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# Difference between an income tax and a CFT[26]

The most famous neutrality proposition with respect to a tax on economic income is Samuelson’s (1964) “invariant valuations” theorem: if, and only if, changes in asset values are accounted for as they accrue for tax purposes will the discounted present value of a stream of cash flows be independent of the tax rate. In other words, the maximum amount that an individual is prepared to pay for an income-producing asset will only be independent of (invariant with respect to) the tax rate on income if the tax base is economic income.

To see this, define the following terms:

is an asset’s value at the end of period t, where the superscript i=N if the asset is not taxed and G if it is taxed;

Nt is the sum of cash flows attributable to the asset over period t;

rt is the opportunity cost of capital (i.e, the discount rate) in period t;

t is the tax rate.

In the absence of taxes, the asset’s value at the end of period t is

(3)

Since (3) will also hold in period t+1, we can re-express it as

(3’)

which we can rearrange to obtain the following expression for Vt:

(4)

where .

The expression in the numerator of (4) is economic income in period t+1, while the denominator is simply the pre-tax discount rate in that period. Introducing a tax on economic income into the model, such that individual i faces the tax rate τi, will have two effects. It will reduce the numerator of (4) by a factor of (1-τi) and it will reduce the denominator by the same factor. Since, by assumption, the same tax rate applies to all alternative investments available to individual i, it follows that the opportunity cost of investing in this particular asset is the after-tax rate of return foregone by not investing in the next best alternative asset. Thus (4) becomes

(5)

Equation (5) is Samuelson’s result. It shows that an economic income tax results in the valuation of an asset with a given stream of receipts being independent of the tax rate an individual faces, the extent to which returns to the asset comprise capital gains, depreciation, or cash[27], the extent to which returns to the asset comprise normal returns to capital or rents[28] and whether tax rates are constant or change over time.

In combination, these represent powerful results. They mean that an economic income tax will not create opportunities for arbitrage between taxpayers who face different rates, will not affect relative asset values, and will therefore not distort the composition of investment, will not affect absolute asset values, and will therefore not affect the level of investment[29] and will not affect decisions about when to acquire or divest an asset.

## A tax on cash flows

Because a cash flow tax is just that - a tax on cash flows - we do not need explicitly to incorporate changes in an asset’s value over time into the analysis, and can therefore commence our analysis with equation (3) rather than (3’).

The base of a CFT comprises all incoming and outgoing cash flows. Consequently, the first step in incorporating taxes in (3) is to multiply each Nt by (1-τ).

The slightly more difficult problem is figuring out what effect a CFT has on the discount rate. In fact, the CFT leaves the discount rate unchanged from its no-tax value. To see this, consider a firm that can borrow at interest rate r to finance an investment of \$1 that will return (for certain) in one period’s time. Under the CFT, the asset’s after-tax present value is:

(6)

which will equal zero when .[30] The intuition behind this result is straightforward: a CFT effectively exempts marginal assets from tax.

Since the CFT reduces cash flows by a factor of 1-τ, but has no effect on the discount rate, it follows that the tax’s introduction results in the asset’s value being:

(7)

From (7), for a non-zero tax rate, a CFT will result in asset values being independent of the tax rate if the asset’s value is zero. That is, a CFT is “tax-rate invariant” only with respect to marginal assets, unlike the economic income tax, which is tax-rate invariant with respect to both marginal and infra-marginal assets. Under the CFT, taxpayers on lower tax rates will value positive NPV assets more highly than taxpayers on lower tax rates,[31] creating incentives for arbitrage and potentially resulting in distorted portfolio allocation decisions.

Unlike the economic income tax, a CFT is also not necessarily neutral in its impact on decisions about when to acquire or divest an asset. To see this, suppose the tax rate in period 1 is τ1 but that in period 2 it is going to change to τ2, and that the cash flows received or payable in each period after the asset’s acquisition would be identical whichever period the asset were purchased. If the asset is acquired in period 0, its value is:

(8)

but if it is acquired in period 1, its value is:

.

Thus:

(9)     .

If first-period cash flows are positive, and the tax rate is expected to increase, ; that is, the CFT creates an incentive to invest earlier rather than later. (The intuition is simply that the taxpayer benefits from having at least one period’s returns taxed at the lower rate.)

But if first-period cash flows are negative, and the tax rate is expected to increase, the CFT creates an incentive to defer investment - deferral earns the taxpayer a larger tax saving in respect of the first period’s outgoings.

To summarise the results presented in this Annex, a CFT is “neutral” in that it does not affect any taxpayer’s relative valuations of different assets and it does not affect the sign of asset valuations, meaning it will not affect the level of investment. But unlike the economic income tax, the CFT will create opportunities for arbitrage between taxpayers who face different rates (albeit only with respect to infra-marginal assets) and will affect decisions about when to acquire or divest an asset in the presence of expectations that future tax rates will change.

In assessing the practical significance of these apparent disadvantages of a CFT relative to an income tax, it should be borne in mind that they arise in the comparison of two theoretically pure bases. A real world income tax that deviates from the tax considered above will also be non-neutral in the ways described in these last two dot points.

### Notes

• [26]This part of the annex is mostly taken from Goss (1995).
• [27]Since there are no restrictions on either the relative magnitudes or the signs of either of the terms in the numerator of (4) .
• [28]Since (5) obtains irrespective of the magnitude of V.
• [29]At least so long as the tax does not affect the discount rate, r, say by decreasing the supply of funds available to finance investment.
• [30]A further step is required to show that the pre-tax discount rate is the appropriate discount rate, since the same result can be obtained for an income tax: under both bases, the required rate of return on a marginal asset equals the pre-tax discount rate. The difference in the case of the cash flow tax is that, unlike the income tax, it raises no revenue from the marginal asset, as shown in the final equation.
• [31]This result can be contrasted with that arising under an income tax with selective concessions, which results in higher rate taxpayers valuing tax-favoured assets more highly than lower rate taxpayers.
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