## _wp11117 ## Source details **Canonical URL:** [_wp11117](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11117.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11117.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11117.pdf.json) --- ### Introduction - Context - Real business cycle (RBC) models impose strong discipline on consumption and hours; optimal hours equate the marginal rate of substitution between consumption and leisure (mrs) and the marginal productivity of labor (mpl). - Empirical evidence shows a substantial wedge between mrs and mpl that co-varies with the cycle; Chari Kehoe and McGrattan (2007) (CKM) find that, along with the efficiency wedge, the labor wedge accounts for most output fluctuations. - Motivation - CKM’s finding is interpreted as indicating misspecification of the prototype RBC model regarding the labor market. - Search and matching frictions (Mortensen and Pissarides 1994, Pissarides 2000) introduce wedges between the wage and both mpl and mrs, providing a framework to address labor-market misspecification. ### Model and Mechanisms - Model setup - Embeds search and matching frictions into a standard RBC framework, nesting a prototype RBC model a la CKM. - Incorporates Nash bargaining between the firm and the marginal worker over hours and wage. - Mechanism insights - Nash bargaining alters the firm’s perceived benefit of an additional hour at the intensive margin (hours per employed worker), valued via additional marginal output per worker. - Search frictions, internalized at the wage bargaining stage, do not directly alter the static equation that determines the labor wedge. - Fluctuations in total hours are decomposed into intensive (h_t) and extensive (n_t) margins; the mrs and thus the measured labor wedge differ from the prototype RBC implication. ### Wage, Wedges, and Key Relations - Wage per hour (compact form, notation preserved) - w_t = (1 - ϵ) mpl_t + ϵ mrs_t / (1 - _{l,t}) ϕ_t + β_t / h_t (equation 27 form) - Alternative highlighting wedges: w_t = (1 - ) mpl_t +  mrs_t / (1 - _{l,t}) + _t / h_t (equations 28–29 context) - Static intratemporal relation in search model - mrs_t = mpl_t [1 - _{l,t}] s^L_t (equation 30) - s^L_t = 1 + f_{ll} l_t / f_l ∈ [0,1] - Measured wedges: prototype vs. search model - Prototype: [1 - "_{l,t}] = mrs_t(l_t) / mpl_t = - mpl_t U_c(c_t) / G_h(n_t h_t) (equation 33) - Search model: [1 - _{l,t}] = mrs_t(h_t) / mpl_t × 1/(1 - β) = - mpl_t U_c(c_t) / G_h(h_t) × 1/(1 - β) (equation 34) - Mapping under constant Frisch elasticity G(x) = -x^{(1+ε)}/(1+ε): [1 - "_{l,t}] = (1 - _{l,t}) (1 - β) n_t^{ε} (equation 35) - Conceptual point - Prototype wedge procyclicality may partly reflect procyclical employment (extensive margin) rather than a pure labor-tax-like distortion. ### Quantitative Calibration and Parameters (values preserved) - General/calibration notes - Discount factor: = 0.99 (simulation calibration). - Depreciation:  = 0.025. - Production curvature (Cobb-Douglas labor exponent): = 0.35 (baseline) and = 0.36 (simulation calibration). - Target steady-state tax wedge for "_{l} (or _{l}) = 0.4. - Frisch-elasticity cases - High Frisch elasticity: Frisch ≈ 2.8 (prototype ε = 1 limiting case; search model ε^e = 0.6 to match steady-state). - Low Frisch elasticity target: average Frisch = 0.5 → requires ε = 5.54 in (36) for prototype and ε^e = 3.33 in (37) for search model. - Search-friction calibration (simulation) - Vacancy posting cost share:  v / y = 0.015 (1.5 percent). - Worker search-cost share: c(e)(1 - n) / y = 0.005 (0.5 percent). - Employment n = 0.7074. - Hours per worker h = 0.3752. - Vacancy-fill probability q = 0.9. - Matching elasticity ρ = 0.5; Hosios condition enforced: ρ = . - Quarterly job-destruction rate: ϵ = 0.15. - Convexity parameter in search-cost function implied  = 3. ### Main Quantitative Findings (exact magnitudes and comparisons) - Directional and magnitude findings - For a Frisch elasticity of 2.8, variability of the labor wedge can decline by up to 20 percent. - For Frisch elasticities more consistent with micro estimates, the decline can be as large as 40 percent. - In a numerical exercise treating the search model as the data generating process, about 15 percent of the relative variation in the labor wedge and all its comovement with output and total hours could be explained by the misspecification that ignores search frictions and the intensive/extensive margin distinction. - Empirical/calibrated moments and comparisons - Low-frequency (HP-detrended): the search labor wedge is about 23 percent less volatile than the prototype labor wedge in the baseline high-Frisch calibration. - mrs_search is about 40 percent less volatile than mrs_proto (low-frequency). - Business-cycle frequency (HP-filtered except L_t) with high Frisch (≈ 2.78): the search labor wedge is 20 percent less volatile than the prototype wedge. - Correlations reduced in high-Frisch case: correlation of wedge with total hours reduced by about 8 percent; with output reduced by about 20 percent. - Low-Frisch case (Frisch ≈ 0.5 target): labor wedge volatility falls by more than 40 percent relative to prototype; reduction in correlation with output about 22 percent at business-cycle frequency. - Lower Frisch elasticity increases overall standard deviations across wedges; mrs volatility in low-elasticity search model is 1.46 times the mrs volatility in the prototype high-elasticity case. - Simulated-data experiment (search model as DGP; only productivity Z_t active; [1 - _{l,t}] constant at 0.6) - Simulated results show an econometrician applying the prototype measurement will recover a strongly procyclical labor wedge even when no exogenous labor wedge movements exist. ### Simulation Evidence on Procyclicality (exact figures preserved) - Simulation results (as reported) - corr([1" l;t ]; y t ) = 0.85 (actual data: 0.51) - corr([1" l;t ]; l t ) = 0.96 (actual data: 0.87) - std([1" l;t ]) = std(y t ) = 0.15 (actual data: 1.00) - Interpretation - A search-frictions extension is consistent with a significantly procyclical labor wedge measured under prototype methods. - Endogenous movements in the extensive margin (employment) and intensive margin (hours per worker) can generate observed procyclicality. - Business cycle accounting as in CKM can falsely detect a labor wedge when the true model is a labor search model. ### Mechanisms Driving Measurement Differences and Main Conclusions - Measurement and mechanism insights - Search frictions do not directly alter the static wedge between mpl and mrs; bargaining internalizes search frictions via wages rather than hours. - The mrs must be measured in terms of hours per worker; conflating hours per worker with total hours produces substantial mismeasurement of the mrs and a spurious labor wedge. - At business-cycle frequency, about 20 percent of the observed volatility and most of the procyclicality of the labor wedge can be attributed to fluctuations in the extensive margin (employment) through their effect on the mrs. - Implication for Frisch-elasticity debate - Using total hours in mrs measurement exacerbates divergence between macro and micro estimates of labor supply elasticity. - The search model can accommodate much lower Frisch elasticity values without dramatically increasing mrs volatility, explaining why macro estimates imply high labor supply elasticity. ### Policy-Relevant Implications and Modeling Guidance - Measurement caution - Interpreting the prototype labor wedge [1 - "_{l,t}] as a pure tax or distortion can be misleading when search frictions and extensive-margin employment fluctuations matter. - Modeling guidance - Macro accounting exercises that do not separate intensive and extensive margins risk attributing endogenous employment responses to exogenous wedges. - Empirical practice - The choice of Frisch elasticity materially affects the inferred quantitative importance of search frictions for explaining the measured labor wedge. - Simulated-economy exercises recovering wedges via prototype equilibrium conditions can produce spurious procyclical wedges even when none exist exogenously. ### Data and Empirical Implementation (exact details) - Data sources and sample - Real output (y_t), consumption (c_t), and government expenditures ("g;t) constructed from NIPA tables; government expenditures include net exports. - Employment (n_t) and average hours per worker (h_t) from Cociuba, Prescott and Ueberfeldt (2009). - Sample period: 1959:Q2 through 2010:Q3. - Series seasonally adjusted at an annualized rate when relevant. - Output and some components deflated by the GDP deflator; real output y_t defined as quarterly GDP net of sales taxes; consumption c_t is sum of non-durable goods purchases and services; "g;t lumps government consumption and net exports. - Military hours and employment incorporated into totals following Cociuba, Prescott and Ueberfeldt (2009). - Data pulled from Haver Analytics database. ### Household and Firm First-Order Conditions (selected equations preserved) - Representative household FOCs (selected) - Consumption Euler: U_c(c_t) [1 + _{x,t}] = E_t[ U_c(c_{t+1}) (r_{t+1} + (1 - )[1 + _{x,t+1}]) ] (equation 17) - Search-effort Euler: U_c(c_t) c_e(e_t) = p_t = E_t[ U_c(c_{t+1})[ w_{t+1} (1 - _{l,t+1}) h_{t+1} + c(e_{t+1}) ] + G(h_{t+1}) + U_c(c_{t+1}) c_e(e_{t+1}) p_{t+1} (1 - ϵ - p_{t+1} e_{t+1}) ] (equation 18) - Firm FOCs (selected) - Capital: "_{z,t} f_k(k_t, n_t h_t) - r_t = 0 (equation 21) - Vacancy posting:  / q_t = ~ _t E_t[ "_{z,t+1} f_l(k_{t+1}, n_{t+1} h_{t+1}) h_{t+1} - w_{t+1} h_{t+1} + (1 - ϵ)  / q_{t+1} ] (equation 22) - Employment contract and bargaining - Worker surplus: W^h_n_t equations (45); firm surplus: W^f_n_t equations (46). - Nash bargaining FOCs determine wage and hours; algebra yields explicit wage equation (26) and labor wedge equation (30). *Source: _wp11117 - Section 4 and Appendix material (IMF working paper content)._ ### 1. Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 ### 1. Appendix 1 ### Introduction - Context - Real business cycle (RBC) models (i.e. Kydland and Prescott, 1982) impose a strong discipline on the choice of consumption and hours, both intertemporally and intratemporally. - In these models, the optimal choice of hours is determined in equilibrium such that the marginal rate of substitution between consumption and leisure (mrs) is equal to the marginal productivity of labor (mpl). - Empirical evidence shows a substantial wedge between these two quantities that strongly co-varies with the economic cycle; Chari Kehoe and McGrattan (2007) (CKM) conclude that, along with the e¢ ciency wedge, the labor wedge accounts for most of the áuctuations in output. - Motivation - The paper interprets CKM’s finding as an indication of a signiÖcant misspeciÖcation of the prototype RBC model as it relates to the labor market. - Search and matching frictions (Mortensen and Pissarides 1994, Pissarides 2000) introduce a wedge between the wage and both the mpl and the mrs, providing a natural framework to address misspeciÖcation related to labor market imperfections. ### Model and Mechanisms - Model setup - The paper presents a model with labor market frictions in the form of search and matching that nests a prototype RBC model a la CKM. - Nash bargaining between the Örm and the marginal worker over hours and wage is incorporated. - Mechanism insights - Nash bargaining alters the Örmís perceived beneÖt of an additional hour at the intensive margin (i.e. hours per employed worker), which is valued in terms of additional marginal output per worker. - Search frictions, internalized at the wage bargaining stage, do not affect the equation that determines the labor wedge directly or explicitly. - Since fluctuations in total hours can now be attributed to both the intensive and extensive margin, the marginal rate of substitution, and thus the labor wedge, di§er from the one implied by the prototype RBC model. ### Key Findings and Quantitative Results - Directional effects - The modification due to search and matching is in the right direction: the labor wedge obtained is less variable and less procyclical than the prototype labor wedge. - The result is sensitive to the parameterization of labor supply elasticity (Frisch elasticity). - Numerical magnitudes preserved exactly as reported - For a Frisch elasticity of 2.8, as in most macro models, variability of the labor wedge can decline by up to 20 percent. - For Frisch elasticities more consistent with micro estimates, the decline can be as large as 40 percent. - In a numerical exercise treating the search model as the data generating process, about 15 percent of the relative variation in the labor wedge and all its comovement with output and total hours could be explained by the misspeciÖcation that ignores search frictions and the explicit distinction between the intensive and the extensive margin. - Interpretation - Even though theoretically there is no labor wedge in the simulated data, an econometrician who ignores search frictions and the intensive/extensive margin distinction can falsely measure a signiÖcantly procyclical and variable labor wedge. ### Simulation Exercise and Business Cycle Accounting Context - Purpose - The simulation treats the search model as the data generating process and examines the behavior of the labor wedge that an econometrician would recover in business cycle accounting. - Outcome - Demonstrates that ignoring search frictions and margins of adjustment yields spurious measurement of the labor wedge, explaining a nontrivial share of its measured variability and comovement with output and total hours. - Relation to CKM framework - Places the exercise within CKM’s business cycle accounting approach, which identifies four wedges: the e¢ ciency, labor, investment and government consumption wedge; CKM consider the labor and e¢ ciency wedge the most important for explaining real macro áuctuations. *Source: _wp11117 - 1. Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25* ### Section 4 presents an extension of the prototype RBC model with search frictions and ### _wp11117 - Section 4 presents an extension of the prototype RBC model with search frictions and ### Extension overview and relation to literature - The paper embeds search and matching frictions into a standard RBC framework (building on Andolfatto (1996) and Merz (1995, 1999)) and analyzes implications for the measured labor wedge. - Key literature connections: - Business-cycle accounting and the labor wedge (CKM lineage; Hall 1997; Shimer 2009). - Alternative explanations for labor-market distortions: labor market institutions, competitive structure, price-wage markup, regulation, tax policy (Cole and Ohanian 2002; Gali, Gertler and Lopez-Salido 2007; Mulligan 2002; Rotemberg and Woodford 1991, 1999). - Aggregation/heterogeneity explanations (Chang and Kim 2007; Arseneau and Chugh 2010). - Recent work stressing matching efficiency as a driver of labor wedge variation (Cheremukhin and Restrepo-Echavarria 2010). - The authors favor a good-intensive recruitment technology (firms respond to productivity shocks by increasing recruitment during booms), unlike formulations that imply neutrality (Blanchard and Gali 2010; Shimer 2010). ### Prototype RBC model (competitive labor market) - Representative household problem (separable utility, complete markets): - Lifetime utility: E0 Σ_{t=0}^∞ ^t [U(c_t) + G(h_t)]; 0 < < 1. - Budget constraint includes wedges/taxes on investment and labor earnings: [1 + "_{x,t}] and [1 - "_{l,t}]. - Competitive firm problem: - Production: "_{z,t} f(k_t, l_t) with l = h × n (hours per worker × employment). - Competitive equilibrium conditions (selected): - Consumption Euler: U_c(c_t) [1 + "_{x,t}] = E_t[ U_c(c_{t+1})["_{z,t+1} f_k(k_{t+1}, l_{t+1}) + (1 - )] (1 + "_{x,t+1}) ] (equation 6) - Labor market (intratemporal) condition: mrs_t(l_t) = -G_h(l_t) / U_c(c_t) = mpl_t [1 - "_{l,t}] (equation 7) - Resource constraint: "_{z,t} f(k_t, l_t) = c_t + k_{t+1} - (1 - ) k_t + "_{g,t} (equation 8) - Measured prototype labor wedge: [1 - "_{l,t}] = mrs_t / mpl_t (equation 33). - Empirical observation: the prototype labor wedge [1 - "_{l,t}] is procyclical (falls during recessions) and positively correlated with per capita hours worked (Figure 1, 1959:I to 2010:III). ### Search-frictions extension: structure and first-order conditions - Key features: - Aggregate matching function M_t = _{m,t} V_t [(1 - N_t) E_t]^{1-ρ}, 0 < ρ < 1, with matching efficiency _{m,t}. - Unemployment probability for workers: p_t = M_t / (1 - N_t) E_t. - Law of motion for employment: N_{t+1} = (1 - ϵ) N_t + M_t, ϵ ∈ [0,1]. - Aggregate representative household objective: E0 Σ ^t [U(c_t) + n_t G(h_t)] (equation 9). - Search costs: c(e_t) per unemployed worker exerting effort e_t; search efforts and vacancies create an extensive margin. - Representative household FOCs (selected): - Consumption Euler: U_c(c_t) [1 + _{x,t}] = E_t[ U_c(c_{t+1}) (r_{t+1} + (1 - )[1 + _{x,t+1}]) ] (equation 17) - Search-effort Euler (optimal search): U_c(c_t) c_e(e_t) = p_t = E_t[ U_c(c_{t+1})[ w_{t+1} (1 - _{l,t+1}) h_{t+1} + c(e_{t+1}) ] + G(h_{t+1}) + U_c(c_{t+1}) c_e(e_{t+1}) p_{t+1} (1 - ϵ - p_{t+1} e_{t+1}) ] (equation 18) - Firms: - Vacancy posting cost per vacancy:  units of output; free-entry drives value of a vacancy to zero. - Firm FOCs (selected): - Capital: "_{z,t} f_k(k_t, n_t h_t) - r_t = 0 (equation 21) - Vacancy posting condition:  / q_t = ~ _t E_t[ "_{z,t+1} f_l(k_{t+1}, n_{t+1} h_{t+1}) h_{t+1} - w_{t+1} h_{t+1} + (1 - ϵ)  / q_{t+1} ] (equation 22), where q_t is vacancy fill probability and ~ _t = E_t[ U_c(c_{t+1}) / U_c(c_t) ]. - Nash bargaining over the marginal surplus of a match determines (w_t, h_t). Generalized Nash bargaining problem yields: - Sharing condition linking marginal surpluses (equation 24). - Hours FOC: U_c(c_t) (1 - _{l,t}) [ "_{z,t} f_l + "_{z,t} f_{ll} n_t h_t ] + G_h(h_t) = 0 (equation 25). ### Wage equation, wedges, and interpretation - Wage per hour (compact form): - w_t = (1 - ϵ) mpl_t + ϵ mrs_t / (1 - _{l,t}) ϕ_t + β_t / h_t (equation 27 form; notation preserved in text). - Rewritten highlighting wedges: - w_t = (1 - ) mpl_t +  mrs_t / (1 - _{l,t}) + _t / h_t (equations 28–29 context). - Static relation between mrs and mpl in search model: - mrs_t = mpl_t [1 - _{l,t}] s^L_t (equation 30) - s^L_t = 1 + f_{ll} l_t / f_l ∈ [0,1] (output elasticity to total hours; approximates labor share). - Key conceptual point: - Search frictions introduce time-varying wedges between the wage and both mpl and mrs, but when equating supply and demand the explicit search-frictions terms (and bargaining-power parameter  and term ϕ) cancel, yielding equation (30). Thus the static intratemporal labor wedge is shaped by (1 - _{l,t}) and s^L_t rather than directly by search frictions. - Distinction in measurement: prototype labor wedge uses total hours l_t (no separation of intensive/extensive margins), whereas search model labor wedge uses hours per worker h_t (separates intensive n_t and extensive margins). The two measured wedges: - Prototype: [1 - "_{l,t}] = mrs_t(l_t) / mpl_t = - mpl_t U_c(c_t) / G_h(n_t h_t) (equation 33) - Search model: [1 - _{l,t}] = mrs_t(h_t) / mpl_t × 1/(1 - β) = - mpl_t U_c(c_t) / G_h(h_t) × 1/(1 - β) (equation 34) - Mapping relationship under constant Frisch elasticity G(x) = -x^{(1+ε)}/(1+ε): - [1 - "_{l,t}] = (1 - _{l,t}) (1 - β) n_t^{ε} (equation 35) - Implication: procyclicality in the prototype wedge may partly reflect procyclical employment (extensive margin) rather than a true labor-tax-like distortion. ### Quantitative experiments and main empirical/statistical findings - Two-part quantitative strategy: 1. Use U.S. data to recover prototype labor wedge ([1 - "_{l,t}]) and search labor wedge ([1 - _{l,t}]) from equations (33) and (34) respectively; calibrate G(·), β, and Frisch-elasticity parameters to match steady-state and elasticities. 2. Simulate artificial data from calibrated search model (shut down all exogenous wedges except productivity Z_t, so [1 - _{l,t}] constant at 0.6) and then compute the prototype labor wedge as an econometrician would (CKM style). - Calibration notes (as used in mapping to U.S. data and simulations): - β (discount factor) used in general modelic contexts: = 0.99 (simulation calibration). - Depreciation:  = 0.025. - Production curvature: β (notation for Cobb-Douglas exponent) = 0.35 or 0.36 depending on context; baseline = 0.35 to match average labor share in prototype model, = 0.36 in simulation calibration. - Target steady-state tax wedge for "_{l} (or _{l}) = 0.4 as in Prescott (2004) when calibrating preference parameter that pins down steady-state. - Frisch-elasticity cases considered: - High Frisch elasticity (limiting case) ε = 1 in prototype → steady-state Frisch ≈ 2.8; search model ε^e = 0.6 to match steady-state elasticity. - Low Frisch elasticity target: average Frisch = 0.5 → requires ε = 5.54 in (36) for prototype and ε^e = 3.33 in (37) for search model (sample averages). - Search-friction calibration (simulation): - Vacancy posting cost share:  v / y = 0.015 (1.5 percent). - Worker search-cost share: c(e)(1 - n) / y = 0.005 (0.5 percent). - Employment n = 0.7074, hours per worker h = 0.3752, vacancy-fill probability q = 0.9. - Matching elasticity ρ set to 0.5; Hosios condition enforced: ρ =  (workers’ bargaining power equals matching elasticity). - Quarterly job-destruction rate: ϵ = 0.15. - Convexity parameter in search-cost function implied  = 3 (from search/recruitment cost ratios). - Main quantitative findings (empirical moments and comparative statements): - Using U.S. data and focusing on the single intratemporal equilibrium condition: - The search model can account for somewhere between 15 to more than 40 percent of the fluctuations in the prototype labor wedge (text summary). - Low-frequency (HP-detrended) comparison: - The search labor wedge is about 23 percent less volatile than the prototype labor wedge in the baseline high-Frisch calibration (Figure 4; Table 1). - This reduction is driven by mrs measurement: mrs_search is about 40 percent less volatile than mrs_proto. - Business-cycle frequency (HP-filtered except L_t): - With high Frisch elasticity (≈ 2.78), the search labor wedge is 20 percent less volatile than the prototype wedge (Table 1 business-cycle results). - Correlation of the wedge with output and total hours is reduced when using the search model: correlation with total hours reduced by about 8 percent; with output reduced by about 20 percent (high-Frisch case). - Low-Frisch-elasticity case (Frisch ≈ 0.5 target): - Search frictions amplify the reduction: labor wedge volatility falls by more than 40 percent relative to prototype (Figures 6–7; Table 2). - Reduction in correlation with output about 22 percent at business-cycle frequency. - Lower Frisch elasticity increases overall standard deviations across wedges but narrows the gap between macro-implied elasticities and micro estimates: mrs volatility in low-elasticity search model is 1.46 times the mrs volatility in the prototype high-elasticity case (text summary). - Simulated-data experiment (null: no exogenous labor wedge movements; [1 - _{l,t}] constant at 0.6): - The authors simulate the calibrated search model (only productivity Z_t shocks active) and then compute the prototype labor wedge as an econometrician would using consumption, output, and labor. - The simulation shows that even when the true economy has no exogenous labor wedge movements, the econometrician measuring [1 - "_{l,t}] from the prototype-equation will still find a strongly procyclical labor wedge (Figure 8; standard-deviation comparison referenced but truncated at document end). - Interpretation: - Much of the observed procyclicality and volatility of the prototype labor wedge may be an artifact of aggregation and of conflating intensive and extensive margins (total hours vs. hours per worker). - Search frictions alter the measurement of mrs primarily through the intensive/extensive margin decomposition and the endogenous allocation of surplus via wages and employment decisions. - The intratemporal labor wedge (as typically measured) is inherently intratemporal; search frictions are primarily intertemporal and their effects are absorbed into wages and the extensive margin rather than directly appearing in the static labor wedge expression. ### Policy-relevant implications and avenues for interpretation - Measurement caution: - Interpreting the prototype labor wedge [1 - "_{l,t}] as a pure tax or a pure distortion can be misleading when search frictions and extensive-margin employment fluctuations are important. - Modeling guidance: - Macroeconomic accounting exercises that do not separate intensive and extensive margins risk attributing endogenous employment responses to exogenous wedges. - Calibration and empirical practice: - The choice of Frisch elasticity materially affects the inferred quantitative importance of search frictions for explaining the measured labor wedge (high vs. low Frisch cases produce large differences in wedge volatility). - Simulated-economy exercises that recover wedges using the prototype equilibrium conditions can produce spurious procyclical wedges even when none are present exogenously. *Italicized source: IMF working paper section content (_wp11117 - Section 4 presents an extension of the prototype RBC model with search frictions and).* ### 0.15 which makes the null hypothesis of no áuctuations clearly rejected. In other words, ### _wp11117 - 0.15 which makes the null hypothesis of no áuctuations clearly rejected. In other words, ### Simulation evidence on the labor wedge and procyclicality - Simulation results indicate a strongly procyclical labor wedge: - corr([1" l;t ]; y t ) = 0.85 (actual data: 0.51) - corr([1" l;t ]; l t ) = 0.96 (actual data: 0.87) - std([1" l;t ]) = std(y t ) = 0.15 (actual data: 1.00) - Interpretation: - A simple extension of the prototype RBC model with search frictions is consistent with a significantly procyclical labor wedge. - Endogenous movements in both the extensive margin (employment) and the intensive margin (hours per worker) can generate the observed procyclicality. - Doing business cycle accounting as in CKM would falsely detect the presence of a labor wedge when reality is well described by the labor search model. ### Main conclusions on mechanisms driving the labor wedge - Labor-market search frictions per se do not directly alter movements in the wedge between mpl and mrs as observed in the data; bargaining internalizes search frictions through the wage decision rather than the hours decision. - Key measurement and mechanism insights: - The mrs must be measured in terms of hours per worker. - Confounding hours per worker with total hours leads to substantial mismeasurement of the mrs and a misspecification that appears as a labor wedge. - At business cycle frequency, about 20 percent of the observed volatility and most of the procyclicality of the labor wedge can be attributed to fluctuations in the extensive margin (employment) through their effect on the mrs. - Implication for labor supply elasticity debate: - Using total hours in the measurement of the mrs exacerbates the divergence between macro and micro estimates of the labor supply elasticity. - The search model can accommodate much lower values for the Frisch elasticity without dramatically increasing the volatility of the mrs, explaining why macro estimates give high values of the labor demand elasticity. ### Data used in the exercises - Data sources and sample: - Real output (y t ), consumption (c t ), and government expenditures ( " g;t ) constructed from NIPA tables; government expenditures include net exports. - Labor market variables employment (n t ) and average hours per worker (h t ) taken from Cociuba, Prescott and Ueberfeldt (2009). - Sample period: 1959:Q2 through 2010:Q3. - Seasonally adjusted at an annualized rate when relevant. - Output and some components deflated by the GDP deflator; real output y t defined as quarterly GDP net of sales taxes; consumption c t is the sum of non-durable goods purchases and services; " g;t lumps government consumption and net exports. - Military hours and employment are incorporated into total hours and total employment figures following Cociuba, Prescott and Ueberfeldt (2009). - Data pulled from Haver Analytics database. ### Household decision problem — first-order conditions and implications - First-order necessary conditions and costates (equations as in source): - x t : U c (c t ) [1 +  x;t ] + E t W h k t+1 ( ! h t+1 j ! h t ) @k t+1 @x t = 0 (38) - e t : U c (c t ) c e (e t ) (1n t ) + E t W h n t+1 ( ! h t+1 j ! h t ) @n t+1 @e t = 0 (39) - k t : W h k t ( ! h t j ! h t1 ) = U c (c t ) r t + E t W h k t+1 ( ! h t+1 j ! h t ) @k t+1 @k t (40) - n t : W h n t ( ! h t j ! h t1 ) = U c (c t ) [w t [1 l;t ] h t + c(e t )] + G(h t ) + E t W h n t+1 ( ! h t+1 j ! h t ) @n t+1 @n t (41) - Combining (38) and (40) yields the consumption Euler equation (17) in the text. - Equation (39) implies E t W h n t+1 ( ! h t+1 j ! h t ) = U c t (c t ) c e t (e t ) p t. Substituting and taking expectations yields the household's choice of effort (18). ### Firms' decision problem — first-order conditions and implications - First-order necessary conditions and costates (equations as in source): - k t :  z;t f k (k t ; n t h t )r t = 0 (42) - v t :  + q t ~ E t W f ( ! f t+1 j ! f t ) = 0 (43) - k t+1 : W f n t ( ! f t j ! f t1 ) =  z;t f k (k t ; n t h t ) h t w t h t + (1) ~ t E t W f n t+1 ( ! f t j ! f t1 ) (44) - Equation (42) gives the equilibrium rental rate on capital (21). - Equation (43) implies ~ t E t W f n t+1 ( ! f t+1 j ! f t ) =  q t; substitution and expectations yield the firm’s choice of vacancies (22). ### Employment contract, bargaining, and the labor wedge - Marginal surpluses for worker and firm: - W h n t ( ! h t ) = U c (c t ) [w t [1 l;t ] h t + c(e)] + G(h t ) + (1p t e) E t W h n t+1 ( ! h t+1 j ! h t ) (45) - W f n t ( ! f t ) =  z;t f l t (k t ; n t h t ) h t w t h t + (1) ~ t E t W f n t+1 ( ! f t+1 j ! f t ) (46) - Nash bargaining first-order conditions determine optimal wage and hours: - w t :  W f nw ( ! f t ) / W f n ( ! f t ) + (1) W h nw ( ! h t ) / W h n ( ! h t ) = 0 - h t :  W f nh ( ! f t ) / W f n ( ! f t ) + (1) W h nh ( ! h t ) / W h n ( ! h t ) = 0 - Using (45)-(46) and algebra yields the explicit wage equation (26) in the text and the labor wedge equation (30). *Source: _wp11117 - 0.15 which makes the null hypothesis of no áuctuations clearly rejected. In other words,* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11117.pdf_