Solving the DMP Model Accurately

Transcription

1 Solving the DMP Model Accurately Nicolas Petrosky-Nadeau Carnegie Mellon University Lu Zhang The Ohio State University and NBER December 23 Abstract An accurate global algorithm is crucial for quantifying the dynamics of the Diamond- Mortensen-Pissarides model. Loglinearization understates the mean and volatility of unemployment, overstates the unemployment-vacancy correlation, and ignores impulse responses that are an order of magnitude larger in recessions than in booms. Although improving on loglinearization, the second-order perturbation in logs also induces large approximation errors. We demonstrate these insights within the context of Hagedorn and Manovskii (28). Our quantitative results highlight the extreme importance of accurately accounting for nonlinear dynamics in quantitative macroeconomic studies. JEL Classification: E24, E32, J63, J64. Keywords: Search and matching, the unemployment volatility puzzle, global approximation methods, projection, loglinearization, nonlinear impulse response functions Tepper School of Business, Carnegie Mellon University, 5 Forbes Avenue, Pittsburgh PA 523. Tel: (42) and npn@cmu.edu. Fisher College of Business, The Ohio State University, 76A Fisher Hall, 2 Neil Avenue, Columbus OH 432; and NBER. Tel: (64) and zhanglu@fisher.osu.edu. We are grateful to Hang Bai, Andrew Chen, Daniele Coen-Pirani, Steven Davis, Wouter Den Haan, Paul Evans, Lars-Alexander Kuehn, Dale Mortensen, Paulina Restrepo-Echavarria, Etienne Wasmer, Randall Wright, and other seminar participants at The Ohio State University and the 23 North American Summer Meeting of the Econometric Society for helpful comments. Nicolas Petrosky-Nadeau thanks Stanford Institute for Economic Policy Research and the Hoover Institution at Stanford University for their hospitality. The Matlab programs are available from the authors upon request. All remaining errors are our own.

2 Introduction Since the foundational work of Diamond (982), Mortensen (982), and Pissarides (985), the search model of unemployment has become the dominant framework for studying the labor market. Shimer (25) argues that the unemployment volatility in the baseline Diamond- Mortensen-Pissarides (DMP) model is too low relative to that in the data. A large subsequent literature has developed to address this quantitative puzzle. Given its quantitative focus, an accurate approximation to the model s equilibrium is of paramount importance. However, this literature has mostly relied on loglinearization to solve the model. Our key message is that an accurate global approximation is crucial for characterizing the quantitative properties of the DMP model. We demonstrate the importance of nonlinearity within the context of Hagedorn and Manovskii (28). They argue that under their calibration, the DMP model produces realistic labor market volatilities. The volatility of labor market tightness is.292 in the model, which is not far from.259 in the data. The unemployment volatility is.45 in the model, which is close to.25 in the data. However, using an accurate projection algorithm and the Hagedorn-Manovskii (28) calibration, we calculate the unemployment volatility to be.258, which is about twice as large as.25 in the data. The unemployment-vacancy correlation is also lower in magnitude,.567, versus.724 from loglinearization. Finally, the stochastic mean of the unemployment rate using projection, 6.7%, is almost one percentage point higher than its deterministic steady state rate, 5.28%. These results cast doubt on the validity of calibration Hall (25) uses sticky wages and Mortensen and Nagypál (27) and Pissarides (29) use fixed matching costs to help explain the volatility puzzle. Gertler and Trigari (29) propose a staggered wage bargaining framework to generate wage rigidity. Hagedorn and Manovskii (28) show that a calibration with small profits and a low bargaining power for workers can produce realistic volatilities. Hall and Milgrom (28) replace the Nash bargaining wage with a credible bargaining wage. Finally, Petrosky-Nadeau and Wasmer (23) use financial frictions to increase labor market volatilities.

3 that relies exclusively on steady state relations. More generally, these results question loglinearization as a solution method for the DMP model (despite its simplicity and speed). 2 What explains the differences between loglinearization and projection? The crux is that the unemployment dynamics in the DMP model with the Hagedorn-Manovskii calibration are highly nonlinear. In recessions unemployment rises rapidly, whereas in booms it declines only gradually. The projection algorithm fully captures the nonlinearity. In contrast, by focusing only on local dynamics around the deterministic steady state, loglinearization ignores completely the unemployment spikes in recessions. Because of this omission, loglinearization underestimates the volatility of unemployment but overestimates its correlation with vacancies. The two algorithms also differ dramatically in impulse responses. With projection, impulse responses are stronger in recessions than in booms. In response to a negative onestandard-deviation shock to log labor productivity, the unemployment rate increases only by.8% if the economy starts at the 95 percentile of the model s stationary distribution of employment and productivity (with an unemployment rate of 4%). In contrast, the unemployment rate shoots up by.83% (an order of magnitude larger) if the economy instead starts at the 5 percentile of the bivariate distribution (with an unemployment rate of.5%). In contrast, such nonlinearity in impulse responses is entirely missed by loglinearization. Since Merz (995) and Andolfatto (996), the DMP model has been embedded into the dynamic stochastic general equilibrium framework to study business cycles. While the Hagedorn-Manovskii calibration calls for a quarterly persistence of.88 for the log productivity, business cycle studies typically calibrate the persistence to be.95 as in, for example, Cooley and Prescott (995). We show that the higher persistence induces even larger differ- 2 It should be noted that Hagedorn and Manovskii (28) use a global approximation algorithm, not loglinearization, to solve their model. However, due to their Markov-chain approximation to the highly persistent log labor productivity, their results are quantitatively similar to those from loglinearization (see Section 5.). 2

4 ences between loglinearization and projection (even with a lower unconditional volatility for the log productivity). The unemployment volatility is only.8 from loglinearization but.285 from the projection algorithm. Intuitively, because job creation is forward looking, a low realization of productivity reduces job creation more if it is expected to last longer. As such, the high persistence strengthens nonlinear unemployment dynamics, thereby exacerbating the deviation of loglinearization from the projection algorithm. Although improving on loglinearization, the second-order perturbation in logs also fails to deliver accurate results. The unemployment volatility is.64, which, although higher than.33 from loglinearization, is still substantially lower than.25 from projection. Similarly, the unemployment-vacancy correlation is.79, although lower in magnitude than.848 from loglinearization, is still substantially higher than.564 from projection. 3 We also perform accuracy tests on the different numerical algorithms. The results show that the projection algorithm provides an extremely accurate solution, whereas loglinearization and the second-order perturbation in logs induce large approximation errors. In particular, dynamic Euler equation errors in consumption are in the order of 4 for the projection algorithm, but are around 7% for the other two algorithms. Our work suggests that many prior conclusions in the macro labor literature might need to be reexamined with an accurate global solution. Although the Hagedorn-Manovskii (28) calibration seems somewhat extreme in its implications on the magnitude of profits, it provides a parsimonious setting with realistic labor market volatilities in loglinear approxi- 3 We have also experimented with the third-order perturbation in logs. However, as discussed in Kim, Kim, Schaumburg, and Sims (28) and Den Haan and Wind (22), high-order perturbations often produce explosive simulated sample paths. In our practice, after confirming this phenomenon in the DMP model, we opt not to pursue the third- or higher-order perturbations. As such, although casting doubt on lower-order perturbations, our results should not be interpreted as a critique on high-order perturbations. On the contrary, our results indicate that the higher order terms are in fact quantitatively important. 3

5 mations. As such, their setup offers a natural starting point for our analysis. Many other economic mechanisms have been proposed to generate realistic labor market volatilities. Our quantitative insights should apply as a consequence of nonlinear unemployment dynamics. However, we emphasize that our results should not be interpreted as a critique on Hagedorn and Manovskii. On the contrary, we show that once their model is solved accurately, a less extreme calibration is sufficient to match the unemployment volatility (albeit with a lower volatility for the labor market tightness than that in the data). More broadly, our insights apply to a wide body of academic and policy research as the DMP model has been adopted throughout macroeconomics. 4 Characterizing dynamics accurately is crucial for deriving quantitative properties. The capability to perform policy counterfactuals requires a macroeconomic model to be estimated (e.g., Smets and Wouter (23, 27)). However, inferences can be biased if an approximate equilibrium ignores nonlinear dynamics (e.g., Fernández-Villaverde and Rubio-Ramírez (27)). The projection method for solving dynamic equilibrium models is proposed by Judd (992). Christiano and Fisher (2) show how to incorporate occasionally binding constraints into a projection algorithm. Den Haan (2a) compares perturbations with projection in solving the incomplete markets model. Finally, we adapt the projection algorithm in Petrosky-Nadeau, Zhang, and Kuehn (23), who study asset prices in the DMP model. We instead examine how different algorithms affect labor market moments. The rest of the paper is organized as follows. Section 2 sets up the model. Section 3 describes the projection algorithm. Section 4 shows how the labor market moments with loglinearization deviate quantitatively from with the projection algorithm. Section 5 recon- 4 Recent examples include Merz (995) and Andolfatto (996) in the real business cycle literature, Gertler and Trigari (29) in the New Keynesian literature, Blanchard and Gali (2) on monetary policy, and Monacelli, Perotti, and Trigari (2) on fiscal policy. 4

6 ciles our global results with those in Hagedorn and Manovskii (28). Section 6 reports the results with the Cobb-Douglas matching function. Finally, Section 7 concludes. 2 The Model We follow closely the setup in Hagedorn and Manovskii (28, HM). There exist a representative household and a representative firm that uses labor as the single productive input. Following Merz (995), we use the representative family construct, which implies perfect consumption insurance. The household has a continuum with a unit mass of members who are, at any point in time, either employed or unemployed. The fractions of employed and unemployed workers are representative of the population at large. The household pools the income of all the members together before choosing per capita consumption and asset holdings. The household is risk neutral with a time discount factor β. The representative firm posts a number of job vacancies, V t, to attract unemployed workers, U t. Vacancies are filled via a constant returns to scale matching function, G(U t,v t ): G(U t,v t )= U t V t, () (Ut ι + Vt ι /ι ) in which ι> is a constant parameter. This matching function, from Den Haan, Ramey, and Watson (2), implies that matching probabilities fall between zero and one. Define θ t V t /U t as the vacancy-unemployment (V/U) ratio. The probability for an unemployed worker to find a job per unit of time (the job finding rate), f(θ t ), is: f t = f(θ t )= G(U t,v t ) U t = ( ) +θ ι /ι. (2) t 5

7 The probability for a vacancy to be filled per unit of time (the vacancy filling rate), q(θ t ), is: q t = q(θ t )= G(U t,v t ) V t =. (3) ( + θ ι /ι t) An increase in the scarcity of unemployed workers relative to vacancies makes it harder to fill a vacancy, q (θ t ) <. As such, θ t is labor market tightness from the firm s perspective. The firm takes aggregate labor productivity, X t, as given. We specify x t log(x t )as: x t+ = ρx t + σɛ t+, (4) in which ρ (, ) is the persistence, σ> is the conditional volatility, and ɛ t+ is an independently and identically distributed (i.i.d.) standard normal shock. The firm uses labor to produce output, Y t, with a constant returns to scale production technology, Y t = X t N t. (5) As in HM, the representative firm incurs costs in posting vacancies with the unit cost: κ t = κ K X t + κ W X ξ t, (6) in which κ K,κ W,andξ are positive parameters. Once matched, jobs are destroyed at a constant rate of s per period. Employment, N t, evolves as: N t+ =( s)n t + q(θ t )V t, (7) in which q(θ t )V t is the number of new hires. The population is normalized to be unity, U t = N t. As such, N t and U t are also the rates of employment and unemployment, respectively. 6

8 The dividends to the firm s shareholders are given by: D t = X t N t W t N t κ t V t, (8) in which W t is the wage rate. Taking q(θ t )andw t as given, the firm posts an optimal number of job vacancies to maximize the cum-dividend market value of equity, denoted S t : S t max {V t+τ,n t+τ+ } τt= [ ] E t β τ [X t+τ N t+τ W t+τ N t+τ κ t+τ V t+τ ], (9) τ= subject to equation (7) and a nonnegativity constraint on vacancies: V t. () Because q(θ t ) >, this constraint is equivalent to q(θ t )V t. As such, the only source of job destruction in the model is the exogenous separation of employed workers from the firm. Let λ t denote the multiplier on the constraint q(θ t )V t. From the first-order conditions with respect to V t and N t+, we obtain the intertemporal job creation condition: [ [ [ ]]] κ t q(θ t ) λ κt+ t = E t β X t+ W t+ +( s) q(θ t+ ) λ t+. () Intuitively, the marginal costs ofhiring at time t (with the nonnegativity constraint accounted for) equal the marginal value of employment to the firm, which in turn equals the marginal benefit of hiring at period t +, discounted to t with the discount factor, β. Themarginal benefit at t+ includes the marginal product of labor, X t+, net of the wage rate, W t+, plus the marginal value of employment, which equals the marginal costs of hiring at t +,netof separation. Finally, the optimal vacancy policy also satisfies the Kuhn-Tucker conditions: q(θ t )V t, λ t, and λ t q(θ t )V t =. (2) 7

9 The wage rate is from the sharing rule per the outcome of a generalized Nash bargaining process between the employed workers and the firm. Let η (, ) be the workers relative bargaining weight and b the workers flow value of unemployment activities. The wage rate is: W t = η (X t + κ t θ t )+( η)b. (3) Let C t denote consumption. In equilibrium, the goods market clearing condition says: C t + κ t V t = X t N t. (4) 3 A Projection Algorithm We approximate the model solution with a projection algorithm. The state space of the model consists of employment and log productivity, (N t,x t ). The goal is to solve for the optimal vacancy function, V t = V (N t,x t ), and the multiplier function, λ t = λ(n t,x t )fromthe intertemporal job creation condition given by equation (). We work with the job creation condition because the competitive equilibrium is in general not Pareto optimal. In addition, V (N t,x t )andλ(n t,x t ) must also satisfy the Kuhn-Tucker condition (2). The standard projection method would approximate V (N t,x t )andλ(n t,x t ) directly to solve the job creation condition, while obeying the Kuhn-Tucker condition. However, with the vacancy nonnegativity constraint, these kinked functions can cause problems in the approximation with smooth functions. To deal with this issue, we follow Christiano and Fisher (2) to approximate the conditional expectation in the right-hand side of equation () as E t E(N t,x t ). A mapping from E t to policy and multiplier functions then eliminates the need to parameterize the multiplier function separately. Specifically, after obtaining E t,we first calculate q(θ t ) κ t /E t. If q(θ t ) <, the nonnegativity constraint is not binding, we 8

10 set λ t =andq(θ t )= q(θ t ), and then solve θ t = q ( q(θ t )), in which q ( ) is the inverse function of q( ) from equation (3). We then set V t = θ t ( N t ). If q(θ t ), the constraint is binding, we set V t =,θ t =,q(θ t ) =, and λ t = κ t E t. We approximate the log productivity process, x t,inequation(4)basedonthediscrete state space method of Rouwenhorst (995). We use 7 grid points to cover the values of x t, which are precisely within four unconditional standard deviations from the unconditional mean of zero. For the N t grid, we set the minimum value to be. and the maximum.99 (N t never hits the bounds in simulations). For each grid point of x t, we use piecewise linear splines with, nodes on the N t space to approximate E(N t,x t ). To obtain an initial guess, we use the model s loglinear solution. We implement the loglinear solution (and the second-order perturbation in logs) using Dynare (e.g., Adjemian et al. (2)). Because Dynare is well known, we do not discuss the details. Instead, we report our Dynare program in Appendix A. Two comments on the implementation are in order. First, in the loglinear solution, we ignore the nonnegativity constraint of vacancy by setting the multiplier, λ t,tobezeroforallt. Doing so is consistent with the common practice in the literature. Second, following Den Haan s (2) recommendation, we substitute out as many variables as we can and use only a minimum number of equations in the Dynare program. We use only three equations (the employment accumulation equation, the job creation condition, and the law of motion for log productivity) with three primitive variables (employment, log productivity, and consumption). The solutions to all the other variables are obtained using the actual nonlinear equations of the model, equations that connect them to the three variables in the perturbation system. 9

11 4 How Loglinearization Deviates from Projection In this section, we quantify in depth how the results based on loglinearization deviate significantly from those based on the global projection algorithm. 4. Labor Market Volatilities It is customary to detrend variables in log-deviations from the HP-trend with a smoothing parameter of,6. In contrast, we use the HP-filtered cyclical component of proportional deviations from the mean with the same smoothing parameter. We cannot take logs because vacancies can be zero in simulations when the V t constraint is binding The HM Calibration To solve and simulate from the model, we use exactly the same parameter values from the HM s weekly calibration. Specifically, the time discount factor, β,is.99 /2. The persistence of log labor productivity, ρ, is.9895, and its conditional volatility, σ, is.34. The workers bargaining weight, η, is.52, and their flow value of unemployment activities, b, is.955. The job separation rate, s, is.8. The elasticity of the matching function, ι, is.47. Finally, for the vacancy cost function, the capital cost parameter, κ K, is.474, the labor cost parameter, κ W, is., and the exponential parameter in the labor cost, ξ, is.449. With these parameter values, we first reach the model s stationary distribution by simulating the economy for weekly periods from the initial condition of zero for log productivity and.947 for employment (the deterministic steady state). From the station- 5 In the data, the two detrending methods yield quantitatively similar results. We use the same data sources as HM (28) and calculate the standard deviations of unemployment, vacancy, and the labor market tightness to be.9,.34, and.255, respectively, with our detrending method (untabulated). Also, unemployment and vacancy have a correlation of.93, indicating a downward-sloping Beveridge curve. These moments are close to those in HM s Table 3 based on log-deviations.

12 ary distribution, we repeatedly simulate 5, artificial samples, each with weekly periods. We take the quarterly averages of the weekly unemployment, vacancy, and labor productivity to obtain 26 quarterly observations, matching HM s sample length from the first quarter in 95 to the fourth quarter in 24. We then calculate the model moments for each artificial sample and report the cross-simulation averages. Table reports the labor market moments from the model. Panel A is borrowed from Table 4 in HM. In Panel B, we attempt to replicate their results with loglinearization (see the Dynare code in Appendix A.). Although the moments are not identical, the results in Panel B are close to those in Panel A. We detrend all the variables in Panel B as log-deviations from the HP-trend as in HM (The V t constraint is never binding with loglinearization). The standard deviation for the labor market tightness is.284, which is close to.292 in HM. The unemployment volatility is.46, which is close to.45 in HM. Finally, the unemployment-vacancy correlation is.777, which is not far from.724 in HM s Table 4. As noted, because vacancies can hit the zero lower bound (albeit infrequently) in the model solved with the projection algorithm, we cannot take logs. Panel C reports the results from loglinearization while detrending the variables as HP-filtered proportional deviations from the mean. Comparing Panels B and C shows that the detrending method does not materially affect the results. The standard deviation of the labor market tightness becomes.327, which is not far from.284 with the log-deviations. The unemployment volatility is.33, which is close to.46 with the log-deviations. However, the unemployment-vacancy correlation is.848, which is somewhat higher in magnitude than.777 with the log-deviations. Panel D reports the key message of the paper. The projection results differ from the loglinearization results in quantitatively important ways. Most important, the unemployment

13 volatility is.25, which is almost twice as large as that from loglinearization,.33. Also, the unemployment-vacancy correlation is.564 in the projection solution and is substantially lower in magnitude than.848 with loglinearization. As such, loglinearization understates the unemployment volatility but overstates the unemployment-vacancy correlation The Impact of Higher Persistence in Labor Productivity A higher persistence of the log productivity gives rise to even larger quantitative differences between loglinearization and projection. This result holds even when the unconditional standard deviation of productivity is lower. We recalibrate the model by setting the persistence of log productivity to.9957 and the conditional volatility to.2. These weekly values correspond to the quarterly values of.95 and.7, respectively, which are standard in the business cycles literature (e.g., Cooley and Prescott (995)). From Panels E and F of Table, the volatility of labor productivity,.9, is lower than.3 in the HM calibration. Despite the lower productivity volatility, the differences between loglinearization and projection increase in magnitude with the high persistence calibration. The unemployment volatility from projection,.285, is almost three times as large as.8 from loglinearization. The unemployment-vacancy correlation is.52 from projection in contrast to.86 from loglinearization. While the volatility of the market tightness from loglinearization is (relatively) close to that from projection with the HM calibration,.327 versus.27, the gap widens to.29 versus.25 in the high persistence calibration Second-order Perturbations in Logs Loglinearization is the first-order perturbation in logs. Can the second-order perturbation in logs (also popular in the business cycle literature) eliminate the quantitative differences between projection and loglinearization? The answer is not affirmative, unfortunately. 2

14 From Panel G of Table, although improving on loglinearization, the labor market moments from the second-order perturbation still deviate significantly from those from the projection algorithm. Relative to loglinearization, the unemployment volatility from the second-order perturbation increases slightly from.33 to.64, which still differs greatly from.25 with projection. Similarly, the unemployment-vacancy correlation drops in magnitude from.848 to.79, once we move from loglinearization to the second-order perturbation. However,.79 still deviates significantly from.564 with projection. Panel H shows that the deviations are further exacerbated with the high persistence calibration. 4.2 Intuition: Nonlinearity Matters Why does loglinearization differ so much from projection? The crux is that the unemployment dynamics in the DMP model are highly nonlinear. In recessions unemployment rises rapidly, whereas in booms unemployment declines only gradually. The distribution of unemployment is highly skewed with a long right tail. The projection solution fully captures these nonlinear dynamics. In contrast, by focusing only on local dynamics around the deterministic steady state, loglinearization ignores the large unemployment dynamics in recessions altogether, thereby understating its volatility but overstating its correlation with vacancies Stationary Distribution Figure plots unemployment against labor productivity using one million weekly periods simulated from the model s stationary distribution. From Panel A, the projection-based unemployment-productivity relation is highly nonlinear under the HM calibration. When the labor productivity is high, unemployment falls gradually, whereas when the productivity is low, unemployment rises drastically. The unemployment rate can reach above 65% in simulations. The correlation between unemployment and labor productivity is.69. Panel 3

15 B plots the empirical cumulative distribution function for unemployment in the simulated data. Unemployment is positively skewed with a long right tail. The mean unemployment rate is 6.2%, the median is 5.34%, and the skewness is The 2.5 percentile, 3.8%, is close to the median, whereas the 97.5 percentile is far away, 3.43%. In contrast, Panel C shows that the unemployment-productivity relation from loglinearization is linear. Their correlation is nearly perfect,.96. The maximum unemployment rate in simulations is only.38%. Panel D also shows that the empirical cumulative distribution for unemployment from loglinearization is largely symmetric. The mean unemployment rate is 5.28%, which is close to the median of 5.29%. The skewness is almost zero,.5. The 2.5 and 97.5 percentiles, 2.72% and 7.77%, respectively, are distributed symmetrically around the median of 5.29%. Also, the stochastic mean of the unemployment rate from projection, 6.2%, is almost one percentage point higher than its deterministic steady state rate, 5.28%. From Panels E and F, unlike loglinearization, the second-order perturbation in logs captures some nonlinear dynamics. However, the nonlinearity is not nearly as strong as that from the projection algorithm. As such, although improving on loglinearization, the labor market moments from the second-order perturbation are closer to those from loglinearization than those from the projection algorithm (Panel G in Table ). Panels G and H in Figure report the unemployment-productivity relation and the empirical distribution of unemployment under the high persistence calibration. Comparing with Panels A and B shows that this calibration exhibits stronger nonlinear dynamics than the HM calibration. The unemployment rate can reach above 9% in simulations in contrast to 65% under the HM calibration. Intuitively, job creation is a forward looking decision, in which the persistence of labor productivity plays a determining role. A low realization 4

16 reduces job creation more if it is expected to last a longer time. As such, nonlinear dynamics are stronger under the high persistence calibration, and loglinearization deviates further from the projection algorithm (Panels E and F in Table ) An Illustrative Example To further illustrate the differences between projection and loglinearization, we contrast simulated sample paths of unemployment from the algorithms. The underlying labor productivity series is identical. As such, the only difference is the algorithm. Figure 2 shows that the two sample paths differ in two important ways. First, in recessions the unemployment rates from the projection algorithm spike up frequently to more than % (the blue solid line). In contrast, no such spikes are visible from the sample path based on loglinearization (the red broken line). For instance, toward the end of the sample path, the unemployment rate from the projection algorithm spikes up to more than 25%, whereas the corresponding unemployment rate from loglinearization is only about 8.5%. Second, in booms the unemployment rates from the projection algorithm are often higher than those from loglinearization. In particular, around the 3,th weekly period, the unemployment rate from the projection algorithm reaches a lower level of about 3.2%. However, the corresponding unemployment rate from loglinearization is about %. Intuitively, these nonlinear dynamics are driven by a combination of matching frictions and small profits due to high and inelastic wages. The matching frictions induce a congestion externality in the labor market. In booms, many vacancies compete for a small pool of unemployed workers. An extra vacancy can cause a large drop in the vacancy filling rate, causing the marginal costs of hiring to increase rapidly. The increasing marginal costs of hiring mitigate the impact of rising profits on the incentives of hiring, slowing down job 5

17 creation flows. As such, the unemployment rates fall gradually in booms. In contrast, many unemployed workers compete for a small pool of vacancies in recessions. An extra vacancy is quickly filled and hardly reduces the vacancy filling rate. As such, the marginal costs of hiring hardly decline in recessions (downward rigidity), and a given decline in vacancies leads to a disproportionately large drop in hiring. Consider a large, negative shock to labor productivity. Profits drop disproportionately more due to the high and inelastic wages. The downward rigidity of the marginal costs of hiring further exacerbates the impact of falling profits on the incentives of hiring, stifling job creation flows. As such, as vacancies become scarce, the unemployment rates spike up in recessions (e.g., Petrosky- Nadeau, Zhang, and Kuehn (23)). As noted, loglinearization misses these dynamics Nonlinear Impulse Response Functions We consider three different initial points, bad, median, and good economies. The bad economy is the 5 percentile of the model s bivariate stationary distribution of employment and log productivity under the projection algorithm, the median economy is the 5 percentile, and the good economy is the 95 percentile. (The unemployment rates are.54%, 5.34%, and 3.96%, and the log productivity levels are 3.82,, and 3.82, across the bad, median, and the good economies, respectively.) We calculate two sets of responses. The first set is the responses to a one-standard-deviation shock to log productivity, both positive and negative, starting from all three initial points. The second set is the responses to a two- and a threestandard-deviation negative shock to log productivity, starting from the median economy. Figure 3 reports the first set of impulse responses. Several nonlinear patterns emerge. First, the responses fromthe projection algorithmareclearly stronger in recessions than those in booms. For instance, in response to a negative impulse, the unemployment rate shoots up 6

18 by.83% in the bad economy (Panel A). In contrast, the response is only.8%, which is an order of magnitude smaller, in the good economy (Panel C). The response in the median economy is.8%, which is closer to that in the good economy than in the bad economy. Second, the responses to a negative impulse from the projection algorithm are somewhat larger in magnitude than those to a positive impulse. The response to a positive impulse is.72% in the bad economy, in contrast to the response to a negative impulse,.83%. Third, and more important, although close in the median economy, the responses in unemployment from loglinearization diverge significantly from the projection responses. In particular, the responses from loglinearization are substantially weaker in the bad economy. For instance, the response to a negative impulse under loglinearization is.5%, which is less than 2% of the response under the projection solution,.83%. However, in the good economy, the loglinearization responses are somewhat stronger than the projection responses. The loglinearization response to a negative impulse is.6%, which is twice as large as the projection response,.8%. The intuition is that loglinearization misses the gradual nature of economic expansions captured by projection as illustrated in Figure 2. Panels D to F present quantitatively similar results for the impulse responses in the labor market tightness. From Panels B and E in Figure 3, the projection-based responses and the loglinearizationbased responses to one-standard-deviation shocks to log productivity are quantitatively similar in the median economy. It is natural to ask to what extend this finding holds up quantitatively in the face of larger impulses. To this end, Figure 4 reports the responses to negative two- and three-standard-deviation shocks to log productivity starting from the median economy. Panel A shows that, for the unemployment rate, the projection-based responses are larger than the loglinearization-based responses:.39% versus.32% as the maximum 7

19 response to a two-standard-deviation impulse and.6% versus.47% to a three-standarddeviation impulse. However, for the labor market tightness (Panel B), the two algorithms produce quantitatively similar responses even to large shocks in the median economy. 4.3 Accuracy Tests While it is natural to expect that projection would be more accurate than loglinearization, we are not aware of any prior attempt to quantify the approximation errors for loglinearization relative to an accurate global solution in the DMP model. In this subsection, we fill this gap Static Euler Equation Errors Following Judd (992) and Den Haan (2b), we calculate Euler equation errors, both static and dynamic. The static Euler equation error is defined as: e S t E t [β [ [ ]]] ( ) κt+ κt X t+ W t+ +( s) λ t+ λ t. (5) q t+ q t If an algorithm is accurate, e S t should be zero on all points in the state space. We calculate the static error on a fine employment-log productivity grid. To compute the conditional expectation in equation (5), we discretize x t via the Rouwenhorst (995) method and calculate the conditional expectation via matrix multiplication. 6 The grid size remains at 7. We create an evenly spaced grid of N t with, points (in contrast to the original,-point grid). We use linear interpolation for N t values that are not on the original grid. 6 We have experimented with the Gauss-Hermite quadrature in the continuous state space of x t when calculating the conditional expectation. However, when x t is extreme, implementing this method involves large extrapolation errors in the policy functions. For instance, to calculate the expectation conditional on the minimum value of the x t grid, which is precisely four unconditional standard deviations below the mean of zero, we need to put another grid of x t around this extreme x t value. As such, the lower half of the second grid lies outside the original x t grid, on which we solve the policy functions. Extrapolating the policy functions outside the original x t grid can induce large errors. However, these errors are irrelevant because the original grid is large enough to cover x t values within four unconditional standard deviations from the mean. 8

20 To calculate interpretable Euler equation errors per Den Haan (2), we compute two consumption (employment) values on each grid point. The first consumption (employment) value, C t (N t ), is computed from a given numerical solution method. The second consumption (employment) value, C S t (N S t ), is the implied consumption (employment) value from an accurately calculated conditional expectation, which is the right-hand side of the Euler equation. Denoting the deterministic steady state levels of consumption and employment as C ss and N ss, we define the static consumption and employment errors, respectively, as: e S Ct C t C S t C ss, and e S Nt N t N S t N ss. (6) Figure 5 reports these errors. From Panels A to C, the projection algorithm offers an extremely accurate solution to the model. In particular, e S t is in the magnitude of 3,andthe two interpretable errors are in the magnitude of 4. Panels D to F show that loglinearization exhibits large approximation errors: e S t is in the magnitude of % and can potentially go up to 2%. Even at the deterministic steady state (employment =.947 and log productivity = zero), e S t is 3.63%. The consumption error, e S Ct, ranges from 8% to 2% and remains at 2.3% at the steady state. The employment error, e S Nt, ranging from slightly below.3% to 3.64%, is the smallest among the three errors, and its level is.79% at the steady state. Finally, Panels G to I show that the second-order perturbation in logs can be even more inaccurate than loglinearization. Although at the deterministic steady state, the three errors are close to those under loglinearization, the errors can be larger once the economy wonders away from the steady state. The static Euler equation error varies from 75% to 25%, the consumption error from 2% to %, and the employment error from.5% to 8.5%. 9

21 4.3.2 Dynamic Euler Equation Errors The static errors are only one-period ahead forecast errors, which ignore the accumulation of small errors over time. To allow for this possibility, we follow Den Haan (2) to calculate dynamic errors. We simulate two series for C t and N t. First, we use a given algorithm in simulations. Second, we simulate alternative series using the following steps in each period: (i) Use a given algorithm to calculate the conditional expectation accurately; (ii) use this conditional expectation to calculate implied consumption value, Ct D ; and (iii) compute employment, Nt D, from this implied consumption value and the resource constraint. The dynamic consumption and employment errors are defined, respectively, as: e D Ct C t C D t C ss and e D Nt N t N D t N ss. (7) Figure 6 shows that the differences in accuracy between projection and perturbations are even starker in dynamic Euler equation errors. We simulate one million weekly periods from the model s stationary distribution under a given algorithm and report the empirical cumulative distribution functions of e D Ct and ed Nt. From Panels A and B, the dynamic errors from the projection algorithm are extremely small (in the magnitude of 4 ). In contrast, Panels C and D report very large dynamic errors for loglinearization. For the consumption errors, the mean is 73.78%, and the median is 73.5%. For the employment errors, the mean is 74.2%, and the median is 73.79%. Finally, the dynamic errors from the second-order perturbation in logs are only slightly smaller (Panels E and F). 2

22 5 Reconciling with the HM Results As noted, HM in fact use a global algorithm to solve their model. In this section, we address two issues. First, why the results based on their global method differ from those from our projection algorithm (Section 5.). Second, we ask whether there exists a different small surplus calibration (but in the same spirit of HM) that can explain the Shimer puzzle when solved accurately with our projection algorithm (Section 5.2). 5. The Importance of Choosing Markov-chain Approximations to the Highly Persistent Productivity Process Our projection algorithm differs from the HM algorithm in a crucial aspect. While we follow Kopecky and Suen (2) in using the Rouwenhorst (995) method with a grid of 7 points to approximate the log productivity, HM use the Tauchen (986) method with 35 grid points. We show that this difference goes a long way in reconciling our results with theirs. In Table 2, we report labor market volatilities under alternative Markov-chain approximations to the log productivity, x t. Panel A shows the results under the Rouwenhorst approximation but with a small grid of only five points. The five-point grid covers a range of x t that is precisely two unconditional standard deviations above and below the unconditional mean of zero. (The range is four standard deviations with the 7-point grid.) We choose the five-point grid to be consistent with the HM implementation of the Tauchen method that also covers a range of two standard deviations above and below the mean. Panel A shows that the quantitative results from the Rouwenhorst method are (relatively) robust with respect to the range of the grid. Even with only five points, the unemployment volatility is.22, which is not far from.25 with the 7-point grid (Panel D in Table ). 2

23 The volatility of the market tightness is.267, which is close to.27 from the large grid. Finally, the unemployment-vacancy correlation is.66, which is not far from.564 from the large grid. These results on the DMP model echo those reported in Kopecky and Suen (2) in the context of the stochastic growth model. Panels B to D in Table 2 show that the quantitative results from the Tauchen method are extremely sensitive to the range of the grid chosen. From Panel B, when the range of the grid covers two standard deviations above and below the conditional mean as in HM, the results from the Tauchen method are largely similar to those reported in their Table 4. For instance, the unemployment volatility is.54, which is close to.45 in their Table 4. We calculate the unemployment-vacancy correlation as.698, which is close to their estimate of.724. The volatility of the market tightness is.246, which is not far from their.292. Panels C and D shows that enlarging the range of the Tauchen grid raises the unemployment volatility but dampens the unemployment-vacancy correlation. As we vary the grid range from two to three and four unconditional standard deviations above and below the mean, the unemployment volatility increases from.54 to.267 and.39, and the unemployment-vacancy correlation falls in magnitude from.698 to.562 and.52, respectively. The only difference across Panels B to D is the range of the Tauchen grid, as all three panels use our projection algorithm. As such, these results cast doubt on the Tauchen method in approximating highly persistent autoregressive processes. 7 7 Kopecky and Suen (2) also note that the performance of the Tauchen (986) method is extremely sensitive to the choice of a free parameter that determines the range of the discrete state space. Their results are based on the stochastic growth model. We reinforce their conclusion in the context of the DMP model. 22

24 5.2 Restoring the Small Surplus Calibration Table shows that when solved accurately, the HM small surplus calibration matches the volatility of the labor market tightness but overshoots the unemployment volatility by almost 5%. It is natural to ask whether there exists a different small surplus calibration (but in the same spirit of HM) that can explain the Shimer puzzle, when we solve the model accurately with our projection algorithm. The glass is half full, at least. When we reduce the flow value of unemployment slightly from.955 in HM to.94, the unemployment volatility falls to.25, which is identical to that in the data (see HM s Table 3). As such, the key insight that small profits increase the unemployment volatility survives our computational scrutiny. The volatility of the market tightness is only.9 in the model, which is about 36% lower than that in the data,.259. However, to the extent that the quality of the unemployment data might be higher than the quality of the available vacancy data, the succuss of the HM calibration in matching the unemployment volatility seems more important than its failure in matching the volatility of the market tightness. 6 Cobb-Douglas Matching Function HM use the matching function first introduced by Den Haan, Ramey, and Watson (2) with the property that the job finding rate, f t, and the vacancy filling rate, q t, both fall between zero and one. As such, no additional constraints are necessary to ensure that the two matching probabilities are properly bounded. In contrast, the Cobb-Douglas matching function does not have this property. We show that loglinearization still diverges from the projection algorithm in terms of labor market moments in the DMP model with the Cobb- Douglas matching function. However, the differences are smaller than those with the Den 23

25 Haan-Ramey-Watson matching function, at least in some specifications. 6. The Model with the Cobb-Douglas Matching Function The Cobb-Douglas matching function is given by: G(U t,v t )=χu α t V α t, (8) in which χ> is the level parameter, and α (, ) is the constant elasticity of the matching process to unemployment, U t. Equation (8) implies that the job finding rate is: f(θ t )=χθ α t (9) and the vacancy filling rate is: q(θ t )=χθ α t. (2) The matching probabilities are bounded below by zero but not bounded above by one. To impose the upper bound, we define an upper bound for the market tightness, denoted θ, such that f(θ) =. Equation (9) implies θ = χ /(α ). We also define a lower bound for the market tightness, θ, such that q(θ) =. Equation (2) implies θ = χ /α. The corresponding lower and upper bounds for vacancy, V t, are then given by V t θu t and V t θu t. As such, f t is equivalent to V t V t and q t is equivalent to V t V t. Again because q t >, in terms of new hires, these constraints are equivalent to q t V t q t V t and q t V t q t V t. Let λ t and λ t be the Lagrangian multipliers associated with q t V t q t V t and q t V t q t V t, respectively. The intertemporal job creation condition is given by: [ ( ( ))] κ κ q(θ t ) λ t + λ t = E t β X t+ W t+ +( s) q(θ t+ ) λ t+ + λ t+, (2) 24

26 in which we simplify HM s procyclical vacancy costs, κ t,tobeaconstant,κ. The optimal vacancy policy should also satisfy the Kuhn-Tucker conditions: V t V t, λ t, and λ t V t = ; (22) V t V t, λ t, and λ t V t =. (23) We continue to approximate the conditional expectation in the right-hand side of equation (2) as E t = E(N t,x t ). After obtaining E t, we calculate q t = κ/e t. There are three scenarios in regard to q t. First, if q t q(θ), then V t V t is binding. We set q t = q(θ),θ t = θ, and V t = θu t. The lower bound for vacancies is not binding. As such, λ t =andλ t = E t κ/q t. Second, if q(θ) < q t <, then neither constraint is binding for vacancies. We set λ t = λ t = and q(θ t )= q t.wethensolveforθ t = q ( q t ), in which q ( ) is the inverse function of q( ) from equation (2). We then set V t = θ t ( N t ). Finally, if q t, then the lower bound V t V t is binding. We set λ t =,θ t = θ,v t = θu t,q t =,andλ t = κ E t. 6.2 Quantitative Results We calibrate the model with the Cobb-Douglas matching function in weekly frequency as in HM. The general strategy is to match the mean unemployment rate and labor market volatilities under loglinearization and then use the projection algorithm with the same parameter values to quantify its differences in quantitative results. As common in business cycle studies, we calibrate the persistence of the labor productivity process to be ρ =.9957 and its conditional volatility to be.2. Following Pissarides (29), we set the curvature parameter, α, in the matching function to be.5 and the level parameter, χ, to be.7. For the constant vacancy cost, κ, we set it to be.584, which is HM s average procyclical vacancy cost. All the other parameter values are from the HM calibration. 25

27 6.2. Weekly, Both V t Constraints Under these parameter values, Panel A of Table 3 reports that the model solved with loglinearization produces labor market volatilities that come close to those in the data. The unemployment volatility is., and the volatility of the market tightness is.288. The mean unemployment rate is 6.46% (untabulated). Comparing Panels A and B shows that, relative to the projection algorithm, loglinearization still underestimates the unemployment volatility (. versus.5) but overestimates the magnitude of the unemployment-vacancy correlation (.86 versus.68). However, the biases are smaller in magnitude than those with the Den Haan-Ramey-Watson matching function (.8 versus.285 for the unemployment volatility and.86 versus.52 for the unemployment-vacancy correlation, see Table ). In untabulated results, we show that the second-order perturbation in logs brings the unemployment volatility to.24, which is closer to its accurate value of.5. However, the unemployment-vacancy correlations remains largely unchanged at.826. To dig deeper behind these quantitative results, Figure 7 reports the scatter plots of unemployment versus labor productivity in the DMP model with the Cobb-Douglas matching function. From Panel A, the model exhibits similar but weaker nonlinear dynamics than those with the Den Haan-Ramey-Watson matching function (see Panel G of Figure ). The unemployment rate seems capped at a level slightly higher than 2%, when the V t V t constraint (i.e., q t ) starts to bind. The frequency of this constraint binding is.% in the models s simulations. The other constraint, V t V t (i.e., f t ), is never binding Daily, Both V t Constraints Despite the low frequencies with which the q t constraint is binding, it might be possible that the effective cap imposed by the constraint on the unemployment rate dampens 26