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

2. 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

3. 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).

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

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
Household discount rate
Household depreciation

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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 /

17. Equilibrium

18. Equilibrium

19. 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

20. 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

21. 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

22. 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

23. 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.

24. 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…

25. 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…

26. Externalities with households

27. 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

28. Equilibrium when households provide liquidity support

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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)]

36. 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)

37. 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

38. Thank you! ☺

39. 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

40. 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

41. 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

42. Graphs: leverage