Editing Minimum wage (section)

=== Mathematical models of the minimum wage and frictional labor markets ===
The following mathematical models are more quantitative in orientation, and highlight some of the difficulties in determining the impact of the minimum wage on labor market outcomes.<ref>{{Cite book|url=https://mitpress.mit.edu/books/labor-economics-second-edition|title=Labor Economics|last1=Cahuc|first1=Pierre|last2=Carcillo|first2=Stéphane|last3=Zylberberg|first3=André|publisher=The MIT Press|year=2014|isbn=9780262027700|edition=2nd|location=Cambridge, MA|pages=796–799}}</ref> Specifically, these models focus on labor markets with frictions and may result in positive or negative outcomes from raising the minimum wage, depending on the circumstances.

==== Welfare and labor market participation ====
{{technical|section|date=September 2022}}
Assume that the decision to participate in the labor market results from a trade-off between being an unemployed job seeker and not participating at all.&nbsp;All individuals whose expected utility outside the labor market is less than the expected utility of an unemployed person <math>V_{u}</math> decide to participate in the labor market.&nbsp;In the basic search and [[Matching theory (economics)|matching model]], the expected utility of unemployed persons <math>V_{u}</math> and that of employed persons <math>V_{e}</math> are defined by:

<math display="block">\begin{aligned}
rV_{e} &= w + q(V_{u}-V_{e}) \\
rV_{u} &= z + \theta m(\theta) (V_{e}-V_{u})
\end{aligned}</math>Let <math>w</math> be the wage, <math>r</math> the interest rate, <math>z</math> the instantaneous income of unemployed persons, <math>q</math> the exogenous job destruction rate, <math>\theta</math> the labor market tightness, and <math>\theta m(\theta)</math> the job finding rate.&nbsp;The profits <math>\Pi_{e}</math> and <math>\Pi_{v}</math> expected from a filled job and a vacant one are:<math display="block">r\Pi_{e} = y-w+q(\Pi_{v}-\Pi_{e}), \quad r\Pi_{v} = -h + m(\theta)(\Pi_{e}-\Pi_{v})</math>where <math>h</math> is the cost of a vacant job and <math>y</math> is the productivity.&nbsp;When the ''free entry condition'' <math>\Pi_{v} = 0</math> is satisfied, these two equalities yield the following relationship between the wage <math>w</math> and labor market tightness <math>\theta</math>:
{{Labor|expanded=rights|sp=us}}
<math display="block">{h\over{m(\theta)}} = {y-w\over{r+q}}</math>If <math>w</math> represents a minimum wage that applies to all workers, this equation completely determines the equilibrium value of the labor market tightness <math>\theta</math>.&nbsp;There are two conditions associated with the matching function:<math display="block">m'(\theta) < 0, \quad [\theta m(\theta)]' > 0</math>This implies that <math>\theta</math> is a decreasing function of the minimum wage <math>w</math>, and so is the job finding rate <math>\alpha = \theta m(\theta)</math>.&nbsp;A hike in the minimum wage degrades the profitability of a job, so firms post fewer vacancies and the job finding rate falls off.&nbsp;Now let's rewrite <math>rV_{u}</math> to be:<math display="block">rV_{u} = {(r+q)z + \theta m(\theta) w\over{r+q + \theta m(\theta)}}</math>Using the relationship between the wage and labor market tightness to eliminate the wage from the last equation gives us:
<math display="block">rV_{u} = {\theta m(\theta)y + (r+q)z - \theta(r+q)h\over{r+q + \theta m(\theta)}}</math> By maximizing <math>rV_{u}</math> in this equation, with respect to the labor market tightness, it follows that:<math display="block">{[1-\eta(\theta)](y-z)\over{r+q+\eta(\theta)\theta m(\theta)}} = {h\over{m(\theta)}}</math>where <math>\eta(\theta)</math> is the [[Elasticity (economics)|elasticity]] of the matching function:<math display="block">\eta(\theta) = -\theta{m'(\theta)\over{m(\theta)}} \equiv -\theta {d\over{d\theta}}\log m(\theta)</math>This result shows that the expected utility of unemployed workers is maximized when the minimum wage is set at a level that corresponds to the wage level of the decentralized economy in which the bargaining power parameter is equal to the elasticity <math>\eta(\theta)</math>.&nbsp; The level of the negotiated wage is <math>w^{*}</math>.

If <math>w < w^{*}</math>, then an increase in the minimum wage increases participation ''and'' the unemployment rate, with an ambiguous impact on employment.&nbsp;When the bargaining power of workers is less than <math>\eta(\theta)</math>, an increase in the minimum wage improves the welfare of the unemployed – this suggests that minimum wage hikes can improve labor market efficiency, at least up to the point when bargaining power equals <math>\eta(\theta)</math>.&nbsp;On the other hand, if <math>w \geq w^{*}</math>, any increases in the minimum wage entails a decline in labor market participation and an increase in unemployment.{{disputed-inline|Math in the Welfare and labor market participation section|date=April 2025}}

==== 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.