Computation of Equilibria in OLG Models with Many
Heterogeneous Households
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Citation
Rausch, Sebastian, and Thomas F. Rutherford. “Computation of
Equilibria in OLG Models with Many Heterogeneous
Households.” Computational Economics 36.2 (2010) : 171-189.
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http://dx.doi.org/10.1007/s10614-010-9229-8
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Springer Netherlands
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Author's final manuscript
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Thu May 26 23:27:44 EDT 2016
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http://hdl.handle.net/1721.1/67034
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Detailed Terms
Computation of Equilibria in OLG Models with
Many Heterogeneous Households ∗
Sebastian Rausch†
Joint Program on the Science and Policy of Global Change, Massachusetts Institute of Technology
Thomas F. Rutherford
Swiss Federal Institute of Technology (ETH), Zürich
May 2010
(shortened version for re-submission to Computational Economics)
Abstract
This paper develops a decomposition algorithm by which a market economy with many households may be solved through the computation of equilibria for a sequence of representative agent
economies. The paper examines local and global convergence properties of the Sequential Recalibration (SR) algorithm. The SR algorithm is then demonstrated to efficiently solve AuerbachKotlikoff OLG models with a large number of heterogeneous households. We approximate equilibria in OLG models by solving a sequence of related Ramsey optimal growth problems. This
approach can provide improvements in both efficiency and robustness as compared with integrated complementarity-based solution methods.
Keywords: Computable general equilibrium; Overlapping generations; Microsimulation; Sequential recalibration
JEL Classification: C68; C81; D61; D91
∗ Without
implication, we would like to thank Volker Clausen and participants at the 2007 Far Eastern Meeting of Econo-
metric Society for helpful comments and discussion. Financial support by the Alfried Krupp von Bohlen und Halbach
Foundation and the Ruhr Graduate School in Economics is gratefully acknowledged.
† Contact
information: rausch@mit.edu or Joint Program on the Science and Policy of Global Change, Massachusetts
Institute of Technology, 77 Massachusetts Ave, Bldg E19-411, Cambridge, MA 02139, USA.
1
Introduction
Over the past twenty years infinite horizon general equilibrium models with overlapping generations
(OLG) have become an important tool for policy analysis, and these have been fruitfully applied in
fields such as macroeconomics and public finance (see, e.g., Auerbach and Kotlikoff [1987], and Kotlikoff [2000] for an overview). OLG models naturally involve a large number of variables and equations that describe the equilibrium behavior of economic agents. As a consequence, the development of large-scale OLG models is often limited by the computational capacity of available numerical solution methods. In particular, models that exhibit a rich household side including a variety
of household-specific effects, a large number of heterogeneous households, and realistic agent lifetimes typically require “customized solution methods” which may be both costly to implement and
difficult to validate.
This paper develops a decomposition algorithm based on “off the shelf” numerical tools1 for solving general equilibrium models with many households, of which OLG models are a special case. The
presented approach is primarily appropriate for computing equilibria in models in which the number of agents is so large that simultaneous solution methods that operate directly on the equilibrium
system of equations are infeasible due to the high dimensionality related to income and householdspecific effects. The proposed “Sequential Recalibration” (SR) algorithm is based on the solution of
a sequence of nonlinear complementarity problems.2
The key idea of the SR algorithm is to solve a market economy with many households through
the computation of equilibria for a sequence of representative agent economies. In an OLG context,
the SR algorithm approximates the equilibrium allocation by solving a sequence of Ramsey optimal
growth problems. To overcome dimensionality issues, the numerical problem is decomposed into
two sub-problems. First, a “related” Ramsey optimal growth problem, which is constructed by replacing the OLG demand system with an infinitely-lived representative agent, is solved to obtain a
candidate equilibrium price vector. Second, candidate prices are used to solve a partial equilibrium
relaxation of the OLG economy by simply evaluating demand functions of OLG households. Our iterative procedure between the general equilibrium and household model is based on a successive
recalibration of preferences of the representative agent, and ensures that prices and quantities converge to the true equilibrium allocation of the OLG economy.
The close connection between the allocation of a competitive market economy and the optimal
solution to a representative agent’s planning problem is well known and widely cited in the eco1
GAMS code for the presented applications is available at http://www.mpsge.org.
2
In special cases the same procedure may be implemented by solving a sequence of convex nonlinear programming
problems. Rutherford [1995b] and Mathiesen [1985] have shown that a complementary-based approach is convenient,
robust, and efficient. A characteristic of many economic models is that they can be cast as a complementarity problem.
The complementarity format embodies weak inequalities and complementarity slackness, relevant features for models
that contain bounds on specific variables, e.g. activity levels which cannot a priori be assumed to operate at positive
intensity. Such features are not easily handled with alternative solution methods.
1
nomic literature. The use of an optimization problem to characterize equilibrium allocations in a
general equilibrium framework goes back to Negishi [1960]. Negishi’s original paper was primarily
concerned with optimization as a means of proving existence. Dixon [1975] developed the theory
and computational effectiveness of “joint maximization algorithms” for multi-country trade models.
Rutherford [1999] presented the “sequential joint maximization algorithm” (SJM) which provides a
simple recursive version of Negishi’s method. Similar to the SJM algorithm, the SR approach solves
subproblems representing relaxations of the equilibrium conditions. The SR algorithm employs a
loose representation of individual consumer’s demand systems by omitting both income constraints
and global properties of the individual utility functions. The omission of global characteristics of
preferences considerably simplifies the model but can also hinder convergence.
This paper demonstrates that the SR algorithm can be used to effectively solve large-scale Auerbach-Kotlikoff OLG models.3 We consider a prototype Auerbach-Kotlikoff model which includes a
large number of heterogeneous households within each generation that differ with respect to labor
productivity over the life cycle and other behavioral parameters. We compare the performance of
the SR algorithm with an integrated complementarity-based solution method by Rasmussen and
Rutherford [2004]. We find that the SR approach can provide improvements in both efficiency and
robustness, and demonstrate that it can routinely solve high-dimensional OLG models.
The rest of the paper is organized as follows. Section 2 introduces the SR algorithm for the case of
a static economy and illustrates its basic logic by means of graphical analysis. Section 3 develops a
local convergence theory for a simple exchange economy, and uses numerical methods to characterize global convergence properties. Section 4 applies the SR algorithm to solve a prototype AuerbachKotlikoff OLG model, and evaluates its performance relative to an integrated complementarity-based
solution method. Section 5 concludes.
2
CGE with Many Households: A Decomposition Approach
This section presents the decomposition algorithm by which a market economy with many heterogeneous households may be solved through the computation for equilibria for a sequence of representative agent economies. While the primary interest is in dynamic models, it is advantageous to
introduce the algorithm for the case of a simple static economy with two factors of production and
two commodities for which we can provide a graphical description to illustrate the basic logic.
3
Rutherford and Tarr [2008] employ the SR algorithm to incorporate all 55,000 households from the Russian Household
Budget Survey in a static computable general equilibrium (CGE) model to assess the impact of Russia’s WTO accession
on income distribution and the poor.
2
2.1
A Simple Static Economy
Consider the following static economy which is populated by a large number of heterogeneous households h = 1, . . . , H that are endowed with K h and L h units of capital and labor, respectively. Households earn income M h = r K h + w L h from supplying their factor endowments inelastically at respective market prices r and w . The utility maximization problem for household h is given by

2
X h
max U h (ch ) = 
θi
h h
c 1 ,c 2
i =1
c ih
c hi
!ρ h 1/ρ h


s .t .
2
X
p i c ih = M h ,
i =1
where utility is represented by a constant elasticity of substitution (CES) function which is written in
calibrated share form [Rutherford, 1995a]. θih and c hi denote the observable benchmark value share
and the observable benchmark consumption of good i for household h. Households are heterogeneous with respect to θih , ρ h , K h , and L h .
The production side is described by a representative firm which uses capital and labor services to
produce two consumption goods, denoted by X i , i , j = 1, 2, according to a constant returns to scale
production function. All goods and factor markets are perfectly competitive.
2.2
The Sequential Recalibration Algorithm
The main challenge for computing equilibria in a model in which H is very large is dimensionality.
Simultaneous solution methods that operate directly on the system of equations that defines the
general equilibrium can become infeasible if the number of variables and equations is too large. To
overcome restrictions of dimensionality, the proposed algorithm decomposes the numerical problem into two subproblems and employs an iterative procedure between them to find the general
equilibrium of the underlying model.
More specifically, the first subproblem solves a representative agent version of the underlying
model which is obtained by replacing the heterogeneous households by a single representative agent
(RA). Hence the H utility maximization problems as described in (2.1) are replaced by the following
optimization problem:

ρ 1/ρ
2
X
C

 i 
max U k (C) = 
Θki  k  
C 1 ,C 2
Ci
i =1

s .t .
2
X
i =1
pi Ci = w
H
X
h=1
Lh + r
H
X
Kh.
h=1
k
Note that upper case letters refer to variables associated with the RA. Θki and C i denote the benchmark value share and consumption of good i , respectively. k represents an iteration index.
The second subproblem solves a partial equilibrium relaxation of the underlying model by evaluating household demand functions taking equilibrium prices from the first subproblem as given.
Prices from the RA problem and households’ quantity choices from the second subproblem typically
3
do not represent a general equilibrium allocation as we have suppressed interactions between producers and “real” households. Thus, an iterative procedure between both subproblems is needed to
reconcile both sides of the economy.
The key departure from the routine use of calibration in our decomposition algorithm is the idea
that the calibration of preferences occurs more than once. The first iteration of the algorithm is based
on observable benchmark data, but in subsequent iterations the preferences of the RA are sequentially recalibrated to values determined in the iterative process. More specifically, in each iteration we
re-benchmark preferences of the RA such that—given candidate prices—the consumption demand
by the RA is equal to the sum of individual household demands. The algorithm is termed “Sequential
Recalibration (SR)” on the basis of this idea.
More specifically, the SR algorithm comprises the following iterative procedure.
SR Algorithm :
1. Given benchmark data on household demand c hi and prices p i , initialize the representative
agent model such that consumption demand by the representative agent (RA) is consistent with
the aggregate of benchmark household demands, i.e.
0
Ci =
H
X
c hi
h=1
0
Θ0i
=P
pi Ci
0
i0 pi0 C i0
,
i 6= i 0 ,
0
where C i and Θ0i denote initial consumption by the RA and the aggregate value share of good i
in iteration k = 0, respectively.
2. Solve the RA model for a generic economic policy, i.e., an exogenous shock which changes relative
prices, and obtain a candidate equilibrium price vector pk .
3. Solve a partial equilibrium relaxation of the underlying model by evaluating household demand
functions c ih (pk , M hk ).
4. Recalibrate preferences U k (C) of the RA based on households’ quantity choices by updating benchmark consumption
k +1
Ci
=
X
c ih (pk , M hk )
h
and benchmark value shares
Θki +1
P
p ik h c ih (pk , M hk )
=P kP h
.
k
k
j pj
h c j (p , M h )
5. Exit if stopping rule (e.g., 1-norm of price differences between one iteration and the next) is satisfied, else go to Step 2.
4
Graphical Illustration
2.2.1
This section provides a graphical illustration for each of the steps outlined in the preceding description of the SR algorithm. For ease of exposition, we adopt the example of the static economy presented in Section 2.1.4
0
Step 1. The initial consumption point C i of the RA in the benchmark equilibrium is represented by point A in Figure 1 where initial goods prices are denoted by P 0 . Benchmark prices and an
arbitrary elasticity are used to extrapolate preferences in the neighborhood of the benchmark point
to the global preferences of the RA, as indicated by the indifference curve which is tangent to the
benchmark budget constraint at point A. The key limitation of the RA model on the demand side is
that the “community indifference curve” represented by this indifference curve does not truthfully
portray the response of household demand to a comprehensive change in both goods and factor
prices.
Step 2. Now consider an exogenous policy shock that results in increased factor earnings and
a reduction in the price of X 1 relative to X 2 . This new price situation is denoted by P 1 . The RA
model, based on the assumed community indifference curve and the associated change in factor
and commodity prices returns point B as the optimal consumption point.
Step 3. In the solution program we read equilibrium prices from the RA model and evaluate
the household demand vector. This produces a different point on the same budget constraint. The
household demand model is based on compensated demand functions so the aggregate budget constraint for the household demand system is equivalent to the budget constraint which applies to the
RA. Hence point C corresponds to aggregated individual household demand obtained for the price
situation P 1 . The extent to which C differs from B depends on both the difference in implicit substitution elasticities and differences in income effects.
Step 4. The next step in the algorithm consists of specifying a different set of preferences for
the RA model. After having solved one RA model we construct a new RA model based on a set of
preferences which are locally calibrated to the aggregate consumption quantities at point C and the
associated relative prices. This ensures that given prices P 1 the optimal consumption point C of the
new RA is consistent with the aggregated choices by households. Preferences of the RA in iteration
k , are based on household demands at the prices returned in iteration k − 1. The indifference curves
tangent at A and B are based on identical preferences, but the indifference curve tangent at point C is
based on a new set of community indifference curves, hence it may intersect the indifference curves
from the previous iteration. Note that the preferences of the “real” households are never altered.
Step 5. When the RA model is recalibrated at point C, both the representative agent and all households are in equilibrium at C with prices P 1 , but at these prices firms will only supply quantities
4
The application to a framework with many goods and factors of production is straightforward. For empirical applications see Rutherford and Tarr [2008] and Rausch [2009].
5
X2
C
A
B
P0
P1
X1
FIGURE 1: STEPS 1-4 IN SR ALGORITHM
given by point B—hence production plans at point C are not consistent with a general equilibrium.
To illustrate this point, it is convenient to portray the supply side of the economy by a production
possibility frontier (PPF). Assume that the policy shock produces an expansion from PPF to PPF 0
and a substantial change in relative prices from point A to B. The next step in the solution procedure
is to compute the general equilibrium of a new RA model with recalibrated preferences at point C.
Point C in Figure 2 becomes therefore interpreted as point A in the next iteration. The solution of this
RA model is then characterized by a new optimal consumption point, here depicted by point D, and
prices P 2 .
Subsequent iterations involve carrying out Steps 2 to Steps 4 (Step 1 solely initializes the solution
procedure). We stop if some convergence metric, e.g., the 1-norm of the difference between the price
vectors from one iteration to the next, is satisfied. Note that subsequent iterations of the algorithm
only involve refinements of the demand system and result in much smaller changes in relative prices,
as indicated here by the change from C to D as compared with A to B.
3
Convergence Properties
This section evaluates the performance of the algorithm for an economy in which the exact equilibrium is known and where the computed allocation can be compared to the true equilibrium allocation. We develop local convergence theory for the proposed algorithm and also examine conditions
under which the adjustment process may fail to converge.
Note that our objective is not to develop a comprehensive convergence theory for the SR algorithm. We rather aim to demonstrate by counterexample that the SR algorithm belongs to a large
class of algorithms commonly used in computational economics which are robust, efficient and yet
fail to provide global convergence.
6
X2
C
D
A
B
PPF
PPF'
P1
P2
X1
FIGURE 2: ITERATIVE ADJUSTMENT (STEP 5)
To exemplify, we consider a pure exchange economy due to Scarf [1960] which is characterized by
an equal number of n consumers and goods. Consumer h is endowed with one unit of good h and
demands only goods h and h + 1. Let d i ,h denote demand for good i by consumer h. Preferences are
represented by CES utility functions with the following structure
‹ σ−1

σ
σ
σ
σ−1
σ−1
+ (1 − θ ) d h+1,h
.
Uh (d ) = θ d h,h
Scarf [1960] demonstrates that this economy has a unique equilibrium in which all prices are equal
to unity.5
3.1
Local Convergence
As explained in Section 2.2, the SR algorithm iteratively adjusts the preference parameters C i and
P
Θi (C i ) in the utility function of the representative agent. These may be normalized so that i C i = n.
Market clearing commodity prices for the RA economy are determined given the baseline level parameters. Let p i (C ) denote the price of good i consistent with C = (C 1 , . . . ,C n ). In this exchange econ€
Š
omy, let ζi p (C ) denote the market excess demand function for good i that is obtained from evaluating household demand functions. Given the special structure of preferences and endowments,
this function has the form
€
Š
ζi p (C ) = d i ,i + d i ,i −1 − 1 .
€ Š
€ Š
Furthermore, let ξi C denote the value of market excess demand for good i at prices p i (C ): ξi C =
€
Š
∗
p i (C ) ζi p (C ) . Of course, in equilibrium it must be true that ξi (C ) = 0, ∀ i . Let the initial estimate
P 0
0
C be selected on the n-simplex, i.e.
C i = n . Walras’ law ensures that the adjustment process
iP
Š
€
Š
€
P
d
d Ci
i C i (t )
= ξi C remains on the n -simplex
= i p i (C (t )) ζi p (C (t )) = 0.
dt
dt
5
See Lemma 1 and Lemma 2 [Scarf, 1960, p.164]. The parameters of this utility function correspond to Scarf’s parame1
b
ters a and b [Scarf, 1960, p.168] as: σ = 1+a
and θ = 1+b
.
7
Local convergence concerns properties of the Jacobian matrix evaluated at the equilibrium point,
” —
∗
∇ξ(C ) = ξi j . This Jacobian has entries defined as follows

€
Š
∂ ξi
ξi j ≡
∗
∂ Cj
”
∂ ζi p (C )
=


i =j
∂€C i Š
∂ ζi p (C )
.
i 6= j
∂ Cj
—
If all principal minors of ∇ξ(C ) = ξi j are negative, the adjustment process is locally convergent.
If, however,
€
Š
∂ ζi p (C )
∂ Ci
> 0, the process is locally unstable. When an equilibrium is unique and the
process is uninterrupted, then local stability implies global stability.
For this model, the tâtonnement price adjustment process is unstable (in the case of n = 3) when
θ
1−θ
>
1
1−2σ
[Scarf, 1960]. In the following, we show that the same condition implies instability for
the adjustment process of the SR algorithm. Furthermore, it is shown numerically that while the
tâtonnement and SR price adjustment processes are locally identical, they may be quite different at
points in the price space far from the equilibrium.
€
Š
Given the special structure of ζi p (C ) , we have
∂ d i ,i ∂ d i ,i −1
∂ ζi
=
+
,
∂ pi
∂ pi
∂ pi
∂ ζi
∂ d i ,i −1
=
,
∂ p i −1
∂ p i −1
∂ ζi
∂ d i ,i
=
.
∂ p i +1 ∂ p i +1
€
Š 1
1−σ
.
Defining the “unit-utility” expenditure function for consumer i as e i (p ) = θ p i1−σ + (1 − θ ) p i1−σ
+1
Demand functions are then given by
d i ,i =
Evaluating
Š
€
∂ ζi p (C )
∂ Ci
at p ∗ = 1, yields
Š
€
∂ ζi p (C )
θ pi
,
1−σ σ
ei pi
d i +1,i =
(1 − θ ) p i
e i1−σ p iσ+1
.
Š
∂ pi €
(2σ − 2)θ 2 + (3 − 2σ)θ − 1
∂ Ci
∂ Ci
∂ p i +1
(−θ (1 − θ )(1 − σ))
+
∂ Ci
∂ p i −1
(−θ (1 − θ )(1 − σ) + 1 − θ ) .
+
(1)
∂ Ci
€
Š
€
Š
e p,C = 0 where ζ
e p,C denotes the vecThe function p (C ) is defined implicitly by the equation ζ
=
tor of market excess demand functions from the representative agent economy. Its i -th element is
€
Š
ei p,C = P C i −1 with αi 0 = PC i and where σ
e denotes the elasticity of substitugiven by ζ
e
1−σ
e
σ
i0
αi 0 p
i0
pi
i0 C i0
e at C ∗ , we make a first-order Taylor series
tion for the representative agent. In order to evaluate ∇p ζ
expansion
€
Š
€
Š
e p,C d p + ∇ ζ
e p,C d C = 0
∇p ζ
C
which gives




e
∇p ζ = 


dp
dC
∗
∗
e
e
= −∇−1
p ζ ∇C ζ . Evaluating gradients at p = C = 1 yields



e
−(2σ+1)
3
e
−(1−σ)
3
e
−(1−σ)
3
e
−(1−σ)
3
e
−(2σ+1)
3
e
−(1−σ)
3
e
−(1−σ)
3
e
−(1−σ)
3
e
−(2σ+1)
3





−1 e 
 , −∇p ζ = 




8
e
σ+2
e
3σ
e
σ−1
e
3σ
e
σ−1
e
3σ
e
σ−1
e
3σ
e
σ+2
e
3σ
e
σ−1
e
3σ
e
σ−1
e
3σ
e
σ−1
e
3σ
e
σ+2
e
3σ



 1 0 0 






e
 , ∇C ζ =  0 1 0  .






0 0 1
p3
•
p1
p2
FIGURE 3: COMPARISON OF SR AND TÂTONNEMENT FIELDS
Hence, in the neighborhood of the equilibrium
∂ p i (C )
∂ Ci
=
e +2
σ
> 0,
e
3σ
∂ p i (C )
∂ Cj
= 0.
From (1) it therefore follows that the adjustment process in C is locally unstable if
(2σ − 2)θ 2 + (3 − 2σ)θ − 1 > 0
(2)
which is equivalent to the condition for instability of a simple price tâtonnement adjustment process
as demonstrated by Scarf [1960, p.169].
3.2
Global Convergence
Although the local behavior of the price tâtonnement and the SR algorithm adjustment processes
are identical, they produce different search directions away from a neighborhood of the equilibrium.
This is apparent in Figure 3 where the two vector fields are superimposed. Only local to the equilibrium where price effects dominate income effects do the two fields coincide exactly, as indicated by
the stability condition (2). As one moves further away from the center of the simplex, the vector fields
become more divergent. We find that there are cases in which the SR algorithm does not converge
even though the price tâtonnement is globally stable. This convergence failure is a manifestation
of the simplifying nature of the adjustment process. By solving a sequence of representative agent
economies the SR algorithm omits both income constraints and global properties of the individual
utility functions. While the omission of global characteristics of preferences reduces the dimensionality of the model significantly, this may at the same time hinder convergence.
9
16
12
8
4
non-convergent
2.00
1.75
1.50
1.25
s
1.00
0.75
0.50
0.25
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
q
FIGURE 4: GLOBAL CONVERGENCE BEHAVIOR FOR DIFFERENT CONFIGURATIONS OF θ
AND
σ
To assess the global convergence properties of the SR algorithm, we perform a grid search over
the behavioral parameters σ and θ . We let the algorithm start from a disequilibrium point p =
(0.2, 0.2, 0.6) where local equilibrium dynamics are absent. Figure 4 reveals that convergence of the
SR algorithm fails for combinations of small values for σ and high values for θ .6 For these parameter configurations, income effects are relatively strong vis-à-vis substitution effects. In cases where
convergence is achieved, the presence of significant income effects means that more iterations are required to find the true equilibrium. If, however, income effects are relatively weak, the SR algorithm
only requires a modest number of iterations. The appropriateness of the presented solution method
therefore depends on the characteristics of the underlying model.
One last remark is in order. To guarantee convergence of the SR algorithm, it is necessary to select
e the elasticity of substitution of the representative agent. If σ
e is too low,
a sufficiently large value for σ,
convergence may fail even if income effects are relatively weak. Non-convergent behavior, however,
e The choice of σ
e is
that occurs in the bottom right corner of Figure 4 is robust with respect to σ.
entirely innocuous since this parameter bears no economic significance for the behavior of “real”
households in the underlying economic model. Computational experience suggests to use values of
e ≥ 1.
order σ
4
Solving OLG Models with Many Households
This section presents a decomposition approach for solving overlapping generations models with
many heterogeneous households which relies on the SR algorithm presented in the last section. We
approximate the equilibrium allocation of an OLG economy by computing equilibria for a sequence
6
e = 1, and allow for a maximum of 1000 iterations.
For both parameters, we choose a grid resolution of 0.05, set σ
The adjustment process is said to converge if the 1-norm of differences between a computed price vector and the
equilibrium point drops below some metric δ, i.e. kp i −p i∗ k1 < δ, where p i∗ denotes the analytical equilibrium solution.
We set δ = 0.01.
10
of related Ramsey growth problems which are re-benchmarked in each iteration based on an evaluation of OLG household demand. Figure 5 provides a schematic exposition of the steps involved in
the decomposition procedure. Below we provide detail on each of the involved steps specific to the
OLG context.
The algorithm is demonstrated for a simple prototype Auerbach-Kotlikoff OLG economy with
production activities, intra-cohort heterogeneity, a labor-leisure choice, and a government sector.
The otherwise standard OLG model is augmented by introducing many heterogenous households
with in each age group. Intra-cohort heterogeneity refers to differences in labor productivity and the
intertemporal elasticity of substitution. A full description of the OLG model can be found in Rausch
and Rutherford [2008].7 We solve for the effects of a tax reform that is introduced unexpectedly in
year zero, and evaluate the performance of the SR algorithm against an integrated complementaritybased approach as presented in Rasmussen and Rutherford [2004].
4.1
Decomposition of the OLG Problem
We decompose the underlying OLG economy into the following two sub-problems: (1) a related Ramsey growth problem, and (2) a partial equilibrium relaxation of OLG demand.
1. Related Ramsey Growth Problem. We define the related Ramsey growth problem as a lowdimensional variant of the underlying OLG economy obtained by replacing the OLG demand system
with a single infinitely-lived Ramsey agent. For the OLG economy described in Rausch and Rutherford [2008], the related Ramsey growth problem is given by:8
b
σ
1−1/σb  σ−1
b
T

X
 Zt 

max U (Z t ) = 
Θkt  k 


C t , Lt
Zt
t =0



ν  ν1
€
Š
 Lt  
  Ct 
= ∆kt  k  + 1 − ∆kt  k  
Lt
Ct

s .t .
Zt
Ct + It + Gt
K t +1
Lt
C t , Lt

=
ν
F (K t , Ωkt − Lt )
≤ (1 − δ) K t + I t
≤ Ωkt
≥ 0
K0
≤
K0
K T +1
=
K̂ T +1 .
(3)
7
While we investigate a single-sector closed economy here, the logic can be readily extended to a multi-region and
multi-sector framework.
8
We write the nested lifetime utility function U in calibrated share form [Rutherford, 1995a]. We monotonically transform preferences for OLG households to obtain a linear homogenous CES representation. This does not alter the
underlying preference orderings and hence optimization yields the same demand functions.
11
Here k denotes an iteration index, and C t , Lt , Z t , K t , I t , G t , and Ωkt denote consumption, leisure
time, full consumption, the capital stock, investment, government demand, and the time endowb ν , Θk , and
ment, respectively. Variables with a bar denote the respective reference parameters. σ,
∆k denote the intertemporal elasticity of substitution, a labor supply elasticity parameter, and share
parameters, respectively.
Step 2:
Evaluate household
demand functions
Step 1:
Solve related Ramsey
growth problem
Step 3:
Recalibrate preferences
of the Ramsey agent
FIGURE 5: SOLVING OLG BY RAMSEY:
Partial equilibrium relaxation of
the underlying OLG model
Ramsey growth problem of the
underlying OLG model
General equilibrium prices
Partial equilibrium
quantity choices
STEPS IN THE DECOMPOSITION ALGORITHM
To approximate the infinite-horizon Ramsey economy by a finite-dimensional complementarity problem defined over t = 0, . . . , T , we use state-variable targetting to endogenously solve for the
post-terminal capital stock (K̂ T +1 ) such that that investment grows at the steady-state rate in the last
period [Lau, Pahlke, and Rutherford, 2002].
2. Partial Equilibrium Relaxation of OLG Demand. Given CES preferences for OLG households,
we can obtain closed-form demand functions for generation g and household type h:
€
Š
x g ,h,t = D Ψ g ,h , p
where x represents a vector of demands for consumption (c ) and leisure (`), Ψ g ,h a vector of household parameters, and p a vector of candidate equilibrium prices obtained from the related Ramsey
growth problem in (3).9 Assuming a steady-state closure, we can evaluate demand functions for terminal households, i.e. households living beyond T , by using a steady state projection of terminal
period prices.
3. Updating the related Ramsey growth problem. In each iteration, we recalibrate preferences of
the Ramsey agent in (3) based on quantity choices by OLG households. Re-benchmarking involves
updating:
k +1
Ct
=
t
H
X
X
c g ,h,t (pkg ,h , M kg ,h )
g =t −N h=1
9
For an explicit representation of the derived OLG demand functions see Rausch and Rutherford [2008].
12
(4)
k +1
Lt
=
t
H
X
X
` g ,h,t (pkg ,h , M kg ,h )
(5)
g =t −N h=1
k
k
k
Zt =Ct +L t
(6)
k
∆kt +1
=
p ck,t C t
k
k
Θkt +1
=
(7)
k
p ck,t C t + p lk,t L t
k
p ck,t C t + p lk,t L t
,
P
k
k
k
k
C
L
+
p
p
0
0
0
0
0
t
t
t
l ,t
c ,t
(8)
where p denotes respective present value prices obtained from the Ramsey model, M g ,h is lifetime
income, and N + 1 denotes the number periods a generation lives in the model. The aggregate time
endowment Ωkt +1 is updated based on aggregate productivity-weighted labor supply and aggregate
leisure time. Correct initialization of the related Ramsey growth problem therefore requires calibrating (4)-(8) based on baseline values for OLG demand, and setting the initial aggregate capital stock
P0
PH
equal to initial aggregate capital in the OLG model, i.e. K 0 = g =−N h=1 k g ,h,0 , where k g ,h,0 denotes capital holdings of households alive in t = 0.10
4.2
Algorithmic Performance
The OLG economy presented above has no analytical solution. In order to evaluate the algorithm,
we therefore compare its performance to an integrated complementarity-based solution method by
Rasmussen and Rutherford [2004].
4.2.1
Illustrative Application: Fundamental Tax Reform
This section presents an illustrative application of the decomposition algorithm which solves for the
effects of a policy change that in year zero unexpectedly and permanently reduces the capital income
tax and introduces a consumption tax to endogenously balance the government budget. The capital
income tax is reduced from a benchmark value of 28.4% to 22.9%.
In the numerical analysis, we test the performance of the algorithm for a different number of
household types H and also allow for various degrees of intra-cohort heterogeneity. For simplicity,
we assume that σh is generated by random draws from a uniform distribution with support [σ, σ].
Likewise, differences in labor productivity are modeled by randomly drawing a g ,h from a uniform
distribution with support [a , a ].11
10
See Rasmussen and Rutherford [2004] for a balanced growth path calibration procedure specific to the OLG context. A
non-steady state calibration for OLG modes is more involved, and discussed in Wendner [1999].
11
Detailed information about the values we use to parameterize the household and production side of economy can be
found in in Rausch and Rutherford [2008] as well as in the computer programs for this paper which are available at
http://www.mpsge.org.
13
8
0.015
7.5
0.012
Iteration 2
1-norm of price changes
Deviation from steady state (in %)
Iteration 1 (= Ramsey)
Iteration 3
4
5
7
6
7
...
6.5
0.009
0.006
0.003
OLG = Iteration 34
6
0
20
40
60
80
100
120
0
140
1
Time period
FIGURE 6: SOLVING OLG BY RAMSEY:
5
10
15
20
25
30
Iteration
SEQUENCE OF INVESTMENT TIME PATHS AND APPROXIMATION ERROR
We start out by considering a case where H = 1, σh = 1.2, and a g ,h = 0.04. Figure 6 shows the
sequence of time paths for investment that emerges from the iterative procedure of the algorithm.
The true transition path to a new steady state of the OLG economy as computed by the benchmark
Rasmussen and Rutherford [2004] method is labeled “OLG”. The curve "Iteration 1" refers to the impact of the tax reform scenario after the first iteration of the SR algorithm. The observed impact
is equivalent to what would be obtained from a Ramsey growth model. Each subsequent iteration
of the algorithm produces a new time path for investment that eventually converges to the true solution. We terminate the search process if kp ck,t −p c∗,t k1 < 10−6 . For the current model this convergence
tolerance implies 34 iterations until the search process is interrupted.
We use the following two measures to assess the quality of approximation. First, the approximation error e k reports the 1-norm of differences between computed consumption prices and true
equilibrium prices as calculated by the benchmark method: e k = kp ck,t − p c∗,t k1 . As e k constitutes a
summary statistic which is defined over the entire model horizon, it says little about whether price deviations of the computed from the true price path lie within a tolerable bandwidth. As a second measure, we therefore report the maximum distance error τk which is defined as: τk = max{|p ck,t − p c∗,t |}.
Figure 6 plots e k as a function of the number of iterations. The approximation error quickly
decreases and then converges to zero. After the first few iterations the decomposition technique
only involves refinements of the demand system, and consequently, subsequent changes in relative
prices are small.
4.2.2 Robustness and Accuracy
In order to explore the capacity of the algorithm to solve large-scale OLG models, we examine its
performance for a different number of household types and various degrees of intra-cohort heterogeneity. We look again at the effects of the tax reform scenario as described above. We vary the degree
of heterogeneity, which we denote by Γ, by changing the support for the distributions from which σh
14
TABLE 1: CONVERGENCE PERFORMANCE AND APPROXIMATION ERROR
H
(Γ = Γ1 )
Number of iterations
Approx. error e k
(last iteration)
Max. distance τk
(last iteration)
CPU computing
time
1
10
50
100
500
1000
2000
37
36
35
35
34
34
29
10−6
0.002
0.005
—
—
—
—
10−7
10−4
10−4
—
—
—
—
0 min 13 s (3.78)
0 min 21 s (1.17)
1 min 10 s (0.18)
2 min 06 s ( × )
6 min 14 s ( × )
10 min 48 s ( × )
30 min 31 s ( × )
Note: Figures in parentheses denote the running time of the SR algorithm expressed as a fraction of the running time of
the benchmark integrated solution method. A “×” indicates that the benchmark method is infeasible.
TABLE 2: CONVERGENCE BEHAVIOR IN THE PRESENCE OF STRONG INCOME EFFECTS
ν,α
(for H = 50)
Number of iterations
Approx. error e k
(last iteration)
Max. distance τk
(last iteration)
CPU computing
time
0.50 , 0.80
0.25 , 0.80
0.80 , 0.50
0.80 , 0.25
0.50 , 0.50
0.25 , 0.25
33
30
42
53
42
47
0.015
0.035
0.014
0.042
0.117
0.171
10−3
10−3
0.013
0.037
0.011
0.015
1 min 06 s (0.17)
1 min 05 s (0.18)
1 min 27 s (0.19)
1 min 33 s (0.25)
1 min 15 s (0.23)
1 min 42 s (0.25)
Note: Figures in parentheses denote the running time of the SR algorithm expressed as a fraction of the running time of
the benchmark integrated solution method.
and a g ,h are drawn.12
Table 1 reports results from a series of runs where the number of households within each generation is increased while holding fixed the degree of intra-cohort heterogeneity. The quality of
approximation is excellent (τk is around 10−4 ). As the number of household types increases, the
proposed decomposition procedure become advantageous.13 Most importantly, it is shown that the
algorithm can provide improvements in robustness as compared to the benchmark simultaneous
solution method which becomes infeasible for models in which H ≥ 100.14
To examine the performance of the algorithm in the presence of a substantial degree of heterogeneity among households, we report results for different configurations of Γ. We set H = 50 so that
the benchmark solution method is feasible and the calculation of approximation errors is available.
Not surprisingly, the approximation quality of the method is decreasing in the degree of heterogeneity. The overall quality of approximation, however, is still execllent: computed prices fall within a
12
We consider the following sets of choices for (σ, σ), (a , a ) in ascending order of their implied degree of heterogeneity:
Γ1 = {(1.00, 1.50), (0.2, 0.3)}, Γ2 = {(1.00, 1.50), (0.2, 0.4)}, Γ3 = {(0.25, 0.75), (0.2, 0.3)}, Γ4 = {(0.25, 0.75), (0.2, 0.4)}, Γ5 =
{(0.25, 1.25), (0.2, 0.3)}, Γ6 = {(0.25, 1.25), (0.2, 0.4)}, Γ7 = {(0.25, 2.00), (0.2, 0.3)}, Γ8 = {(0.25, 2.00), (0.2, 0.4)}.
13
All reported running times refer to an implementation on a 2 GHz processor machine.
14
If the number of agents is small, the SR algorithm is slightly more costly in terms of computing time as the number
of outer iterations is relatively large. We do not view this as a drawback of the proposed method which is designed to
solve models with many households.
15
TABLE 3: APPROXIMATION ERRORS FOR DIFFERENT Γ
Γ
(H = 50)
Number of iterations
Approx. error e k
(last iteration)
Max. distance τk
(last iteration)
CPU computing
time
Γ1
Γ2
Γ3
Γ4
Γ5
Γ6
Γ7
Γ8
35
35
79
74
58
55
41
37
0.005
0.005
0.001
0.004
0.005
0.011
0.011
0.021
10−4
10−4
10−5
10−4
10−4
10−3
10−3
10−3
1 min 10 s (0.18)
0 min 56 s (1.14)
2 min 14 s (0.39)
2 min 23 s (0.42)
2 min 14 s (0.29)
1 min 44 s (0.26)
1 min 10 s (0.18)
1 min 15 s (0.22)
Note: Figures in parentheses denote the running time of the SR algorithm expressed as a fraction of the running time of
the benchmark integrated solution method.
reasonably small interval around the true equilibrium price path (τk is around 10−5 − 10−3 ).15
5
Concluding Remarks
This paper develops a decomposition approach which can be applied to solve high-dimensional
static and dynamic general equilibrium models with many households. We demonstrate its effectiveness for computing equilibria in high-dimensional OLG models which are infeasible for existing
complementarity-based integrated methods. We find that the proposed algorithm provides an efficient and robust way to approximate general equilibrium in models with a large number of heterogeneous agents if income effects remain sufficiently weak. The appropriateness of the solution method
therefore depends on the characteristics of the underlying model and the nature of the implemented
policy shock.
We believe that the approach can be beneficial for a wide range of economic applications. In
particular, it is advantageous for modeling tasks which necessitate to economize on the dimensionality of the corresponding numerical problem. Potential applications may include multi-country and
multi-sectoral OLG models, and analyses of relevant policy issues—such as, e.g., population aging,
trade policy, and poverty—which require detailed account of the distributional effects on a household level while at the same time taking into account general equilibrium effects. Moreover, the
decomposition approach may prove useful for the further development of fully-integrated static and
dynamic microsimulation models that incorporate the essential micro-macro linkages required for
a comprehensive policy analysis.
15
Motivated by the discussion of the potential convergence failure of the SR algorithm in the presence of significant income effects in Section 3, we also conduct a number of sensitivity analyses for key behavioral parameters governing
intra-period and intertemporal substitution and income effects (see Table 2). Looking first at the intra-period dimension, we find that combinations of too small ν and α can pose serious problems for the decomposition approach.
Although the search process is terminated within a modest number of iterations, both approximation measures indicate a rather poor quality of approximation for α ≤ 0.5. If α is too small, the equilibrium behavior over the life cycle
of a household displays periods with zero labor supply in old ages, i.e. there is endogenous retirement. This happens
because the shadow price of time exceeds the market wage rate. In the presence of such corner solutions, it is harder to
portray the choices of OLG households by using a representative agent which in turn explains why the approximation
error increases.
16
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17