Lecture Notes # 1 Tito Boeri

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1 Lecture Notes # 1 Tito Boeri 1 The Aggregate Matching Function: Theory and Empirical Evidence 1.1 Motivations For policy evaluation purposes, important to look at flows. stocks can hardly be discerned. Changes in Facts: availability of longitudinal data on firms-jobs and workers reveals that gross job market flows are many times larger than net flows. Denote by e employment in a statistical unit and by X + (X )thesetofunitsfor which e t+1 >e t (e t+1 <e t ). Then define gross job creation (POS) and destruction (NEG) as POS X X + (e t+1 e t )/E t and NEG X X e t+1 e t /E t Charts from Davis and Haltiwanger (US) and Boeri and Cramer (Germany). Concepts of Job Turnover (JT POS+NEG)andNet employment change(net POS NEG). Excess Job Reallocation (EJR = JT NET ) as a measure of the extent of job reallocation required to accommodate a given change in employment. In which sense are gross flows large? They signal large heterogeneity of plant-level employment outcomes. In fact JT X g t s t X where g denote the plant-level employment growth and s the employment share of each unit. From JT to Labour Turnover (LT). Lower bound (LT = max{pos,neg}) and upper bound (LT = POS + NEG = JT) of labour turnover associated with job reallocation. In practice JT is about 30% of LT, hence at most about one-third of labour turnover is originated by changes in the distribution of posts across plants and excess labour reallocation ELR ' 3JT NET. Analysis of cyclical properties of gross worker flows (Burda and Wyplosz): they tend to be positively correlated over the cycle. Unemploymentinflows and outflows are countercyclical. Employment inflows and outflows are procyclical. Out-of-the labour force inflows and outflows are a-cyclical. 1

2 Questions raised by these observations: Why so large? correlated across time (and space)? Why Gross job and worker flows originate from heterogeneities and frictions in the labour market. Heterogeneity: distribution of the stocks of vacancies (V) and unemployment (U) across space traditionally used to buildup aggregate measures of labour market mismatch (Jackman and Roper; Padoa-Schioppa, Attanasio). Underlying question: how relevant is this heterogeneity? If low substitutability then no purpose to add up unemployment and vacancies, defined over different types of jobs-workers. How to aggregate them up? Matching function. Frictions: dynamics of U and V over time or distance of the Beveridge Curve (BC) from the origin as a measure of matching inefficiencies creating (monopoly) rents in competitive markets. Beveridge function can be rationalised as equilibrium unemployment condition (inflows=outflows) in a matching framework. EJR cannot be attached a normative meaning. More broadly, an analytical framework is needed. 1.2 Microeconomic foundations Rationale for the matching function in coordination failures (Pissarides 1979; Blanchard and Diamond, 1984) and heterogeneities. Uncoordinated applications to existing stocks of vacancies. Let V denote the stock of unfilled job vacancies and U the number of unemployed individuals (for simplicity, we rule out on-the-job search, hence U equals the number of job applicants). Then, the number of contacts between U and V occurring in any period will be given by H = m(u, V )=m(l U L,LV L ) where L is the labour force. The matching function is defined in the domain {(U, V ) m(u, V ) min (U, V )}. Functional form, specialisation of matching technologies. Black box. Closest form you can get from theory is with random matching (whilst generally vacancy are filled via sequential screening). In particular, suppose random placement of balls (workers, U) intourns(firms, V ). Frictions arise because of lack of co-ordination: not of all vacancies are filledevenwhenthereareas many balls as urns. Two variants: i) search effort and ii) heterogeneity across jobseekers. Random matching with search effort. Each applicants sends e applications to e vacant posts. No co-ordination among job seekers. If a vacancy receives more than one application, it takes one of those randomly (non-sequential recruitment-matching) and throw away all others. Each vacancy receives a given worker with probability (e/v ) and conversely the probability that the vacancy does receive that applicant is (1 e/v ). The probability that a vacancy 2

3 does not receive any application is then (1 e/v ) eu and the probability that it receives at least one application is therefore 1 (1 e/v ) eu. As then random draw from all applications received, total matches occurring at any round of applications will be V [1 (1 1/V ) eu ]. For large V with respect to e, a good approximation of (1 1/V ) eu is the exponential e eu/v. Thus total matches are given by H = M(V,eU)=V [1 (1 1/V ) eu ] ' V (1 e eu/v ) (1) and the probability that a typical jobseeker finds a job is eh/eu. Heterogeneity across jobseekers. A fraction s of the unemployed (balls) apply randomly to firms (urns). This captures heterogeneities across workers. As above, if a vacancy receives more than one application, it takes one of those randomly (non-sequential recruitment-matching) and throw away all others. Each vacancy receives a worker with probability (1/V )andtherearesu applicants. Thus, the probability that a vacancy receives no applicant at all is (1 1/V ) su. Total matches occurring at any round of applications will be V (1 1/V ) su.for large V, a good approximation of (1 1/V ) su is the exponential e su/v. Thus total matches are given by H = M(V,sU) =V [1 (1 1/V ) su ] ' V (1 e su/v ) (2) A way to rationalise s is in hetoregeneities in reservation wages across individuals so that only a fraction of the unemployed (e.g., those having a reservation wage lower than the average) apply for a job. Optimal stopping implies that jobs are accepted if w R i where w is the wage offer drawn from a know distribution G(w) andr i is the idyosincratic reservation productivity. Thus s = 1 G(R) wherer is the average reservation productivity. This interpretation is useful when dealing with policies or shocks which affect both matching technologies and the reservation wage or productivity of individuals. Notice that both (1) and (2) are homogeneous of degree one in their arguments. Suppose now that heterogenous jobseekers choose to apply without knowing whetherornotafirm has unfilled vacancies and therefore send randomly one application to one firm. Now each vacancy-firm receives an application with probability (1/(V +E)) where E is the employment level. Thus, the probability that a vacancy receives no applicants at all is (1 1/(V + E)) su. Total matches occurring at any round of applications can therefore be approximated by H = V (1 1/(V + E)) su ' V (1 e su/(v +E) ) which has constant returns for fixed E. However, E is not independent of U. In particular, when the labour force (L) isfixed, E = L U,then H = V [1 (1 1/(V + L U)) su ] ' V (1 e su/(v +L U) ) which has increasing returns to scale. 3

4 Notice that the early matching literature (Blanchard and Diamond, 1989; Pissarides, 1992 and Storer, 1994) assumed the matching function to have CRS. H = m(u, V )=ml( U L, V L ) This is consistent with constant unemployment along a balanced growth path and with unemployment independent of the size of a country. However, there are also arguments for increasing returns such as search externalities, lags between matching and hiring and heterogeneity among the unemployed (measurement problem) (Coles and Smith, 1994; Mortensen, 1997). Thin labour market externalities and spatial aggragation may reconcile IRS at the local level with CRS at the aggregate level. Returns to scale are important to establish the possible presence of multiple equilibria. IRS are a necessary condition for multiple equilibria. 1.3 Empirical estimation Returns to scale are, in any event, an empirical issue. Generalities. Typically panel data (allow to identify dis-embodied technological progress in matching). Time dimension: the matching function in discrete time reads H t,t+1 = m(u t,v t ) Need to have high-frequency data (see time aggregation below). Otherwise, we cannot impose H t,t+1 min(u t,v t ) Introducing search intensity, c, and dis-embodied technical progress A (measuring the efficiency in the matching process): H t,t+1 = m(a t,c t U t,v t ) Technological progress can only be identified combing time-series with crosssectional observations Testing returns to scale and the functional form Typically assumed to be Cobb-Douglas H t,t+1 = A t (U t ) α V β t taking logs log(h t,t+1 )=a t + α log(u t )+βlog(v t ) When the matching function is the Cobb-Douglas we can easily test whether it takes place under CRS. When this is the case H t,t+1 = A t (U t ) α V (1 α) t 4

5 hence log(h t,t+1 )=a t + α log(u t )+(1 α)log(v t ) It follows that we can run log(h t,t+1 )=a t + α[log(u t ) log(v t )] + log(v t ) the test of CRS being provided by whether or not the coefficient for V is significantly different from unity. A better way to proceed (but rarely implemented in estimating the matching function) is to adopt a general and flexible specification and then test the restrictions involved by the standard static Cobb-Douglas specification. Suppose to have panel data (across regions, i, and time,t). Then the translog form h it = a t + µ i + α 1 u it + α 2 v it + γ 1 (u it v it ) 2 + ε it where µ captures fixed region effects. Testing the Cobb-Douglas specification involves testing whether γ 1 = Aggregation over time Temporal aggregation problems arise because estimating a matching function involves estimating flows from stock conditioning variables. Suppose that estimates rely in the following (log-linear) empirical specification of the matching function (leave aside search intensity for notational ease and use small caps to denote logs) h t = a t+1 + α 1 u t + α 2 v t + ε t where the two stocks (unemployment and vacancies) are measured at some point intheperiodinwhichflows are registered. As both v and u are reduced by the left-hand-side variable, the estimated coefficients, α 1 and α 2 will be biased downwards. To deal with this problem, usually the two stocks are measured at the beginning of the period and, if the error term is not serially correlated, then also Cov(u, ε) = Cov(u, ε) = 0, that is, the two regressors qualify as good instruments. However in the period in which flows are measured also the stock changes. Put another way unemployment outflows will include not only flows out of the initial stock, but also from the inflow. At still relatively high frequencies (e.g., when using quarterly stock-flow data) one may actually have flows being larger than (at least of the two) initial stocks, that is, matching probabilities exceeding one. At relatively high frequencies, this problem can be dealt with by considering the bias induced by time aggregation as a simple linear function of its length. This amounts to comparing estimates at different frequencies, using the estimated bias in the low frequency case as an adjustment factor to be applied the coefficients in order to approximate higher frequency estimates. 5

6 1.3.3 Aggregation over Space Usually the matching function is estimated combining cross-section (regional) and time-series observations. Insofar as the cross-sectional units have a different size and matching does NOT take place under constant returns to scale, estimates may be affected by a spurious scale effect. Suppose that matching technologies are the same across all regions. Then stocks and flows would be proportional to national aggregates, as follows: H i = s i H U i = s i U V i = s i V where s i is the labour force share of region i, that is s i = Li L and variables without subscripts the national values. Substituting in the matching function: and taking logs s i H t+1 = A t (s i U t ) α1 (s i V t ) α2 log(s i )=(α 1 + α 2 )log(s i )+(a t h t+1 + α 1 u t + α 2 v t ) Estimating this equation cross-section amounts to regressing the (log of the) labour force share against itself plus a constant term. It follows that it will generate constant-returns to scale (α 1 + α 2 =1)andazeroconstantterm (a t = h t+1 α 1 u t α 2 v t ), while in reality we know that matching does not occur at CRS. A remedy to this is to adjust the variables by the size of the region, that is use h i l i = a t + α 1 ( u i l i )+α 2 ( v i li) where the have dropped time-subscripts for notational ease, or h i = a t + α 1 u i + α 2 v i +(α 1 + α 2 1) li Comparing this with the un-adjusted model shows that the two specifications are identical iff α 1 + α 2 1=0 that is, matching takes place under CRS or ρ(l i,u i )=ρ(l i,v i )=0 Clearly both assumptions are violated under IRS or DRS. The above is done assuming a Cobb-Douglas specification. The adjustment is still possible, but more complex, under a translog specification. 6

7 1.4 Extensions Dynamic specifications The economic rationale behind the matching functions is the presence of frictions in the labour market. Thus, it would seem natural to adopt a dynamic specifications allowing for partial adjustment in the matching process, e.g.: (1 c(l))h t,t+1 = a t + α log(u t )+(1 α)log(v t ) where lower case variables denote logs and c(l) is a polynomial in the lag operator. Partial adjustment in the matching process may be brought about by the fact that the acceptance of a job offer precedes by several days or weeks the reported date of a match with no search taking place in the interim period. Another possibility is that information on vacancies is not circulated immediately to all jobseekers (likely to happen when there are no on-line vacancy registers). The presence of lagged dependent variables may create problems of endogeneity. Suppose for instance that vacancies include posts created by the Public Employment Service (e.g., within direct job creation schemes or subsidised employment programmes). Suppose further that the offer of slots into such schemes is correlated with the underlyining labour market conditions (e.g., under periods of low outflows from unemployment, more slots are offered). Then OLS estimates of the matching function would be biased (downwards). It is then necessary to find instruments (correlated with vacancies but not with the error term in the matching function) Non-Random Matching Ranking Lack of realism of random matching. Strategic considerations are important. Models of ranking (Blanchard and Diamond, 1994). Short-term unemployment and long-term unemployment may have different search intensity and effectiveness, due to the presence of stigma, discouragement effects and skill obsolence. The empirical literature points indeed to significant differences in the elasticity of job findings with respect to unemployment of different duration. This suggest breaking down the unemployment stock into its short-term component (e.g., inflows into unemployment from t 1tot) and its long-term component (the remaining unemployed at t). It is necessary in this case to impose restrictions as to the relation between the two (or more) unemployment types and vacancies. A convenient assumption is that there is weak separability between the various kind of unemployment pools and vacancies, that is H t,t+1 = f[g(ut s,ut),v l t ] where superscripts s and l denote short-term and long-term unemployment respectively. In particular, one can model matching as a translog function of 7

8 vacancies and of an aggregate of unemployment types (a sort of sub-matching function), e.g. H t.t+1 = A t V α t [wu s( ρ) t +(1 w)u l( ρ) t ] ν/ρ which, by taking logs and linearising around ln(ρ) = 0, reduces to h t,t+1 = a t + αv t + γ 1 u s t + γ 2 u l t + γ 3 (u s t u l t) 2 We can then recover the original parameters as follows: w = γ 1 γ 1 + γ 2,ν = γ 1 + γ 2,ρ= 2γ 3 (γ 1 + γ 2 )/γ 1 γ 2 This general specification allows to test a number of restrictions. First, we can test whether there is strong separability between short-term and long-term unemployment, that is, whether g(u s t,u l t)=m(u s t) n(u l t) A test of this hypothesis amounts to testing whether γ 3 = 0. Another test concerns the heterogeneity itself in the unemployment pool. When unemployment is a homogeneous input, then γ 1 = γ 2. Allowing for disco effects: flows matching stocks The matching process is one of the least investigated fields of economics. Coles and Smith (1998) have investigated the non-random components of matching and modelled a matching function from jobseekers having full information about job offers and applying only to those acceptable to them. The resulting matching process is one where the unmatched stock of jobseekers will match only with the flow of vacancies or vice versa. Intuitively, if a vacancy (an unemployed) has been looked into and rejected, then the employer (jobseeker) will less likely apply for it in the future than for a new jobseeker (vacancy). The matching function is in this case H t+1 = V t+1 (1 α U t )+U t+1 (1 α V t ) where α is the probability that a random pairing is unacceptable. This form of the matching function does not involve congestion externalities. It actually involves increasing returns to scale (in the stock) as the stock-flow matching probability increases with the stock Introducing institutions Boeri and Burda (1996) found effects of ALMPs on job matching. Other authors did not find effects of UBs on matching (do they act more on job destruction?). 8