A Macroeconomic Model with a Financial Sector

1. A Macroeconomic Model with a Financial Sector Markus K. Brunnermeier and Yuliy Sannikov

2. Motivation • Current financial turmoil • 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).

3. Modeling financial frictions • Central idea: heterogeneous agents • have special skills/more productiveBernanke-Gertler-Gilchrist, He-Krishnamurthy, Kiyotaki-Moore • less risk-averse  Geanakoplos, Adrian-Shin, Garleanu-Pedersen • more optimistic  Geanakoplos type 1: financially constrained type 2: unlevered

4. Modeling financial frictions • Central idea: heterogeneous agents • have special skills/more productiveBernanke-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

5. Loss of capital Modeling financial frictions • Central idea: heterogeneous agents • have special skills/more productiveBernanke-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 Fire sales shock to net worth Precaution + tighter margins volatility price 

6. Loss of capital Modeling financial frictions • Central idea: heterogeneous agents • have special skills/more productiveBernanke-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 Fire sales shock to net worth Precaution + tighter margins volatility price

7. Preview of results • Unified model, replicates dynamics near steady state (e.g. Bernanke-Gertler-Gilchrist) … but unstable dynamics away from steady statedue 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

8. 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

9. 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

10. 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 capitalis 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

11. 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 - tpt ≤ 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 Insident= αtet Outside(1-αt)et et

12. Balance sheets • 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

13. Evolution of balance sheets • Asset side – long maturity d(ktpt) = kt (gpt + 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

14. Equilibrium • 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 (risk premium) * pt

15. 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 /

16. Equilibrium

17. Equilibrium

18. 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

19. Loss of capital System dynamics and instabilities • 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 Fire sales Zt-shock on kt Precaution + tighter margins amplification tp pt

20. 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

21. 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 zlcorrelation (ptjktj, ptlktl) is (σ + σp)/(σ + σp + σ’) which is increasing in σp • xxx aggregate shock uncorrelated shock

22. 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.

23. 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…

24. 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…

25. Externalities with households

26. 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

27. Equilibriumwhen households provide liquidity support

28. Modification 3: Idiosyncratic losses dkti = g kti dt +  ktidZt + 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

29. 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

30. Modification 3: Idiosyncratic losses dkti = g kti dt +  ktidZt + 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

31. Equilibrium • Experts borrow at rate larger than r • Rate depends on leverage, price volatility • dt = diffusion process (without jumps) because losses cancel out in aggregate

32. 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

33. Contracting • 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 - tpt≤ 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 Insident= αtet Outside(1-αt)et et

34. 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 equitydnt = 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)]

35. 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)

36. 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

37. Thank you! ☺

38. 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

39. Differences to Kiyotaki-Moore 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 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) 40

40. Differences to He-Krishnamurthy 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 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

41. Graphs: leverage