I. The Makin Credit Model

The argument is widely accepted that indexing the tax treatment of interest to inflation would lower nominal interest rates and the government deficit.² To illustrate the argument, it is assumed that the after-tax real interest rate is 3.0 percent, the tax rate is one third, and inflation is 6.0 percent. Under these assumptions, a nominal interest rate of 13.5 percent, given a one-third tax rate, amounts to a 9.0 percent after-tax interest rate. Subtracting 6.0 percent inflation from this leaves a 3.0 percent after-tax real interest rate. With the introduction of indexing, the nominal rate would fall to 10.5 percent. A 10.5 percent nominal interest rate less 6.0 percent inflation equals a 4.5 percent before-tax real interest rate that, with a one-third tax rate, leaves lenders the same 3.0 percent after-tax real interest rate as before. According to the argument, the government deficit would be reduced because the nominal interest rate would have fallen with the introduction of indexing the tax treatment of interest income and expense.

One question about this analysis is that it fails to account for the effects on government revenue of indexing interest taxation and of lower tax rates. In Section V it is shown that indexing might reduce tax revenue, so that a tax rate increase—not a decrease— could be required to maintain revenue neutrality. A second question is that the underlying credit market model used to derive the above results is not general but is based on an implicit assumption that investment returns of borrowers are not taxed, as Section II makes clear.

Table 1 formally presents this tax-adjusted credit market.³ The model is linear, with lending and borrowing measured in logarithms. Taxes are introduced on the assumptions that borrowers can deduct interest payments and that lenders are taxed on gross interest earnings. The after-tax real interest rate is i(l − τ) − π without indexing and (i − π)(l − τ) with indexing. Such an aftertax real rate is specified as the interest rate variable that would increase the supply of, and decrease the demand for, credit as recorded in Table 1. Other factors that affect the credit market are assumed to be reflected in the constants.

According to the lending equation, an increase in inflation would reduce lending unless nominal interest rates were to increase by 1/(1 − τ) times as much. An increase in the tax rate would lower the after-tax interest rate and reduce lending.

Table 1.

Makin Tax-Adjusted Credit Market With and Without Interest Taxation Indexed to Inflation

Note: With indexing, ∂B/∂i = β₁(i − π) > 0; without indexing, ∂B/∂τ = β₁i > 0. For comparisons, π > 0 is assumed throughout the paper. L is real lending (logarithm); B is real borrowing (logarithm); r is the after-tax real interest rate; α and β are nonnegative parameters; i is the nominal interest rate; τ is the tax rate; π is expected inflation; i − π is the before-tax real interest rate.

Table 1.

Makin Tax-Adjusted Credit Market With and Without Interest Taxation Indexed to Inflation

With Indexing		Without Indexing
[r = (i−π)(l−τ)]		[r = i(l−τ) −τ]
	Equilibrium values
r =(α0 + β0)/(α1 + β1)	=	same
i − π [(α0 + β0)/(α1 + β1)]/(1 − τ)	<	[(α0 + β0)/(α1 + β1) + τπ-]/(I − τ)
i = [(α1 + β0)/(α1 + β1)]/(1−τ) + π	<	[(α0 + β0)/(α1 + β1) + π]/(l− τ)
L = (α1 β0 − α0 β1)/(α1 + β1)	=	same
	Derivatives
dr/d π = 0	=	same
d(i − π)/dπ = 0	<	τ/(1 − τ)
di/d π = 1	<	1/(1 −τ)
dL/dπ = 0	=	same
dr/dτ = 0	=	same
d(i − π)/dτ = (i − π)/(l − τ)	<	i/(1 − τ)
di/dτ =(i − π)/(l − τ)	<	i/(1 − τ)
dL/dτ = 0	=	same

Note: With indexing, ∂B/∂i = β₁(i − π) > 0; without indexing, ∂B/∂τ = β₁i > 0. For comparisons, π > 0 is assumed throughout the paper. L is real lending (logarithm); B is real borrowing (logarithm); r is the after-tax real interest rate; α and β are nonnegative parameters; i is the nominal interest rate; τ is the tax rate; π is expected inflation; i − π is the before-tax real interest rate.

View Table

Table 1.

Makin Tax-Adjusted Credit Market With and Without Interest Taxation Indexed to Inflation

With Indexing		Without Indexing
[r = (i−π)(l−τ)]		[r = i(l−τ) −τ]
	Equilibrium values
r =(α0 + β0)/(α1 + β1)	=	same
i − π [(α0 + β0)/(α1 + β1)]/(1 − τ)	<	[(α0 + β0)/(α1 + β1) + τπ-]/(I − τ)
i = [(α1 + β0)/(α1 + β1)]/(1−τ) + π	<	[(α0 + β0)/(α1 + β1) + π]/(l− τ)
L = (α1 β0 − α0 β1)/(α1 + β1)	=	same
	Derivatives
dr/d π = 0	=	same
d(i − π)/dπ = 0	<	τ/(1 − τ)
di/d π = 1	<	1/(1 −τ)
dL/dπ = 0	=	same
dr/dτ = 0	=	same
d(i − π)/dτ = (i − π)/(l − τ)	<	i/(1 − τ)
di/dτ =(i − π)/(l − τ)	<	i/(1 − τ)
dL/dτ = 0	=	same

Note: With indexing, ∂B/∂i = β₁(i − π) > 0; without indexing, ∂B/∂τ = β₁i > 0. For comparisons, π > 0 is assumed throughout the paper. L is real lending (logarithm); B is real borrowing (logarithm); r is the after-tax real interest rate; α and β are nonnegative parameters; i is the nominal interest rate; τ is the tax rate; π is expected inflation; i − π is the before-tax real interest rate.

According to the borrowing equation, an increase in inflation would affect borrowing through a reduction in the after-tax interest rate, which would tend to increase borrowing unless nominal interest rates increased by 1/(1 − τ) times as much. According to the tax-adjusted credit market of Table 1, if the real balance and relative price uncertainty effects of inflation on lending and borrowing are ignored, the nominal interest rate would increase by the full 1/(1 — τ) times an increase in expected inflation, which is the hypothesis of the tax-adjusted Fisher equation.

A clue that raises a question about the borrowings equation of Table 1 is that an increase in the tax rate would reduce after-tax real interest rates and increase borrowings. Increase borrowings? This implication of the model is not intuitive. Even though firms occasionally borrow to pay taxes, higher tax rates would normally reduce expected returns. The demand for credit as represented by the borrowing equation, however, is not specified to include any such direct effect of tax rates on investment but only the indirect effects of tax rates on the after-tax real interest rate at which expected returns are discounted, as will be shown in the next section. Furthermore, equilibrium credit and after-tax real rates in Table 1 are independent of tax rates and, thus, of indexing. These results merit scrutiny.

II. Present Values Under Alternative Tax Regimes

Under a tax code such as in the Netherlands, Switzerland, or the United States, interest income is taxable, and interest expense by both individuals and businesses is deductible in calculating taxable income. The tax on interest income effectively taxes saving and lending as would an excise tax, except for that portion which escapes because of evasion and the exemption of interest income from particular credit instruments (such as savings deposits in Switzerland and state and local government securities in the United States). But does interest deductibility subsidize spending and investment? The answer is partly yes and partly no. Borrowing for consumption that does not yield taxable income is actually subsidized because interest expense is deductible in calculating taxable income. Borrowing to finance investment in houses or consumer durables for personal use is also subsidized because the services of such assets are not taxed, but interest expense is deductible. Unrealized capital gains may also escape taxation temporarily. Nevertheless, consumer credit, home mortgages, and government obligations totaled only about two thirds of the funds raised in U.S. credit markets by nonfinancial sectors in 1984 (Board of Governors of the Federal Reserve System (1985)). Therefore, it is important to recognize that, for a large part of total borrowing, an increase in tax rates not only may lower the interest rate at which investment returns are discounted but also may raise the tax burden on the income that is thereby generated. It will be shown that failure to account for this heavier tax burden is what explains the result of the standard models, that a change in a uniform tax rate would not affect real variables in the credit market.

Tax rates can be introduced directly into present-value calculations on which investment decisions depend. Investments whose present values are positive would be undertaken, others abandoned. It is assumed that all investment is financed by borrowing. Hence, positive present values would increase the demand for borrowing. The question is how tax rates and inflation would affect present values under alternative tax regimes.

It is easily shown that present values would be independent of tax rates if the tax rate applicable to investment returns is the same as the tax rate applicable to interest cost deductions from taxable income. First consider an investment expected to return 1 + r dollars at the end of one period. If i is the market rate of interest, the present value of the return is

Suppose that the cost of the investment is c. The investment would be made as long as net profit a is not negative:

Equivalently, the present value must be no less than the cost of an investment:

The introduction of a uniform tax rate on income, with interest as a deductible expense, would not affect this condition. After-tax profit would simply be the before-tax profit less taxes:

As before, the investment would be made as long as the before-tax profit—the term in brackets—is not negative. Thus, the investment decision would be independent of the tax rate in this simple case, as Stiglitz (1973), Auerbach (1983), and others have argued. Increasing the tax rate would reduce after-tax profits but would not affect investment or borrowing to finance investment.

Table 2 records present-value formulas under alternative tax regimes with and without inflation. Capital is assumed to yield a marginal physical product r in perpetuity and with no depreciation. Results would not be affected if the nominal interest rate is thought of as including the depreciation rate (Gandolfi (1982)). According to the table, in all cases in which taxes on real or nominal returns are matched by comparable deductions of real or nominal interest expense, present values are independent of tax rates. Regime C in Table 2 corresponds to the one-period illustration previously discussed. The after-tax real return is r(l − τ). It grows in nominal terms at a rate of only π(1 − τ) rather than π because of the tax on the inflationary gain. Discounting this nominal after-tax return at the nominal after-tax interest rate i(1 − τ) yields the same present value as under regime A, with no taxes, or regime B, with indexation of both taxable rates of return and deductible interest rates. Thus, tax rates would not influence present values, investment, or borrowing to finance investment as long as both investment returns and deductible interest expenses influence tax liabilities symmetrically.

Table 2.

Present Values Under Alternative Tax Regimes

Note: r, i, t, and π are, respectively, the marginal physical product of capital, the nominal interest rate, the uniform tax rate on taxable income, and the inflation rate. Interest expense is deductible in calculating taxable income except in regime F. Gandolfi (1982) incorporates a comparable taxonomy including depreciation.

^ar = i − π/(l − τ) as in Feldstein (1976a).

^br = i(1− τ)-π as in Makin (1984).

^cr = i/(l− τ) −π as in Fisher (1906).

Table 2.

Present Values Under Alternative Tax Regimes

	One Dollar of Present Value
Tax Regime	Without inflation	With inflation
(A) No taxes	${\displaystyle {\int }_0^{\infty }r{e}^{-it}dt=r/i}$	${\displaystyle {\int }_0^{\infty }r{e}^{\pi t}{e}^{-it}dt=r/\left(i-\pi \right)}$
(B) Tax on real return, interest taxation indexed	${\displaystyle {\int }_0^{\infty }\left(i-\pi \right)r{e}^{-i\left(i-\tau \right)t}dt=r/i}$	$\begin{array}{l}{\displaystyle {\int }_0^{\infty }\left(i-\tau \right)r{e}^{-\left(i-\pi \right)\left(i-\tau \right)t}dt}\hfill \\ =r/\left(i-\pi \right)\hfill \end{array}$
(C) Tax on nominal return	${\displaystyle {\int }_0^{\infty }\left(i-\tau \right)r{e}^{-i\left(i-\tau \right)t}dt=r/i}$	$\begin{array}{l}{\displaystyle {\int }_0^{\infty }\left(i-\tau \right)r{e}^{\pi \left(i-\tau \right)t}{e}^{-i\left(i-\tau \right)t}dt}\hfill \\ =r/\left(i-\pi \right)\hfill \end{array}$
(D) Tax on nominal return, capital gains from inflation excluded	${\int }_0^{\infty }\left(i-\tau \right)r{e}^{-i\left(i-\tau \right)t}dt=r/i$	$\begin{array}{l}{\displaystyle {\int }_0^{\infty }\left(i-\tau \right)r{e}^{\pi t}{e}^{-i\left(i-\tau \right)t}dt}\hfill \\ =\left(i-\tau \right)r/{\left[i\left(i-\tau \right)-\pi \right]}^{\text{a}}\hfill \end{array}$
(E) Untaxed returns	${\int }_0^{\infty }r{e}^{-i\left(1-\tau \right)t}dt=r/\left[i\left(1-\tau \right)\right]$	$\begin{array}{l}{\displaystyle {\int }_0^{\infty }r{e}^{\pi \pi }{e}^{-i\left(1-\tau \right)t}dt}\hfill \\ =r/{\left[i\left(1-\tau \right)-\pi \right]}^{\text{b}}\hfill \end{array}$
(F) Tax on nominal return, interest cost not deductible	${\int }_0^{\infty }\left(1-\tau \right)r{e}^{-it}dt=\left(1-\tau \right)r/i$	$\begin{array}{l}{\displaystyle {\int }_0^{\infty }\left(1-\tau \right)r{e}^{\pi \left(1-\tau \right)t}{e}^{-it}dt}\hfill \\ =\left(1-\tau \right)r/{\left[i-\pi \left(1-\tau \right)\right]}^{\text{c}}\hfill \end{array}$

Note: r, i, t, and π are, respectively, the marginal physical product of capital, the nominal interest rate, the uniform tax rate on taxable income, and the inflation rate. Interest expense is deductible in calculating taxable income except in regime F. Gandolfi (1982) incorporates a comparable taxonomy including depreciation.

^ar = i − π/(l − τ) as in Feldstein (1976a).

^br = i(1− τ)-π as in Makin (1984).

^cr = i/(l− τ) −π as in Fisher (1906).

View Table

Table 2.

Present Values Under Alternative Tax Regimes

	One Dollar of Present Value
Tax Regime	Without inflation	With inflation
(A) No taxes	${\displaystyle {\int }_0^{\infty }r{e}^{-it}dt=r/i}$	${\displaystyle {\int }_0^{\infty }r{e}^{\pi t}{e}^{-it}dt=r/\left(i-\pi \right)}$
(B) Tax on real return, interest taxation indexed	${\displaystyle {\int }_0^{\infty }\left(i-\pi \right)r{e}^{-i\left(i-\tau \right)t}dt=r/i}$	$\begin{array}{l}{\displaystyle {\int }_0^{\infty }\left(i-\tau \right)r{e}^{-\left(i-\pi \right)\left(i-\tau \right)t}dt}\hfill \\ =r/\left(i-\pi \right)\hfill \end{array}$
(C) Tax on nominal return	${\displaystyle {\int }_0^{\infty }\left(i-\tau \right)r{e}^{-i\left(i-\tau \right)t}dt=r/i}$	$\begin{array}{l}{\displaystyle {\int }_0^{\infty }\left(i-\tau \right)r{e}^{\pi \left(i-\tau \right)t}{e}^{-i\left(i-\tau \right)t}dt}\hfill \\ =r/\left(i-\pi \right)\hfill \end{array}$
(D) Tax on nominal return, capital gains from inflation excluded	${\int }_0^{\infty }\left(i-\tau \right)r{e}^{-i\left(i-\tau \right)t}dt=r/i$	$\begin{array}{l}{\displaystyle {\int }_0^{\infty }\left(i-\tau \right)r{e}^{\pi t}{e}^{-i\left(i-\tau \right)t}dt}\hfill \\ =\left(i-\tau \right)r/{\left[i\left(i-\tau \right)-\pi \right]}^{\text{a}}\hfill \end{array}$
(E) Untaxed returns	${\int }_0^{\infty }r{e}^{-i\left(1-\tau \right)t}dt=r/\left[i\left(1-\tau \right)\right]$	$\begin{array}{l}{\displaystyle {\int }_0^{\infty }r{e}^{\pi \pi }{e}^{-i\left(1-\tau \right)t}dt}\hfill \\ =r/{\left[i\left(1-\tau \right)-\pi \right]}^{\text{b}}\hfill \end{array}$
(F) Tax on nominal return, interest cost not deductible	${\int }_0^{\infty }\left(1-\tau \right)r{e}^{-it}dt=\left(1-\tau \right)r/i$	$\begin{array}{l}{\displaystyle {\int }_0^{\infty }\left(1-\tau \right)r{e}^{\pi \left(1-\tau \right)t}{e}^{-it}dt}\hfill \\ =\left(1-\tau \right)r/{\left[i-\pi \left(1-\tau \right)\right]}^{\text{c}}\hfill \end{array}$

Note: r, i, t, and π are, respectively, the marginal physical product of capital, the nominal interest rate, the uniform tax rate on taxable income, and the inflation rate. Interest expense is deductible in calculating taxable income except in regime F. Gandolfi (1982) incorporates a comparable taxonomy including depreciation.

^ar = i − π/(l − τ) as in Feldstein (1976a).

^br = i(1− τ)-π as in Makin (1984).

^cr = i/(l− τ) −π as in Fisher (1906).

Regimes D, E, and F show three tax regimes under which tax rates would affect present values and hence investment. In regime D, only real investment returns are taxed, but total nominal interest is deductible. In this case, when there is inflation an increase in the tax rate would increase the present value of a given investment because the increased tax on the real return would be more than offset by the increased deduction of nominal interest payments. Regime D corresponds to Feldstein’s model (1976a) in that Feldstein explicitly assumed that real but not inflationary returns are taxed. Under regime E, investment returns are not taxed at all. With inflation, an increase in the tax rate would therefore increase the present value of a return because of the effect of the higher tax rate in increasing deductible nominal interest expense. Regime E is the one implicitly assumed in much of the literature on the tax-adjusted Fisher equation (including Makin (1984), reported in Table 1). Regime F, under which nominal returns are taxed but interest is not deductible, seems pathological today, but when this case was discussed by Fisher (1906, pp. 398–400) there was no income tax, so that Fisher was in effect evaluating the effect of property taxes on present values.⁴ Tax regime F is not considered further in this paper.

III. The Feldstein Credit Model

Table 3 modifies the Makin model of Table 1 to make it consistent with Feldstein (1976a), who assumed that only inflationary capital gains are untaxed. The two models are similar to the extent that they imply that di/dτ = 1/(1 − τ). The Makin model specifies that borrowers are influenced by the same after-tax real interest rate as lenders, i(l − τ) − π (Table 2, tax regime E). The Feldstein model specifies that borrowers are influenced by i − τ/(l − τ) (Table 2, tax regime D). In the latter case, the after-tax rate earned by lenders is therefore less than that earned by borrowers by a factor of 1 − τ. Nevertheless, in both the Makin and Feldstein models, an increase in the tax rate would increase borrowing. In the Makin model, present values and hence borrowing demand would increase with an increase in the tax rate because the higher tax rate would reduce the after-tax interest rate used to discount returns, which are assumed to be untaxed. In the Feldstein model, such a decrease in the after-tax interest rate used to discount returns would only increase the present value of inflation returns, which are assumed to be untaxed.

Table 3.

Feldslein Tax-Adjusted Credit Market With and Without Interest Taxation Indexed to Inflation

Note: With indexing. ∂B/∂τ = 0; without indexing, ∂B/∂τ = β₁ π(l − τ)² > 0. Δ = (1 − τ)α₁ + β₁

Table 3.

Feldslein Tax-Adjusted Credit Market With and Without Interest Taxation Indexed to Inflation

(L = −α0 + α1rL; B = β0 − β1rB; L = B)
With Indexing		Without Indexing
[rL = (i − π)(l − π);rB = i− π]		[rL = i(l − τ) − τ; rB = i − π/(1 − τ)]
	Equilibrium values
rL = (l − τ)(α0 + β0)/Δ	=	same
rB = (α0 + β0)/Δ	=	same
i = (α0 + β0)/Δ + π	<	(α0 + β0)/Δ + π/(l − τ)
L = [(l − τ)α1 β0 − α0β1]/Δ	=	same
	Derivatives
drL/dπ = 0	=	same
drB/d π = 0	=	same
di/dπ = 1	<	1/(1 − τ)
dL/dπ = 0	=	same
drL/dτ = −β1,(α0 + β0)/Δ2	=	same
drB/dτ = α1(α0 + β0)/Δ2:	=	same
dr/dτ = α1(α0 + β0)/Δ2	<	α1(α0 + β0)/Δ2 + π/(l − τ)2
dL/dτ = −α1β1{α0+ β0)/Δ2;	=	same

Note: With indexing. ∂B/∂τ = 0; without indexing, ∂B/∂τ = β₁ π(l − τ)² > 0. Δ = (1 − τ)α₁ + β₁

View Table

Table 3.

Feldslein Tax-Adjusted Credit Market With and Without Interest Taxation Indexed to Inflation

(L = −α0 + α1rL; B = β0 − β1rB; L = B)
With Indexing		Without Indexing
[rL = (i − π)(l − π);rB = i− π]		[rL = i(l − τ) − τ; rB = i − π/(1 − τ)]
	Equilibrium values
rL = (l − τ)(α0 + β0)/Δ	=	same
rB = (α0 + β0)/Δ	=	same
i = (α0 + β0)/Δ + π	<	(α0 + β0)/Δ + π/(l − τ)
L = [(l − τ)α1 β0 − α0β1]/Δ	=	same
	Derivatives
drL/dπ = 0	=	same
drB/d π = 0	=	same
di/dπ = 1	<	1/(1 − τ)
dL/dπ = 0	=	same
drL/dτ = −β1,(α0 + β0)/Δ2	=	same
drB/dτ = α1(α0 + β0)/Δ2:	=	same
dr/dτ = α1(α0 + β0)/Δ2	<	α1(α0 + β0)/Δ2 + π/(l − τ)2
dL/dτ = −α1β1{α0+ β0)/Δ2;	=	same

Note: With indexing. ∂B/∂τ = 0; without indexing, ∂B/∂τ = β₁ π(l − τ)² > 0. Δ = (1 − τ)α₁ + β₁

A critical similarity between these models is that neither a change in inflation nor the introduction of indexing the tax treatment of interest income and expense would have any effect on the real interest rates influencing borrowers or lenders and, thus, on the volume of credit. The introduction of indexing interest taxation would also unambiguously decrease the nominal interest rate in the Feldstein model, a result that lends support to his argument that such indexing would reduce interest payments on the government debt.⁵ Nevertheless, it is ironic to note that, in either the Makin or Feldstein models of Tables 1 and 3, the introduction of indexing the tax treatment of interest would not in itself influence the real rate that lenders earn. As Section V shows, this is also the real rate at which the government borrows; thus, indexing in the framework of the Makin and Feldstein models would not in general reduce the government deficit, as these authors have argued.

IV. The Generalized Credit Model

In the standard models of Tables 1 and 3, an increase in the tax rate or the inflation rate would not affect the after-tax real interest rate or the amount of credit. These results depend on the assumption that borrowers are not taxed on inflationary investment returns. By modifying the specification of the borrowing relationship to include fully taxed borrowers, not only tax rates but also indexing the tax treatment of interest to inflation are shown to affect both real interest rates and the amount of credit. A model yielding these results is discussed in the following subsection without indexing interest taxation, and in the next subsection with indexing. In the third subsection the model is generalized to incorporate three categories of borrowing that reflect the alternative tax regimes.

Inflation, Taxes, and Interest Rates when Nominal Investment Returns Are Taxed

Table 4 modifies the borrowing relationships of Tables 1 and 3. Borrowing is specified to be a function of the before-tax real interest rate and, thus, to be independent of the tax rate. Think of the function as involving the demand for credit, net of borrowing both for consumption and for investment not subject to taxation of inflationary capital gains. Government borrowing will be introduced in Section V.

Focusing on the results without indexing of the tax treatment of interest in Table 4, one finds that an increase in the tax rate would raise the real after-tax rate to borrowers, even as it would lower the real after-tax rate that lenders earn, and would reduce the amount of credit. Figure 1 depicts how an increase in the tax rate would both shift the lending function up to the left and increase its slope. Table 4 also shows that an increase in inflation would in general increase the real rate to borrowers, decrease the real after-tax rate to lenders, and decrease the amount of credit. Figure 2 depicts this case. The supply of credit shifts up by 1/(1 − τ) times the increase in inflation. The demand for credit shifts up by only the increase in inflation. Hence, the nominal interest rate must rise, and the amount of credit must fall. The magnitude of the effect wouid depend on the interest elasticities of the supply of and demand for credit.

Table 4.

Modified Tax-Adjusted Credit Market With and Without Interest Taxation Indexed to Inflation

Note: r^B = i − π; Δ = (1 − τ)α₁ + β₁.

Table 4.

Modified Tax-Adjusted Credit Market With and Without Interest Taxation Indexed to Inflation

(L = −α0 + α1rL; B = β0 − β1rB; L = B)
With Indexing		Without Indexing
[rL = (i − π(l − τ)]		[rL = i(l − τ) − π]
	Equilibrium values
rB = (α0 + β0)/Δ	<	(α0 + β0 + α1 τπ)/Δ
rL = rB (l − τ)	>	[(l − τ)(α0 + β0) α β1 τπ]/Δ
i = rB + π	<	[α0 + β0 + (α1 + β1}π]/Δ
L, = [(l − τ)α1 β0 − α0 β1]/Δ	>	[(1 −τ)α1 β0 − α0β1 − α1 β1 τπ]/Δ
	Derivatives
drB/dπ = 0	<	α1 τ/Δ
drL/dπ = 0	>	−β1 τ/Δ
di/dπ = 1	<	(α1 + β1)/Δ ≤ 1/(1 − τ)
dLldπ = 0	>	−α1 β1 τ/Δ
drB/dτ = α1rB/Δ	<	α1i/Δ >0
drL/dτ= −β1rB/Δ	>	−β1i/Δ < 0
di/dτ = α1rB/Δ	<	α1i/Δ > 0
dL/dτ = − α1 β1rB/Δ	>	−α1 β1i/Δ <0

Note: r^B = i − π; Δ = (1 − τ)α₁ + β₁.

View Table

Table 4.

Modified Tax-Adjusted Credit Market With and Without Interest Taxation Indexed to Inflation

(L = −α0 + α1rL; B = β0 − β1rB; L = B)
With Indexing		Without Indexing
[rL = (i − π(l − τ)]		[rL = i(l − τ) − π]
	Equilibrium values
rB = (α0 + β0)/Δ	<	(α0 + β0 + α1 τπ)/Δ
rL = rB (l − τ)	>	[(l − τ)(α0 + β0) α β1 τπ]/Δ
i = rB + π	<	[α0 + β0 + (α1 + β1}π]/Δ
L, = [(l − τ)α1 β0 − α0 β1]/Δ	>	[(1 −τ)α1 β0 − α0β1 − α1 β1 τπ]/Δ
	Derivatives
drB/dπ = 0	<	α1 τ/Δ
drL/dπ = 0	>	−β1 τ/Δ
di/dπ = 1	<	(α1 + β1)/Δ ≤ 1/(1 − τ)
dLldπ = 0	>	−α1 β1 τ/Δ
drB/dτ = α1rB/Δ	<	α1i/Δ >0
drL/dτ= −β1rB/Δ	>	−β1i/Δ < 0
di/dτ = α1rB/Δ	<	α1i/Δ > 0
dL/dτ = − α1 β1rB/Δ	>	−α1 β1i/Δ <0

Note: r^B = i − π; Δ = (1 − τ)α₁ + β₁.

increase in Tax Rate (di/dτ = α₁i/Δ≥; dL/dτ= −α₁ β₁i/Δ ≤ 0)

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

View Full Size

increase in Tax Rate (di/dτ = α₁i/Δ≥; dL/dτ= −α₁ β₁i/Δ ≤ 0)

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

increase in Tax Rate (di/dτ = α₁i/Δ≥; dL/dτ= −α₁ β₁i/Δ ≤ 0)

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

Increase in Inflation [1 ≤, di/dπ = (α₁ + β₁)/Δ ≤ 1/(1 − τ); dL/dπ = −α₁ β₁τ/Δ ≤ 0]

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Increase in Inflation [1 ≤, di/dπ = (α₁ + β₁)/Δ ≤ 1/(1 − τ); dL/dπ = −α₁ β₁τ/Δ ≤ 0]

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

Increase in Inflation [1 ≤, di/dπ = (α₁ + β₁)/Δ ≤ 1/(1 − τ); dL/dπ = −α₁ β₁τ/Δ ≤ 0]

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

If the interest elasticity of credit supply is low compared with the interest elasticity of credit demand, the nominal interest rate would change by approximately the change in inflation—the hypothesis of the Fisher equation. Figures 3 and 4 show two extreme cases. In Figure 3, lending is totally insensitive to the interest rate. In Figure 4, borrowing is perfectly elastic with respect to the interest rate. In both cases the borrowing relationship shifts up by the increase in inflation, which is fully reflected in the nominal interest rate.

If the interest elasticity of credit demand is low compared with the interest elasticity of credit supply, however, a change in inflation would be reflected in an increase in the nominal interest rate of approximately 1/(1 − τ) times as much—the hypothesis of the tax-adjusted Fisher equation. Figures 5 and 6 show two extreme cases. In Figure 5, borrowing is totally insensitive to the interest rate; in Figure 6, lending is perfectly elastic with respect to the interest rate. In both cases the lending relationship shifts up by 1/(1 − τ) times the increase in inflation, which is fully reflected in the nominal interest rate.

Lending Insensitive to Interest Rate (di/dπ = 1;dL/dπ = 0)

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Lending Insensitive to Interest Rate (di/dπ = 1;dL/dπ = 0)

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

Lending Insensitive to Interest Rate (di/dπ = 1;dL/dπ = 0)

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

Borrowing Perfectly Elastic with Respect to Interest Rate (di/dπ = 1;dL/dπ < 0)

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Borrowing Perfectly Elastic with Respect to Interest Rate (di/dπ = 1;dL/dπ < 0)

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

Borrowing Perfectly Elastic with Respect to Interest Rate (di/dπ = 1;dL/dπ < 0)

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

Borrowing Insensitive to Interest Rate [di/dπ = 1/(1 − τ); dL/dπ = 0]

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Borrowing Insensitive to Interest Rate [di/dπ = 1/(1 − τ); dL/dπ = 0]

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

Borrowing Insensitive to Interest Rate [di/dπ = 1/(1 − τ); dL/dπ = 0]

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

Because empirical evidence indicates that the interest elasticity of saving is comparatively low, the tax-adjusted credit market of Table 4 could be interpreted as almost reinstating the hypothesis of the Fisher equation, that nominal interest rates would change proportionately with inflation. Thus, di/dπ would equal unity not only if the tax rate is zero, as in the original Fisherian formulation, but also if either the interest elasticity of lending is zero or the interest elasticity of borrowing becomes infinitely large. That investment might be infinitely elastic with respect to the interest rate has been argued by elasticity optimists, such as Knight (1933) and Friedman (1976) for a closed economy and Dornbusch (1976) and Harberger (1980) for an open economy. According to Knight (1933, p. 336),

The supply of capital goods is the demand for savings, and is indefinitely elastic, because capital goods are indefinitely producible at substantially constant cost. Likewise the demand for capital goods is the amount of saving, which depends on individual choices between saving and spending. The supply of savings, conceived in the only proper way, as the rate of flow into the investment market, in response to various rates of interest, is certainly inelastic, quite possibly inversely elastic, under some conditions, if not on the whole.

Ironically, in light of the controversy between Fisher and Knight regarding interest rate determination (with Fisher arguing that saving behavior is a critical factor determining interest rates), a Knightian world (with the real interest rate determined by investment demand regardless of the propensity to save) would reinstate the original Fisher equation relating nominal interest rates to inflation.

In the empirical literature on the tax-adjusted Fisher equation, it has frequently been observed that nominal interest rates tend to increase by even less than increases in expected inflation—let alone by more, as is the hypothesis of the tax-adjusted Fisher equation (see, for example, the studies included in Tanzi (1984)).

Lending Perfectly Elastic with Respect to Interest Rate [di/dπ = 1/(1 − τ); dL/dπ < 0]

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Lending Perfectly Elastic with Respect to Interest Rate [di/dπ = 1/(1 − τ); dL/dπ < 0]

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

Lending Perfectly Elastic with Respect to Interest Rate [di/dπ = 1/(1 − τ); dL/dπ < 0]

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

One theoretical possibility to account for this phenomenon is an inverse relationship between credit supply and the after-tax real interest rate. Such a modified lending equation is introduced in Figures 7 and 8, on the assumption (for dynamic stability) that the interest elasticity of borrowing remains more negative than that of lending. Figure 7 depicts how an increase in the tax rate in this case would decrease the nominal interest rate as the lending function shifts up and flattens. Because lending is now assumed to be related negatively to the after-tax real interest rate, an increase in the tax rate would reduce the after-tax real interest rate and increase lending—a paradox of thrift. Figure 8 shows how an increase in inflation would increase the nominal interest rate less than proportionately. The lending function shifts up more than proportionately, but the borrowing function only proportionately, to an increase in inflation. Hence the nominal interest rate would increase less than proportionately, and the after-tax real interest rate would fall.

Increase in Tax Rare Under Modified Lending Equation (di/dτ = −α₁i/Δ ≤ 0; dL/dτ = α₁β₁i/Δ ≤ 0)

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Increase in Tax Rare Under Modified Lending Equation (di/dτ = −α₁i/Δ ≤ 0; dL/dτ = α₁β₁i/Δ ≤ 0)

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

Increase in Tax Rare Under Modified Lending Equation (di/dτ = −α₁i/Δ ≤ 0; dL/dτ = α₁β₁i/Δ ≤ 0)

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

• Download Figure
• Download figure as PowerPoint slide

Increase in influiion Under Modified Lending Equation [0 ≤ di/dπ = (−α₁ + β₁)/Δ ≤ 1; dL/dπ =α₁β₁τ/Δ ≥ 0]

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

Note: (α₁ + β₁)/Δ = 1 + α₁ τ/Δ ≤ 1; since β₁ > −α₁(l − τ), both (α₁ + β₁) and Δ are nonnegative; thus, di/dπ ≥ 0.

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Increase in influiion Under Modified Lending Equation [0 ≤ di/dπ = (−α₁ + β₁)/Δ ≤ 1; dL/dπ =α₁β₁τ/Δ ≥ 0]

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

Note: (α₁ + β₁)/Δ = 1 + α₁ τ/Δ ≤ 1; since β₁ > −α₁(l − τ), both (α₁ + β₁) and Δ are nonnegative; thus, di/dπ ≥ 0.

• Download Figure
• Download figure as PowerPoint slide

Increase in influiion Under Modified Lending Equation [0 ≤ di/dπ = (−α₁ + β₁)/Δ ≤ 1; dL/dπ =α₁β₁τ/Δ ≥ 0]

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

Note: (α₁ + β₁)/Δ = 1 + α₁ τ/Δ ≤ 1; since β₁ > −α₁(l − τ), both (α₁ + β₁) and Δ are nonnegative; thus, di/dπ ≥ 0.

• Download Figure
• Download figure as PowerPoint slide

Indexing Interest Taxation when Nominal Investment Returns Are Taxed

Figure 9 compares credit market results from Table 4 with and without indexing interest taxation. The borrowing function remains the same with or without indexing. Lending is again assumed to vary directly with interest rates. Indexing interest taxation would increase lending because it would raise the after-tax real return to tenders. Consequently, the nominal interest rate would decrease with the introduction of such indexing. Reducing the tax rate and, in the limit, eliminating taxes would further increase lending and reduce the nominal interest rate.

Figure 9 also shows that indexing interest taxation can lower r^B, the real rate that fuliy taxed borrowers pay, and simultaneously raise r^L, the after-tax real rate that lenders earn.⁶ Without indexing, the after-tax real rate i₀(l − τ) − τ induces L₀, lending. The corresponding real interest rate facing fully taxed borrowers is i₀ − π, which exceeds the lenders’ rate by τi₀−the tax paid by lenders per dollar of interest income. This real tax wedge depends directly on the tax on the inflation premium in the nominal interest rate τπ. With indexing, the lending function shifts down by τπ/(1 − τ)—the tax on the π/(1 − τ) increase in the nominal interest needed to maintain the after-tax real rate to lenders in the presence of inflation. Lending increases to L₁, reducing the nominal rate to i₁. The after-tax real interest rate earned by lenders rises to r^L₁. With indexing, the real rate facing fully taxed business borrowers can simultaneously fall. The gap between it and the after-tax real rate earned by lenders is τ(i, − π) with indexing. which is smaller than τi₀ (the gap without indexing) not only because the inflation premium is not taxed but also because the nominal interest rate is reduced. Thus, indexing interest taxation to inflation not only would lower nominal and real interest rates paid by fully taxed borrowers but also would raise the after-tax real rate earned by lenders. Both lenders and fully taxed borrowers would be better off because the government would have removed an artificial wedge in the pricing of credit, allowing lenders and fully taxed borrowers to increase their use of credit to mutual advantage.

Modified Tax-Adjusted Credit Market With and Without interest Taxation Indexed to Inflation

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

Note: B = β₀ − β₁(i − π); L = −α₀ + α₁[i(l − τ) − τ] (without indexing); L = −α₀ + α₁[(i − π)(1 − τ)] (with indexing); L = B. See also Table 4.

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Modified Tax-Adjusted Credit Market With and Without interest Taxation Indexed to Inflation

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

Note: B = β₀ − β₁(i − π); L = −α₀ + α₁[i(l − τ) − τ] (without indexing); L = −α₀ + α₁[(i − π)(1 − τ)] (with indexing); L = B. See also Table 4.

• Download Figure
• Download figure as PowerPoint slide

Modified Tax-Adjusted Credit Market With and Without interest Taxation Indexed to Inflation

Citation: IMF Staff Papers 1986, 002; 10.5089/9781451946956.024.A003

Note: B = β₀ − β₁(i − π); L = −α₀ + α₁[i(l − τ) − τ] (without indexing); L = −α₀ + α₁[(i − π)(1 − τ)] (with indexing); L = B. See also Table 4.

• Download Figure
• Download figure as PowerPoint slide

Credit Market with Three Tax Regimes Combined

Table 4 demonstrated the distortionary effect of inflation when lenders are taxed on their nominal interest income and borrowers are taxed on both real and inflationary investment returns. Table 5 generalizes the credit market to incorporate borrowers in three tax categories: (1) untaxed investment returns (Makin (1984)); (2) untaxed inflationary investment returns (Feldstein (1976a)); and (3) fully taxed returns. Qualitatively, the results are exactly the same as for the case of fully taxed returns in Table 4. The reason is that as long as there are some fully taxed borrowers (β₁ ≠ 0), then the distortion of the taxation of inflationary premiums in nominal interest rates would have real effects. In particular, indexing interest taxation to inflation would have real effects because, in the presence of inflation, indexing interest taxation would reduce the tax wedge between the real rate that fully taxed borrowers pay and the real after-tax rate that lenders earn.

Table 5.

Generalized Tax-Adjusted Credit Market With and Without interest Taxation Indexed to Inflation

Note: Δ = α₁(1 − τ) + β₁ + β₂(l − τ) + β₃ ; $r_1^B=i-\pi$ (real rate when both nominal and real returns are taxed); $r_2^B=r^L$ (real rate when returns are untaxed); $r_3^B=r^L/\left(1-\tau \right)$ (real rate when only real returns are taxed).

Table 5.

Generalized Tax-Adjusted Credit Market With and Without interest Taxation Indexed to Inflation

$\left(L=-{\alpha }_0+{\alpha }_1{r}^L;B={\beta }_0-{\beta }_1{r}_1^B-{\beta }_2{r}_2^B-{\beta }_3{r}_3^B;L=B\right)$
With Indexing	Without Indexing
[rL = (i − π)(l − τ)]	[rL = i(l − τ) − π]
	Equilibrium values
$r_1^B=\left({\alpha }_0+{\beta }_0\right)/\Delta$	< (α0 + β0)/Δ + [α1, + β2+ β3/(l − τ)]τπ/Δ
rL = (α0 + β0)(l − τ)/Δ	> (α0 + β0)(l − τ)/Δ − β0 τπ/Δ
i = (α0 + β0)/Δ + π	< (α0 + β0)/Δ + [α1 + β1 + β2 + β3/(I − τ)]π/Δ
L = {(1 − τ)α1 β0	> {(1 − τ) α1 β0
−α0[β1 + β2(l − τ) + β3]}/Δ	− α0[β1 + β2(l − τ) + β3 ] − α1 β1 τπ}/Δ
	Derivatives
$d{r}_1^B/d\pi =0$	< [α1 + β2 + β3/(1 − τ)]τ/Δ > 0
drL/dπ = 0	> − β1 τ/Δ < 0
di/dπ = 1	< [α1 + β1 + β2 + β3/(l − τ)]/Δ ≤ l/(l − τ)
dL/dπ = 0	> − α1 β1τ/Δ < 0

Note: Δ = α₁(1 − τ) + β₁ + β₂(l − τ) + β₃ ; $r_1^B=i-\pi$ (real rate when both nominal and real returns are taxed); $r_2^B=r^L$ (real rate when returns are untaxed); $r_3^B=r^L/\left(1-\tau \right)$ (real rate when only real returns are taxed).

View Table

Table 5.

Generalized Tax-Adjusted Credit Market With and Without interest Taxation Indexed to Inflation

$\left(L=-{\alpha }_0+{\alpha }_1{r}^L;B={\beta }_0-{\beta }_1{r}_1^B-{\beta }_2{r}_2^B-{\beta }_3{r}_3^B;L=B\right)$
With Indexing	Without Indexing
[rL = (i − π)(l − τ)]	[rL = i(l − τ) − π]
	Equilibrium values
$r_1^B=\left({\alpha }_0+{\beta }_0\right)/\Delta$	< (α0 + β0)/Δ + [α1, + β2+ β3/(l − τ)]τπ/Δ
rL = (α0 + β0)(l − τ)/Δ	> (α0 + β0)(l − τ)/Δ − β0 τπ/Δ
i = (α0 + β0)/Δ + π	< (α0 + β0)/Δ + [α1 + β1 + β2 + β3/(I − τ)]π/Δ
L = {(1 − τ)α1 β0	> {(1 − τ) α1 β0
−α0[β1 + β2(l − τ) + β3]}/Δ	− α0[β1 + β2(l − τ) + β3 ] − α1 β1 τπ}/Δ
	Derivatives
$d{r}_1^B/d\pi =0$	< [α1 + β2 + β3/(1 − τ)]τ/Δ > 0
drL/dπ = 0	> − β1 τ/Δ < 0
di/dπ = 1	< [α1 + β1 + β2 + β3/(l − τ)]/Δ ≤ l/(l − τ)
dL/dπ = 0	> − α1 β1τ/Δ < 0

Note: Δ = α₁(1 − τ) + β₁ + β₂(l − τ) + β₃ ; $r_1^B=i-\pi$ (real rate when both nominal and real returns are taxed); $r_2^B=r^L$ (real rate when returns are untaxed); $r_3^B=r^L/\left(1-\tau \right)$ (real rate when only real returns are taxed).

Note that in long-run equilibrium the before-tax real returns from investment in each of the three categories would be adjusted so as to exploit any opportunity to increase utility or profits by switching from one kind of investment to another. In other words, the real effects of the differential taxation of investment returns would be reflected not in different borrowing rates but in investors’ selecting investments on the basis of the tax structure, thus affecting the composition of the capital stock.

V. Government Deficits in the Credit Market

It has thus far been assumed that the government intervened in the credit market only by defining taxable income to include interest income and certain investment returns and by allowing interest expense as a deduction. But the government borrows and lends too. It will be assumed that the government finances deficits solely by issuing interest-bearing securities and not by issuing money. Thus, the deficit is defined as the change in the amount of interest-bearing government debt outstanding:

where

P = the price level

γ = real government purchases of goods and services i = the nominal interest rate

D_−i = outstanding nominal government debt in the previous period

T = nominal government tax receipts less transfers other than interest payments.

The nominal interest rate is assumed to be the same for private borrowers as for the government, although actually the government would borrow at the lowest after-tax interest rate because of its unique powers to tax or issue money (or both) to pay its obligations. To avoid the complication of interest rates affecting the value of outstanding government debt, all such debt is assumed to be of one-period maturity. P can be thought of as a function of the amount of government demand debt outstanding.

Taxable income, defined as national income plus interest earned on government debt, is assumed to be subject to a uniform tax rate; both the tax rate and real taxable national income are considered to be autonomous:

where τ is the tax rate,⁷ and Y is real national income.

The deficit defined in equation (5) is in nominal terms. It must be expressed in real terms to fit into the credit market framework of the previous section. The real government deficit, Z, is simply the change in the real government debt per period (see Barro (1984a)):

The real deficit depends on whether and how interest taxation is indexed to inflation. Consider first the case in which nominal interest income is taxed, as in the present U.S. tax system. Substitute equation (6) into equation (5) to obtain the government budget constraint:

Deflate both sides of the equation by P and collect terms to get

Using the definition that P/P₋₁ = 1 + π, where π is the inflation rate, yields the real deficit as defined in equation (8):

Note well that, without indexing of interest taxation, the real deficit to be financed in the credit market is a function of the very same r^L, the after-tax real interest rate without indexing that influences lenders in the credit market. By definition, the primary deficit is γ − τY, and the financing deficit is r^L D₋₁/P. The latter would increase with an increase in r^L.

Consider next the case in which interest taxation is indexed to inflation. Only real interest income is taxed, so that the tax base is defined as PY + (i − π)D₋₁,. Therefore, the tax equation is

Substituting equation (6a) into the budget equation (8) yields the real deficit under indexation of the tax treatment of interest:

In the case of full inflation indexing, including not only indexing interest taxation but also the outstanding government debt, the real government budget deficit is