optimal taxation and economic growth: a comment

Optimal Taxation and Economic Growth: A CommentAuthor(s): Roderick HillSource: Public Choice, Vol. 134, No. 3/4 (Mar., 2008), pp. 419-427Published by: SpringerStable URL: http://www.jstor.org/stable/27698234 .

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Public Choice (2008) 134 419^27

DOI 10 1007/sl 1127 007 9235 0

Optimal taxation and economic growth: a comment

Roderick Hill

Received 13 December 2006 / Accepted 12 September 2007/Published online 20 October 2007

Springer Science+Business Media LLC 2007

Abstract Scully (Public Choice 115 299-312, 2003) claims that for the United States in

1960-1990, the growth maximizing size of the state was about 19% of GDP However if an

error in the model specification is corrected and if 2001 vintage data is used (instead of 1996

vintage data), estimates of the growth-maximizing size of the state range between 9% and

29% of GDP Further, the method spuriously identifies a 'growth maximizing tax rate' even

if no relationship exists between growth and the size of the state The model cannot address

reliably the question it attempts to answer

Keywords Taxation Government expenditures Economic growth

1 Introduction

In an article in this journal, Scully (2003) employed a method of modelling and estimating a country's 'optimal tax rate,' meaning the tax rate that maximizes the rate of economic

growth In his view, the relationship between the growth rate and the tax rate should take the

form of an inverted-U, reminiscent of the Laffer curve

This method had earlier been applied to New Zealand (Scully 1996, 1999), the United States (Scully 1995), and Canada (Chao and Grubel 1998) For the United States using 1960-1990 data, Scully (2003, p 305) reported that "there is empirical confidence in the result that the optimal or growth-maximizing tax rate for the United States, over this period, is 19 3 percent" Estimated growth at this tax rate would be 6 97% (Scully 2003, p 309) compared with 3 38% in his data This paper replicates Scully's study and finds that his econometnc specification did not conform to the theoretical model When this error is cor

rected, the results change significantly The results are also shown to be very sensitive to the

vintage of data used As well, the model itself is fundamentally flawed The functional form

used produces spurious estimates of the relationship between economic growth and the size

of the state

R Hill (El) Department of Social Science University of New Brunswick PO Box 5050 Saint John

NB E2L 4L5 Canada

e mail rhill@unbsj ca

Springer



420 Public Choice (2008) 134: 419^27

2 Estimates of the 'optimal tax rate' for the United States

The model used in Scully (2003) was based on Scully (1996, 1999). Government produces public goods in period t - 1, G,_i, using labour and capital. Public goods are paid for

with taxes Tt_\ on that period's gross domestic product (GDP) Yt-\. The tax-to-GDP ra

tio is xt-\. The budget is assumed balanced "for mathematical convenience" (Scully 1999,

p. 99), so G = xY for each year. The remainder of production consists of private goods:

yr_i -

Gt-\ =

(1 -

rt-i)Yt-\. "These public goods and private goods then are used to pro

duce national output at time r" (Scully 1996, p. 173). The production function is specified in Cobb-Douglas form:

Yt=a(Gt^)b[(l-xt.x)Yt^f (1)

No attempt was made to justify this peculiar production function, although Sieper (1997) and Kennedy (2000) had already shown that it embodies the implausible assumption that all

capital was completely used up every year.

Substituting the balanced-budget assumption, G = rY, into the production function, di

viding by F,_i and assuming that b + c= 1, we obtain:1

(l+gt)=ax?_l(l-xt-l)l-b (2)

where g is the real GDP growth rate. Estimation of the model requires data for real GDP, and total government expenditures or tax revenues as a share of nominal GDP. Scully used

data published in 1996 that are described and reproduced in the Appendix. Table 1 reproduces in columns 1 and 6 the estimates reported in Scully (2003). The

estimates using the alternative independent variables, the tax-to-GDP and expenditure-to GDP ratios, were both successfully replicated, as reported on columns 2 and 7, but only by

using the following model:2

(l+gt) = ax >(l-xt)l-h

The use of xt instead of rr_i is inconsistent with the stated model shown in (2) and repeats a similar mistake in Scully (1996) that was pointed out by Chapple (1997, p. 8).

If this specification error is corrected so that lagged values of the independent variables are used, the results are those shown in columns 4 and 9 of Table 1. These imply that the

'optimal tax rate' is about 16%, while what might be called the 'optimal expenditure rate'

is about 28%. This poses something of a paradox, given that the two are supposed to be the

same! As is also shown in the table, the predicted growth rates in these two scenarios differ

greatly.

What happens to the estimates if the regressions use the most recent data available when

Scully (2003) was likely written? Anderson (2006, p. 81) has stressed the importance of not only carrying out replication studies, but also what he terms "real-time data studies"

which examine the robustness of findings to subsequent data revisions. The data available at

a particular time is termed 'vintage' data and is distinguished from data for the same time

1 Constant returns are necessary for the model to have the constant growth rate charactenstic of an equihbnum

growth path (McCann 1998, Phillips 1998) Phillips (1998, p 9) wrote "The use of an equihbnum growth model may be convenient, but it is not clear that it is empincally relevant A preferable theoretical

approach that seems more suited to the empincal conditions would have been to construct a cyclical growth or disequihbnum growth model

"

2Data, program and output files are available from the author on request

Springer



Table 1 Estimates of Scully's model, (1 + g) = axb(\ - r)

\l-b

Parameter

Regression equation

(1) Scully

(2003,

P 311)

using (T/Y)t

(2)

Replication

using

(?7 n,

and 1996 data

(3) Estimate

using

(T/Y)t and 2001

data

(4)

Estimate using

oyy)/-i

and 1996 data

(5)

Estimate

using

(T/r),-i

and 2001 data

(6) Scully (2003,

p 305)

using

(G/Y)t

(7)

using (G/Y)t and 1996

data

(8)

Replication Estimate

using (G/Y)t

and 2001 data

(9) Estimate

using

(G/r),-i

and 1996 data

(10) Estimate

using

(G/n,-i and 2001 data

b (the 'optimal tax rate')

DW

Predicted growth at 'optimal tax rate' (b)

171 (0 080) 0 18 (0 052) 011 177 not stated

1710 (0 084) 0 180 (0 054) 011 177

0 0671

1627 (0 109) 0 140 (0 066) 0 10 1 81

0 0846

1678 (0 073) 0 160 (0 048) 0 19 166 0 0809

1551 (0 090) 0 094 (0 057) 0 24 167 0 1354

175 (0 055) 0 193 (0 038) 0 22 158 0 0697

1747 (0 057) 0 193 (0 040) 0 22 158 0 0698

1676 (0 727) 0 152 (0 049) 0 22 157 0 0963

1881 (0 069) 0 284 (0 044) 0 01 160 0 0358

1833 (0 052) 0 291 (0 055) 0 16 157

0 0357

Note Standard errors in parentheses Estimation by non-linear least squares in TSP 4 2B Predicted growth rates were calculated from actual values of estimated coefficients, not

the rounded values reported here



422 Public Choice (2008) 134: 419-427

period that is available at a later date (Anderson 2006, p. 83). In this case, in the Appendix Table 2 compares the 1996 vintage data used by Scully with the February 2001 vintage data that would have been available before the paper's acceptance for publication in April 2002.

Examination of Tables 2 and 3, in the Appendix, shows that revisions of 1959-1990 data made between 1996 and 2001 were significant. Revisions to nominal data were made

that covered the entire sample period right back to 1959, with the result that the independent variables, the tax-to-GDP and expenditure-to-GDP ratios, decreased an average of 0.022 and

0.013 respectively. In some years, the growth rate of real GDP changed by half a percentage point.

Columns 3 and 8 in Table 1 report the results of using the 2001 vintage data with Scully's original specification. It would lower the estimated growth-maximizing size of the state

to 14-15% of GDP while raising the predicted real growth rate to between 8 and 9.6%. Columns 5 and 10 summarize the results if the model-consistent specification is used: the

'optimal tax rate' falls to 9.4% (with a predicted real GDP growth rate of an incredible

13.5%), while the 'optimal expenditure rate' rises to 29% (with predicted real GDP growth of only 3.6%). Clearly, Scully's claim (2003, p. 305) that "there is empirical confidence in the result that the optimal or growth-maximizing tax rate for the United States, over this

period, is 19.3 percent" is not correct, nor was it correct to claim (Scully 2003, p. 304) that

the conclusions were unaffected by the choice of whether to use the tax share of GDP or the

share of government expenditures as the independent variable.3

As well, no confidence can be placed in growth rates that are predicted using values of the

independent variables that lie far outside the sample range observed in the data. This was the

case for Scully's original 'optimal tax rate' of 18% and holds for all of the other regressions

reported in Table 1 with the exception of the last two. (See Table 3 in the Appendix for the values of the independent variables over the sample period.)

A separate and serious problem is the fact that the regressions themselves can produce

spurious results, a point raised by Chapple (1997), Easton (1999, pp. 85-86) and Kennedy (2000) concerning an earlier New Zealand study (Scully 1996). The basic model, 1 + g =

axh(l -

x)l~h, implies that dg/dx =

axh~l(l -

x)~b(b -

x), as Scully pointed out (2003,

p. 304). Following Kennedy (2000, p. 90), "suppose that g is a random variable unaffected

by x, so that there is no meaningful growth-maximizing tax rate." Then dg/dx = 0, implying

that (b -

t) = 0. In this case, while the model would appear to be producing estimates of an

'optimal tax rate,' the estimated b would be just the average r in the sample, something that

Kennedy confirmed using randomly-generated simulation data for g and r.

A similar exercise to Kennedy's can carried out here by replacing the actual data on

real GDP growth with sets of random numbers generated from a uniform distribution with

bounds [-0.015, 0.065] that approximate the range of annual growth rates of real U.S. GDP

in 1960-1990. 1000 sets of such random numbers were generated and a regression was

done for each. For the regressions using the lagged share of taxes in GDP, the average

estimated 'optimal tax rate' was 0.281. This is very close to the mean tax-to-GDP ratio of

0.288. Similarly, for regressions using the lagged share of government expenditures in GDP,

the average estimate of the 'growth-maximizing expenditure rate' was 0.302, as compared

3The model has other problems such as the assumed exogeneity of the ratio of government revenues and ex

penditures to GDP, an assumption that Phillips (1998, p 5) observed is "unlikely to be satisfied empirically It is not mentioned in the paper and is not justified there" While he was referring to Scully's New Zealand

study, it applies equally well to Scully (2003), as does his comment that "several important channels of taxa tion effects on economic growth, primarily in terms of infrastructure spending and human capital investment

expenditures by the public sector" are omitted "However, since the tax ratio is not exogenous and exogeneity is not tested, it cannot be argued that these equations are reduced forms

"

Springer



Public Choice (2008) 134 419^127 423

with the average value in the data of 0 306 With no systematic relationship between the

randomly-generated 'growth rate' and the tax and expenditure vanables, such results cannot

be interpreted as indicating any meaningful relationship between them 4

Scully (2000, pp 94-95) attempted to address the concern about spunous estimates in his

reply to Kennedy (2000) He asserted that Kennedy had offered "no formal proof (although Kennedy had clearly set out the theoretical reason to expect the possibility of spunous esti

mates), but conceded that Kennedy's simulation result was "disturbing "

He tried to address

this by pointing out that his estimates for the New Zealand 'optimal tax rate' were not, in

fact, the average tax rate (as indeed the estimates reported here in Table 1 are not the average

tax or expenditure rates for the United States) This is beside the point, however, because

as Kennedy himself wrote "A little algebra shows that whenever g and r are related, as is

clearly the case for Scully's New Zealand application, this estimation equation produces a

growth-maximizing tax rate estimate that departs from the average r Unfortunately, how

ever, it does not produce automatically a measure that can be used to check whether g

and r are actually related" (Kennedy 2000, p 90)5

3 Conclusions

When an error in the econometric specification in Scully (2003) is corrected, the estimate

of the growth-maximizing size of government in the United States in the period 1960-1990

broadens from 18-19% to 16-28% The results were also sensitive to the vintage of the data

when data from 2001 are used, the estimate broadens to 9-29% of GDP with implied real

growth rates of between 3 6 and 13 5% As well, the model can produce spunous estimates

of an 'optimal tax rate ' Taken together, these features render moot the calculations in the

rest of Scully (2003) that purport to show that implementing the 'optimal tax rate' would

only raise inequality by a modest amount

Acknowledgements I am grateful to Trevor Breusch, Lars Osberg, and an anonymous referee for com ments on an earlier draft and to George Stoica and Rameshwar Gupta for helpful discussions I thank Robin Oliver (Inland Revenue Department New Zealand), Jessica Matthewson (New Zealand Institute of Economic

Research) and Jo Harris (The Treasury, New Zealand) for their assistance in obtaining unpublished reports

Appendix

The two sets of 1959-1990 data used in this study are reproduced in Table 2 Data for U S real GDP in 1992 dollars is taken from Appendix B of the Economic Report of the President 1996 (Washington United States Government Pnnting Office), Table B-2 (These reports

4The spurious nature of the regression results can also be easily seen if actual U S GDP growth rates are

replaced by a constant, such as 4 The 'production function' becomes 1 4- 4 = a xb ( 1 - r )1 ~b When estimated

by non-linear least squares using tax/GDP data, the estimate of b is 0 282 with a standard error of 0 0009 and

R2 = 0 It would be absurd to interpret this as showing the tax rate that maximizes the number 4

5Scully (2000, p 95) also presented further econometric evidence to show that in the New Zealand case there was an inverted-U shaped relationship between growth rates and tax rates, something that Kennedy did not

dispute and that was also irrelevant to the critique of the possibility of spunous estimates resulting from the use of Scully's method As with New Zealand, the relationship between growth rates and the T/ Y or G/Y ratio can be investigated empincally using a quadratic equation, 1 + gt

- a + bxt-\ 4- cz2_l

For the United

States, no stable inverted-U relationship exists (Econometric results are available from the author on request )

%1 Springer



I&

Table 2

Year

Real GDP (billions of

1992 dollars, 1996 data)

Real GDP

(billions

of

1996 dollars, 2001 data)

Nominal government consump tion and

investment,

billions

(1996 data)

Nominal transfers,

billions

(1996 data)


investment,

billions

(2001 data)

Nominal transfers,

billions

(2001 data)

Nominal

taxes,

billions

(1996 data)

Nominal

taxes,

billions

(2001 data)

Nominal

GDP, billions

(1996 data)

Nominal

GDP,

billions (2001 data)

1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978

2212 3 22617 2309 8 2449 1 2554 0 2702 9 2874 8 3060 2 3140 2 3288 6 3388 0 3388 2 3500 1 3690 3 3902 3 3888 2 3865 1 4081 1 4279 3 4493 7

2319 0 2376 7 2432 0 2578 9 2690 4 2846 5 3028 5 3227 5 3308 3 3466 1 35714 3578 0 3697 7 3898 4 4123 4 4099 0 4084 4 43117 45118 4760 6

1120 1132 120 9 1314 137 7 144 4 153 0 173 6 194 6 212 1 223 8 236 1 249 9 268 9 287 6 323 2 362 6 385 9 416 9 457 9

27 5 29 3 33 6 34 7 36 6 38 1 41 1 45 8 54 5 62 6 69 3 83 8 99 4 1109 126 6 150 5 189 2 206 5 220 9 238 6

1125 1138 1215 132 2 138 5 145 1 153 7 174 3 195 3 212 8 224 6 237 1 2510 2701 287 9 322 4 361 1 384 5 415 3 455 6

24 7 26 3 30 2 30 9 32 4 33 4 36 0 39 7 47 5 54 9 60 6 73 5 87 5 97 0 1105 1315 166 4 180 4 192 0 206 1

128 8 138 8 144 1 155 8 167 5 172 9 187 0 210 7 226 4 260 9 293 9 299 6 3196 364 8 408 8 4518 468 4 535 9 603 9 678 5

122 1 1312 135 8 147 0 157 9 162 1 175 4 197 8 212 1 245 3 276 3 279 6 295 9 338 1 380 3 419 6 430 5 492 6 552 8 626 0

507 2 526 6 544 8 585 2 6174 663 0 719 1 787 8 833 6 910 6 982 2 1035 6 1125 4 1237 3 1382 6 1496 9 1630 6 18190 2026 9 22914

507 4 527 4 545 7 586 5 618 7 664 4 720 1 789 3 834 1 9115 985 3 1039 7 1128 6 1240 4 1385 5 1505 0 1635 2 1823 9 20314 2295 9

n ST O



o

Table 2 (Continued) Year Real GDP Real GDP

(billions of (billions of 1992 dollars, 1996 dollars,

1996 data) 2001 data)

Nominal government consump

tion and investment,

billions

(1996 data)

Nominal transfers,



investment, billions

(2001 data)

Nominal transfers,

billions

(2001 data)

Nominal

taxes, billions

(1996 data)

Nominal

taxes, billions

(2001 data)

Nominal

GDP,


Nominal

GDP, billions

(2001 data)

I

1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990

4624 0 46119 4724 9 4623 6 4810 0 5138 2 5329 5 5489 9 5648 4 5862 9 6060 4 6138 7

4912 1 4900 9 50210 4919 3 5132 3 5505 2 5717 1 5912 4 6113 3 6368 4 65918 6707 9

507 1 572 8 633 4 684 8 735 7 796 6 875 0 938 5 992 8 1032 0 1095 1 1176 1

266 9 3176 360 7 403 3 434 4 448 2 480 9 5109 533 7 568 3 616 3 679 8

503 5 569 7 6314 684 4 735 9 800 8 878 3 942 3 997 9 1036 9 1100 2 11814

230 2 275 0 3118 348 5 376 4 387 4 414 2 440 4 458 0 486 5 529 6 583 1

761 1 834 2 952 2 9715 1028 6 1144 5 1239 7 1313 1 1429 4 15173 1642 1 1726 4

702 7 767 1 877 6 890 3 944 5 1047 8 1135 8 1206 7 1322 5 1410 9 1530 9 1607 7

2557 5 2784 2 31159 3242 1 3514 5 3902 4 4180 7 4422 2 4692 3 5049 6 5438 7 5743 8

2566 4 2795 6 31313 3259 2 3534 9 3932 7 4213 0 4452 9 4742 5 5108 3 5489 1 5803 2

Sources See the Appendix text

IE&



426 Public Choice (2008) 134 419^127

Table 3

Year Real GDP Real GDP Tax to GDP Tax to GDP Government Government

growth growth ratio ratio expenditures expenditures

(percent) (percent) (1996 data) (2001 data) to GDP ratio to GDP ratio

( 1996 data) (2001 data) ( 1996 data) (2001 data)

1959 - - 0 254 0 241 0 275 0 270

1960 2 23 2 49 0 264 0 249 0 271 0 266

1961 2 13 2 33 0 265 0 249 0 284 0 278

1962 6 03 6 04 0 266 0 251 0 284 0 278

1963 4 28 4 32 0 271 0 255 0 282 0 276

1964 5 83 5 80 0 261 0 244 0 275 0 269

1965 6 36 6 39 0 260 0 244 0 270 0 263

1966 6 45 6 57 0 267 0 251 0 278 0 271

1967 2 61 2 50 0 272 0 254 0 299 0 291

1968 4 73 4 77 0 287 0 269 0 302 0 294

1969 3 02 3 04 0 299 0 280 0 298 0 289

1970 0 01 0 18 0 289 0 269 0 309 0 299

1971 3 30 3 35 0 284 0 262 0 310 0 300

1972 5 43 5 43 0 295 0 273 0 307 0 296

1973 5 74 5 77 0 296 0 274 0 300 0 288

1974 -0 36 -0 59 0 302 0 279 0 316 0 302

1975 -0 59 -0 36 0 287 0 263 0 338 0 323

1976 5 59 5 57 0 295 0 270 0 326 0 310

1977 4 86 4 64 0 298 0 272 0 315 0 299

1978 5 01 5 51 0 296 0 273 0 304 0 288

1979 2 90 3 18 0 298 0 274 0 303 0 286

1980 -0 26 -0 23 0 300 0 274 0 320 0 302

1981 2 45 2 45 0 306 0 280 0 319 0 301

1982 -2 14 -2 03 0 300 0 273 0 336 0 317

1983 4 03 4 33 0 293 0 267 0 333 0 305

1984 6 82 7 27 0 293 0 266 0 319 0 302

1985 3 72 3 85 0 297 0 270 0 324 0 307

1986 3 01 3 42 0 297 0 271 0 328 0 311

1987 2 89 3 40 0 305 0 279 0 325 0 307

1988 3 80 4 17 0 300 0 276 0 317 0 298

1989 3 37 3 51 0 302 0 279 0 315 0 297

1990 129 176 0 301 0 277 0 323 0 304

are available at www gpoaccess gov/eop/ ) The nominal expenditures of federal, state, and

local (FSL) governments are the sum of government consumption and gross investment

expenditures from Table B-16 and government transfers from Table B-79, which are shown

separately in the table Nominal receipts of FSL governments are from Table B-79 Nominal

GDP is from Table B-l

The second parallel set of data for 1959-1990 is taken from Economic Report of the President 2001 (Washington United States Government Pnnting Office) U S real GDP in

1996 dollars is from Table B-2 Nominal expenditures of FSL governments are the sum of

4 Springer



Public Choice (2008) 134 419-427 427

government consumption and gross investment expenditures from Table B-20 and govern

ment transfers from Table B-83. Nominal receipts of FSL governments are from Table B-83.

Nominal GDP is from Table B-l. Table 3 shows the resulting values for the real GDP growth rate and the ratios of taxes

and government expenditures to GDP from the data of different vintages.

References

Anderson, R G (2006) Replicability, real-time data, and the science of economic research FRED, ALFRED, and VDC Federal Reserve Bank of St Louis Review, 88(1), 81-93

Chao, J C P,&Grubel, H (1998) Optimal levels of spending and taxation in Canada In H Grubel (Ed ), How to use the fiscal surplus (pp 53-68) Vancouver Fraser Institute

Chapple, S (1997) One for the X-files A critical assessment of Professor Scully's 'Taxation and economic

growth in New Zealand' Report to the Periodic Report Group New Zealand Institute of Economic

Research, Wellington, November 1997

Easton, B (1999) The whimpering of the state Policy after MMP Auckland University of Auckland Press

Kennedy, P E (2000) On measunng the growth-maximizing tax rate Pacific Economic Review, 5(1), 89-91

McCann, E (1998) An error in Professor Scully's WP14 Appendix 2 m Supplementary briefing papers

(Vol 2) Report on research commissioned by Inland Revenue, New Zealand, November 1999 Available at www taxpohcy ird govt nz/pubhcations/files/volume2 pdf

Phillips, P C B (1998) Report on E McCann (1998) 'An error in Professor Scully's WP14' Appendix 4 in Supplementary briefing papers (Vol 2) Report on research commissioned by Inland Revenue, New

Zealand, November 1999 Available at www taxpohcy ird govt nz/pubhcations/files/volume2 pdf

Scully, G W (1995) The 'growth tax' in the United States Public Choice, 85(1-2), 71-80

Scully, G W (1996) Taxation and economic growth in New Zealand Pacific Economic Review, 1(2), 169 177

Scully, G W (1999) Unfinished reform taxation and economic growth in New Zealand Journal of Private

Enterprise, 14(2), 92-114

Scully, G W (2000) The growth-maximizing tax rate Pacific Economic Review, 5(1), 93-96

Scully, G W (2003) Optimal taxation, economic growth, and income inequality Public Choice, 115, 299 312

Sieper, E (1997) Review by E Sieper of Gerald W Scully, 'Taxation and economic growth in New Zealand '

Revised IRD working paper No 14,28 March 1996 Unpublished study commissioned for The Treasury, Government of New Zealand

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