Unemployment

According to the standardised definition of the International Labour Organization (ILO):

• Employment is the number of people who have a job.
• Unemployment is the number of people who do not have a job but are looking for one.
  • an unemployed person is a person aged 15 or over:
  • without a job during a given week;
  • available to start a job within the next two weeks;
  • actively having sought employment at some time during the past four weeks or having already found a job that starts within the next three months.

Terminology

• \(N\) is the working-age population.
• \(\mathcal{Q}\) is the labour force (employed + unemployed).
• \(U\) is the number of unemployed.

\[ \text{Unemployment Rate} = \frac{U}{\mathcal{Q}} \]

\[ \text{Participation Rate} = \frac{\mathcal{Q}}{N} \]

• the employment/population ratio

\[ \frac{\text{Employment}}{\text{Population}} = \frac{Q-U}{N} \]

How to measure?

• Difficult!
• Many countries rely on large surveys of households to compute the unemployment rate.
• The U.S. Current Population Survey (CPS) relies on interviews of 60,000 households every month.
• The two measures of unemployment used in the UK are the Claimant count and the Labour Force Survey. Between the two, the Labour Force Survey is considered a more accurate measure of unemployment.
• A person is unemployed if he or she does not have a job and has been looking for a job in the last four weeks.
• Those who do not have a job and are not looking for one are counted as not in the labour force.
• Discouraged workers are those persons who give up looking for a job and so no longer count as unemployed.
• The participation rate is the ratio of the labour force to the total population of working age.
• Because of discouraged workers, a higher unemployment rate is typically associated with a lower participation rate.

US Unemployment Rate – 1950-2019

US Unemployment Rate – 1950-2022

Unemployment and Happiness

To give you a sense of scale, other studies suggest that this decrease in happiness is close to the decrease triggered by a divorce or a separation.

Unemployment and Happiness

Failure of Walrasian models of the labour market

• Recall that in the RBC and New Keynesian models we have considered so far the labour markets are Walrasian:
  • Wage always clears the market.
  • Individuals are always on their optimal labour supply.
  • There is no unemployment, only choices to consume more leisure.
• Fluctuations in employment in these models reflect the willingness of people to substitute labour and leisure across periods.
  • Yet empirical studies find little evidence of significant intertemporal substitution, and point to inelastic individual labour supply.
  • This mechanism predicts counterfactually large fluctuations in the real wage, but much less volatility in employment than in the data.
• Key question: Why do shifts in labour demand lead to large movements in employment, and only small changes in the real wage?

Setup

• No capital for simplicity; labour is the only factor of production.

• There is a large number \(N\) of firms and a representative firm maximizes profits:

  \[ \pi = Y - wL, \]

  where \(Y\) is the firm’s output, \(w\) is the wage that it pays, and \(L\) is the amount of labor it hires.

• Output depends on the number of workers and on their effort; that is, \(Y = F(eL)\).

• Thus, the representative firm’s output is given by:

  \[ \pi = F(eL) - wL, \quad F'(\bullet) > 0,\ F''(\bullet) < 0, \]

  where \(e\) is workers’ effort (so that \(eL\) is effective labour).

Setup (continued)

• Assume effort \(e\) is an increasing function of the wage:

  \[ e = e(w), \quad e'(\bullet) > 0. \]

• For now, we are interested in the implications (not the precise reasons).

• There are \(\bar{L}\) workers, each supplying 1 unit of labour inelastically
  \(\Rightarrow\) i.e., they are prepared to work at any wage.

The firm’s problem (continued)

• In either case, the firm is free to choose its employment level.
  The first-order condition (FOC) with respect to \(L\) yields:

  \[ F'(e(w)L) = \frac{w}{e(w)}. \quad \text{(Equation $\ref{eq:foc_L}$)} \]

• That is, the marginal product of effective labour equals its unit cost \(\frac{w}{e(w)}\).

Note: When a firm hires a worker, it obtains \(e(w)\) units of effective labour.

The efficiency wage

• When the firm is unconstrained and sets its wage freely, the FOC with respect to \(w\) from the profit maximization yields:

  \[ F'(e(w)L)\, L\, e'(w) - L = 0. \]

• Substituting for \(F'(e(w)L)\) from the earlier condition and rearranging, we obtain:

  \[ \frac{w\, e'(w)}{e(w)} = 1. \quad \text{(Equation $\ref{eq:efficiency_wage}$)} \]

• That is, at the optimum, the elasticity of effort with respect to wage is 1.

Intuition:

• The firm wants to hire effective labour \(eL\) as cheaply as possible.

• Equation \(\frac{w\, e'(w)}{e(w)} = 1\) defines the efficiency wage that minimizes the unit cost of effective labour (i.e. it solves \(\min_{w} \frac{w}{e(w)}\))

• Put differently, the optimal \(w\) maximizes effort per dollar spent, \(e(w)/w\).

The efficiency wage (continued)

Equilibrium

• Let \(L^*\) and \(w^*\) denote the values that satisfy Equations \(\ref{eq:foc_L}\) and \(\ref{eq:efficiency_wage}\).

• Since all firms are identical, the total labour demand at \(w^*\) is \(N L^*\).

• If \(N L^* < \bar{L}\):

  • There is positive unemployment.
  • Firms are free to set wages; hence the equilibrium wage is \(w^*\).
  • Employment equals \(N L^*\), and \(\bar{L} - N L^*\) workers remain unemployed.

• If \(N L^* > \bar{L}\):

  • There is full employment.
  • At \(w^*\), labour demand would exceed supply, so the wage is bid up above \(w^*\) until \(N L(w) = \bar{L}\).
  • Firms become wage constrained and cannot reduce the wage to \(w^*\).

Case study: Henry Ford and efficiency wages

A big BUT

• Unchanged Real Wage Over Time:
  • In the model, as labor demand grows, the real wage remains constant for an extended period.
  • Eventually, as unemployment falls to zero, any further increase in labor demand must raise the real wage.
• Mismatch with Observed Unemployment Trends:
  • In reality, unemployment does not steadily trend toward zero.
  • The model’s prediction of a declining unemployment rate over time conflicts with long-run data that shows no clear trend in unemployment.

Unemployment and GDP

Unemployment and GDP

Empirical Puzzle

• Short-Run vs. Long-Run Behavior:
  • Short run: Labor-demand shifts primarily affect employment rather than the real wage.
  • Long run: The same shifts should eventually move the real wage more than employment, but this is not observed.
• Unanswered Questions:
  • Why do real wages not remain constant as demand grows over long periods?
  • How can we reconcile the absence of a clear trend in unemployment with a model that implies a downward trend?
  • The efficiency wage framework alone does not resolve these issues—additional mechanisms or explanations may be needed.

A More General Efficiency-Wage Framework

Effort Function
\[ e \;=\; e\bigl(w,\,w_a,\,u\bigr), \]

subject to \[ \frac{\partial e}{\partial w} \;>\; 0,\quad \frac{\partial e}{\partial w_a} \;<\; 0,\quad \frac{\partial e}{\partial u} \;>\; 0. \]

Intuition: 1. Higher own wage (\(w\)) raises a worker’s effort. 2. Higher outside wage (\(w_a\)) lowers effort (because alternative jobs look better). 3. Higher unemployment (\(u\)) raises effort (fear of job loss).

The Representative Firm’s Problem

Production Side:
Let the firm’s production be
\[ F\bigl(e(w,w_a,u)\,L\bigr). \]

Profit Maximization:
Choose \(w\) to balance wage costs against effort gains.

When the firm is price-taking in both product and labor markets, the key first-order conditions can be rearranged to: \[ F'\!\bigl(e(w,w_a,u)\,L\bigr) \;=\; \frac{w}{\,e(w,w_a,u)}, \] \[ w \;\frac{\partial e/\partial w}{\,e(w,w_a,u)} \;=\; 1. \]

Interpretation:
- The marginal product of effective labor, \(F'\), must equal the ratio of the wage to effort.
- The elasticity of effort with respect to the firm’s own wage is exactly 1 in equilibrium.

A Specific Example (Summers, 1988)

\[ e = \begin{cases} \left(\frac{w - x}{x}\right)^{\beta} & \text{if } w > x, \\[5pt] 0 & \text{otherwise} \end{cases} \]

where \[ x = (1 - b\,u)w_{a}. \]

Parameters: - \(0 < \beta < 1\). - \(b > 0\) measures how strongly unemployment, \(u\), affects a worker’s perceived outside options.

Mechanics: - If \(w \le x\), workers exert no effort. - For \(w > x\), effort increases less than proportionally with \((w - x)\).

Shapiro-Stiglitz (1984) model: an overview

• So far we have simply been assuming that workers’ effort is an increasing function of the wage in \(\eqref{eq:effort_function}\).
• In the highly influential paper, Shapiro and Stiglitz (1984) provide a microeconomic rationale for this assumption.
• Idea: If firms have a limited ability to monitor their workers, they are forced to provide them with enough incentives to exert high effort.
• In the model such incentive arises from the risk of being fired and losing a well-paid job if not working hard (shirking).
• Not only does the model provide a logical justification for the efficiency wage, but it also does so in a spectacularly elegant fashion.

Assumptions of the Model

• The economy has:

  • A large number of workers \(\bar{L}\) and firms \(N\).

  • Workers maximize lifetime utility:

    \[ U = \int_{0}^{\infty} e^{-\rho t} u_{t} \, dt,\quad \rho > 0 \] where \(u_{t}\) is the instantaneous utility at time \(t\) and \(\rho\) is the discount rate.

• Instantaneous utility:

  \[ u_{t} = \begin{cases} w_{t} - e_{t}, & \text{if employed, effort exerted} \\ 0, & \text{if unemployed} \end{cases} \]

• Effort \(e_{t}\) has two levels: \(e_{t} = 0\) (shirking) or \(e_{t} = \bar{e} > 0\) (effort).

Worker States

\[ u_{t} = \begin{cases} w_{t} - e_{t}, & \text{if employed, effort exerted} \\ 0, & \text{if unemployed} \end{cases} \]

At any moment in time, a worker can be in one of three states:

• Employed and exerting effort \(E\): gets utility \(w_{t} - \bar{e}\).
• Employed and shirking \(S\): gets utility \(w_{t}\).
• Unemployed (\(U\)): gets utility \(0\).

Transitions between states follow simple Poisson processes.

Job Ends: Exogenous Reasons – \(b\) – From \(E\) to \(U\)

• Jobs end randomly (exogenously) at a rate \(b\) with \(b>0\).
• The probability of the job surviving at some later time \(t\) is:

Equation Example

\[ P(t) = e^{-b(t - t_0)} \]

• Equation \(\eqref{eq:job_ends}\) shows that the survival probability decays exponentially (faster for higher \(b\)).
• In particular, \(P(t+\tau)/P(t) = e^{-b\tau}\), meaning the job survival is independent of \(t\) (i.e. memoryless).

Job Ends: Shirking Detection – \(q\) – From \(S\) to \(U\)

• \(q\) is the probability per unit time that a shirker is detected.

• \(q\) is exogenous and independent of the job separation rate \(b\).

• Firms detect shirkers and fire them.

• The probability that a shirker is still employed after time \(\tau\) is \(e^{-q\tau} \times e^{-b\tau}\).

• Combined, the survival probability (no detection and job still exists) is:

  \[ P(t) = e^{-(b+q)\tau}. \]

• Can \(q \rightarrow \infty\)?

  • \(q\) measures how frequently shirkers are monitored.
  • With very effective monitoring, \(q\) can become arbitrarily large.
  • Then \(e^{-q\tau} \rightarrow 0\), so the probability of being detected within \(\tau\) approaches 1.

Job Finding: \(a\) – From \(U\) to \(E\) (and/or to \(S\))

• Definition: Probability per unit time that an unemployed worker finds a job is \(a\).

  • If a worker is unemployed at time \(t\), the probability that they are employed at \(t+dt\) is \(a\,dt\).
  • \(a\) is determined endogenously by labor market conditions.

• Steady-State Condition:

  \[ b \cdot E = a \cdot U \]

• Expression for \(a\):

  \[ a = \frac{NL b}{\overline{L} - NL} \quad \text{(Equation $\ref{eq:equilibrium_a}$)} \]

• Economic Implications:

  • Reflects labor market tightness. An increase in \(a\) means jobs are easier to find.
  • Influences worker incentives: Higher \(a\) leads to lower effort \(e\).
  • Guides policy interventions aimed at reducing unemployment.

Firm’s Profit Function

\[ \pi(t) = F\bigl(\bar{e}\,L(t)\bigr) - w(t)\bigl[L(t) + S(t)\bigr] \quad \text{(Equation $\ref{eq:firm_profit}$)} \]

where \(F'(\bullet) > 0\) and \(F''(\bullet) < 0\).

• \(F(\bar{e}\,L(t))\): Total output (with \(\bar{e}\) as the effort level and \(L(t)\) as the number of workers exerting effort).
• \(w(t)\bigl[L(t) + S(t)\bigr]\): Total wage cost (all employed workers are paid).

Firm’s Objective: - Maximize instantaneous profits \(\pi(t)\) by choosing: - A wage \(w\) high enough to deter shirking. - The employment level \(L(t)\) at the profit-maximizing point. - Higher wages reduce shirking (i.e. \(S(t) \to 0\)) but increase wage costs.

Final Assumption and Full Employment

\[ \bar{e}\,F'\!\left(\frac{\bar{e}\,\overline{L}}{N}\right) > \bar{e}, \quad \text{or} \quad F'(\bar{e}L/N) > 1. \]

• Here, \(\bar{e}\,F'(x)\) is the marginal product of labor times effort, and \(\bar{e}\) is the cost of exerting effort.
• Condition: The marginal product exceeds the cost of effort, ensuring full employment.

Economic Insight: - Without shirking, firms hire until the marginal product equals the cost of hiring. - Imperfect monitoring forces firms to pay efficiency wages to deter shirking, creating equilibrium unemployment. - Higher wages reduce shirking but result in fewer hires, thereby causing involuntary unemployment.

Intuitive Explanation: - The marginal product of labor exceeds the effort cost, which would ensure full employment without monitoring issues. - However, because of imperfect monitoring, firms must pay a premium, leading to unemployment.

Dynamic Programming

Key Idea:
Use value functions to summarize the future in dynamic programming.

State Values: - Let \(V_i\) denote the value of being in state \(i\). - States include: - \(E\): Employment - \(U\): Unemployment - \(S\): Shirking - \(V_i\) represents the expected discounted lifetime utility from the present onward for a worker in state \(i\).

Why Constant? - Transitions among states follow Poisson processes. - In steady state, \(V_i\) is independent of the time a worker has been in a state.

Deriving \(V_E\) – Mathematically (1/3)

• In continuous time, time is divided into infinitesimally small intervals of length \(\Delta t\).
• The Shapiro-Stiglitz model provides a framework to:
  • Evaluate trade-offs between wages, effort, and unemployment risk.
  • Predict worker behavior (e.g., shirking).
• Let \(V_E(\Delta t)\) and \(V_U(\Delta t)\) denote the value of being employed and unemployed at the beginning of an interval.
• Two components:
  • Flow utility: Utility from earning wages during \(\Delta t\): \(w - \epsilon\).
  • Future utility: After \(\Delta t\), accounting for:
    • The probability of remaining employed, \(e^{-b\Delta t}\).
    • The probability of becoming unemployed, \(1 - e^{-b\Delta t}\).

Deriving \(V_E\) – Mathematically (3/3)

If we compute the integral in Equation \(\ref{eq:Value_E}\), we get:

\[ V_E(\Delta t) = \frac{1}{\rho + b} \Bigl(1 - e^{-(\rho + b)\Delta t}\Bigr)(w - \epsilon) + e^{-\rho \Delta t} \Bigl[e^{-b \Delta t} V_E(\Delta t) + \Bigl(1 - e^{-b \Delta t}\Bigr)V_U(\Delta t)\Bigr] \]
(Equation \(\ref{eq:Value_E\_11.25}\))

Solving for \(V_E(\Delta t)\) then gives:

\[ V_E(\Delta t) = \frac{1}{\rho + b}(w - \epsilon) + \frac{1}{1 - e^{-(\rho + b)\Delta t}} e^{-\rho \Delta t} \Bigl[e^{-b \Delta t} V_E(\Delta t) + \Bigl(1 - e^{-b \Delta t}\Bigr)V_U(\Delta t)\Bigr]. \]
(Equation \(\ref{eq:Value_E\_11.26}\))

Taking the limit as \(\Delta t \to 0\) yields:

\[ V_E = \frac{1}{\rho + b}(w - \epsilon) + \frac{b}{\rho + b}V_U. \]
(Equation \(\ref{eq:Value_E\_11.27}\))

Deriving \(V_E\) – Intuitively (1/3)

Think of employment as an “asset” that pays a stream of dividends over time: - Dividends: While employed, the worker earns a utility per unit time given by \(w - \bar{e}\). - When unemployed, no dividends are received. - \(V_E\) is the fair price of such an asset, representing the expected present value of all future dividends (discounted at rate \(\rho\)). - The expected return is \(\rho V_E\) per unit time. - Additionally, there is a probability \(b\) per unit time of a “capital loss” \((V_E - V_U)\) if the worker becomes unemployed.

Thus, we have: \[ \rho V_E = (w - \bar{e}) - b(V_E - V_U) \]
which is equivalent to Equation \(\ref{eq:Value_E\_11.27}\).

Deriving \(V_E\) – Intuition Summary (2/3)

Key Insights:

• Employment is analogous to an asset that provides:

  • Dividends: $ (w - ) $
  • Capital Loss: $ b(V_E - V_U) $
  • Required Return: $ V_E $

• Balancing these gives:

  \[ \rho V_E = (w - \epsilon) - b(V_E - V_U) \]

• Rearranging yields:

  \[ V_E = \frac{1}{\rho + b}(w - \epsilon) + \frac{b}{\rho + b}V_U. \]

Deriving \(V_S\) and \(V_U\) – Intuitively (3/3)

• For a worker who is shirking, the “dividend” is \(w\) per unit time, but the rate of job loss is higher at \(b+q\):

  \[ \rho V_S = w - (b+q)(V_S - V_U). \] (Equation \(\ref{eq:V_S}\))

• For an unemployed worker (receiving no dividend) but who gets a job at rate \(a\):

  \[ \rho V_U = a(V_E - V_U). \] (Equation \(\ref{eq:V_U}\))

• We assume that if an unemployed worker finds a job, they exert effort (as in equilibrium).

Firms’ Problem

• A representative firm’s profit per unit time is given by:

  \[ \pi(t) = F\bigl(\bar{e}L(t)\bigr) - w(t)\bigl[L(t) + S(t)\bigr], \quad F'(\bullet)>0,\quad F''(\bullet)<0. \] (Equation \(\ref{eq:firm_profit}\))

• Here, \(L\) and \(S\) are the numbers of workers exerting effort and shirking, respectively.

The firm’s challenge: - It must pay enough so that \(V_E \ge V_S\); otherwise, workers will prefer shirking. - The firm optimally chooses \(w\) so that the incentive constraint is just met, i.e. \(V_E = V_S\).

• By setting \(V_S = V_E\) in the equation for \(V_S\) and subtracting the equation for \(V_E\), one obtains:

  \[ V_E - V_U = \frac{\bar{e}}{q} > 0. \] (Equation \(\ref{eq:incentive_condition}\))

• This implies that workers must strictly prefer employment to unemployment; hence, firms pay a premium over the cost of effort \(\bar{e}\).

Wage Level that Induces Effort

• Subtract \(\rho V_U\) (from Equation \(\ref{eq:V_U}\)) from \(\rho V_E\) (from Equation \(\ref{eq:Value_E\_11.27}\)) to obtain:

  \[ \rho(V_E - V_U) = (w - \bar{e}) - (a+b)(V_E - V_U). \]

• Substitute \(V_E - V_U = \frac{\bar{e}}{q}\) (from the incentive condition) and solve for \(w\).

Wage Level that Induces Effort (Result)

\[ w = \bar{e} + (a+b+\rho)\frac{\bar{e}}{q}. \]
(Equation \(\ref{eq:effort_inducing_wage}\))

• This wage level:
  • Exceeds the cost of effort \(\bar{e}\) by a positive amount.
  • Increases with the cost of effort \(\bar{e}\), the ease of finding jobs \(a\), the job separation rate \(b\), and the discount rate \(\rho\).
  • Decreases with the shirking detection rate \(q\).
• Firms will pay this wage to deter shirking in equilibrium.

The Aggregate No-Shirking Condition (NSC)

• In steady state, the number of unemployed is constant; hence, flows into and out of unemployment balance.

  • \(NLb\) workers become unemployed per unit time (with \(L\) as employment per firm and \(NL\) as aggregate employment).
  • \(a(\bar{L}-NL)\) unemployed workers find jobs per unit time.

• Equating flows yields:

  \[ a = \frac{NLb}{\bar{L} - NL}. \] (Equation \(\ref{eq:equilibrium_ax}\))

The No-Shirking Condition (NSC) – Reminder

• Substitute Equation \(\ref{eq:equilibrium_ax}\) into the wage-inducing condition to get the NSC:

  \[ w = \bar{e} + \left( \rho + \frac{\bar{L}}{\bar{L} - NL} \, b \right) \frac{\bar{e}}{q}. \] (Equation \(\ref{eq:nsc}\))

• The no-shirking wage is an increasing function of aggregate employment:

  • As \(NL\) increases, unemployed workers find jobs more easily, so the cost of being fired falls and the wage must rise to prevent shirking.

Equilibrium – (1/2)

Equilibrium – (2/2)

The Effect of a Rise in \(q\)

With Turnover: \(b=0\)

Search and Matching

Search and Matching

Terminology

• \(N\) is the working-age population.
• \(\mathcal{Q}\) is the labour force (employed + unemployed).
• \(U\) is the number of unemployed.

\[ \text{Unemployment Rate} = \frac{U}{\mathcal{Q}} \]

\[ \text{Participation Rate} = \frac{\mathcal{Q}}{N} \]

• The employment/population ratio:

\[ \frac{\text{Employment}}{\text{Population}} = \frac{Q-U}{N} \]

Search and matching models: an overview

• A modern approach to model labour markets.
• Starting point: Workers and jobs are highly heterogeneous.
• Therefore, one should not think of the labour market as a single market with traditional supply and demand.

\(\Rightarrow\) Instead, matching of workers and jobs occurs through a complex process of search and matching.

Search and matching models: an overview (continued)

• Workers and firms meet one-on-one and engage in a costly process to match idiosyncratic preferences, skills, and needs.
• Wages are determined by bargaining between workers and firms, rather than by a Walrasian auctioneer.
• As the process is not instantaneous, some unemployment is inevitable.

Search and matching models are relatively complicated, so we will focus only on the basic framework and main issues.

Setup

• Time is continuous. The economy consists of workers and firms/jobs.
• There is a continuum of workers of mass 1.
• Agents are risk-neutral with discount rate \(r > 0\).
• Each worker can be either employed (\(E\)) or unemployed (\(U\)):
  • An employed worker produces \(y\) per unit time.
  • An unemployed worker receives a benefit \(b\) per unit time.

Setup (Jobs)

• A job can be either filled (\(F\)) or vacant (\(V\)):
  • A filled job generates \(y\) and pays the worker wage \(w_t\) per unit time.
  • If a job is vacant, there is neither output nor labour cost.
  • Both filled and vacant jobs incur a maintenance cost \(c\) per unit time.
  • Assume \(y > b + c\), so filled jobs generate positive value.
• Free entry: Jobs can be created freely, but incur cost \(c\) once created.
• Absent search frictions, there would be full employment:
  • $ $ Exactly a unit mass of filled jobs and no vacant jobs.

Search frictions and matching

• The central feature of the model is search frictions.
  • $ $ Unemployed workers and vacant jobs cannot find each other costlessly.
• Stocks of unemployed workers \(U_t\) and vacancies \(V_t\) yield a flow of matches via the matching function:

\[ M_t = M(U_t, V_t), \quad M_U > 0,\quad M_V > 0. \]

• This function proxies recruitment, worker search, and evaluation.
• We assume that all matches lead to hirings.
  • Filling a job creates a positive surplus, divided between the worker and the firm according to their relative bargaining power.
• Positive turnover: There is an exogenous separation rate \(\lambda\) at which jobs end (similar to the Shapiro-Stiglitz model).

Job finding and vacancy filling rates

• Assuming the matching function \(M\) has constant returns to scale (CRS), we can write:

\[ M(U_t,V_t) = U_t\, M\left(1,\frac{V_t}{U_t}\right) = U_t\, m(\theta_t), \]

where \(\theta \equiv \frac{V_t}{U_t}\) is the labour market tightness and \(m(\theta) \equiv M(1,\theta)\).

• The job finding rate (probability per unit time that an unemployed worker finds a job) is:

\[ a_t = \frac{M(U_t,V_t)}{U_t} = m(\theta_t) \quad \text{(increasing in $\theta$)}. \]

• Similarly, the vacancy-filling rate is:

\[ \alpha_t = \frac{M(U_t,V_t)}{V_t} = \frac{m(\theta_t)}{\theta_t} \quad \text{(decreasing in $\theta$)}. \]

The Beveridge Curve

• In steady state, flows into and out of unemployment balance:
  • \(\lambda (1-U)\) workers become unemployed per unit time.
  • \(a\, U\) workers find employment.

Thus,

\[ \lambda (1-U) = a\, U. \]

• With \(a = m(\theta)\) and \(\theta = \frac{V}{U}\), rearranging yields:

\[ U = \frac{\lambda}{\lambda + m(V/U)}. \]

• This equation defines the Beveridge Curve (or \(UV\) curve), showing a negative relationship between unemployment and vacancies.

The Beveridge Curve (with columns)

The Beveridge Curve(s) in the US in recent years

Dynamic Programming for \(V_E(t)\)

• Let \(V_E(t)\) denote the value of being employed at time \(t\) — the expected lifetime utility from \(t\) onward, discounted to \(t\).

Intuitively: - The return from employment includes: - Dividend: \(w(t)\). - Capital gain: \(\dot{V}_E(t)\). - Capital loss: \(-\lambda\bigl(V_E(t)-V_U(t)\bigr)\).

• \(r\) is the discount rate and \(\lambda\) is the job separation rate.

Mathematically:

\[ rV_E(t) = w(t) + \dot{V}_E(t) - \lambda\bigl(V_E(t)-V_U(t)\bigr). \]

In Steady State:

\[ rV_E = w - \lambda (V_E - V_U). \]

Key Intuitions from \(rV_E = w - \lambda (V_E - V_U)\)

\[ rV_E = w - \lambda (V_E - V_U) \]

• Economic Insights:
  • Employment provides utility through wages and potential capital gains.
  • The risk of unemployment creates a capital loss of \(-\lambda(V_E - V_U)\).
• Dynamic Adjustment:
  • Higher wages or improved job security (lower \(\lambda\)) increase \(V_E\).
  • A higher separation rate (\(\lambda\)) decreases \(V_E\).
• Policy Implications:
  • Policies reducing separation rates increase the value of employment.
  • Wage subsidies enhance the lifetime utility from employment.

Equilibrium Conditions (1/2)

Value Functions for Workers: - For \(V_E(t)\):

\[ rV_E(t) = w(t) + \dot{V}_E(t) - \lambda\bigl(V_E(t)-V_U(t)\bigr). \]

• For \(V_U(t)\):

\[ rV_U(t) = b + \dot{V}_U(t) + \alpha(t)\bigl(V_E(t)-V_U(t)\bigr). \]

Value Functions for Firms: - For \(V_F(t)\):

\[ rV_F(t) = \bigl[y - w(t) - c\bigr] + \dot{V}_F(t) - \lambda\bigl(V_F(t)-V_V(t)\bigr). \]

• For \(V_V(t)\):

\[ rV_V(t) = -c + \dot{V}_V(t) + \alpha(t)\bigl(V_F(t)-V_V(t)\bigr). \]

Equilibrium Conditions – In Steady State (2/2)

For workers:

• For \(V_E\):

\[ rV_E = w - \lambda (V_E - V_U). \]

• For \(V_U\):

\[ rV_U = b + a(V_E - V_U). \]

For firms:

• For \(V_F\):

\[ rV_F = (y - w - c) - \lambda (V_F - V_V). \]

• For \(V_V\):

\[ rV_V = -c + \alpha (V_F - V_V). \]

Wage determination: Nash bargaining

• The wage is determined by Nash bargaining:
  • A fraction \(\phi\) of the total surplus goes to the worker and \(1-\phi\) to the firm, where \(\phi \in (0,1)\).
• Subtracting the equation for \(V_U\) from that for \(V_E\) gives:

\[ V_E - V_U = \frac{w - b}{a + \lambda + r}. \]

• Similarly, the firm’s surplus is:

\[ V_F - V_V = \frac{y - w}{\alpha + \lambda + r}. \]

• Nash bargaining implies:

\[ V_E - V_U = \phi \Bigl[(V_E - V_U) + (V_F - V_V)\Bigr]. \]

Wage determination: Nash bargaining (cont.)

• Combining these equations and solving for \(w\) yields:

\[ w = b + \frac{(a+\lambda+r)\,\phi}{\phi\,a + (1-\phi)\,\alpha + \lambda + r} (y-b). \]

• When vacancies and unemployment are equal (\(\theta = 1\), so \(a = \alpha\)), this simplifies to:

\[ w = b + \phi(y-b). \]

• That is, a fraction \(\phi\) of the surplus \(y-b\) goes to the worker.
• If \(a\) exceeds \(\alpha\), workers’ bargaining power improves and \(w\) increases; conversely, \(w\) decreases when \(\alpha\) exceeds \(a\).

Free entry and the value of a vacancy

• Because entry is free, in equilibrium the value of a vacancy must be zero (\(V_V = 0\)); otherwise, firms would adjust vacancy posting.
• Substituting for \(V_F - V_V\) in the firm’s value function gives:

\[ rV_V = -c + \alpha \frac{y-w}{\alpha+\lambda+r}. \]

• Replacing \(w\) from the Nash bargaining solution and simplifying, we obtain:

\[ rV_V = -c + \frac{(1-\phi)(y-b)\,\alpha}{\phi a + (1-\phi)\,\alpha + \lambda + r} = 0. \]

Vacancies supply

• Recall:
  • \(a = m(\theta)\)
  • \(\alpha = \frac{m(\theta)}{\theta}\), where \(\theta \equiv \frac{V}{U}\).
• Substituting these into the free entry condition and rearranging, we obtain the equilibrium condition for vacancies supply:

\[ \boxed{ \theta \Bigl( \phi + \frac{\lambda + r}{m(\theta)} \Bigr) = \frac{1-\phi}{c}(y-b-c) } \]

(Equation \(\ref{eq:VS}\))

• This condition uniquely determines the labour market tightness \(\theta\).
• In the \((U,V)\) space, it appears as a straight line with slope \(\theta\), known as the Vacancies Supply (VS) or Job Creation Curve.

The Job Creation Curve

Equilibrium vacancies and unemployment

Welfare

• Firms’ entry decisions have externalities for workers and other firms: 1. Entry makes it easier for unemployed workers to find jobs, and improves their bargaining position when they do. 2. It also makes it harder for other firms to find workers, and also worsens their bargaining position when they do.\[7pt]

• As a result, the decentralised equilibrium is generally . I.e., social welfare \(Ey + (1-E)b-(E+V)c\) is not necessarily maximized.

• However, depending on which effect dominates, equilibrium employment can be either inefficiently high, or inefficiently low.

• Determining which of these cases is correct is an important policy-relevant question.

Comparative statics: a shift in labour demand

• We are interested in how matching frictions affect the cyclical behaviour of the labour market.

• I.e., does a shift in labour demand have a larger effect on employment and smaller effect on the wage compared to the Walrasian case?\[7pt]

• Shortcut: model cyclical change as a shift in \(y\) in the steady state.

• An increase in \(y\) to \(y'\) does not affect the Beveridge Curve.

• However, the vacancies supply condition \(\eqref{eq:VS}\) implies that the equilibrium labour market tightness increases from \(\theta\) to \(\theta'\).

• The Vacancies Supply curve thus rotates up.

• The new equilibrium is at point E’ with higher level of vacancies and lower unemployment.

• Alas, the model , however.

• When \(\theta\) increases, \(a\) rises and \(\alpha\) falls.

• From equation \(\eqref{eq:wage}\), wage thus increases substantially with \(y\).

• Large increase in the wage reduces incentives to create new vacancies.

• Thus employment effects from shifts in demand are typically small.

A shift in labour demand: illustration

Where are we?

• Successfully solved a modern search and matching model. \[7pt]
• Explains normal unemployment as the result of continually matching workers and jobs in the complex real world.
• This is known as .
• Evidence indeed points to a considerable role for matching frictions.
• Yet even if unemployment to result from search and matching frictions, it may have deeper underlying causes.
• E.g. Difficulty of finding parking in NY may look like searching for a spot, yet ultimately the search is long because there are more cars than spots.
• Evidence suggests that significant part of unemployment isn’t frictional.
• Thus considerations like efficiency wages likely play an important role.
• The basic model also does not generate substantial wage rigidity to explain the cyclical behaviour of the labour market.
• For these reasons, introducing wage rigidity into search and matching models (e.g. via efficiency wages) is an important research agenda.