Lecture Notes. Petrosky-Nadeau, Zhang, and Kuehn (2015, Endogenous Disasters) Lu Zhang 1. BUSFIN 8210 The Ohio State University

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1 Lecture Notes Petrosky-Nadeau, Zhang, and Kuehn (2015, Endogenous Disasters) Lu Zhang 1 1 The Ohio State University BUSFIN 8210 The Ohio State University

2 Insight The textbook Diamond-Mortensen-Pissarides model of equilibrium unemployment gives rise endogenously to rare disasters

3 Outline 1 The Model 2 Quantitative Results 3 Home Production 4 Capital 5 Recursive Utility

4 Outline 1 The Model 2 Quantitative Results 3 Home Production 4 Capital 5 Recursive Utility

5 Model Search and matching The representative rm posts job vacancies, V t, to attract unemployed workers, U t, via a CRS matching function: The job lling rate: G(U t, V t ) = q(θ t ) G(U t, V t ) V t = U t V t (U ι t + V ι t ) 1/ι 1 (1 + θ ι t) 1/ι in which θ t = V t /U t is labor market tightness: q (θ t ) < 0

6 Model The unit costs of vacancy The representative rm incurs costs of vacancy posting, κ t V t, with the unit costs, κ t, given by: in which κ 0 : The proportional costs κ 1 : The xed costs κ t κ 0 + κ 1 q(θ t ) Fixed matching costs (paid after a hired worker arrives): Training, interviewing, negotiation, and administrative setup costs of adding a worker to the payroll, etc., as in Pissarides (2009)

7 Model An occasionally binding constraint on vacancy Once matched, jobs are destroyed at a constant rate s: N t+1 = (1 s)n t + q(θ t )V t in which V t 0 The rm uses labor to produce via a CRS production function: Y t = X t N t in which log(x t+1 ) = ρ log(x t ) + σɛ t+1

8 Model The representative rm Dividends to shareholders: in which W t is the wage rate D t = X t N t W t N t κ t V t Taking the stochastic discount factor, M t+1 = β(c t /C t+1 ), as given, the rm maximizes the market value of equity, S t : [ ] S t max E t M t+τ D t+τ {V t+τ,n t+τ+1 } τ=0 τ=0 subject to N t+1 = (1 s)n t + q(θ t )V t and V t 0

9 Model The intertemporal job creation condition and the stock return Let λ t be the multiplier on the q(θ t )V t 0 constraint: κ 0 q(θ t ) + κ 1 λ t = [ [ ]]] κ 0 E t M t+1 [X t+1 W t+1 + (1 s) q(θ t+1 ) + κ 1 λ t+1 The Kuhn-Tucker conditions: V t 0, λ t 0, λ t V t = 0

10 Model Wages divide rents between workers and the rm Wages as the endogenous outcome of a generalized Nash bargaining process between a worker and the rm: W t = η (X t + κ t θ t ) + (1 η)b η: Relative bargaining weight of the worker X t : Marginal product of labor κ t θ t = κ t V t /U t : Vacancy costs per unemployed worker b: The ow value of unemployment activities η and b govern wage elasticity to labor productivity

11 Model Equilibrium The goods market clearing condition: C t + κ t V t = X t N t The recursive competitive equilibrium consists of vacancies, V t 0; multiplier, λ t 0; consumption, C t ; and indirect utility, J t : V t and λ t satisfy the intertemporal job creation condition and the Kuhn-Tucker conditions, while taking the wage equation and the representative household's SDF as given; C t and J t satisfy the optimality condition 1 = E t [M t+1 R t+1 ]; the goods market and the nancial market clear

12 Model Parameterized expectations per Christiano and Fisher (2000) Solve for λ t λ(n t, X t ) from κ 0 q(θ t ) + κ 1 λ t = [ ( )]] κ0 E t M t+1 [X t+1 W t+1 + (1 s) q(θ t+1 ) + κ 1 λ t+1 while obeying the Kuhn-Tucker conditions log(x t ) discretized with 17 grid points; cubic splines (50 basis functions) in N for each log(x )-level

13 Model Parameterized expectations per Christiano and Fisher (2000) Parameterizing the conditional expectation, E t E(N t, X t ) eliminates the need to parameterizing λ t separately: κ 0 q(θ t ) + κ 1 λ t = E t After obtaining E t, calculate q(θ t ) = κ 0 / (E t κ 1 ): If q(θ t ) 1 (binding constraint): set V t = 0, θ t = 0, q(θ t ) = 1, and λ t = κ 0 + κ 1 E t If q(θ t ) < 1 (nonbinding constraint): set λ t = 0, q(θ t ) = q(θ t ) θ t = q 1 ( q(θ t )), V t = θ t (1 N t )

14 Model Monthly calibration Rate of time preference, β Aggregate productivity persistence, ρ Gertler and Trigari (2009) Conditional volatility of shocks, σ 0.01 Gertler and Trigari (2009) Workers' bargaining weight, η 0.04 Hagedorn and Manovskii (2008) Job destruction rate, s 0.04 Davis et al. (2006) Elasticity of matching function, ι 1.25 Den Haan et al. (2000) Value of unemployment activities, b 0.85 Hagedorn and Manovskii (2008) The proportional costs, κ The xed costs, κ 1 0.5

15 Outline 1 The Model 2 Quantitative Results 3 Home Production 4 Capital 5 Recursive Utility

16 Quantitative Results Basic moments Panel A: Log output growth Panel B: Log consumption growth data mean 5% 95% p-value data mean 5% 95% p-value σ Y σ C S Y S C K Y K C ρ Y ρ C ρ Y ρ C ρ Y ρ C ρ Y ρ C Panel C: Unemployment data mean 5% 95% p-value data mean 5% 95% p-value E[U] S U K U σ U

17 Quantitative Results Empirical cumulative distribution functions: U t, Y t, and C t Probability Probability Probability Unemployment Output Consumption

18 Quantitative Results Moments of rare disasters Data Model Mean 5% 95% p-value Panel A: Output Probability Size Duration Panel B: Consumption Probability Size Duration

19 Quantitative Results Distributions of disasters by size 5 5 Number of disasters Number of disasters Cumulative decline Cumulative decline

20 Quantitative Results Distributions of disasters by duration Number of disasters Number of disasters Duration Duration

21 Quantitative Results Comparative statics Baseline b = b = 0.4 s = κ t = 0.7 ι = 1.1 η = 0.05 Panel A: Output Probability Size Duration Panel B: Consumption Probability Size Duration

22 Quantitative Results Downward rigidity in marginal hiring costs Employment Employment

23 Outline 1 The Model 2 Quantitative Results 3 Home Production 4 Capital 5 Recursive Utility

24 Home Production Model Let C mt be market consumption, C ht home consumption, and the composite consumption bundle: in which e (0, 1] and a [0, 1] The stochastic discount factor: C t [ac e mt + (1 a)c e ht ]1/e, M t+1 = β ( Cmt+1 C mt ) e 1 ( Ct C t+1 ) e

25 Home Production Model Home production technology: C ht = X h U t, with X h > 0 The equilibrium Nash-wage becomes: W t = η(x t + κ t θ t ) + (1 η)z t, with z t X h ( 1 a a ) ( ) 1 e Cmt + b C ht The market clearing condition: C mt + κ t V t = X t N t

26 Home Production Quantitative Results Data Model Data Model σ σ a a e e σ Y σ C S Y S C K Y K C ρ Y ρ C ρ Y ρ C ρ Y ρ C ρ Y ρ C Prob Y Prob C Size Y Size C Dur Y Dur C E[U] S U K U σ U

27 Outline 1 The Model 2 Quantitative Results 3 Home Production 4 Capital 5 Recursive Utility

28 Capital Model Production Y t = X t K α t N 1 α t, α (0, 1), and x t = log(x t ): x t+1 = (1 ρ) x + ρx t + σɛ t+1 Capital accumulation: K t+1 = (1 δ)k t + Φ(I t, K t ), in which δ is the capital depreciation rate, I t is investment, and [ ) ] 1 1/ν Φ(I t, K t ) a 1 + a 2 1 1/ν ( It K t K t

29 Capital Model The investment Euler equation: ( ) ( 1/ν 1 It = E t [M t+1 α Yt a 2 K t+1 a 2 K t ( It+1 K t+1 ) 1/ν (1 δ + a 1) + 1 ν 1 I t+1 K t+1 )] The intertemporal job creation condition: κ 0 +κ1 λt = Et q(θ t) The equilibrium wage: [M t+1 ( (1 α) Yt+1 N t+1 W t+1 + (1 s) W t = η [ ] (1 α) Yt + κ tθ t + (1 η)b N t ])] κ 0 + κ1 λt+1 q(θ t+1) [ The goods market clearing condition: C t + I t + κ tv t = Y t

30 Capital Quantitative results Data Model Data Model σ σ ν ν σ Y σ C S Y S C K Y K C ρ Y ρ C ρ Y ρ C Prob Y Prob C Size Y Size C Dur Y Dur C σ I E[U] S I S U K I K U ρ I σ U ρ I

31 Outline 1 The Model 2 Quantitative Results 3 Home Production 4 Capital 5 Recursive Utility

32 Preferences: J t = max {C t} [ (1 β)c 1 1 ψ t [ + β (E t Recursive Utility J 1 γ t+1 Model ] 1 ]) 1 1/ψ 1 1/ψ 1 γ Optimality condition: 1 = E t [M t+1 R t+1 ]: M t+1 = β ( Ct+1 C t ) 1 ψ J t+1 E t [J 1 γ ] 1 1 γ t+1 1 ψ γ R t+1 = X t+1 W t+1 + (1 s) [κ 0 /q(θ t+1 ) + κ 1 λ t+1 ] κ 0 /q(θ t ) + κ 1 λ t

33 Recursive Utility Quantitative results Data Model Data Model γ γ ψ ψ σ Y σ C S Y S C K Y K C ρ Y ρ C ρ Y ρ C ρ Y ρ C ρ Y ρ C Prob Y Prob C Size Y Size C Dur Y Dur C E[U] E[R R f ] S U E[R f ] K U σ R σ U σ R f

34 Recursive Utility Predicting excess returns and consumption growth with log price-to-consumption H 1y 2y 3y 4y 5y 1q 4q 8q 12q 16q 20q U.S. annual data, U.S. quarterly data, 1947q22013q4 β R t R RR β C t C RC Recursive utility Recursive utility β R t R RR β C t C RC

35 Conclusion Endogenous disasters The textbook Diamond-Mortensen-Pissarides model of equilibrium unemployment gives rise endogenously to rare disasters