Computation of Equilibria in OLG Models
with Many Heterogeneous Households
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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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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 house-holds 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 Recal-ibration (SR) algorithm. The SR algorithm is then demonstrated to efficiently solve Auerbach-Kotlikoff OLG models with a large number of heterogeneous households. We approximate equi-libria 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 inte-grated complementarity-based solution methods.
Keywords: Computable general equilibrium; Overlapping generations; Microsimulation;
Sequen-tial 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: [email protected]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 Kot-likoff[2000] for an overview). OLG models naturally involve a large number of variables and
equa-tions that describe the equilibrium behavior of economic agents. As a consequence, the develop-ment of large-scale OLG models is often limited by the computational capacity of available numer-ical 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 life-times 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 tools1for
solv-ing 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 num-ber 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 household-specific 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 re-placing 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 it-erative 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 con-verge 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
eco-1 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] andMathiesen[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.
nomic literature. The use of an optimization problem to characterize equilibrium allocations in a general equilibrium framework goes back toNegishi[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 Auer-bach-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. Section2introduces the SR algorithm for the case of a static economy and illustrates its basic logic by means of graphical analysis. Section3develops a local convergence theory for a simple exchange economy, and uses numerical methods to character-ize global convergence properties. Section4applies the SR algorithm to solve a prototype Auerbach-Kotlikoff OLG model, and evaluates its performance relative to an integrated complementarity-based solution method. Section5concludes.
2 CGE with Many Households: A Decomposition Approach
This section presents the decomposition algorithm by which a market economy with many hetero-geneous households may be solved through the computation for equilibria for a sequence of repre-sentative 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.
2.1 A Simple Static Economy
Consider the following static economy which is populated by a large number of heterogeneous house-holds h= 1,...,H that are endowed with Khand Lh units of capital and labor, respectively. House-holds earn income Mh= r Kh+ w Lhfrom supplying their factor endowments inelastically at respec-tive market prices r and w . The utility maximization problem for household h is given by
max ch 1,ch2 Uh(ch) = 2 X i=1 θh i cih chi !ρh 1/ρh s .t . 2 X i=1 picih= Mh,
where utility is represented by a constant elasticity of substitution (CES) function which is written in calibrated share form[Rutherford,1995a]. θihand chi denote the observable benchmark value share and the observable benchmark consumption of good i for household h. Households are heteroge-neous with respect toθih,ρh, Kh, and Lh.
The production side is described by a representative firm which uses capital and labor services to produce two consumption goods, denoted by Xi, 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 prob-lem into two subprobprob-lems 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: max C1,C2 Uk(C) = 2 X i=1 Θk i Ci Cki ρ 1/ρ s .t . 2 X i=1 piCi = w H X h=1 Lh+ r H X h=1 Kh.
Note that upper case letters refer to variables associated with the RA.Θki andCki denote the bench-mark 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 eval-uating 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
do not represent a general equilibrium allocation as we have suppressed interactions between pro-ducers 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 sequen-tially 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 chi and prices pi, 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.
C0i = H X h=1 chi Θ0 i = piC0i P i0pi0C 0 i0 , i6= i0,
where C0i 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 cih(pk, Mk h).
4. Recalibrate preferences Uk(C) of the RA based on households’ quantity choices by updating
bench-mark consumption
Cki+1=X
h
cih(pk, Mkh)
and benchmark value shares
Θk+1 i = pikP hchi (pk, Mhk) P jpkj P hchj(pk, Mkh) .
5. Exit if stopping rule (e.g., 1-norm of price differences between one iteration and the next) is
2.2.1 Graphical Illustration
This section provides a graphical illustration for each of the steps outlined in the preceding descrip-tion of the SR algorithm. For ease of exposidescrip-tion, we adopt the example of the static economy pre-sented in Section2.1.4
Step 1. The initial consumption point C0i of the RA in the benchmark equilibrium is repre-sented by point A in Figure1where initial goods prices are denoted by P0. 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 X1 relative to X2. This new price situation is denoted by P1. 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 con-straint for the household demand system is equivalent to the budget concon-straint which applies to the RA. Hence point C corresponds to aggregated individual household demand obtained for the price situation P1. The extent to which C differs from B depends on both the difference in implicit
substi-tution 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 P1the 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
house-holds are in equilibrium at C with prices P1, 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 applica-tions seeRutherford and Tarr[2008] andRausch[2009].
A C P1 P0 X2 X1 B
FIGURE1: STEPS1-4INSR 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 PPF0
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 Figure2becomes 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 P2.
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 equilib-rium is known and where the computed allocation can be compared to the true equilibequilib-rium alloca-tion. 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 algo-rithm. 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.
B A C P1 X2 X1 D PPF' PPF P2
FIGURE2: ITERATIVE ADJUSTMENT(STEP5)
To exemplify, we consider a pure exchange economy due toScarf[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 di ,hdenote demand for good i by consumer h. Preferences are represented by CES utility functions with the following structure
Uh(d ) = θ dσ−1σ h,h + (1 − θ )d σ σ−1 h+1,h σ−1σ .
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 Section2.2, the SR algorithm iteratively adjusts the preference parameters Ci and Θi(Ci) in the utility function of the representative agent. These may be normalized so that PiCi = n. Market clearing commodity prices for the RA economy are determined given the baseline level pa-rameters. Let pi(C ) denote the price of good i consistent with C = (C1, . . . ,Cn). In this exchange econ-omy, letζi
p(C )denote the market excess demand function for good i that is obtained from eval-uating household demand functions. Given the special structure of preferences and endowments, this function has the form
ζi
p(C ) = di ,i+ di ,i−1− 1 .
Furthermore, letξi
Cdenote the value of market excess demand for good i at prices pi(C ): ξiC = pi(C )ζi
p(C ). Of course, in equilibrium it must be true thatξi(C
∗
) = 0, ∀i . Let the initial estimate
C0be selected on the n -simplex, i.e. Pi C0i = n. Walras’ law ensures that the adjustment process
d Ci
d t = ξi
Cremains on the n -simplexd
P iCi(t )
d t = Pi pi(C (t ))ζi
p(C (t )) = 0.
5 See Lemma 1 and Lemma 2[Scarf,1960, p.164]. The parameters of this utility function correspond to Scarf’s parame-ters a and b[Scarf,1960, p.168] as: σ = 1
1+aandθ = b 1+b.
Local convergence concerns properties of the Jacobian matrix evaluated at the equilibrium point, ∇ξ(C∗) =ξi j. This Jacobian has entries defined as follows
ξi j≡ ∂ ξ i ∂ Cj = ∂ ζi p(C ) ∂ Ci i= j ∂ ζi p(C ) ∂ Cj i6= j .
If all principal minors of∇ξ(C∗) =ξi jare negative, the adjustment process is locally convergent. If, however, ∂ 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.
Given the special structure ofζi p(C ), we have ∂ ζi ∂ pi = ∂ di ,i ∂ pi + ∂ di ,i−1 ∂ pi , ∂ ζi ∂ pi−1 = ∂ di ,i−1 ∂ pi−1 , ∂ ζi ∂ pi+1 = ∂ di ,i ∂ pi+1.
Defining the “unit-utility” expenditure function for consumer i as ei(p) =θ pi1−σ+ (1 − θ )pi1−σ+1
1 1−σ.
Demand functions are then given by
di ,i = θ pi ei1−σpiσ, di+1,i = (1 − θ )pi ei1−σpσi+1. Evaluating∂ ζi p(C ) ∂ Ci at p ∗= 1, yields ∂ ζi p(C ) ∂ Ci = ∂ pi ∂ Ci (2σ − 2)θ2+ (3 − 2σ)θ − 1 +∂ pi+1 ∂ Ci (−θ (1 − θ )(1 − σ)) +∂ pi−1 ∂ Ci (−θ (1 − θ )(1 − σ) + 1 − θ ) . (1) The function p(C ) is defined implicitly by the equation eζp,C = 0 whereζe
p,Cdenotes the vec-tor of market excess demand functions from the representative agent economy. Its i -th element is given by eζi p,C = Ci P i 0αi 0p1−ei 0σ pσe i −1 with αi0=PCi i 0Ci 0
and whereσ denotes the elasticity of substitu-e tion for the representative agent. In order to evaluate∇pζ at Ce
∗
, we make a first-order Taylor series expansion ∇pζe p,Cd p+ ∇Cζe p,Cd C= 0 which givesd p d C = −∇ −1 p ζ∇e Cζ. Evaluating gradients at pe ∗= C ∗ = 1 yields ∇pζ =e −(2eσ+1) 3 −(1−σ)e 3 −(1−eσ) 3 −(1−eσ) 3 −(2eσ+1) 3 −(1−eσ) 3 −(1−eσ) 3 −(1−σ)e 3 −(2eσ+1) 3 ,−∇−1p ζ =e e σ+2 3eσ σ−1e3σe e σ−1 3eσ e σ−1 3eσ σ+2e3σe e σ−1 3eσ e σ−1 3σe e σ−1 3σe e σ+2 3σe ,∇Cζ =e 1 0 0 0 1 0 0 0 1 .
•
p3
p1 p2
FIGURE3: COMPARISON OFSRAND TÂTONNEMENT FIELDS
Hence, in the neighborhood of the equilibrium
∂ pi(C ) ∂ Ci = eσ + 2 3σe > 0, ∂ pi(C ) ∂ Cj = 0.
From (1) it therefore follows that the adjustment process inC is locally unstable if
(2σ − 2)θ2+ (3 − 2σ)θ − 1 > 0 (2)
which is equivalent to the condition for instability of a simple price tâtonnement adjustment process as demonstrated byScarf[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 Figure3where the two vector fields are superimposed. Only local to the equilib-rium 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 dimension-ality of the model significantly, this may at the same time hinder convergence.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 4 8 12 16 q s non-convergent
FIGURE4: GLOBAL CONVERGENCE BEHAVIOR FOR DIFFERENT CONFIGURATIONS OFθANDσ
To assess the global convergence properties of the SR algorithm, we perform a grid search over the behavioral parametersσ and θ . We let the algorithm start from a disequilibrium point p = (0.2,0.2,0.6) where local equilibrium dynamics are absent. Figure4reveals that convergence of the SR algorithm fails for combinations of small values forσ and high values for θ .6 For these
parame-ter 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 re-quired 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 a sufficiently large value forσ, the elasticity of substitution of the representative agent. Ife σ is too low,e convergence may fail even if income effects are relatively weak. Non-convergent behavior, however, that occurs in the bottom right corner of Figure4is robust with respect toσ. The choice ofe σ ise 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 orderσ ≥ 1.e
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 For both parameters, we choose a grid resolution of 0.05, set e
σ = 1, and allow for a maximum of 1000 iterations. 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 metricδ, i.e. kpi−p∗ik1< δ, where pi∗denotes the analytical equilibrium solution. We setδ = 0.01.
of related Ramsey growth problems which are re-benchmarked in each iteration based on an evalu-ation of OLG household demand. Figure5provides 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 inRausch 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 complementarity-based approach as presented inRasmussen 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 Ram-sey 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
low-dimensional 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 inRausch and Ruther-ford[2008], the related Ramsey growth problem is given by:8
max Ct,Lt U(Zt) = T X t=0 Θk t Zt Zkt 1−1/bσ b σ b σ−1 s .t . Zt = ∆ k t Ct Ckt ν + 1− ∆kt Lt Lkt ν 1 ν Ct+ It+Gt = F (Kt,Ωkt − Lt) Kt+1 ≤ (1 − δ) Kt+ It Lt ≤ Ωkt Ct,Lt ≥ 0 K0 ≤ K0 KT+1 = ˆKT+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 trans-form 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.
Here k denotes an iteration index, and Ct,Lt, Zt, Kt, It, Gt, andΩkt denote consumption, leisure time, full consumption, the capital stock, investment, government demand, and the time endow-ment, respectively. Variables with a bar denote the respective reference parameters. σ, ν, Θb
k, and ∆k denote the intertemporal elasticity of substitution, a labor supply elasticity parameter, and share parameters, respectively.
General equilibrium prices
Ramsey growth problem of the underlying OLG model
Partial equilibrium relaxation of
the underlying OLG model
Partial equilibrium quantity choices Step 1:
Solve related Ramsey growth problem
Step 3:
Recalibrate preferences of the Ramsey agent
Step 2:
Evaluate household demand functions
FIGURE5: SOLVINGOLGBYRAMSEY:STEPS IN THE DECOMPOSITION ALGORITHM
To approximate the infinite-horizon Ramsey economy by a finite-dimensional complementar-ity problem defined over t = 0,...,T , we use state-variable targetting to endogenously solve for the post-terminal capital stock ( ˆKT+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:
xg ,h,t = DΨg ,h, p
where x represents a vector of demands for consumption (c ) and leisure (`), Ψg ,ha vector of house-hold 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
ter-minal households, i.e. households living beyond T , by using a steady state projection of terter-minal 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: Ckt+1= t X g=t −N H X h=1 cg ,h,t(pkg ,h, Mkg ,h) (4)
Lkt+1= t X g=t −N H X h=1 `g ,h,t(pkg ,h, Mkg ,h) (5) Zkt = Ckt + Lkt (6) ∆k+1 t = pk c ,tC k t pk c ,tC k t + p k l ,tL k t (7) Θk+1 t = pk c ,tC k t + p k l ,tL k t P t0 pkc ,t0C k t0+ pkl ,t0L k t0 , (8)
where p denotes respective present value prices obtained from the Ramsey model, Mg ,h is lifetime income, and N+ 1 denotes the number periods a generation lives in the model. The aggregate time endowmentΩkt+1is updated based on aggregate productivity-weighted labor supply and aggregate leisure time. Correct initialization of the related Ramsey growth problem therefore requires calibrat-ing (4)-(8) based on baseline values for OLG demand, and setting the initial aggregate capital stock equal to initial aggregate capital in the OLG model, i.e. K0= P0g=−NPHh=1kg ,h,0, where kg ,h,0 de-notes 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 thatσh is generated by random draws from a uniform distribution with support[σ,σ]. Likewise, differences in labor productivity are modeled by randomly drawing ag ,h from a uniform distribution with support[a,a].11
10 SeeRasmussen 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 inWendner[1999].
11 Detailed information about the values we use to parameterize the household and production side of economy can be found in inRausch and Rutherford[2008] as well as in the computer programs for this paper which are available at http://www.mpsge.org.
6 6.5 7 7.5 8 0 20 40 60 80 100 120 140
Deviation from steady st
ate (in %) Time period Iteration 1 (= Ramsey) Iteration 2 Iteration 3 4 5 6 7 OLG = Iteration 34 ... 0 0.003 0.006 0.009 0.012 0.015 1
1-norm of price changes
Iteration
5 10 15 20 25 30
FIGURE6: SOLVINGOLGBYRAMSEY:SEQUENCE OF INVESTMENT TIME PATHS AND APPROXIMATION ERROR
We start out by considering a case where H = 1, σh = 1.2, and ag ,h = 0.04. Figure6shows 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
im-pact of the tax reform scenario after the first iteration of the SR algorithm. The observed imim-pact 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 solu-tion. We terminate the search process ifkpk
c ,t−p∗c ,tk1< 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 approxi-mation error ek reports the 1-norm of differences between computed consumption prices and true equilibrium prices as calculated by the benchmark method: ek = kpk
c ,t− pc ,t∗ k1. As ek constitutes a
summary statistic which is defined over the entire model horizon, it says little about whether price de-viations of the computed from the true price path lie within a tolerable bandwidth. As a second mea-sure, we therefore report the maximum distance errorτk which is defined as:τk= max{|pk
c ,t− pc ,t∗ |}. Figure6 plots ek 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 hetero-geneity. We look again at the effects of the tax reform scenario as described above. We vary the degree of heterogeneity, which we denote byΓ, by changing the support for the distributions from which σh
TABLE1: CONVERGENCE PERFORMANCE AND APPROXIMATION ERROR H Number of iterations Approx. error ek Max. distanceτk CPU computing (Γ = Γ1) (last iteration) (last iteration) time
1 37 10−6 10−7 0 min 13 s (3.78) 10 36 0.002 10−4 0 min 21 s (1.17) 50 35 0.005 10−4 1 min 10 s (0.18) 100 35 — — 2 min 06 s (× ) 500 34 — — 6 min 14 s (× ) 1000 34 — — 10 min 48 s (× ) 2000 29 — — 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.
TABLE2: CONVERGENCE BEHAVIOR IN THE PRESENCE OF STRONG INCOME EFFECTS
ν , α Number of iterations Approx. error ek Max. distanceτk CPU computing
(for H= 50) (last iteration) (last iteration) time
0.50 , 0.80 33 0.015 10−3 1 min 06 s (0.17) 0.25 , 0.80 30 0.035 10−3 1 min 05 s (0.18) 0.80 , 0.50 42 0.014 0.013 1 min 27 s (0.19) 0.80 , 0.25 53 0.042 0.037 1 min 33 s (0.25) 0.50 , 0.50 42 0.117 0.011 1 min 15 s (0.23) 0.25 , 0.25 47 0.171 0.015 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 ag ,hare drawn.12
Table1reports results from a series of runs where the number of households within each gen-eration 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.13Most 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 hetero-geneity among households, we report results for different configurations ofΓ. 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 heterogene-ity. 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.
TABLE3: APPROXIMATION ERRORS FOR DIFFERENTΓ Γ Number of iterations Approx. error ek Max. distanceτk CPU computing (H= 50) (last iteration) (last iteration) time
Γ1 35 0.005 10−4 1 min 10 s (0.18) Γ2 35 0.005 10−4 0 min 56 s (1.14) Γ3 79 0.001 10−5 2 min 14 s (0.39) Γ4 74 0.004 10−4 2 min 23 s (0.42) Γ5 58 0.005 10−4 2 min 14 s (0.29) Γ6 55 0.011 10−3 1 min 44 s (0.26) Γ7 41 0.011 10−3 1 min 10 s (0.18) Γ8 37 0.021 10−3 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 effec-tiveness 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 effi-cient and robust way to approximate general equilibrium in models with a large number of heteroge-neous 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 dimension-ality 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 house-hold 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 in-come effects in Section3, we also conduct a number of sensitivity analyses for key behavioral parameters governing intra-period and intertemporal substitution and income effects (see Table2). Looking first at the intra-period dimen-sion, we find that combinations of too smallν 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 indi-cate a rather poor quality of approximation forα ≤ 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.
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