Business cycles, unemployment insurance, and the calibration of matching models

TL;DR: The authors theoretically and empirically document a puzzle that arises when an RBC economy with a job matching function is used to model unemployment, and show that either sticky wages or match-specific productivity shocks can improve the model's performance by making the firm's flow of surplus more procyclical.

About: This article is published in Journal of Economic Dynamics and Control.The article was published on 2008-04-01 and is currently open access. It has received 412 citations till now. The article focuses on the topics: Unemployment & Efficiency wage.

Summary

2 The model

• The authors general model is a version of the standard RBCM model, as spelled out in Pissarides (2000) and elsewhere.
• The authors simplify by ignoring physical capital; including it would be likely to reinforce their “puzzle”, since capital can more easily adjust to long term policy changes than to short term business cycle fluctuations.

2.1 Values and surpluses

• The authors consider a labor productivity process y that allows the output of a match to depend on its vintage: y(z, Z) = 1 + αZZ + ζ(1− αZ)z (1) In the usual RBC specification (αZ = 1), aggregate productivity fluctuates because technology shocks immediately affect all matches.
• The authors also simplify by ignoring two other generalizations that are unlikely to resolve the dilemma at hand.
• Without mentioning unemployment, the authors can write transition probabilities in terms of labor market tightness, which in turn depends on productivity.
• To save on notation, the authors immediately impose these restrictions by writing the value and policy functions in terms of their appropriate state variables.

2.2 The labor market

• Matching probabilities are thus functions of tightness and search: p(S, θ) = M U = γθ1−λS (10) and pF (S, θ) = M V = p(S, θ) θ (11) Equ. (10) implicitly provides a metric for search effort, saying that the individual probability of finding a job is proportional to search.
• The authors then distinguish between the fraction of the labor force in matches with low productivity, NLOt , and the fraction matched with high productivity, N HI t .

3 Unemployment volatility: cycles and policies

• For this special case, the authors can characterize the dynamics explicitly, and calculate how the cyclical variability of labor market aggregates relates to their response to UI policy.
• Thus when search is exogenous, (17) and (18) suffice to determine total surplus Σt and tightness θt.
• (Now tildes signify log deviations from steady state, and unadorned variables are steady state values or constants.).
• Intuitively, the model says that a permanent increase in UI or taxes should have exactly the same effect on the surplus process, and therefore on hiring, as a permanent decrease in productivity by the same amount.
• The relative standard deviation of log unemployment to log output, and the long-run semielasticity of unemployment with respect to the replacement ratio ξ, have the following ratio: σU/σQ ²Uξ = (σy/σQ)(σU/σy).

4 Empirical evidence

• The authors have shown that the RBCM framework implies a tight relationship between cyclical and policy-related variation in unemployment and other labor market variables.
• Next, the authors briefly discuss labor market fluctuations (which have been extensively reviewed elsewhere recently), and then explore the effects of labor market policies in greater detail.

4.1 Unemployment over the business cycle

• For evidence on cyclical fluctuations, the authors consider US data from 1951:1 to 2006:2 from the St. Louis Fed’s FRED database, either using quarterly series, or monthly series aggregated to quarterly frequency.
• All series discussed below are seasonally adjusted, logged, and detrended with the HP filter, unless otherwise specified.
• Another striking labor market fact is the robust negative correlation between the cyclical components of log unemployment and log vacancies, -0.884 in their data.
• Given this correlation, the tightness ratio θ = V/U is even more volatile than the two series separately: σθ = 0.3736.
• This robust finding has been discussed in many other studies; see for example Merz (1995), Cole and Rogerson (1999), and Greenwood, Gomes, and Rebelo (2001) for similar second moments.

4.2 Literature on labor market policy and unemployment

• The authors paper’s implications for labor market policy effects are general equilibrium predictions, like its implications for business cycles.
• Since large policy changes are relatively rare, this strategy typically involves case studies of major reforms that act as “natural experiments”.9.
• Thus, by process of elimination, the authors believe that the best evidence on these issues comes from international cross-sectional or panel-data studies.
• Some recent studies are more ambitious, addressing higher-frequency data and attempting to identify interaction terms.
• While these papers’ estimates vary widely, the important point for their purposes is that none of them find substantially larger effects of unemployment benefits than those the authors estimate.

4.3 Possible problems with cross-country estimates

• Cross-country regressions to measure the impact of policy are frequently criticized,12 and concern is justifiable on at least three grounds.
• First, the number of countries and periods is inevitably rather small, and data on institutions and policy variables may 10LN99 regress log unemployment on the replacement rate and other labor market policies and controls for 20 OECD countries.
• The issues of data inconsistency between countries, and possibly omitted time series or cross-sectional regressors, make it essential to check how results change when the authors control for time effects and country effects.
• The authors use a much longer time sample than LN99, and they compare regressions for two different data sets.
• Moreover, since budgetary pressures tend to push benefits and taxes in opposite directions, the coefficient on their sum τ + b is less likely to suffer endogeneity bias than those on taxes and benefits separately.

4.4 Impact of benefits and taxes on unemployment

• The authors now run a variety of cross-country regressions to estimate the effect of the UI replacement ratio and the tax rate on log unemployment.
• Third, the authors test and then impose their model’s restriction that the coefficients on taxes and benefits should be equal, thus improving the stability of their estimates.
• Nonetheless, their reading of Table 1 and also Table 2, which reports the coefficient on τ + b for many alternative specifications, is that there is strong cross-sectional and time series evidence for a semielasticity of approximately two.
• In summary, their estimate of the semielasticity of unemployment with respect to benefits is somewhat larger than LN99 found: two instead of 1.3.
• Third, LN99 ran a GLS regression with random country effects and fixed time effects, which in their estimates yields a lower coefficient.

5 Variations on the standard model

• Tinkering with parameters will not help, since the upper bound in Prop. 2 is independent of calibration.
• Some generalization of the model might fit better, so the authors turn next to numerical simulations of the general model from Sec. 2.

5.1 Benchmark parameters

• The authors benchmark numerical calibration is as follows.
• Vacancy duration is just a normalization: doubling it would mean doubling vacancies, reducing κ by half, and adjusting γ to keep total matches, total vacancy costs, and job finding probabilities unchanged.
• As the authors mentioned above, a higher b can increase unemployment variability, by making the surplus smaller and proportionally more volatile.
• The unemployment semielasticity ²Uξ falls to 0.82, and the cyclical volatility of unemployment falls to σU/σQ = 0.62.

5.2 Variable separation and variable search

• Davis, Haltiwanger, and Schuh (1996) argue that job destruction is strongly countercyclical.
• Therefore, the authors need to ask whether variation in separation rates might change their results.
• The usual model of variable separation (Mortensen and Pissarides 1994) posits a match-specific productivity shock, so that workers and firms separate when their joint surplus becomes negative.
• Second, although unemployment becomes more variable, the probability of job finding now varies less: the ratio σp/σQ falls from 1.61 in the numerical benchmark to 1.40 with variable separation.
• 1 and 2 indicated, endogenous search only makes the tradeoff worse.

5.3 Finite UI benefit duration

• Another issue that might matter for their results is their assumption that UI benefits continue as long as unemployment lasts.
• Here, for the first time, the authors must distinguish the actual UI benefit b−b0 (which expires at rate φ) from the disutility of working b0.
• The results are similar to those of the numerical benchmark: benefits have a reasonable effect on unemployment, but the cyclical variability of unemployment is much too small, so the key ratio (σU/σQ)/².
• Otherwise there would be a fourth labor market state, employed without benefits, with a lower outside option and thus lower wages.
• Thus considering finite benefit duration reinforces their claim that the standard RBCM framework understates cyclical volatility relative to the effects of policies.

5.4 Sticky wages

• The authors have seen that higher b means higher percentage variation in the firm’s surplus over the cycle, increasing the variability of hiring and unemployment.
• The authors assume that workers’ bargaining power varies negatively with the technology shock, so that workers get a larger share of surplus in recessions.
• This raises σU/σQ to 5.67, roughly consistent with the data.
• The authors own duration regressor is the fraction of benefits remaining after the first year, which is harder to interpret in terms of (30)-(31).
• Imposing a constant surplus for the worker makes hiring incentives fall sharply with the replacement ratio, so that their efficiency wage model fits less well than their ad hoc sticky wage model, in which wages adjust flexibly to long run changes in UI.

5.5 Cohort-specific technology shocks

• Finally, the authors show that a form of embodied technological change could also help solve the puzzle that concerns us.
• Also, since employment now varies more relative to output, and high and low productivity matches coexist, the authors now find that aggregate productivity varies less relative to output than it did with disembodied productivity: σy/σQ falls from 0.92 in the model of line 1 to 0.54 in the cohort-specific benchmark of line 15.
• This also improves the model’s fit, though it goes somewhat too far, overshooting the ratio σy/σQ = 0.65 the authors calculate from the FRED data.
• A potential problem with the embodied technology specification is that wages become much more volatile: σw/σQ more than triples from its benchmark value in line 1, which is already too high.
• The authors should emphasize here that matching models do not actually tie down the wage process.

6 Matching in business cycle models with capital

• The authors have argued that their model’s lack of physical capital is probably inessential for their results.
• But to be sure, the authors finish by reexamining the models of Merz (1995) and Andolfatto (1996), which include capital.
• While these papers reported some success in modeling labor market fluctuations, when the authors recalculate their steady states to measure the effects of UI benefits, they find that they suffer from the same problem as their benchmark model: insufficient cyclical volatility compared with the impact of policy.

6.3 Other models with capital

• Den Haan et. al. (2000) study an RBCM model with endogenous separations.
• This is consistent with their finding that variable separation can make matching volatile.
• Their calculations suggest that their model will fail to generate a Beveridge curve.
• 29 Gomes et. al. (2001) simulate a business cycle model in which individuals search for jobs.
• They state that raising the replacement ratio from 0.5 to 0.7 increases unemployment from 6.1% to 13.9%, which is a semielasticity of 6.49, exceeding their estimate by a factor of three.

7 Conclusions

• A model of real business cycles and matching implies that job creation depends on the surplus available to the matched pair.
• The authors findings suggest that modeling labor market fluctuations by calibrating a very small match surplus, as Hagedorn and Manovskii (2006) advocate, is unhelpful because it is inconsistent with robust observations about the effects of labor market policy.
• This equation, together with the formula (37) for x, and the formula (23) for the steady state comparative statics, gives us Proposition 2.
• The LMIDB database is constructed from OECD data on institutional and labor market characteristics of 20 countries for 1960-94.
• This variable, available in the BGHS data but not the LMIDB, represents the fraction of GDP, per unemployed worker, spent by the government on job training and job matching.