an approximation technique for computing optimal dynamic paths

Journal of Economic Dynamics and Control 11 (1987) 405-423. North-Holland

AN APPROXIMATION TECH N IQ U E FOR C O M P U T I N G OPTIMAL DYNAMIC PATHS*

Hector SIERRA

Analytic Concepts, Inc.

Timothy CONDON

The World Bank, Washington, DC 20433, USA

Received November 1985, final version received April 1987

This paper describes a method for approximating optimal dynamic paths when model dimensionality prohibits the solution of the exact optimal trajectory. The method reduces the dimensionality, and hence the computational cost, of the problem by approximating the 'key' relationships of the model. The methodology is tested by computing the actual and the approximated optimal borrowing paths using a dynamic computable general equilibrium model of Thailand.

1. Introduction

The ability to solve complex, non-linear dynamic models numerically is relatively recent. In his discussion of development planning models, Taylor (1975) lists only a handfull of such models. Recent attempts have taken advantage of advances in computing software to examine such issues as optimal borrowing [Martin and Selowsky (1984), Kharas and Shishido (1985)] and the optimal policy response to oil windfalls [Gelb (1985b)].

The solution of dynamic optimizing models, however, remains something of an art and requires an intensive use of computing resources. Moreover, software advances have not reached the point where large models can be solved. Non-linear. models typically contain several sectors and are solved for about as many periods) Compared to the size of dynamic linear-programming models, dynamic non-linear optimizing models are quite small. Computing software and cost continue to be important constraints on the type of questions one can address in applied dynamic optimizing models.

This paper presents an approximation technique for overcoming the size constraint. The technique is based on solving an approximated problem. This

*The World Bank does not accept responsibility for the views expressed herein which are those of the authors and should not be attributed to the World Bank or to its affiliated organizations.

1Gelb (1985b) solves a six-sector computable general equilibrium model for ten periods, Kharas and Shishido (1985) a five-sector model for seven periods.

0165-1889/87/$3.50©1987, Elsevier Science Publishers B.V. (North-HoUand)

406 H. Sierra and T. Condon, Computing optimal dynamic paths

stands in contrast to other approximation techniques, linearization for example, that approximate the relationships in the underlying model, then solve the exact optimization problem. The shortcomings of the linearization approach are well known. 2 By approximating the problem on the 'objective function' space instead of the 'model' space, the dimensionality, and hence the computational cost of the problem is substantially reduced. The results of using this approach suggest that in general it will be sufficient to find an 'almost' optimal path by solving a much simpler approximated problem.

In the following section we outline with a simple example the main approach of the proposed method. Section 3 presents the general formulation and the theory underlying the approach. Section 4 then outlines the method of approximating the relevant functions. In section 5 the methodology is applied to a dynamic computable general equilibrium (CGE) model originally developed by Kharas and Shishido (1985). Section 6 presents the results of a sensitivity analysis experiment and is followed by a concluding section.

2. Approximating the optimal path: An example

Insight into the method may be gained by a simple example based on the neoclassical one-sector growth model. Consider the following problem in which investment resources are allocated over time, t. The objective is to maximize a welfare function, F:

T

m a x F = E U(Ct)( 1 + P ) - t + ~ r + l K r + l , (1) t ~ 0

where C represents consumption, U is the utility function, and K is the capital stock. Apart from the discounted utility of consumption, the final value of capital stock has been added to the maximand to reflect the fact that life does not cease after the time horizon, T. 3 It is assumed that U(-) is twice continuously differentiable, and U' > 0, U" < 0, the latter reflecting declining marginal within period utility. The following constraints, representing the macroeconomic relations of the model, must also be satisfied:

Q,=I~+C, ,

Qt---Q(Lt, Kt, t),

L t = t t ,

Kt+ 1 = (1 - d ) K t + l t , K o = K 0.

(2)

(3)

(4)

(5)

2See Baumol (1982) for an account of the operational and technical difficulties that arise when linearization is applied.

3See Taylor (1975) for a discussion of terminal conditions in dynamic models.

H. Sierra and T. Condon, Computing optimal dynamic paths 407

The material balance equation (2) states that, in equilibrium, gross output must be equal to total final demand. Qt and 1 t denote, respectively, output and investment levels. The production function (3) depends on capital, labor, and time (technical progress). The full employment assumption is represented in (4). Finally, given some initial capital stock, capital accumulation (5) is set equal to investment minus depreciation, where the parameter d accounts for the rate of depreciation of the stock of capital.

Given an initial value of the capital stock we can solve for the welfare-maxi- mizing level of investment and the capital stock, that also satisfy the accounting relation (5). The complexity of the problem increases rapidly with the number of sectors. Each variable contains not only a time subscript, but also a sector subscript, and intersectoral linkages arise. Adding n sectors increases the dimensionality of the problem by at least a multiple of n. In the case of computable general equilibrium models, the fact that prices as well as quanti- ties have to be simultaneously determined compounds the dimensionality problem. Interesting multi-sectoral problems can easily become too large even for numerical analysis and one may begin to search for ways to simplify the problem.

In general, the complexity of the optimization problem arises from the non-linearities of the underlying economic model. Thus, eq. (3), representing production technology, may be highly non-linear and costly to compute for a large number of sectors. If, for every period t, we are able to find a good approximation Q, of the output level Qt, we may solve an approximated problem in which eq. (3) is replaced by

Q.t=~_(Lt, Kt, t). (3')

The approximated problem retains the same dimensionality^ as the original, but it is preferable if it is less costly to compute (3') and if Q is a reasonable approximation of Q. Most approximation methods use this approach by linearizing all or part of the non-linear equations in the model, thus allowing the application of powerful linear programming techniques. As Baumol (1982) points out, however, there are problems associated with this approach.

If we perform some simple substitutions we may reformulate the optimization problem as follows:

T

m a x F = E U [ Q ( g t , Zt, l ) - I t ] ( l + p ) - t + ~ T + l g T + l , (6) t=0

subject to

K , + I = ( 1 - d ) K , + I ', K o = K o. (7)

The problem has been formulated as a typical control-theory problem. In the terminology of control theory, the variables in a problem are divided into

408 H. Sierra and 7". Condon, Computing optimal dynamic paths

those which represent the condition or state of the objective functional (the so-called state variables), and those which guide or control the state variables (the so-called control variables). As formulated in (6), the problem has only one state variable, the capital stock, and one control variable, investment.

The two optimization problems (1) and (6) are equivalent, except that they have been formulated differently. Notice, however, that if in (6) we approximate U(-) in terms of I and K, we solve a much smaller problem than the original since during the optimization we do not have to compute the values Qt, Lt and Cr In an n-sector model the dimensionality of the problem is reduced by a factor of 3n.

In contrast to the techniques that approximate the technological relationship in (3), the technique developed in this paper directly focuses on obtaining an approximation of the objective function (6). That is, the approximation is made on the 'objective function' space, rather than on the 'model' space. This enables us to solve a model similar to (6):

T

m a x F = E ~ ( K , , I,)(1 + p)-'+~T+xKT+I, (8) t = 0

subject to (7). The difference between (6) and (8) is that in (8) the utility function is approximated by ~ , which depends only on the control variable 1, and its associated state variable g r The underlying economic model is used to generate the information necessary to compute polynomials to approximate U(.). Part of the problem is to find the 'best' functions to approximate the relevant relationships, that is the relationships between the maximand and the control and state variables.

In the next section we discuss in more precise terms, and for the general case, the implications of solving an 'approximate' problem in which the original maximand has been replaced by an approximation. Later, the method is applied to a dynamic computable general equilibrium model of Thailand.

3. General formulation and theory

Consider a general structural macroeconomic model that can be represented in discrete time as a system of simultaneous non-linear equations m(.). Such a model is intended to explain the determination of an n-vector of current endogenous variables, y, given a vector of predetermined or exogenous variables, x. We assume there are n behavioral equations, each representing a specified functional relationship between y and x involving known parameters contained in the /-vector a. In the typical formulation, observations corresponding to distinct points in time, t, are obtained for y and x, and the model can be represented as a system of equations,

y ,=

H. Sierra and T. Condon, Computing optimal @namic paths 409

Let x = (x,) represent the vectors of exogenous variables, y = ( y , ) the vectors of endogenous variables, and a = (at) the vector of parameters over the period t = 1, 2, . . . , T. Suppose that we are interested in solving the following dynamic optimization problem, denoted as (P):

(P) maximize F(x, y) such that

y=m(x la ), x ~ X ,

where F is a real-valued continuous function of x and y with x taking values on the set X. If we apply this formulation to the example in the last section, the x-variables correspond to investment and the capital stock, 1 = (I,), K = ( K t ) , with I, K ~ R r+!. The y-variables correspond, respectively, to gross output Q, consumption C, and labor L, which are computed by solving system (2)-(4) over the period 1,2 . . . . . T, for a given value of x = (1, K). The feasible set X includes all the non-negative vectors ! and K for which the capital accumulation relation, eq. (5) in the last section, is satisfied. In general the set X on which the vector x takes values will include the equations of motion of the variables x, and other constraints imposed on the control variables. Finally, the maximand F includes the welfare summation and the terminal value of the capital stock.

For simplicity, we assume that X is a compact set and that there is at least one feasible solution to problem (P). If m(-[a) is continuous on the set X for a given set of parameters a, these assumptions imply that there is always a solution to (P).

If we use the relationship y = m(xla), then the problem above may be transformed into an equivalent problem:

(P) mardmize f (x )=F(x ,m(x la) ) , such that

X E X,

where f is also a continuous function the computation of which requires solving the model y = m(xla) for the period 1, 2, . . . , T.

Many of the techniques seeking to simplify (P) compute an approximation rh of the model m(.). The original problem (P) can then be reformulated as

(~') maxJmize f ( x ) = F ( x , rh(xia)), such that

X E X,

where f is identical to f , but m(.) has been replaced by the approximation rh(.). If f is a continuous approximation of f on X then, under the


assumptions that X is non-empty and compact, there is also a solution to problem (P). Thus, the existence of a solution is invariant under continuous approximations.

In general, the goodness of the approximation technique is determined by estimating a measure of the discrepancy between the original and the approximated models. For example, if I1" II is some vector norm, then the value of

[[m(x[a) - r~(xla) [1

over the set X provides a criterion of the closeness of the approximation. (If the model has been linearized this will be a measure of the non-linearities of the model being approximated.) Note, however, that a close approximation of m is sufficient but not necessary for solving (P). A good approximation refers to how 'close' f is from f on X. If the model is approximated so that f is 'close' to f on X, then a solution of (P) is 'almost' a solution of the original problem (P). To make this statement more precise, define:

Definition 1. Let x* be an optimal solution of (P). Then we say that x ~ X is e-optimal if we have If(x*) - f ( x ) [ < e.

Given this definition it is now easy to prove the following proposition:

Proposition 1. Let e= maxx~xl f (x ) - f ( x ) [ . Then, if ~* solves (P), ~* is 2e-optimal.

This proposition shows that, for the purpose of solving (P), it is enough to concentrate on the discrepancy between f and f on X, not between 6z and m. This is an important point since, as we shall see later, it will allow us to simphfy the approximation procedure by directly concentrating on the estimation of the function f and not on the entire set of model equations in m. Clearly, the value of e for which a solution of (P) is 'almost' a solution of (P), varies from problem to problem and will depend on the units in which f is measured and on the specific criterion adopted by the modeller.

In practice, it is very difficult to obtain an accurate estimate of e, which represents a global measure of discrepancy between f and the original f . If the function f is 'smooth' and depending on the cost, an estimate of e may be computed by randomly ~lecting values x over the set X and then comparing the values of f ( x ) and f(x) . In general, however, given the cost incurred in computing the model in every period, we will be able to compare the values of f and f at only one point, namely, ~* the solution of the simpler problem (P). The question then is: What information is gained by comparing the actual and the approximated values of the objective function at the point ~*, the optimal solution of (P)?

4 1 1

i ii

f

^

f ° - -

f

for"

1t. Sierra and T. Condon, Computing optimal dynamic paths

X*

Fig. 1

~(° X

If the objective function f evaluated at the optimal path is larger than the approximated optimal value of (P), then we have

f * = max f> f (~*) > f * = mEa~/. x E X

In this case the actual optimal, denoted as f*, is at least as large as the approximated optimal, but we do not know how far, or close, f* lies from f(~*). If, however, the approximate optimal f* is larger than the actual optimal f*, then it is possible to determine how far the approximated solution lies from f* by looking at the difference,

],

Formally, this argument is stated in the following proposition:

Proposition 2. Let ~* be an optimal solution of ( P) and suppose f* = maxx ~ x f > maxx ~ x f . Then if f* - f ( ~*) < e, the solution ~* is e-optimal.

A geometrical representation of the proposition is shown in fig. 1. Thus, according to these results, we cannot really know where the optimal solution lies unless we obtain an estimate of the global discrepancy e, or by solving both (P) and (P), either of which may be an infeasible or expensive option. If,

J.E.D.C.-- E

412 H. Sierra and 7". Condon, Computing optimal dynamic paths

however, f (~*) is very close to f*, then the optimal value of f on X will provide a lower bound for the optimal value of f on the same set X.

Let us suppose, for example, that the model m is initially evaluated at some reference or historical path. That is, suppose that the value of f(.~) is known, where .~ ~ X represents the historical path over the period 1, 2 . . . . , T. Suppose also that an approximation f of f on X has been computed which satisfies the condition

f ( x ) = f ( x ) .

If f and f coincide at the base path ~, it is not hard to show that this implies f* > f (~) , and if f* is close to f(~*), then we will generally be able to improve on the historical performance. If, on the other hand, f* and f(~*) are not 'close' enough, then a refinement of the approximation may be introduced. In the next section we concentrate on computing approximations of f that satisfy the condition f ( ~ ) = f ( ~ ) .

A more general problem than the one originally proposed may have the form

(P) maximize F(x, y), such that

y=m(xla), x~X, y~Y,

where Y is a set that represents constraints imposed on the vector of state variables y. We still may be able to apply the propositions above if the constraint y ~ Y can be substituted out by adding a 'penalty' function to the objective function F. For example, suppose that y ~ Y is of the form

g(y) = b ,

where g(.) is some continuous function and b is a vector of parameters. Then, the following term can be added to F:

M(g(y) - b ) z,

where M is a large negative number. In this case, the vector of y-variables in the approximate solution of the problem will 'approximately' satisfy the constraint y ~ Y.

4. Approximating the objective function

In the last section the advantage of having a good approximation of the objective function f ( . ) - without necessarily having to approximate the ex-


plicit relationships between endogenous and exogenous variables - was estab- lished. The success of the approach will depend on how well we are able to approximate the objective function f.

For a given continuous function f, defined in some set X, the central problem of all approximation and interpolation techniques is to find an approximating function depending on a fixed finite number of parameters. Thus, consider a real and continuous function f(x) and an approximating function f(A, x) which may depend on the parameters A in a linear or non-linear way. The objective is then to find those parameters A* which make f(A*, x) closest to f(x) by some measure of closeness. Much has been written in the literature about the existence, uniqueness, and characteristic properties of the solution of the problem. [See, e.g., Rice (1964).] Thus far, however, there is no scientific method for selecting the 'best' approximating function, and the criteria of 'closeness' is highly subjective.

The first item is then to select the type of approximating function to be used; next, a way of measuring the 'goodness' of the approximation is needed. In the linearization approach, for example, the value of the gradient v f of f is evaluated, or approximated, at some given point ~, and the function is approximated by the linear expression

f ( A, x) = f ( f f ) + vf(£)(x - £),

with the relevant parameters being A = (f (~) , vf(~)). The goodness of the approximation is obtained by estimating the size of the non-linearities of f .

In general, the objective function f will be highly non-linear, and there will be an inherent difficulty in obtaining an analytic, or numerical, expression of its gradient. Furthermore, the solution point obtained by linearizing f depends entirely on the choice of the initial path from which the linear approximation is constructed, and may have no connection whatsoever with the location of the true optimum [Baumol (1982)]. This makes linear approximation a poor choice for our purposes.

There are in fact many different methods available to interpolate continuous functions using polynomials or other types of functions. For example, suppose that we are given m data points:

(xi,fi), i = 1 , 2 . . . . ,m,

where we may think of the x i as the independent (control and state) variables and of f~ as the dependent variables, satisfying the relationship

fi =f(xi).

Suppose further that f is to be approximated by a linear combination of k


given basis functions ~j,

k

/(x)--- E 9¢j(x). j = l

The problem is then to choose the 'best' basis functions ~j(.) and the k coefficients cj that best fit the data in some sense. Of the many criteria for determining the coefficients cj, the method of least squares is used most frequently. For any choice of the c j, the residual of the ith data point is

k , ,= E

j = l

Assuming that the basis functions are linearly independent at the data points, the least-squares criterion specifies that the cj be chosen to minimize the sum of the squares of the residuals, that is,

minimize ~ r~ 2. cj i=1

Thus, by choosing the x i ~ X in some given range, we can use the underlying model y,. = m(xila ) to generate the observations f~ =f(x i ) , and then use the least-squares method to obtain an approximation f of the function f . The more observations f~ are generated, the greater the accuracy obtained. Thus, the balance to be considered is that between the increase in the cost and the accuracy obtained for an extra observation fi.

If f ( x ) is a one-dimensional continuous function, then we can choose the basis functions to be ~j(x) = x j - l , in which case f will be a (k - 1)-degree polynomial. The use of polynomials as basis functions has many advantages; they can be easily evaluated, and they can be summed, multiplied, or differen- tiated. 4 The estimation problem, however, becomes increasingly difficult with higher dimensionality. This is essentially due to the fact that a function of two or more variables presents more information than a function of one variable and of comparable 'smoothness'. Thus if a 'smooth and well-behaved' one- dimensional function f ( x ) is adequately approximated by, say, a cubic polynomial with four coefficients, then we must consider a two-dimensional function f ( x , y) as smooth and well-behaved if it is adequately approximated by a bi-cubic polynomial with sixteen coefficients. It is then apparent that

'*According to the Weierstrass approximation theorem [see, e.g., Rice (1964)], there is always a polynomial that approximates a one-dimensional continuous function to arbitrary closeness. Thus far, however, there is not an efficient method of finding the polynomial, and indeed the resulting polynomial may be of such a high degree as to make its use impractical.

H. Sierra and T. Condon, Computing optimal dynamicpaths 415

anything more than the simplest problems in three or more dimensions will require a sizeable computation.

In the case of a multi-period model, the approximating function will be multi-dimensional. Unless we take advantage of the particular structure of the problem, in general it will be very expensive to generate information to approximate f with enough accuracy. Consider, for example, a problem of the general form

T

(P) maximize F ( y ) = E U~(y,)(1 + o ) - t , t = l

such that

y, = m(x,l~,), x = (xt) ~ X.

By using the equation y = re(x), we can substitute the endogenous variables to formulate the equivalent control-theory problem

T

(V) maximize f ( y ) = E u t ( x t ) ( 1 + p ) - t , t = l

such that

We can concentrate on estimating the individual components ut (x t ) of F. If ~t represents the function u, evaluated at some historical or reference path, ~, then at every period t we want to obtain an approximation )~ such that

u , -

where ~ ( 0 ) = 0 for all t. This condition insures that the original and the approximated functions coincide at the reference path. If the x-vector is

1 n-dimensional, that is, if x t - ( x t . . . . . xT) ~ R" for all t, then at every period t an n-dimensional function has to be approximated.

The relationship to be approximated is how many extra 'units' of u t result i from an extra 'unit' of x, over the historical path level, x[. For concreteness,

suppose that this relationship refers to the additional borrowing needed for an additional dollar of investment over the historical level. Suppose then that, in period t, b t dollars have to be borrowed for an extra dollar of investment. In an actual economy, this relationship will be affected by factors such as technological progress, specific government policies, etc. However, if in the model these are parameters fixed for every period, the relationship is affected only by the levels of other x-variables that are being considered.


For example, the stock of capital in period t will increase over its historical value if investment exceeds its historical level in earlier periods. The increased capital stock may result in less borrowing being necessary for the same amount of investment in period t. In other words, given an increase in the capital stock, an extra dollar spent on investment in period t may require b* dollars of borrowing, with b* < b t. If, however, investment represents just a small fraction of capital stock and only a few time periods are being considered, then this 'cross-term' effect may be small, and its effect on the key relationship can be safely ignored.

If all the 'cross-term' interactions are negligible, then a polynomial of the form

k

Z(x, E E ' ' / i -- = Ct ~X t - ~t]Ji~ i=l j = l

will provide a good approximation. The parameters in the polynomial may be computed using the method of least squares. The goodness of the approximation will depend on the number of observations f,. considered and on the size of the 'cross-term' effects. The latter may be estimated empirically by computing the cross-partial derivatives or by plotting the difference Ut(Xi(1 + 8), X j ) - -Ut (X i, x j ) for fixed x; and 8, and for different levels of the control variables xj.

5. An application

Proposition 2 implies that if the exact and the approximated functions coincide at the approximated path, the solution to the approximated problem is a lower bound to the solution of the exact problem. The goodness of the approximation, however, must refer to the exact optimal solution. In this section an empirical example is used to test the goodness of the approximation.

For this purpose, we use a five-sector dynamic computable general equilibrium (CGE) model. The model was originally developed by Kharas and Shishido (1985) to examine the issue of optimal borrowing in Thailand. Investment and borrowing are the control variables, and the objective is to find the investment level for each period of time, t, that maximizes the discounted utility stream over a period of known length, T, subject to terminal conditions on the levels of investment and borrowing. Three different types of borrowing are distinguished: private borrowing, and public borrowing from private and official institutions.


Utility in any period depends on sectoral consumption in that period. Households derive current income from endogenously determined wages, capital income, and exogenous other sources. Income can be either consumed or saved, and savings are used to purchase increments to the capital stock. Foreign borrowing must be repaid, and the stream of interest and amortiza- tion payments required to repay the debt affects household income and consumption.

The strategy we will pursue in reformulating the problem is to reduce the dynamic CGE model by approximating the relationship between utility and the levels of the control variables (investment and borrowing) and the corresponding state variables (total capital stock and outstanding debt). Thus, at every point in time we will be estimating a four-dimensional polynomial.

Starting from an historical path, ~ , in the first step we fix three of the four variables, then perturb the remaining one around the historical path. Succes- sive perturbations of the state and control variables yield pairs of observations on the variables and the value of the maximand, utility. These simulations of the exact model (as distinct from optimization runs) generate the information needed to compute the polynomial approximations.

Care is required when determining the range in which a variable is perturbed. It is very important that the range include all likely values that the variable could assume. This is obviously a greater difficulty with flow variables; stock variables are less likely to record large percentage changes from historical or reference path levels. Flow variables, however, may double or change sign, and the perturbation range should reflect such possibilities. Our experience indicates that the approximation is quite robust within the perturbation range, but it breaks down when state or control variables take a value outside the bounds.

In the multi-sector model it is important to distinguish between sectoral capital stocks. Output in all sectors is determined through a production function formulation, and capital is one of the factors of production. For the approximation, however, it is the relationship between the aggregate capital stock and utility that matters. Thus in computing that polynomial approximation, sectoral capital stocks are perturbed proportionately. A similar remark applies to the treatment of public foreign borrowing from official and private sources. (Private foreign borrowing is exogenous.)

In the second step, the observations on the perturbed variables and the associated utility levels are used to compute the coefficients of the polynomial approximations. In practice, the coefficients were obtained by solving a linear programming problem in which the sum of residuals is minimized, subject to a non-negativity constraint on the residuals. For the numerical example, this involved solving four simple linear programming problems. Good approximations were obtained using third- and fourth-degree polynomials. There is no

4 1 8

Z

o t ~ r,.,

0 t ~

0

- 5

-10

H. Sierra and T. Condon, Computing optimal dynamic paths

35

30 - 1 t t , , . , ~ J

20 - / , ' ' . \ \

10- ~"

5 - ~....- \ \

t I I I I I 2 3 4 5 6 7

PERIOD

- - B A S E . . . . EXACT . . . . . . . APPROX.

Fig. 2a. Historical and optimal borrowing.

rule for determining a priori the order of the polynomial approximation for a given relationship.

Once the coefficients c,, i. j are computed, the approximation can be formed as

. , - = E j. (9) i j

Eq. (9) embodies the relationship between utility and the control and state variables of the CGE model. It is the new maximand in the approximated problem, which consists of that relationship and the relevant accounting and technical equations, the equations of motion of the state variables and the terminal conditions constraining investment and borrowing. The CGE has been stripped of all of the technological equations except for the equations of motion of the state variables, and we are left with a 'reduced form.'

A graphical depiction of the 'closeness' of the approximation is given in figs. 2a and 2b. Before discussing those, however, we must decide in what units to measure closeness. The value of the objective function is measured in utility units, but a more objective measure is needed. Gelb (1985a) also faced this problem. His solution, which we have also adopted, was to arbitrarily impose an ordering by standardiT.ing a 'util'. Thus, we say that the change in utility that results from a ten-percent terms-of-trade deterioration in all periods represents one util.


190

170

150

130

110

90

70

50

30

/ f SS ssS S \ \

/ / j / \\ / s j

ss S / •

~ Tssssss ~ ss

I I I 2 3 4 5

PERIOD

,,,/

A

B A S E . . . . EXACT . . . . . . . APPROX.

Fig. 2b. Historical and optimal investment.

419

The value of the exact optimal, f(x*), is 87.7 utils compared to 74.7 along the reference path, a difference of about 17.4%. The approximate optimal, f (2*) , is 86.5 utils, for an approximation error of 1.4 percent. The exact problem, evaluated at the approximate optimal solution, f(2*), results in a level of 87.2 utils. As Proposition 2 indicates, the approximate optimal solution provides a lower bound for the exact optimal. Figs. 2a and 2b compare historical (labelled BASE), exact and approximate optimal trajecto- ries of borrowing and investment. As a practical matter, the comparison of either optimal with the historical level would lead the investigator to similar conclusions about the performance of investment and foreign borrowing in Thailand.

Finally a note on computing costs is in order. The exact optimal is expensive to compute. The model is programmed in GAMS. 5 Running on a Control Data Corporation Cyber 750 computer, the model required 7895 resource units. The approximate model, also programmed in GAMS and running on the same computer, used 158 resource units. The polynomial approximations required about 25 resource units apiece, and the perturbations used about 140 units. There were 28 perturbations in all, and four polynomial approximations, resulting in start-up costs of about 4000 resource units, less

5 GAMS stands for General Algebraic Modelling System and was developed by A. Meeraus and associates at the World Bank. See Meeraus (1983) for details.

J.E.D.C.-- F


- 0 . 1 0 4

- 0 . I 0 5

- 0 . 1 0 6

- 0 . I 0 7

- 0 . 1 0 8

- 0.109

- 0 . 1 1 0

-0 .111 , ,

0 1 2 3

PERTURBATION

. . . . . . . CAPITAL STOCK DEBT STOCK .... BORROWING

Fig. 3. Cross effects.

than the total for one optimizing run of the exact model. Thus, even if start-up costs are included, substantial savings of computer resources are obtainable from the approximation techni~que.

Recall that the function f is computed assuming that the cross-effects between the key variables are negligible. The closeness of the approximation supports this assumption. Still, to indicate the size of the cross-effects, we have computed a measure of their size in utils:

Au=f(Sj(l + 8), .:i(I +a))-f(~j,~i(l +a)), (10)

j = l , i = 2,3,4.

If cross-effects are important, we expect that as a increases so will Au. We perturb the variable xj, investment, by a standard ten percent (8--0.1). We then perturb the flow variable, foreign borrowing, by 25, 50, 75 and 100 percent and the two stock variables, capital and external debt, by 5, 10, 15 and 20 percent. The results appear in fig. 3 where the x-axis measures the

m

60

50

40

30

20

10

0

- 1 0


@ z

t ' ~ t ' v

O r n

I I I I I 2 3 4 5 6

PERIOD . . . . . . . . . . . EXACT ~ APPROX.

Fig. 4a. Exact and approximate optimal borrowing.

421

perturbations of a in eq. (10). The first is equal to zero corresponding to a Au of -0 .109 utils, followed by the forementioned percentage shocks to the stock and flow variables. As expected, the cross-effects are small, with the largest being the effect of increasing the capital stock on the marginal utility of investment.

6. An e x p e r i m e n t

To test the sensitivity of the approximation we perform an experiment. The experiment fixes the level of the interest rate on public debt at its initial period level. This amounts to a significantly lower interest rate profile in the final periods.

The optimal solution of the exact problem yields a level of 88.4 utils; as one would expect, this exceeds the optimal associated with the historical (higher) pattern of interest rates. The approximate optimal solution is 87.0 utils, for an approximation error of 1.6 percent. The solution f(~*) results in a level of 87.8 utils. Figs. 4a and 4b again compare the borrowing and investment levels of the actual and the approximate optimal solutions.

7. Final remarks

We have presented a proposition that justifies replacing a problem with an approximated problem. In the numerical example examined, the solution of

422

180

160

140

120

100

80

60

H. Sierra and 7". Condon, Computing optimal dynamic paths

. , . o .

I I I I I 2 3 4 5

PERIOD ........... EXACT ~ APPROX.

Fig. 4b. Exact and approximate optimal investment.

6

the approximated problem is very close to the solution of the exact problem. Further, the results of performing an experiment with the model indicate that the approximation is quite robust. This is not, of course, a proof of the general applicability of the method. However, our experience with dynamic CGE models leads us to believe that the technique can be applied to a wide range of interesting questions that, because of software limitations, have not previously been analyzed in computable models.

The technique is very powerful and in principle can be applied to any problem. The attractiveness of the technique depends on the relation between marginal costs and benefits of employing it. 6 For small problems it may not be an attractive option. There are also some side benefits from employing the technique. For example, one possible use could be to obtain a feasible solution that is closer to the exact optimal than starting from the historical path. Further, the information gathered from the exercise is useful for understanding what are generally very complex models. For example, the importance of general equilibrium or cross-effects between variables is small in our example. This is useful knowledge for understanding what 'drives' a particular model result.

6Costs could be reduced substantially by automating the steps required to produce the polynomial approximations.


References

Baumol, W.J., 1982, Planning and dual variables of linearized nonlinear problems: A gothic tale, in: M. Gersovitz, ed., The theory and experience of economic development (Allen and Unwin, Boston, MA).

Gelb, A.H., 1985a, The impact of oil windfalls: Comparative statics with an Indonesia-like model, Report no. DRD133, Oct. (World Bank, Washington, DC).

Gelb, A.H., 1985b, Are oil windfalls a blessing or a curse? Policy exercises with an Indonesia-like model, Report no. DRD135, Nov. (Word Bank, Washington, DC).

Kharas, H. and H. Shishido, 1985, Thailand: An assessment of alternative foreign borrowing strategies, CPD discussion paper no. 1985-29, May (World Bank, Washington, DC).

Martin, R. and M. Selowsky, 1984, Energy prices, substitution, and optimal borrowing in the short run, Journal of Development Economics 14, 331-350.

Meeraus, A., 1983, An algebraic approach to modelling, Journal of Economic Dynamics and Control 5, 81-108.

Rice, J.R., 1964, The approximation of functions, Vol. 1 (Addison-Wesley, Reading, MA). Taylor, L., 1975, Theoretical foundations and technical implications, in: C. Blitzer, P. Clark and

L. Taylor, eds., Economy-wide models and development planning (Oxford University Press, New York).

an approximation technique for computing optimal dynamic paths

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