# Which is the correct derivation for HH value of employment in search and matching

Hi. I’m studying DSGE models with labor search frictions. In general, taking only the labor issue, particularly the optimal unemployment (U_t) choice the problem of the household is to solve:

\max_{\{\cdot, U_t\}}E_0\sum_{t=0}^{\infty}\beta^{t}u(\cdot, U_t)

s.t.

n_{t}=(1-s)n_{t-1}+q_{t}U_{t}

Where n_t is labor. And more importantly, q_{t}\equiv m(U_t,V_t)/U_t, is the probability of an unemployed worker of finding a job, where m(U_t,V_t) is a constant returns to scale aggregate matching function (V_t are vacancies posted).

Most of the times it’s stated explicitly in words (Gertler, Sala and Trigari (2009)) or one may implicitly assume it by checking FOCs presented (Galí (2010)) that q_t is given for the representative household, and similarly p_t\equiv m(V_t,U_t)/V_t is given for the firm. (say m(V_t,U_t)= U_t^{\gamma}V_t^{1-\gamma})

The mathematical consequences of that subtlety (or not), are that for example if one is going to take the FOC of the problem stated above with respect to U_t yield:

u_{U}-\lambda_t^Uq_{t}+g(\cdot)=0

Where \lambda_t^U is a Lagrange multiplier associated with the problem, and g(\cdot) is just representing other terms that may entry in the FOC.

Note that indeed algebraically q_{t}U_{t}=m(V_t, U_t). For the literature I’ve reviewed I thought that it’s always assumed that agents take as given those ratios, and that’s the reason for the matching function m not appearing directly in the labor law of motion equation. Yet recently I checked Thomas (2008), and even though the author also defines q_t and p_t, when setting the problem he uses (simplified):

n_{t}=(1-s)n_{t-1}+m(V_t, U_t)

In which case the FOC would change to:

u_{U}-\lambda_t^U \gamma U_t^{\gamma-1}V_t^{1-\gamma}+g(\cdot)=0
\equiv u_{U}-\lambda_t^U \gamma q_t+g(\cdot)=0

Which is correct and why? Is it really relevant the difference?

In a very particular case that I have assessed it (in a very superficially manner), the main relevant consequence is that the calibration of some parameter would change (in this case the parameter governing the marginal disutility of U_t, assuming u is additively separable in U_t), beyond from that I didn’t see any obvious harmful consequence in the model.

PD: I’m aware Galí (2010) is not a “full” search model, but taking the analogies it keeps with search models, one can assumed for the equations presented that the author also takes those probabilities as given.

Usually, the idea is that the search friction creates an externality. Because each agent is too small, their decisions do not affect aggregate labor market tightness, i.e. the probability of getting a match. But because every agent is the same, the decision will actually have an effect. This setup is reflected in the FOCs you mentioned for the first papers. Agents do not take into account the effect of their actions on aggregate variables.

This may be different in papers studying optimal policy. When studying the central planner’s solution, it is often assumed that he/she internalizes such decisions. That seems to be the case for the last paper cited.

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That’s a good point, I’ll have to check that difference cheerfully, since I’m planning to make an optimal monetary policy analysis. In summary (roughly speaking in most of the cases) if I’m solving for the individual (say i\in[0,1]) household/firm problem assuming that each one takes probabilities as given, then I compute FOCs as in the first papers case, in the other hand if I’m analyzing analytically optimal policy from the view of the CB/Govt. I should not then take probabilities as given, then placing matching function directly, am I right with the general idea?

Thanks!!

That is my understanding. But note that you as the model builder can decide which frictions you allow your planner to address.

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