Congressional Budget OfficeSkip Navigation
Home Red Bullet Publications Red Bullet Cost Estimates Red Bullet About CBO Red Bullet Press Red Bullet Careers Red Bullet Contact Us Red Bullet Director's Blog Red Bullet   RSS
Two Papers on Fundamental Tax Reform
October 1997		 PDF
APPENDIX B
THE FULLERTON AND ROGERS GENERAL-EQUILIBRIUM MODEL

The Fullerton-Rogers model uses measures of lifetime income based on longitudinal data, and classifies households according to lifetime-income categories. By specifying functions that describe consumer utility and industrial production, the model is able to calculate the general-equilibrium effects of tax changes on the prices and quantities of goods and factors. It also measures the subsequent effects on economic efficiency and the welfare of each income category.

 
LIFETIME INCOMES

The Fullerton-Rogers model incorporates data on lifetime incomes, requiring longitudinal data for many individuals over many years. Although no data set spans the entire lifetimes of individuals, the University of Michigan's Panel Study of Income Dynamics (PSID) has been asking the same questions of the same individuals now for over 20 years. From the PSID, Fullerton and Rogers drew a sample of 500 households that included 858 adult individuals, with information on wages, taxes, transfers, and various demographic variables for the years from 1970 through 1987. They included heads of households and wives in the sample, and for simplicity in defining the lifetime of a "household," they excluded households whose marital status varied over the sample period. For heads of households and wives separately, they estimated the wage rate as a nonlinear function of age. As a result, for each individual in the sample they were able to predict the wage rate for the years that come after as well as before the sample period; multiply the actual or estimated wage rate by a total number of hours per year (for example, 4,000) to get the value of the individual's labor endowment; and calculate the present value of this endowment over the individual's lifetime.

Thus, the level of well-being in the Fullerton-Rogers model is defined by potential earnings, including the value of leisure. Those levels are used to classify individuals into twelve groups according to lifetime ability-to-pay, in which an individual's lifetime income is defined to be the average of the head's and wife's (if any) lifetime incomes. The groups are constructed by starting with the 10 deciles, but the poorest 2 percent is separated from the next poorest 8 percent, and the richest 2 percent from the next richest 8 percent.

For a given level of lifetime income, the timing of income matters: the shape of an individual's profile for lifetime income determines savings and therefore the composition of any year's annual income. Therefore, Fullerton and Rogers reestimate the profiles of wages by age separately for each of the 12 groups. In addition, they estimate the time paths of personal income taxes paid and transfers received. In that way, they set up a consistent benchmark data set with a path of consumer spending out of total available after-tax income.
 
 
MODEL STRUCTURE AND NUMERICAL SPECIFICATION OF PARAMETERS

The general-equilibrium approach to tax analysis accounts for behavioral effects and excess burdens caused by taxes. It can capture the important influences of taxes on diverse household choices about labor supply, savings, and the consumption of different commodities. Consumers supply labor and capital and purchase goods and services in a way such that well-being is maximized. The assumption that producers will maximize profits determines the demands for labor and capital and the effects of taxes on those demands. As the model solves for the prices establishing general equilibrium, it captures the net impact of taxes when those consumer and producer behaviors are considered simultaneously.

In the Fullerton-Rogers model, consumer decisions maximize the lifetime economic well-being of individuals. To begin, the individual calculates the present value of potential lifetime earnings. That endowment is then supplemented by government transfers, reduced by taxes, discounted at the after-tax interest rate, and augmented by a fixed initial inheritance. For computational simplicity, the model assumes "myopic" expectations about future prices--in other words, the consumer expects the current interest rate to prevail in all future periods.
 
One part of the lifetime endowment must be saved for a bequest upon death. Fullerton and Rogers avoid the many possible motivations for individual bequests, or the many ways in which taxes might affect the size of those bequests. Instead, the Fullerton-Rogers model simply acknowledges that life-cycle saving by itself can only explain about half of the observed capital stock. In the model, part of the capital stock is attributable to individuals receiving a fixed level of inheritances and then being required to leave comparable bequests at the end of life. The incidence of capital taxes thus depends on the differences in those inheritances among groups. To achieve balanced growth, each group must add some additional savings to their inheritance before they make their bequest.
 
The rest of the present value of income is available for spending. Decisions are made in stages. In the context of fundamental tax reform, the first two stages are the most important because they define the saving and labor-supply responses.
 
At the first stage, the consumer chooses how much to spend each period. That choice depends on assumptions about the form of lifetime utility and the values of certain key parameters. Lifetime utility is specified as a "constant-elasticity-of-substitution" (CES) function:

lifetime utility is a function of xt and epsilon1, summed over all periods t=1 through T
where T=60 (chronological age 79) is the individual's certain date of death, epsilon1, is the intertemporal elasticity of substitution, and xt is the amount of "composite commodity" (a combination of a composite consumption good and leisure) at economic age t. The weighting parameter, at reflects the consumer's subjective rate of time preference, which is set at 0.005.

Although the Fullerton and Rogers study used a central-case intertemporal elasticity equal to 0.5, that elasticity is varied from a low of 0.15 up to 0.5 in the present study's examination of efficiency gains. The consumer's choice about how much to spend each period is also affected by changes in the net rate of return (which is set at 0.04 in the central case).(2)

At the second stage, the consumer allocates one period's "spending" between leisure and other consumption goods, according to the CES function:

for each period t, x is a function of cbar, l, and epsilon2
where cbart is the amount of composite consumption good consumed at t, lt is the amount of leisure taken at t, and epsilon2 is the elasticity of substitution between consumption and leisure. The decision about how much labor to supply depends on what is assumed about the value of this consumption-leisure elasticity of substitution. Fullerton and Rogers set that elasticity at 0.5 in their central case, but for the purposes of this study that elasticity is varied from 0.15 to 0.5 (just as the intertemporal elasticity is varied). In the general-equilibrium model, individuals can "buy" more leisure at a price equal to the forgone after-tax wage, instead of buying other goods. Both taxes and age affect that choice. Individuals in that model never fully retire. The weight on leisure increases with age after they reach 60 in a way that reflects actual choices.

In the third stage, individuals decide how to allocate current consumption spending among 17 particular goods (such as food, alcohol, tobacco, utilities, housing, and so forth), according to the function:

for each period t, cbar is a function of ci minus bi and the parameter betai, multiplied over all goods i=1 through N
where N is the number of consumer goods (=17), and cit is the amount of consumer good i consumed at age t. That function is of the "Stone-Geary" form, which means that a consumer at a given age has to buy a set of "minimum required purchases" (b) and then allocates remaining spending according to a set of "marginal expenditure shares" (beta). In this model, those 34 (17 x 2) parameters are estimated for each of 12 age categories using data from the Consumer Expenditure Survey, as described thoroughly in the Fullerton and Rogers book.

The Stone-Geary framework has several important implications. By making a portion of spending nondiscretionary, it reduces the sensitivity of total consumption and saving to the net rate of return. In addition, because discretionary income may be spent in proportions different from minimum requirements, the proportion of total income spent on any particular good will vary with total income. Required spending is relatively high for housing and gasoline, while discretionary spending is relatively high for clothing, services, and recreation. Thus, the rich and the poor buy different bundles, and bear different tax burdens because of those differences in how they spend their incomes.(3)

In the fourth stage of the consumer's allocation process, the expenditure on each consumer good is divided by fixed coefficients among components drawn from a list of industries. No real "decision" is made here, but that step allows the matching up of consumption data using one definition of commodities with production data using a different definition. For example, expenditures on the consumer good "appliances" is composed of portions from metals and machinery, transportation, and the trade industry.

Then, in the fifth and final stage of the decision process, the consumer takes the spending on each industry output and allocates it between the corporate sector and the noncorporate sector, according to the CES function:

for each producer good j, Q bar is a function of QC and QNC and the parameters epsilon3 and gamma
where Qjc is the amount of corporate production of producer good j, Qjnc is the amount of noncorporate production of producer good j, and epsilon3 is the elasticity of substitution between corporate and noncorporate outputs in consumption. Corporate output is assumed to be slightly different from the noncorporate output in the same industry. Hand-carved furniture, for example, is not the same as manufactured furniture. The consumer chooses the amount of each, using a weighing parameter gamma based on initial corporate and noncorporate shares of production within each industry (as observed in the data), and using another elasticity of substitution (epsilon3, which is set to 5.0 in the central case). That specification is consistent with the observed coexistence of both sectors within an industry, despite different tax treatments. If the outputs were identical, then a higher tax rate would drive one sector out of production. The elasticity of substitution reflects the degree of similarity. The other purpose of that specification is to capture ways in which changes in corporate taxes affect relative product prices and quantities demanded of the outputs of each sector.
 
A similar process characterizes the behavior of producers in each sector of each industry. Many competitive firms produce each output according to multistage production functions with constant returns to scale. Also, to keep the computation simple, the model assumes no externalities, no adjustment costs, and no uncertainty.
 
In the first stage of production, output is composed of a fixed-coefficient combination of value added and intermediate inputs. Each of the 19 industries uses the outputs of all other industries in fixed proportions. Thus, changes in the price of one product affect many other product prices. In the second stage, value added is a function of labor and composite capital, according to the function:
 
value added is a function of L and K bar and the parameters sigma1 and zeta
The weighting parameters (zeta) are based on observed labor L and capital K bar in each industry, and the elasticity of substitution (sigma1) varies by industry (between 0.68 and 0.96 in the central case). Thus, a tax on labor can induce the firm to use more capital instead, and vice versa. It also raises the cost of production, and thus the price of output in any industry that uses a high proportion of the taxed factor.
In the third and final stage of production, composite capital is a CES function of five asset types (Kk)--equipment, structures, land, inventories, and intangibles:
K bar is a function of the different K sub k capital types and the parameters sigma2 and psi sub k

Those types are defined by important tax differences such as the investment credit for equipment and the expensing of new intangible assets created through advertising or research and development. The weighting shares psi sub k are again based on observed use of assets in each industry, and the response to tax differences is again specified by an elasticity of substitution (sigma2 1.5 in the central case).

Government in the model conducts several functions. It pays transfers to individuals according to the estimated lifetime transfer profiles discussed above. It produces an output for sale through an industry called "government enterprises," and it also produces a free public good by a combining its use of labor, capital, and purchases of each private industry output. The weights in that combination are based on observed government purchases, and the elasticity of substitution is one. The level of that public good is held fixed in all simulations, as any tax change involves an adjustment that ensures a constant yield of real revenues. A final government function, of course, is to collect taxes. Simplifying assumptions of the model are that the government balances its budget in each period, and that only one level of government exists (that is, no distinctions are made among federal, state, and local levels).

Each tax instrument enters the model as a wedge between the producer's price and the consumer's price. The payroll tax, for example, applies at an ad valorem rate to each producer's use of labor. Consequently, the gross wage paid by the producer is higher than the net wage received by the worker. Similarly, sales and excise taxes appear as an ad valorem rate on each consumer good. Therefore, the gross price paid by the consumer exceeds the net price received by the seller.

The modeling of the personal income tax is a bit more complicated when used to capture that tax's progressive structure of burdens. The actual U.S. personal income tax system imposes higher effective tax rates on higher incomes through a graduated rate structure with a changing marginal tax rate.

Ideally, one would calculate the effects of individual choices at each different possible marginal tax rate to determine the behavior that would maximize utility. For computational tractability, however, Fullerton and Rogers use a set of linear tax functions that approximate the U.S. system with a negative intercept for each group and a single marginal tax rate (0.25 in the 1993 benchmark). Although all individuals face the same marginal tax rate, average tax rates still increase with income because of the negative intercepts. Fullerton and Rogers do not model the myriad exemptions and deductions. Those simpler, linear tax functions can replicate the observed data on personal taxes actually paid by each group.

Property taxes and income taxes at all levels of government raise the producer's gross cost of capital for each type of asset compared with the investor's net rate of return. The cost of capital corresponding to each type of asset depends on the statutory corporate tax rate (set at 0.395 to reflect federal and state taxes in the 1993 benchmark), depreciation allowances at historical cost, how the real value of those allowances is eroded by the rate of inflation (set at 0.04), the rate of investment tax credit (set at zero after the Tax Reform Act of 1986), and the required net rate of return for the firm. That required rate of return depends, in turn, on the going market rate and the personal taxation of interest (at rate 0.246), dividends (0.292), and capital gains (0.13).

The simulations described in this study assume the "old view" of taxing dividends, in which the personal-level taxation of dividends affects the cost of capital for marginal investments.(4) A similar cost of capital formula applies to the noncorporate sector. That treatment allows the producer's choice among assets to depend on relative tax rules, and the price of output in each industry to depend on the relative use of assets with different tax treatments.

Other assumptions help to close the model in a way that accounts for all flows and that helps facilitate computation. The model ignores international mobility of labor or capital, but allows for the trade of industrial outputs. Also, the value of imports must match the value of exports; the government's expenditures and transfer payments must match tax revenue; and the value of personal savings must match the value of expenditures for investment. Producer investment is not the result of a firm's decisions about the timing of investment, but instead results from the levels of personal saving that consumers choose. The amount of personal saving is growing over time because consumers' earnings from labor are growing as a result of population and technical change. On the steady-state growth path, the capital stock grows at exactly the same rate as the effective labor supply.

Data used within the Fullerton-Rogers model derive from many sources, adjusted to represent 1993 as the base year.(5) In addition to the survey data used to estimate wage profiles and preference parameters, they use the national income and product accounts for an input-output matrix, labor compensation by industry, government purchases, and international trade. Those published data are combined with other unpublished data on capital allocations and inheritances.

For some parameters, such as the elasticities of substitution, particular values are assumed. For other parameters, such as the Stone-Geary preferences, econometric estimates are used. Finally, some remaining parameters are "calibrated" from data on actual allocations. Demand functions and all initial prices and observed quantities are used to solve backward for the value of the parameter that would make that quantity the desired one. That procedure establishes a "benchmark" equilibrium, with existing tax rules and prices. As a result, all consumers are buying the desired quantities and supplying the desired amounts of each factor, while producers are using their desired amounts of factors to produce the desired output.

Thus, using all of those parameters together, one can solve for an equilibrium with unchanged tax rules that replicates the benchmark's consistent data. That ability provides an important check on the procedure for solution. Then starting from that verified benchmark, any particular tax rule can be altered and one can determine how much more or less that consumers want to buy of each good. The model's algorithm then raises the price of any good in excess demand, and lowers the price of any good in excess supply, until it finds a set of prices at which the quantity supplied equals the quantity demanded for every good and factor. It simulates the effect of the tax change to calculate all new prices, quantities, and levels of consumer utility. The measure of the change in tax burden is the "equivalent variation," the dollar value of the change in utility measured in terms of benchmark prices. Gains in efficiency from a tax change are calculated as the present value of equivalent variations added over all income groups and all generations relative to the present value of lifetime incomes.

Results characterized as "short run" or "initial" correspond to an equilibrium immediately after the simulated policy change. Results characterized as "long run" or "steady state" reflect allocations and prices after 30 equilibria are achieved, calculated five years apart from each other. Although the 30th equilibrium is 145 years after the time of the tax change, that equilibrium is virtually identical (in terms of allocation of resources and relative prices) to an equilibrium that is 35 to 50 years out, at least in terms of the simulations discussed in this study.


1. Don Fullerton and Diane Lim Rogers, Who Bears the Lifetime Tax Burden? (Washington, D.C.: Brookings Institution, 1993).

2. Fullerton and Rogers, Who Bears the Lifetime Tax Burden?, Chapter 8. The book discusses the sensitivity of calculations of incidence to these parameter values. The current study emphasizes the importance of the intertemporal elasticity in determining the efficiency gains from a switch to consumption-based taxation.

3. This framework also allows Fullerton and Rogers to use the same utility function for everyone in the model. In previous efforts, rich and poor individuals spend in different proportions because they have different preferences. But then the rich and the poor differ in fundamental characteristics and not just by the amount of income they receive. With differences in utility functions, if the poor were to receive additional income, they would still spend it as if they were poor, according to their unchanged proportions. Fullerton and Rogers argue that it seems more natural that a poor person with more money would begin to behave like a rich person. That is, the primary distinction between rich and poor is the amount of income they receive. Therefore, in their model, everyone has the same preference parameters. The poor spend more on goods with high minimum required expenditures, because they are poor, and the rich spend more on goods with relatively high marginal expenditure shares.

4. See Fullerton and Rogers, Who Bears the Lifetime Tax Burden, pp. 210-213, for discussion of how adopting the alternative "new view" affects the efficiency and distributional effects of the various U.S. taxes.

5. The benchmark specified in the Fullerton-Rogers book is based on earlier (1984) data.

Previous Page Table of Contents Next Page