Household Debt, Financial Intermediation, and Monetary Policy Shutao Cao 1 Yahong Zhang 2 1 Bank of Canada 2 Western University October 21, 2014
Motivation The US experience suggests that the collapse of house price can be the leading factor for the recent great recession shape decline in house price delinquency rate rises both households and banks balance sheet deteriorate rise in mortgage risk premium further worsens the problem This calls for a model that take into account both banks and households balance sheets Existing works on household debt in DSGE literature are silent on the role of banks exception: Iacoviello (2014) however, it does not has a micro-foundation for why banks face the limitation of raising deposits 2 / 46
220 S&P/Case-Shiller 20-City Composite Home Price Index 200 (Index January 2000 = 100) 180 160 140 120 100 80 2000 2002 2004 2006 2008 2010 2012 2014 Source: S&P Dow Jones Indices LLC Shaded areas indicate US recessions - 2014 research.stlouisfed.org 3 / 46
12 11 10 9 8 Delinquency Rate On Single-Family Residential Mortgages, Booked In Domestic Offices, All Commercial Banks (Percent) 7 6 5 4 3 2 1 2000 2002 2004 2006 2008 2010 2012 2014 Source: Board of Governors of the Federal Reserve System Shaded areas indicate US recessions - 2014 research.stlouisfed.org 4 / 46
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Our Contribution This paper introduces a micro-founded banking sector to a DSGE model with agents with heterogenous desires to save and borrow Key features: Two types of agents: patient agents: depositors impatient agents: borrowers enforcement problem between bankers and borrowers: collateral constraint Bankers channel funds between borrowers and depositors Gerter and Karadi (2010), agency problem between bankers and depositors bankers have incentive to divert funds as long as the constraint binds, bankers demand excess return (risk premium) 6 / 46
Key Mechanism an Example An unexpected risk in housing demand leads to a rise in house price Household debt rises while collateral constraint for household borrowers tightens Rise in house price increases bankers asset values Net worth of the bankers rise Bankers incentive constraint becomes less tight, leading a decline in risk premium The other way around for the financial crisis case 7 / 46
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Closely Related Papers Financial intermediaries in business cycle: Gertler and Kiyotaki (2013), Gertler and Karadi (2011), Iacoviello (2014) Household debt and business cycle: Iacoviello (2005), Iacoviello and Neri (2010), and Justiniano, Primiceri and Tambalotti (2013) 9 / 46
The Model 10 / 46
Set Up of the Model Households: patient and impatient patient: depositors impatient: borrowers (subject to a collateral constraint) Bankers: provide funds to firms and impatient households, issue deposits to patient households incentive of diverting funds demand excess return Producers competitive house producers competitive capital producers: provide capital to wholesale goods wholesale goods producers: use labour and capital, borrow from bankers for capital acquisition retailers Government and monetary authority traditional Taylor rule credit policy 11 / 46
Two types of households Patient households are depositors to the financial intermediaries β p Impatient households are borrowers from the financial intermediaries β ip β ip < β p 12 / 46
Depositors Maximize life-time utility by choosing consumption, housing and capital accumulation, and labour supply E 0 t=0 βp(ε t c t log(c p,t bc p,t 1 ) + ε h t φ log h p,t ψ n1+η p,t 1 + η ), subject to a budget constraint P t c p,t + P h t i h p,t + D t <= W p,t n p,t + R t 1 D t 1 T p,t + Π t, where Π t includes the rest of resources/costs: the revenue from of the existing bankers, the transfer to the new bankers housing stock law of motion h p,t = (1 δ h )h p,t 1 + i h p,t 13 / 46
First Order Conditions for Depositors Consumption decision Housing decision λ p 1,t = E t β p R t λ p 1,t+1 π t+1 ε c t εh t φ h p,t + β p λ p 1,t+1 q t+1(1 δ h ) = q t λ p 1,t Labour supply ϕn η p,t = λ p 1,t w p,t 14 / 46
Borrowers Maximize by choosing consumption, housing and labour supply E 0 t=0 βip t (εc t log(c ip,t bc ip,t 1 ) + ε h t φ log h ip,t ψ n1+η ip,t 1 + η ), subject to housing stock law of motion a budget constraint h ip,t = (1 δ h )h ip,t 1 + i h ip,t, P t c ip,t + P h t i h ip,t + R t 1 l L t 1 <= W ip,t n i,t + L t T ip,t, and a collateral constraint L t <= E tθp h t+1 h ip,t(1 δ h ) R l t+1 15 / 46
First Order Conditions for Borrowers Consumption decision Housing decision ε c t ε h t λ ip 1,t = E t β ip R l t λ ip 1,t+1 π t+1 + λ ip 3,t Rl t, φ h ip,t λ ip 2,t +E tλ ip 2,t+1 β ip(1 δ h )+λ ip 3,t E tθε θ t q t+1 π t+1 (1 δ h ) = 0. Labour supply decisions is similar 16 / 46
Financial Intermediaries Fund capital investment to capital producers k t qt k and lend mortgage loans L t to the impatient households Issue deposits D t to patient households, In real terms, total assets are financed by deposits and their own net worth n t A t = k t qt k + l t = d t + n t Banker s net worth evolves as n t+1 = R l t(k t q k t + l t ) R t d t 17 / 46
Bankers Incentive Constraint A moral hazard problem, following Gertler and Kiyotaki (2013), Gertler and Karadi (2010). At the end of the each period a banker can choose to divert the fraction of κ of assets for personal use. For the patient households to be willing to lend to the banker, the following incentive constraint must be satisfied: V t κa t Present value of payout from operating the bank, V t, must exceed the gain of diverting assets. 18 / 46
Bankers Value Function In each period, probability σ of surviving, and a probability of 1 σ of exiting Bankers are risk neutral and only consume (their net worth) when they exit V t can be expressed recursively as V t = E t [β(1 σ)n t+1 + βσv t+1 ] 19 / 46
Value Function and Incentive Constraint Gertler and Kiyotaki (2013) show that V t is a linear function of assets and deposit V t = µ t A t + υ dt n t µ t is the excess marginal value of assets over deposits. The incentive constraint can be written as µ t A t + υ dt n t κa t (1) The maximum feasible ratio between assets to net worth is φ t = A t n t = υ dt κ µ t. the agency problem between the banker and depositors leads to an endogenous capital constraint. It binds if and only if when the excess marginal value from honestly managing assets µ t is positive but less than the marginal gain from diverting assets κ. 0 < µ t < κ 20 / 46
Risk Premium µ t and υ dt can be further expressed as µ t = βe t [(R l t+1 R t+1 )Ω t+1 ], with υ dt = βe t [R t+1 Ω t+1 ], Ω t+1 = 1 σ + σ(υ dt+1 + φ t+1 µ t+1 ), Ω t+1 can be thought of as a weighted average of marginal values of net worth for the exiting bankers (1 σ) and surviving bankers (σ). 21 / 46
Risk Premium and Incentive Constraint Let λ b t be the Lagrangian multiplier for the incentive constraint equation (1) λ b t = µ t. κ µ t When the incentive constraint is not binding, λ b t = 0, µ t = 0 and (Rt+1 l R t+1) = 0 As long as the constraint is binding, limits to arbitrage will lead to a positive expected excess return, R l t+1 > R t+1. The tighter the constraint is, the larger λ b t is, and the higher the excess return is In the model, a decline in asset value caused by house prices will tighten the incentive constraint of the bankers, and in turn the excess return is much higher in the crisis period. 22 / 46
Aggregate Net Worth At aggregate level, banker s net worth evolves as n = n n t + n e t with and n e t = σ[(r l t R t )φ t + R t ]n e t 1 n n t = ωa t 1 23 / 46
Capital Producers The capital producer solves for i k t to maximize max E t β p Λ t,t+1 {P k t [τ k t where Λ t,t+1 = c p,t /c p,t+1. S k ( ik t it 1 k )]it k P t it k } S(i t, i t 1 ) is investment adjustment costs with S = S = 0 and S = χ k > 0, and χ k > 0 is an investment adjustment cost parameter 24 / 46
Wholesale Goods Producers The wholesale goods are produced by using the following production technology y t = A t k α t ((n p,t ) γ (n ip,t ) 1 γ ) 1 α. Firms borrow from the financial intermediaries for the capital acquisition at the rate R l t. Firms are competitive and earn zero profits, and at the end of period they pay out the realized return to capital to the intermediaries. E t Rt+1 l = E t[rt+1 k + qk t+1 (1 δ)], q k t 25 / 46
Wholesale Goods Producers Each period, firms maximize the profit by choosing k t, n p,t and n ip,t max pt w Y t rt k k t w p,t n p,t w ip,t n ip,t and the first-order conditions are p w t α Y t k t = r k t, and where p w t p w t αγ Y t n p,t = w p,t, p w t α(1 γ) Y t n p,t = w ip,t is the price for the wholesale goods. 26 / 46
Retailers There are continuum of retailers of mass 1, indexed by j. They buy intermediate goods from intermediate goods producers at pt w in a competitive market and differentiate the goods at no costs into y t (i), and sell y t (j) at the price p t (j). The final goods y t is the composite of individual variety, [ 1 y t = 0 ] ε y t (j) ε 1 ε 1 ε dj. Each period, only a fraction 1 ν of retailers reset their prices The retailer chooses p i,t to maximize its expected real total profit over the periods during which its prices remain fixed: [( ) ] E t Σ i=0ν p pi,t i,t+i y t+i mc t+i y t+i (i), mc t is the real marginal cost p i,t+i 27 / 46
House Producers The production of new house h n t = (τ h t S h ( ih t it 1 h ))it h, The house producer solves for i h t to maximize where Λ t,t+1 = c p,t /c p,t+1. max E t β p Λ t,t+1 [P h t h n t P t i h t ] 28 / 46
Government and Monetary Authority Government expenditures are financed by lump sum tax G t = T t where G t follows an AR(1) process, log G t = (1 ρ g ) log G ss + ρ g log G t 1 + ɛ g t, ɛg t i.i.d.n(0, σɛ 2 ). g The central bank operates according to the standard Taylor Rule. R t R = (R t 1 R )ρr (( π t π )ρπ ( Y t Y )ρy ) 1 ρr e ɛ m t, 29 / 46
Resource Constraint Goods market: c p,t + c ip,t + q t (i h p,t + i h ip,t) + q k t i k t = y f t, Housing market: h ip,t + h p,t = h t, 30 / 46
Data and Estimation 31 / 46
Table : Calibrated Parameter Values Household Discount rate for patient households β p 0.99 Discount rate for impatient households β ip 0.94 Relative utility weight of labour ψ 1 Relative utility weight of housing φ 0.25 Inverse Frisch elasticity of labour supply η 1.01 Collateral constraint parameter θ 0.85 Loan persistence parameter ρ l 0.95 Financial intermediaries Fraction of assets that can be diverted λ 0.381 Survival rate of the bankers σ 0.97 Intermediate good producers Capital share α 0.33 Share of patient households labour γ 0.64 32 / 46
Table : Calibrated Parameter Values Continued Capital producers Capital depreciation rate δ k 0.025 House producers Housing investment adjustment cost parameter χ h 30 Housing depreciation rate δ k 0.01 Retailers Elasticity of substitution across different goods ε 11 Government Steady-state government expenditure g 0.2 33 / 46
Data Used output, consumption, business investment, government spending nominal interest rate, inflation mortgage risk premium, mortgage loan 34 / 46
Data Summary Table : Standard Deviation of Key Variables y c i k i h q l r n π s Data 1 0.87 3.9 11.6 6.7 2.9 0.33 0.14 0.06 35 / 46
Data Summary Continued Table : Some Key Correlations y, s y, q -0.4 0.85 q, s q, l -0.48 0.22 l, s -0.21 36 / 46
Shocks Estimated technology, monetary policy, preference, investment specific and government spending shocks housing preference shock, LTV shock bank net worth shock 37 / 46
Preliminary Estimation Results Table : Estimation Results Parameter Prior Distribution Mode ρ r beta 0.75 0.2 0.7351 ρ π gamm 1.5 0.25 1.9248 ρ y norm 0.125 0.15 0.0617 ρ b beta 0.8 0.1 0.9356 υ beta 0.75 0.2 0.8149 b beta 0.8 0.1 0.8462 χ k gamm 4 1.5 3.5194 ρ A beta 0.6 0.2 0.5931 ρ c beta 0.6 0.2 0.5819 ρ h beta 0.6 0.2 0.9767 ρ ltv beta 0.6 0.2 0.2455 ρ n beta 0.6 0.2 0.4328 ρ g beta 0.6 0.2 0.8681 ρ i beta 0.6 0.2 0.5423 38 / 46
Preliminary Estimation Results Continued Table : Estimation Results Parameter Prior Distribution Mode ɛ a t invg 0.005 2 0.017 ɛ r t invg 0.005 2 0.0016 ɛ c t invg 0.005 2 0.0375 ɛ h t invg 0.005 2 0.1981 ɛ ltv t invg 0.005 2 0.1069 ɛ n t invg 0.005 2 0.0164 ɛ g t invg 0.005 2 0.0079 ɛ τ k t invg 0.005 2 0.0444 39 / 46
Model Performance Table : Standard Deviation Model vs. Data y c i k i h q l r n π s Data 1 0.87 3.9 11.6 6.7 2.9 0.33 0.14 0.06 Model 1 1.75 3.6 11.7 10.8 15 0.13 0.1 0.18 40 / 46
Model Performance Table : Standard Deviation Model vs. Data Data Model Data Model y, s y, q -0.4-0.37 0.85 0.21 q, s q, l -0.48-0.05 0.22 0.61 l, s -0.21 0.014 41 / 46
Model Performance Table : Variance Decomposition Business Cycle Frequency mon. pol. tech agg. dem housing ltv bank net gov inv output 18.01 1.28 23.93 16.29 0.47 17.25 1.62 21.15 consumption 0.23 0.36 79.11 12.27 0.07 7.4 0.04 0.5 business inv. 14.35 0.04 0.5 0.75 0.43 30.76 0.06 53.11 housing inv. 0.01 0.05 0.01 99.92 0 0 0 0.01 house price 0.03 0.5 0.18 99.09 0 0.07 0.02 0.11 mortgage 0.44 0.13 0.22 70.39 28.81 0.01 0 0.01 policy rate 30.99 40.89 7.24 3.63 0.13 0.25 0.49 16.37 inflation 7.93 64.6 4.35 8.26 0.68 0.82 0.3 13.06 risk premium 31.3 3.91 5.4 10.38 0.96 43.78 0.27 4 42 / 46
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Conclusions We introduces a micro-founded banking sector to a DSGE model with agents with heterogenous desires to save and borrow In the model collateral constraint and risk premium co-exist Preliminary results show that model generates counter cyclical risk premium bank net worth shocks are important in explaining output, mortgage risk premium and business investment Loan-to-Value ratio shocks are important in explaining mortgage 46 / 46