Czech labour market through the lens of a search and matching DSGE model
Daniel Němec
Abstract. This contribution reveals some important structural properties of the Czech labour market in the last fifteen years and evaluates possible changes within this period. A small search and matching model incorporated into standard macroeconomic dynamic stochastic general equilibrium model is estimated using Bayesian techniques. The results show that search and matching aspect provides satisfactory description of employment flows in the Czech economy. Model estimates provide convincing evidence that wage bargaining process is determined mainly by the power of the unions and that the institutional changes of the Czech labour market in the last fifteen years had only little real impact on the matching effectiveness.
Keywords: search and matching model, Bayesian estimation, DSGE model, structural changes
JEL classification: C51, E24, J60 AMS classification: 91B40, 91B51
1 Introduction
The goal of my contribution is to reveal some interesting and important structural properties of the Czech labour market in the last fifteen years and to evaluate possible changes within this period. For this purpose, I use a small search and matching model incorporated into standard macroeconomic dynamic stochastic general equilibrium model (DSGE). Search and matching model is an important tool to model labour market dynamics. This model is a log-linear version of the model originally developed by Lubik [4]. Using real macroeconomic data I am able to estimate some key labour market indicators: the wage bargaining power of unions, the match elasticity of unemployed and the efficiency of the matching process.
The structure of my contribution is as follows. The next section provides a short description of the small search and matching DSGE model which is used for my analysis. Section 3 discusses used data, priors and estimation techniques. Section 4 presents the main results. Section 5 provides a deeper insight into model properties and its ability to match observed data. Section 6 concludes this contribution with some ideas regarding the possibilities of further research in this area.
2 The model
As mentioned previously, I use the model developed by Lubik [4]. It is a simple search and matching model incorporated within a standard DSGE framework. The labour market is subject to friction because a time-consuming search process for workers and firms. The wages are determined by the outcome of a bargaining process which serves as a mechanism to redistribute the costs of finding a partner.
Households   Representative household maximizes its expected utility function
^-Masaryk University, Faculty of Economics and Administration, Department od Economics, Lipova 41a, 602 00 Brno, Czech Republic, e-mail: nemecd@econ.muni.cz
i=l
(1)
1
where C is aggregate consumption, n G [0,1] is a fraction of employed household members (determined by the matching labour market), ft G (0,1) is the discount factor and a > 0 is the coefficient of relative risk aversion. Variable xt represents an exogenous stochastic process which may be taken as a labour shock. The budget constraint is defined as
Ct + Tt = wtnt + (l-nt)b + ILt, (2)
where b is unemployment benefit financed by a lump-sum tax, T. Variable Ht are profits from ownership of the firms and w is wage. There is no explicit labour supply because it is an outcome of the matching process. The first-order condition is thus simply
= At, (3) where At is the Lagrange multiplier on the budget constraint.
Labour market The labour market is characterized by search frictions captured by a standard Cobb-Douglas matching function
m(ut,vt) = ntufvl't, (4)
where unemployed job seekers, ut, and vacancies, ft, are matched at rate m(ut, ft)- Parameter 0 < £ < 1 is a match elasticity of the unemployed and /it is stochastic process measuring the efficiency of the matching process. Aggregate probability of filling a vacancy may be defined as
q(6t) = m(ut,vt)/vt, (5)
where 0t — — is a standard indicator of the labour market tightness. The model assumes that it takes one period for new matches to be productive. Moreover, old and new matches are destroyed at a constant separation rate, 0 < p < 1, which corresponds to the inflows into unemployment. Evolution of employment, nt — 1 — ut, is given by
nt = (1 - p) [nt-i + ut-iq(6t-i)] . (6)
Firms As a deviation from the standard search and matching framework, the model assumes monopolistic firms. Demand function of a firm is defined by
Vt= (%)  1 Vt, (7)
where yt is firm's production (and its demand), Yt is aggregate output, pt is price set by the firm, Pt is aggregate price index and e is demand elasticity which will be not treated as a stochastic process in my empirical application. Production function of each firm is
Vt
(8)
where At is an aggregate technology (stochastic) process and 0 < a < 1 introduces curvature in production. Capital is fixed and firm-specific. The firm controls the number of workers, nt, number of posted vacancies, ft, and its optimal price, pt, by maximizing the inter-temporal profit function
i=i
Pi
WjTlj
4
(9)
subject to the employment accumulation equation (7) and production function (8). Profits are evaluated in terms of marginal utility Xj. The costs of vacancy posting is ^vf, where k > 0 and ip > 0. For 0 < ip < 1, posting costs exhibit decreasing returns. For tp > 1, the costs are increasing while vacancy costs are fixed for tp — 1. The first-order conditions are
Vt
ntl + e
nvf^1 = (1 - p)q(0t)Etftt+1Tt+1
- wt + (1 - p)Etftt+1Tt+1,
(10)
(11)
where ftt+i — ft-^- is a stochastic discount factor and rt is the Lagrange multiplier associated with employment constraint. The first condition represents current-period marginal value of a job. The second condition is a link between the cost of vacancy and the expected benefit of a vacancy in terms of the marginal value of a worker (adjusted by the job creation rate, q(0t)).
Wage bargaining Wages are determined as the outcome of a bilateral bargaining process between workers and firms. Both sides of the bargaining maximize the joint surplus from employment relationship:
_ /1 dwt(nt)Y fdjMy-"
At dnt
V dnt
(12)
where n G [0,1] is the bargaining power of workers,
household's welfare and -1 is the marginal value of a worker to the firm. The term — Tt is
d^Ora"'"1 18 *ne marginal value of a worker to the
dJt{nt)
given by the first-order condition (10). Recursive representation for "yVg^'tH> is derived as
dWt(nt) dnt
Using employment equation (6), it holds
AtWt - At 6 -dnt+i
Xt + ßEt
dWtjnt) dnt
dWt+i(nt+i) dnt+i
dnt+i
dnt
(13)
dnt
All real payments are valued at
the marginal utility At. Standard optimality condition for wages may be derived as
(1-7/)
1 dWt(nt) djt{nt)
At    dnt dnt Expression for the bargained wage is given after some algebra as
wt — n
Vt e t—--
nt 1 + e
+ (1-V) [b + XtC°t
(14)
(15)
Closing the model The model assumes, that unemployment benefits, b, are financed by lump-sum taxes, T, where a condition of balanced budget holds, i.e. Tt — (1 — nt)b. Social resource constraint
is thus Ct
4,
Yt. The technology shock At, the labour shock \t and the matching shock p,t are
assumed to be independent AR(1) processes (in logs) with coefficients pi, i G (A,^,n) and innovations ej~iV(0,<7?).
Log-linearised model For estimation purposes, I did not use the non-linear form of the model mentioned in the previous section (of course, this form is important to understand the meaning of the key structural model parameters). Instead of that, I use a log-linear version of the model based on my own derivations.1 In the following equations, the line over a variable means its steady-state value (derived simply from the non-linear equations).2 The variables with a tilde represent the gaps from their steady-states.
At
-oCt
qt=mt- vt ü
nt ~-
mt = fit + £üt + (1 - O^t
1
1 — vi
yt = (-l-e)(pt-Pt) + Yt
nt
n + vq yt — At + aht
ufit-i + qv{vt-\ + qt-i
1
(ip - l)Pt = qt + Et (ßt+i + n+i
Ät + Ät-i 1
ßt
: (yt - nt) - wwt + (l - p)rßEt ßt+i + h+i
w
1
1
— {Vt- nt) + n^-xe ((V> - l)vt + 6t) ) + (1 - n)xC" {Xt + crCt) + en * ' '
CCt + nv^h
"'"Log-linear version is not a part of the original contribution of Lubik [4].
2Initial steady-state values are calibrated as follows: fi* = A* = \* = 1, /3* = 0.99, u* Remaining steady-states are computed using these values and the prior means of all parameters.
= 0.0763, v* = 0.0127.
Ät = pAÄt-i +ef   xt = pxXt-i + e*   fit = PßPt-i + <h   % = pyYt-i + <h
The last equation results from the fact that variable Y is an observed variable. We have thus four shocks (e\ for four observed variables - ü, v,w and Y). The model consists of 17 endogenous variables (variable ipt ~ Pt) is a single variable in my application), four shocks and 14 parameters.
3   Data and priors
The model for the Czech economy is estimated using the quarterly data set covering a sample from 1996Q1 to 2010Q4. The observed variables are real output (GDP, in logs), hourly earnings (in logs), unemployment rate and rate of unfilled job vacancies. All data are seasonally adjusted and de-trended (excluding vacancies) using Hodrick-Prescott filter (with the smoothing parameter A — 1600). The rate of unfilled job vacancies was demeaned prior estimation. The variables used are expressed as corresponding gaps. The original data are from databases of the OECD and the Czech Statistical Office (CZSO).3 In the following parts, I will not use the mark ~ to explicitly express the appropriate gaps.
Description	Parameter	Density	Mean	Std. Dev.
Discount factor	ß	Fixed	0.99	—
Labor elasticity	a	Fixed	0.67	—
Demand elasticity	e	Fixed	10	—
Relative risk aversion	a	Gamma	1.00	0.50
Match elasticity	i	Gamma	0.70	0.10
Separation rate	p	Gamma	0.10	0.05
Bargaining power of the workers	V	Uniform	0.50	0.3
Unemployment benefits	b	Beta	0.20	0.15
Elasticity of vacancy creation cost		Gamma	1.00	0.50
Scaling factor on vacancy creation cost	K	Gamma	0.10	0.05
AR coefficients of shocks	P{x,A,ß,Y}	Beta	0.8	0.2
Standard deviation of shocks		Inv. Gamma	0.05	oo
Standard deviation of measurement errors	{"}	Uniform	0.001	0.0006
Standard deviation of measurement errors		Uniform	0.001	0.0003
Table 1: Parameters description and prior densities
Parameters are estimated using Bayesian techniques combined with Kalman filtering procedures. All computations have been performed using Dynare toolbox [2] for Matlab. Table 1 reports the model parameters and the corresponding prior densities. The priors (and calibrations) are similar to those used by Lubik [4]. On the other hand, the standard deviations are rather uninformative.
4   Estimation results
Table 2 presents the posterior estimates of parameters and 90% highest posterior density intervals. It may be seen (in comparison with the Table 1) that most of the parameters are moved considerably from their prior means. The data seems to be strongly informative. There are some remarkable results which should be emphasized:
• The first surprising estimate is the bargaining power of workers, rj. The mean value of this parameter is almost 0.9 with a 90 percent coverage region that is shifted considerably away from the prior density. This implies that the workers can gain the most of their entire surplus. The firms are thus not willing to create vacancies. This result is in sharp contrast to the results of Lubik [4] or Yashiv [6] who aimed to model the U.S. labour market.
3GDP at purchaser prices, constant prices 2000, s.a., CZSO, millions of CZK; index of hourly earnings (manufacturing), 2005=100, s.a., OECD; registered unemployment rate, s.a., OECD; unfilled job vacancies, level (transformed to ratio of unfilled vacancies to labour force), s.a., OECD.
Posterior mean       90% HPDI Posterior mean       90% HPDI
(7	1	0399	0	6036	1	4894	Pß	0	8281	0	7298	0	9525
	0	6459	0	5823	0	7099	PY	0	8695	0	8197	0	9241
p	0	0268	0	0086	0	0486	°X	0	0162	0	0108	0	0214
V	0	8694	0	8311	0	9082	<JA	0	0080	0	0067	0	0093
b	0	2924	0	1318	0	4688	aß	0	0066	0	0059	0	0073
	3	5706	3	0908	3	9944	UY	0	0095	0	0080	0	0108
K	0	0880	0	0637	0	1199	°u	0	0005	0	0002	0	0009
Px	0	7831	0	7208	0	8555	aw	0	0003	0	0001	0	0004
PA	0	9629	0	9332	0	9996	°v	0	0002	0	0001	0	0004
Table 2: Parameter estimates
• The second interesting result is the estimated separation rate, p. This parameter is considerably lower than the one estimated by Lubik [4]. Its value supports the view of less flexible Czech labour market with limited ability to destroy old and new matches.
• The third remarkable estimate is the vacancy posting elasticity, ip. The posterior mean 3.57 is shifted away from the prior mean. The vacancy creation is thus more costly because of increasing marginal posting costs (increasing in the level of vacancies or labour market tightness, 6). Lubik [4] estimated this parameter at the mean value of 2.53. In this case, the high value of ip may be interpreted as a balancing factor which "restrict" potentially excessive vacancy creation driven by the low bargaining power. In case of the Czech labour market, this higher value provides further evidence of specifically less flexible labour market.
• The estimate of parameter b corresponds to a reasonable value 0.3 which might be in accordance with the real unemployment benefits paid within the Czech social insurance system (30% of average wage).
• The posterior mean of the matching function parameter, £, is in accordance with the common values in literature (see Lubik [4] or Christoffel et al. [1]).
-0.04I-1-1-1-1-1-1-1-1
1996 1998 2000 2002 2004 2006 2008 2010
Figure 1: Trajectories of selected (smoothed) variables
Figure 1 presents the trajectories of selected smoothed variables. We can see a relative smooth development of variable q (probability of filling a vacancy) with a sharp decline at the end of the year 2007. This evidence is in favour of conclusion presented by Němec and Vašíček [5] who stressed the role
of an obvious lack of employees in the Czech economy. This tendency was reverted as a result of the last global economic slowdown starting at the end of 2008. The efficiency of the matching process is strongly correlated with the output gap (correlation coefficient is 0.8). This indicator is thus seemingly independent of the institutional framework of the Czech labour market. It probably means that the changes in labour market institution have been mostly marginal (with little real impacts). The correlation between output gap and the matching variable m is relative small, 0.4. On the other hand, the correlation between actual value of output gap and the lagged value of m is 0.65. Current value of matching function might be thus an useful indicator of future (one-quarter ahead) changes in the real output (output gap).
5   Model evaluation
In order to see how the model fits the data, sample moments, autocorrelation coefficients and cross-correlations are computed. I computed these statistics from simulation of the estimated models with parameters set at their posterior means. All these statistics correspond to the four observed series (unemployment gap, u, gap of vacancies, v, gap of the wages, w, and output gap, Y). The results may be found in the Tables 3 and 4.
Sample moments Lags for autocorrelation coefficients
			Mean	Std. dev.	1	2	3	4
u	data		0.00	0.009	0.91	0.71	0.45	0.16
	model		-0.00	0.010	0.88	0.70	0.51	0.35
	90% HPDI	(-	-0.01,0.01)	(0.007,0.014)	(0.79,0.94)	(0.48,0.83)	(0.11,0.72)	(-0.08,0.62)
V	data		0.00	0.004	0.91	0.71	0.45	0.17
	model		0.00	0.008	0.72	0.54	0.40	0.29
	90% HPDI	(-	-0.01,0.01)	(0.006,0.011)	(0.55,0.87)	(0.25,0.80)	(0.08,0.73)	(-0.09,0.67)
w	data		0.00	0.014	0.80	0.53	0.29	0.14
	model		0.00	0.054	0.72	0.52	0.36	0.24
	90% HPDI	(-	-0.04,0.04)	(0.041,0.071)	(0.57,0.84)	(0.30,0.72)	(0.06,0.61)	(-0.09,0.57)
Y	data		0.00	0.020	0.91	0.74	0.54	0.33
	model		0.00	0.017	0.79	0.62	0.47	0.36
	90% HPDI		-0.01,0.01)	(0.012,0.024)	(0.64,0.88)	(0.33,0.77)	(0.09,0.70)	(0.01,0.63)
Table 3: Sample moments and autocorrelation coefficients
	u	Data v w	Y	u	Model (90% HPDI) v w		Y
u	1.00	-0.90 -0.74	-0.77	1.00	-0.17	-0.12	-0.16
				(1.00,1.00)	(-0.56,0.25)	(-0.53,0.31)	(-0.66,0.34)
V	-0.90	1.00 0.80	0.88	-0.17	1.00	0.63	0.72
				(-0.56,0.25)	(1.00,1.00)	(0.25,0.89)	(0.46,0.87)
w	-0.74	0.80 1.00	0.60	-0.12	0.63	1.00	0.21
				(-0.53,0.31)	(0.25,0.83)	(1.00,1.00)	(-0.26,0.62)
Y	-0.77	0.88 0.60	1.00	-0.16	0.78	0.21	1.00
				(-0.66,0.34)	(0.46,0.87)	(-0.26,0.62)	(1.00,1.00)
Table 4: Matrix of correlation
The model is very successful in matching sample moments and autocorrelation coefficients (they are mostly within the appropriate 90% highest posterior density intervals). This ability is not used to be typical for such a small-scale model. But, there is one exception regarding the fit of sample moments. The model predicts higher volatility in wages. This pattern reveals the necessity of enrichment by a new source of wage rigidity (as suggested by Krause and Lubik [3] or Christoffel et al. [1]).
My results are in accordance with the authors arguing that the model with search and matching frictions in the labour market is able to generate negative correlation between vacancies and unemployment (see Krause and Lubik [3]). Unfortunately, the values of cross-correlation coefficients (see the lowest bounds of HPDI in the Table 4) are not sufficient for the correlations of unemployment and the rest of observable variables. The similar experience may be found in the results for U.S. labour market provided by Lubik [4]. Lubik pointed out that this may be due the presence of matching shock, which can act as a residual in employment and wage equations.
6 Conclusion
In my contribution, I investigated structural properties of the Czech labour market using a simple DSGE framework with labour market rigidities. Two sources of rigidities were implemented: wage bargaining mechanism and "'search and matching"' process matching workers and firms. Estimated model provides satisfactory description of employment flows in the Czech economy. Parameter estimates provide convincing evidence that wage bargaining process is determined mainly by the power of the unions and that the institutional changes of the Czech labour market in the last fifteen years had only little real impact on the matching effectiveness. Unfortunately, the model predicts higher volatility in wages. This pattern reveals the necessity of enrichment by a new source of wage rigidity as proposed by Krause and Lubik [3] or Christoffel et al. [1].
Acknowledgements
This work was supported by MSMT project Research centers 1M0524. References
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[3] Krause, M., and Lubik, T.: The (ir)relevance of real wage rigidity in the New Keynesian model with search frictions, Journal of Monetary Policy 54 (2007), 706-727.
[4] Lubik, T. A.: Estimating a Search and Matching Model of the Aggregate Labor Market, Economic Quarterly 95 (2009), 101-120.
[5] Němec, D., and Vašíček, O.: Estimating structural changes of the Czech economy: How convincing are the symptoms of the economic crisis? In: Proceedings of the Mathematical Methods in Economics 2010 (M. Houda, and J. Fribelov, eds.), České Budějovice, 2010, 469-476.
[6] Yashiv, E. A.: Evaluating the performance of the search and matching model, European Economic Review 50 (2006), 909-936.