Module 7

Labor Market & Unemployment

The competitive labor market model cannot explain involuntary unemployment. The Diamond-Mortensen-Pissarides (DMP) search and matching model provides a microfounded theory of frictional unemployment.

Measuring the Labor Market

Labor Market Stocks

Individuals are classified into three groups:

Employed ($E$): Worked in the past week.
Unemployed ($U$): Not employed, but actively searched in the last 4 weeks.
Not in labor force ($NL$): Neither employed nor unemployed.

Key Rates

$\displaystyle u = \frac{U}{E+U}, \quad p = \frac{E+U}{E+U+NL}, \quad n = \frac{E}{E+U+NL}$

$u$: unemployment rate (most volatile).
$p$: participation rate.
$n$: employment-population ratio.

Vacancies & the Beveridge Curve

Vacancy Rate

The vacancy rate is the ratio of job openings to the total labor force. When the vacancy-to-unemployment ratio is high, the labor market is considered tight.

Beveridge Curve: A strong negative empirical relationship between the vacancy rate and the unemployment rate. High vacancy rates coincide with low unemployment.

Labor Market Flows & Steady States

Job Finding and Job Loss Rates

Workers flow between states each period:

Job finding rate: fraction of unemployed who become employed per month (~30% in the US).
Job loss (separation) rate: fraction of employed who become unemployed per month (~1-2% in the US).

Steady-State Unemployment

In steady state, flows into unemployment equal flows out:

$U \cdot f = E \cdot s \quad\Rightarrow\quad u = \frac{s}{s + f}$

where $f$ is the job finding rate and $s$ is the separation rate. The unemployment rate is determined by the flow rates, even in steady state.

Competitive Labor Market (Benchmark)

Equilibrium

Household: $\max U(c) - V(h)$ subject to $c = wh + \Pi$.

Firm: $\max f(n) - wn$.

Equilibrium:

$\displaystyle \frac{V'(h)}{U'(c)} = w = f'(n)$

MRS between leisure and consumption equals the real wage, which equals the MPL. All workers work the same hours; there is no involuntary unemployment.

With $U(c) = \ln c - \gamma h$ and $f(n) = An$: $n^* = 1/\gamma$, independent of productivity $A$. Income and substitution effects cancel.

Shortcomings

Cannot explain involuntary unemployment — some workers want jobs at the prevailing wage but cannot find them.
All identical workers are either all employed or all unemployed — no dispersion.
Fluctuations in total hours are mostly extensive margin (E/U transitions), not intensive margin (hours per worker).

DMP Model: Matching Function

The central friction: recruiting is costly and matching between workers and firms is not instantaneous.

Matching Technology

Matching Function: $\quad m = M(u, v) = \bar{\mu} v^\eta u^{1-\eta}, \quad 0 < \eta < 1$

where $u$ is the measure of unemployed workers and $v$ is the number of job vacancies. The matching function captures coordination failures — if workers and firms could perfectly coordinate, all jobs would be filled.

Labor Market Tightness

Tightness: $\quad \theta \equiv \frac{v}{u}$

Key derived probabilities:

Vacancy filling rate: $\mu(\theta) = \frac{m}{v} = \bar{\mu}\theta^{\eta-1}$ (decreasing in $\theta$).
Job finding rate: $f(\theta) = \frac{m}{u} = \bar{\mu}\theta^{\eta}$ (increasing in $\theta$).

Intuition: When the labor market is tight (high $\theta = v/u$), it's easier for workers to find jobs ($f(\theta) \uparrow$) but harder for firms to fill vacancies ($\mu(\theta) \downarrow$).

DMP Model: Equilibrium Unemployment

Wage Determination

Wages are determined by Nash bargaining between workers and firms, splitting the match surplus according to bargaining power.

In the simple static model, the equilibrium is determined by:

Job creation (free entry): Firms post vacancies until the expected cost equals the expected benefit: $\kappa = \mu(\theta) \cdot (\text{firm surplus})$.
Wage curve: Workers' outside option (unemployment) and bargaining power determine the wage.

Key Insights

There is equilibrium unemployment even though all workers are identical — some are matched, some are searching.
Unemployment responds to productivity shocks through changes in vacancy posting (labor demand) — consistent with the observation that unemployment is countercyclical.
Fluctuations in employment are driven by the extensive margin (E/U switches), as in the data.

Phillips Curve Connection

The DMP model provides microfoundations for the Phillips curve relationship: when output is above potential ($x_t > 0$), labor markets tighten ($\theta \uparrow$), wages rise, and firms pass higher costs into prices, generating inflation. This is the chain linking the output gap to inflation in the NKPC.