SØK531 Economics of the firm, Diderik Lund, 8 March 2002
Neutrality of cash flow tax, Sandmo (1979)
Different definitions of a neutral tax:
• Most general: Neutral tax does not affect behavior of firms and
individuals compared with a no-tax situation.
• Discussing taxes for one country or sector: Neutral tax does
not affect behavior in sector, compared with a no-tax situation
in sector, for given taxes in other sectors.
Requires theory of behavior of firms and individuals.
Why concern with neutral tax?
• Part of theory of optimal taxation. Under a set of assumptions:
Competitive equilibrium is Pareto optimal. Should not distort
this.
• More pragmatic: Should not distort randomly. Should know
effect of tax system. Neutral system good point of departure
for analysis.
• When assumptions for Pareto optimum violated: Corrective
taxes.
• Neutral tax system as basis for designing corrective taxes.
Welfare theory, theory of optimal tax, typically based on theory
of production and consumption. Tax neutrality also important for
other types of decisions, such as how to finance or organize a firm,
or whether to split a firm in two parts.
1
SØK531 Economics of the firm, Diderik Lund, 8 March 2002
• Sandmo’s paper makes point about a constant tax rate.
• On your reading list because it shows two systems of neutral
taxation of firms.
• Dynamic model in continuous time.
Consider first static model: Firm maximizes
(1 − s) [F (K, L) − wL − qK]
s is rate of proportional tax on profits, other symbols well known,
q is price per unit of capital, output price is unity (— output is
numeraire).
Neutrality result in static model: Firm’s behavior not affected by
tax rate s. Firm will maximize expression in square brackets irrespective of s, as long as s is a constant between zero and one.
2
SØK531 Economics of the firm, Diderik Lund, 8 March 2002
Will rewrite Sandmo’s model in discrete time (— more realistic;
simpler math).
Let t denote time period. There is exponential depreciation, so that
without any reinvestment, capital would decay by same fraction
each period. With gross investment It, capital evolves according to
Kt+1 = Kt(1 − δ) + It
Without taxes first: Each period firm has cash flow
Ct = F (Kt, Lt) − wtLt − qt [Kt+1 − Kt(1 − δ)]
Let i denote interest rate. Firm maximizes
∞
Ct
V =
t
t=0 (1 + i)
through choice of all periods’ Lt and Kt, given K0. Rewrite:
V =
∞
1
{F (Kt, Lt) − wtLt − qt [Kt+1 − Kt(1 − δ)]}
t
t=0 (1 + i)
= F (K0, L0) − w0L0 + q0K0(1 − δ)
∞
1
{F (Kt, Lt) − wtLt − qt−1Kt(1 + i) + qtKt(1 − δ)}
t
t=1 (1 + i)
First-order conditions for all t = 1, 2, . . .:
+
∂F
= wt,
∂Lt
∂F
= qt−1i + qtδ − (qt − qt−1) ≡ rt,
∂Kt
defining rt, the rental rate of capital, playing similar role as wt.
3
SØK531 Economics of the firm, Diderik Lund, 8 March 2002
What is rental rate?
• rt = qt−1i + qtδ − (qt − qt−1) expresses cost of using capital for
one period, given assumptions about depreciation.
• Part of capital remains after the period, has possibly new price
qt, thus complicated.
• qt−1i is cost of financing the capital unit. Would be the whole
rental rate if no depreciation and constant qt = qt−1.
• Remaining terms are net costs due to change in value of capital.
• qtδ is cost due to depreciation, valued at t.
• qt − qt−1 is gain due to increasing price of physical capital (or
−(qt − qt−1) is loss from decrease in same).
The neoclassical investment model: Important assumptions
• Capital can be freely chosen each period.
• Capital invested in different years can be aggregated (subject
to depreciation).
• Disinvestment and investment has same price, qt, per unit capital.
• Equivalent to: Sell all capital each period, and buy new.
Exist alternative models: Putty-clay. Real options.
4
SØK531 Economics of the firm, Diderik Lund, 8 March 2002
Two alternative forms of taxation
• Cash flow taxation
• Income taxation, using standard definition(s) of income
• (Will come back to what definition of income)
Using a constant tax rate s in all periods, after-tax value of firm is
V (1−s) =
1−s
{F (Kt, Lt) − wtLt − qt [Kt+1 − Kt(1 − δ)]}
t
(1
+
i)
t=0
∞
= (1 − s) [F (K0, L0) − w0L0 + q0K0(1 − δ)]
∞ 1−s
{F (Kt, Lt) − wtLt − qt−1Kt(1 + i) + qtKt(1 − δ)}
+
t
t=1 (1 + i)
• First formulation is cash flow taxation.
• Second is (ideal) income taxation.
• The two are equivalent and neutral (when tax rate constant
over time).
• Shows how to define basis for income taxation:
– Deduction for (full) financing cost Ktqt−1i.
– In practice often: Only for debt financed part.
– Deduction for decline in capital value Kt[qtδ − (qt − qt−1)].
– In practice: Approximate δ, and qt − qt−1 only if sold.
5
SØK531 Economics of the firm, Diderik Lund, 8 March 2002
Problem of time-varying tax rates
Cash flow tax no longer neutral if tax rates differ between periods.
After-tax value of firm will then be:
∞ 1 − st
Vs =
{F (Kt, Lt) − wtLt − qt [Kt+1 − Kt(1 − δ)]}
t
t=0 (1 + i)
= (1 − s0) [F (K0, L0) − w0L0 + q0K0(1 − δ)]

∞ 1 − st 
F (Kt, Lt) − wtLt
+
t
(1
+
i)
t=1


1 − st−1
−
qt−1Kt(1 + i) + qtKt(1 − δ)
1 − st
First-order condition for capital will be affected:
∂F
1 − st−1
=
qt−1(1 + i) − qt(1 − δ) = rt
∂Kt
1 − st
This tax is not neutral if st = st−1.
An ideal income tax would be different, and still neutral.
6