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Murat Tasci is Research Economist at the Federal Reserve Bank of Cleveland; the views stated herein are those of the authors and are not necessarily those of the Federal Reserve Bank of Cleveland or of the Board of Governors of the Federal Reserve System. The authors would like to thank David Arseneau, Sanjay Chugh, Charles Carlstrom, Tim Fuerst, and seminar participants at the Federal Reserve Bank of Cleveland, the Federal Reserve Bank of Boston, and the Board of Governors of the Federal Reserve System.
The business cycle accounting research program has the goal of identifying promising modeling avenues for dynamic general equilibrium models by measuring the “discrepancy” between the data and a prototype real business cycle model. CKM identify four wedges: the efficiency, labor, investment and government consumption wedge. The labor and efficiency wedge are considered the most important, suggesting that macro models that would like to explain real macro fluctuations should pay more attention to understanding what type of frictions could manifest themselves as these wedges.
Usual restricions on the functional form apply, that is Uc (c) > 0, Ucc (c) ≤ 0, Gh(h) < 0, and Ghh(h) ≤ 0 as well as usual Inada conditions. In addition, we normalize G(0) = 0 and impose G(h) < 0, ∀h > 0.
Notice that once the marginal productivity of labor is equal to the wage the firm is also indifferent between the extensive and intensive margin.
It is worth noting that the household problem could be recast in terms of a big family that equalizes consumption across its members and maximizes the momentary utility function U (ct)+ ψntG(ht). In the budget constraint labor income would be (1 − εt)wtnth with the additional constraint nt ≤ 1. In general, the full emplyment constraint would be binding and, thus, we would have the same equilibrium conditions shown in the text. In the next section we will make explicit use of this setting.
Notice that we have made use of the normalization G(0) = 0. Hence the term (1 − nt)ψG(0) does not explicitely appear in the objective function.
Cost of search function, c(e), is assumed to be strictly convex and increasing in e.
Given that any single current-period match survives with probability 1 − σ, households’ expected utility will increase simply by reducing expected future recruiting costs by the quantity
From the aggregate matching function in (12), we know that the probability of filling a vacancy must be qt = Mt/Vt.
Implicitly, we assume that the contracted hours apply to every worker. However, given that G(·) is convex, it is easy to prove that homogenous hours minimize the overall worker disutility.
We drop the distinction between the state variables for the households and the firms, since in equilibrium,
There is a possibility of multiple steady state equilibria in this model due to the complementarities between firms’ recruitment effort and workers’ search effort. Intuitively, if firms expect that workers will not search as hard, the returns to firms’ recruitment will diminish, hence the number of vacancies posted. This will in turn provide workers with the incentive to search less, thereby fulfilling firms’ expectations in the first place. However, one can show that given constant returns to scale of the matching function and assuming enough convexity in c(e), we will have a unique steady state equilibrium in this model. We calibrate the model such that c(e) has enough convexity in the numerical exercises.
At least quantitatively for the resource constraint, since most calibrations does not imply big search costs relative to output.
In the text we have implicitely defined
In the search model, due to the search frictions sL is not exactly the labor share—while it is in the perfectly competitive model.
While a different functional form for the production function would imply a time-varying sL, we do not believe that this will much help in explaining the cyclical properties of the labor wedge.
The calibration follows strictly the one of CKM. Series are demeaned for ease of comparison.
The same parameter, e.g., ψ, will take different values when calibrated in one of the two models. With a loose notation, we denote with a
Our results are essentially same when µ =1.5.
Note that, in this quantitative exercise, shocks to the matching efficiency is shut down, as well as τl and g,. The latter two parameters are set to 2/5 and 0.16, respectively, following the avergaes in the data and the measured labor wedge from the previous section. Our targets for n and h are the means for employment and hours per worker respectively. The calibration target for q follows from the average duration of a posted vacancy based on van Ours and Ridder (1992).
We pulled this data from Haver Analytics database.