A Framework for Financial Stability Analysis: Contagion and Policy Measures

Transcription

1 A Framework for Financial Stability Analysis: Contagion and Policy Measures C.A.E. Goodhart, C. Osorio, D.P. Tsomocos XIXth European Workshop on General Equilibrium Theory Cracow University of Economics Special Session: The Current Financial Crisis June 11, 2010

2 Features of the 2007-Present Global Crisis Increased default in the U.S. mortgage market Contagion to securitized products and credit markets world wide Interbank markets fail to act as a conduit for monetary policy Systemically important financial institutions collapse Our finite horizon equilibrium outcome is consistent with these features The infinite horizon version of our model shows the importance of endogenous default: it captures short to medium run dynamics that may render the economy financially unstable

3 What should (Have) Be(en) done? Use the interest rate, instead of the monetary base, as the monetary policy instrument in times of financial distress (See Goodhart, Sunirand and Tsomocos 2009) Include an appropriate measure of housing prices in the targeted CPI Pursue the (Central Banks ) Financial Stability objective through regulatory policies for systemic financial agents

4 What should (Have) Be(en) done? Regulatory Policies Capital requirements reduce leverage in the banking sector, and induce banks to internalize (default) losses without taking a toll on the taxpayer. Margin requirements prevent excess leverage in the housing and derivatives markets, thus reducing/containing the adverse effects of the housing crisis via lower housing deflation rates Liquidity requirements reduce banks exposure to risky assets, thereby promoting lending in times of financial distress and stemming house price deflation

5 Literature Survey Finite Horizon Models Tsomocos 2003 and Goodhart, Sunirand and Tsomocos, 2006 Formal micro-founded banking sector Capital adequacy ratios modeled Interplay between nominal interest rates, banking sector default and allocations Espinoza and Tsomocos, 2007 and Espinoza, Goodhart and Tsomocos, 2008 Cash-in-advance (generalized liquidity constraints) Tighter monetary policy adverse effects on trade and welfare Higher liquidity of goods, vis-a-vis money, improves trade and welfare, and lowers the risk premium Goodhart, Tsomocos and Vardoulakis, 2009 Modelling of debt-deflationary pressures and defaults on collateralized assets to analyze housing crises Contractionary monetary policy and higher risk appetite promote financial instability.

6 Literature Survey DSGEs Woodford, 2003: Representative agent model, with no banking sector, liquidity or default Bernanke, Gertler and Gilchrist, 1999 Representative agent model with asymmetric information Partial equilibrium model for the credit market is embedded into the standard new Keynesian framework However, banks do not play an active role, and equilibrium outcomes are constrained efficient Woodford and Curdia, 2009: Exogenous risk premia, and no liquidity Leao and Leao, 2007: Include default but ignore liquidity and agent heterogeneity de Walque et. al, 2008: Include default and heterogeneity in the banking sector but treat money as a veil Iacoviello and Neri, 2007: have agent heterogeneity and a (non-active) banking sector, but no default risk

7 Literature Survey Finite Horizon Monetary Models An equilibrium outcome whereby fiat money has positive value is difficult to justify (Hahn, 1965). Money can have positive value if: It has a store value (Grandmont and Younes, 1972; Hool, 1976; infinite horizon or OLG models) An external agent (e.g. Government) is ready to exchange goods for money (e.g., Lerner 1974) If it has a transaction role and there is a competitive banking sector (Dubey and Geanakoplos, 1992)

8 Literature Survey Infinite Horizon Monetary Models New Keynesian framework with Calvo pricing: No money price process assumed and price level indeterminacy Money in the utility function: questionable assumption and yields neutrality with separable utility Helicopter Money (Cooley and Hansen, 1989) fiscal policy, thus the liquidity puzzle New monetarism : search models (e.g. Lagos and Wright, 2005), where money can be neutral Goodhart-Osorio-Tsomocos: Cash in advance models (Dubey and Geanakoplos; 1992) with inside (OMOs) and outside (helicopter) money.

9 Our Model Monetary General Equilibrium Model with Commercial Banks, Collateral, Securitisation and Default (MEBCSD) Non-trivial quantity theory of money Term structure of interest rates depends on aggregate liquidity and default risk Fisher effect Financial fragility is an equilibrium outcome Constrained inefficient equilibrium allocations Assessment of various policies for crisis management and prevention

10 Our Model Extend the Goodhart, Sunirand and Tsomocos and Goodhart (2006), Tsomocos and Vardoulakis (2008) model to: Introduce non-bank financial institutions (NBFI) Separate the interbank from the repo market Monetary Policy can be properly analyzed Model two types of default Discontinuous default in mortgages (Geanakoplos, 2003) Continuous default in credit markets (Shubik and Wilson, 1977 and Dubey et al.,2005)

11 The Economy Endowment economy 2 periods (t T = {0, 1}) First period: a single state Second period: S possible states S = {0} S = {0, 1, 2} 2 goods: Consumption goods basket (1) Housing (2): a durable good, but infinitely divisible Agents Households: h H = {α, θ}, CRRA preferences Commercial Banks: j J = {γ, δ}, quadratic preferences Investment Banks: ψ, risk neutral Hedge Fund: φ, risk neutral The Central Bank/Government/FSA: strategic dummies

12 The Economy 10 Markets Real sector goods housing Short-term (Default-free) credit markets Repo Consumer Long-term credit markets deposit (Default-free) mortgage (Default) interbank (Default) wholesale money markets (Default) Derivatives markets Mortgage Backed Securities (MBSs) Credit Debt Obligations (CDOs)

13 Time Structure

14 Nominal Flows of the Economy

15 Money and Collateral Money Introduced by a cash-in-advance (liquidity) transaction technology Enters the system as outside or inside money Collateral Household α pledges purchased housing as collateral when he takes out the mortgage If α defaults on the mortgage, the bank seizes the collateral and offers it for sale in the next period (US s walk away option)

16 Default Two types of Default: Discontinuous mortgage default. Household α defaults on his mortgage if (collateral s worth) (mortgage debt) Continuous default in the interbank and wholesale money markets: agents choose a repayment rate satisfying the On the Verge Condition (for k = {δ, ψ, φ} and j = {γ, δ}): marginal utility of default = bankruptcy penalty

17 Securitisation Scarcity of collateral incentivizes agents to strech it by using it many times. The investment bank (ψ) buys the mortgage from bank γ at a price p α in the MBS s market The investment bank (ψ) structures a CDO by attaching a Credit Default Swap (CDS) to the MBS The hedge fund (φ) purchases the CDO at a price q α CDO s gross returns: [ ] R CDO (1 + r = γα ) / q α 1 The investment bank bears the mortgage and CDS risk

18 Definition of Equilibrium 1 All agents maximize given their budget sets 2 All markets clear. 3 Expectations are rational.

19 Credit spreads, and the term structure of interest rates Credit spreads proposition Since ex-ante interest rates are considered, in the presence of default, borrowing rates have to be at least as high as lending rates to preclude arbitrage opportunities.

20 Credit spreads, and the term structure of interest rates Credit spreads proposition Since ex-ante interest rates are considered, in the presence of default, borrowing rates have to be at least as high as lending rates to preclude arbitrage opportunities. The term structure of interest rates proposition The term structure of interest rates is affected by aggregate liquidity and default, because interest rates price-in anticipated default rates (default premium). Aggregate ex-post interest rate payments to commercial banks adjusted by default equal the economy s total amount of outside money

21 Money Non-Neutrality, the Quantity Theory of Money and the Fisher Effect Price Wedge Lemma Since agents must borrow money to purchase goods and interest rates are positive, there is a wedge between selling and purchasing prices.

22 Money Non-Neutrality, the Quantity Theory of Money and the Fisher Effect Price Wedge Lemma Since agents must borrow money to purchase goods and interest rates are positive, there is a wedge between selling and purchasing prices. Monetary Policy Non-Neutrality Proposition If nominal interest rates are positive, then monetary policy is non-neutral.

23 Money Non-Neutrality, the Quantity Theory of Money and the Fisher Effect Price Wedge Lemma Since agents must borrow money to purchase goods and interest rates are positive, there is a wedge between selling and purchasing prices. Monetary Policy Non-Neutrality Proposition If nominal interest rates are positive, then monetary policy is non-neutral. Quantity Theory of Money Proposition If nominal interest rates are positive, the real velocity of money is endogenous and nominal changes affect both prices and quantities.

24 Money Non-Neutrality, the Quantity Theory of Money and the Fisher Effect Price Wedge Lemma Since agents must borrow money to purchase goods and interest rates are positive, there is a wedge between selling and purchasing prices. Monetary Policy Non-Neutrality Proposition If nominal interest rates are positive, then monetary policy is non-neutral. Quantity Theory of Money Proposition If nominal interest rates are positive, the real velocity of money is endogenous and nominal changes affect both prices and quantities. The Fisher Effect Proposition Nominal long term interest rates are approximately equal to real interest rates plus expected inflation and a risk premium.

25 Discussion of the Equilibrium: Assumptions Parameters determine the structure of the economy, the degree of agent heterogeneity = the relationships and interconnectedness between the different agents, markets and sectors of the economy. Agents believe the probability of a housing and financial crisis is relatively small α is poorer than θ in monetary endowments at t = 0 Bank γ is more capitalized than bank δ and NBFIs Commercial banks are more risk averse than the NBFIs Economy experiences an adverse productivity shock: moderate at s = 1 and severe at s = 2

26 Discussion of the Equilibrium: Assumptions Exogenous Variables also describe the action set of authorities: the Central Bank, the Government and the Financial Supervisory Agency. Central Bank responds by contracting its monetary policy stance (more in at s = 2 than in s = 2) FSA sets default penalties so as to protect the average consumer (bank γ faces the highest default penalty, and the NBFI is more severely penalized than bank δ) At t = 1 the Government injects capital to commercial banks as hazardous times prevent them from raising capital in the equity market

27 Exogenous Variables Risk Aversion Goods Housing Monetary Default Others Coefficients Endowments Endowments Endowments Penalties c α 1.30 e 01 α 30 e 02 θ 20 e m,0 α c θ 1.30 e 11 α 20 e m,1 α c γ 0.03 e 21 α 4 e m,2 α c δ 0.03 e m,0 θ e m,1 θ e m,2 θ e γ 0 e γ 1 e γ 2 e 0 δ e 1 δ e 2 δ e ψ 0 e ψ 1 e ψ 2 10 τ 1 δ 1 τ 2 δ 1 τ ψ 1 60 τ ψ 2 1 τ φ 1 1 τ φ M 0 CB M 1 CB M 2 CB ω ω

28 Discussion of the Equilibrium: Results Endogenous Variables Housing deflation and goods inflation Negative productivity (supply) shock increases goods prices House prices fall due to α s lower demand for housing Relative house prices fall smoothly in s = 1 and plummet in s = 2, thus At s = 1 no mortgage default, hence interbank default rate is low At s = 2, α defaults on his mortgage

29 Discussion of the Equilibrium: Results (continued) In the bad state (s = 2) Mortgage s effective return drops Significant losses in non-banking financial sector CDS contract executed: bank δ delivers collateral to NBFI ψ in exchange for its initial investment NBFI ψ assumes write-off loss Collateral sales exacerbate the housing crisis Economy becomes financially unstable at s = 2: Default increases in the mortgage and interbank markets Banks profits fall

30 Equilibrium Households Financial Sector Trade and Spending Prices Lending Lending Repayment Borrowing Borrowing Rates Goods Housing Derivatives p µ α d Gγ 0 p µ α d Gγ 1 p µ α d Gγ 2 p µ α m γ v α v α v δ v δ 2 100% q 01 α 85.3% q 11 α 98.5% q 21 α 58.6% b 01 θ q 02 θ 9.29 q 12 θ 1.60 q 22 θ b 02 α 4.47 m α ˆm α p µ θ m γ v ψ 1 100% b θ b α p µ θ m γ v ψ % b θ b α r γ µ θ m α 9.81 v φ 1 100% r γ 0.28 d µ γ d r γ 0.53 dγ v φ 2 r 0 δ 0.44 µ Gδ r 1 δ 0.28 µ Gδ r 2 δ 0.53 µ Gδ r γ d 0.44 m 0 δ r γα 2.47 m 1 δ ρ CB m 2 δ ρ CB m ρ CB µ δ ρ 0.55 µ ψ r 0.56 µ φ p α 1.44 q α %

31 Comparative Statics: Optimal Monetary Policy Instrument Increase Increase Money Decrease Money Decrease Supply Repo Rate Supply Repo Rate s = 2 s = 2 s = 2 s = 2 p µ ψ - - p v ψ r γα - - µ φ - - r - v φ ρ - U α r γ d U θ - + d γ - + π γ v 2 δ + + π 2 δ - - Monetary Base Instrument Households welfare (θ credit constrained) Default in mortgage, v δ 2, v ψ 2 and v φ 2 Banks profits Localized liquidity trap FF improves partially Interest Rate Instrument Households welfare (no credit constraints) Mortgage default, and v δ 2, v ψ 2 and v φ 2 Banks profits (insufficient lending) Undistorted transmission mechanism of M.P. FF improves partially Interest rate instrument is preferable to the monetary base instrument in times of financial distress

32 Inflation vs. Asset Price Targeting Decrease Decrease Decrease Decrease Repo Rate Repo Rate Repo Rate Repo Rate t = 0 s = 2 t = 0 s = 2 p µ ψ + - p v ψ r γα - - µ φ + - r - - v φ ρ - - U α + r γ d - U θ + + d γ + + π γ v 2 δ + π 2 δ - Expansionary monetary policy at t = 0 Improves households welfare α and ψ default more and π γ 2 as ( ρ r γ ) d Mortgage crisis exacerbated Leverage procyclicality Financial Fragility (FF) Expansionary monetary policy at s = 2 Households welfare (no credit constraints) Mortgage default, and v δ 2, v ψ 2 and v φ 2 Banks profits (insufficient lending) Undistorted transmission mechanism of M.P. FF improves partially Targeting goods inflation alone makes the economy Financially Unstable. The targeted price index should include a measure of house prices

33 Bankruptcy Code and Prudential Regulation (proxy) Tighter ψ s Increase γ s Tighter ψ s Increase γ s Default Risk Default Risk Penalty Aversion Penalty Aversion s = 2 Coefficient s = 2 Coefficient p µ ψ - - p v ψ r γα - µ φ - - r - + v φ + 2 ρ + U α + r γ d - - U θ d γ - - π γ v 2 δ + π 2 δ - Default Penalties for ψ Weak improvement of households welfare Default in mortgage, v δ 2, v ψ 2 and v φ 2 Banks profits Countercyclical leverage FF Pareto improvement γ becomes more prudent Households welfare (credit conditions ease) Mortgage default, and v δ 2, v ψ 2 and v φ 2 Banks profits as ( ρ r γ ) d Countercyclical leverage and FF Financial Stability objective should be primarily achieved by regulating systemic financial agents

34 Macroprudential Regulation: simplified model

35 Capital Requirements Households Banking Sector Repayment Welfare Profits Rates U α - π γ - v α 2 - U θ + π δ - v δ 2 + U α 2 + π γ 2 - v ψ 2 - U θ 2 - π δ 2 - v 2 - Banks finance their portfolios with less debt Lower deposit rate induces savers to anticipate consumption p 02 p 02 higher mortgage default rates default rates in the NBFI, but not the BFI sector Consumers welfare improves: households smooth consumption efficiently (household α via default) Commercial banks internalize (default) losses without taking a toll on the taxpayer

36 Margin Requirements Households Banking Sector Repayment Housing Welfare Profits Rates Prices U α + π γ + v α 2 + p 02 + U θ - π δ + v δ 2 + p 22 + U α 2 + π γ 2 + v ψ 2 + (p 22/p 02) + U θ 2 - π δ 2 - r γα 2 + r γα - Higher LTV and haircut constraints induce home buyers and NBFIs to leverage less Lower demand for mortgages r γα reduces derivatives demand commercial banks debt-to-equity ratios Lower house deflation an mortgage rates collateral value mortgage default Thus, interbank repayment rates rise lending to households at s = 2 house deflation Margin requirements improve home buyers welfare as well as financial stability via lower BFI, NBFI and household leverage

37 Liquidity Ratios Households Banking Sector Repayment Housing Lending Welfare Profits Rates Prices at s = 2 U α + π γ + v α 2 + (p 22/p 02) + m γ 2 + U θ - π δ + v δ 2 + r γα 2 + m δ 2 + U α 2 + π γ 2 + v ψ 2 + r γα + U θ 2 - π δ 2 - Commercial banks make fewer (more) credit extensions in the interbank and mortgage (short term consumer) markets NBFI reduces its leverage and bank δ makes smaller CDO investments As commercial banks reduce their exposure to derivatives beginning of t = 1 revenue lending to households at s = 2 Higher demand by home buyers at s = 2 stems collapse of housing prices and value of collateral Hence, default rates in the mortgage and interbank markets decrease Liquidity requirements improve home buyers welfare as well as financial stability by reducing credit exposure of the BFI sector

38 The Economy Infinite horizon (t T = {0, 1, 2,..., t 1, t, t + 1,...}) 1 consumption good CRS Production Function Two factors of production: labour and capital No technological progress Agents Household α; consumes, rents out capital, works and saves Firm: hires labour, rents capital, and produces and sells goods final goods Commercial Banks: j J = {δ, ψ}, risk averse The Central Bank/Government/FSA: strategic dummies 6 Markets: goods, labour, capital, consumer deposit, interbank, and corporate credit markets.

39 Nominal Flows of the Economy

40 Financial Frictions Default Agents are allowed to default partially: they choose the fraction of outstanding debt they repay Default choice trade-offs the benefit of defaulting (more consumption) and its pecuniary cost (bankruptcy costs). Money Introduced by a cash-in-advance (liquidity) transaction technology Enters the system as outside or inside money

41 Outside Money: Money-financed Fiscal Transfers Aggregate outside money (G t ) enters the system free and clear of any offsetting obligation is accumulated (stock) Thus, following Schorfheide (2000) and Nason and Cogley (1994), we stationarize the model with respect to G t 1 An outside money shock is a perturbation to its growth rate g G,t = Gt G t 1 ( ) ( ) ln (g G,t ) = ρ G ln ( g G ) + 1 ρ G ln (g G,t 1 ) + e G,t ; e G,t N 0, σ 2 G We assume that outside money is always distributed to agents on the same pro-rata basis G t = m θ t + m γ t + m ψ t + m δ t m i t = ω i G t, where ω i = 1, ω i > 0 i I = {θ, γ, ψ, δ} i

42 Inside Money: Open Market Operations Inside money represents the interventions of the Central Bank in the interbank market every period These liquidity injections exit the system when borrowing commercial banks repay their obligations An inside money shock is a perturbation to the inside to outside money (interest to non-interest bearing money) ratio; i.e. Open Market Operations. ( ln η CB t ˆM t = η CB t ) ( ) ( ) ( ) ( ) = ρ CB ln η CB + 1 ρ CB ln η CB t 1 + e CB,t ; e CB,t N 0, σ 2 CB ˆM

43 Simulations Objectives Describe how the endogenous variables of the model respond to several shocks Assess the impact of introducing agent heterogeneity, liquidity and default into a DSGE model Therefore, we analyze the effects of monetary and fiscal policy shocks in three different models: New Keynesian Our model in the absence of default frictions (MGE) The complete version of our model Regulatory policy has non-trivial effects only in the presence of default frictions

44 Expansionary Monetary Policy in NKM and MGE In the NKM model Monetary Policy is neutral in the long run In the short run monetary policy has real effects due to sluggishly adjusting prices (Calvo pricing) In the MGE model interest rates (equally) borrowing, spending, trade, and prices Banking sector profits due to interest rates Variables converge gradually to steady state as shock disappears

45 Impulse Response Functions

46 Expansionary Monetary Policy in the presence of Default interest rates borrowing, spending, trade, and prices v d, v IB due to lower credit costs v F due to higher sales revenues Banking sector profits due to wider credit spreads and lending Due to E t { v d t+s } in the first few periods after the shock r d r IB, r F : Built-in stabilizer medium run destabilizer Variables oscillate around the steady state as shock disappears Economy becomes financially unstable in the medium run

47 Impulse Response Functions

48 Expansionary Fiscal Policy in NKM and MGE In the NKM model aggregate demand positive output gap prices start rising prices real money balances output gap: Built-in stabilizer Economy experiences a short recession in the medium run (prices remain high after shock disappears) Aggregate demand = consumption demand no crowding-out effect unlike the IS-LM model In the MGE model agents capacity to make interest payments demand for credit and bids interest rates up (equally) Firm sells fewer goods labour, trade and wages, but prices These effects remain for a few periods as credit conditions remain tight (despite prices starting to fall) Household θ is worst-off Banking sector profits due to interest rates Variables converge gradually to steady state as shock disappears

49 Impulse Response Functions

50 Expansionary Fiscal Policy in the presence of Default agents capacity to make interest payments demand for credit and bids interest rates up Higher monetary endowments allow agents to afford more credit costs (SR) Expected Default interest rates more than in default-free case profits in banking sector due to significant interest rates Sharp interest rates default a few periods after the shock: Built-in stabilizer medium run destabilizer Variables oscillate to steady state as shock disappears Economy becomes financially unstable in the short to medium run

51 Impulse Response Functions

52 Regulatory Policy Proportional Increase of all Default Penalties credit costs repayment rates in all markets Thus, interest rates credit demand, spending, trade, prices and wages Proportionate increase of default penalties more pronounced effects in more regulated markets credit spreads narrow and banking sector profits Changes in interest rates affect credit costs such that, repayment rates oscillate as shock disappears This policy promotes financial stability in the medium run

53 Impulse Response Functions

54 Concluding Remarks We propose finite and infinite horizon models to analyze Financial Stability, which (should) include: Heterogeneous agents Endogenous Default An essential role for money Heterogeneous and active banks In our finite horizon model, we introduced frictions such as collateral and securitisation, to examine and obtain an equilibrium outcome consistent with the global /10 crisis Our infinite horizon model reveals the importance of allowing for endogenous default for the economy s short to medium run dynamics and its impact on financial stability

55 Concluding Remarks Our comparative statics and simulations suggest that: In times of crisis, monetary policy conducted by means of the interest rate instrument is a more effective than using the monetary base instrument. CPI should include an appropriate measure of housing prices Optimal regulatory policies should target systemic financial agents and induce them to behave more prudently Capital requirements on their own do not promote financial stability Macro-prudential regulation should be concerned with margin and liquidity ratios as well Margin requirements reduce banks leverage, whereas liquidity requirements lower their risk exposure

56 Concluding Remarks Extensions Microfound the Government s budget constraint Endogenize the Central Bank/Government/Regulator s best response policy actions THANK YOU

57 Household α s Optimisation Problem max q α s 1,bα s 2,µα s, µα Uα = u ( e α 01 qα 01 + s S α 1 ( ) b α + u 02 p 02 ( b α ) 02 ω s u + bα s2 + p 02 p s2 ) + ω s u ( e α s1 ) qα s1 s S ( b α ) s2 ω s u s / S 1 α p s2 s.t. b α 02 µ α (1 + r γα ) + µ α 0 ( 1 + r γ) + e α m,0 0 i.e. housing expenditure at t=0 mortgage loan + short-term borrowing + private monetary endowments at t=0 µ α 0 p01qα 01 i.e. short term loan repayment at t=0 goods sales revenues at t=0

58 Household α s Optimisation Problem b α s2 + µα µα s (1 + r γ s ) + eα m,s for s S α 1 i.e. housing expenditure at s S α 1 + mortgage repayment short-term borrowing+private monetary endowments at s S α 1 b α s2 µα s (1 + r γ s ) + eα m,s for s / S α 1 i.e. housing expenditure at s / S α 1 endowments at s / S α 1 short-term borrowing+private monetary µ α s p s1q α s1 i.e. short term loan repayment goods sales revenues at t=0 q α s 1 eα s 1 i.e. quantity of goods sold at s S goods endowments at s S

59 Household θ s Optimisation Problem max q s θ 2,bθ s 1,µθ s, d θ Uθ = u ( ) b θ ( ) 01 + u e θ 02 p qθ 02 + ω s u 01 s S + s S ω s u ( ) e θ 02 qθ s0 qθ s2 ( ) b θ 02 p 02 s.t. b θ 01 + d θ µθ r0 δ + e θ m,0 i.e. goods expenditure at t=0 + inter-period deposits short-term borrowing + private monetary endowments at t=0 µ θ 0 p02qθ 02 (i.e. short term loan repayment at t=0 housing sales revenues at t=0)

60 Household θ s Optimisation Problem b θ s1 µθ s + d θ ( 1 + r γ ) θ 1 + rs δ d + e m,s for s S i.e. goods expenditure at s S short-term borrowing + deposits and interest payment+private monetary endowments at s S µ θ s p s2q θ s2 i.e. short term loan repayment at s S housing sales revenues at s S q θ s 2 eθ s2 qθ 02 i.e. number of housing units sold at s S endowment of housing at t=0 - units of housing sold at s S

61 Bank γ s Optimisation Problem s.t. max Π γ = ω s (π γ m γ s 2, mα,d Gγ s, d γ,πs γ s c γ ( π γ ) ) 2 s s S d Gγ 0 + m γ 0 + mα + d γ e γ 0 + ( µγ d /1 + r γ d ) i.e. deposits in the repo market + short-term lending +mortgage extension + interbank lending capital endowment at t=0 + consumer deposits

62 Bank γ s Optimisation Problem d Gγ s + m γ s + µ γ d + eγ s + π γ 0 + R δ s d γ (1 + ρ) i.e. short-term lending + deposits in the repo market at s S + deposits repayment capital endowment at s S + accumulated profits + interbank loan repayments at s S π γ 0 = mγ 0 ( 1 + r γ 0 ) ( Gγ + d ρ CB 0 ) + p α m α i.e. profits at (t=0) = short term loan repayment + repo deposits and interest payment at t=0 + MBS s sales revenues π γ s = m γ s ( 1 + r γ s ) ( Gγ + d s 1 + ρ CB s ) i.e. profits at s S = short term loan repayment + repo deposits and interest payment at s S

63 Bank δ s Optimisation Problem s.t. max Π δ = ω s (π ( ) ) δ m s 2 δ, m,µgδ s, µδ,µ δ s, vδ s,πγ s cδ π δ 2 s [ ] ω s τ δ s D δ + s s s S s S m δ 0 + m eδ 0 + µgδ ρ CB + µδ ρ i.e. short-term lending at t=0 + wholesale money market credit extension capital endowment + short-term borrowing in the repo market at t=0 + interbank borrowing

64 Bank δ s Optimisation Problem ( ) µ Gδ 0 m δ r δ 0 i.e. repo loan repayment at t=0 short-term loan repayment at t=0 m δ s + v δ s µδ e δ s + µgδ s + R 1 + ρ CB s m (1 + r) s i.e. short-term lending + interbank loan repayment at s S capital endowment + wholesale money market loan repayment short-term loan repayment at s S π δ s = m δ s ( ) 1 + r δ s µ Gδ s i.e. profits at s S = short term loan repayment - repo loan repayment at s S

65 Investment Bank (ψ) s Optimisation Problem s.t. max Π ψ = ω s π ψ m α, µ ψ, v s ψ s ω s τ ψ s, s S s S [ ] + Dψ s m α e ψ 0 + µψ 1 + r i.e. expenditure in MBS s capital endowments at t=0 + wholesale money market borrowing v ψ s µ ψ mα p α qα for s S α 1 i.e. whole sale money market loan repayment at s S α 1 revenues + capital endowments at s S α 1 CDO s sales ( b v ψ s µ ψ e ψ α ) 02 s + p22 m α for s / S α m α p 02 p α 1 i.e. CDS settlement payment + wholesale money market loan repayment at s / S α 1 endowment at s / S α 1 + CDO s sales revenues + collateral sales revenues capital

66 Hedge Fund (φ) s Optimisation Problem s.t. max Π φ = ω s π φ µ φ, ˆm α, v φ s ω s τ φ s s s S s S [ ] + Dφ s ˆm α µφ 1 + r i.e. expenditure in the CDO s market wholesale money market borrowing v φ s µ ψ ˆmα q α ( 1 + r γα ) for s S α 1 i.e. wholesale money market loan repayment CDO s payoffs at s S α 1 v φ s µ ψ ˆm α for s / S α 1 i.e. wholesale money market loan repayment CDO s payoffs at s / S α 1

67 Market Clearing Conditions Goods Market p 01 = bθ 01 q α 01 p s1 = bθ s1 q α s1 for s S Housing Market p 02 = bα 02 q θ 02 p s2 = bα s2 q θ s2 b α s2 p s2 = qs2 θ + bα 02 /p 02 for s S α 1 for s / S α 1

68 Market Clearing Conditions Mortgage Market (1 + r γα ) = µα m α Clearing conditions for effective returns on mortgages (1 + r γα ) for s S1 α (1 + r s γα ( ) = p22 b02 α ) ( ) µ α r γα for s / S1 α p 02 Short-term Consumer Markets (1 + r γ s ) = µα s m γ s ( 1 + r δ s ) = µ θ s m δ s

69 Market Clearing Conditions Consumer Deposit Market (1 + r γ d ) = µγ d d θ Wholesale Money Market Repo Market (1 + r) = µψ + µ φ m ( ) 1 + ρ CB µ Gδ s = s M CB s + d Gγ s Interbank Market MBS s Market CDO s Market (1 + ρ) = µδ d γ p α = mα m α q α = ˆmα m α

70 Conditions on Expected Delivery Rates (Rational Expectations) Wholesale Money Market v s ψ µ ψ + v s φ µ φ µ R ψ + µ φ if µ ψ + µ φ > 0 s = arbitrary if µ ψ + µ φ = 0 s S Interbank Market v s δ µ δ µ R δ = v s δ if µ δ > 0 s δ = arbitrary if µ δ = 0 s S

71 Extensions: Macro-Prudential Regulation Are Capital Adequacy Requirements welfare improving? ks γ es γ + π γ 0 = ( ) ξ 1 ms γ (1 + rs γ ) + ds Gγ (1 + ρ CB s ) + ξ 2 Rδ s d γ (1 + ρ), ξ 1 ξ 2 e δ s ks δ = ξ 1 ms δ (1 + rs δ ) + ξ 3 Rs m (1 + r), ξ 1 ξ 3

72 Extensions: Macro-Prudential Regulation Are Margin Requirements welfare improving? µ α 1 + r γα χltv b α 02, 0 χ LTV 1 (i.e. mortgage lending Loan-to-value housing s worth at t=0) µ ψ (1 + r) ( 1 χ MG,ψ) b α 02 p 22 p 02, 0 χ MG,ψ 1 (i.e. borrowing by ψ (1- margin ratio) housing collateral at s / S α 1 ) µ φ (1 + r) ( 1 χ MG,φ) ˆm α (i.e. borrowing by φ (1- margin ratio) CDO s repayment at s / S α 1 )

73 Household θ s Optimisation Problem { max d t θ } {, L θ } { t, b t θ } E 0 β t t=0 ln ( ) b θ ( ) t + ln N θ L θ t p t s.t. b θ t + d θ t m θ t + w θ t Lθ t + R d t d θ t 1 ( ) 1 + r d t 1 + π ψ t + π δ t, t T (1) i.e. goods expenditures + deposits at time t private monetary endowments + labour income at time t + deposits repayment + banks dividend payments

74 Household θ s FOCs [ b θ t ] : g G,t U ( c θ t {( ) ) 1 + r d = t R d t+1 βet U ( c θ ) } t+1 (2) ˆp t+1 /ˆp t [ L θ t ] : The budget constraint binds because: ( ) ŵ t = U N θ L θ t ˆp t U ( ) ct θ (3) Money is fiat and agents do not derive utility from holding it agents do not hold idle cash, they lend it out

75 Yeoman Farmer γ s Optimisation Problem s.t. { max b γ },{q γ L,t t },{µ γ { t }, v t F } E 0 β t t=0 Y γ t ln ( Y γ t ( b γ ) α L,t = A t t T w t q γ ) t v F t µγ t 1 mγ t + p t 1q γ t 1 t T with v F 0 = µγ 1 = 0 (4) i.e. corporate loan repayment at time t private monetary endowments at time t + sales revenues carried forward from period (t-1) b γ L,t µ γ t ( 1 + r F t ) τ γ t ( ) 1 v F t µ γ t 1 t T with v F 0 = µγ 1 = 0 (5) i.e. payroll payment at time t corporate borrowing at time t - credit costs on outstanding corporate debt ln (A t) = ρ A ln ( Ā ) ( + 1 ρ A) ( ) ln (A t 1) + e A,t ; e A,t N 0, σ 2 A ln ( τ γ t ) = ρ γ ln ( τ γ ) + ( 1 ρ γ) ( ) ( ) ln τ γ t 1 + e γ,t; e γ,t N 0, σ 2 γ

76 Yeoman Farmer γ s FOCs [ ] b γ L,t : {( ) 1 + r g G,t U (ct γ F ) = βe t τ γ t t U ( c γ ) } t+1 (6) ŵ t+1 /ŵ t [q γ t ] : Budget constraints bind ( ) ( ) ŵ t 1 + r F t = U L θ t ˆp t U (qt γ ) (7)

77 Commercial Bank ψ s Optimisation Problem s.t. π ψ t { µ d t } {, d IB t max } {, v t d } {, π t ψ } E0 ˆβ t t=0 ( ) ln π ψ t ( ) = R IB t 1 + r IB t 1 d IB t 1 v d t µd t 1 t T with v d 0 = µd 1 = d IB 1 = 0 (8) d IB t m ψ µ d t t + ( ) τ ψ 1 + r d t t ( ) 1 v d t µ d t 1 t T with v d 0 = µd 1 = d IB 1 = 0 (9) i.e. interbank credit extensions at time t monetary endowments + household deposits - credit costs on outstanding debt with households at time t ( ) ( ) ( ln τ ψ t = ρ ψ ln τ ψ + 1 ρ ψ) ( ) ( ) ln τ ψ t 1 + e ψ,t ; e ψ,t N 0, σ 2 ψ

78 Commercial Bank ψ s FOCs [ µ d t ] : (ˆπ ) g G,t Π t ψ = β ( 1 + rt d ) { (ˆπ )} Et τ ψ t+1 Π ψ t+1 (10) [ d IB t ] : g G,t Π (ˆπ ψ t ) = β ( 1 + rt IB ) Et {R (ˆπ )} t+1τ IB ψ t+1 Π ψ t+1 (11) From equations (10) and (11) ( 1 + r d t ) ( ) { } = 1 + r IB t Et R IB t+1 r IB t rt d

79 Commercial Bank δ s Optimisation Problem { d F t } {, µ IB t max } {, v IB t } {, π t δ } E 0 ˆβ t t=0 ( ) ln π δ t s.t. π δ t = R F t ( ) 1 + r F t 1 d F t 1 v IB t µ IB t 1 t T with v IB 0 = µ IB 1 = d F 1 = 0 (12) d F t m δ µ IB ( t t + ( ) τ δ 1 + r IB t t 1 v IB t ) µ IB t 1 t T with v IB 0 = µ IB 1 = d F 1 = 0 (13) (i.e. corporate credit extensions at time t monetary endowments + interbank borrowing - credit costs on outstanding interbank debt at time t) ( ) ln τ δ t ( ) ( = ρ δ ln τ δ + 1 ρ δ) ( ) ( ) ln τ δ t 1 + e δ,t ; e δ,t N 0, σ 2 δ

80 Commercial Bank δ s FOCs [ µ IB t ] : g G,t Π (ˆπ t δ ) ( ) { = β 1 + r IB t Et τ δ t+1 Π (ˆπ t+1 δ )} (14) [ d F t ] : g G,t Π ( πt δ ) ( ) { = β 1 + r F t Et R F t+1 τt+1π δ ( πt+1 δ )} (15) From equations (14) and (15) ( 1 + rt IB ) ( ) { } = 1 + r F t Et R F t+1 r F t rt IB