Stock-Flow Matching with Heterogeneous Workers and Firms: Theory and Evidence from the U.K.

Transcription

1 Stock-Flow Matching with Heterogeneous Workers and Firms: Theory and Evidence from the U.K. Carlos Carrillo-Tudela (Essex) William Hawkins (Yale) Essex Dec. 7, 2017

2 Stock-flow matching 101 Unemployed workers and vacant jobs coexist because they cannot form sufficiently productive matches (not due to search frictions) Matching pattern: Stock of unemployed workers matches with inflow of vacant jobs Stock of vacant jobs matches with inflow of unemployed workers Theory: Taylor 95, Coles Smith 98, Coles 99, Ebrahimy Shimer 10 Natural explanation for several labor market phenomena: 1. long-duration unemployed match with newly posted vacancies Kettemann Mueller Zweimüller applications are concentrated at newly posted vacancies Banfi Villena-Roldán 16, Davis Samaniego de la Parra 17, Bélot Kircher Muller matching prob. for newly unemployed workers depends on stock of vacancies; matching rate for long-duration unemployed depends more on vacancy flow Coles Smith 98, Gregg Petrongolo 05, Kuo Smith duration dependence in unemployment and vacancy hazards Ebrahimy Shimer 10 Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 1

3 Stock-flow matching 101 Unemployed workers and vacant jobs coexist because they cannot form sufficiently productive matches (not due to search frictions) Matching pattern: Stock of unemployed workers matches with inflow of vacant jobs Stock of vacant jobs matches with inflow of unemployed workers Theory: Taylor 95, Coles Smith 98, Coles 99, Ebrahimy Shimer 10 Natural explanation for several labor market phenomena: 1. long-duration unemployed match with newly posted vacancies Kettemann Mueller Zweimüller applications are concentrated at newly posted vacancies Banfi Villena-Roldán 16, Davis Samaniego de la Parra 17, Bélot Kircher Muller matching prob. for newly unemployed workers depends on stock of vacancies; matching rate for long-duration unemployed depends more on vacancy flow Coles Smith 98, Gregg Petrongolo 05, Kuo Smith duration dependence in unemployment and vacancy hazards Ebrahimy Shimer 10 Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 1

4 Stock-flow matching 101 Unemployed workers and vacant jobs coexist because they cannot form sufficiently productive matches (not due to search frictions) Matching pattern: Stock of unemployed workers matches with inflow of vacant jobs Stock of vacant jobs matches with inflow of unemployed workers Theory: Taylor 95, Coles Smith 98, Coles 99, Ebrahimy Shimer 10 Natural explanation for several labor market phenomena: 1. long-duration unemployed match with newly posted vacancies Kettemann Mueller Zweimüller applications are concentrated at newly posted vacancies Banfi Villena-Roldán 16, Davis Samaniego de la Parra 17, Bélot Kircher Muller matching prob. for newly unemployed workers depends on stock of vacancies; matching rate for long-duration unemployed depends more on vacancy flow Coles Smith 98, Gregg Petrongolo 05, Kuo Smith duration dependence in unemployment and vacancy hazards Ebrahimy Shimer 10 Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 1

5 What we do 1. Document vacancy-filling and job-finding hazards for low-skill segment of U.K. labor market using administrative data for Aggregate Vacancy Hazard 0.04 Aggregate Daily Unemployment Hazard Weeks Days 2. Develop a stock-flow matching model to account for these hazards Arbitrary ex ante heterogeneity of workers and firms Endogenize set of acceptable matches 3. Application: how does the unemployment duration distribution change with labor market conditions? Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 2

6 When you referee this paper, say It makes stock-flow matching as easy as falling off a bicycle. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 3

7 U.K. Data

8 Plan: Data 1. Vacancy durations: data construction; hazards 2. Unemployment durations: data construction; hazards 3. Motivating a stock-flow model Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 4

9 Vacancy Duration Data Jobcentre Plus (JCP) administers Jobseekers Allowance (UI). Many vacancies targeting the unemployed are posted with JCP. During , JCP collected higher quality vacancy duration information, using the following process: 1. Firm posts vacancy with regional call center 2. Becomes visible to applicants immediately 3. Follow-up contact by Vacancy Services Manager (VSM) after 2-3 days, then every 7-10 days Vacancy ends either when agreed limit of applicants met or at follow-up if firm indicates vacancy no longer open. Data available Apr : {notified vacancies, inflow, outflow} by {industry, postcode, whether filled by JCP} and duration (binned: 1 week, 1 2, 2 4, 4 8, 8 13, 13 26, ). Coverage: 39 percent of all UK vacancies More Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 5

10 Vacancy hazards: filled by Jobcentre Plus vs. all outflows Note: hazard is shown as constant within each bin The two hazards are qualitatively similar; sharper initial drop for vacancies filled by JCP Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 6

11 Vacancy hazards: by occupation 0.16 Aggregate Vacancy Hazard 0.16 Low Skill Occupations Sales PPMO Elementary Weeks Weeks 0.16 Medium Skill Occupations 0.16 High Skill Occupations Admin Managers Skill Trade Professional 0.14 Personal 0.14 Assoc Prof Weeks Weeks Qualitatively similar; initial drop more pronounced in low-skill occupations Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 7

12 Vacancy hazards: by occupation and geographic region Low Skill Occupations Sales and Customer Service occupations Sales PPMO Elementary East East Midlands London North East North West South East South West West Midlands Yorkshire Wales Scotland Weeks Process, Plant and Machine Operatives Weeks Elementary Occupations East East 0.2 East Midlands London North East 0.2 East Midlands London North East North West North West South East South East 0.15 South West West Midlands Yorkshire 0.15 South West West Midlands Yorkshire Wales Wales Scotland Scotland Weeks Weeks Substantial heterogeneity in vacancy hazard shape across occupations and regions. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 8

13 Unemployment Duration Data Data: duration of active claims for Jobseekers Allowance (UI) Advantage: this is the population who are obliged to search using Jobcentre Plus job listings. Disadvantage: not all unemployed individuals are JSA claimants. (Take-up around 60%.) Process to claim: First Contact Financial Assessment Work-Focused Interview sign Jobseekers Agreement begin benefits 3-day waiting period for benefits Undercount short-duration spells? Spell duration (usually) measured from First Contact to first day of either starting work or otherwise becoming ineligible Common reasons for claim end: found work determined to be working 16 hours per week; failed to attend biweekly interview; unknown. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 9

14 Unemployment hazards: daily vs. weekly 0.04 Aggregate Daily Unemployment Hazard - restricted 0.04 Aggregate Daily Unemployment Hazard Days Days Aggregate Weekly Unemployment Hazard - restricted Aggregate Weekly Unemployment Hazard Weeks Weeks Top panels: daily duration data. Bottom panels: aggregated to weeks Left panels: only spells ending in found work or works 16 hours / wk. Right panels: also unknown reason and failed to attend interview Majority of these likely found work: Sweeney 96 L-shaped hazard in only one specification; clear duration dependence Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 10

15 Unemployment hazards: by occupation 0.1 Aggregate Weekly Unemployment Hazard 0.1 Low Skill Occupations Sales PPMO Elementary Weeks Medium Skill Occupations Admin Skill Trade Personal Weeks High Skill Occupations Managers Professional Assoc Prof Weeks Weeks Show only found work and working 16 hours from now on Fall over first 5-10 weeks more pronounced in low-skill occupations Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 11

16 Unemployment hazards: by occupation and geographic region Low Skill Occupations Sales and Customer Service occupations Sales PPMO Elementary East East Midlands London North East North West South East South West West Midlands Yorkshire Wales Weeks Weeks Process, Plant and Machine Operatives Elementary Occupations 0.14 East 0.14 East East Midlands East Midlands 0.12 London North East North West 0.12 London North East North West 0.1 South East South West West Midlands 0.1 South East South West West Midlands 0.08 Yorkshire Wales 0.08 Yorkshire Wales Weeks Weeks Again, substantial heterogeneity across occupations and regions. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 12

17 Summary Low Skill Occupations Sales PPMO Elementary Weeks Vacancies Low Skill Occupations Sales PPMO Elementary Weeks Unemployment Large initial drop in vacancy hazard; not unemployment hazard Vacancy hazard constant thereafter; unemployment hazard declines steadily However, benefit waiting period and bureaucratic burden associated with application may cause low-duration unemployment spells to be under-counted. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 13

18 Testing for stock-flow matching: vacancies Regress matching rates for new and old vacancies on the stocks and flows of unemployed workers and the stock of vacancies. Coles Smith 98 (aggregate), Andrews et al. 13 (Lancashire) Labor market = 25 occupations 39 NUTS-2 region pairs. Monthly observations. Predictions of stock-flow model: orange > 0, blue < 0, light gray = 0. Outflow hazard Hazard difference 1 wk 4+ wks 1 wk 4+ wks Stock U (0.04) (0.01) (0.04) Inflow U (0.29) (0.08) (0.29) Stock V (0.02) (0.00) (0.01) R Obs 18,648 37,975 18,607 OLS regression. Includes dummies for 2-digit occupation and 39 NUTS-2 regions. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 14

19 Testing for stock-flow matching: unemployment Outflow hazard Hazard difference 0-2 wks 4-8 wks 0-4 wks 4+ wks Stock U (0.02) (0.02) (0.02) Stock V (0.02) (0.02) (0.03) Inflow V (0.10) (0.09) (0.12) R Obs 26,134 32,326 30,647 Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 15

20 Model: Homogeneous Firms and Workers

21 Setup Agents and islands Measure 1 of identical islands, each in steady state. Each has measure N w workers, N f jobs Stock-flow matching N w = U + M and N f = V + M A worker arriving on island ( in the flow ) contacts k Poisson(V) jobs. Microfoundation for Poisson: a measure 1 of locations within the island, with workers and jobs continually reallocated across them. For each job contacted, draw match-specific productivity x F( ). {x i } k i=1 common knowledge Worker can match with one of the k jobs. If not: (1) she enters the stock; (2) contacted jobs remain in stock; (3) contacts dissolve. Job in flow contacts k Poisson(U) of workers, Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 16

22 Setup Agents and islands Measure 1 of identical islands, each in steady state. Each has measure N w workers, N f jobs Stock-flow matching N w = U + M and N f = V + M A worker arriving on island ( in the flow ) contacts k Poisson(V) jobs. Microfoundation for Poisson: a measure 1 of locations within the island, with workers and jobs continually reallocated across them. For each job contacted, draw match-specific productivity x F( ). {x i } k i=1 common knowledge Worker can match with one of the k jobs. If not: (1) she enters the stock; (2) contacted jobs remain in stock; (3) contacts dissolve. Job in flow contacts k Poisson(U) of workers, Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 16

23 Setup Agents and islands Measure 1 of identical islands, each in steady state. Each has measure N w workers, N f jobs Stock-flow matching N w = U + M and N f = V + M A worker arriving on island ( in the flow ) contacts k Poisson(V) jobs. Microfoundation for Poisson: a measure 1 of locations within the island, with workers and jobs continually reallocated across them. For each job contacted, draw match-specific productivity x F( ). {x i } k i=1 common knowledge Worker can match with one of the k jobs. If not: (1) she enters the stock; (2) contacted jobs remain in stock; (3) contacts dissolve. Job in flow contacts k Poisson(U) of workers, Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 16

24 HJB equations [ ] Matched agents: rw m (w) = w + s W f W m (w) Flow wage rj m (x, w) = x w Flow profit Worker reallocation [ ] + s J f J m (x, w) } {{ } Worker reallocation [ + δ ] W f W m (w) } {{ } Job destruction δj m (x, w) Job destruction [ Stock agents: rw s = b + f v G s + g u W f W s] Unemployment Gain from contacts income by jobs in flow Gain from returning to flow rj s = y + f u H s Output Gain from contacts when vacant by workers in flow Flow agents: W f = W s + G f Stock value Value of sampling V stock J f = J s + H f Stock value Value of sampling U stock δj s Job destruction [ + g v J f J s] } {{ } Gain from returning to flow Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 17

25 Multilateral meetings Value for worker in stock of being contacted by a vacancy in the flow: G s e U U k 1 1 = G s (x) F k (x). (k 1)! k k=1 X k Gain best worker gets Prob. of k 1 Prob. of being when x is drawn competing workers highest x x Value for worker in flow of sampling the vacancy stock G f e V V k = G f (x) F k (x). k! k=1 X k Gain when Prob. of contacting draw x k vacancies Private efficiency: we assume highest-x agent matches. Functional forms G f (x), G s (x) depend on how wages are determined Agent in flow runs a second-price auction? Nash bargaining between worker and highest-x job with outside option the value of being unmatched? with outside option for worker to negotiate with the next-highest x job,? Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 18

26 Multilateral meetings Value for worker in stock of being contacted by a vacancy in the flow: G s e U U k 1 1 = G s (x) F k (x). (k 1)! k k=1 X k Gain best worker gets Prob. of k 1 Prob. of being when x is drawn competing workers highest x x Value for worker in flow of sampling the vacancy stock G f e V V k = G f (x) F k (x). k! k=1 X k Gain when Prob. of contacting draw x k vacancies Private efficiency: we assume highest-x agent matches. Functional forms G f (x), G s (x) depend on how wages are determined Agent in flow runs a second-price auction? Nash bargaining between worker and highest-x job with outside option the value of being unmatched? with outside option for worker to negotiate with the next-highest x job,? Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 18

27 Multilateral meetings Value for worker in stock of being contacted by a vacancy in the flow: G s e U U k 1 1 = G s (x) F k (x). (k 1)! k k=1 X k Gain best worker gets Prob. of k 1 Prob. of being when x is drawn competing workers highest x x Value for worker in flow of sampling the vacancy stock G f e V V k = G f (x) F k (x). k! k=1 X k Gain when Prob. of contacting draw x k vacancies Private efficiency: we assume highest-x agent matches. Functional forms G f (x), G s (x) depend on how wages are determined Agent in flow runs a second-price auction? Nash bargaining between worker and highest-x job with outside option the value of being unmatched? with outside option for worker to negotiate with the next-highest x job,? Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 18

28 Balanced reallocation We say that there is balanced reallocation if g u = s + δ and g v = s. Implications: The Poisson rate at which any worker returns to the flow is g u = s + δ. The Poisson rate at which any job returns to the flow is g v = s. The flows of workers and jobs, f u = (s + δ)(m + U) = (s + δ)n w f v = (s + δ)(m + V) = (s + δ)n f do not depend on the unemployment and vacancy rates. We assume balanced reallocation for the rest of the talk. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 19

29 Balanced reallocation We say that there is balanced reallocation if g u = s + δ and g v = s. Implications: The Poisson rate at which any worker returns to the flow is g u = s + δ. The Poisson rate at which any job returns to the flow is g v = s. The flows of workers and jobs, f u = (s + δ)(m + U) = (s + δ)n w f v = (s + δ)(m + V) = (s + δ)n f do not depend on the unemployment and vacancy rates. We assume balanced reallocation for the rest of the talk. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 19

30 Reservation match quality x Surplus S(x) = [W m (w) W s ] + [J m (x, w) J s ] is well defined and increasing in match quality x. Reservation match quality satisfies S( x) = 0, that is, x = b + y + f v G s + f u H s Forgone Forgone contacts flow output while in stock x is independent of which agent is in the flow: if no match forms, both agents go to the stock. Rates of returning to flow don t depend on match status (balanced reallocation), so G f and H f don t show up. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 20

31 Matching probabilities and rates Matching prob. for unemployed worker in flow: p u = k=0 e V V k k! [ 1 F( x) k] = 1 e V[1 F( x)] = p u ( V (+), x ( ) Poisson matching rate for unemployed worker in stock (without reallocation): ϕ u = 1 U fv p v = 1 U fv p v ( U, x ) = ϕ u ( U, x ) ( ) ( ) ( ) ( ) Similar for vacancies: p v = p v ( U, x ), ϕ v = ϕ v ( V, x ). (+) ( ) ( ) ( ) ). Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 21

32 Matching probabilities and rates Matching prob. for unemployed worker in flow: p u = k=0 e V V k k! [ 1 F( x) k] = 1 e V[1 F( x)] = p u ( V (+), x ( ) Poisson matching rate for unemployed worker in stock (without reallocation): ϕ u = 1 U fv p v = 1 U fv p v ( U, x ) = ϕ u ( U, x ) ( ) ( ) ( ) ( ) Similar for vacancies: p v = p v ( U, x ), ϕ v = ϕ v ( V, x ). (+) ( ) ( ) ( ) ). Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 21

33 Matching probabilities and rates Matching prob. for unemployed worker in flow: p u = k=0 e V V k k! [ 1 F( x) k] = 1 e V[1 F( x)] = p u ( V (+), x ( ) Poisson matching rate for unemployed worker in stock (without reallocation): ϕ u = 1 U fv p v = 1 U fv p v ( U, x ) = ϕ u ( U, x ) ( ) ( ) ( ) ( ) Similar for vacancies: p v = p v ( U, x ), ϕ v = ϕ v ( V, x ). (+) ( ) ( ) ( ) ). Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 21

34 Existence of equilibrium Characterization Natural definition of equilibrium with free entry. More Characterized by three equations in (M, x, µ): 1. Flow balance 2. Optimal reservation x 3. Free entry Equations If wages are regular and reallocation is balanced, then: Proposition: Existence and Uniqueness with Exogenous Entry For any N f, there exists a unique pair ( x, M ) that satisfies the flow and reservation equations jointly. Proposition: Endogenizing Entry If job entry is profitable when N f = 0, there exists a stock-flow equilibrium with positive and finite entry. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 22

35 Existence of equilibrium Characterization Natural definition of equilibrium with free entry. More Characterized by three equations in (M, x, µ): 1. Flow balance 2. Optimal reservation x 3. Free entry Equations If wages are regular and reallocation is balanced, then: Proposition: Existence and Uniqueness with Exogenous Entry For any N f, there exists a unique pair ( x, M ) that satisfies the flow and reservation equations jointly. Proposition: Endogenizing Entry If job entry is profitable when N f = 0, there exists a stock-flow equilibrium with positive and finite entry. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 22

36 Existence of equilibrium Characterization Natural definition of equilibrium with free entry. More Characterized by three equations in (M, x, µ): 1. Flow balance 2. Optimal reservation x 3. Free entry Equations If wages are regular and reallocation is balanced, then: Proposition: Existence and Uniqueness with Exogenous Entry For any N f, there exists a unique pair ( x, M ) that satisfies the flow and reservation equations jointly. Proposition: Endogenizing Entry If job entry is profitable when N f = 0, there exists a stock-flow equilibrium with positive and finite entry. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 22

37 The shapes of the hazards Hazard h u p u + ψ u ψ u Time, t Proposition Suppose δ = 0. Then the shapes of the unemployment and vacancy hazards are uniquely determined by the unemployment and vacancy rates u = U/N w and v = V/N f and the churn rate s. Intuitively, conditional on u, a market with higher v has higher job-finding prob. for workers in flow, lower job-finding rates for workers in stock. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 23

38 Heterogeneity

39 Empirical hazards Model without heterogeneity can (roughly) match vacancy hazard, not close for unemployment hazard Solution: allow for workers and jobs to be heterogeneous Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 24

40 Heterogeneity Heterogeneous contact intensity Flow measure of meetings λ i µ j f u i V j between type-i workers in flow and type-j jobs in stock λ i µ j f v j U i between type-i workers in stock and type-j jobs in flow Implies that a type-i worker in the flow, contacts a number of type-j jobs Poisson(λ i µ j V j ) for each j in the stock, is contacted by a type-j job in the flow at rate λ i µ j f v j. Heterogeneous match surpluses Type-i worker earns b i when unmatched Type-j job produces y j when unmatched. Problem for individual agent remains McCall-like: type-specific reservation wages w i and profits π j. Given match quality draws x = (x 1,..., x k ), a type-i worker in flow matches only with the highest surplus x κ π jκ draw; and only if that value exceeds w i. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 25

41 Heterogeneity Heterogeneous contact intensity Flow measure of meetings λ i µ j f u i V j between type-i workers in flow and type-j jobs in stock λ i µ j f v j U i between type-i workers in stock and type-j jobs in flow Implies that a type-i worker in the flow, contacts a number of type-j jobs Poisson(λ i µ j V j ) for each j in the stock, is contacted by a type-j job in the flow at rate λ i µ j f v j. Heterogeneous match surpluses Type-i worker earns b i when unmatched Type-j job produces y j when unmatched. Problem for individual agent remains McCall-like: type-specific reservation wages w i and profits π j. Given match quality draws x = (x 1,..., x k ), a type-i worker in flow matches only with the highest surplus x κ π jκ draw; and only if that value exceeds w i. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 25

42 Value Functions with Heterogeneity Agents in the Stock: Workers rw s i = b i Unemployment income Agents in the Stock: Firms rj s j = y j Output when vacant J [ ] + λ i µ j f v j G s i,j + g u W f i Ws i j=1 Gain from returning to flow Gain from contacts by jobs in flow I + µ j λ i f u i H s i,j i=1 Gain from contacts by workers in flow δj s j Job destruction ] [ + g v J f j Js j Gain from returning to flow Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 26

43 Exponential distribution of match-specific shocks Suppose unemployed worker of type i 0 in the stock is contacted by job of type j in the flow. The probability a match forms is { [ p u,s i 0,j = I ] e λ i µ j U i(λ i µ j U i ) k [ i I F ( ] } ) ki x + w i w i0 f(x) x k i! k 1,...,k I =0 i=1 w i0 + π j i=1 Prob. k 1 type-1 workers, Prob. this worker k 2 type-2, also contacted is the one who matches How does the answer depend on i 0? j? The realization of (k 1,..., k I )? In general these questions are hard to answer. Key assumption: F(x) = 1 exp( ρx). Implications: An increase from x to x + lowers log f(x) by an amount that depends only on and not on x Matching patterns exhibit no sorting: on meeting a job of type j, a high-b worker is less likely to match than a low-b worker. The ratio of these probabilities is the same no matter what the job type j is. Math Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 27

44 Exponential distribution of match-specific shocks Suppose unemployed worker of type i 0 in the stock is contacted by job of type j in the flow. The probability a match forms is { [ p u,s i 0,j = I ] e λ i µ j U i(λ i µ j U i ) k [ i I F ( ] } ) ki x + w i w i0 f(x) x k i! k 1,...,k I =0 i=1 w i0 + π j i=1 Prob. k 1 type-1 workers, Prob. this worker k 2 type-2, also contacted is the one who matches How does the answer depend on i 0? j? The realization of (k 1,..., k I )? In general these questions are hard to answer. Key assumption: F(x) = 1 exp( ρx). Implications: An increase from x to x + lowers log f(x) by an amount that depends only on and not on x Matching patterns exhibit no sorting: on meeting a job of type j, a high-b worker is less likely to match than a low-b worker. The ratio of these probabilities is the same no matter what the job type j is. Math Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 27

45 Model equations With exogenous numbers of firms and workers, the model equilibrium is characterized by the following set of equations. p u i Ū = I λ i e ρ w iu i V = i=1 = 1 exp ( ) λ i e ρ w i V I µ j e ρ π jv j i=1 p v j = 1 exp ( ) µ j e ρ π jū γ u i = λ ie ρ w i Ū γ v j = µ je ρ π j V J I ϕ u i = (s + δ)γ u i N f j pv j ϕ v j = (s + δ)γ v j N w i pu i j=1 i=1 J I N w i (1 pu i ) = U i + γ u i U i p v j Nf j j=1 N f j (1 pv j ) = V j + γj v V j p u i Nw i i=1 w i = b i + 1 ρ ϕu i π j = y j + 1 ρ ϕv j Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 28

46 Existence and uniqueness Proposition: Existence with Exogenous Entry For any distribution of heterogeneity (λ i, b i ) I i=1 and (µ j, y j ) J j=1, there exists an equilibrium. When there is not too much heterogeneity, the equilibrium is unique. Sketch of proof: 1. Given reservation wages ( w i ) I i=1 and profits ( π j ) J j=1, flow balance uniquely determines unemployment and vacancy stocks (U i ) I i=1 and (V j ) J j=1. 2. Reservation wages and profits depend only on worker s own characteristics, other exogenous parameters, and on effective unemployment and effective vacancies Ū = I λ i U i e ρ w i and V = i=1 J µ j V j e ρ π j 3. The resulting fixed-point problem for (Ū, V) has a solution. 4. For uniqueness, use the implicit function theorem in the neighborhood of the (unique) homogeneous-agent equilibrium. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 29 j=1

47 Existence and uniqueness Proposition: Existence with Exogenous Entry For any distribution of heterogeneity (λ i, b i ) I i=1 and (µ j, y j ) J j=1, there exists an equilibrium. When there is not too much heterogeneity, the equilibrium is unique. Sketch of proof: 1. Given reservation wages ( w i ) I i=1 and profits ( π j ) J j=1, flow balance uniquely determines unemployment and vacancy stocks (U i ) I i=1 and (V j ) J j=1. 2. Reservation wages and profits depend only on worker s own characteristics, other exogenous parameters, and on effective unemployment and effective vacancies Ū = I λ i U i e ρ w i and V = i=1 J µ j V j e ρ π j 3. The resulting fixed-point problem for (Ū, V) has a solution. 4. For uniqueness, use the implicit function theorem in the neighborhood of the (unique) homogeneous-agent equilibrium. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 29 j=1

48 Heterogeneity and Durations Matching probabilities and matching rates For worker types i and i, the relative matching probabilities when in the flow and matching rates when in the stock satisfy log(1 p u i ) log(1 p u i 0 ) = λ ie ρ wi λ i0 e ρ w i 0 = ϕu i ϕ u i 0. Interpretation: The opportunity to match while in the flow generates the same probability of matching as T units of time in the stock, where 1 p u i = e ϕu i T This duration is independent of worker type i. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 30

49 Heterogeneity and Durations Matching probabilities and matching rates For worker types i and i, the relative matching probabilities when in the flow and matching rates when in the stock satisfy log(1 p u i ) log(1 p u i 0 ) = λ ie ρ wi λ i0 e ρ w i 0 = ϕu i ϕ u i 0. Interpretation: The opportunity to match while in the flow generates the same probability of matching as T units of time in the stock, where 1 p u i = e ϕu i T This duration is independent of worker type i. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 30

50 Heterogeneity and the Survivor Function S(t) 1 1 p u i Time, t Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 31

51 Heterogeneity and the Survivor Function S(t) 1 1 p u i 1 p u i Time, t Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 31

52 Heterogeneity and the Survivor Function S(t) 1 1 p u i 1 p u i Time, t Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 31

53 Heterogeneity Low Skill Occupations Sales PPMO Elementary Weeks Vacancies Low Skill Occupations Sales PPMO Elementary Weeks Unemployment Aggregating across heterogeneous types delivers declining hazard for agents in stock. Identification (intuition) 1. Shape of declining hazard for t > 1 identifies amount of heterogeneity. 2. Excess job-finding at short durations (p v ) then identifies T. 3. Can impute unemployment spells not measured at short durations using flow balance. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 32

54 Efficiency The planner chooses {U i (t), V j (t), w i (t), π j (t)} t=0 to maximize output, taking stock-flow matching frictions as technological. Planner s Problem Proposition - Mortensen Rule with Fixed Entry If wages are determined by second-price auctions, then the decentralized economy is constrained efficient conditional on N f. Proposition - Inefficient Entry If wages are determined using second-price auctions, the agent in the flow receives his/her marginal contribution to the surplus. But efficient entry requires the firm in the flow should get the entire surplus Coles 99: trading externality. Under budget balance both cannot be achieved at the same time. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 33

55 Efficiency The planner chooses {U i (t), V j (t), w i (t), π j (t)} t=0 to maximize output, taking stock-flow matching frictions as technological. Planner s Problem Proposition - Mortensen Rule with Fixed Entry If wages are determined by second-price auctions, then the decentralized economy is constrained efficient conditional on N f. Proposition - Inefficient Entry If wages are determined using second-price auctions, the agent in the flow receives his/her marginal contribution to the surplus. But efficient entry requires the firm in the flow should get the entire surplus Coles 99: trading externality. Under budget balance both cannot be achieved at the same time. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 33

56 Efficiency The planner chooses {U i (t), V j (t), w i (t), π j (t)} t=0 to maximize output, taking stock-flow matching frictions as technological. Planner s Problem Proposition - Mortensen Rule with Fixed Entry If wages are determined by second-price auctions, then the decentralized economy is constrained efficient conditional on N f. Proposition - Inefficient Entry If wages are determined using second-price auctions, the agent in the flow receives his/her marginal contribution to the surplus. But efficient entry requires the firm in the flow should get the entire surplus Coles 99: trading externality. Under budget balance both cannot be achieved at the same time. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 33

57 Extensions

58 Stock-flow vs. random matching Low Skill Occupations Sales PPMO Elementary Weeks Vacancies Low Skill Occupations Sales PPMO Elementary Weeks Unemployment Random (stock-stock) matching model with heterogeneous matching efficiencies for both workers and jobs can rationalize this data too (Heckman Singer 84; Ahn Hamilton 16) But with a lot more heterogeneity in vacancy types (need a type that matches very rapidly) Does it matter which approach we choose? Potentially yes in the nature of responses to comparative static changes in the environment. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 34

59 Stock-flow vs. random matching Low Skill Occupations Sales PPMO Elementary Weeks Vacancies Low Skill Occupations Sales PPMO Elementary Weeks Unemployment Random (stock-stock) matching model with heterogeneous matching efficiencies for both workers and jobs can rationalize this data too (Heckman Singer 84; Ahn Hamilton 16) But with a lot more heterogeneity in vacancy types (need a type that matches very rapidly) Does it matter which approach we choose? Potentially yes in the nature of responses to comparative static changes in the environment. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 34

60 Stock-flow and stock-stock matching Vacancy spells are recorded as either filled by JobCentre Plus-provided candidate or withdrawn. The figure shows (for elementary occupations) the hazards required to rationalize the duration distribution of both. Around 25% of vacancies fill using JCP-provided candidates We do not have (much) analogous information by duration and channel for the unemployed. Carrillo-Tudela What and Hawkins to do about this? Stock-Flow Matching with Heterogeneous Workers and Firms 35

61 The composition of unemployment: boom vs. recession Ahn 16, Ahn Hamilton 16 argue that the evolution of U.S. unemployment durations is accounted for by shifts in composition of unemployment entrants. In our model, changes in {p u i } change effective inflow rates (1 pu i )(s + δ)nw i. Proposition. Suppose γu i γ u = λ i e ρ w i i λ i e ρ w = ϕu i i ϕ u > 1. i A change in aggregate conditions which raises ϕ u lowers ϕ u i i /ϕ u i. The effect on 1 pu i 1 p u i is ambiguous. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 36

62 The composition of unemployment: boom vs. recession Ahn 16, Ahn Hamilton 16 argue that the evolution of U.S. unemployment durations is accounted for by shifts in composition of unemployment entrants. In our model, changes in {p u i } change effective inflow rates (1 pu i )(s + δ)nw i. Proposition. Suppose γu i γ u = λ i e ρ w i i λ i e ρ w = ϕu i i ϕ u > 1. i A change in aggregate conditions which raises ϕ u lowers ϕ u i i /ϕ u i. The effect on 1 pu i 1 p u i is ambiguous. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 36

63 The composition of unemployment: boom vs. recession Ahn 16, Ahn Hamilton 16 argue that the evolution of U.S. unemployment durations is accounted for by shifts in composition of unemployment entrants. In our model, changes in {p u i } change effective inflow rates (1 pu i )(s + δ)nw i. Proposition. Suppose γu i γ u = λ i e ρ w i i λ i e ρ w = ϕu i i ϕ u > 1. i A change in aggregate conditions which raises ϕ u lowers ϕ u i i /ϕ u i. The effect on 1 pu i 1 p u i S(t) 1 is ambiguous. 1 p u i 1 p u i Time, t Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 36

64 The composition of unemployment: boom vs. recession Ahn 16, Ahn Hamilton 16 argue that the evolution of U.S. unemployment durations is accounted for by shifts in composition of unemployment entrants. In our model, changes in {p u i } change effective inflow rates (1 pu i )(s + δ)nw i. Proposition. Suppose γu i γ u = λ i e ρ w i i λ i e ρ w = ϕu i i ϕ u > 1. i A change in aggregate conditions which raises ϕ u lowers ϕ u i i /ϕ u i. The effect on 1 pu i 1 p u i S(t) 1 p u i 1 is ambiguous. 1 p u i Time, t Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 36

65 The composition of unemployment: boom vs. recession Ahn 16, Ahn Hamilton 16 argue that the evolution of U.S. unemployment durations is accounted for by shifts in composition of unemployment entrants. In our model, changes in {p u i } change effective inflow rates (1 pu i )(s + δ)nw i. Proposition. Suppose γu i γ u = λ i e ρ w i i λ i e ρ w = ϕu i i ϕ u > 1. i A change in aggregate conditions which raises ϕ u lowers ϕ u i i /ϕ u i. The effect on 1 pu i 1 p u i S(t) 1 p u 1 i 1 p u i is ambiguous Time, t Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 36

66 Conclusion Summary New evidence on stock-flow nature of vacancy-filling process in UK, focusing on vacancy and unemployment hazards Developed an equilibrium stock flow model with endogenous match formation and firm entry. In progress Structural estimation of model Policy implications. Carrillo-Tudela and Hawkins Stock-Flow Matching with Heterogeneous Workers and Firms 37

67 Wage and Profit Determination Definition Wages and profits are regular if there exist continuous functions G f, H f, G s, H s : R 2 (X) [0, ) such that for all parameter constellations, and such that G f 1 = r + δ + s Gf ( x, V, F), G s 1 = r + δ + s Gs ( x, U, F), (1) H f 1 = r + δ + s Hf ( x, U, F), and H s 1 = r + δ + s Hs ( x, V, F), (2) 1. G s ( ), H s ( ), G f ( ), and H f ( ) are all strictly decreasing in the first argument; 2. G s ( ) and H s ( ) are strictly decreasing and G f ( ) and H f ( ) are strictly increasing in the second argument; 3. for any U, V 0, lim x x G s ( x, U) = lim x x H s ( x, V) = lim x x G f ( x, V) = lim x x H f ( x, U) = 0; Proposition Back

68 Wage and Profit Determination Proposition 1 Suppose that there are continuous functions ŵ f, ŵ s, ˆπ f, ˆπ s : X [0, ) such that w f (x) = ŵ f (x x) + w, w s (x) = ŵ s (x x) + w, π f (x) = ˆπ f (x x) + π, and π s (x) = ˆπ s (x x) + π, where (x 1, x 2,..., x k ) y = (x i y i ) i x y, with the properties that 1. the entirety of the match surplus be split between the worker and the job 2. an increase in x reduces the excess of both wages and profits over the reservation values. 3. an increase in the x of a formed match increases wages and profits. 4. an improvement in the set of other match quality draws by an agent in the flow raises this agent s payoff and reduces the payoff of the agent in the stock. Then wages and profits are regular. Back

69 Stock-Flow Matching Equilibrium Definition A SFM equilibrium is a set of values W s, W m, W f, J s, J m and J f ; a set of wages w s (x), w f (x) and profits π s (x), π f (x); a measure N f of firms, U of unemployed workers in the stock, V of job vacancies in the stock and M of matches, for all i such that: 1. The value functions satisfy the HJB equations described above. 2. A multilateral meeting leads to a match only with the highest x draw and only if x x, where x solves S( x) = Optimal entry determines µ. 4. The measures U, V and M are the steady state solution to the transition equations described above. Back

70 Firms Participation Decisions Firms can create vacancies or stay inactive. Creating a vacancy requires paying sunk cost c. The number of firms in the economy, N, is determined by c = J f = J s + H f,

71 Characterization Flow balance: [ Ṁ = f v 1 e U(1 F( x))] [ + f u 1 e V(1 F( x))] (s + δ)m Destroyed Worker in stock, Worker in flow job in flow job in stock matches Reservation productivity: x = b + y + f v G s + f u H s Free entry: c = 1 [ y + f u H s + (r + g v + δ)h f]. r + δ In steady state U + Ṁ = 0 and V + Ṁ = 0, so that U = (λ/s) M and V = (µ/δ) M, where N w = λ/s and N f = µ/δ. Here λ and µ are the exogenous inflows. Back

72 Planner s Problem The planner chooses {m( ), u( ), V( ), x( ), µ( )} to maximize aggregate (net) output. This maximization can be represented as max e [bu(t) rt + yv(t) cµ(t) M( ),U( ),V( ), x( ),µ( ) 0 + (µ(t) + χ v V(t) + sm(t)) x(t) + (λ + χ u U(t) + δm(t)) x(t) x r + s + δ U(t)e U(t)[1 F(x)] f(x)dx x r + s + δ V(t)e V(t)[1 F(x)] f(x)dx ] dt subject to the flow constraints. Back

73 Examples of Regular Wages and Profits 1. Nash Bargaining Agent in the flow bargains with highest-x draw in the stock and does not consider the other draws. Outside options for both agents are the values of being in the stock. 2. Second Price Auction Bertrand competition. If x 1 x > x 2, then flow agent gets flow payoff x; stock agent gets x 1 x. If x 1 x 2 x, then flow agent gets flow payoff x 2 ; stock agent gets x 1 x Cahuc, Postel-Vinay, and Robin (2006) on steroids Agent in the flow bargains with highest-x draw in the stock. Outside option for flow agent: bargain with 2nd-best x. The outside option in that bargain: bargain with 3rd-best x, and so on. Outside option for stock agent: remain unmatched Closed-form solutions in a special case. Details

74 Solving for wages as in Cahuc, Postel-Vinay, and Robin To make progress assume F is exponential. Consider the expected gain of finding a job for a worker in the flow. G f = k=0 = = = k=0 V Vk e k! V Vk e k! 1 r + s + δ [ ] max W m (w f (x)) W s, 0 df k (x) β x 1 < <x k k (1 β) k j x x r + s + δ df(x 1)... df(x k ) j=1 V(1 F( x)) [V(1 F( x))]k e β 1 k 1 (1 β) j k! a j k=0 j=1 1 Ein(β(1 F( x))v), a(r + s + δ) where z 1 e t Ein(z) dt. 0 t is the entire exponential integral function.

75 Implications Deriving G f, H s and H f in a similar way as above and using the exponential distribution yields G s 1 = (r + s + δ)au H s 1 = (r + s + δ)av G f = H f = 1 ( (r + s + δ)a Ein βve a x) 1 ( (r + s + δ)a Ein βue a x) [ ( Ein Ue a x) Ein (βue a x)] [ ( Ein Ve a x) Ein (βve a x)] G s and H s are decreasing in u and V, respectively. G f and H f are increasing in V and u, respectively. Back

76 Jobcentre Plus Primary channel for 39 percent of all UK vacancies Newspapers 35%, internet 14%, recruiters 13% Most common occupations: Sales and customer service (sales assistants, cashiers, phone sales): 22% Elementary occupations (bar staff, kitchen assistants, cleaners, waiters): 20% Most common industries: Banking, finance, insurance: 44%; distribution, hotels, restaurants: 23% Back

77 Flow equations Flow equations: Back [ Ṁ = f v 1 e U(1 F( x))] [ + f u 1 e V(1 F( x))] (s + δ)m Destroyed Worker in stock, Worker in flow job in flow job in stock matches U [ = f v 1 e U(1 F( x))] + f u e V(1 F( x)) (s + δ)u Workers in flow who fail to match Workers who Workers in stock who match return to flow V = f v e U(1 F( x)) Jobs in flow that fail to match [ f u 1 e V(1 F( x))] Jobs in stock that match (s + δ)v Jobs reallocated

78 Exponential distribution delivers tractability Suppose unemployed worker of type i 0 in the stock is contacted by job of type j in the flow. What is the probability a match forms? { [ p u,s i 0,j = I e λ i µ j U ] i(λ i µ j U i ) k [ i I F ( ] } ) ki x + w i w i0 f(x) x k k 1,...,k I =0 i=1 i! w i0 + π j i=1 Prob. k 1 type-1 workers, Prob. this worker k 2 type-2, also contacted is the one who matches Ii=1 =... = e µ j λ i U i µ k [ I k j λ i U i F(x + w i w i0 )] f(x) x. k! k=0 w i0 + π j i=1 If F( ) is exponential, then f(x + ) = ρe ρ(x+ ) = e ρ f(x), so we can simplify: Ii=1 p u,s i 0,j = e µ j λ i U i µ k j e ρ w [ i 0 I ] k+1 [ I ] k+1 (k + 1)! I k=0 i=1 λ λ iu i e ρ w i i U i λ i U i F( x ij ) i=1 i=1 e ρ w [ ( i 0 1 I [ =... = I i=1 λ 1 exp λ iu i e ρ w i µ j U i 1 F( xij ) ])]. (3) i µ j i=1 What delivers tractability is that an increase from x to x + lowers the log density by an amount that depends only on, and not on x. Back

79 Exponential distribution delivers tractability Suppose unemployed worker of type i 0 in the stock is contacted by job of type j in the flow. What is the probability a match forms? { [ p u,s i 0,j = I e λ i µ j U ] i(λ i µ j U i ) k [ i I F ( ] } ) ki x + w i w i0 f(x) x k k 1,...,k I =0 i=1 i! w i0 + π j i=1 Prob. k 1 type-1 workers, Prob. this worker k 2 type-2, also contacted is the one who matches Ii=1 =... = e µ j λ i U i µ k [ I k j λ i U i F(x + w i w i0 )] f(x) x. k! k=0 w i0 + π j i=1 If F( ) is exponential, then f(x + ) = ρe ρ(x+ ) = e ρ f(x), so we can simplify: Ii=1 p u,s i 0,j = e µ j λ i U i µ k j e ρ w [ i 0 I ] k+1 [ I ] k+1 (k + 1)! I k=0 i=1 λ λ iu i e ρ w i i U i λ i U i F( x ij ) i=1 i=1 e ρ w [ ( i 0 1 I [ =... = I i=1 λ 1 exp λ iu i e ρ w i µ j U i 1 F( xij ) ])]. (3) i µ j i=1 What delivers tractability is that an increase from x to x + lowers the log density by an amount that depends only on, and not on x. Back