1

Draft Paper

A model of the optimal tax mix including capital

taxation

Abstract

This paper uses a three-tax linear model to study numerically the optimal tax mix. The

three aggregate taxes relate to labour, capital and commodities respectively. The most striking

result is that with a linear expenditure system (LES) based on econometric estimates, indirect

(i.e. commodity) taxation dominates the optimal tax mix, with over 80% of tax revenue coming

from indirect taxes. Although some analytical justification is presented for this surprising result,

it should be tested numerically using other econometrically estimated utility-demand systems.

Comprehensive computational modelling of the optimal tax mix has been so far missing from

the literature. Considering the results presented here, there seems to be considerable scope for

further research in this area. From a theoretical perspective, a major innovation in this paper is

the introduction of capital taxation into a static tax mix model. The analytical part of the paper

provides approximate predictions for optimal linear tax rates (including capital tax) called the

modified inverse elasticity rule. Using these formulas, it is possible to provide some

explanations for the dominance of indirect taxation in the optimal tax mix.

keywords: Optimal taxation, capital taxation, income taxation, indirect taxation

JEL Classification: H21, H23, H26, C63

Introduction

This paper uses a three-tax linear model to study numerically the optimal tax mix. The

three aggregate linear taxes relate to labour, capital and commodities respectively. The most

striking result is that with a linear expenditure system (LES) based on econometric estimates,

indirect (i.e. commodity) taxation dominates the optimal tax mix, with over 80% of tax revenue

coming from indirect taxes. From a theoretical perspective, a major innovation in this paper is

the introduction of capital taxation into a static tax mix model. The analytical part of the paper

provides approximate predictions for optimal linear tax rates (including capital tax), called the

modified inverse elasticity rules. Using these formulas, it is possible to examine the reasons for

the dominance of optimal indirect taxation.

The three-tax model studied here is a fairly comprehensive one, incorporating labour,

capital and indirect taxes. It is assumed that there is only one linear labour income tax rate,

denoted 𝜏𝐿 , and a single capital income tax rate, denoted 𝜏𝑘. Labour income is denoted zi = wiLi, where wi is the fixed wage rate for household i and Li is the corresponding labour supply.

Capital income is capital (k) multiplied by a fixed average pre-tax real rate of return (r), that is

kr. r and wi are assumed to be fixed. The net after-tax rate of return is 𝑟(1 − 𝜏𝑘). ̀ r´ is measured after taking away depreciation and the remuneration of self-employed.

2

It is assumed that capital income is taxed separately from labour income. 𝜏𝑐 is a uniform proportional indirect tax rate on all commodities, 𝑝𝑗 are constant producer prices and 𝑞𝑗 are

quantities. Total consumption is: C= ∑ 𝑝𝑗𝑞𝑗𝑗 .

This model treats capital as merely an income generating tool, which means that unlike

commodities and leisure, it does not enter directly into the social welfare function (U). It is

assumed that there is only one type of capital (k), with a uniform rate of return (r). However,

wages and commodity prices can be heterogeneous. Then the linear model is given as:

maximize U = ∑ 𝑢𝑖(𝐪𝒊, 𝐿𝑖)𝑖 (1)

s.t. the budget constraint ∑ (𝜏𝐾𝑖 𝑘𝑖𝑟 + 𝜏𝐿𝑤𝑖𝐿𝑖 + 𝜏𝐶 ∑ 𝑝𝑗𝑞𝑗𝑖 𝑗 ) ≥ 𝐻𝑏 + 𝐺 (1a)

Income redistribution is carried out by providing a uniform demogrant (b) to every member of

the population. G is fixed public goods expenditure requirement. H is total population. In the

numerical work a linear expenditure (LES) utility function was used.1 Note that capital is

missing from the social welfare function in (1) but enters constraints (1a).

Aggregate capital supply is an increasing function of the after-tax rate of return to

capital, 𝑟(1 − 𝜏𝐾), and total real income M.

Symbolically: 𝐾 = 𝜑(𝑟(1 − 𝜏𝐾), 𝑀) (2)

A specific form of this function will be presented in eq. (18).

By definition:

rK + wL = 𝑀 = ∑ 𝑝𝑗𝑞𝑗𝑗 + 𝐺 = 𝐶 + 𝐺 (3)

In these equations: K is the aggregator of capital stock; L is the aggregator of labour supply. C

is total consumption. 𝑀 is price-indexed total income.

Defining lump-sum income as y = rK, then from (3):

𝜕𝐶

𝜕𝑦 = ∑ 𝑝

𝑗

𝜕𝑞𝑗

𝜕𝑦𝑗= 1 + 𝑤

𝜕𝐿

𝜕𝑦+ 𝑟

𝜕𝐾

𝜕𝑦 (4)

The social marginal utility of income is: 𝑔𝑖 = 𝜕𝑢𝑖

𝜕𝑚𝑖 =

𝜕𝑢𝑖

𝜕𝑦 (5)

Essentially, 𝑔𝑖 reflects political value judgments concerning income distribution. The relevant utility transformation parameter is referred to in the literature as the inequality aversion rate.

In the present model an inequality aversion rate of one corresponds to LES utility. A rate of

zero corresponds to equal marginal utilities for everyone, i.e. the absence of distributional

objectives. A rate between zero and one is a convex combination of the two.

1 This model is not zero homogeneous due to the presence of an endogenously determined lump-sum grant

(b). The indeterminacy of the optimum with zero homogeneous utility is discussed in Mirrlees (1976, pp. 330-

332)

3

The model described above, excluding capital taxation, is close to the standard income-

commodity tax models in the optimal taxation literature. Like other optimal taxation models, it

can highlight some aspects of the equity-efficiency trade-off. However, it has severe limitations

due to its high level of abstraction. First, population is heterogenous not only due to differences

in wage rates and capital incomes, but because of other features as well. Perhaps the most

important among these is household composition. Also, political preferences are only partly

captured by the single inequality aversion rate and the uniform demogrant. Distributional

preferences cover complex political considerations in relation to incomes, household

composition, place of residence, source of income, and with support grants (b), also in respect

to the provision of education and health services. Altogether, optimal taxation models, like the

present one, can offer some insights, but only a limited guide to policy.

The paper is structured as follows: The next section presents a literature review on the

subject. It is followed by approximate formulas for optimal linear capital, labour and

commodity tax rates, based on the specifications described in eq. (1). Details of the

mathematical analysis are presented in the Online Supplemental Appendix. The next sections

examine the computational model. It starts by explaining the optimisation methodology. This

is followed by numerical results. The interior point numerical optima will be compared with

the predictions from the modified inverse elasticity formulas. Finally, some explanations will

be provided to account for the dominance of indirect taxation in the optimal tax mix.

The final section presents summary and conclusions. The main conclusion is that there

is a need for further computational research, to test the validity of the conjecture about the

dominance of indirect taxation, using other empirically based utility-demand functions.

The location of the computer program used to produce the numerical results appears in

the references under Revesz (2019). The same website also contains the user’s guide. The

interested reader can experiment with this program for checking the numerical results presented

here and for testing results under different parameters and specifications.

Related literature

It is impossible to review in this short paper the large literature pertaining to the tax mix

issue. The discussion here starts with optimal taxation theory. The optimal tax mix model

involving labour and commodity taxes has been introduced into the literature by Atkinson and

Stiglitz (1976) and Mirrlees (1976). An important result is the Atkinson-Stiglitz (1976)

theorem. This theorem states that provided income tax is the control theoretic non-linear

solution and utility is weakly separable between commodities and leisure, there is no need to

apply commodity taxation in conjunction with income tax. Income tax will be sufficient to

generate all tax revenue. This is exactly the opposite to what is coming out from the present

linear model, which suggests that under linear taxation, the bulk of tax revenue should be

generated from indirect taxation rather than income tax.

4

The Atkinson-Stiglitz theorem is based on a number of not too realistic assumptions,

such as the absence of tax evasion, the absence of administration and compliance costs, weakly

separable utility between commodities and leisure, no externalities, no paternalistic concerns

and the assumption that the control theoretic continuous and differentiable income tax function

can be implemented in practice. Because of these assumptions, the Atkinson-Stiglitz (1976)

theorem had been criticised in the literature, particularly in connection with the debate about

indirect tax uniformity. The interested reader can find comprehensive reviews in Tuomala

(2016, Chapter 12) and Revesz (2018 pp. 3-28).

Some of the above-mentioned omissions also apply in the present model, however if tax

evasion, administrative and compliance costs, leisure complements-substitutes, externalities

and paternalistic concerns would have been included, they would have further reinforced the

conclusion in favour of indirect taxation.

Another subject from the optimal taxation literature that has been taken up in the present

model is capital taxation. There is a large and disparate literature on this subject. A recent

literature review on optimal capital taxation and related topics is presented by Bastani and

Waldenström (2018). A shorter literature review is contained in Piketty and Saez (2012, pp. 1-

3). Studies that show some similarities in their mathematical framework to the present model

on capital taxation include Saez and Stantcheva (2018) and Piketty and Saez (2012, 2013).

Note, the present model is largely irrelevant to several issues raised in the capital taxation

literature. These include the treatment of contributions to retirement benefit funds (Diamond,

1999), and the role of capital taxation in alleviating the uncertainties of life (Conesa et al., 2008;

Golosov et al., 2006; Krueger and Ludwig, 2018).

Apart from the Atkinson-Stiglitz theorem and optimal capital taxation, there is little else

in the optimal taxation literature that is closely related to the subject matter of this paper.

A more relevant strand in the literature is what could be described as policy-related

empirical studies on the tax mix. These studies examine empirical observations on the direct-

indirect tax mix combined with economic reasoning and econometric estimates, particularly on

the relationship between the tax mix and economic growth. These reports include Fitoussi

(2005), Martinez-Vazquez et. al (2009, pp. 1-27), Arnold et al. (2011) and European

Commission (2013, pp. 42-51). Generally, these policy-related reports are in favour of indirect

taxation, advocating greater reliance on commodity and property taxation at the expense of

direct taxes. IMF (2014, pp. 36-43) also deals with this subject, but sees a limited role for

indirect taxation in the advisable tax mix, because of its regressivity.

The main arguments advanced against income taxation (both labour and capital) are:

• The labour and saving disincentives related to progressive income taxation, which are absent with indirect taxation.

• The lower evasion propensities of indirect taxes on goods and services produced and/or marketed through large organisations.

• Simpler administration and lower administrative and compliance costs of indirect taxes compared with direct taxes.

5

One may add to these several other factors in favour of indirect taxation, such as the

need for higher taxes to offset negative externalities, more scope for taxing/subsidising leisure

complements and substitutes, as well as more scope for accommodating paternalistic concerns.

None of these factors appear in the present model. Effectively, the argument in favour of

indirect taxation presented here is separate and additional to arguments that appeared elsewhere

in the literature. Generally, the above-mentioned empirical studies are in line with the main

conclusion of this paper, but unlike the present study, they are not based on comprehensive

mathematical modelling in the optimal taxation tradition.

The modified inverse elasticity rules

This section concentrates mainly on capital taxation, but labour and commodity taxation

will be covered as well. The present optimal capital and labour income taxation model is in

some respects a static version of the linear steady-state model developed by Saez and

Stantcheva (2018).

Regarding capital taxation some definitions and explanations are needed. Suppose we

have an international panel data spanning more than 20 years, which links capital income per

person to the average capital income tax rate. From that longitudinal and cross-country

database, it is possible to obtain through linear regression a parameter estimate for the derivative

of real (inflation adjusted) capital income per person (kr) over the average capital income tax

rate (𝜏𝑘), as well as its derivative with respect to real per capita income (m). To the best of my kmowledge, no extensive cross-country research has been carried out so far to estimate the tax

elasticity of capital, although the required data are available in many countries. Such data are

also available for more disaggregated estimates of the tax elasticity of capital supply, according

to type of investment and source of investment. For example: domestic savings, capital inflow

and outflow, foreign direct investment (FDI), investment in machinery and equipment,

bequests, corporate income, etc.

Denoting total capital as K, the following discussion will use 𝜕𝐾

𝜕(1−𝜏𝐾) instead of

𝜕𝐾

𝜏𝐾 ,

in order to produce comparable expressions to Saez and Stantcheva (2018). Assuming that the

central parameter estimates in the model are based on observations over a long period of time,

the static model may be regarded to represent a long-run model. The tax elasticity of capital

supply is defined in respect to the after-tax return to capital as:

𝑒𝐾 = (1−𝜏𝐾 )

𝐾𝑟

𝜕𝐾𝑟

𝜕(1−𝜏𝐾) (6)

Saez and Stantcheva (2018) used three estimates for 𝑒𝐾 in their numerical work. These are 0.25, 0.5 and 1. Perhaps 𝑒𝐾 = 1 is the most realistic, bearing in mind the strong international mobility of capital. In any case, in the present computational work all three estimates are used.

In equation (1a), ∑ 𝜏𝐾𝑖 𝑘𝑖𝑟 represents “real” capital tax revenue and not nominal revenue. The difference between the two is the dead-weight loss associated with raising the

capital tax rate from 0 to 𝜏𝐾. From the theory of excess burden (Hoy et al (2011), given a linear

supply curve, this amounts to the triangle 𝑟𝜕𝐾

𝜕𝜏𝐾 𝜏𝐾

2

2 . Hence, in nominal terms equation eq. (1a)

should appear as:

∑ (𝜏𝐾𝑖 𝑘𝑖𝑟 −𝑟𝜕𝐾

𝜕𝜏𝐾 𝜏𝐾

2

2 + 𝜏𝐿𝑤𝑖𝐿𝑖 + 𝜏𝐶 ∑ 𝑝𝑗𝑞𝑗𝑖 𝑗 ) ≥ 𝐻𝑏 + 𝐺 (1b)

6

It is applied to reduce “real” capital income in the numerical simulations. It is also used to work

out the inverse elasticity rule for capital in the Online Supplemental Appendix.

The specifications described above yield an approximate formula for the optimal

capital income tax rate. This formula can be expanded also to optimal linear labour income tax.

The approximation to the optimal linear capital and labour income tax rates follows the same

procedure described in respect to approximate optimal indirect tax rates in Revesz (2014, pp.

47-51).

The solution for the optimal capital tax formula is based on deriving the welfare function

in (1) in respect to 𝜏𝐾 and then equalizing the sum of resulting components to zero. There are three welfare effects to consider following a change in 𝜏𝐾: (i) the income effect on capital owners, (ii) the effect on social utility of variations in tax revenue and consequently in the

uniform lump-sum grant (b), (iii) the variations in dead-weight costs that arise due to tax related

distortions in the allocation of resources (based on definition (1b)).

Details of the derivations for the approximately optimal capital tax rate are presented in

the Online Supplemental Appendix. The end result, titled the modified inverse elasticity rule of

capital taxation (from A.15 in the Appendix) is:

𝜏𝐾

(1−𝜏𝐾)=

− �̅�𝑘�̅�𝐻

+ �̅�𝑏�̅�𝐻

( 1 − 𝜏𝐶𝜕𝑐

𝜕𝑦 − 𝜏𝐿

𝑤𝜕𝐿

𝜕𝑦)

𝑒𝐾{1

(1− 𝜏𝐾) −

𝜏𝐾2(1− 𝜏𝐾)

(1+ 𝑒𝐾)} (7)

The average social marginal utility of income of capital income recipients is defined as �̅�𝑘, the average for demogrant recipients is �̅�𝑏 , while the average for the entire population is �̅�𝐻. Since

everybody is receiving the demogrant �̅�𝑏 = ∑ 𝑔𝑖𝑖

𝐻⁄ , while �̅�𝐻 is the income-weighted

average: �̅�𝐻 = ∑ 𝑔𝑖𝑖 𝑚𝑖

𝑀⁄

The 𝑤𝜕𝐿𝑖

𝜕𝑦𝑖 term represents the change in labour income in response to variation in lump-

sum income. 𝜕𝑚𝑖

𝜕𝑦𝑖= 𝑤

𝜕𝐿𝑖

𝜕𝑦𝑖 is known in the labour supply literature as the income effect (see

Hum and Simpson (1994, pp. 72-74)). Its estimated value usually ranges between -0.1 to -0.4.

It is not difficult to see that the denominator on the right-hand side is usually close to

𝑒𝐾. When 𝑒𝐾 = 1 then the denominator will be 𝑒𝐾. When 𝑒𝐾 = 2 then the denominator will

be 𝑒𝐾{1 - 𝜏𝐾

2(1− 𝜏𝐾) }. When 𝑒𝐾 = 0.5 then the denominator will be: 𝑒𝐾{1 +

𝜏𝐾

4(1− 𝜏𝐾) }.

Using the methodology described in the Online Supplemental Appendix, it is possible

to generalise the inverse elasticity rule of capital taxation to linear labour income tax as well.

In this case the (compensated) elasticity of labour supply with respect to the net after-tax

return, 𝑤(1 − 𝜏𝐿), is defined as:

𝑒𝐿(u0) = (1−𝜏𝐿 )𝑤

𝑤𝐿

𝜕𝐿(u0)

𝜕(1−𝜏𝐿) . (8)

7

Econometric estimates of eL(u0) usually range between 0.15 and 0.5 (Hum and Simpson, 1994,

pp. 72-74). Higher estimates can be obtained when considering not only changes in working

hours, but also long-term responses related to occupational choice and human capital

accumulation. In the present numerical work, the mean value of eL(u0) is given between 0.40

and 0.48, depending on the scenario.

The average social marginal utility of wage earners is defined as: 𝑔𝐿 =∑ 𝑤𝑖𝐿𝑖𝑔𝑖𝑖

∑ 𝑤𝑖𝐿𝑖𝑖. The

marginal capital-income ratio is 𝜕𝐾

𝜕𝑚 , where total income m = Kr + wL. In this paper

𝜕𝐾

𝜕𝑚 =

𝜕𝐾

𝜕𝑦

defines the net change in capital stock related to a change in static income, after taking away

depreciation. Based on empirical evidence, it is around 3. Using these new variables, and

following the same steps outlined in the Online Supplemental Appendix, will lead to the

following modified inverse elasticity rule for the optimal linear labour income tax:

τL

(1-τL)=

- g̅Lg̅H

+ g̅bg̅H

( 1 - τC∂c

∂y - 𝜏𝐾𝑟

𝜕𝐾

𝜕𝑚)

𝑒𝐿(𝑢0){1

(1− 𝜏𝐿) −

𝜏𝐿

2(1− 𝜏𝐿)(1+ 𝑒𝐿(𝑢0))}

(9)

Deriving the modified inverse elasticity rule in respect to commodity taxation follows

the methodology described in Revesz (2014, pp. 47-51). Eq. (9) there gives the predicted linear

tax rate for a single composite good as:

𝜏𝐶 ≈ 1 + 𝜏𝐶 −

g̅bg̅C

(1 + 𝜏𝐶 + 𝜀�̅�𝐿𝜏𝐶)

𝜀�̅�(𝑢0) (10)

Here g̅C

represents the weighted average social marginal utility of consumers and g̅b that of

demogrant recipients. �̅�𝑐𝐿 =𝜕𝐿

𝜕(1+𝜏𝐶)

(1+𝜏𝐶)

𝐿 is the elasticity of labour supply over the cost of

consumption. Taking zero homogeneous labour supply, where b is an exogeneous variable

the following condition must be satisfied:

𝜕𝐿

𝜕(1+𝜏𝐶)𝑐 +

𝜕𝐿

𝜕(1+𝜏𝐿) 𝐿 +

𝜕𝐿

𝜕(1+𝜏𝐾) 𝐾 +

𝜕𝐿

𝜕𝑦 𝑏 = 0 (11a)

Assuming that the last two terms are relatively small, leads to the following approximation:

𝜀�̅�𝐿 = −𝜖�̅�(1+𝜏𝐶)

(1+𝜏𝐿) , (11b)

where 𝜖𝐿 = 𝜕𝐿

𝜕(1+𝜏𝐿)

(1+𝜏𝐿)

𝐿 is the uncompensated elasticity of labour supply. This approximation

opens the way for estimating the unknown 𝜀�̅�𝐿 using estimates for ∈̅𝐿 . According to empirical estimates (Hum and Simpson, 1994, pp. 72-74), ∈̅𝐿 lies in the range between +0.4 and -0.15, hence 𝜀�̅�𝐿 will be oppositely signed in a similar range. The relatively small value of 𝜀�̅�𝐿 will be of considerable importance when explaining the dominance of indirect taxation.

When τL and τK are also included in the model, they should be added to the effect of

change in labour supply on tax revenue. Consequently, in the present model a better

approximation to (10) will be:

𝜏𝐶 ≈ 1+𝜏𝐶 −

g̅b

g̅H(1+ 𝜏𝐶+ �̅�𝐶𝐿 (𝜏𝐶 + 𝜏𝐿𝑊

𝜕𝐿

𝜕𝑦 + 𝜏𝐾𝑟

𝜕𝐾

𝜕𝑦)

�̅�𝑐(𝑢0) (12)

This is the predictive formula for 𝜏𝐶 used in the numerical analysis.

8

Finally, a few words about the �̅�𝐿(𝑢0) and 𝜀�̅�(𝑢0) terms appearing in the denominators of (9) and (12). According to demand theory, the dead-weight loss of taxation is represented by

the triangle defined by the difference between the post-tax and pre-tax price multiplied by the

change in the quantity demanded. In its turn, the change in quantity is given by the change in

price multiplied by the derivative of compensated demand (not ordinary demand) (see Hines,

1998). In symbolic terms the dead-weight loss (D) following an increase in the labour tax from

zero to 𝜏𝐿 will be:

𝐷(𝐿) = 𝜏𝐿

2(

𝜕𝑊𝐿

𝜕𝜏𝐿(𝑢0)𝜏𝐿) =

𝜏𝐿2

2

𝜕𝑊𝐿

𝜕𝜏𝐿(𝑢0) ≈ −

𝜏𝐿2

2𝑒𝐿(𝑢0)

𝑊𝐿

(1−𝜏𝐿) (13)

Similarly, for consumption tax: 𝐷(𝐶) ≈ − 𝜏𝐶

2

2𝑒𝐶(𝑢0)

𝐶

(1−𝜏𝑐) (14)

where 𝑢0 designates compensated labour supply and compensated commodity demand respectively. In the case of capital supply, the dead-weight loss is defined not by a utility

function, but by constraint (1b). In this case the dead-weight loss is defined as: −𝑟𝜕𝐾

𝜕𝜏𝐾 𝜏𝐾

2

2 , and

there was no need to apply a compensated elasticity approach.

The framework and specifications of the numerical model

There are several utility functions, including the LES, CES, AIDS, NLPS and direct and

indirect addilog, which define explicitly not only utility, but also the corresponding demand

and labour supply functions (see Thomas, 1987) for a given set of post-tax prices and incomes.

In the present computational work Linear Expenditure System (LES) utility-demand

was used. The reason for this choice was that already much earlier, we developed a numerical

tax model dealing with the question whether indirect taxation should be uniform or otherwise

(see Revesz 1997, 2014). The original idea was to extend this model to include also the

optimisation of labour and capital income tax.

Later, it was discovered that the original optimisation method, based on “gradient”

iterations, was not well-suited to this task, hence we turned to the three-tax “grid” search

optimisation method. Nonetheless, the basic model has been retained, including the LES utility

function, 15 taxpayers and 9 or 18 commodities.

LES appears in many publications. For one, see Thomas (1987, pp. 71-74). The present

version uses instead of income (m) the earning parameters w and y, as well as total time (Z)

and leisure (L). The LES utility function is defined as:

𝑢 = ∑ 𝛽𝑖 log (𝑞𝑖𝑖 − 𝛼𝑖) (15)

where qi represents the consumption of good i. 𝛼𝑖 and 𝛽𝑖 are function parameters. The corresponding demand functions for commodities are:

𝑞𝑖 = 𝛼𝑖 + 𝛽𝑖

𝑝𝑖 (𝑍𝑤 + 𝑦 − ∑ 𝑝𝑗 𝛼𝑗 𝑗 ) (16)

In the present model w = W(1 - 𝜏𝐿) and y = kr(1 - 𝜏𝐾) + b

Labour supply (L) is given as:

𝐿 = 𝑍 − 𝑞𝐿 = 𝑍(1 − 𝛽𝐿 ) − 𝛼𝐿 − 𝛽𝐿

𝑤(𝑦 − ∑ 𝑝𝑗𝑗 𝛼𝑗 ) ` (17)

9

where 𝑞𝐿 stands for leisure. Z is the upper limit of time available to work, therefore 𝑞𝐿 = 𝑍 − 𝐿

In the computational model it is assumed that constant elasticity capital supply from (3)

is given in the form: 𝑟𝐾 = 𝑟(𝐾0 +𝜕𝐾

𝜕𝑚(𝑚 𝑃⁄ − 𝑚0) )(1 − 𝜏𝐾)

𝑒𝐾 (18)

where 𝜕𝐾

𝜕𝑚 is the change in net capital (after depreciation) and m/P is price-indexed total income,

given as: m = (wL + rK)/P. From (18) we can obtain a more precise definition of dead-weight

loss than in (1b). The change in dead-weight costs when moving from 0 to 𝜏𝐾 will be then:

∆𝐷(𝐾) = − 𝑟𝑒𝐾(𝐾0 +𝜕𝐾

𝜕𝑚(𝑚 𝑃⁄ − 𝑚0))

(1− 𝜏𝐾)(𝑒𝐾−1)𝜏𝐾

2

2 (19)

The numerical model that produced the results presented in Table 1 has the following

features: Initially an ordinary LES function was used, covering 9 commodities and labour. Later

a two-part LES covering 18 goods was introduced. There are 15 taxpayers in the model. The

LES utility function satisfies quasi-concavity. Combined with a convex budget constraint that

ensures a unique global optimum. The inequality aversion rate was set to 0.7, in order to obtain

an overall tax burden in the range between 40% and 50%, as observed in several advanced

economies. The demand parameter estimates for the 9 goods model were taken from Deaton

and Muellbauer (1980, p 71). The labour and capital income distributions are blended from

several empirical distributions. Usually, expenditure on public goods is fixed, set to 10% of

initial tax-free total income. The computer program related to this paper (Revesz (2019))

displays in the DATA lines all the parameters and income values adopted in the numerical

work.

The search for the optimum was carried out using a “grid” search method. This method

involves checking all tax combinations in order to find the optimum. In the present model

discrete tax rates were separated by intervals of 0.01. With indirect taxation the calculations

were carried out from 0.1 to 1.1, implying 100 discrete taxes. With labour and capital taxation

the range was from 0.1 to 0.7, implying 60 discrete taxes. Altogether, 100*60*60 = 360,000

combinations were examined. At each round, the program compared the utility level of the

present combination with the highest utility recorded earlier. If the utility of the present

combination turned out to be higher than the earlier one, then it replaced the top utility

combination. Otherwise, no change was made.

The model was restricted to only three taxes, because this way the program could be run

in less than an hour on a personal computer. A four-tax model (say with two indirect tax rates,

one for luxuries and one for necessities, or with two income tax brackets) could run on more

powerful computers. Beyond that, the number of combinations will run into billions. It is

possible that less computation-intensive methods could be used to locate the global optimum,

but for the time being this subject has not been taken up.

The specifications in Table 1 are restricted to the choice of taxes, the capital supply

elasticity and the percentage of public goods from total income. These choices can be made

through prompts in the Revesz (2019) program. Other choices can be made in the program by

changing DATA lines, as explained in the user’s guide. Overall, the percentage share results

indicate that indirect taxes dominate the optimal tax mix, usually accounting for over 80% of

total tax revenue.

10

Table1

Numerical results with ordinary LES

No. of

taxes

Elasticity

of capital

% public

goods

Indirect

tax

Labour

tax

Capital

tax

change

welfare

2 - 0 Actual 0.61 0.1

Predicted 0.55 -

% share 86 14

Compared with 40% uniform tax % change 10 10 2

2 - 10 Actual 0.68 0.1

Predicted 0.61 -

% share 86 14

Compared with 40% uniform tax % change 8 8 2

2 1 10 Actual 0.42 0.10

Predicted 0.59 -

% Share 91 9

Compared with 40% uniform tax % change 9 -7 59 5

3 1 10 Actual 0.60 0.10 0.10

Predicted 0.61 - -

% Share 84 11 5

Compared with 40% uniform tax % change 22 4 69 14

3 0.25 10 Actual 0.59 0.10 0.12

Predicted 0.60 - 0.17

% Share 83 11 6

Compared with 40% uniform tax % change 15 10 21 6

3 0.25 0 Actual 0.48 0.10 0.12

Predicted 0.51 - 0.20

% Share 82 11 7

Compared with 40% uniform tax % change 17 14 22 6

Source: Own calculations using the program in Revesz (2019).

Predictions are only presented for cases where the actual tax rate is an interior point,

located above the 0.1 lower boundary line. For optimum values located at the boundaries, it is

not possible to apply predictive formulas, because these only apply at the interior. In all cases

the indirect tax rate is an interior point, but only in a few cases does this apply to labour or

capital tax.

The numbers in the fourth row require further explanations. The first three numbers

relate to quantities corresponding to tax rates, e.g. with indirect tax the corresponding quantity

will be total consumption. Each quantity in the optimal solution is compared with the

corresponding quantity obtained when all taxes are set to 40%. This percentage was chosen for

comparison because it is close the average tax burden in most scenarios. The purpose of the

exercise is to provide the reader a comparative view about the changes involved in moving from

a uniform solution to the highly differentiated optimal solution.

The change in welfare is more complicated. It is intended to reflect the change in utility

in monetary terms. It is based on dividing the change in utility by the Lagrangian using the

following formula: ∆𝑈∗ = 𝑈𝑜𝑝𝑡− 𝑈40%

𝜆𝑀𝑜𝑝𝑡 (20)

where 𝜆 = 𝑈(𝜏, 𝐺 + 1) − 𝑈(𝜏, 𝐺) and 𝑀𝑜𝑝𝑡 is total income in the optimal situation.

11

Equation (20) provides some tentative indication about the proportional change in welfare in

monetary terms.

The first scenario in Table 1 is a two-tax model involving only indirect and labour

taxation and no expenditure on public goods. The second scenario is the same, but with 10%

expenditure on public goods included. The results from both scenarios illustrate the clear

dominance of indirect taxation in the optimal tax mix.

The third scenario shows a situation where indirect taxation is missing. In this case

optimal labour income tax is much higher than the capital tax rate, which does not even move

above the boundary line, i.e. 0.1. This is in line with the inverse elasticity rule. In this model

the compensated elasticity of labour supply is 0.4, whereas that of capital is 1. Therefore, a

much higher optimal tax rate on labour than on capital is to be expected.

It should be noted that the optimums do not represent sharp peaks. They are top solutions

among many other solutions with similar high utilities. These high-utility solutions sometimes

involve tax combinations that are markedly different from the combination at the optimum, but

to a lesser extent, even in these cases the dominance of indirect taxation is maintained.

Notice that predicted tax rates are usually not far away from the actuals. This fairly close

concordance between actual and predicted tax rates, will help to explain later the puzzling

dominance of indirect taxation in the optimal tax mix. The next three scenarios, covering three-

tax models, further reinforce the conjecture about the dominance of indirect taxation.

Table 2

Numerical results with two-part LES

No of

taxes

Elasticity

of capital

% public

goods

Indirect

tax

Labour

tax

Capital

tax

Change

welfare

2 0 Actual 0.64, 1.28 0.10

Predicted 0.74 -

% Share 87 13

Compared with 40% uniform tax % change 5 6 2

2 1 10 Actual - 0.48 0.10

Predicted - 0.60 -

% Share - 92 8

Compared with 40% uniform tax % change 4 -11 54 3

3 1 0 Actual 0.55, 1.10 0.10 0.11

Predicted 0.0.81 -

% Share 84 11 5

Compared with 40% uniform tax % change 18 1 66 13

Source: Own calculations using the program in Revesz (2019).

So far, the discussion was concerned with ordinary LES covering 9 goods. In earlier

work (see Revesz, 2014, p. 18; 2018, p. 39) LES was divided into two functions with different

parameters, one for low wage earners the other for high wage earners. The two-part LES models

cover 18 goods, 9 for necessities and 9 for luxuries. This arrangement was needed in order to

open the way to non-linear Engel curves with LES. Notice that because of the introduction of

9 additional goods, the utility function of two-part LES is not the same as that of ordinary LES.

12

Some simulations were done with two-part LES. In was assumed in these simulations

that the tax rate on the nine luxuries is twice the tax rate on the nine necessities. This way, still

only one consumption tax had to be optimised, the second one was just consequential. As shown

in Table 2, the pattern of results with two-part LES turned out to be very similar to ordinary

LES. The introduction of the luxury tax made indirect taxation even more attractive, hence the

dominance of indirect taxation has increased slightly.

An explanation for the dominance of indirect taxation

In all models that included indirect taxation, it emerged as the dominant source of

revenue. This section will try to explain this phenomenon using the modified inverse elasticity

rules, described in eqs. (7), (9) and (12).

As noted earlier, there is reasonably good agreement between predicted values and

actual values, hence the inverse elasticity rules may help to gain a better understanding of the

mechanics driving the puzzling outcomes. These explanations will look not only at variables,

but also at some of the estimated numbers behind them.

Looking at the modified inverse elasticity rules in (7), (9) and (12) perhaps the most

striking feature is the large difference in the structure of predictive equation for 𝜏𝐶 in (12)

compared with the predictive equations for 𝜏𝐾 in (7) and for 𝜏𝐿 in (9). Not surprisingly, this difference will provide a major clue for the explanation.

In eqs. (7) and (9) the 𝜏𝐶 term is multiplied by 𝜕𝑐

𝜕𝑦. From (4):

𝜕𝑐

𝜕𝑦 = ∑ 𝑝𝑗

𝜕𝑞𝑗

𝜕𝑦𝑗

= 1 + 𝑤𝜕𝐿

𝜕𝑦+ 𝑟

𝜕𝐾

𝜕𝑦

The “income term”, 𝑤𝜕𝐿

𝜕𝑦 , in the program stays constant at -0.24. The interest rate, r, stays

constant at 0.07 and the capital-income ratio 𝜕𝐾

𝜕𝑦 is given as 3. Thus, in numerical terms

𝜕𝑐

𝜕𝑦 = 1

- 0.24 + 0.21 = 0.97. Hence, in the capital and labour formulas negative 𝜏𝐶 is multiplied by nearly 1. Assuming a 20% tax on labour and capital, the other terms inside the brackets are

small: 𝜏𝐿𝑤𝜕𝐿

𝜕𝑦= 0.2 ∗ (−0.24) = −0.048 and 𝜏𝐾𝑟

𝜕𝐾

𝜕𝑚 = 0.2*0.07*3 = 0.042

The situation is markedly different in the consumption tax equation (12). Here negative

𝜏𝐶 is multiplied by the labour cross-elasticity term, 𝜀�̅�𝐿, which is close to the negative of ∈̅𝐿 , according to (11b). In the present model 𝜀�̅�𝐿 ranges between -0.1 and -0.3, depending on the scenario. Hence, there is a situation where 𝜏𝐶 is multiplied by nearly 1 in the labour and capital tax equations, and by a negative number close to zero in the consumption tax equation.

That means that in the labour and capital equations 𝜏𝐶 has a strong negative effect in reducing optimal tax rates, whereas in the consumption tax equation 𝜏𝐶 has a small positive effect. This appears to be the main reason for high optimal tax rates on consumption and low tax rates on

capital and labour.

13

This conjecture is supported by another observation that contradicts a narrow

interpretation of the modified inverse elasticity rules. The compensated consumption elasticity

𝜀�̅�(𝑢0) lies in the range -0.8 to -0.85, depending on the scenario. By contrast, the compensated elasticity of labour 𝜀�̅�(𝑢0) lies in the range of 0.4 to 0.48. From a straight interpretation of the inverse elasticity rules, that implies that consumption should be taxed at a lower rate than labour

income. But in practice exactly the opposite happens, apparently because of the different roles

of 𝜏𝐶 in these predictive equations. This explanation implies that in tax mix models, optimal tax rates might be substantially different from those in non-mix models.

Some policy implications

At this stage the question arises: how do tax mixes look like in the real world? According

to statistics presented by Martinez-Vazquez et al. (2009, p. 5), in 2005 the average worldwide

direct-indirect tax ratio stood at around 0.95. However, there are large variations between

countries. The average for developed countries was around 2.05 while for developing countries

it was 0.7. Developing countries rely much less on direct taxation, because they lack adequate

accounting capabilities. Also, social security contributions, which are classified as direct tax,

are much less important in the developing world.

It appears that little change has occurred in recent years in Organisation for Economic

Cooperation and Development (OECD) countries. OECD (2019, Figure 3.20) reports that the

average standard VAT rate in OECD countries has increased from 18% in 2000 to 19.3% in

2019.

Fitoussi (2005) nominates two main reasons against shifting from income tax to

consumption tax. First such a move may increase, at least temporarily, inflationary pressures

and related disequilibrium problems. Second, such a move is likely to reduce tax progressivity.

The first point seems realistic, even if the tax reform is designed to be revenue neutral. The

second point is more debatable. Even if the current indirect taxation system is regressive, a

reform could be highly progressive, particularly if it involves taxing at higher rates luxury goods

and imposing higher property rates on luxury housing. Such a reform appears administratively

feasible in the computer age, though its political acceptability may be in doubt. All in all, apart

from possible inflationary pressures and adjustment problems, there seems to be no strong

economic reason against replacing part of income tax by progressive indirect taxation.

Summary and conclusions

This paper presented analytical and computational results in relation to a static three-

tax model of labour, capital and indirect taxation. Perhaps, the most significant finding in this

paper, is the dominant role of indirect taxation in the optimal tax mix. It is possible that this

result was partly due to some unique characteristics of the LES utility function.

Although the numerical findings are also supported by some analytical results, there is

a need for further research, to test the validity of the conjecture about the dominance of indirect

taxation in other models, under different utility function and capital supply specifications. The

explicit utility-demand functions that could be tested apart from LES include: CES, AIDS,

NLPS and direct and indirect addilog.

14

Another interesting topic in future numerical research could be, the estimated welfare

losses involved in adopting a “politically acceptable” solution, rather than the mathematically

optimal solution. The presence of many sub-optimal solutions with a similar utility level to the

optimum, suggests that it may be unwise to concentrate exclusively on the mathematical

optimum.

Some attention was given to approximate formulas for optimal linear capital and labour

income taxes, titled the modified inverse elasticity rule. These formulas were used to examine

the optimal linear taxation of capital and labour. They were helpful in explaining the puzzling

numerical results on the dominance of indirect taxation.

The incorporation of capital taxation does not appear to make much difference in the

results. Even in a two-tax model without capital taxation, there is a strong predilection in favour

of indirect taxation. But in this context, let us not forget that low capital taxation is politically

acceptable only in societies that are not strongly averse to inherited wealth or some other socio-

political aspects of high capital concentration, as implicitly assumed in the present model.

It is interesting to note that the modified inverse elasticity rule is an important element

of optimal taxation not only in respect to capital and labour, as indicated earlier, but also in the

field commodity taxation. Originally, the inverse elasticity rule was proposed by Ramsey

(1927) as an efficient method to generate a given amount of tax revenue through commodity

taxation. Apparently, the modified version of it, which takes into account also equity

considerations and the presence of other taxes, is applicable to all major forms of taxation.

However, as indicated in the explanation about the dominance of indirect taxation, the inverse

elasticity rule is not always the most important factor in determining optimal taxes. In a tax mix

model, the size and configuration of other taxes is sometimes more important.

So far, comprehensive numerical modelling of the tax mix has been missing from the

optimal taxation literature. Hopefully, the tantalising results from the present model, will

stimulate further analytical and computational research.

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Appendix

Proof for the optimal capital income tax formula The solution for the optimal capital tax formula is based on deriving the welfare function

in (1) in respect to 𝜏𝐾 and then equalizing the sum total of resulting components to zero. There are three welfare effects to consider following a change in 𝜏𝐾: (i) the income effect on capital owners, (ii) the effect on social utility of variations in the uniform lump-sum grant due to

changes in total tax revenue, (iii) the variations in dead-weight costs that arise due to tax related

distortions in the allocation of resources. The formula developed using this approach will be

examined here in more detail.

First comes the utility effect on capital owners. Assuming 𝜏𝐾 is increased by a small amount ∆𝜏𝐾. Then the change in welfare will be:

− ∑𝜕𝑢𝑖

𝜕𝑦𝑖𝑖 𝑘𝑖𝑟∆𝜏𝐾 = − ∑

𝜕𝑢𝑖

𝜕𝑦𝑖𝑖

𝑘𝑖 ∑ 𝑘𝑖𝑖 𝑟

∑ 𝑘𝑖𝑖 ∆𝜏𝐾 = − ∑ �̅�𝐾𝐾𝑖 𝑟∆𝜏𝐾 (A.1)

where �̅�𝐾 is the capital income weighted average social marginal utility of income of capital owners.

The change in welfare due to a variation in tax revenue is much more complicated than

that. Assuming fixed public goods expenditure requirements (G), all the tax generated above G

will be redistributed, by providing a uniform lump-sum grant (b) to everyone (Hb). The average

social marginal utility of income of this uniform grant will be: ∑𝑔𝑖

𝐻𝑖= �̅�𝑏, (A.2)

This is the unweighted average of the social marginal utilities of everyone.

Total tax revenue is given as:

𝑅 = ∑ (𝜏𝐾𝑘𝑖𝑟 +𝑖 𝜏𝐿𝑤𝐿𝑖 + 𝜏𝐶𝑐𝑖) (A.3)

where ci represents total consumption (in producer prices) of taxpayer i. The variation in total

tax revenue following ∆𝜏𝐾 change in the tax rate can be divided into two parts. First the direct

variation in capital income tax revenue: 𝜕𝑅1

𝜕𝜏𝐾∆𝜏𝐾 = ∑ (𝑘𝑖𝑟𝑖 − 𝜏𝐾

𝜕𝑘𝑖

𝜕(1−𝜏𝐾)𝑟)∆𝜏𝐾 (A.4)

Note, 𝜕𝑘𝑖

𝜕𝜏𝐾= −

𝜕𝑘𝑖

𝜕(1−𝜏𝐾)

https://doi.org/10.1016/j.jpubeco.2017.10.004

17

Second, the change in the tax rate will cause an oppositely signed change in consumer

income, amounting approximately to − ∑ 𝑘𝑖𝑟∆𝜏𝐾𝑖 . This change in household income will cause a change in consumption tax and labour income tax revenue, as shown in (A.5).

𝜕𝑅2

𝜕𝜏𝐾∆𝜏𝐾 = ∑ 𝑘𝑖𝑟𝑖 (− 𝜏𝐶

𝜕𝑐

𝜕𝑦− 𝜏𝐿

𝑤𝜕𝐿𝑖

𝜕𝑦𝑖)∆𝜏𝐾 (A.5)

Combining the two revenue effects leads to:

𝜕𝑅

𝜕𝜏𝐾∆𝜏𝐾 = 𝐾𝑟(1 − 𝜏𝐶

𝜕𝑐

𝜕𝑦− 𝜏𝐿

𝑤𝜕𝐿

𝜕𝑦)∆𝜏𝐾 − 𝜏𝐾

𝜕𝐾

𝜕(1−𝜏𝐾)𝑟∆𝜏𝐾 (A.6)

𝜕𝑐

𝜕𝑦 is defined from (4).

From definition (6), the derivative 𝜕𝐾𝑟

𝜕(1−𝜏𝐾) = 𝑒𝐾

𝐾𝑟

(1−𝜏𝐾) (A.7)

Replacing the derivative by the elasticity term in (A.6), we obtain:

∂R

∂τK∆τK = Kr (1- τC

∂c

∂y- τL

w∂L

∂y) ∆τK - τK𝑒𝐾

Kr

(1-τK)∆τK (A.8)

Combining (A.2) and (A.8), the change in social welfare due to the change in total tax

revenue will be: ∆𝑅 = �̅�𝑏[Kr (1- τC∂c

∂y- τL

w∂L

∂y) ∆τK - τK𝑒𝐾

Kr

(1-τK)∆τK] (A.9)

The third component of welfare change is related to dead-weight loss (D), defined in

eq. (1b). This is redefined in (A.10).

𝐷(𝐾) = − 𝜏𝐾

2(

𝜕𝐾𝑟

𝜕𝜏𝐾𝜏𝐾) = −

𝜏𝐾2

2

𝜕𝐾𝑟

𝜕𝜏𝐾≈

𝜏𝐾2

2𝑒𝐾

𝐾𝑟

(1−𝜏𝐾) (A.10)

Note, the third expression is only valid if the elasticity of capital supply stays at the

constant rate, 𝑒𝐾, along the domain from zero to 𝜏𝐾. If this condition is not satisfied, one can only talk about an approximation. The approximate change in the social welfare value of dead-

weight costs following ∆𝜏𝐾 variation in the tax rate will be then:

∆𝐷(𝐾) = �̅�𝐻𝜕𝐷

𝜕(1−𝜏𝐾)∆𝜏𝐾 = −

�̅�𝐻

2𝑒𝑘𝑟

𝜕(𝜏𝐾

2 𝐾(1−𝜏𝐾)

⁄ )

𝜕𝜏𝐾∆𝜏𝐾 =

= - �̅�𝐻Kr𝑒𝐾𝜏𝐾

(1− 𝜏𝐾){

1

(1− 𝜏𝐾)−

𝜏𝐾

2(1− 𝜏𝐾)(1 + 𝑒𝐾)} ∆𝜏𝐾 (A.11)

𝑔𝐻 represents the income weighted average social marginal utility of income of the entire population. In the absence of more detailed information, we assume that variations in dead-

weight costs are distributed in proportion to the marginal utilities of incomes of consumers.

We can now combine the three welfare variations from (A.1), (A.9) and (A.11) to yield

𝜕𝑈

𝜕𝜏𝐾∆𝜏𝐾 = −�̅�𝑘𝐾𝑟∆𝜏𝐾 + �̅�𝑏 [Kr

1

(1− 𝜏𝐾)( 1 – τC

∂c

∂y – τL

w∂L

∂y) ∆τK] −

− �̅�𝐻𝑒𝐾𝐾𝑟𝜏𝐾

(1− 𝜏𝐾){

1

(1− 𝜏𝐾)−

𝜏𝐾

2(1− 𝜏𝐾)(1 + 𝑒𝐾)} ∆𝜏𝐾 (A.12)

In order to be at the optimum, the terms on the right-hand side must add up to zero.

Eliminating ∆𝜏𝐾 from both sides, dividing all terms by Kr and setting the left-hand side to zero, we have:

0 = −�̅�𝑘 + �̅�𝑏 ( 1 – τC∂c

∂y – τL

w∂L

∂y) − �̅�𝐻𝑒𝐾

𝜏𝐾

(1− 𝜏𝐾){

1

(1− 𝜏𝐾)−

𝜏𝐾

2(1− 𝜏𝐾)(1 + 𝑒𝐾)} (A.13)

18

Transferring the last term to the left-hand side and dividing all terms by �̅�𝐻𝑒𝐾, we get:

𝜏𝐾

(1−𝜏𝐾){

1

(1− 𝜏𝐾)−

𝜏𝐾

2(1− 𝜏𝐾)(1 + 𝑒𝐾)} = −

�̅�𝑘

�̅�𝐻𝑒𝐾+

�̅�𝑏

�̅�𝐻𝑒𝐾 ( 1 - τC

∂c

∂y - τL

w∂L

∂y) (A.14)

Or in a more explicit form:

𝜏𝐾

(1−𝜏𝐾)=

− �̅�𝑘�̅�𝐻

+ �̅�𝑏�̅�𝐻

( 1 − 𝜏𝐶𝜕𝑐

𝜕𝑦 − 𝜏𝐿

𝑤𝜕𝐿

𝜕𝑦)

𝑒𝐾{1

(1− 𝜏𝐾) −

𝜏𝐾2(1− 𝜏𝐾)

(1+ 𝑒𝐾)} (A.15)

This is the approximation for the optimal linear capital income tax rate. It will be

referred to it as the modified inverse elasticity rule for capital taxation.

It is of some interest to compare formula (A.15) with the one presented by Saez and

Stantcheva (2018). The steady-state tax formulas presented in their work are derived from the

following dynamic social welfare function for consumer i:

Vi = 𝛿𝑖 ∫ [𝑐𝑖(𝑡) + 𝑎𝑖(𝑘𝑖(𝑡)) − ℎ𝑖(𝑧𝑖(𝑡))]∞

0𝑒−𝑡𝛿𝑖𝑑𝑡 (A.16)

where 𝛿𝑖 is the time discount rate. In the steady-state solution time discounting disappears, and the problem becomes one of maximizing ∑ 𝑔𝑖[𝑐𝑖 + 𝑎𝑖(𝑘𝑖) − ℎ𝑖(𝑧𝑖)]𝑖 subject to the budget constraint 𝑐𝑖 = 𝑟𝑘𝑖 + 𝑧𝑖 − 𝑇(𝑘𝑖𝑟, 𝑧𝑖), where 𝑇(𝑘𝑖𝑟, 𝑧𝑖) is the combined tax (or support grant) to person i on capital and labour income.

Notice that the social welfare term is strongly separable between consumption and labour

income, is linear in consumption and contains a separate term, 𝑎𝑖(𝑘𝑖(𝑡)), to represent the

intrinsic utility derived from holding wealth. By contrast the utility function defined in (1) does

not aggregate consumption into a single variable, does not assume strong separability between

consumption and labour income, is not linear in consumption, and there is no intrinsic utility

from holding wealth. Wealth only contributes to utility by virtue of generating income.

Moreover, the present model is a static model that does not assume steady-state conditions, as

defined in dynamic growth models. The model of Saez and Stantcheva (2018) excludes ’income

effect’ by virtue of the utility function chosen. By contrast, in the numerical work discussed

here, 𝑤𝜕𝐿𝑖

𝜕𝑦𝑖 is set to -0.24.

From the definition of utility in (A.16), they obtain the following steady-state linear

capital income tax equation: 𝜏𝐾 = 1−�̅�𝑘

1− �̅�𝑘+ 𝑒𝐾 (A.17)

The definitions of �̅�𝑘 and 𝑒𝐾 in (A.17) are exactly the same as in (A.15). The differences between the formulas are mainly due to the dissimilarities between the utility

functions in (1) and (A.16). The presence of indirect taxes in the present model is another

reason. Despite differences in appearance, in polar cases the two formulas lead to similar

results.

When 𝑒𝐾 is very high, both (A.15) and (A.17) suggest that optimal 𝜏𝐾 will approach zero. When 𝑒𝐾 approaches zero, then 𝜏𝐾 in both (A.15) and (A.17) will converge to one, which is the highest permissible value.

When the social marginal utility of all income groups is the same (absence of

distributional objectives) then according to (A.17) �̅�𝑘 equals to 1 and optimal 𝜏𝐾 will be zero. In this case, (A.15) is reduced to the equation:

19

𝜏𝐾

(1−𝜏𝐾)=

− 1 + ( 1 − 𝜏𝐶𝜕𝑐

𝜕𝑦 − 𝜏𝐿

𝑤𝜕𝐿𝑖𝜕𝑦𝑖

)

𝑒𝐾{1

(1− 𝜏𝐾) −

𝜏𝐾2(1− 𝜏𝐾)

(1+ 𝑒𝐾)} (A.18)

Provided G = 0, and 𝜏𝑐 and 𝜏𝐿 are zero, then 𝜏𝐾 = 0, in line with (A.17).

When the social marginal utility of capital owners approaches zero, then according to

(A.17) 𝜏𝐾 = 1

(1 + 𝑒𝐾)⁄ . This is the revenue maximizing rate. Eq. (A.15) is reduced to a

simpler form presented in (A.19):

τK

(1-τK)≈

g̅bg̅H

( 1 - τC∂c

∂y - τL

w∂Li∂yi

)

eK (A.19)

Denoting �̅�𝑏

�̅�𝐻 𝑎𝑠 �̅�𝑏𝐻, when 𝜏𝐶 and 𝜏𝐿 are both zero, then (A.19) is reduced to:

𝜏𝐾

(1−𝜏𝐾)≈

𝑔𝑏ℎ

𝑒𝐾 (A.20)

�̅�𝑏𝐻 is the ratio between the average social marginal utility of demogrant recipients and that of the entire population. The relationship between the two is discussed in Revesz (2014, pp. 21-

22). Adopting an estimated value between 1.1 and 1.6 appears realistic. Numerical

comparisons, using plausible values for �̅�𝑏𝐻 between 1.1 and 1.6 and for 𝑒𝐾 between 0.5 and 2, indicate that optimal 𝜏𝐾 from (A.17) and from (A.20) tend to be fairly close. The similarity of polar outcomes from the present model and the static steady-state model of Saez and

Stantcheva (2018), suggests that the two models are not far apart.

User’s guide to the computer program ‘grid1’

This program carries out optimisation of linear commodity, labour and capital income

tax rates. We used the “grid” method to find the optimum, which means that it examined all

possible combinations of linear taxes. It yielded the numerical results discussed in the paper

titled “A model of the optimal tax mix including capital taxation”. The user can specify values

such as utility function parameters, wage distribution and the inequality aversion of society.

The original program was developed more than 20 years ago and was used in publications

dealing with optimal commodity taxation. (Revesz, 1997, 2014, 2018). The current version of

the program focuses only on the optimal mix of three linear taxes. It contains in addition to

commodity taxation, the optimisation of labour and capital income taxes. Prompts enable the

user to choose any combination of these three broad categories of taxes.

20

Program installation and running

This program, located on the john1revesz.com website, is written in a programming

language called QBasic, which two decades ago was part of Microsoft Windows. The

program is written as a “source code” in a Word file, without being converted into machine

language, as are most computer programs sold on the market. A special compiler is needed to

convert it into machine code. A QBasic compiler for 64 bit machines is available free of

charge from the following website: http://www.qb64.net/ . Note, the QBasic compiler will

accept the source code only in Word format.

Having a QBasic compiler, the installation is simple. Open the file named qb64 and

then Edit and Paste the entire content of the ‘grid’ Word file into the QBasic screen. Next,

click Run and then Start. Less than two minutes later the first prompt will appear on the

screen. The prompts present a menu of options. This menu provides access to various options

such as type of utility function, inequality aversion rate, the selection of taxes to be optimised,

the choice of ordinary or two-part LES, the parameter of constant elasticity capital, as well as

the size of expenditure on public goods as a percentage of total income.

After you finished with the prompts, processing will start automatically and will finish

in less than an hour. The output file, called ‘tax2.txt’, will be located in the same folder as the

QBasic compiler. In case you encounter any problem when responding to prompts, the

quickest way to overcome it is to exit and start again. Click on the cross at the upper right

hand corner of the QBasic screen. After the screen disappears, start running the program

again.

Finally, all the library files associated with the compiler should be kept in the same

folder as the compiler file. Because the program is a source code in Word format, it could be

corrupted accidentally or during data entry. For that reason, it is advisable to keep a couple of

copies in a separate folder.

DATA lines

Virtually all parameters related to consumer demand and labour supply can be

changed by the user, as well as data related to income distribution. All the numbers in the

DATA lines close to the beginning of the source code can be altered. Above the DATA lines

are text lines (marked at the left by asterisk to indicate that it is a comment and not a program

line) explaining what the DATA lines refer to. These include initial wage rates, initial capital

incomes and LES utility-demand parameters. After reading the paper, most of these

parameters are self-explanatory.

Usually there are a number of DATA lines for each item. Those marked with an

asterisk at the beginning of the line are ignored by the program. Only the unmarked DATA

line is currently active. Considerable care must be exercised when entering new numbers. If

the number of entries in the DATA line does not correspond to the number of READs

specified underneath, then the program will be corrupted without warning. If there are too few

entries, the READ instruction will assume that the missing numbers are zero. If there are too

many entries, the surplus numbers will be picked up by the following READ instruction

relating to a different variable or parameter. As a result, the output from the program will be

meaningless.

http://www.john1revesz.com/http://www.qb64.net/

21

Please check the parameter summary tables appearing at the beginning of the printout,

to ensure that correct numbers are used by the program. Always make sure that there is only

one DATA line without an asterisk for each variable.

Note, every time you change DATA lines on the ‘grid1’ Word file, you have to copy

and paste the file again into the QBasic screen

Utility and demand parameters

The current version of the program enables the user to specify LES utility-demand

parameters according to his/her choice, for the two wage groups in the model. The relevant

DATA lines for 9 goods and 18 goods utility-demand appear close to the beginning of the

source code. The 9 goods utility function is allocated to the 8 lower wage earners, the 18

goods utility function is allocated to the 7 higher wage earners. The leisure parameters are

defined under them, through equations with numbers on the right hand side. The program

checks that the betas of commodities and leisure add up to one. If this condition is not

satisfied, a warning will appear at the very start of the printout and program execution will

terminate. To ensure the global quasi-concavity of LES utility, all betas must be positive.

Labour and capital income tax schedules

The constant elasticity of capital supply is entered through a prompt. No such prompt

appears for labour supply, because the compensated elasticity of labour supply is determined

by the program for each taxpayer, according to his/her wage rate, lump-sum income and

relevant LES parameters, specified in DATA lines.

Inequality aversion

The program contains two social welfare functions to describe the inequality aversion

of society, one is LES utility (a logarithmic function) the other is maximin. Maximin

considers only the utility level of the lowest wage group. It leads to very high optimal tax

rates. Even LES leads to fairly high tax rates. To overcome this problem, an inequality

aversion rate (z) was introduced, as explained in Revesz (2018, Appendix 3.3 section C.4).

The inequality aversion rate ranges between zero and one. Zero represents the absence of

distributional objectives, one represents LES utility. Any number between zero and one is a

convex combination of the two. The user can choose this rate in the second prompt.

Sometimes when z is low, the search for the optimum does not converge properly. In these

cases an error message is issued, and you should try a somewhat higher value of z.