three period rollover crisis model - northwestern universitylchrist/...objective: max dt 0 vt, where...
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Three Period Rollover Crisis Model
Lawrence J. Christiano Husnu Dalgic,with Xiaoming Li
September 21, 2019
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Background• Banks:
– borrow short term and acquire long-term assets.– term mismatch.
• Vulnerability:– Banks with term mismatch need creditors to roll over short
term loans (‘deposits’), or else they could be forced to sell theirassets at fire sale prices (see, e.g., Shleifer-Vishny JEP2011).
– Central Banks have solved this problem for commercial banks(deposit insurance, liquidity backstops).
• Many financial institutions lie outside protective umbrella of thecentral bank: Shadow Banks.
• Roll-over Crisis scenario:– All creditors refuse to roll over their short term loans, forcing
banks into asset fire sales.
– If fire sale prices are low enough, banking system could becomeinsolvent and collapse, with damaging consequences for rest ofthe economy.
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Zoltan Pozsar, Tobias Adrian, Adam Ashcraft, and Hayley Boesky, ‘Shadow Banking’, FederalReserve Bank of New York Economic Policy Review , December 2013
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Background• View that run on Shadow Banks was trigger (or, massive
amplifier) of Great Recession:– Gary Gorton Slapped by the Invisible Hand, the Panic of 2007,
Oxford University Press, 2010.– Bernanke, statement before the Financial Crisis Inquiry
Commission, September 2010.– Formal models: Gertler-Kiyotaki, Banking, Liquidity, and Bank
Runs in an Infinite Horizon Economy, in AER2015.
• Here: provide a three period version of Gertler-Kiyotaki model.– Exploit simplicity of setup to explore number of equilibria and
impact of macro-prudential policy.
• Model is as simple as possible– Minimize agent heterogeneity.– Minimize number of periods: need at least three to have
maturity mismatch.– Style of Diamond-Dybvig (JPE1983), Chang-Velasco
(QJE2001).
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Outline: Baseline Equilibrium
t = 0 t = 1 t = 2
Roll-Over
Annihilation
Partial-Run
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Outline: Baseline Equilibrium
t = 0 t = 1 t = 2
No-run
Annihilation
Partial-Run
In period 0, banks issue one-period deposits, d0.Limited by amount of banker net worth, N0.
In period 0, banks use N0+d0 to purchasetwo-period lived capital.
In period 1, two possible equilibrium outcomes:
(i) No-run. Banks roll over their liabilities.
(ii) Annihilation run. Banks cannot roll over.
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Bankers, Workers and Households
• Workers earn exogenous income in each period.
• Bankers acquire long-lived assets financed by their own networth and deposits.
• Workers and bankers live in identical households.
– Simplifies welfare analysis.
• The household instructs bankers to maximize the presentdiscounted value of profits.
– They can do this being good bankers and playing by the rules.
– Or, if bankers have an opportunity to bring home higherpresent value of profits by exploiting opportunities to steal,that’s ok too.
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Bankers• At start of period t banker has assets, kb
t−1, and liabilities,Rt−1dt−1, acquired in previous period:
capital︷︸︸︷kb
t−1 ,
gross interest rate︷︸︸︷Rt−1 ×
deposits︷︸︸︷dt−1 .
• During period t = 0, 1:– Bankers combine deposits, dt, with net worth, Nt, and
purchase assets,Qtkb
t = Nt + dt,
Nt = (Zt + Qt) kbt−1 − Rt−1dt−1.
• In t = 2, Q2 = d2 = 0.• Here,
Zt ∼ exogenous productivity of capital.
Qt ∼ period t market price of capital.
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Bankers, Scaling
• Later, will find it useful to scale variables.
• Banker net worth:
Nt = (Zt + Qt) kbt−1 − Rt−1dt−1
=
Rkt︷ ︸︸ ︷(
Zt + Qt
Qt−1
) φt−1︷ ︸︸ ︷Qt−1kb
t−1Nt−1
Nt−1 − Rt−1
φt−1−1︷ ︸︸ ︷dt−1
Nt−1Nt−1
=[(
Rkt − Rt−1
)φt−1 + Rt−1
]Nt−1
• Here,
φt ∼ banker leverage in period t
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Banker Problem• Objective:
maxdt≥0
Vt,
where Vt denotes the value of banking, t = 0, 1, subject to:
Qtkbt = Nt + dt.
• Banker has the option to ‘run away’ with θQtkbt .
– Assets are diverted to banker’s household and (1− θ)Qtkbt are
destroyed, depositors get nothing.
• Banker must announce dt in advance, so depositors knowwhether banker intends to run away or not.
– As a result, bankers only consider dt for which
θQtkbt ≤ Vt.
• Must ensure banker problem has well-defined solution.
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Rollover Crisis in period t = 1• Suppose all bankers receive zero deposits in t = 1 :
d∗1 = 0.
• Then, to pay period t = 1 liabilities, R0d0, the bankers mustsell capital, kb
0.
• With all bankers selling capital, the only buyers are households.– Households assign relatively low value to capital.– Capital sold at fire sale price, Q∗1.– Suppose the fire sale is sufficiently severe,
(Z1 + Q∗1) kb0 − R0d0 < 0→ N∗1 = 0
then households only get:
R0xd0,recovery ratio︷︸︸︷
x =(Z1 + Q∗1) kb
0R0d0
< 1.
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Possible Events in Period t = 1• Previous slide suggests that collapse of the banking system
could be an equilibrium.– We will parameterize the model so that this is in fact the case.
• There is another equilibrium in period t = 1 when householdsprovide deposits and bankers pay off R0d0 without having tofire sale assets.
• So, have two equilibria:– Run in which bank net worth is wiped out (annihilation run).– No-run equilibrium in which d1 > 0 and Q1 > Q∗1.
• Later, will explore possibility of a third, partial run, equilibrium.
• For now, follow Gertler-Kiyotaki and suppose there are twothings that can happen in period 1:
P = prob[annihilation run]= 1− x1− P = prob[no-run equilibrium]= x.
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Banker Problem in Period t = 0• Problem:
V0 = maxd0≥0
βm1 (1− P) [N1 + V1]+P×value of banking in run state︷︸︸︷
0 ,
s.t.: θQ0kb0 ≤ V0.
• Here,
P ∼ prob of rollover crisis in period 1
βm1 ∼ discount factor for no-run period 1 state
• Banker belongs to representative household and householdrequires discounting by intertemporal marginal utility ofconsumption, βm1:
βmt =βu′ (ct)
u′ (ct−1).
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Scaled Banker Problem in t = 0• Period 0 problem:
V0 = maxd0≥0
βm1 (1− P) [N1 + V1]
s.t.: θQ0kb0 ≤ V0.
• Divide objective by N0 (ψt ≡ Vt/Nt):
ψ0 = maxd0≥0
βm1 (1− P)[
N1
N0+
V1
N1
N1
N0
]= max
φ0≥1βm1 (1− P) [1 + ψ1]
[(Rk
1 − R0
)φ0 + R0
]subject to participation constraint:
θφ0 ≤ ψ0.
• We only consider examples in which Rk1 > R0
– so banker sets φ0 to maximum allowed by participationconstraint.
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Banker Problem in No-run Period t = 1• Period 1 problem:
V1 = maxd1≥0
βm2N2 = maxd1≥0
βm2
[Z2kb
1 − R1d1
],
s.t.: θQ1kb1 ≤ V1
• Scale by N1:
ψ1 = β maxφ1≥1
m2
[(Rk
2 − R1
)φ1 + R1
]subject to
θφ1 ≤ ψ1.
• We only consider examples where Rk2 > R1
– so, banker sets φ1 to the max allowed by participationconstraint.
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Ensuring Banker Problem Well-defined inNo-run Period t = 1
• Because we assume Rkt+1 > Rt, bankers always go to boundary
of participation constraint:
θφt = ψt
• Boundary must be finite for equilibrium to exist.– Infinite leverage incompatible with loan market clearing.
• To ensure ψ1 < ∞ must have (see figure on next slide):
βm2
(Rk
2 − R1
)< θ
• Discussion of household problem below ensures that, inequilibrium
βm2R1 = 1, m2 = u′ (c2) /u′ (c1)
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Figure: Condition for Finite Leverage in No-run Period t = 1
Leverage, !1
Value of bank, ψ1
Slope = "#$ %$& − %(
Value of bank (per unit of net worth), as a function of leverage.
If "m2(Rk2-R1) > )then infinite leverage consistent with participation constraint.
Slope = )
Participation constraint requires that bank have at least this value, ψ1=)!1 , for given leverage, !1.
1
"#$%$& = %$&/%(
)
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Restrictions in no-run t = 1 forParticipation Constraint to be Binding• Previous slide showed must have
βm2
(Rk
2 − R1
)< θ
to ensure ψ1 < ∞.
• Since βm2R1 = 1 this corresponds to
Rk2 < (1 + θ)R1,
so banker selecting leverage so that θφ1 = ψ1 implies
θφ1 = βm2
[(Rk
2 − R1
)φ1 + R1
]or,
φ1 =βm2R1
θ − βm2(Rk
2 − R1) =
R1
(1 + θ)R1 − Rk2< ∞.
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Restrictions in t = 1 Annihilation Run forParticipation Constraint to be Binding• The value of the bank in the period 1 run state (N∗1 = 0):
V∗1 = maxd1
βm∗2N2 = maxd1
βm∗2[Z2kb
1 − R∗1d1
]
= βm∗2
R∗,k2
=d∗1︷︸︸︷Q∗1kb
1−R∗1d∗1
.
• Participation constraint:
θd∗1 ≤ βm∗2[R∗,k2 − R∗1
]d∗1 .
• For this to represent a restriction on d1, need
(1 + θ)R∗1 > R∗,k2 ,
in which case d∗1 = 0.
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Restrictions in t = 0 for ParticipationConstraint to be Binding
• Period 0 problem - maximize value by choice of φ0 :
ψ0 = maxφ0≥1
βm1 (1− P) [1 + ψ1][(
Rk1 − R0
)φ0 + R0
]subject to θφ0 ≤ ψ0.
• Assume Rk1 > R0, so banker wants φ0 big as possible.
• Need slope of firm value in φ0 smaller than θ:
βm1 (1− P) [1 + ψ1](
Rk1 − R0
)< θ
in which case, going to boundary of part. const. implies:
φ0 =βθ m1 (1− P) [1 + ψ1]R0
1− βθ m1 (1− P) [1 + ψ1]
(Rk
1 − R0) < ∞
We also require, φ0 > 1 (i.e., value at φ0 = 1 greater than θ).
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Figure: Condition for Finite Leverage in Period t = 0
Leverage, !0
Value of bank, ψ0
ψ0 = "m1(1-P)(1+ψ1) [(Rk1-R0)!0+R0]
Value of bank (per unit of net worth), as a function of leverage.
If "m1(1-P)(1+ψ1) (Rk1-R0) > #
then infinite leverage consistent with participation constraint.
Slope = #
Value of bank required byparticipation constraint, as afunction of leverage, ψ0=#!0.
1
"m1(1-P)(1+ψ1)R1k
#
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Households
• In period 0, inherit assets from the past, R−1d−1, kh−1.
• Choose deposits, consumption and capital holdings in period 0,period 1 (with probability, P, it is a run state).
• Consume in period 2 and then disappear.
• Bankers live in the households and so social welfare function isjust the utility of the typical household.
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Table: Balance Sheets and Budget Constraints
Bank t Household
Q0kb0 = (Z0 + Q0) kb
−1 − R−1d−1︸ ︷︷ ︸N0
+d0 0 Q0
(kh
0 − kh−1
)+ c0 + d0 + f
(kh
0
)≤ R−1d−1 + Z0kh
−1 + y0
Q1kb1 = (Z1 + Q1) kb
0 − R0d0︸ ︷︷ ︸=N1=
[Rk
1φ0−R0(φ0−1)]N0
+d1 1 Q1
(kh
1 − kh0
)+ c1 + d1 + f
(kh
1
)≤ R0d0 + Z1kh
0 + y1
π2 = Z2kb1 − R1d1︸ ︷︷ ︸
=N2=[Rk
2φ1−R1(φ1−1)]N1
2 c2 ≤ R1d1 + Z2kh1 + π2
,
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Households• utility at date 0:
u (c0) + βP [u (c∗1) + βu (c∗2)] + β (1− P) [u (c1) + βu (c2)]
• first order conditions for deposits (no deposits in run state ofperiod 1):
u′ (c0) = β[(1− P) u′ (c1) + Pu′ (c∗1) x
]R0
u′ (c1) = βu′ (c2)R1
• first order condition for capital decision in period 0:
u′ (c0) = β (1− P) u′ (c1)
(Z1 + Q1
Q0 + f ′(kh
0))
+ βPu′ (c∗1)
(Z1 + Q∗1
Q0 + f ′(kh
0))+ µu′ (c0)
µkh0 = 0, µ, kh
0 ≥ 0
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Households
• first order condition for capital decision in no-run period 1:
u′ (c1) = βu′ (c2)Z2
Q1 + f ′(kh
1
) + ν× u′ (c1)
νkh1 = 0, ν, kh
1 ≥ 0
• first order condition for capital decision in run-state in period 1(when households hold kh
1 = 1) :
u′ (c∗1) = βu′ (c∗2)Z2
Q∗ + f ′ (1)
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Market Clearing and other AggregateConditions
• Capital market:kh
t + kbt = 1
• Resource constraints obtained by combining household budgetconstraint with bank budget constraint in table above.
– Periods t = 0, 1 : ct + f(kh
t)= Zt + yt.
– Period t = 2 : c2 = c∗2 = Z2.• Probability of a run in period 1:
P = 1−min {x, 1} , x =(Z1 + Q∗) kb
0R0d0
.
• Discounting:
m1 =u′ (c1)
u′ (c0), m2 =
u′ (c2)
u′ (c1),
taken as exogenous by banks.
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Solving the Model
• Exogenous variables:
R−1d−1, Z0kb−1, Z0kh
−1, y0 = y1 = y, Z0 = Z1 = Z, Z2
and parameters:α, β, σ, θ
where
u (c) =c1−σ
1− σ, f
(kh)=
α
2
(kh)2
.
• Solving simple in some cases, e.g., c2 = c∗2 = Z2.
• Other equations are set equal to zero, enforcing non-negativityconstraints and constraints about variables lying inside unitinterval (e.g., kh
t ).
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Baseline Parameter Values
Parametersβ 0.9097σ 1.4951α 0.0626θ 0.3626Z 0.1263Z2 0.0908y 0.0878
kh−1 0.1738
Rd−1 0.3502
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Baseline Equilibrium
Baseline Baselineφ0 5.95 x 0.90φ1 3.09 P 0.10ψ0 2.16 R0 1.08ψ1 1.12 R1 0.30c0 0.21 R∗1 0.39
kb0 0.98 Rk
1 1.11
kb1 0.82 Rk
2 0.32c1 0.21 d0 0.30c∗1 0.18 d1 0.16c2 0.09 N0 0.06Q0 0.37 N1 0.08Q1 0.28 σC 4.68Q∗ 0.17 Risk Premium 1.43
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Macro Prudential Analysis
• Let φ̃0 denote leverage in the baseline equilibrium (= 5.95).
– Impose a restriction, φ0 ≤ φ̃0τ
– So, banker chooses φ0 ≤ min{
φ̃0τ, ψθ
}– The best equilibrium is one associated with τ = 0.98.
• Leverage restriction forces banks to internalize impact on P ofhigher leverage.
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Baseline and Leverage Restriction
Baseline Policy Change (%) Baseline Policy Change (%)
φ0 5.95 5.83 -2.00 x 0.90 0.91 1.67
φ1 3.09 3.10 0.58 P 0.10 0.09 -14.54
ψ0 2.16 2.32 7.57 R0 1.08 1.08 0.15
ψ1 1.12 1.13 0.58 R1 0.30 0.31 0.06
c0 0.21 0.21 -0.12 R∗1 0.39 0.39 0.00
kb0 0.98 0.91 -7.56 Rk
1 1.11 1.12 1.24
kb1 0.82 0.82 -0.98 Rk
2 0.32 0.32 0.25
c1 0.21 0.21 -0.04 d0 0.30 0.28 -9.23
c∗1 0.18 0.18 0.00 d1 0.16 0.16 -0.95
c2 0.09 0.09 0.00 N0 0.06 0.06 -6.99
Q0 0.37 0.37 -1.40 N1 0.08 0.07 -1.80
Q1 0.28 0.28 -0.25 σC 4.68 4.36 -7.04
Q∗ 0.17 0.17 0.00 Risk P. 1.43 1.0474 -26.62
Welfare 0.0636Note: ‘welfare’ is the percent increase in period t = 0 consumption in the baseline equilibrium,which makes household indifferent between baseline equilibrium and leverage restrictedequilibrium.
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Implementation Problem
• When we imposed the leverage constraint, we found that therewas another equilibrium:
– The other equilibrium is one with lower welfare (bank capitalfalls a lot).
– In the model, need to do more than just announce a leveragerestriction, to get good results.
• Example:
– Tax capital holdings, τ ×(kh
0 − k̄)
, where k̄ denotes capitalholdings in the desired equilibrium.
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Looking for other Period t = 1 Equilibria
• Two equilibria: ‘no-run’ and ‘annihilation’ equilibrium.
• Can think about the two equilibria using Diamond-Dybvig logic.
– An equilibrium is the fixed point of a best response function.– Each bank chooses its d1 based on a conjecture about D1,
what all the other banks do:
d1 = f (D1) .
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Looking for other Period t = 1 Equilibria
• Annihilation equilibrium:
– Each bank conjectures other banks set D1 = 0.– Each bank understands the other banks are not rolling over
their liabilities, so they must fire-sale their assets:
(Z + Q∗1) kb0 < R0d0.
– In this case, each bank chooses d1 = 0 knowing that it iswiped out:
0 = f (0) ,
fixed point!– This is why D1 = 0 is an equilibrium.
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Looking for other Period t = 1 Equilibria• No-run equilibrium:
– Each bank expects the other banks will set high deposits, D1.– Each bank knows that bankers will be able to roll over their
liabilities and not fire-sale assets.– So, Q1 high and all banks solvent:
(Z + Q1) kb0 > R0d0.
– Each bank responds with d1 = D1, or,
D1 = f (D1)
– Another equilibrium!
• Can search for all equilibria by fixing the date 0 equilibriumallocations and graphing the function, f .
– An equilibrium is a D1 where f crosses the 45 degree line.
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Best Response Function
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
No run equilibrium
Annihilationequilibrium
Partial run equilibrium
Annihilation, partial run and no-run equilibrium.
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Observations
• There are three equilibria: annihilation, partial run and no-runequilibrium.
• One expects these three equilibria to exist, as long as x < 1,i.e., depositors lose money in an annihilation run equilibrium.
• As D1 rises above zero, Q1 rises because banks don’t have tosell so many assets to pay off debt.
– But, initially, N1 remains stuck at zero because x < 1:
N1 = max{0, (Z + Q1) kb0 − R0d0}
– So, f (D1) remains stuck at zero too.– Eventually, D1 rises enough that N1 becomes positive.
• Then, f rises rapidly and cuts 45 degree line from below.• That’s because banks earn a lot per dollar of deposits when net
worth is low (recall Gertler-Karadi).
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Three Period 1 Equilibria
t = 0 t = 1 t = 2
Roll-Over
Annihilation
Partial-Run
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Relationship to Gertler-Kiyotaki (AER2014)
• GK have Infinite horizon model.
• Must drain equity from bankers, or they’ll accumulate toomuch.
– A fraction of bankers die each period.– An equal fraction is born, with a small amount of net worth.
• Newborn bankers must be kept out of business during anannihilation run, because otherwise it would not be anequilibrium.
• If we assume that new-born bankers stay out during anannihilation run and enter in small numbers as D1 rises abovezero, we find the same three equilibria in GK.
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Deviation in State s = t, from GKEquilibrium
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Aggregate Best Response, GK Model
xOne interpretation of GK: aggregate best response functiondiscontinuous at zero.
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Aggregate Best Response, GK Model
xObtain indicated best response function in GK model if assumenewborns coming in very slowly at low levels of D1
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Conclusion• Described simple three-period model with maturity mismatch in
banks.
• Three findings.
1 Limitations on deposits reduces probability of crisis andimproves welfare.
2 Macro prudential policy requires solving a non-trivialimplementation problem.
– Can have multiple equilibria.– Showed how supplementing leverage constraint with a tax on
household capital could solve implementation problem.
3 Gertler-Kiyotaki run/no-run result is fragile.– Under reasonable changes in assumptions about the arrival
rate of new banks, there is a qualitative change in the set ofequilibria.
– One change results in a unique equilibrium with no runs.– Another change introduces a third, partial-run, equilibrium.