Liquid and Illiquid Assets
with Fixed Adjustment Costs
Greg Kaplan and Benjamin Moll
This note describes a continuous-time version of the model in Kaplan and Violante (2014)
in which a household optimally splits his wealth between liquid and illiquid assets. Because
adjustments are subject to a fixed cost, the model has a stopping-time element. We here
describe how such models can be solved numerically using a finite difference method. Based
on ongoing research (Kaplan and Moll, 201?). Note that the problem is different from
that in Kaplan, Moll and Violante (2016). The problem presented here features a nonconvex adjustment cost whereas the problem in Kaplan, Moll and Violante (2016) features
a kinked but strictly convex adjustment cost. Both types of cost functions give rise to an
inaction region. However, the mathematical structure is fundamentally different: a nonconvex adjustment cost results in a stopping-time problem whereas a strictly convex cost
function does not.
1
Model Setup
Households can invest in two assets: an illiquid asset a, and a liquid asset b. The liquid asset
pays a real return rb and can be freely traded subject to a borrowing limit. The illiquid
asset pays a return ra > rb . Deposits and withdrawals can be made into and out of the
illiquid asset only upon payment of a transaction cost κ. The households receives an income
flow z where z can take a finite number of values. Productivity shocks arrive according to a
stochastic process which is either Poisson or a diffusion process.
1.1
Households’ Problem
Households solve the following problem
ˆ
τ
e−ρt u(ct )dt + e−ρτ E0 vi∗ (aτ + bτ )
vi (a, b) = max E0
{ct },τ
0
ȧt = ra at ,
at ≥ 0,
ḃt = rb bt + wzt − ct
bt ≥ 0,
(a0 , b0 , z0 ) = (a, b, zi )
1
(1)
where zt ∈ {z1 , z2 } is a two-state Poisson process with intensities λ1 , λ2 , and where
vi∗ (a + b) = max
vi (a0 , b0 ) s.t. a0 + b0 = a + b − κ,
0 0
a ,b
a0 ≥ 0, b0 ≥ 0.
(2)
For future reference, denote the optimal adjustment decisions conditional on adjustment by
a0 (a, b) and b0 (a, b). Note that these only depend on the total amount of assets a + b (rather
than a and b separately). The HJB equation is
ρvi (a, b) = max u(c) + ∂a vi (a, b)ra a + ∂b vi (a, b)(wzi + rb b − c) + λi (vj (a, b) − vi (a, b)) (3)
c
for i = 1, 2 and j 6= i, and with state-constraint boundary condition
∂b vi (a, 0) ≥ u0 (wzi )
(4)
vi (a, b) ≥ vi∗ (a + b) all a, b.
(5)
and a constraint that
This can also be written compactly as follows
min{ρvi (a, b) − max u(c) − ∂a vi (a, b)ra a − ∂b vi (a, b)(wzi + rb b − c) − λi (vj (a, b) − vi (a, b)),
c
vi (a, b) −
vi∗ (a
(6)
+ b)} = 0
In mathematics, (6) is called a “HJB variational inequality” (HJBVI for short). See e.g.
Barles, Daher and Romano (1995) and Tourin (2013).
2
Numerical Solution
See http://www.princeton.edu/~moll/HACTproject/liquid_illiquid_LCP.m.
2.1
Household’s Problem: Linear Complementarity Problem +
Finite Differences
We use an analogous approach to the simple model of exercising an option as in these notes
http://www.princeton.edu/~moll/HACTproject/option_simple.pdf. Once discretized
the HJBVI (6) becomes
min{ρv − u(v) − A(v)v, v − v ∗ (v)} = 0
2
(7)
The main difference to the “exercising an option” problem laid out above is that the HJB
equation without adjustment (the left branch of (7)) is non-linear in v and hence some
iteration is necessary. Related also the value of having a reoptimized portfolio v ∗ depends
on the value function as can be seen from (2). We proceed as follow:
1. as an initial guess v 0 use the solution to
ρv − u(v) − A(v)v = 0
i.e. the problem without adjustment, i.e. in which the fixed cost κ is infinite.
2. Given v n , find v n+1 by solving
min
v n+1 − v n
n+1
n
n n+1 n+1
∗ n
+ ρv
− u(v ) − A(v )v , v
− v (v ) = 0
∆
Exactly as in http://www.princeton.edu/~moll/HACTproject/option_simple.pdf,
this problem can be written as a linear complementarity problem (LCP).
3. Stop when v n+1 is sufficiently close to v n .
2.2
Household’s Problem: Alternative (Inferior) Approach
We have also tried another approach to solve the household’s problem (6). This approach
combines the finite-difference method with a so-called “splitting method” to take care of the
constraint (5).1 However, this method seems to be inferior to the LCP-algorithm developed
above, particularly in terms of speed.
0
Denote vi,j,` = v` (aj , bi ). The algorithm works as follows. Start with an initial guess vi,j,`
n
and for n = 1, 2, ... compute vi,j,`
as follows
n+ 1
n
1. given vi,j,`
, obtain vi,j,`2 by solving a discretized HJB equation that is the same as if
there were no adjustment decision:
n+ 1
n
vi,j,`2 − vi,j,`
n+ 1
n+ 1
n+ 1
+ ρvi,j,`2 =u(cni,j,` ) + ∂a vi,j,`2 ra aj + ∂b vi,j,`2 (wz` + rb bi − cni,j,` )
∆
+ λ` (vi,j,k − vi,j,` ),
n
).
cni,j,` =(u0 )−1 (∂b vi,j,`
1
Note that this splitting method has nothing to do with the “splitting the drift” trick in http://www.
princeton.edu/~moll/HACTproject/two_asset_nonconvex.pdf.
3
for ` = 1, 2 and k 6= `. Here ∂a vi,j,` and ∂b vi,j,` denote the finite-difference approximation of the partial derivative of v with respect to a and b (either forward- or backwarddifference approximations), and where ∆ is the size of the updating step. In particular,
one can write the corresponding system of equations
1
1 n+ 1
v 2 − v n + ρv n = un + Av n+ 2
∆
(8)
where v n is a vector of length M = I × J × 2 with the stacked value function as
its entries, and A is the M × M transition matrix corresponding to the discretized
process summarizing the evolution of (at , bt , zt ). See http://www.princeton.edu/
~moll/HACTproject/HACT_Numerical_Appendix.pdf for details in a similar model
(but with one asset only).
2. Compute
n+ 12
1
∗
(vi,j,`
)n+ 2 = max
v`
0 0
a ,b
n+ 12
where v`
(a0 , b0 ) s.t. a0 + b0 = aj + bi − κ,
a0 ≥ 0, b0 ≥ 0.
n+ 1
(a0 , b0 ) is vi,j,`2 interpolated at points (a0 , b0 ).
n+ 1
3. given vi,j,`2 by setting
n+ 1
1
n+1
∗
)n+ 2 }.
vi,j,`
= max{vi,j,`2 , (vi,j,`
(9)
n+1
n
4. If vi,j,`
is “close to” vi,j,`
, stop.
The algorithm is called a “splitting algorithm” because the “operator” on v defined by (6)
1
1
is split into two steps: that of going from v n to v n+ 2 and that of going from v n+ 2 to v n+1 . It
can be shown that the algorithm converges if the discretization of the HJB equation in step
1 satisfies the monotonicity, consistency and stability conditions of Barles and Souganidis
(1991). See Achdou et al. (2014) for a discussion in the context of a model without a
stopping-time decision. See e.g. Barles, Daher and Romano (1995) and Tourin (2013) for a
discussion of convergence of numerical schemes for models with a stopping-time decision.
For future reference, denote by a0i,j,` = a0` (aj , bi ) and b0i,j,` = b0` (aj , bi ) the optimal “adjustment targets” conditional on adjustment. Also denote by m0 (m) the corresponding mapping
in the stacked and discretized state space. Finally denote by ι(m) an indicator for whether
1
n+1
∗
a given grid point is in the adjustment region (which happens whenever vi,j,`
= (vi,j,`
)n+ 2 ).
4
2.3
Kolmogorov Forward Equation
[THIS IS OUTDATED AND MAY HAVE TO BE UPDATED] Instead of solving the KF
equation directly, it turns out to be much easier to work with the discretized process as
captured by the matrix A constructed in section 3.2. See section 2 here http://www.
princeton.edu/~moll/HACTproject/numerical_MATLAB.pdf for a more detailed explanation of this logic in a model without an adjustment decision.
If there were no adjustment, things would be very simple. In particular, the stacked finite
difference of g, gm , i = 1, ..., M where M = I × J × 2 would satisfy a linear system
0 = AT g
where AT is the transpose of the transition matrix A from the HJB equation (8). To deal
with the fact that there is entry and exit, define a modified transition matrix C as follows.
In particular, the elements of C, denoted by Cm,k for m = 1, ..., M and k = 1, ..., M are
given by
Cm,k


0,



P
= Am,k + ι(k)=1 Am,k ,



A ,
m,k
if ι(k) = 1
if ι(k) = 0,
m0 (m) = k
if ι(k) = 0,
m0 (m) 6= k
(10)
The first row says that points m and k that lie in the adjustment region are never reached.
The second row says that instead the corresponding elements are moved to the corresponding
adjustment targets. The third row says that the transition matrix is unchanged if the points
m and k lie in the non-adjustment region and are not the target of an adjustment decision.
P
One can check that the modified transition matrix C satisfies M
k=1 Cm,k = 0 (just like A).
Finally we set
Cm,m = −
1
,
∆
if ι(m) = 1
(11)
To see why the −1/∆ term shows up, consider the time-discretized process for g
ġt = CT gt ,
⇒
gt+∆t = (∆C + I)T gt
where I is the identity matrix. The transition matrix C̃ = (∆C + I) needs to have all entries
in the adjustment region C̃mm = 0 and hence ∆Cmm + 1 = 0. Without the adjustment (11),
the matrix C is singular.
5
3
Results
Figure 1 plots the “adjustment region”, i.e. the region of the state space (a, b) in which
individuals adjust their portfolios. The adjustment region is in yello and the non-adjustment
region is in blue.
Low Productivity
6
5
Illiquid Wealth (a)
5
Illiquid Wealth (a)
High Productivity
6
4
3
2
1
4
3
2
1
0
0
0
0.5
1
1.5
2
2.5
3
0
0.5
Liquid Wealth (b)
1
1.5
2
2.5
3
Liquid Wealth (b)
Figure 1: Adjustment Regions
References
Achdou, Yves, Jean-Michel Lasry, Pierre-Louis Lions, and Benjamin Moll. 2014.
“Heterogeneous Agent Models in Continuous Time.” Princeton University Working Papers.
Barles, G., and P. E. Souganidis. 1991. “Convergence of approximation schemes for fully
nonlinear second order equations.” Asymptotic Analysis, 4: 271–283.
Barles, G., Ch. Daher, and M. Romano. 1995. “Convergence Of Numerical Schemes
For Parabolic Equations Arising In Finance Theory.” Mathematical Models and Methods
in Applied Sciences, 05(01): 125–143.
Kaplan, Greg, and Gianluca Violante. 2014. “A Model Of The Consumption Response
To Fiscal Stimulus Payments.” Econometrica. forthcoming.
Kaplan, Greg, Benjamin Moll, and Giovanni L. Violante. 2016. “Monetary Policy
According to HANK.” National Bureau of Economic Research Working Paper 21897.
Tourin, Agnes. 2013. “An Introduction to Finite Difference Methods for PDEs in Finance.”
In Optimal Stochastic Target problems, and Backward SDE. Fields Institute Monographs,
6
, ed. Nizar Touzi. Springer, available online at http://papers.ssrn.com/sol3/papers.
cfm?abstract_id=2396142.
7