IN A CONTRIBUTION to this journal, Tanzi (1983) has presented annual estimates for the underground economy in the United States in the 1930–82 period. The purpose of this comment is twofold:

(a) To present a variant of Tanzi’s method which provides an upper bound for the estimates of the underground economy. This variant is described in section one and the upper bound estimates are provided in Table 1.

(6) Recently, estimates of the average marginal tax rates for the U.S. economy have been constructed for the 1916–80 period. Theoretically, these are the relevant tax rates for Tanzi’s equation (rather than the average tax rates used by him). Utilizing the newly available data, Tanzi’s equations and the dimension of the underground economy are re-estimated (section three). The results are provided in Tables 2 and 3.

Variations on Tanzi’s Method

Tanzi’s approach is a sophisticated variant of the monetary approach suggested by Gutmann (1977) and Feige (1979), and it is based on a modified version of Cagan (1958). Tanzi (1983) regressed the currency to total money ratio (for the 1930–80 period) against various explanatory variables and the average tax rate (T). Positing (1) Z = ln(C/M), where C denotes currency and M is currency plus demand deposits (DD) plus time deposits (TD), Tanzi obtained estimates for Z₁ and Z₂. Z₁ is the predicted value of Z obtained from the regression equation (where all explanatory variables and T are at their historic levels). Z₂ is the corresponding, where tax rates are zero (T = 0). With (2) Z₁ = ln(C/M) and (3) Z₂ = In(CW/M) (where CW is the amount of currency used with T = 0), it follows that (4) C = Me^z1 and (5) CW = Me^Z2. From (4) and (5) Tanzi obtains (6) CB = C-CW = M(e^ZI–e^Z2). CB is a measure of currency holdings which are tax induced (presumably to evade taxes). Tanzi refers to this amount of currency as illegal money. He then proceeds to calculate the income velocity of legal money (V) by dividing GNP by legal money: (7) V = GNP/(M –CB). Assuming that the velocity of illegal money is the same as that of legal money, an estimate of the underground economy (UE) is obtained by multiplying illegal money by the velocity of money: (8) UE = CB V. This is the basis for Tanzi’s results reported in his tables (pp. 298–301).

One may, however, give a different interpretation to Z₂. Since Z₂ corresponds to an economy with T = 0 and currency holdings of CW, it follows that total money holdings with T = 0 equal (9) CW + DD + TD = M — CB (which is legal money in Tanzi’s terminology). Accordingly, the appropriate definition of Z₂ is (10) Z₂ = In[CW/(M - CB)]. Combining (4) and (10) it follows that (11) CB = M(e^z1-e^z2)/(1-e^Z2). The estimates for the underground economy using this method (upper bound method hereafter) are higher than those of Tanzi since (1 - e^Z2) < 1. The estimates of method one are provided in Table 1, and compared with those of Tanzi. They are indeed about 10 percent higher than Tanzi’s estimates (see Table 4).

One may argue that money supply itself is an exogenous variable and is thus given. In this case M does not change, even when T = 0 and equation (3) rather than equation (10) is relevant, as assumed by Tanzi.¹

Since there is no a priori way to argue for either of these methods of calculating the underground economy, it seems that both methods should be used to give lower and upper bounds for the dimension of the underground economy, using Tanzi’s approach. The estimates given in this paper put the upper bound for the U.S. unrecorded economy in 1980 at 6.1 percent of GNP according to the equation, which uses weighted average tax rates, and at 5.0 percent of GNP when average tax rates are used.

^{Table 1.}

U.S. Underground Economy as Percentage of GNP

Note: This table corresponds to Tables 4 and 5 in Tanzi (1983, pp. 298–301).

^{Table 1.}

U.S. Underground Economy as Percentage of GNP

	Using Weighted Average Tax Rates		Using Average Tax Rates
Year	Tanzi’s Method	Upper-Bound Method	Tanzi’s Method	Upper-Bound Method
1930	0.6	0.7	–0.2	–0.3
1931	0.6	0.7	–0.2	–0.2
1932	3.9	4.4	2.9	3.3
1933	2.2	2.5	2.6	3.0
1934	4.3	4.9	2.6	3.0
1935	2.5	2.8	2.1	2.4
1936	3.0	3.3	3.5	4.0
1937	3.0	3.3	1.1	1.2
1938	1.3	1.5	0.5	0.5
1939	2.8	3.1	1.4	1.6
1940	2.4	2.6	1.5	1.6
1941	5.1	5.7	3.5	3.9
1942	4.8	5.4	4.7	5.3
1943	4.1	4.7	4.2	4.9
1944	8.0	9.3	2.9	3.5
1945	6.1	7.2	3.8	4.6
1946	4.5	5.3	2.2	2.7
1947	5.2	6.1	3.3	3.9
1948	5.3	6.2	1.4	1.6
1949	4.9	5.7	2.5	3.0
1950	5.1	5.8	3.3	3.9
1951	4.7	5.4	3.6	4.2
1952	5.2	6.0	3.3	3.9
1953	4.8	5.5	2.9	3.4
1954	4.6	5.3	2.3	2.7
1955	3.2	3.6	2.8	3.2
1956	4.3	4.9	2.9	3.3
1957	4.1	4.7	2.7	3.0
1958	4.4	5.0	2.8	3.2
1959	4.0	4.5	2.8	3.1
1960	4.1	4.6	2.6	3.0
1961	4.2	4.7	2.9	3.3
1962	3.9	4.4	2.6	3.0
1963	4.0	4.5	2.8	3.1
1964	3.8	4.2	2.2	2.4
1965	3.8	4.2	2.5	2.8
1966	4.1	4.5	2.9	3.3
1967	4.5	5.0	3.2	3.5
1968	4.3	4.8	3.6	4.0
1969	4.5	5.0	3.3	3.7
1970	4.6	5.1	2.6	2.9
1971	4.7	5.2	2.8	3.1
1972	4.7	5.2	2.9	3.3
1973	4.4	4,8	3.1	3.4
1974	4.9	5.4	3.4	3.8
1975	5.0	5.5	3.0	3.4
1976	5.5	6.0	3.6	3.9
1977	5.2	5.7	3.7	4.1
1978	5.3	5.8	3.8	4.2
1979	5.4	5.9	3.7	4,1
1980	6.1	6.6	4.5	5.0

Note: This table corresponds to Tables 4 and 5 in Tanzi (1983, pp. 298–301).

View Table

^{Table 1.}

U.S. Underground Economy as Percentage of GNP

	Using Weighted Average Tax Rates		Using Average Tax Rates
Year	Tanzi’s Method	Upper-Bound Method	Tanzi’s Method	Upper-Bound Method
1930	0.6	0.7	–0.2	–0.3
1931	0.6	0.7	–0.2	–0.2
1932	3.9	4.4	2.9	3.3
1933	2.2	2.5	2.6	3.0
1934	4.3	4.9	2.6	3.0
1935	2.5	2.8	2.1	2.4
1936	3.0	3.3	3.5	4.0
1937	3.0	3.3	1.1	1.2
1938	1.3	1.5	0.5	0.5
1939	2.8	3.1	1.4	1.6
1940	2.4	2.6	1.5	1.6
1941	5.1	5.7	3.5	3.9
1942	4.8	5.4	4.7	5.3
1943	4.1	4.7	4.2	4.9
1944	8.0	9.3	2.9	3.5
1945	6.1	7.2	3.8	4.6
1946	4.5	5.3	2.2	2.7
1947	5.2	6.1	3.3	3.9
1948	5.3	6.2	1.4	1.6
1949	4.9	5.7	2.5	3.0
1950	5.1	5.8	3.3	3.9
1951	4.7	5.4	3.6	4.2
1952	5.2	6.0	3.3	3.9
1953	4.8	5.5	2.9	3.4
1954	4.6	5.3	2.3	2.7
1955	3.2	3.6	2.8	3.2
1956	4.3	4.9	2.9	3.3
1957	4.1	4.7	2.7	3.0
1958	4.4	5.0	2.8	3.2
1959	4.0	4.5	2.8	3.1
1960	4.1	4.6	2.6	3.0
1961	4.2	4.7	2.9	3.3
1962	3.9	4.4	2.6	3.0
1963	4.0	4.5	2.8	3.1
1964	3.8	4.2	2.2	2.4
1965	3.8	4.2	2.5	2.8
1966	4.1	4.5	2.9	3.3
1967	4.5	5.0	3.2	3.5
1968	4.3	4.8	3.6	4.0
1969	4.5	5.0	3.3	3.7
1970	4.6	5.1	2.6	2.9
1971	4.7	5.2	2.8	3.1
1972	4.7	5.2	2.9	3.3
1973	4.4	4,8	3.1	3.4
1974	4.9	5.4	3.4	3.8
1975	5.0	5.5	3.0	3.4
1976	5.5	6.0	3.6	3.9
1977	5.2	5.7	3.7	4.1
1978	5.3	5.8	3.8	4.2
1979	5.4	5.9	3.7	4,1
1980	6.1	6.6	4.5	5.0

Note: This table corresponds to Tables 4 and 5 in Tanzi (1983, pp. 298–301).

Marginal Tax Rates

Tanzi’s approach is based on the assumption that higher tax rates increase the incentive to evade taxes. Tanzi (1980, pp. 440–41) argues that “it is the marginal tax rate on a taxpayer’s income—rather than the average rate—that is more likely to determine whether he evades the tax on the marginal dollar.” However, in his estimates he used two measures for the tax variable, which are based on average taxes. This was done because of a lack of data regarding marginal tax rates. How much does this affect Tanzi’s estimates?

Recently, Barro and Sahasakul (1983) and Seater (1985), have provided estimates for the average marginal tax rates in the United States for the 1916–80 period. Using these measures, Tanzi’s equations were re-estimated. The regression results are provided in Table 2. The results based on Seater’s tax measures (equations 2.3–2.6), are surprising. All of these equations have an that is lower than in Tanzi’s equations (2.1 and 2.2). Moreover, in two equations (2.3 and 2.6) the income variable is significant only at the 10 percent level (vs. a 5 percent level in Tanzi’s equations).

^{Table 2.}

Regression Results with Various Tax Measures

Note: The equations have been corrected with a first-order Cochrane-Orcutt correction for serial correlation. All variables are significant at α = 1 percent, unless marked by * (α = 5 percent) or + (α = 10 percent). The tax measures are T1—Weighted average tax rate on interest income (Tanzi, 1983, Table 1, pp. 291–92). T2—Ratio of total income tax payments after credit to adjusted gross income (Tanzi, 1983, Table 1). S1—S4 adjusted for the fraction of GNP subject to the personal income tax (Seater, 1985, Table 1). S2—Overall average marginal tax rate on household income (Seater, 1985). S3—Overall average marginal tax rate on labor’s marginal product (Seater, 1985). S4—Average marginal federal personal income tax rate on adjusted gross income (Seater, 1985). BS—Average marginal federal income tax (Barro and Sahasakul, 1983, Table 2). Definitions of other variables are provided in Tanzi (1983, p. 290).

^{Table 2.}

Regression Results with Various Tax Measures

Equation	Constant	In T	In WS/NI	In R	In Y	D-W	Measure of Tax
(2.1)	–5.02	.248	1.73	–.155	–.203	.950	Tl
	3.61	5.81	5.33	3.66	1.90*	1.58
(2.2)	–4.20	.310	1.58	–.160	–.280	.947	T2
	2.93	5.26	4.76	3.37	2.22*	1.68
(2.3)	–9.73	.231	1.83	–.191	–.189	.940	S1
	6.03	4.43	4.87	3.98	1.48+	1.52
(2.4)	–9.78	.253	1.84	–.195	–.229	.942	S2
	6.13	4.63	4.96	4.25	1.76*	1.51
(2.5)	–9.90	.264	1.87	–.203	–.248	.943	S3
	6.29	4.77	5.10	4.56	1.90*	1.48
(2.6)	–9.58	.252	1.74	–.181	–.161	.943	S4
	6.30	4.65	4.91	3.69	1.33+	1.64
(2.7)	–9.05	.284	1.64	–.165	–.338	.950	BS
	6.22	5.59	4.83	3.50	2.65	1.56

Note: The equations have been corrected with a first-order Cochrane-Orcutt correction for serial correlation. All variables are significant at α = 1 percent, unless marked by * (α = 5 percent) or + (α = 10 percent). The tax measures are T1—Weighted average tax rate on interest income (Tanzi, 1983, Table 1, pp. 291–92). T2—Ratio of total income tax payments after credit to adjusted gross income (Tanzi, 1983, Table 1). S1—S4 adjusted for the fraction of GNP subject to the personal income tax (Seater, 1985, Table 1). S2—Overall average marginal tax rate on household income (Seater, 1985). S3—Overall average marginal tax rate on labor’s marginal product (Seater, 1985). S4—Average marginal federal personal income tax rate on adjusted gross income (Seater, 1985). BS—Average marginal federal income tax (Barro and Sahasakul, 1983, Table 2). Definitions of other variables are provided in Tanzi (1983, p. 290).

View Table

^{Table 2.}

Regression Results with Various Tax Measures

Equation	Constant	In T	In WS/NI	In R	In Y	D-W	Measure of Tax
(2.1)	–5.02	.248	1.73	–.155	–.203	.950	Tl
	3.61	5.81	5.33	3.66	1.90*	1.58
(2.2)	–4.20	.310	1.58	–.160	–.280	.947	T2
	2.93	5.26	4.76	3.37	2.22*	1.68
(2.3)	–9.73	.231	1.83	–.191	–.189	.940	S1
	6.03	4.43	4.87	3.98	1.48+	1.52
(2.4)	–9.78	.253	1.84	–.195	–.229	.942	S2
	6.13	4.63	4.96	4.25	1.76*	1.51
(2.5)	–9.90	.264	1.87	–.203	–.248	.943	S3
	6.29	4.77	5.10	4.56	1.90*	1.48
(2.6)	–9.58	.252	1.74	–.181	–.161	.943	S4
	6.30	4.65	4.91	3.69	1.33+	1.64
(2.7)	–9.05	.284	1.64	–.165	–.338	.950	BS
	6.22	5.59	4.83	3.50	2.65	1.56

Note: The equations have been corrected with a first-order Cochrane-Orcutt correction for serial correlation. All variables are significant at α = 1 percent, unless marked by * (α = 5 percent) or + (α = 10 percent). The tax measures are T1—Weighted average tax rate on interest income (Tanzi, 1983, Table 1, pp. 291–92). T2—Ratio of total income tax payments after credit to adjusted gross income (Tanzi, 1983, Table 1). S1—S4 adjusted for the fraction of GNP subject to the personal income tax (Seater, 1985, Table 1). S2—Overall average marginal tax rate on household income (Seater, 1985). S3—Overall average marginal tax rate on labor’s marginal product (Seater, 1985). S4—Average marginal federal personal income tax rate on adjusted gross income (Seater, 1985). BS—Average marginal federal income tax (Barro and Sahasakul, 1983, Table 2). Definitions of other variables are provided in Tanzi (1983, p. 290).

The equation that is based on Barro and Sahasakul’s measure of the marginal tax rate (equation 2.7) gives an that is the same as Tanzi’s highest (equation 2.1). One slight improvement in equation (2.7) is that all variables in this equation (including income) are significant at the 1 percent level. Since equation (2.7) gives empirical results at least as good as Tanzi’s equations, while using a theoretically superior tax variable, it is interesting to derive the estimates of the underground economy from this equation. These estimates are provided in Table 3. The diagram shows the size of the underground economy as a percentage of GDP using Tanzi’s method and his tax variables and the Barro and Sahasakul’s tax variable.

^{Table 3.}

U.S. Underground Economy as Percentage of GNP, Using Average Marginal Tax Rates

Note: For average marginal tax rates, see Barro and Sahasakul (1983; equation (2.7)).

^{Table 3.}

U.S. Underground Economy as Percentage of GNP, Using Average Marginal Tax Rates

Year	Tanzi’s Method	Upper-Bound Method
1930	0.0	0.1
1931	0.3	0.3
1932	3.5	3.9
1933	2.1	2.4
1934	2.1	2.3
1935	2.0	2.2
1936	3.0	3.4
1937	1.4	1.6
1938	0.6	0.7
1939	1.9	2.1
1940	3.0	3.4
1941	4.9	5.4
1942	5.4	6.1
1943	3.8	4.4
1944	5.1	5.9
1945	4.6	5.4
1946	3.4	4.0
1947	4.3	5.0
1948	2.6	3.0
1949	3.5	4.1
1950	4.1	4.7
1951	4.6	5.2
1952	4.6	5.3
1953	4.2	4.9
1954	3.6	4.2
1955	4.0	4.6
1956	4.1	4.7
1957	3.9	4.5
1958	4.0	4.6
1959	4.0	4.6
1960	3.9	4.4
1961	4.1	4.7
1962	4.0	4.6
1963	4.1	4.6
1964	3.4	3.8
1965	3.5	3.9
1966	3.8	4.2
1967	4.1	4.5
1968	4.9	5.4
1969	4.7	5.3
1970	4.1	4.6
1971	4.5	5.0
1972	4.7	5.2
1973	4.5	4.9
1974	4.7	5.2
1975	4.9	5.5
1976	5.3	5.8
1977	5.3	5.9
1978	5.8	6.3
1979	4.6	5.0
1980	5.4	6.0

Note: For average marginal tax rates, see Barro and Sahasakul (1983; equation (2.7)).

View Table

^{Table 3.}

U.S. Underground Economy as Percentage of GNP, Using Average Marginal Tax Rates

Year	Tanzi’s Method	Upper-Bound Method
1930	0.0	0.1
1931	0.3	0.3
1932	3.5	3.9
1933	2.1	2.4
1934	2.1	2.3
1935	2.0	2.2
1936	3.0	3.4
1937	1.4	1.6
1938	0.6	0.7
1939	1.9	2.1
1940	3.0	3.4
1941	4.9	5.4
1942	5.4	6.1
1943	3.8	4.4
1944	5.1	5.9
1945	4.6	5.4
1946	3.4	4.0
1947	4.3	5.0
1948	2.6	3.0
1949	3.5	4.1
1950	4.1	4.7
1951	4.6	5.2
1952	4.6	5.3
1953	4.2	4.9
1954	3.6	4.2
1955	4.0	4.6
1956	4.1	4.7
1957	3.9	4.5
1958	4.0	4.6
1959	4.0	4.6
1960	3.9	4.4
1961	4.1	4.7
1962	4.0	4.6
1963	4.1	4.6
1964	3.4	3.8
1965	3.5	3.9
1966	3.8	4.2
1967	4.1	4.5
1968	4.9	5.4
1969	4.7	5.3
1970	4.1	4.6
1971	4.5	5.0
1972	4.7	5.2
1973	4.5	4.9
1974	4.7	5.2
1975	4.9	5.5
1976	5.3	5.8
1977	5.3	5.9
1978	5.8	6.3
1979	4.6	5.0
1980	5.4	6.0

Note: For average marginal tax rates, see Barro and Sahasakul (1983; equation (2.7)).

Comparing Tables 1 and 3 and looking at Diagram 1, one sees that the estimates of the underground economy based on equation (2.7) are much closer to Tanzi’s estimates based on equation (2.1) than to estimates based on equation (2.2). The Barro and Sahasakul tax variable provides estimates of the underground economy that are, on average, 11 percent lower than Tanzi’s (see Tables 1, 3, and 4). However, this difference is significant in the years 1944–55. If only the 1956–80 period is compared, the difference between these estimates is a mere 3 percent (see Table 4).

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The Underground Economy as a Percentage of GNP

Citation: IMF Staff Papers 1986, 004; 10.5089/9781451930696.024.A007

Note: For T1, T2, and BS tax variables, see the note to Table 2.

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Conclusions

This note introduces a variant of Tanzi’s method for estimating the underground economy. While retaining Tanzi’s basic approach, this variant provides an upper bound for Tanzi’s estimates, and therefore it is recommended that it be used along with Tanzi’s method. The upper-bound estimates are, on average, 12.5 percent higher than Tanzi’s.

The use of marginal tax rates is theoretically superior to the use of average tax rates. Using recent estimates for marginal tax rates it is shown that

(a) Alternative measures of the tax variable do not alter Tanzi’s basic results: all coefficients have the predicted signs and are (with the exclusion of y, in some cases) highly significant.

(b) Using a marginal tax rate, estimates of the underground economy exhibit patterns similar to Tanzi’s estimates based on weighted average tax rates. The difference between these estimates is 11 percent on average, but in the more recent period (1956–80) this difference is quite small (3 percent).

^{Table 4.}

Average Size of the Underground Economy

Note: For T1, T2, and BS tax variables see the note to Table 2.

^{Table 4.}

Average Size of the Underground Economy

	1930–80	1947–80	1956–80	1960–80	1970–80
Tanzi’s method
T1	4.24	4.62	4.56	4.62	5.07
T2	2.77	3.00	3.07	3.13	3.37
BS	3.79	4.29	4.42	4.50	4.90
Upper-bound method
T1	4.77	5.17	5.05	5.10	5.56
T2	3.14	3.39	3.43	3.48	3.72
BS	4.27	4.83	4.92	4.99	5.40

Note: For T1, T2, and BS tax variables see the note to Table 2.

View Table

^{Table 4.}

Average Size of the Underground Economy

	1930–80	1947–80	1956–80	1960–80	1970–80
Tanzi’s method
T1	4.24	4.62	4.56	4.62	5.07
T2	2.77	3.00	3.07	3.13	3.37
BS	3.79	4.29	4.42	4.50	4.90
Upper-bound method
T1	4.77	5.17	5.05	5.10	5.56
T2	3.14	3.39	3.43	3.48	3.72
BS	4.27	4.83	4.92	4.99	5.40

Note: For T1, T2, and BS tax variables see the note to Table 2.