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**A Macroeconomic Model with a Financial Sector**

Markus K. Brunnermeier and Yuliy Sannikov

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Motivation Financial crises happen every now and then, spill over into real economy Once about every 10 years in the 400 years in Western Europe - Kindleberger (1993) Great Depression and the failure of the financial system Fisher (1933), Keynes (1936) Current financial turmoil Current macro approach Most DSGE models use representative agents, ignore financing frictions Models with a financial sector (e.g. Bernanke-Gertler-Gilchrist) log-linearize near steady state, miss instability below steady state

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**Motivation Current financial turmoil Current macro approach**

Spirals and adverse feedback loops Spillovers Across financial institutions To real economy Current macro approach Representative agent analysis ignores wealth distribution (including leveraged lending) spillovers (externalities) and hence, financial stability Steady-state log-linearization ignore many dynamic effects (like precautionary hoarding due to higher volatility).

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**Modeling financial frictions**

Central idea: heterogeneous agents have special skills/more productive Bernanke-Gertler-Gilchrist, He-Krishnamurthy, Kiyotaki-Moore less risk-averse Geanakoplos, Adrian-Shin, Garleanu-Pedersen more optimistic Geanakoplos type 1: financially constrained type 2: unlevered

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**Modeling financial frictions**

Central idea: heterogeneous agents have special skills/more productive Bernanke-Gertler-Gilchrist, He-Krishnamurthy, Kiyotaki-Moore less risk-averse Geanakoplos, Adrian-Shin, Garleanu-Pedersen more optimistic Geanakoplos type 1: financially constrained type 2: unlevered Diamond (1984) type 3: intermediary financing capital inside equity Debt financing capital Equity inside equity

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**Modeling financial frictions**

Central idea: heterogeneous agents have special skills/more productive Bernanke-Gertler-Gilchrist, He-Krishnamurthy, Kiyotaki-Moore less risk-averse Geanakoplos, Adrian-Shin, Garleanu-Pedersen more optimistic Geanakoplos type 1: financially constrained type 2: unlevered Amplification (e.g. Brunnermeier-Pedersen) Shocks affect the distribution of assets between agents of types 1 and 2, total output Loss of capital shock to net worth Precaution + tighter margins volatility price Fire sales

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**Modeling financial frictions**

Central idea: heterogeneous agents have special skills/more productive Bernanke-Gertler-Gilchrist, He-Krishnamurthy, Kiyotaki-Moore less risk-averse Geanakoplos, Adrian-Shin, Garleanu-Pedersen more optimistic Geanakoplos type 1: financially constrained type 2: unlevered Amplification (e.g. Brunnermeier-Pedersen) Shocks affect the distribution of assets between agents of types 1 and 2, total output But… Existing models study only steady-state dynamics or local dynamics two-three period models Loss of capital shock to net worth Precaution + tighter margins volatility price Fire sales

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Preview of results Unified model, replicates dynamics near steady state (e.g. Bernanke-Gertler-Gilchrist) … but unstable dynamics away from steady state due to (nonlinear) liquidity spirals 2. Welfare: Fire-sale externalities within financial sector, externalities b/w financial sector and real economy… When levering up, institutions ignore that their fire-sales depress prices for others Inefficient pecuniary externality in incomplete market setting 3. Securitization can lead to excessive leverage volatility prices expert net worth

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**Overview Start with simple model Show non-linearities in equilibrium**

Add households, capital producers and direct investment to show externalities Add idiosyncratic shocks to show effects of securitization

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**Model: production of output and capital**

Experts hold capital, which produces output yt = a kt Investment of ι(g) kt makes capital grow at rate g δ: expert depreciation ι(-δ) = 0 Liquidation value: ι’(-∞) = a/(r + δ*), δ* > δ Household discount rate Household depreciation

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**Model: production of output and capital**

Experts hold capital, which produces output yt = a kt Investment of ι(g) kt makes capital grow at rate g δ: expert depreciation ι(-δ) = 0 Liquidation value: ι’(-∞) = a/(r + δ*), δ* > δ Holding capital is risky dkt = g kt dt + kt dZt ρ ≥ r: expert discount rate (may “pay out” too much) Household discount rate Household depreciation Brownian macro shock

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**Financing through experts**

Agency problem: experts can increase depreciation/cause losses Get benefit b per unit of capital Assume: effort, kti not contractible, but market value of assets ptkti is Incentive constraint b - t pt ≤ 0 t b/pt Solvency constraint: debt holders liquidate when ktpt drops to dt Expert capital depends on aggregate risk, amplified through prices Later: contracting on aggregate risk separately Assets Liabilities Experts borrow and may unload some risk dt = ktipt – et ktipt Inside nt = αtet Outside (1-αt)et et

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**Balance sheets Assets Liabilities**

For simplicity, take α = 1 for most of this talk Assets Liabilities ptkt dpt= tp dt+tp dZt dkt = gkt dt+kt dZt Debt dt Equity = net worth nt = ptkt - dt d(ktpt) = kt (g pt + tp + tp) dt + kt (pt + tp) dZt ddt = (r dt - a kt + ι(g) kt) dt - dct

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**Evolution of balance sheets**

Asset side – long maturity d(ktpt) = kt (g pt + tp + tp)dt + kt (pt + tp)dZt Liability side Debt – overnight maturity ddt = (r dt - a kt + ι(g) kt) dt - dct 2. Equity dnt = d(ktpt) - ddt = rnt + kt [(a-ι(g)-(r-g)pt+tp+tp)dt+(pt+tp)dZt] -dct

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**Equilibrium f(t)nt = maxk,g,c E[dc+d(f(t)nt)] = maxk,g,c**

Aggregate: Nt, Kt State variable t = Nt/Kt Price p(t) Expert value function f(t)nt Bellman equation f(t)nt = maxk,g,c E[dc+d(f(t)nt)] = maxk,g,c {dct+μtfnt + f(t) (rnt + kt(a-ι(g)-(r-g)pt+tp+tp)) + σtfkt(pt+tp)} FOC: ι’(g) = pt, a-ι(g)-(r-g)pt+tp+tp = - σtf/f(t) (pt+tp) dct = 0 unless f(t)=1 Bellman equation: ( - r) f(t) =μtf Ito’s lemma: tp = tη p’ + ½(tη)2p’’ differential equations Note that risk premium goes to zero as eta goes to eta* SINCE σtf goes to zero as \eta goes to \eta*. Why? σtf = f’ σteta and f’ goes to zero at \eta*. (risk premium) * pt

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**Stochastic Discount Factor**

Experts’ SDF: m0,t = e-ρt f(ηt)/f(η0) Households’ SDF: m0,tHH= e-rt Note that m0,t = m0,tHH, since δ*> δ time preference agency constraint /

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Equilibrium

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Equilibrium

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**Full vs. steady-state dynamics**

Log-linearized impulse response functions Stationary distribution around steady-state But.. below steady state system gets unstable, volatility Liquidity spiral prices volatility expert net worth time ηt

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**System dynamics and instabilities**

Loss of capital Zt-shock on kt Precaution + tighter margins tp pt Fire sales Volatility effects Macro shocks Higher volatility exacerbates precautionary hoarding motive, amplifying price drops Outside equity shrinks (+deleveraging) Dynamical system spends most time near steady state, but has occasional volatile destructive episodes Sensitivity of price to ηt amplification

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**Asset pricing (time-series)**

Predictability For low η values price is (temporarily) depressed Price-earnings ratio predicts future prices Current earnings: ak0 Current price: p0k0 =E0[e-ρt f(ηt)/f(η0) (ptkt+divt)] Excess volatility Volatility of ktpt per dollar is σ + σtp/pt Cash flow volatility amplified by SDF movements Stochastic volatility See ση plot

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**Asset pricing (cross section)**

Correlation increases with σp Extend model to many types j of capital dkj/kj = g dt + σ dz + σ' dzj Experts hold diversified portfolios Equilibrium looks as before, but Volatility of ptkt is σ + σp + σ’ For uncorrelated zj and zl correlation (ptjktj, ptlktl) is (σ + σp)/(σ + σp + σ’) which is increasing in σp xxx aggregate shock uncorrelated shock

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**Externalities so far there are no externalities…**

Proposition. The competitive equilibrium in this economy is equivalent to the optimal policy by a monopolist expert. Sketch of proof. (1) Write Bellman equation for monopolist. (2) Define price pt = ι’(g). (3) Show that prices etc. are as in competitive eq. Intuition: In competitive equilibrium experts do affect prices by their choices (compensation and investment), but they are isolated from prices because they don’t trade given equilibrium prices.

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**Optimal payout policy of monopolist**

Debt dDt = (rDt - a Kt + ι(g) Kt) dt – dCt, take ρ > r Solvency constraint: Dt ≤ LKt Value function h(ωt)Kt where ωt = -Dt/Kt Bellman equation…

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**Modification 1: add labor sector**

Fixed labor supply L Production function a’ Kt kt1- lt Workers get wages wt = a’ Kt L-1 Experts get ak = (1 - ) a’ L k Worker welfare depends on Kt Experts do not take that into account when choosing leverage (investment and payout decisions) Household value function…

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**Externalities with households**

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**Modification 2: speculative households**

So far fixed liquidation value at a/(r + δ*)… now households can sell back to experts Break even for HH a- rpt - δ*pt + tp+ tp ≤ 0, equality when experts hold fraction ψt < 1 of assets depreciation rate is δ* > δ pt ≥ a/(r + δ*) In equilibrium households pick up assets when financial sector suffers losses, i.e. ηt becomes small Fire sale externalities (within financial sector) – when levering up, experts hurt prices that other experts can sell to households in the event of a crisis Capital gains/losses, E[d(ktpt)] financing cost earnings

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**Equilibrium when households provide liquidity support**

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**Modification 3: Idiosyncratic losses**

dkti = g kti dt + kti dZt + kti dJti Jti is an idiosyncratic compensated Poisson loss process, recovery distribution F and intensity λ(σtp) Vt = ktpt drops below Dt, costly state verification by debt

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**Review: costly state verification**

Developed by Townsend (1979), used in Diamond (1984), Bernanke-Gertler-Gilchrist Time 0: principal provides funding I to agent Time 1: agent’s profit y ~ F[0, y*] is his private information but principal can verify y at cost Optimal contract (with deterministic verification) is debt with face value D: agent reports y truthfully and pays D if y ≥ D, triggers default and pays y if y < D In our context: expert can cause losses (reduce Vt for private benefit); debtholders verify if Vt falls below Dt

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**Modification 3: Idiosyncratic losses**

dkti = g kti dt + kti dZt + kti dJti Jti is an idiosyncratic compensated Poisson loss process, recovery distribution F and intensity λ(σtp) V = ktpt drops below Dt, costly state verification by debt Debtholders’ loss rate Verification cost rate Leverage bounded not only by precautionary motive, but also the cost of borrowing Asset Liabilities Dt = ktpt – Et Vt = ktpt Equity

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**Equilibrium Experts borrow at rate larger than r**

Rate depends on leverage, price volatility dt = diffusion process (without jumps) because losses cancel out in aggregate

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Securitization Experts can contract on shocks Z and L directly among each other, contracting costs are zero In principle, good thing (avoid verification costs) Equilibrium experts fully hedge idiosyncratic risks experts hold their share (do not hedge) aggregate risk Z, market price of risk depends on tf (pt + tp) with securitization, experts lever up more (as a function of t) and pay themselves sooner financial system becomes less stable

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**Contracting Assets Liabilities**

Agency problem: experts can lower growth rate/cause losses Get benefit b per unit of capital Assume: effort, kti not contractible, but market value of assets ptkti is Incentive constraint b - t pt ≤ 0 t b/pt Expert capital depends on aggregate risk, amplified through prices Contracting on aggregate risk separately: consider that in an extension, but only within financial sector Assets Liabilities dt = ktipt – et ktipt Inside nt = αtet Outside (1-αt)et et

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**Evolution of balance sheets**

Asset side – long maturity d(ktpt) = kt (g pt + tp + tp)dt + kt (pt + tp)dZt Liability side Debt – overnight maturity Outside equity: 1-t Inside equity dnt = nt + kt [(a - pt +g pt + tp+ tp)dt + t (pt+tp) dZt] All results (nonlinear dynamics, externalities, excessive leverage under securitization) hold in this expanded model Capital appreciation, E[d(ktpt)]

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**Conclusion Incorporate financial sector in macromodel**

Higher growth Exhibits instability – due to non-linear liquidity spirals (away from steady state) Leverage/payouts chosen without taking into account externalities Towards households (labor provision) Within financial sector: possible fire sales compromise others’ balance sheets Securitization avoids inefficiencies, but can increase instabilities (higher leverage/payout rate)

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**To do list Introducing risk aversion**

Asset pricing implication – time-varying risk premia Time-varying risk-free interest rate Incorporating instabilities into general DSGE model Optimal regulation Introducing money/assets with different liquidity Exploring maturity mismatch

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Thank you! ☺

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**Differences to Bernanke-Gertler-Gilchrist**

Brunnermeier-Sannikov Focus on (large) aggregate shocks (idiosyncratic shocks not essential), explore nonlinearities using Bellman equation Asset price drops also due to fire sales Expert’s rent depends on state t Incentive to keep “dry powder” (liquidity) Procyclical leverage: Experts reduce position after drop in net worth Liquidity spirals Securitization (debt, inside + outside equity) Fire-sale externality (rationale for regulation) BGG 1. “small” aggregate shocks, log- linearization around steady state 2. Price dynamics driven by idiosyncratic shocks and default risk Higher state verification costs when expert capital goes down 3. With small aggregate noise, expert incentives to keep “dry powder” (liquidity) are negligible 4. Countercyclical leverage Experts take on same position after drop in net worth Leverage increases after drop in net- worth 5. Debt vs. Equity 6. No fire-sale externality

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**Differences to Kiyotaki-Moore**

KM – (Kiyotaki version) Zero-prob. temporary shock Persistent (dynamic loss spiral) Amplified through collateral value Non- vs. productive (leveraged) sector Dual role of durable asset Production Collateral Exogenous contract One period contract Debt is limited by collateral value Durable asset doesn’t depreciates (capital, fully) BruSan Permanent TFP shocks Margin/haircut spiral (leverage) Loss spiral Investment through leveraged financial sector Dual role of durable asset Production Securitization Optimal contract Dynamic contract Debt is limited due idiosyncratic risk and costly state verification δ-depreciation rate 40

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**Differences to He-Krishnamurthy**

Endowment economy GDP growth is exogenously fixed No physical investment No direct investment in risky asset by households Limited participation model Contracting Only short-run relationship (t to t+dt) Fraction of return, fee Asset composition (risky vs. risk-free) is not contractable Non-effort lowers return by xdt x is exogenous,not linked to fundamental Private benefit from shirking No benchmarking Pricing Implications When experts wealth declines, their market power increases, and so does their fee Price impact depends on assumption that household have larger discount rate than experts Procyclical Leverage In H-K calibration paper No fee, households are rationed in their investment As expert wealth approaches 0, interest rate can go to –∞ Heterogeneous labor income for newborns of lDt Non-log utility function BruSan Production economy GDP growth depends on net-wealth Physical investment Direct investments by all households Contracting (Potential) long-run relationship Fraction of return, fee, size of asset pool Effort increases fundamental growth to gdt Monetary benefit from shirking No benchmarking Pricing Implication Price drop with state variable Countercyclcial Leverage Entrepreneur take on same position after drop in networth Leverage increases after drop in net-worth

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Graphs: leverage

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