A Stock-Flow Theory of Unemployment with Endogenous Match Formation

Transcription

1 A Stock-Flow Theory of Unemployment with Endogenous Match Formation Carlos Carrillo-Tudela and William Hawkins Univ. of Essex and Yeshiva University February 2016 Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 1

2 INTRODUCTION Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 2

3 Motivation What drives the changes of the unemployment duration distribution when aggregate conditions change? How are these changes related to changes in measured matching efficiency and mismatch? Elsby et al. 14, Barnichon and Figura 14, Kroft et al. 15, Şahin et al. 14, among others. Graphs Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 3

4 Motivation What drives the changes of the unemployment duration distribution when aggregate conditions change? How are these changes related to changes in measured matching efficiency and mismatch? Elsby et al. 14, Barnichon and Figura 14, Kroft et al. 15, Şahin et al. 14, among others. Graphs Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 3

5 Objective What we do Develop a stock-flow matching model with endogenous match formation. Study implications for [changes in] duration distribution, measured matching efficiency, and mismatch. Key ingredient Results Firms and workers jointly decide which matches are acceptable, given aggregate state. Model delivers realistic unemployment duration distribution. Match formation margin important in accounting for shifts over time. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 4

6 Objective What we do Develop a stock-flow matching model with endogenous match formation. Study implications for [changes in] duration distribution, measured matching efficiency, and mismatch. Key ingredient Results Firms and workers jointly decide which matches are acceptable, given aggregate state. Model delivers realistic unemployment duration distribution. Match formation margin important in accounting for shifts over time. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 4

7 Why stock-flow matching? Main friction Workers are unemployed and jobs vacant because the right match is not currently in the market. Some Suggestive Evidence Exit rates of unemployed workers at short durations depend on the stock of vacancies; while exit rates at long durations depend on the flow of vacancies Coles Smith 99, Coles Petrongolo 08 Vacancy hazard Our mechanism generates more short spells and more long-term unemployment than a (DMP) search model. nb: homogeneous workers Spell duration CPS ( ) Const. hazard rate Stock-flow < 4 weeks to 26 weeks > 26 weeks Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 5

8 Why stock-flow matching? Main friction Workers are unemployed and jobs vacant because the right match is not currently in the market. Some Suggestive Evidence Exit rates of unemployed workers at short durations depend on the stock of vacancies; while exit rates at long durations depend on the flow of vacancies Coles Smith 99, Coles Petrongolo 08 Vacancy hazard Our mechanism generates more short spells and more long-term unemployment than a (DMP) search model. nb: homogeneous workers Spell duration CPS ( ) Const. hazard rate Stock-flow < 4 weeks to 26 weeks > 26 weeks Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 5

9 Why stock-flow matching? Main friction Workers are unemployed and jobs vacant because the right match is not currently in the market. Some Suggestive Evidence Exit rates of unemployed workers at short durations depend on the stock of vacancies; while exit rates at long durations depend on the flow of vacancies Coles Smith 99, Coles Petrongolo 08 Vacancy hazard Our mechanism generates more short spells and more long-term unemployment than a (DMP) search model. nb: homogeneous workers Spell duration CPS ( ) Const. hazard rate Stock-flow < 4 weeks to 26 weeks > 26 weeks Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 5

10 Why stock-flow matching? Main friction Workers are unemployed and jobs vacant because the right match is not currently in the market. Some Suggestive Evidence Exit rates of unemployed workers at short durations depend on the stock of vacancies; while exit rates at long durations depend on the flow of vacancies Coles Smith 99, Coles Petrongolo 08 Vacancy hazard Our mechanism generates more short spells and more long-term unemployment than a (DMP) search model. nb: homogeneous workers Spell duration CPS ( ) Const. hazard rate Stock-flow < 4 weeks to 26 weeks > 26 weeks Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 5

11 Contribution to the stock-flow literature Previous work Closest ancestors: Coles 99, Shimer 07 Heterogeneity is captured by an exogenous probability that determines whether a worker and a firm can form a match. Agents do not make acceptance decisions. Our model Agents acceptance sets depend on the stock and flow of unemployment and vacancies and the distribution of match quality. Encompasses different wage determination mechanisms. Quantitative analysis. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 6

12 In progress as of Feb Model can account for arbitrary unemployment duration distribution given arbitrary heterogeneity. Empirically, main challenge is in teasing out whether changes in shape of unemployment duration distribution are accounted for by plausible changes in underlying distribution of heterogeneity. With high-quality distribution on not just unemployment but also vacancy hazard, we could tease these apart better. We are in the process of doing this using German administrative data on vacancy durations, as well as the more standard Socio-Economic Panel (GSOEP) data for unemployment durations. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 7

13 THEORY Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 8

14 Basic framework Population Measures N w of workers and N f of firms. Workers can either be unemployed or employed. Flow payoffs: b, w (more on wages later). Firms have only one job. It can be either vacant or filled. Flow payoffs: γ, x w. All agents risk neutral, discount at rate r. Labor Markets (Islands) Aggregate steady state. Continuum of islands, indexed by i [0, 1], each also in steady state. Continua of U i unemployed workers, V i vacancies and M i matches. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 9

15 Heterogeneity and matching Assumptions Agent in flow can potentially match with finite number (k) of agents on other side of market. # jobs encountered by worker in flow Poisson(V i ) # workers encountered by job in flow Poisson(U i ) For each agent encountered, match-specific productivity x F ( ) is drawn. Restrictions {x i } k i=1 is common knowledge to the k + 1 agents involved. No re-sampling No stock-stock matching. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 10

16 Heterogeneity and matching Assumptions Agent in flow can potentially match with finite number (k) of agents on other side of market. # jobs encountered by worker in flow Poisson(V i ) # workers encountered by job in flow Poisson(U i ) For each agent encountered, match-specific productivity x F ( ) is drawn. Restrictions {x i } k i=1 is common knowledge to the k + 1 agents involved. No re-sampling No stock-stock matching. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 10

17 Value Functions Employed workers rw m i (w) = w }{{} Flow wage [ + s i Ei Wi f W m i (w) ] [ + δ i W f i }{{} Reallocation to other island W m i (w) ] } {{ } Job destruction Filled jobs rj m i (π) = π }{{} Flow profit [ + s i J f i Ji m (π) ] }{{} Worker reallocation δ i J m i (π) }{{} Job destruction Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 11

18 Value Functions Agents in the Flow: Workers W f i = Wi s }{{} Stock value + G f i }{{} Gain from being in flow Agents in the Flow: Firms Ji f = Ji s }{{} + Hi f }{{} Stock value Gain from being in flow Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 12

19 Value Functions Agents in the Stock: Workers rw s i = b i }{{} Unemployment income + (µ i + χ v i V i + s i M i ) G s i } {{ } Gain from contacts by jobs in flow [ + s i Ei Wi f W s ] i + χ u [ i W f i W s ] i }{{}}{{} Reallocation Return to flow to other island on this island Agents in the Stock: Firms rji s = γ }{{} i + (λ i + χ u i U i + δ i M i ) Hi s δ i Ji s + χ v [ i J f }{{}}{{} i J s ] i }{{} Vacancy Gain from contacts Job Return to flow posting cost by workers in flow destruction on this island Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 13

20 Match Formation Assume matching is privately efficient Given x = (x 1,..., x k ), a match forms only with the highest x draw and only if the associated surplus is 0. Surplus Reservation Match Quality Surplus is increasing in match quality. There is a reservation match quality x i such that S i ( x) = 0, that is, x i = b i γ i + (µ i + χ v i V i + s i M i ) Gi s + (λ i + χ u i U i + δ i M i ) Hi s + (χ u i δ i ) Gi f + (χ v i s i ) Hi f. x is independent of which agent is in the flow since if no match forms, both agents go to the stock. There exist a reservation wage w i and a reservation profit, π i, with x i = w i + π i. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 14

21 Wage and Profit Determination Definition Wages and profits are regular if the expected gain terms G f, H f, G s, H s have the following properties: r+δ i +s i G f and 1 r+δ i +s i G s and 1 r+δ i +s i H s depend only on x i and U i ; 1 r+δ i +s i H f depend only on x i and V i ; 3 Gains for agents in the flow are increasing in the measure of agents on the other side of the market; 4 Gains for agents in the stock are decreasing in the measure of agents on the same side of the market 5 All gains are decreasing in x i 6 Plus continuity and asymptotic conditions Formalism We assume wages and profits are regular from now on. Proposition Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 15

22 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 Equivalently, 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 2. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 16

23 Examples of Regular Wages and Profits Cahuc, Postel-Vinay, and Robin (2006) 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. Flow agent gets share β of surplus above outside options. β k ( ) (1 β) j 1 x[j] x max r + s + δ, 0. j=1 Here x [j] denotes the jth reverse-order statistic of the vector x, with x [1] being the maximum and x [k] the minimum. Closed-form solutions for G and H assuming x is distributed exponential, F (x) = 1 exp( ax). Details Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 17

24 Flow equations On island i, stocks evolve according to U = (µ + χ v U(1 F V + sm) (1 e ( x))) } {{ } New matches by workers in stock V (1 F ( x)) + (λ + χ u U + δm)e }{{} Workers in flow who fail to match ( V = (µ + χ v V + sm)e U(1 F ( x)) (λ + χ u U + δm) ( Ṁ =(µ + χ v V + sm) 1 e U(1 F ( x))) + (λ + χ u U + δm) (s + δ) M (s + χ u )U }{{} Workers who return to flow 1 e V (1 F ( x))) (δ + χ v ) V (1 F (1 e ( x))) The measures U, V and M are the steady state solutions to the above equations. Endogenous matching function that depends on stock, flows and x. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 18

25 Flow equations On island i, stocks evolve according to U = (µ + χ v U(1 F V + sm) (1 e ( x))) } {{ } New matches by workers in stock V (1 F ( x)) + (λ + χ u U + δm)e }{{} Workers in flow who fail to match ( V = (µ + χ v V + sm)e U(1 F ( x)) (λ + χ u U + δm) ( Ṁ =(µ + χ v V + sm) 1 e U(1 F ( x))) + (λ + χ u U + δm) (s + δ) M (s + χ u )U }{{} Workers who return to flow 1 e V (1 F ( x))) (δ + χ v ) V (1 F (1 e ( x))) The measures U, V and M are the steady state solutions to the above equations. Endogenous matching function that depends on stock, flows and x. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 18

26 Flow equations On island i, stocks evolve according to U = (µ + χ v U(1 F V + sm) (1 e ( x))) } {{ } New matches by workers in stock V (1 F ( x)) + (λ + χ u U + δm)e }{{} Workers in flow who fail to match ( V = (µ + χ v V + sm)e U(1 F ( x)) (λ + χ u U + δm) ( Ṁ =(µ + χ v V + sm) 1 e U(1 F ( x))) + (λ + χ u U + δm) (s + δ) M (s + χ u )U }{{} Workers who return to flow 1 e V (1 F ( x))) (δ + χ v ) V (1 F (1 e ( x))) The measures U, V and M are the steady state solutions to the above equations. Endogenous matching function that depends on stock, flows and x. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 18

27 Firms Participation Decisions Firms can create vacancies or stay inactive. Creating a vacancy requires paying sunk cost c. Vacancy is randomly allocated on an island i. The number of firms in the economy, N, is determined by c = E i J f i = E i [J s i + H f i ], where the expectation is taken with respect to the distribution of islands characteristics. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 19

28 Existence of Equilibrium Characterization Natural definition of equilibrium. More Characterized by three equations in (M, x, µ). Equations Uniform Reallocation Reallocation is uniform if χ u = δ and χ v = s. Assume this from now on. Proposition - Single Island For any N f, there exists a unique pair ( x i, M i ) that satisfies the flow and reservation equations jointly. Proposition - Aggregate Economy 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 with Endogenous Matching 20

29 Existence of Equilibrium Characterization Natural definition of equilibrium. More Characterized by three equations in (M, x, µ). Equations Uniform Reallocation Reallocation is uniform if χ u = δ and χ v = s. Assume this from now on. Proposition - Single Island For any N f, there exists a unique pair ( x i, M i ) that satisfies the flow and reservation equations jointly. Proposition - Aggregate Economy 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 with Endogenous Matching 20

30 Existence of Equilibrium Characterization Natural definition of equilibrium. More Characterized by three equations in (M, x, µ). Equations Uniform Reallocation Reallocation is uniform if χ u = δ and χ v = s. Assume this from now on. Proposition - Single Island For any N f, there exists a unique pair ( x i, M i ) that satisfies the flow and reservation equations jointly. Proposition - Aggregate Economy 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 with Endogenous Matching 20

31 Existence of Equilibrium Characterization Natural definition of equilibrium. More Characterized by three equations in (M, x, µ). Equations Uniform Reallocation Reallocation is uniform if χ u = δ and χ v = s. Assume this from now on. Proposition - Single Island For any N f, there exists a unique pair ( x i, M i ) that satisfies the flow and reservation equations jointly. Proposition - Aggregate Economy 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 with Endogenous Matching 20

32 COMPARATIVE STATICS Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 21

33 Comparative Statics in Aggregate Steady States All islands are identical. We change b on all islands and allow the entry rate of jobs µ to adjust consistent with free entry. Main characteristics of the economy In recessions: Higher unemployment rates and lower vacancy rates. Higher long-term unemployment and lower short-term unemployment. More congestion lower probability of leaving the stock. Measured matching efficiency decreases (but there is non-monotonicity). More Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 22

34 Changes in Aggregate Flow Surplus F( x) b Short- and long-term unemployment < 4 weeks > 26 weeks b Probabilities for worker in flow Match No meeting Meet but no match b Probabilities for job in flow Match No meeting Meet but no match b Poisson matching rate for agent in stock Worker Job b 3.3 Matching efficiency, M/[u0.5 v 0.5 ] b Decomposition Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 23

35 EFFICIENCY Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 24

36 Efficiency The planner chooses {M( ), U( ), V ( ), x( ), µ( )} to maximize GDP. Planner s Problem Proposition - Single Island (Mortensen Rule) If wages are determined by a second price auction, then the planner s reservation match productivity is the same as the one in the decentralized economy conditional on N f and N w. Proposition - Aggregate Economy If wages are determined using second price auction the agent in the flow receives its marginal contribution to the surplus. 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 with Endogenous Matching 25

37 Efficiency The planner chooses {M( ), U( ), V ( ), x( ), µ( )} to maximize GDP. Planner s Problem Proposition - Single Island (Mortensen Rule) If wages are determined by a second price auction, then the planner s reservation match productivity is the same as the one in the decentralized economy conditional on N f and N w. Proposition - Aggregate Economy If wages are determined using second price auction the agent in the flow receives its marginal contribution to the surplus. 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 with Endogenous Matching 25

38 Efficiency The planner chooses {M( ), U( ), V ( ), x( ), µ( )} to maximize GDP. Planner s Problem Proposition - Single Island (Mortensen Rule) If wages are determined by a second price auction, then the planner s reservation match productivity is the same as the one in the decentralized economy conditional on N f and N w. Proposition - Aggregate Economy If wages are determined using second price auction the agent in the flow receives its marginal contribution to the surplus. 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 with Endogenous Matching 25

39 DURATION DISTRIBUTIONS Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 26

40 Homogeneous Islands: Duration Distributions Stark implications for unemployment and vacancy duration distributions Proposition - Matching Probabilities in the Flow Matching probabilities for workers and jobs in the flow are determined solely by the unemployment and vacancy rates u = U/(U + M) and v = V /(V + N). Proposition - Poisson Matching Rates in the Stock Poisson matching rates for workers and jobs when in the stock are determined solely by u, v, and the match destruction rate s + δ. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 27

41 Homogeneous Islands: Duration Distributions Stark implications for unemployment and vacancy duration distributions Proposition - Matching Probabilities in the Flow Matching probabilities for workers and jobs in the flow are determined solely by the unemployment and vacancy rates u = U/(U + M) and v = V /(V + N). Proposition - Poisson Matching Rates in the Stock Poisson matching rates for workers and jobs when in the stock are determined solely by u, v, and the match destruction rate s + δ. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 27

42 Homogeneous Islands: Duration Distributions Stark implications for unemployment and vacancy duration distributions Proposition - Matching Probabilities in the Flow Matching probabilities for workers and jobs in the flow are determined solely by the unemployment and vacancy rates u = U/(U + M) and v = V /(V + N). Proposition - Poisson Matching Rates in the Stock Poisson matching rates for workers and jobs when in the stock are determined solely by u, v, and the match destruction rate s + δ. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 27

43 Comparative Statics: Changing u or v Effect of changing u Effect of changing v Difficult to interpret since u and v are endogenous. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 28

44 Comparative Statics: Changing u or v Effect of changing u Effect of changing v Difficult to interpret since u and v are endogenous. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 28

45 Comparative Statics: Changing N f p u and φ u Survivor function Increase in N f, holding constant F ( x), lowers u and raises v. Fall in u occurs via a shift in the survivor function at all durations. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 29

46 Comparative Statics: Changing 1 F ( x) p u and φ u Survivor function Increase in 1 F ( x), holding constant N f, lowers both u and v Fall in u occurs particularly at longer durations. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 30

47 Baseline: Model vs Data (Farber and Valletta) Survivor Hazard We target the unemployment and vacancy rates and let s + δ = 0.1. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 31

48 Baseline: Model vs Data (Farber and Valletta) Survivor Hazard We target the unemployment and vacancy rates in each period keeping s + δ = 0.1. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 32

49 Changes in Match. Eff. or Entry Changes in Matching Efficiency Changes in Firm Entry We target the unemployment rate in both periods keeping s + δ = 0.1. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 33

50 Combining the Exercises We target the unemployment rate in both periods keeping s + δ = 0.1. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 34

51 Allowing for heterogeneous islands Model heterogeneity by assuming that the economy consists of K archipelagos, unrelated to each other Aggregate u, v, duration distribution arise from aggregation across archipelagos. When islands are heterogeneous, duration-biased sampling delivers a declining hazard rate over time for agents in stock Analogous to random matching model: Heckman and Singer (1984), etc Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 35

52 Heterogeneity and Duration Distribution Survivor Functions Hazards Compare three cases with same aggregate (u, v): 1 all islands identical; 2 two types of islands, with Corr(u k, v k ) > 0 α 1 = α 2 = 0.5, u 1 /u = v 1 /v = 0.8, and u 2 /u = v 2 /v = 1.2; 3 two types of islands, with Corr(u k, v k ) < 0 α 1 = α 2 = 0.5, u 1 /u = v 2 /v = 1.5, and u 2 /u = v 1 /v = 0.5. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 36

53 Targeting Empirical Duration Distribution Survivor Functions Hazards Assuming two types of islands, choose {(u i, v i )} and shares {α i } to match empirical hazard rates. Fit is good Parameterization: (α 1, u 1, v 1 ) = (0.611, 0.063, 0.044); (α 2, u 2, v 2 ) = (0.389, 0.039, ). Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 37

54 Using Cross-Industry (u, v) Dispersion Survivor Functions Hazards Figure uses data from JOLTS and CPS to measure cross-sectional dispersion across 17 major industries cf Şahin, Song, Topa, and Violante (2014) Data for Mar 2002 Feb 2004 Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 38

55 Boom vs Recession Survivor Functions Hazards Using same industry data, changes in model-generated duration distribution and hazard rates follow what was actually observed Data for Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 39

56 CONCLUSION Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 40

57 Conclusion and Further Work Summary To do We have developed a tractable equilibrium stock flow model with endogenous match formation and firm participation decisions. Explored the implications for efficiency and of different bargaining protocols (only shown one here). Both match formation and participation margins are needed to explain unemployment duration changes. Further exploration of the role of heterogeneity. Vacancy data to construct hazard functions. Policy implications. Framework could also be applied to other markets e.g. housing market. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 41

58 APPENDIX Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 42

59 Elsby et al (2014): LTU and Measured Matching Efficiency Back Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 43

60 Coles and Smith (1998): Vacancy hazard MARKETPLACES AND MATCHING r 0,15- > I () I)II'.ti rati 11 1);I% S) FIGURE 2 Back MEANS OF VACANCY HAZARD RATES 5. CONCLUSION This paper describes a particularly simple model of marketplace matching that predicts a certain type of matching behaviour-that the stock of traders on one side of the market matches with the flow on the other side. Using matching data available from U.K. Job Centres, we find that the matching behaviour of workers Carrillo-Tudela and Hawkins does change systematically with their Stock-Flow duration of with unemployment. Endogenous Furthermore, Matchingthe 44

61 Surplus Define the surplus associated with forming a match with productivity x to be S i (x) = [Wi m (w) Wi s ] + [Ji m (x w) Ji s ] ; Substituting out the above equations (r + s i + δ i )S i (x) = x b i + γ i (µ i + χ u i U i + δ i M i )Gi s (λ i + χ v i V i + s i M i )Hi s (χ u i δ i )Gi f (χ v i s i )Hi f. The surplus is well defined, since the wage w cancels out. Higher-x matches deliver higher surplus and S i (x) is an increasing affine function of x. Back Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 45

62 Gains from Match Formation Worker in the Flow Since the agents will form a match with the max(x) as along as max(x) > x, then the expected gain for a worker in the flow is given by Gi f e V i V k i = Gi f (x)dfi k (x), k! X k k=1 where Gi f (x) = [Wi m (wi f (x)) Wi s ] if max{x} x i and 0 otherwise. Firm in the Stock The expected gain for a firm in the stock is given by Hi s e V i V k 1 i 1 = Hi s (x)dfi k (x), (k 1)! k X k where H s i (x) = [J m i k=1 (π s i (x)) J s i ] if max{x} x i and 0 otherwise. Similar expressions for G s and H f. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 46

63 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 i and for all parameter constellations {ω i } 1 i=0, G f i = H f i = and such that 1 G f ( x i, V i, F i ), Gi s = r + δ i + s i 1 H f ( x i, U i, F i ), and Hi s = r + δ i + s i 1 G s ( x i, U i, F i ), r + δ i + s i (1) 1 H s ( x i, V i, F i ), r + δ i + s i (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 Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 47

64 Wage and Profit Determination Proposition 1 Suppose that there are continuous functions ŵ f, ŵ s, ˆπ f, ˆπ s : X [0, ) such that for all i and for all parameter constellations {ω i } 1 i=0, w f i (x) = ŵ f (x x i ) + w i, w s i (x) = ŵ s (x x i ) + w i, π f i (x) = ˆπ f (x x i ) + π i, and π s i (x) = ˆπ s (x x i ) + π i, where (x 1, x 2,..., x k ) y = (x i y) i xi 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, reduced 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 Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 48

65 Stock-Flow Matching Equilibrium Definition A SFM equilibrium is a set of values Wi s, Wi m, Wi f, Ji s, Jm i and Ji f ; a set of wages wi s(x), w i f (x) and profits πs i (x), πf i (x); a measure Nf of firms, U i of unemployed workers in the stock, V i of job vacancies in the stock and M i 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 i, where x i solves S i ( x) = 0. 3 Optimal entry determines µ. 4 The measures U i, V i and M i are the steady state solution to the transition equations described above. Back Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 49

66 Endogenous Matching Function On each island i, the (reduced form) matching function is given by ( M = [µ + χ v v + sm] 1 e U(1 F ( x))) ( + [λ + χ u U + δm] 1 e V (1 F ( x))), Back It depends on the acceptance strategies of workers and firms, summarized by x. It depends on the flows of jobs, (µ + χ v V + sm), and the flows of workers, (λ + χ u U + δm). It depends on the stock of unemployed workers, U, vacant jobs, V and matches M. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 50

67 Characterization Flow balance: (s + δ)m = [ ] ( µ + χ v V + sm 1 e U(1 F ( x))) [ ] ( + λ + χ u U + δm 1 e V (1 F ( x))). Reservation productivity: x b + γ = [ ] [ ] µ + χ v V + sm G s + λ + χ u U + δm H s + (χ u δ)g f + (χ v s)h f. Back Free entry: c = E i [ 1 r + δ { [ ] } ] γ + λ + χ u U δm H s + (r + χ v + δ)h f. 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. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 51

68 Comparative Statics in the Cross-section Islands differ only on the size of the flow surplus through difference in workers opportunity cost b. Other parameters are constant and common to all islands. In particular, inflow rates λ and µ Main characteristics of the economy Relative to islands with higher surplus, islands with lower surplus have a higher x. This implies that the latter have: A higher unemployment and vacancy rate. A higher proportion of LTU and a smaller proportion of STU. More congestion lower probability of leaving the stock. Lower measured matching efficiency. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 52

69 Comparative Statics in the Cross-section Islands differ only on the size of the flow surplus through difference in workers opportunity cost b. Other parameters are constant and common to all islands. In particular, inflow rates λ and µ Main characteristics of the economy Relative to islands with higher surplus, islands with lower surplus have a higher x. This implies that the latter have: A higher unemployment and vacancy rate. A higher proportion of LTU and a smaller proportion of STU. More congestion lower probability of leaving the stock. Lower measured matching efficiency. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 52

70 Cross-Sectional Heterogeneity in Flow Surplus F( x) b Short- and long-term unemployment < 4 weeks > 26 weeks b Probabilities for worker in flow Match No meeting Meet but no match b Probabilities for job in flow Match No meeting Meet but no match b Poisson matching rate for agent in stock Worker Job b 7 Matching efficiency, M/[u0.5 v 0.5 ] b Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 53

71 Cross-Sectional Heterogeneity Heterogeneity in island-level surplus across islands generates perfectly positively correlated (U i, V i ). This is by construction: U i = λ s M i, V i = µ δ M i. Allowing for endogenous match formation affects the cross-sectional dispersion in (U i, V i ) but not the correlation. To relax the correlation, need heterogeneity in λ, µ, s, or δ. Main characteristics of the economy when changing µ Relative to islands with higher µ, islands with lower µ have (mostly) a lower x. This implies that the latter have: Higher unemployment rates and lower vacancy rates. Higher long-term unemployment and lower short-term unemployment. More congestion lower probability of leaving the stock. Higher measure of matching efficiency. Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 54

72 Cross-Sectional Heterogeneity in µ F( x) Short- and long-term unemployment 1 1 Probabilities for worker in flow < 4 weeks > 26 weeks Match No meeting Meet but no match µ µ µ 1 Probabilities for job in flow Poisson matching rate for agent in stock Matching efficiency, M/[u 0.5 v 0.5 ] Match No meeting Meet but no match Worker Job µ µ µ Back Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 55

73 Planner s Problem The planner chooses {m( ), u( ), V ( ), x( ), µ( )} to maximize aggregate (net) output. This maximization can be represented as max M( ),U( ),V ( ), x( ),µ( ) 0 e rt [bu(t) γv (t) cµ(t) + (µ(t) + χ v V (t) + sm(t)) x(t) x r + s + δ U(t)e U(t)[1 F (x)] f (x)dx + (λ + χ u x U(t) + δm(t)) x(t) r + s + δ V (t)e V (t)[1 F (x)] f (x)dx ] dt subject to the flow constraints described above. Back Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 56

74 Efficiency Single Island (χ u = δ and χ v = s) The FOC for the planner s x where x b + γ = Ĝ s 1 ( x, U) r + s + δ Ĥ s ( x, V ) 1 r + s + δ λ(s + δ) Ĥ s ( x, V ) + s x x µ(s + δ) Ĝ s ( x, U). δ (x x) [1 U(1 F (x))] e U[1 F (x)] f (x)dx (x x) [1 V (1 F (x))] e V [1 F (x)] f (x)dx. Back Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 57

75 Efficiency Aggregate Economy (χ u = δ and χ v = s) From the FOC for the planner s µ, we find that entry will be efficient in the decentralized economy if J s = η v, where η v is the co-state variable of the law of motion for V. For this to happen, however, both H s ( x, v) = Ĥs ( x, V ) and H f ( x, U) = Ĥf ( x, U), where Ĥ f ( x, U) = 1 r + s + δ x (x x)ue U[1 F (x)] f (x)dx. If wages are determined using second price auction H s ( x, V ) = Ĥ s ( x, V ), but S( x, U) = H f ( x, U) < H f ( x, U). Back Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 58

76 Implications where 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 e V V k k! e V V k k! 1 r + s + δ k=0 [ max W m i β x 1 < <x k ] (wi f (x)) Wi s, 0 dfi k (x) k (1 β) k j x x r + s + δ df (x1)... df (x k) j=1 e V (1 F ( x)) [V (1 F ( x))] k β 1 k! a 1 Ein(β(1 F ( x))v ), a(r + s + δ) Ein(z) is the entire exponential integral function. z 0 1 e t dt. t k 1 (1 β) j Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 59 j=1 j

77 Implications Deriving G f, H s and H f distribution yields in a similar way as above and using the exponential G s 1 [ ( = Ein Ue a x ) Ein ( βue a x)] (r + s + δ)au H s 1 [ ( = Ein Ve a x ) Ein ( βve a x)] (r + s + δ)av G f 1 = (r + s + δ)a Ein ( βve a x) H f 1 = (r + s + δ)a Ein ( βue 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 Carrillo-Tudela and Hawkins Stock-Flow with Endogenous Matching 60