Deterministic Systems Functional Analysis Stochastic Systems
Numerical MethodsLecture 2: Dynamic Programming
Zachary R. Stangebye
University of Notre Dame
Fall 2017
Deterministic Systems Functional Analysis Stochastic Systems
Deterministic Systems
• Multi-period models are workhorse of macroeconomics• Agents make decisions each period, rationally forecasting
behavior in future periods
• Example: Multi-Period Consumption-Saving Model
maxctTt=0≥0,bt+1T−1
t=0
T∑t=0
βtu(ct)
s.t. ct + bt+1 = yt + (1 + r)bt for t < T
cT = yT + (1 + r)bT
• Note b0 and ytTt=0 are given
• Cannot save or borrow in last period
Deterministic Systems Functional Analysis Stochastic Systems
Solution Method: Backward Induction
1. Find optimal behavior in last period• Note this will be a function of bT
2. Find behavior in penultimate period, using optimal behaviorin next period• Again, function of bT−1
3. Find behavior in period before that . . .
4. Continue until period 0
Deterministic Systems Functional Analysis Stochastic Systems
Value Functions
• Helpful tool is to define Value Functions along the way
• For any t < T , we can define it as follows:
Vt(bt) = maxcsTs=t ,bs+1T−1
s=t
T∑s=t
βs−tu(cs)
s.t. cs + bs+1 = ys + (1 + r)bs for s < T
cT = yT + (1 + r)bT
• Note that V0(b0) delivers utility from our original problem
• If c?s (bt)Ts=t , b?s+1(bt)T−1s=t is solution, then
Vt(bt) =T∑s=t
βs−tu(c?s (bt))
Deterministic Systems Functional Analysis Stochastic Systems
Recursive Representation
• With value functions, we can conveniently write the problemin each period as follows:
Vt(bt) = maxbt+1
u (yt + (1 + r)bt − bt+1) + βVt+1(bt+1)
• Once we know VT−1(bT−1), it’s easy to find VT−2(bT−2)and so forth
• At each step, we attain b?t+1(bt). Can find consumption with
c?t (bt) = yt + (1 + r)bt − b?t+1(bt)
• Solution can be found from V0(b0): Can compute policiesalong entire trajectory following this
• This general approach to solving dynamic problems is calledDynamic Programming
Deterministic Systems Functional Analysis Stochastic Systems
Solution Method 1: Grids
• Simplest approach
1. Make a grid over bt for each t: B = b0, b1, . . . , bN2. For each bt on the grid, search over the grid to find the bt+1
that maximizes the recursive value function i.e. ∀bt ∈ B
Vt(bt) = maxbt+1∈B
u (yt + (1 + r)bt − bt+1) + βVt+1(bt+1)
3. Yields solution vector b?t ∈ BN i.e. easy to store
4. Continue until b?0.
• Accuracy (and solution time) increase with N
Deterministic Systems Functional Analysis Stochastic Systems
Example 1: Three Periods
maxc0,c1,c2≥0,b1,b2
u(c0) + βu(c1) + β2u(c2)
• Subject to same constraints as before
• Try two methods
1. Simultaneous solution2. Backward induction/dynamic programming
• Notice when β(1 + r) = 1, it should be that c0 = c1 = c2
• Backward induction scales up more easily than simultaneoussolution as T grows
Deterministic Systems Functional Analysis Stochastic Systems
Solution Method 2: Interpolation
• Grid can sometimes be restrictive, lead to inaccuracies
• Want to be able to pick points off grid
• Linear Interpolation Method
1. Solve period T − 1 on a grid of bT−1, but choosing any bT2. For bT−1 off grid, define the value function as straight line
connecting nearest grid points3. Solve period T − 2 on a grid of bT−2, but choosing any bT−1
4. . . .
• Formula: If bt+1 ∈ (bn, bn+1)
Vt+1(bt+1) = Vt+1(bn)+
[Vt+1(bn+1)− Vt+1(bn)
bn+1 − bn
]×(bt+1−bn)
Deterministic Systems Functional Analysis Stochastic Systems
Linear Interpolation
Deterministic Systems Functional Analysis Stochastic Systems
Extrapolation
• How to deal with points above/below whole grid?
• Two approaches
1. Restrict them in the problem2. Continue first/last interior interpolation lines
• (1) is nice when possible!
• Often (2) is necessary; example later
Deterministic Systems Functional Analysis Stochastic Systems
Other Interpolations
• More sophisticated technique is a spline interpolation
• Splines are piecewise polynomials of order 2 or higher• Idea: Keep function smooth/differentiable at kinks
• Cubic spline most popular: If we have a set (xi , yi )ni=0,
approximate each [xi , xi+1] interval with a cubic
y = ai + bix + cix2 + dix
3
• 4n coefficients: Point conditions and 1st/2nd derivativeconditions =⇒ Linear system
1. yi = ai + bixi + cix2i + dix
3i (i = 1, . . . , n)
2. yi = ai+1 + bi+1xi + ci+1x2i + di+1x
3i (i = 0, . . . , n − 1)
3. bi + 2cixi + 3dix2i = bi+1 + 2ci+1xi + 3di+1x
2i (i = 1, . . . , n− 1)
4. 2ci + 6dixi = 2ci+1 + 6di+1xi (1, . . . , n − 1)
• Two conditions free: Often chosen setting s ′(x0) = s ′(xn) = 0
Deterministic Systems Functional Analysis Stochastic Systems
Life Lessons
• In solving models, always look for model-specific shortcuts
• Often more useful in speeding up computation than anysupercomputer
• In our case, exploit the Euler Equation when β(1 + r) = 1:
u′(ct) = β(1 + r)u′(ct+1) =⇒ ct = c ∀t
• Put this together with the lifetime-budget constraint to getthe solution:
c =1
1− (1 + r)−T−1
(r
1 + r
)×
[(1 + r)b0 +
T∑t=0
(1
1 + r
)t
yt
]
• Faster and better objective than any other method so far!
Deterministic Systems Functional Analysis Stochastic Systems
Extending the Horizon
• Take limit as T →∞...
maxct∞t=0≥0,bt+1T−1
t=0
∞∑t=0
βtu(ct)
s.t. ct + bt+1 = yt + (1 + r)bt
• What happens to last period condition? Need something totie it down• Need to prevent Ponzi schemes: Rolling over borrowed money
ad infinitum• Can achieve infinite utility with Ponzi schemes if not ruled
out...
Deterministic Systems Functional Analysis Stochastic Systems
Ruling Out Ponzi Schemes
• Look at Lifetime BC of finite-horizon case:
NPV (ctTt=0) = (1+r)b0 +NPV (ytTt=0)+
(1
1 + r
)T
BT+1
• To get a ‘sensible’ Lifetime BC, need only that
limT→∞
(1
1 + r
)T
BT+1
i.e. BT+1 cannot grow exponentially at a rate faster than r
• Call this the No-Ponzi Condition• Can be satisfied even if limT→∞ BT+1 > 0
• Past problem plus No-Ponzi Condition is well-defined
Deterministic Systems Functional Analysis Stochastic Systems
Solving
• How do we solve the infinite-horizon problem?• Backward induction directly no good...no terminal period• Not entirely true...more on this later!
• Last finite-horizon method will still work for some utilities e.g.log-utility
EE : ct+1 = β(1 + r)ct
=⇒ NPV (ct∞t=0) = c0
∞∑t=0
βt =c0
1− β
=⇒ c0 = (1− β)× [(1 + r)b0 + NPV (yt∞t=0)]
• Does not require that β(1 + r) = 1
Deterministic Systems Functional Analysis Stochastic Systems
Solving: Shooting Algorithm
• If it doesn’t happen that we can solve as above, we needanother strategy
• Try to solve it forward : Bisection over c0
1. Conjecture a solution for the optimal c0
2. Iterate forward with the budget constraint/Euler Equation3. Check it at a large T : See if No-Ponzi Condition holds
(approximately)• If it holds, we’re done• If bT too large, increase c0 in next guess• If bT too large (negatively), reduce c0 in next guess
• Starting from bt , conjecture a ct and iterate forward with
bt+1 = yt + (1 + r)bt − ct
ct+1 = u′−1
(u′(ct)
β(1 + r)
)Equation (2) may need to be done numerically (not always)
Deterministic Systems Functional Analysis Stochastic Systems
Alternate Model: NCG Model
• Endogenize yt process; save in capital instead of bt
• Solve social planner’s problem (economy efficient)
• yt = f (kt)
1. f (0) = 02. f ′(kt) > 03. f ′′(kt) < 0
• Capital depreciates at rate δ. k0 given
maxct∞t=0,kt+1∞t=0
∞∑t=0
βtu(ct)
s.t. ct + kt+1 = f (kt) + (1− δ)kt
Deterministic Systems Functional Analysis Stochastic Systems
NCG Model: Solution Characterization
1. Euler Equation
u′(ct) = β[1− δ + f ′(kt+1)]u′(ct+1)
2. Resource Constraint
ct + kt+1 = f (kt) + (1− δ)kt
• Can iterate like before on (c0, k0)
• Terminal condition? Transversality condition
limt→∞
kt+1βtu′(ct) = 0
• Like a FOC ‘at t =∞’• Prevents sub-optimal ‘hoarding’ of capital
Deterministic Systems Functional Analysis Stochastic Systems
NCG Model: SS Lines and Trajectories
𝒄𝒄
𝒌𝒌 𝒌𝒌
• Steady state lines: Set EE = 0 and BC = 0
• Trajectories easy to derive on either side of SS line
Deterministic Systems Functional Analysis Stochastic Systems
NCG Model: Shooting Algorithm
• Only SS solution will satisfy TVC in long-run
• Search for c0(k0) that sends system ratcheting to steady state
• This trajectory is unique (saddle-path stable)
• This problem is recursive! Delivers time-invariant solutionalong an endogenous grid:
c?(kt) = ct(kt) ∀kt
• Could interpolate in between endogenous grid points forapproximation to full-solution
Deterministic Systems Functional Analysis Stochastic Systems
The NCG Model Revisited
• Backward induction? How do we do it without a terminalperiod?
• Recall finite-horizon approach. For any t < T , equivalentproblem:
Vt(kt) = maxkt+1≥0
u(f (kt) + (1− δ)kt − kt+1) + βVt+1(kt+1)
• In infinite-horizon model, t should be irrelevant• Should be the case that
V (k) = maxk′≥0
u(f (k) + (1− δ)k − k ′) + βV (k ′)
• Solution k ′(k) should exactly be k1(k0) in infinite-horizon
Deterministic Systems Functional Analysis Stochastic Systems
Thinking Recursively
• How to solve this model? Try something crazy...
1. Guess any value function, V i (k)2. Derive V i+1(k) by solving (for every k)
V i+1(k) = maxk′≥0
u(f (k) + (1− δ)k − k ′) + βV i (k ′)
3. Continue until supk ||V i+1(k)− V i (k)|| < ε for a small ε > 0
• For this model, it will work every time! Regardless of V 0
• This is called Value Function Iteration
• If it converges in I periods, you’ll have approximations of
1. The value function V (k) = V I (k)2. The policy function k ′(k) = k I (k)
Deterministic Systems Functional Analysis Stochastic Systems
Analysis
• Provable results: k ′(k)
1. k ′(0) > 02. limk→∞ k ′(k) < k
3. ∂k′(k)∂k < 1
• Implies the model converges to a steady state (from eitherside)
Deterministic Systems Functional Analysis Stochastic Systems
Why?
• Why would we expect VFI to work?
• Answer lies in Functional Analysis
• A Functional Space, J, is a set of functions on a givendomain, Ω ⊂ RN i.e. if f ∈ J, then
f : Ω→ R
• A Functional Equation maps functions into functions i.e.H : J1 → J2
• We can write most recursive economic problems as finding afunction, d ∈ J1 such that
H(d) = 0
where 0 is the zero function, not the number zero
Deterministic Systems Functional Analysis Stochastic Systems
Example
• In NCG example, J1 is all functions mapping R+ into R.
1. d = V2. H(d) = 0 ⇐⇒ for all k ≥ 0,
V (k)−[
maxk′≥0
u(f (k) + (1− δ)k − k ′) + βV (k ′)
]= 0
• Can write it equivalently by using Euler Equation
1. d = k ′
2. H(d) = 0 ⇐⇒ for all k ≥ 0,
u′(f (k) + (1− δ)k − k ′(k)
)−
β×[1−δ+f ′(k ′(k))]×u′(f (k ′(k))+(1−δ)k ′(k)−k ′(k ′(k))
)= 0
Deterministic Systems Functional Analysis Stochastic Systems
Fixed Points
• Equivalency to fixed points. If T : J → J, an eq’m can oftenbe described as a d ∈ J such that
Td = d
Notation: Often functional equations are just written as ‘Td ’instead of T (d)
• Completely equivalent to saying Hd = Td − d and sayingHd = 0
• Fixed point approach lends itself to iterative approaches
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Metric Spaces
• To characterize things cleanly, zoom out a little bit
• Reference: Stokey, Lucas, and Prescott (1989) Chapter 3
DefinitionA Metric Space is a set S , together with a metric (distancefunction) ρ: S × S → R such that for all x , y , z ∈ S :
1. ρ(x , y) ≥ 0, with equality iff x = y
2. ρ(x , y) = ρ(y , x)
3. ρ(x , z) ≤ ρ(x , y) + ρ(y , z)
• S could be a functional set e.g. f , g ∈ J
• Common metric is the sup-norm:ρ(f , g) = supx∈Ω |f (x)− g(x)|
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Completeness
• A few more definitions
DefinitionA sequence xn∞n=0 in S is a Cauchy sequence if for every ε > 0,∃Nε such that
ρ(xn, xm) < ε ∀n,m ≥ Nε
DefinitionA metric space (S , ρ) is complete if every Cauchy sequence in Sconverges to an element in S .
• Complete metric space ≈ A closed interval on the real line (asopposed to an open one)
Deterministic Systems Functional Analysis Stochastic Systems
The Contraction Mapping Theorem
DefinitionLet (S , ρ) be a metric space and T : S → S . T is a contractionmapping with modulus β if for some β ∈ (0, 1),ρ(Tx ,Ty) ≤ βρ(x , y) for all x , y ∈ S .
• If I apply my mapping to any two items in the set, those twoelements always get closer together
Theorem (Contraction Mapping Theorem)
If (S , ρ) is a complete metric space and T : S → S is a contractionmapping with modulus β, then
1. T has exactly one fixed point, ν ∈ S
2. For any ν0 ∈ S , ρ(T nν0, ν) ≤ βnρ(ν0, ν) for n = 0, 1, 2, . . .
Deterministic Systems Functional Analysis Stochastic Systems
Application
• If we have a contraction mapping...great!
• We know that limn→∞ βnρ(ν0, ν) = 0 since β < 1
• Implies limn→∞ T nν0 = ν• i.e. repeated iteration will eventually get us to the fixed
point/equilibrium
• How do we know if it’s a contraction?...
Deterministic Systems Functional Analysis Stochastic Systems
Application
Theorem (Blackwell’s Sufficiency Theorem)
Let X ⊂ Rl and let B(X ) be a space of bounded functionsf : X → R with the sup-norm. Let T : B(X )→ B(X ) be anoperator satisfying
1. (Monotonicity) f , g ∈ B(X ) and f (x) ≤ g(x) for all x ∈ Ximplies (Tf )(x) ≤ (Tg)(x) for all x ∈ X
2. (Discounting) There exists some β ∈ (0, 1) such that
[T (f + a)](x) ≤ (Tf )(x) + βa, ∀f ∈ B(X ), a ≥ 0, x ∈ X
Then T is a contraction with modulus β
Deterministic Systems Functional Analysis Stochastic Systems
Back the NCG Model
• Define TV as follows: For any k ∈ R+
(TV )(k) = maxk ′≥0
u(f (k) + (1− δ)k − k ′) + βV (k ′)
• Check Blackwell’s conditions
1. Let VL(k) ≤ VH(k) for all k . Easy to see thatTVL(k) ≤ TVH(k) for all k
2. [T (V + a)](k) ≤ (TV )(k) + βa for any positive a
• Both conditions satisfied! Repeatedly applying T i.e. VFI isguaranteed to converge to unique solution
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Example 2: Adding Shocks
• Real Business Cycle Model• NCG model with new production function
yt = At f (kt)
• At is total factor productivity: At = exp(zt) and
zt = ρzt−1 + εt
• Recursive representation:
V (A, k) = maxk′≥0
u(Af (k) + (1− δ)k − k ′) + βEA|A[V (A, k ′)]
• Two states now: Capital stock and productivity
• Still satisfies Blackwell’s sufficiency conditions
Deterministic Systems Functional Analysis Stochastic Systems
RBC Model: Solution Approach 1
1. Tauchenize zt shock: z ∈ z1, z2, . . . , zN2. Cubic spline interpolation across k for each zi
• Delivers a collection of continuous, differentiable functions,Vi (k)Ni=1
• Vi (k) ≈ V (exp(zi ), k) for i = 1, 2, . . . ,N• Monotonicity: Vi (k) ≤ Vi+1(k) when zi < zi+1
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RBC Model: Solution Approach 2
1. Create a grid over k ; grid-search maximum
2. Gaussian quadrature expectations over zt (continuous)
• Must be careful...• Policy functions look sensible• Value function not monotone in zt !
• Limitation of grid-search• Grid-search cannot extrapolate• When zt is high, optimal solution almost always k ′ > k• If k is highest grid point...best you can do is k ′ = k
• Sub-optimal behavior =⇒ Low values• Translates to lower k levels since convergence fast and process
persistent
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Speeding Things Up 1
• VFI linearly at a rate β
• A little slow for large scale problems
• Speeding up: Many tricks devised over the years
• Approach 1 (Judd [1998]): Policy Function Iteration Hybrid
• Start with guess V i (A, k)
1. Find corresponding policy, k i (A, k) via maximization2. Fix policy function: Apply Bellman (without maximization) M
times• Delivers new V i+1(A, k)
• Takes more iterations than VFI, BUT...• Tends to converge faster than linearly in β...helpful!• Can often speed things up by 10x or more• Still a contraction (convergence guaranteed)
Deterministic Systems Functional Analysis Stochastic Systems
Speeding Things Up 2
• Endogenous Grid Method (Barillas andFernandez-Villaverde [2007])
• Observe FOC:
u′(c?(A, k)) = βEA|A
[Vk(A, k?(A, k))
]• Notice that if k? was fixed,
c?(k?) = u′−1(βEA|A
[Vk(A, k?)
])No maximization/root-finding required!
• Exploit this to speed things up (a lot)• Fix grid over optimal capital choice• Derive grid over current capital endogenously
Deterministic Systems Functional Analysis Stochastic Systems
Endogenous Grid Method: Requirements
• Store two different value functions:1. V i (At , kt+1) = βEAt+1|At
[V istandard(At+1, kt+1)]
• Expected discounted value of having kt+1 tomorrow if shock iszt today
2. V i (At ,Yt) =
maxkt+1 u(Yt − kt+1) + βEAt+1|At[V i
standard(At+1, kt+1)]
• Value function as a function of available market resourcesYt = f (kt) + (1− δ)kt
• Store three different grids
1. Tauchenized grid over zt , Gz
2. Capital grid over investment decisions, Gk
3. Market resources grid, Gy : Yt ∈ Gy if ∃(k , z) ∈ Gk × Gz suchthat Yt = ezt f (kt) + (1− δ)kt
Deterministic Systems Functional Analysis Stochastic Systems
Endogenous Grid Method: Procedure
• Begin with a guess V i (At , kt+1)
1. Compute approximate derivative V ik(At , kt+1) by averageing
slopes of linear interpolation
2. Compute c?(At , kt+1) = u′−1(V ik(At , kt+1))
3. Compute necessary market resourcesY (At , kt+1) = c?(At , kt+1) + kt+1
4. ComputeV i (At ,Y (At , kt+1)) = u(c?(At , kt+1)) + V i (At , kt+1)
5. Compute V i+1(At ,Yt) by interpolating V i (At ,Y (At , kt+1))on GY
6. Compute V i+1(At , kt+1) = βEAt+1|At[V i+1(At ,Yt)]
7. Stop if supi ,j |V i+1(Ai , kj)− V i (Ai , kj)| < ε
Deterministic Systems Functional Analysis Stochastic Systems
Endogenous Grid Method
• Once done, use root-solver to find endogenous capital grid i.e.k ∈ Gk,state if Y (At , kt+1) = At f (k) + (1− δ)k for some At
and kt+1
• Value function is Vstandard(At , k(At ,Yt)) = V (At ,Yt) for allYt such that Yt = Y (At , kt+1)
• Policy function is the original grid! (with domain beingendogenous grid)
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Non-contractions
• Without the CMT...
• Cannot always guarantee a process will converge• Sometimes it does anyway! (count your blessings)• Other times, exploit other features
• Example: Monotonicity• Instead of working with distances in a metric space...• Work with a notion of inequality in an ordered set• Different fixed point theorem, but still works!
Deterministic Systems Functional Analysis Stochastic Systems
Partially Ordered Sets
DefinitionA Partially Ordered Set, (X ,≤), is a set taken together with apartial order i.e.
1. For any a ∈ X , a ≤ a (Reflexivity)
2. For any a, b ∈ X , a ≤ b and b ≤ a implies a = b(Antisymmetry)
3. For any a, b, c ∈ X , a ≤ b and b ≤ c implies a ≤ c(Transitivity)
Deterministic Systems Functional Analysis Stochastic Systems
Complete Lattices
DefinitionA partially ordered set, (L,≤), is called a Complete Lattice ifevery subset has a least upper bound and a greatest lower boundin L i.e. for any M ⊂ L,
1. supM ∈ L
2. inf M ∈ L
• Akin to the notion of closedness/boundedness, but there is nodistance metric
Deterministic Systems Functional Analysis Stochastic Systems
Tarski’s Fixed Point Theorem
DefinitionLet (L,≤) be a complete lattice, and suppose T : L→ L is amonotone function i.e. for any x , y ∈ L, the following holds
x ≤ y =⇒ Tx ≤ Ty
Then the set of all fixed points in L for the function T is also acomplete lattice.
• Cool theorem! Some interesting implications
1. There exists a greatest (u) and a least (u) fixed point(possibly the same; fixed point set non-empty)
2. If x ≤ Tx , then x ≤ u
3. If x ≥ Tx , then x ≥ u
Deterministic Systems Functional Analysis Stochastic Systems
Applying Tarski’s: The Eaton-Gersovitz/ArellanoModel
• Based on NCG/RBC model, but a few simple differences
1. Gov’t controls all consumption decisions2. No investment (endowment economy)3. Foreigners buy debt4. Gov’t monopolist in debt market/foreign lenders competitive
• Internalizes price changes from debt issuance
5. Gov’t cannot commit to repay debt• Will default if ex-post optimal• If default, excluded from credit markets forever i.e. ct = yt
and pay default cost
Deterministic Systems Functional Analysis Stochastic Systems
The Arellano Model: Sovereign
• Sovereign wants to solve similar Bellman equation
V (y , b) = maxb′
u(y − b + q(y , b′)b′) + βE [V (y , b′)]
whereV (y , b) = maxV (y , b),X (y)
and X (y) is the utility value of defaulting in state y
X (y) = u(y × [1− φ(y)]) + βE [X (y)]
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The Arellano Model: Lenders
• Foreign lenders are risk-neutral, deep-pocketed
• Competitively price debt =⇒• Can get a risk-free return, r• Can invest in risky sovereign debt (may get defaulted on)
q(y , b′) =Ey |y [1V (y , b′) ≥ X (y)]
1 + r
Deterministic Systems Functional Analysis Stochastic Systems
Solving the Arellano Model
• That’s it! Pretty simple, BUT...• VFI is no longer a contraction• Discounting in Blackwell sufficiency no longer holds, since V
enters into q
• Alternative approach via Tarski: Iterate on qi
1. Define Q to be a complete lattice of decreasing functions i.e.
q ∈ Q then q : Y × B →[
0,1
1 + r
]2. Take our order to be the absolute order i.e. if q1, q2 ∈ Q
q1 ≤ q2 ⇐⇒ q1(y , b) ≤ q2(y , b) ∀(y , b) ∈ Y × B
Deterministic Systems Functional Analysis Stochastic Systems
Solving the Arellano Model
• Iterative operator T defined as follows• Fixing q, the sovereign’s Bellman is a contraction: qi =⇒ V i
• Update step
(Tqi )(y , b) =Ey |y
[1V i (y , b′) ≥ X (y)
]1 + r
Proposition
The operator T is a monotone self-map on Q
• Easy to show Tq must be decreasing, self-map
• Monotonicity follows from fact that if q1 ≤ q2, then V1 ≤ V2
since prices are always higher in world 2
Deterministic Systems Functional Analysis Stochastic Systems
Applying Tarski’s
• T is a monotone operator on a complete lattice, so Tarski’stheorem applies• Get that an equilibrium exists (may be many)
• To find it is easy: Repeatedly apply T from either top orbottom• Set q0 = 0; know q0 ≤ qeq, which implies Tq0 ≤ Tqeq = qeq
limn→∞
T nq0 ≤ T nqeq = qeq
• Gets bigger with each iteration• Must converge to something: Will converge to lowest fixed
point (worst eq’m)• Same logic applies starting from best i.e. q0 = 1
1+r
• Will converge to best