2.1 Specification

2.1.4 Unemployment

Separation between workers and jobs can occur either because of low quality of the match with probability q, or because the (filled) job is destroyed with probability d. After a separation occurs we assume that the worker stays in the labor force. We do not model explicitly job-to-job movement of workers and therefore q should be thought of as the quitting rate plus iring rates that result in unemployment. We also assume that workers who move into the labor force first become unemployed before they find a job. Finally, unemployment falls one-to-one with new hires. We therefore have:

$\dot{U}=-H+(q+d)E+\dot{L}$

Substituting for E, $\dot{L}$ and H using (1), (4), and (5), we get:

$\begin{array}{cc}\dot{U}=-(1-a)\alpha m(U,V)+(1-a)(q+d)(L-U)& \left(7\right)\end{array}$

Equations (3), (6), and (7) provide a system in unemployment, U, vacancies, V, and the labor force, L, that characterizes their evolution in equilibrium. Figure 2 summarizes the lows in and out the different states in the labor market.

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The Flow of Workers and Jobs

Citation: IMF Working Papers 2015, 058; 10.5089/9781484360569.001.A001

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2.2 Steady State

In steady state $\dot{V}=\dot{U}=0$, therefore by (6) and (7):

and from (3):

Substituting for L_ss and then for V_ss gives:

Equation (9) solves for U_ss, and given the solution for U_ss, we can recover V_ss and L_ss.

Notice that since m (U, V) is increasing in both arguments the left-hand-side (LHS) of equation (9) is an increasing function of U_ss. The right-hand-side (RHS) is falling with U_ss. For U_ss = 0 the LHS equals zero while the RHS is positive. For U_ss = Φ the LHS is positive while the RHS equals zero. Therefore, by continuity and monotonicity of both sides, a unique steady state exists.

2.3 Dynamics

We now turn to characterizing the dynamics of the system using a phase diagram in the U − V space.

First note that unemployment is bounded above by Φ: if E = 0 then L = U = Φ, and in this case a rise in employment by one worker has a net effect of reducing unemployment by 1 − a workers; this is because the employed person has a direct effect of reducing unemployment by 1 and raising the labor force by a causing unemployment to fall by 1 − a. We therefore conclude that U cannot exceed Φ.

Substituting for L using equation (3) into (6) and (7) results in a dynamic system in unemployment and vacancies:

Equating the left hand sides of both equations to zero characterizes the loci $\dot{U}=0$ and $\dot{V}=0$. Notice that:

Which suggests:

That is, both loci $\dot{U}=0$ and $\dot{V}=0$ are downward sloping, and the locus $\dot{U}=0$ is steeper than the locus $\dot{V}=0$ at the point of intersection, i.e. at the steady state, as depicted in Figure 3.

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Phase Diagram of Unemployment and Vacancies

Citation: IMF Working Papers 2015, 058; 10.5089/9781484360569.001.A001

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Suppose we are at a point somewhere on the locus $\dot{U}=0$ and V rises. A rise in V creates more matches and therefore raises the outflow from unemployment making $\dot{U}$ negative. Algebraically for a given level of U a rise in V reduces the right hand side of (10). Therefore, above the locus $\dot{U}=0$ unemployment falls and similarly below it unemployment rises.

Now suppose we are at a point on the locus $\dot{V}=0$ and U rises. A rise in U affects vacancies through two channels: first, more unemployed workers create more matches and therefore the outflow from the pool of vacant jobs to that of filled jobs rises, making $\dot{V}$ negative. Second higher unemployment implies a lower level of employment which reduces the inflow of jobs to the vacant state due to separation, this also makes $\dot{V}$ negative. Algebraically for a given level of V a rise in U reduces the right hand side of (11). Therefore, at points located to the right of the locus $\dot{V}=0$ vacancies fall, and at points located to its left vacancies rise.

These dynamic effects are represented by the arrows plotted in Figure 3.

2.4 The Effect of Matching Efficiency (α and q)

Improvements in matching efficiency are captured by a rise in matching technology, α, and/or a fall in separation rate due to quitting and firing, q. Suppose the economy starts from steady state and is suddenly hit by a once and for all positive matching efficiency shock. Higher matching efficiency implies that, for a given level of U and V, either more matches are created, increasing the outflow from the states of unemployment and vacancy, and/or the inflow into these states falls due to fewer break-ups of matches. Both effects imply that U and V fall.

More formally, from equations (10) and (11) a rise in α and/or a fall in q shift both $\dot{U}=0$ and $\dot{V}=0$ loci inward, suggesting that U and V fall together as the old steady state is now located at the NE quadrant of the phase diagram.

To find the exact evolution of U and V over time notice that equation (8) characterizes the relationship between V_ss and U_ss for any value of α and q. From this relation it follows that all steady states lie on a straight line with slope $\frac{1}{1-a}$ (for fixed values of c, d, a and Φ); therefore, both U_ss and V_ss fall.

The dynamics of the system when the economy is located somewhere along the locus define by (8), even if it is out of steady state, is found by evaluating (10) and (11) at a point that satisfies $V=\frac{c}{d}-\frac{1}{1-a}(\mathrm{\Phi }-U)$. In that case we get:

and therefore:

This suggests that the system moves along a straight line (with slope $\frac{1}{1-a}$) towards the new steady state, as depicted in Figure 4.

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The Dynamic Effect of a Rise in Matching Efficiency, Higher α and/or Lower q

Citation: IMF Working Papers 2015, 058; 10.5089/9781484360569.001.A001

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For future reference notice that we have:

which using (3) suggests:

This equation suggests that as matching efficiency improves the rise in employment, ΔL_ss − ΔU_ss, equals the fall in vacancies, −ΔV_ss. We will use the long-run relations between unemployment, vacancies and the labor force as identifying restrictions later in our SVAR estimation.

2.5 The Effect of Aggregate Activity (c and d)

Movements in aggregate activity are captured by changes in job creation and job destruction parameters, c and d, respectively. In our setting we are agnostic about whether these shocks originate in aggregate demand or aggregate supply, all that matters in the model is that firms open jobs and shut them down. This section studies the effect of movements in the supply of jobs on vacancies and unemployment.

Suppose the economy starts from steady state and suddenly the supply of jobs rises through a rise in c. Equation (11) suggests that a higher c implies that more vacancies are created, i.e. $\dot{V}$ becomes positive. Equation (10) suggests that momentarily U is unchanged as it is independent of c, but as V increases more matches are created making $\dot{U}$ negative, as a result unemployment falls.

Furthermore, these equations suggest that the locus $\dot{U}=0$ is unchanged while the locus $\dot{V}=0$ shifts outward, locating the old steady state on the boundary between the SW and SE quadrants of the phase diagram as depicted in Figure 5, and in the new steady state there are more vacancies and fewer unemployed.

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The Dynamic Effect of A Rise In Job Creation, Higher c - Positive Aggregate Activity Shock

Citation: IMF Working Papers 2015, 058; 10.5089/9781484360569.001.A001

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Now suppose the economy starts from steady state and suddenly the supply of jobs falls through a rise in d. Equation (10) suggests that a higher d raises unemployment as more employed workers lose their jobs, making $\dot{U}$ positive. Equation (11) suggests that as the job destruction rate increases more vacancies are shut down which reduces their stock and $\dot{V}$ becomes negative. After the first instant the rise in unemployment increases the stream of matches causing V to fall even further although moderating the rise in U; similarly, the fall in vacancies reduces the stream of new matches causing U to rise even further although moderating the fall in V. Equations (10) and (11) also imply that in response to a rise in d the locus $\dot{U}=0$ shifts outward and the locus $\dot{V}=0$ shifts inward, suggesting that U rises and V falls as the old steady state is now located at the NW quadrant of the phase diagram as depicted in Figure 6.

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The Dynamic Effect of A Rise In Job Destruction, Higher d - Negative Aggregate Activity Shock

Citation: IMF Working Papers 2015, 058; 10.5089/9781484360569.001.A001

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The negative comovement of unemployment and vacancies in response to aggregate activity shocks gives rise to the Beveridge relation in the model.

2.6 The Effect of Labor Supply (Φ)

Changes in the labor supply are captured by movements in Φ; as equation (2) makes clear shifts in Φ alter the labor force for any level of employment. Suppose the economy starts from steady state and suddenly the supply of labor rises through a rise in Φ. On impact, the rise in labor supply raises unemployment one-to-one, as we assume that new entrants to the labor force first become unemployed before moving to employment. As unemployment rises more matches are created which in turn reduces vacancies and offsets the initial jump in unemployment. That is, on impact unemployment overshoots its long-run level and then, along the convergence path, vacancies and unemployment gradually move towards a new steady state with less vacancies and more unemployed relative to the old steady state.

More formally, we start the analysis by establishing the shift in steady state. From (9) we derive:

suggesting that $0<\frac{d{U}_{ss}}{d\mathrm{\Phi }}<1$, and from (8):

and since $0<\frac{d{U}_{ss}}{d\mathrm{\Phi }}<1$ it follows that $\frac{d{V}_{ss}}{d\mathrm{\Phi }}<0$. That is, in the new steady state there are fewer vacancies and more unemployed workers relative to the old one, as suggested earlier.

On impact, the rise in Φ raises unemployment by ΔΦ and leaves vacancies unchanged, moving the system to $(U_{ss}^{old}+\mathrm{\Delta }\mathrm{\Phi },V_{ss}^{old})$.⁹ Also, from (10) and (11) both loci, $\dot{U}=0$ and $\dot{V}=0$, shift outward, and at the point $(U_{ss}^{old}+\mathrm{\Delta }\mathrm{\Phi },V_{ss}^{old})$ both $\dot{U}$ and $\dot{V}$ become negative, which suggests that after the initial jump both U and V gradually fall as the effect of the rise in matches dominates their movement. It turns out that the system converges to the new steady state monotonically along a straight line as depicted in Figure 7.¹⁰

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The Dynamic Effect of A Rise In Labor Supply, Higher Φ

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Notice that although the model gives a clear pattern for the evolution of U and V in response to movements in labor supply it is unclear what is the overall correlation between these variables. On impact U rises while V is unchanged, suggesting no correlation; in the transition period both variables fall, suggesting a positive comovement; and a comparison of the new steady state to the old one suggests a negative comovement.

2.7 Difference from Blanchard and Diamond

As mentioned, our model is heavily based on the work of Blanchard and Diamond (1989). Nevertheless, we deviate from their model in two aspects.

First, and most importantly, we capture fluctuations in aggregate activity through changes in the processes of job creation and job destruction. In our model these movements give rise to the Beveridge curve, i.e. a negative comovement between unemployment and vacancies. In contrast, the BD model combines the probability of an unproductive job to become productive (π₁), i.e. job creation probability, and the probability of a productive job to become unproductive (π₀), i.e. job destruction probability, to generate shocks with two very different outcomes. (1) They consider changes in the potential stock of productive jobs, $\frac{{\pi }_1}{{\pi }_0+{\pi }_1}$ (after normalizing the sum of productive and unproductive jobs to unity), as representing cyclical fluctuations in aggregate activity that generate the Beveridge relation. This works in a similar way to aggregate activity shocks in our model. And (2) they analyze changes in the low of reallocation, in BD terminology, from productive to unproductive jobs, ${\pi }_0\frac{{\pi }_1}{{\pi }_0+{\pi }_1}$. In their model this shock results in a positive comovement between unemployment and vacancies. This is different from our results where fluctuations associated with job creation and job destruction parameters generate the Beveridge relation.

Furthermore, BD analyze movements in aggregate activity keeping reallocation constant, and vise versa. Given that both shocks depend on the probabilities of job creation and job destruction these exercises seem improbable. More specifically, stimulating aggregate activity by raising $\frac{{\pi }_1}{{\pi }_0+{\pi }_1}$ while keeping reallocation, ${\pi }_0\frac{{\pi }_1}{{\pi }_0+{\pi }_1}$, constant implies a negative comovement between π₁ and π₀; however, in considering the reverse exercise, i.e. raising reallocation while keeping aggregate activity constant, a positive comovement is implied. Since the data display negative comovement between job creation and job destruction, we view BD’s first impulse exercise, that is impulsing aggregate activity while keeping reallocation constant, as more relevant.¹¹ Since reallocation shocks result in counterfactual implications, at least in the specification of BD, we chose to focus on matching efficiency fluctuations as the main driver of positive comovement between unemployment and vacancies.

Second, BD chose to model the process of job creation as a rate out of the stock of unproductive jobs, that is the total of potential jobs minus filled jobs and vacancies. Under their specification the locus $\dot{V}=0$ need not be downward sloping throughout the region 0 ≤ U ≤ Φ; assuming lim_U→0 m_U → ∞, as is standard in the literature, the curve may have a positive slope for high enough U under some combinations of parameter values. Nevertheless, all our results go through unchanged under BD’s specification, with only one exception: in response to movements in the job destruction rate, π₀, the system may generate a positive comovement between U and V. This happens only in the special case where the locus $\dot{V}=0$ has a positive slope for high enough levels of unemployment and the steady state happens to be located far enough in that region. In that case a rise in job destruction results in a rise in vacancies, suggesting a counterfactual positive correlation between the series.¹² The reason behind this result is that as jobs are being destroyed the pool of unproductive jobs increases, which in turn creates more new jobs that re-enter the market as vacancies. Our assumption of an exogenous low of job creation shuts down this counterfactual channel, and therefore we feel comfortable with simplifying the BD model along this dimension.

3.1 Empirical Specification

As discussed in the previous section we distinguish between three types of shocks: (1) aggregate activity shocks, denoted by u^aa in the empirical model, (2) matching efficiency shocks, u^me, and (3) labor supply shocks, u^ls. We assume that the vector of structural innovations, u ≡ [u^aa, u^me, u^ls]′, is iid − Normal over time, its elements are contemporaneously uncorrelated, and we normalize the variance of each innovation to unity, that is:

We use insights from the model of the previous section to impose identifying restrictions on the empirical SVAR model:

where y_t is a 3 × 1 vector of data on unemployed workers, business sector vacancies, and the labor force. A₀, A₁,…,A_p and B are 3 × 3 coefficient matrices, and C is a 3 × 1 vector of coefficients. Note that since we assume standard normal structural shocks the standard deviations of the underlying disturbances are embedded in the matrix B.

We use data in logs and since the levels of at least some variables are non-stationary (see section 4.1 below) we estimate the model in log first differences. Specifically we define:

Assuming A₀ is invertible the reduced form representation is given by:

where:

The reduced form representation, equation (15), is estimated with no restrictions, and for the identification of the structural parameters we impose restrictions on the matrices A₀ and B as guided by our theoretical model.

3.2 Short Run Restrictions

First, the diagonal elements of A₀ are normalized to unity. Next, we use the theoretical model to impose identifying restrictions; the model is summarized by equations (3), (6) and (7), which are rewritten here for convenience:

The model suggests that an increase in aggregate activity, i.e. higher c and lower d, reduces unemployment and increases vacancies. The labor force is affected only indirectly through movements in unemployment. We will therefore assume that u^aa affects U and V contemporaneously, and its effect on the labor force is delayed.

The model suggests that an increase in matching efficiency, i.e. higher α and lower q, reduces both unemployment and vacancies. The labor force is affected only indirectly through movements in unemployment. We will therefore assume that u^me affects U and V contemporaneously, and its effect on the labor force is delayed.

The model suggests that a rise in labor supply, i.e. higher Φ, increases the labor force, and that unemployment and vacancies are affected only indirectly through movements in the labor force. Nevertheless, since by definition a rise of the labor force must be immediately reflected in the sum of unemployed and employed workers we will assume that u^ls affects L contemporaneously, that L affects U contemporaneously, and that the effect on vacancies is delayed.

These restrictions imply the following structure of the matrices A₀ and B:

Three comments are in order. First, the matrices A₀ and B affect the likelihood function only through the product $A_0^{-1}B$ which is given by:

This suggests that our specification is observationally equivalent to setting A₀ to the identity matrix and allowing a non-zero entry at the 1, 3 position (first row, third column) of the matrix B.

Second, notice that under our specification the 2, 3 (and 3, 2) elements of the matrix $\begin{array}{c}{A}_0^{-1}B{B}^{’}{A}_0^{-1^{’}}\end{array}$ equals zero; that is, the structural model implies that the reduced form innovations of vacancies and the labor force are uncorrelated, i.e. Ω₂₃ = 0. Since there are no restrictions on the estimation of the model in its reduced form the point estimate of this element is different from zero and we will be able to test whether the data reject this restriction.

Finally, notice that in the current specification there is nothing that distinguishes aggregate activity shocks from matching efficiency shocks, suggesting that the upper-left 2 × 2 block of the matrix B is not identified. Formally, although the order condition for identification is met, the rank condition is not. We complete the identification of the matrix B by imposing long-run restrictions.

3.3 Long-Run Restrictions

To complete the identification we need to impose restrictions that distinguish aggregate activity fluctuations from matching efficiency shocks. To that end we use the long-run relation between the variables after a shock to matching efficiency, as derived in Section 2.4. Specifically, we use equation (13):

and verify that ΔV_ss, ΔU_ss < 0 and ΔL_ss > 0, as suggested by (12) and (13).

The equation above was derived from our stock-flow accounting model and therefore it hinges on the exact assumptions of that model. If, for example, we relax the assumption that the destruction rate of vacant jobs equals that of filled jobs the equality does not hold, although the inequalities are still valid. As a robustness check we will therefore relax the long-run identifying restrictions and use only sign restrictions. That is, we will impose that after an improvement in matching efficiency unemployment and vacancies cannot rise, and the labor force cannot fall.

Using the methodology of Blanchard and Quah (1989), these restrictions imply various constraints on the coefficients of the matrix $A_0^{-1}B$. Let C (1) denote the matrix of cumulative impulses to the reduced form disturbances:

Let D (1) denote the cumulative impulses of the structural disturbances:

The long run effect of a matching efficiency shock is given by the second column of:

where C_ij is the i, j element of the matrix C (1), and we used the result for $A_0^{-1}B$ in (16).

We therefore get that the long-run effects of a matching efficiency shock are given by:

We now specify the exact restriction we used for each case.