Editing Minimum wage (section)

==== Job search effort ====
{{technical|section|date=September 2022}}
In the model just presented, the minimum wage always increases unemployment.&nbsp;This result does not necessarily hold when the search effort of workers is [[Endogeneity (econometrics)|endogenous]].

Consider a model where the intensity of the job search is designated by the scalar <math>\epsilon</math>, which can be interpreted as the amount of time and/or intensity of the effort devoted to search.&nbsp;Assume that the arrival rate of job offers is <math>\alpha\epsilon</math> and that the wage distribution is degenerated to a single wage <math>w</math>.&nbsp;Denote <math>\varphi(\epsilon)</math> to be the cost arising from the search effort, with <math>\varphi' > 0, \; \varphi'' > 0</math>.&nbsp;Then the discounted utilities are given by:<math display="block">\begin{aligned}
rV_{e} &= w + q(V_{u}-V_{e}) \\
rV_{u} &= \max_{\epsilon} \; z - \varphi(\epsilon) + \alpha \epsilon(V_{e}-V_{u})
\end{aligned}</math>Therefore, the optimal search effort is such that the marginal cost of performing the search is equation to the marginal return:<math display="block">\varphi'(\epsilon) = \alpha(V_{e}-V_{u})</math>This implies that the optimal search effort increases as the difference between the expected utility of the job holder and the expected utility of the job seeker grows.&nbsp;In fact, this difference actually grows with the wage.&nbsp;To see this, take the difference of the two discounted utilities to find:<math display="block">(r+q)(V_{e}-V_{u}) = w-\max_{\epsilon}\left[z - \varphi(\epsilon) + \alpha \epsilon(V_{e}-V_{u}) \right]</math>Then differentiating with respect to <math>w</math> and rearranging gives us:<math display="block">{d\over{dw}}(V_{e}-V_{u}) = {1\over{r+q+\alpha\epsilon^{*}}} > 0</math>where <math>\epsilon^{*}</math> is the optimal search effort.&nbsp;This implies that a wage increase drives up job search effort and, therefore, the job finding rate.&nbsp;Additionally, the unemployment rate <math>u</math> at equilibrium is given by:<math display="block">u = {q\over{q+\alpha\epsilon}}</math>A hike in the wage, which increases the search effort and the job finding rate, decreases the unemployment rate.&nbsp;So it is possible that a hike in the minimum wage ''may'', by boosting the search effort of job seekers, boost employment.&nbsp;Taken in sum with the previous section, the minimum wage in labor markets with frictions can improve employment and decrease the unemployment rate when it is sufficiently low.&nbsp;However, a high minimum wage is detrimental to employment and increases the unemployment rate.