Office of Tax Analysis 

U. S. Treasury Department

Washington, D.C. 20220 

Issued: June, 1976 

The Policy Uses of A Computational

General Equilibrium Algorithm 

John B. Shoven 

Stanford University 

OTA Paper 9 February, 1976 
Table of Contents 

Paae 

I. Introduction .................... 1 

11. The Algorithm and Extensions. ........... 2 

A. Modifications and Additions to 

the Algorithm ................. 2 

(1) Continuous Production Functions ...... 2 

(2) Government and Taxes ............ 3 

(3) Multiple Governments and Tax Systems. ... 3 

(4) Equal Yield Tax Replacements. ....... 3 

(5) Calibrating and Parameterizing Model. ... 4 

(6) Newton-Type Termination Routines ...... 4 

B. A Description of the Results. ......... 5 

111. Advantages of the General Equilibrium Algorithm . . 6 

IV. Applications of the Scarf-Shoven-Whalley Approach . 8 

V. Potential Future Applications ........... 12 

VI. Conclusion. .................... 16 
I. INTRODUCTION 

Economists have long recognized the conceptual superiority 

of general equilibrium models over partial equilibrium analysis. 

However, this consensus has not been matched with widespread 

application of the general equilibrium approach. Many empiricists 

have felt that the theoretical beauty of the general equilibrium 

approach was matched only by its empirical uselessness. However, 

this position is no longer valid. The problem of solving relatively 

large scale general equilibrium systems has been mitigated within 

the past six to eight years. In fact, most of the computational 

techniques described in this paper converge to a solution which 

is an approximate competitive equilibrium. Because of this, and 

due to an increasing awareness of the inadequacies of highly 

aggregate or partial equilibrium studies for evaluating some 

important policy proposals, the practical utility of general 

equilibrium analysis needs to be reassessed. Therefore, this 

paper will briefly describe the current general equilibrium 

computational capabilities, review the applications made to date, 

and preview the possible uses for these techniques. 
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11. THE ALGORITHM AND EXTENSIONS 

The seminal work on computational general equilibrium 

algorithms was done by Herbert Scarf and culminated in his recent 

book (Scarf, 1973). In this monograph, Scarf describes a solution 

procedure for general equilibrium models involving an arbitrary 

number of consumers and commodities and an activity analysis 

description of production. Each consumer or consumer class is 

endowed with a vector of assets, and income is determined by 

valuing these assets at the prevailing set of prices. Each 

consumer also exhibits a set of continuous demand functions which 

aggregate such that total income is exactly exhausted. The 

algorithm finds a price vector and a set of production levels 

such that supply approximately equals demand and such that production activities in use break even while those not in use lose 

money. In this sense, the price vector and activity levels 

represent an approximate equilibrium. 

A. Modifications and Additions to the Algorithm. In our 

respective dissertations and in several subsequent articles, John 

Whalley of the London School of Economics and I have extended 

Scarf's algorithm in directions which make it a more useful tool 

for policy evaluation purposes. The modifications we have made 

can briefly be listed. 

(1) Continuous Production Functions. Continuous 

production functions for the various production sectors (Shoven, 

1973) allow an evaluation of factor substitution effects of 

various governmantal policy changes. Such analytic functions as 

the Cobb-Douglas, constant-elasticity-of-substitution (CES) , and 
-3-

Sato-Uzawa nested CES functions have been included; however, 

others could be analyzed as long as they do not imply increasing 

returns to scale. 

(2) Government and Taxes. The second important 

addition to the model involves the introduction of a government 

and ad valorem producer and consumer taxes. The taxes can be 

discriminatory in the extreme--each producer and each consumer 

can face different tax rates on each commodity. The government 

may simply redistribute the tax proceeds with a system of transfer 

payments, or it may retain some revenue for the purchase of goods 

and services. In the latter case, the government is treated as 

a consumer class and assigned a set of demand functions. The most 

general proof that an equilibrium exists with an arbitrary set 

of tax rates and a government is given in Shoven (1974), and a 

constructive proof and computational procedure is described in 

Shoven and Whalley (1973). 

(3) Multiple Governments and Tax Systems. The ability 

to analyze several governments and tax systems bas been achieved, 

making it possible to evaluate such alternatives as: a system of 

nested governments (for example the U.S. federal, state, and local 

systems); revenue sharing; and multi-country international trade 

models with tariffs and quotas applying to inter-country trade. 

The general theoretical model is presented in Shoven (1974), while 

the international trade aspects are described in Shoven and 

Whalley (1974). 

(4) Equal Yield Tax Replacements. A common stipulation 

made when comparing alternative tax systems is that they generate 
-4-

the same real tax yield. A method of calculating the rates 

necessary for two tax systems to match yields (as well as calculating the two equilibria) is described in Shoven and Whalley 

(1975). 

(5) Calibrating and Parameterizing Model. Undue criticism 

has been made concerning the difficulty of estimating the number 

of parameters involved for a general equilibrium model. The 

economy modeled with a partial equilibrium approach involves a 

similar number of parameters--most of them are just arbitrarily 

taken to be zero. Therefore, a technique that involves parameterizing general equilibrium models has been developed 

(Shoven, 1973; Whalley, 1973). By constraining the model to 

reproduce the observed economy when faced with the observed (tax) 

environment, the number of free or independent parameters is 

substantially reduced, often by a factor of about three. The 

remaining parameters must be extraneously derived. Heuristically, 

given some extraneous parameter estimates and the observed 

economic variables, we have solved the model "backwards" to 

determine the necessary values of the remaining parameters--that 

is, what they must equal to give such results. Then, with all 

of the parameter values known, a policy change can be considered 

and the model can be solved ''forwards" in order to predict the 

new equilibrium. Its biases and other econometric properties 

are not completely established, but when faced with a policy 

problem, it offers a reasonable way of proceeding. 

(6) Newton-Type Termination Routines. The algorithm 

provides an approximate equilibrium. Because many general 
-5-

equilibrium problems are locally stable for gradient-type solution 

procedures, we have developed several local termination routines 

which greatly improve the degree of approximation. In fact, supply 

has been equated to demand to the level of fifteen significant 

figures. In many cases, these routines are all that is necessary 

for solution. Two useful termination routines are examined in 

Shoven (1973). 

B. A Description of the Results. These additions and 

modifications to the algorithm have greatly expanded the applicability of general equilibrium analysis. The output provides 

the policy maker with a complete description of the (model) economy 

under each proposed tax change. The fundamental information is 

the vector of prices, but from this each consumer's income, 

leisure, tax payments, and commodity demands can be determined. 

If one is willing to accept a cardinal utility measure, consumer 

utilities may be calculated and, in some sense, the ultimate 

impact of the tax policy is its affect on these utilities. Each 

producer's output, factor usages, and tax payments are available, 

as is the government revenue, transfer payments, and expenditures. 

The descriptions are so complete, that in order to satisfy the 

policy maker's demand for one or a few numbers to characterize 

the efficiency or incidence of a tax program, information must 

be aggregated using index numbers. 
-6-

111. ADVANTAGES OF THE GENERAL EQUILIBRIUM ALGORITHM 

To highlight the advantages of a general equilibrium approach, 

the shortcomings of traditional partial equilibrium analysis 

should be mentioned. First, it is difficult or impossible to 

evaluate the effect of output taxes on factor markets or factor 

taxes on output markets with a partial equilibrium approach. 

That is, the product and factor markets, which are linked by 

production technology, cannot be evaluated simultaneously. 

Second, any tax of significant magnitude will involve some important cross effects which cannot be dealt with appropriabely using 

a market by market approach. Therefore, this type of analysis 

also creates inconsistencies in the aggregate relationships. 

Arnold Harberger introduced general equilibrium analysis 

into public finance with his evaluation of the incidence of the 

corporation income tax (Harberger, 1962). Actually, the model 

was a close derivative of the already familiar two production 

sector, two factor, one consumer model developed by Harry 

Johnson (1956) and James Meade (1955) in the international 

trade context. The major advantage of this approach was in its 

explicit linking of factor and product markets. 

Relative to the two by two approach of the Harberger model, 

the Scarf-Shoven-Whalley general equilibrium approach has 

several important advantages. Its ability to deal with an 

arbitrary number of commodities, production sectors, and consumers 

not only permits the dynamic extension of these basically 

static models, but also makes it possible to simultaneously 

evaluate distortions, such as taxes and tariffs. 
-7-

Computational time does rise as these dimensions increase, but 

a model with 20 commodities, 20 production sectors, 5 factors, 

and 20 consumers would present no particular difficulty. 

A second major advantage of our approach is the absence 

of local or "small change" assumptions. These assumptions are 

made repeatedly in the Harberger calculus model, and are particularly bothersme when the purpose of the analysis is to estimate 

the magnitude of the impacts of a tax policy alteration. The 

algorithmic approach simply involves comparing two equilibria-­

no calculus or local assumptions are necessary. The "new" 

equilibria is not calculated from the ''old"one; rather the 

model is completely resolved. 

The ability to include several consumers in the analysis 

creates a third advantage, for it is now possible to determine 

the incidence of the tax policy on the personal and on the 

functional distributions of income. This is valuable information for a real world which does not confine capitalistF and 

laborers to their respective roles as savers and workers. 

Finally, the relaxation of the assumption of fixed factor 

supplies enables a labor-leisure choice to be included in the 

analysis. Saving and investment behavior should prove feasible 

in the dynamic extensions. 
-8-

IV. APPLICATIONS OF THE SCARF-SHOVEN-WHALLEY APPROACH 

Despite the many desirable features of the approach, general 

equilibrium algorithmic techniques have not been widely used, 

since they are fairly new and require a relatively large investment in computer routines. Gradually the computer programs will 

become more "off the shelf" items so that these techniques should 

experience increasing application. The techniques are often 

superior for analyzing problems involving large policy changes 

or differential impacts on several sectors of the economy. 

Certain policy applications may be described to illustrate 

appropriate problems to which general equilibrium algorithmic 

techniques have been applied. I 

The first policy evaluation made using Scarf's algorithm 

was a study by Marcus Miller and John Spencer of the effects of 

Britain joining the Common Market (Miller and Spencer, 1973). 

Unfortunately, many of the capabilities mentioned above were not 

fully developed (the programs for a few of the features are still 

not available) and the analysis may have suffered on this account. 

The study involved four groups of countries, with two goods and 

two factors of production in each. The general result was that 

it was not in Britain's interest to join the EEC if this involved 

the loss of her favorable trade agreements with the Commonwealth 

countries which previously supplied most of her agricultural 

products. 

Harberger's model deals with the problem of the incidence 

and efficiency effects of taxes on income from capital in the 

United States (Harberger, 1959; 1962; 1966), and enables a direct 
-9-

comparison of the algorithmic approach with two sector analysis 

(Shoven, 1975). 

Harberger, in his 1966 article evaluating the inefficiency 

of these levies, made at least two severe mistakes--a simple 

arithmetic mistake documented in Shoven(1975) and an inconsistency 

in his definition of units. Both mistakes seriously affect 

Harberger's loss estimate results, with the correct numbers ranging 

from 32 to 61 percent of those he published. By comparison, the 

loss results obtained using the algorithmic approach are 35 to 

50 percent larger than those correctly obtained from Harberger's 

model for the most comparable cases. The results, of course, are 

far more detailed giving information regarding both the personal 

and functional incidence of the taxes as well as the inefficiency 

estimates. The most recent paper (Shoven, 1975) also reports on 

the effects of disaggregating to twelve production sectors, which 

approximately doubles the loss estimates of the corrected Harberger model. 

One might speculate why the comparable loss estimates are 

larger using the algorithmic technique. First, the income effect 

of the loss in real output is included in the algorithmic 

estimate. Second, the loss estimate probably increases in a 

nonlinear manner as the distortionary tax rate is increased. 

Since Harberger's model is really a first-order approximation 

method, the ignored second-order terms may be significant. 

Lastly, and perhaps most importantly, the collapsing to two 

sectors hides a great deal of the inefficiencies of the capital 

tax situation in the United States. Capital is not taxed in a 
- 10 -

neutral manner within the "corporate" and the "noncorporate" 

sectors. 

Two additional lessons were gained from the capital income 

tax study. First, Harberger's iriefficiency results are commonly 

characterized as amounting to a rather insignificant 0.5 to 1.0 

percent of GNP. This presentation seems somewhat misleading; a 

more meaningful interpretation may involve taking the ratio of the 

inefficiency cost of the distortionary taxes to their revenue. 

My best estimate of the dead weight loss of these taxes is 

between 10 and 15 percent of their revenue. Secondly, in a model 

permitting a labor-leisure choice, it is even possible that GNP 

is higher with the distortionary taxes than in their absence. 

That is, the inefficiency may show up entirely as reduced leisure. 

Another application of the general equilibrium algorithm 

evaluated the impact of the 1973 U.K. tax reforms (Whalley, 1973). 

The tax changes included abolition of the purchase tax and the 

selective employment tax and introduction of the value added tax 

and the PAYE system of personal income tax withholding. The 

U.K. corporation income tax was also revised in 1973. This 

package of tax changes is evaluated with a nine sector, two 

country, and seven consumer general equilibrium model. Whalley's 

study (Whalley, 1975a) indicates that the reform package will 

result in a rather significant change in the personal distribution of income, but will have very small efficiency 

consequences. 

Another study undertaken by Whalley estimates the impact 

of fiscal harmonization among countries of the European Economic 
- 11 -

Community (EEC). This work is continuing, but preliminary results 

have been generated by solving a large 61 dimensional model 

(Whalley, 1975b). 
- 12 -

V. POTENTIAL FUTURE APPLICATIONS 

One of the most promising areas of future research would 

involve an up-to-date evaluation of U.S. capital income taxes. 

The studies previously mentioned have all used Rosenberg's 

1953-59 data (Rosenberg, 1969); thus, a similar analysis using 

1970 data would be useful for today's policy decisions. To extend 

the work by incorporating other major taxes (such as the personal 

income tax and social security taxes) and by making the analysis 

dynamic would prove valuable. 

One of the most pressing questions today is the adequacy of 

the U.S. capital stock and the impact on saving of the heavy tax 

burden placed on corporate capital income. By examining a model 

which included, say, four five-year periods, one could estimate 

for example, the dynamic impact of removing the corporation income 

tax, the tax on that portion of capital gains which merely reflects 

inflation, or the impact of exempting all savings from income 

taxation. 

The study of capital income taxes, both in the comparative 

static approach and in the dynamic analysis, would naturally 

lead to the development of a small, but comprehensive, general 

equilibrium model of the Uqited States which could be extended 

with further time and effort. Foreign trade could be dealt with 

and both the domestic and global impact of raising or lowering 

import barriers examined. Detailed consumer data could be 

incorporated, as could state and local taxes and the consumer's 

decision regarding where to locate. Each of these extensions 
- 13 -

would involve a major amount of work both in developing the 

analytical framework and in collecting and interpreting the 

relevant data. However, once the small core model of the U.S. 

economy had been developed (which might involve approximately 

10 consumer classes, 15 commodities, a combination of inputoutput and continuous production function technologies, and 

Personal and corporate income taxes) each extension could he 

worked on separately. While initially the results might not be 

numerically precise, they would give a picture of the general 

equilibrium adjustments implied by a particular policy. 

Previous studies which made use of Harberger's model could 

be reexamined using this basic model. This would include the 

effects of labor unions (Johnson and Mieszkowski, 1970) and the 

effects of the tax free nature of non-market economic activity 

(Boskin, 1975). One could investigate whether there is important 

interaction between these and other distortions by analyzing 

such phenomena as the corporation income tax, labor unions, and 

minimum wage laws simultaneously. 

Somewhat further removed examples of the policy evaluation 

capability of a general equilibrium model are: (a) replacing 

part or all of the corporation income tax (or property taxes) 

with a VAT, (b) financing social security out of general revenues 

rather than with a payroll tax, (c) substituting a comprehensive 

negative income tax for the several separate aid-to-poor programs 

now in existence. 

I have mainly concentrated on the evaluation of tax 

distortions, but this type of modeling is also relevant when 
- 14 -

studying non-tax related problems. For instance, the effects 

of monopoly power on the allocation of resources and the distribution of income may be investigated. Industries can require a 

"profit markup" in their prices and these profits can be distributed to the owners of the industries. The mechanics of including 

such markup factors is identical to the inclusion of an excise tax 

with the revenues granted to the stockholders. While this captures some aspects of monopoly power, it omits others. For 

instance, it is not possible to incorporate the increasing returns 

to scale technology of natural monopolies. 

A second non-tax area of application would involve extremely 

large scale cost-benefit 8tudies. These evaluations are typically 

partial equilibrium in that all prices are taken as given and 

unchanged. However, large proposals such as the administration's 

1975 suggestion of a $100 billion energy project can only be 

properly considered in a general equilibrium setting. The 

approach I would recommend for such studies involves the comparison 

of equilibria under different technologies. Such a procedure 

would be appropriate if the task at hand was predicting the 

value and impact of a fusion capability. 

The final non-tax application concern issues of international 

trade. The approach can readily evaluate a system of tariffs 

and export and import quotas. It also would be an appropriate 

framework to investigate the long run effects of a permanent and 

universal oil embargo, or the raising of import barriers by the 

EEC, or the refusal of some sources to trade with the United 

States. 
- 15 -

One common feature of these various problems is that they 

involve large changes and therefore can be expected to have substantial indirect or secondary effects. A second property is 

that most of them require more detail than is possible with a 

two sector bifurcation of the economy.-1/ 
- 16 -

VI. CONCLUSION 

Many policy problems are best evaluated using a general 

equilibrium model of economics. The feasibility of using such 

models has been greatly enhanced within the past five years. 

This paper has reviewed the algorithmic techniques and their 

applications to date, and has previewed a variety of topics which 

could be examined with the general equilibrium algorithmic approach. 

One qualification should be emphasized, despite the general 

positive tone of the paper. Each of the studies undertaken so 

far has involved an ambitious effort. The appropriate data must 

be collected and interpreted, and the algorithms must be suitably 

modified for each application. However the amount of effort 

needed for each study could be substantially reduced if there 

existed a small general equilibrium model of the United States 

to use as a point of departure. Building such a model and 

increasing the availability and flexibility of the general 

equilibrium computer routines is the prerequisite for promoting 

this potentially powerful and important tool of economic 

analysis. 
- 17 -

FOOTNOTES 

1 

In addition to policy applications, further research on 

techniques is desirable. For example, the recent work of 

Diewert (1971) on a generalized Leontief system might be 

incorporated in the algorithmic technique. 
- 18 -

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