).

You could do this in VFI Toolkit, but it is a bit of a work-around. Intuitively, the idea is to assume a ‘threshold accept rule’ and solve the model using this. Then add a later step that makes this assumed rule actually match the accept/reject decision you want.

  • First, parameterize s^*(k), say as polynomial s^*(k)=c0+c1*k+c2*k^2+c3*k^3 (use any functional form you like, what matters is just to turn s^*(k) into a vector of parameters, here [c0,c1,c2,c3]).
  • Second, set up (e,s) [employment state and productivity] as a joint semi-exogenous markov process, and use the threshold together with the matching probability in here to get the transitions to reflect the accept/reject choices.
  • Third, either as a calibration or general eqm condition, setup that the threshold is set to match when V(k,e,s)>V(k,u,s).

In a general eqm model this would be low cost way to do things (is just adding a little bit more general eqm conditions). But in a life-cycle model this will suck, because you have to do this third optimization step which a bit of custom coding would make trivial. In either case, this is obviously a second-best way to solve the problem. Happily Krusell, Mukoyama, Rogerson & Sahin (2010) is anyway a general eqm model, so won’t cost much there.

My feeling is that if you did ‘endogenous search effort’ instead of ‘endogenous accept/reject’, you’d end up with much the same model. Only difference will be that the former is based on s while the later is on s’, but given s is a highly persistent markov you are going to get almost the same answer. Later will be slightly stricter at eliminating poor matches, but I don’t know if this is good or bad empirically. Maybe there is something nice about depending on s’ vs s, or vice-versa, that fits reality, but it is not obvious to me what it is.

PS. Some minor notational abuse above. I used e as employment state, and then used e & u as the two values of this state.
PPS. Actually, endogenous search effort does have the major advantage that you get employed-unemployed-inactive. While exogenous search just has employed-inactive (or employed-unemployed, depending how you want to interpret). So if you are just looking at employment rate then whichever is fine, but if you want the unemployment rate you need endogenous search effort.

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Very interesting way of thinking about this model, thanks!