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Economic effects of land taxes in an inflationary economy
Journal
of Public
Economics
ECONOMIC
42 (1990)
EFFECTS
OF
195-211.
LAND TAXES ECONOMY
Toshihiro Faculty
qf Economics,
Received
North-Holland
IN AN INFLATIONARY
IHORI*
Osaka University,
April 1989, revised version
Toymaka,
received
Osaka, 560 Japan
December
1989
We investigate the incidence of land taxes using a monetary overlapping generations model. The real price of land may be raised with a land tax even when the elasticity of saving with respect to the interest rate is positive. We may have the ‘crowding-out’ case, which is induced by the substitution from capital and land to money. The capital loss suffered by older generations holding land during the transition to a land tax may be partially offset by a gain in money balances as inflation abates.
1. Introduction In recent years many studies of fiscal incidence in a general equilibrium framework have used the overlapping generations model [see Diamond (1965), Atkinson and Sandmo (1980), and Ihori (1984)]. A standard assumption of these studies is the perfect substitutability between consumption goods and investment goods as the outputs of the production technology. In reality, however, a significant fraction of savings is invested in assets that have a very long life and are not easily substitutable with consumption. Feldstein (1977) modifies the standard stylized framework of a two-period, overlapping generations model to include both capital and a fixed factor (land), and obtains the ‘surprising’ result that a land tax may raise the land price in the long run. Recently, under the assumption of perfect foresight, Charnley and Wright (1987), Eaton (1988), and Fried and Howitt (1988) ask if the inclusion of capitalization effects through fixed assets such as land may importantly affect the incidence of fiscal policy. Their dynamic approach is a necessary complement to a general comparative statics analysis of Feldstein’s model for it furnishes the correct stability conditions which are different from those in the conventional literature. Fried and Howitt analyze the effects of fiscal deficits on welfare, interest *This research was supported in part by the Tokyo Center Century Foundation. I thank the referee for helpful comments. 0047-2727/90/$03.50
0
199kElsevier
Science Publishers
of Economic
Research
B.V. (North-Holland)
and the 21
196
T. Ihori, Economic effects of land taxes
rates, and the balance of payments by assuming capital is fixed. Charnley and Wright analyze the impact of the price of land using the technology of a fixed factor (land), malleable capital, and labor. They show that a land tax may initially raise land values, but the upper bound is less than one-half of the tax revenues. Using a small open economy, Eaton shows that a permanent increase in net foreign investment can reduce steady-state welfare if a consequence is higher land values. In this paper we extend and explore their analyses in the following sense. We introduce monetary considerations so as to produce capital gains in the long run. Inflation will lead to capital gains for the holder of land, so that we can investigate the long-run economic effects of a capital gains tax on land. As money is used for reducing transaction costs, money and land are not perfect substitutes. In a model with money, capital, and land we may also have the ‘surprising’ case as in the previous studies where an increase in land taxes will raise the real price of land. The results presented in this paper, however, differ from the previous studies in the following sense. The nominal price of land may rise in response to a land tax even when the elasticity of saving with respect to the interest rate is positive. We may have the ‘crowding-out’ case where the interest rate rises with a land tax, which is induced by the substitution from capital and land to money in the portfolio of savings, leading to a decline in the capital stock. The existing old generation gets capital gains from money holdings although he suffers from the unexpected capital losses from land holdings. Section 2 presents our analytical framework. Section 3 investigates the effects of land taxes. And finally, section 4 concludes this paper.
2. Analytical
framework
In order to explore the problems of intertemporal equilibrium that are involved when considering a fixed asset, it is necessary to consider individuals’ intertemporal utility-maximizing behavior explicitly. We develop an overlapping generations model of identical individuals in which every individual lives for two periods. In the first period of his life an individual works, consumes, and saves. Savings are held in the form of money and capital. In the second period he retires and consumes the fruits of his firstperiod savings. A member of generation t has a standard utility function: tt=U(C:,C:,m’),
(1)
consumpwhere c; is his first-period consumption, ci is his second-period that he holds at the tion, and m’( =M (+ i/pt+ i) is real money balances
T. Ihori, Economic
effects of land taxes
197
beginning of the second period of his life.’ pt is the price of goods in period t, and M, is the total stock held at the beginning of period t. It is assumed that the necessary curvature and continuity properties hold. The government is assumed to purchase in each period a required quantity, G, of the consumption good. Its expenditure is financed by (a) a land rent tax, a; (b) a land value tax, /?; (c) a capital gains tax, 8; and (d) a lump-sum tax, L, and monetary expansion.’ These land taxes are levied on the basis of nominal income. The assets are called to land, capital, and money. A unit of fixed asset, land, is a promise to pay 4 units of the consumption good as a rent each period forever. The real price of a unit of land is a,. The nominal (after-tax) interest rate on an asset purchased in period t and sold t + 1, R,, ,, is defined as il-r)rlP,+1+(l-P)P,+1n,+,-t)(P,+,n,.~~8,cr;)=1+R
ftl’
4Pt
Using
the rate of inflation,
The real rate of interest
rc,+ i( = (p,, 1-p,)/p,),
from period
(l+r,+l)(l+~,+l)=l+R,+l.
(2) may be rewritten
t to t + 1, rt + 1, is defined
as
as (3)
Given perfect foresight without uncertainty, capital and land are perfect portfolio substitutes and have the same rate of return, r. (2) may be regarded as an arbitrage equation. Thus, in the long run substituting rr,, 1= n and a,, 1=ut=u into (2’) and considering (3), we have:
‘As money is not injected into the economy by means of transfers, he does not hold money at the beginning of the first period. Therefore, the service of money is relevant only in the second period. Our formulation is consistent with Weiss (1980) and Ihori (1985). We do not think that putting money into the utility function is good monetary theory. However, until fully satisfactory micro foundations of the demand for money are developed, the procedure we are following is a useful analytical device. ‘In the real world there are few countries that raise more than a trivial amount of revenue from land taxes and inflationary taxes, although the Latin American countries have both high inflation rates and a large endowment of potentially taxable land. However, in this paper we investigate the marginal changes in land taxes and inflationary taxes. Thus, the policy implications of this paper may be relevant to developed countries as well.
T. Ihori, Economic
198
effects of land taxes
(*-4q+(1-m~=1+r, a
A capital inflation nominal measure From
(4)
l+Tc
gains tax will not vanish in the long run so is positive. This is because a capital gains capital gains (na). Changes in the nominal of the capital gains or losses incurred by current (4) we have: a-
long as the rate of tax is imposed on asset prices are a asset holders.
(1 -a)q
r+B+Hrt/(l
(4’)
+7t)’
If there are no taxes (a =/I= Q=O), a=q/r; the price of land is given by the present value of future rental income, and the discount rate is the real rate of interest. (4’) implies that an increase in a land rent tax will reduce the net rental income. An increase in a land value tax or a capital gains tax will increase the discount rate. Overall, land taxes will reduce the long-run price of land for given I’ and 7~.These results are compatible with the conventional literature [see Trestrail (1969)]. As will be investigated in section 3, however, the partial equilibrium approach might be invalid if r and n were to change, especially if r were to decrease with land taxes. The representative individual’s consumption, saving, and money holding programs are restricted by the following first- and second-period budget constraints:
c: =w-(1 c\=m’+(l
+rr,+,)m’-ss’,
(5)
+rt+&‘,
(6)
where w is the constant wage income and s is real savings Output is produced by the technology:
for land.
where y is output, K is capital, L is labor, and D is land. Land is fixed and there is a population of constant size, L= D= 1. From the factor price frontier the marginal return on capital stock and output are given by a decreasing function of r, respectively.3 K,=K(r,),
K’
3For simplicity we have assumed independent of r: jKL=.fKD=O.
that
the marginal
returns
on land and labor
are fixed and
T. Ihori, Economic effects of land taxes
Y, =
y’
&-A
The decision problem lifetime budget constraint:
C: +--
1
199
Then the above decision
1
l+x,+l-------
c:+
l+r,+l
t is to maximize
for generation
(1) subject
to the
m’=w.
1 +r,+r >
( problem
yields the demand
functions: (7)
c: =cI(rt+r,nt+r),
(8)
mr=m(rt+l,.t+l),
(9)
and the real saving function: (10)
s’=s(r,+1,r5+1), where c2, = &Jar > 0, m, = amfdr < 0, and m, = am/h assumption. By the budget constraint, we have:4
< 0 from the normality
An increase in r will lead to an increase in c2, and demand for consumption, c1 +c,, will be increased. The government budget constraint is, for period t:
hence
the aggregate
T+[(l+n,+,)m’-mtp’]+L=G,
(11)
where
means the total revenue considering (4) we have:
crq+/la+85
the
a
=(q-ra).
1+7c
4Cle+C2e=(l--e)CZ.-(cZ-m)-(1+n-e)mp.
m,>O, c,,+c2,<0.
from
three
land
where e=l/(l+r).
taxes.
In
the
long
run,
(12)
As nr0,
Otecl,
c,,
and
T. Ihori, Economic effects of land taxes
200
Land tax revenues are equal to rental income minus net returns on land. Land taxes (T>O) imply that the price of land, a, is lower than the discounted value of future rental income, q/r. Given a sequence of fiscal actions and an initial stock of money supply, an equilibrium is a sequence in which markets clear under perfect foresight starting at t= 1. More formally, it is a sequence {a,,r,+,,~,+,},“=, that satisfies definition (2), the government budget constraint (ll), and the goods market-clearing condition: c~+c~~‘=y(r,)+K(r,)-K(r,+,)-G,
for all tzl,
(13)
where c:, c:, and m’ are given by the demand functions (7), (S), and (9) for t2_1.5 The consumption c;+l of the initial old is not given by the demand function (8) because our definition of equilibrium does not require that that generation’s expectations be f-uifilled. Unexpected movements in a, and p1 confer capital gains or losses on the initial old generation. In equilibrium the asset market-clearing condition,
s’=a,+K,+,,
(14)
holds for all t 2 1. Appendix A investigates the stability condition of economy.6 The following set of conditions will satisfy instability condition: m+(2+7c)m,>O,
clr
Ml+4
(m + nm,)cI,
O
cln=O,
>--
our inflationary the saddle-point
4x 1.
(clr +________ CZJ (m + (2II+ n)m )
The second condition implies that the elasticity of saving with respect to the interest rate is positive. The fourth condition means that m is more substitutable with c2 than with c1 and the degree of substitutability is not 5We implicitly assume that money supply is endogenous so as to meet the government budget constraint. As is well known, one of the policy instruments is given by the government budget constraint. Since we control government expenditures G, lump-sum taxes L, and the total tax revenue from land taxes T, money supply must be endogenous. ‘%alvo (1978) has shown that if the system is stable, then there exists a continuum of equilibrium paths that converge asymptotically to the stationary equilibrium without reaching it in finite time. In such a case one cannot even locally determine equilibrium prices and interest rates. As noted by Fried and Howitt (1988), in a monetary economy without productive capital, if savings is a decreasing function of the rate of interest, the system is stable. Hence, in order to rule out such a possibility, they assume that saving is an increasing function of the rate of interest.
T. Ihori, Economic effects of land taxes
201
very large. Under the saddle-point instability condition the economy is unstable except only one convergent path. Perfect foresight means that the economy always chooses this convergent path. As shown in appendix A, the dynamic paths of rt and rc, are independent of land tax parameters (a, p, 0) so long as the total revenue from land taxes is fixed. In other words, changes in the land rent tax, the land value tax, or the capital gains tax will not affect the real equilibrium of the economy so long as the total tax revenue from land taxes is constant. An increase in the land rent tax will have the same effect on the real economy as an increase in the land value tax or the capital gains tax if the resulting increase in the land tax revenue is the same.
3. Impacts of land taxes We now consider the impact of fiscal action which would change the total revenue from the land taxes. The conceptual experiment by which we investigate the effects of land taxes is the following. Suppose the economy before period 1 is in a stationary equilibrium. In period 1 the government unexpectedly raises land taxes. From period 1 on the total tax revenue from land taxes is maintained at the higher level.’ This requires the government to increase its government expenditure (the balanced budget incidence) or to reduce lump-sum taxes (the differential incidence).8 After the initial fiscal action, there will be no more surprises, so that the economy will be on a new undisturbed convergent path from period 1 forward.
3.1. An increase in government 3.1.1. Long-run The long-run
effects equilibrium
expenditure
is given by
cl@, 4 + c2(r,n) = y(r) - G,
(15)
nm(r,z)+T+L=G.
(16)
We are interested in the case where T+ L < G and hence 7~> 0. The balanced budget incidence means dT = dG > 0. As shown in appendix B, we have dr/dT < 0; an increase in the government expenditure financed by land taxes will reduce the real rate of return ‘r2 and x2 are not the same as rl and n,. The actual parameters of a, /?, and 0 may well be changed in period 2 so as to maintain the higher level of the total land tax revenue. *Our liscal action assumes M, =M,, although M, (t 22) will be changed so as to satisfy the government budget constraint.
202
T. Ihori, Economic effects of land taxes
on land. An intuitive explanation is as follows. An increase in government expenditures will lead to an excess demand for the goods. We know that the aggregate demand increases with the real rate of return on land and capital. Also, production decreases with r. In order to restore the equilibrium in the goods market, the real rate of return must be decreased. The increase in the land tax induces an increase in the capital stock. So far we have explored the effect of an increase in G on the goods market. This ‘crowding-in’ effect may also be explained in terms of the effect of an increase in T on the portfolio behavior. The arbitrage equation (4) implies that by an increase in land taxes the rate of return on land is less than the rate of return on capital for given r and n. Thus, a substitution from land to capital occurs in the portfolio of savings, and this is similar to the Tobin (1965) effect of inflation, as noted by Charnley and Wright (1987). Similarly, we have dn/dT ~0; an increase in the government expenditure financed by land taxes will reduce the long-run rate of inflation (i.e. the expansion rate of monetary growth) and hence the nominal price of land. A decrease in r means an increase in real money demand, m. Inflationary taxes would be raised at the initial rc, so that the rate of inflation must be reduced to maintain the government budget. This effect is induced by the substitution from land to money in the portfolio of savings. How is the real price of land, a, affected in the long run? Remember that s = a + K = w - ci - (1 + n)m. Considering the government budget constraint, the balanced budget incidence means d(nm) = 0. Hence, we have: da=dw-dK-dci-dm=(-K-K’)dr-(c,,dr+c,.dn) -(m,dr+m,dz).
(17)
We know m, < 0, m, CO, dr CO, and drc 0, da ~0. Even if K + K’ ~0, it is still possible to have da ~0. It follows that an increase in the government expenditure financed by land taxes will normally reduce the real value of land in the long run. In the partial equilibrium analysis r and 7c are assumed to be given. Then from (4) an increase in land taxes will reduce a. Our general equilibrium approach shows that such a partial equilibrium analysis will produce the same qualitative result concerning changes in a.’ 3.1.2. Transitionary effects We now consider the transitionary
effects. Let us investigate
the dynamic
9We implicitly assume that an increase in G will not affect the marginal choice of the consumption-saving behavior. This is satisfied if the utility function is additively separable between consumption and government expenditures.
203
T. lhori, Economic effects of land taxes
0
r Fig. 1
properties of this economy using a phase diagram. To analyze the behavior of rl, we find the locus of (z, v) where T,= r’t+ r. We call this locus the YYcurve. To find the behavior of ret, we investigate the locus of (n,r) where 7~,=rc,+ 1. We call this locus the rrrc curve. As shown in appendix A, both curves are downward sloping, and the (absolute) slope of the nn curve is greater than that of the rr curve. From the long-run comparative statics analysis we know of the initial that the new equilibrium point E* is to the south-west equilibrium point E, in fig. 1. Thus, in period 1 rc jumps from E, to E,, so that in period 1 the economy is on the new convergent path a’a’ toward E”. How are a, and c0 affected? In period 1 we have c: -I-ci = y, + K, -K, -G and (1 + nJm’ -m” + T = G. Since r2 and rc2 are less than the initial long-run equilibrium values, c: and m1 of generation 1 are greater than the initial G is, by assumption, raised. equilibrium values, cy and m”, respectively. Hence, c! of generation 0 must be lower than the (expected) old equilibrium value. The existing old generation will suffer from unexpected capital losses on land holdings. This aspect has also been explored in Charnley and Wright (1987) and Fried and Howitt (1988). Note that m” is defined by M,/p,, and M, is exogenously given. Changes in m” are associated with changes in pr. As rci is lower than the old equilibrium value, p1 is lower and hence m” is higher than the old equilibrium value. Intuitively, an increase in ml means an increase in inflationary taxes at the initial money holding, m”. In order to maintain the government budget, the government must ‘transfer’ these revenues to the existing older generation as a form of unexpected capital gains on money holdings. This is a new aspect, which occurs only in the monetary economy.
204
T. Ihori, Economic effects of land taxes
From (2’), when rcl is reduced, u1 must be reduced so long as 1 >fi+ 8. Thus, a, will normally be reduced. The existing old generation suffers from unexpected capital losses from land holdings, but he gets unexpected capital gains from money holdings. As for the effect on second-period consumption, the former effect is stronger than the latter effect: second-period consumption decreases unexpectedly. However, it is possible that older generations lind it beneficial to impose a land tax if they evaluate the service of money significantly.
3.2. A decrease in lump-sum 3.2.1. Long-run effects In this case the long-run
taxes
equilibrium
cl@,n, L) + cz(r, 7t, L) = y(r) TC~(~,TC,L)+T+L=G,
is given by G,
(18) (19)
where L appears in the consumption and money holdings functions explicitly. The differential incidence means dT+ dL = 0. As shown in appendix C, we can easily derive drr/dT 0) will lead to an excess demand for the goods market. If this happens, as in subsection 3.1, dr/dT is likely to be negative. However, as the rate of inflation declines, the goods market might be in a state of excess supply if cln+czrr is large. If c2 and m are highly substitutable, this would be the case. Since a decreae in L directly raises m, rc will decrease more than in subsection 3.1. Appendix C shows that if (~~~+c~~)(rn+71m,)/?lrn,
T. Ihori,
Economic
effects of land taxes
205
0 Fig. 2
Let us investigate the effect on the real price of land, a. If L is levied when the consumer is old, da = dw -dK -(dc, +dm), as in the previous subsection; a will normally be decreased. If a lump-sum tax is levied when the consumer is young, we have ds/dLO) dominates the substitution effect due to changes in r and rc (dc, +dm>O), we would have da/dT>O. The real value of land may increase in the long run. Using the differential incidence, Feldstein (1977) and Charnley and Wright (1987) show that if s,O. Nevertheless, the real price of land may be raised if the income effect due to a reduction in lumpsum taxes in the younger generation is dominant.
3.2.2. Transitionary effect We now consider the transitionary effects in fig. 2. As 7~ is reduced in the long run, the new equilibrium point E* is to the south of E,. 71~ will be reduced. As in subsection 3.1 we may derive that ct is reduced. The existing old generation will suffer from unanticipated capital losses on land holdings. The effect on m” is the same as before. 7~~ is reduced, p1 is reduced, and hence m” is increased. The existing old generation will get unexpected capital
T. Ihori, Economic effects of land taxes
206
Table Balanced r 77 a cl =2 m
_ _ _
4 m0 =I a,
budget
1
incidence
Differential
incidence
-
?” _
+ _
?b + _
+
+
_
_
+ _ _
+ _
“If
_ hr+c2,Nm+(2+W4J v sign will blf the stronger sign may
be positive. income effect due to a reduction of lump-sum taxation is than the substitution effect due to changes in r and n, this be positive.
gains from money holding. As in subsection reduced. Our results are summarized in table 1.
3.1 we may derive
that
u1 is
4. Conclusion We have examined the effects of land taxes on the price of land and the rate of inflation when factors of production are fixed in supply. We explicitly considered the general equilibrium effects of land rent taxes, land value taxes, and capital gains taxes in an inflationary situation. Land taxes may affect the real equilibrium only through changes in the total revenue from the land taxes. It is shown that land taxes will normally reduce the real rate of return on land, the real price of land, and the rate of inflation in the case of the balanced budget incidence. Land taxes may increase the long-run real price of land in the case of differential incidence if the income effect due to a reduction in lump-sum taxes on the younger generation is stronger than the substitution effects due to an increase in land taxes. Feldstein (1977) and Charnley and Wright (1987) obtained similar ‘surprising’ results. However, our dynamic incidence results are quite different from previous results in the following sense.
207
T. Ihori, Economic effects of land taxes
First, by incorporating monetary considerations, we have shown that the nominal price of land will normally be reduced by land taxes. Second, our instability condition in an inflationary economy implies that real saving is an increasing function of the real rate of return. Nevertheless, the real price of land may be raised if the income effect due to a reduction in lump-sum taxes in the younger generation is dominant. Third, it seems likely that the real price of land will initially decrease even if the long-run real price of land were to rise in the ‘surprising’ case. Furthermore, we may have the crowding-out case where the land taxes will increase the real rate of return on capital. This effect is induced by the substitution from land and capital to money in the portfolio of savings. Finally, the existing old generation will suffer from unexpected capital losses from land holding, but they get unexpected capital gains from money holding. It is possible that older generations find it beneficial to impose a land tax if they evaluate the service of money significantly. These aspects have been explored only in the three-asset inflationary economy.
Appendix A Let us investigate the stability system may be summarized by
condition
of our economy.
The dynamic
c1(rt+l~71,+1)+c2(r,,71,)= Y(r,,r,+J-G,
(A.1)
(l+~t+~Mr,+l,~,+l
(A.4
)-m(r,,n,)+T+L=G,
where Y(rt, rr+l)= y(r,) +K(r,)-K(r,+ J. We assume that G, T, and fixed. From (A.1) rt+l is given as a function of n,,,, rt, and rr,: r t+1 =R(n (+ 17rt3 d.
From
(A.3)
(A.2) x*+1 is given as a function
of r, + I, rt, and rr,:
71I+ I = Mb-,+ 1yIt, 4 Substituting
M(r f+ lr rl, 41
+ cZ(rt7 4 = W,, rt+J - G.
(A.9
(A.3) into (A.2), we have:
(1+%+1hCN~ The dynamic (A.6).
(A.4)
(A.4) into (A.l), we have:
crCrtfl, Substituting
L are
evolution
t+l,rt,7t,),n,+ll--m(r,,n,)+T+L=G. of the model
is entirely
determined
(‘4.6) by (A.5) and
208
T. Ihori, Economic effects of land taxes
The local stability of the equilibria can be analyzed mathematically. (A.5) and (A.@ we obtain at the neighbourhood of an equilibrium:
where dr = r - r*, drr = rc--71*, and expression: CllrMrO+c2rA=----------
(r*, n*)
is an equilibrium.
From
In the above
Yro
Clr+ClnMrl - r,l ’
B=
C_----_
ClnMrr+Czn clr+ClnMrl- YT’ (I+ 4mA0 - mL m+(l+7r)(m,+m,R,,)’ (1 + 7c)m,R,, -m,
DC___
m+(l where
Cir = &/dr,
+n)(m,+m,R,,)’
Gin = &Jaq
M,, = dM/&,,
M,, = aMjar,+
1, M, = aMlap,
R,,=aR/ar,+,, R,o=aR/ant, R,,=aRla71~+~ (i=1,2), Yro= Y,, =aY/&,+,. Following Charnley and Wright (1987), we disregard all the stable equilibria and concentrate on the saddle-point unstable case. Let us analyze the sign of the characteristic polynomial of the matrix R,,=aRpr,, dY/drt, and
i,G(A)=l’-(A+D)A+(AD-BC).
There exists a unique dynamic II/(-l)>O. In the long run we have:
path
near
the steady
state
if $( 1)
T. Ihori, Economic @cts
C=-
l-D= Suppose
209
of land taxes
- m,(c,, + c2J - 71172rc2, 3 c,,m+(l +~)kw,,--m,cLJ
-(l
+7C)**+Czn)+cIr(m+7TJ cl,m+(1+7t)(m,c,,--m,c,,)
for simplicity
cIn=O.
.
Then, we have:
$(l)=(l-A)(l-D)-BC
-
Y’CCl,h+ 714 -( 1+ 4w2JL
$(-l)=(l+A)(l+D)-BC
-c,~c,,rm+l(l+71)ml{-(Clr+CZ*)ClrCm+(2+n)m,l n
We know cl,.+ c2,>0, m,O,
O
clr
(m + 7m4cI, [ m,(l+n)
set of conditions
cln=O,
’
nh.71 1,
_ (cI,+c2r)(m+(2+4mz)
Let us investigate the dynamic properties of this economy diagram. From (A.5) the slope of the rr curve is given by
using
a phase
drrldr = (1 - A)/B. From
(A.6) the slope of the rcrc curve is given by drt/dr=C/(l-D).
From the instability condition both curves are downward sloping and the (absolute) slope of the 7~7~curve is greater than that of the rr curve. From (A.5):
T. Ihori, Economic effects of land taxes
210
which implies From (A.6):
that above
the rr curve rt < r,+ 1, and below the locus rt > rt+ 1.
an, +,/ar,= c > 0, which implies that on the right-hand side of the rrrr curve rr,+ 1 >IT,, and on the left-hand side of the rcrt curve rr,, 1
Appendix B Totally
differentiating
(15) and (16) with respect
to r and n, we have:
(B.1) Considering
dG=dT:
dr
dT
we have:
= - i (m-t 7cm,),
U3.2)
where A is the determinant of the matrix of LHD of (B.l) Let us investigate the sign of A. We know that clr+czr-y’>O. From the instability condition we have m +nm,>O and c~~+c~~=c~~>O. It is plausdr/dT
drr dT=z
1
7cm,< 0.
(B.3)
Appendix C Totally
differentiating
(18) and (19), we have:
“‘If dG =O, from (B.l) we know dn/dTO. In such a case an increase in land tax revenues is associated with a decrease in inflationary taxes. Thus, A>0 is intuitively plausible.
T. Ihori,
Economic
effects of land taxes
211
Considering dT + dL = 0, we have:
cc.21
CC.31
References Atkinson, A. and A. Sandmo, 1980, Welfare implications of the taxation of savings, Economic Journal 90, 529-547. Calvo, G.A., 1978, On the indeterminacy of interest rates and wages with perfect foresight, Journal of Economic Theory 19, 321-337. Charnley, C. and B.D. Wright, 1987, Fiscal incidence in an overlapping generations model with a fixed asset, Journal of Public Economics 32, 3-24. Diamond, P. A., 1965, National debt in a neoclassical growth model, American Economic Review 55, 1126-l 150. Eaton, J., 1988, Foreign-owned land, American Economic Review 78, 7688. Feldstein, M., 1977, The surprising incidence of a tax on pure rent: A new answer to an old question, Journal of Political Economy 85, 348-360. Fried, J. and P. Howitt, 1988, Fiscal delicits, international trade, and welfare, Journal of International Economics 24, l-22. Ihori, T., 1984, Partial welfare improvements and capital income taxation, Journal of Public Economics 24, 101-109. Ihori, T., 1985, On the welfare cost of permanent inflation, Journal of Money, Credit, and Banking 17, 22&231. Tobin, J., Money and economic growth, Econometrica 33, 671-684. Trestrail, R., 1969, Forests and the property tax - unsound accepted theory, National Tax Journal 22, 347-356. Weiss, L., 1980, The effects of money supply on economic welfare in the steady state, Econometrica 48, 565-576.
of Public
Economics
ECONOMIC
42 (1990)
EFFECTS
OF
195-211.
LAND TAXES ECONOMY
Toshihiro Faculty
qf Economics,
Received
North-Holland
IN AN INFLATIONARY
IHORI*
Osaka University,
April 1989, revised version
Toymaka,
received
Osaka, 560 Japan
December
1989
We investigate the incidence of land taxes using a monetary overlapping generations model. The real price of land may be raised with a land tax even when the elasticity of saving with respect to the interest rate is positive. We may have the ‘crowding-out’ case, which is induced by the substitution from capital and land to money. The capital loss suffered by older generations holding land during the transition to a land tax may be partially offset by a gain in money balances as inflation abates.
1. Introduction In recent years many studies of fiscal incidence in a general equilibrium framework have used the overlapping generations model [see Diamond (1965), Atkinson and Sandmo (1980), and Ihori (1984)]. A standard assumption of these studies is the perfect substitutability between consumption goods and investment goods as the outputs of the production technology. In reality, however, a significant fraction of savings is invested in assets that have a very long life and are not easily substitutable with consumption. Feldstein (1977) modifies the standard stylized framework of a two-period, overlapping generations model to include both capital and a fixed factor (land), and obtains the ‘surprising’ result that a land tax may raise the land price in the long run. Recently, under the assumption of perfect foresight, Charnley and Wright (1987), Eaton (1988), and Fried and Howitt (1988) ask if the inclusion of capitalization effects through fixed assets such as land may importantly affect the incidence of fiscal policy. Their dynamic approach is a necessary complement to a general comparative statics analysis of Feldstein’s model for it furnishes the correct stability conditions which are different from those in the conventional literature. Fried and Howitt analyze the effects of fiscal deficits on welfare, interest *This research was supported in part by the Tokyo Center Century Foundation. I thank the referee for helpful comments. 0047-2727/90/$03.50
0
199kElsevier
Science Publishers
of Economic
Research
B.V. (North-Holland)
and the 21
196
T. Ihori, Economic effects of land taxes
rates, and the balance of payments by assuming capital is fixed. Charnley and Wright analyze the impact of the price of land using the technology of a fixed factor (land), malleable capital, and labor. They show that a land tax may initially raise land values, but the upper bound is less than one-half of the tax revenues. Using a small open economy, Eaton shows that a permanent increase in net foreign investment can reduce steady-state welfare if a consequence is higher land values. In this paper we extend and explore their analyses in the following sense. We introduce monetary considerations so as to produce capital gains in the long run. Inflation will lead to capital gains for the holder of land, so that we can investigate the long-run economic effects of a capital gains tax on land. As money is used for reducing transaction costs, money and land are not perfect substitutes. In a model with money, capital, and land we may also have the ‘surprising’ case as in the previous studies where an increase in land taxes will raise the real price of land. The results presented in this paper, however, differ from the previous studies in the following sense. The nominal price of land may rise in response to a land tax even when the elasticity of saving with respect to the interest rate is positive. We may have the ‘crowding-out’ case where the interest rate rises with a land tax, which is induced by the substitution from capital and land to money in the portfolio of savings, leading to a decline in the capital stock. The existing old generation gets capital gains from money holdings although he suffers from the unexpected capital losses from land holdings. Section 2 presents our analytical framework. Section 3 investigates the effects of land taxes. And finally, section 4 concludes this paper.
2. Analytical
framework
In order to explore the problems of intertemporal equilibrium that are involved when considering a fixed asset, it is necessary to consider individuals’ intertemporal utility-maximizing behavior explicitly. We develop an overlapping generations model of identical individuals in which every individual lives for two periods. In the first period of his life an individual works, consumes, and saves. Savings are held in the form of money and capital. In the second period he retires and consumes the fruits of his firstperiod savings. A member of generation t has a standard utility function: tt=U(C:,C:,m’),
(1)
consumpwhere c; is his first-period consumption, ci is his second-period that he holds at the tion, and m’( =M (+ i/pt+ i) is real money balances
T. Ihori, Economic
effects of land taxes
197
beginning of the second period of his life.’ pt is the price of goods in period t, and M, is the total stock held at the beginning of period t. It is assumed that the necessary curvature and continuity properties hold. The government is assumed to purchase in each period a required quantity, G, of the consumption good. Its expenditure is financed by (a) a land rent tax, a; (b) a land value tax, /?; (c) a capital gains tax, 8; and (d) a lump-sum tax, L, and monetary expansion.’ These land taxes are levied on the basis of nominal income. The assets are called to land, capital, and money. A unit of fixed asset, land, is a promise to pay 4 units of the consumption good as a rent each period forever. The real price of a unit of land is a,. The nominal (after-tax) interest rate on an asset purchased in period t and sold t + 1, R,, ,, is defined as il-r)rlP,+1+(l-P)P,+1n,+,-t)(P,+,n,.~~8,cr;)=1+R
ftl’
4Pt
Using
the rate of inflation,
The real rate of interest
rc,+ i( = (p,, 1-p,)/p,),
from period
(l+r,+l)(l+~,+l)=l+R,+l.
(2) may be rewritten
t to t + 1, rt + 1, is defined
as
as (3)
Given perfect foresight without uncertainty, capital and land are perfect portfolio substitutes and have the same rate of return, r. (2) may be regarded as an arbitrage equation. Thus, in the long run substituting rr,, 1= n and a,, 1=ut=u into (2’) and considering (3), we have:
‘As money is not injected into the economy by means of transfers, he does not hold money at the beginning of the first period. Therefore, the service of money is relevant only in the second period. Our formulation is consistent with Weiss (1980) and Ihori (1985). We do not think that putting money into the utility function is good monetary theory. However, until fully satisfactory micro foundations of the demand for money are developed, the procedure we are following is a useful analytical device. ‘In the real world there are few countries that raise more than a trivial amount of revenue from land taxes and inflationary taxes, although the Latin American countries have both high inflation rates and a large endowment of potentially taxable land. However, in this paper we investigate the marginal changes in land taxes and inflationary taxes. Thus, the policy implications of this paper may be relevant to developed countries as well.
T. Ihori, Economic
198
effects of land taxes
(*-4q+(1-m~=1+r, a
A capital inflation nominal measure From
(4)
l+Tc
gains tax will not vanish in the long run so is positive. This is because a capital gains capital gains (na). Changes in the nominal of the capital gains or losses incurred by current (4) we have: a-
long as the rate of tax is imposed on asset prices are a asset holders.
(1 -a)q
r+B+Hrt/(l
(4’)
+7t)’
If there are no taxes (a =/I= Q=O), a=q/r; the price of land is given by the present value of future rental income, and the discount rate is the real rate of interest. (4’) implies that an increase in a land rent tax will reduce the net rental income. An increase in a land value tax or a capital gains tax will increase the discount rate. Overall, land taxes will reduce the long-run price of land for given I’ and 7~.These results are compatible with the conventional literature [see Trestrail (1969)]. As will be investigated in section 3, however, the partial equilibrium approach might be invalid if r and n were to change, especially if r were to decrease with land taxes. The representative individual’s consumption, saving, and money holding programs are restricted by the following first- and second-period budget constraints:
c: =w-(1 c\=m’+(l
+rr,+,)m’-ss’,
(5)
+rt+&‘,
(6)
where w is the constant wage income and s is real savings Output is produced by the technology:
for land.
where y is output, K is capital, L is labor, and D is land. Land is fixed and there is a population of constant size, L= D= 1. From the factor price frontier the marginal return on capital stock and output are given by a decreasing function of r, respectively.3 K,=K(r,),
K’
3For simplicity we have assumed independent of r: jKL=.fKD=O.
that
the marginal
returns
on land and labor
are fixed and
T. Ihori, Economic effects of land taxes
Y, =
y’
&-A
The decision problem lifetime budget constraint:
C: +--
1
199
Then the above decision
1
l+x,+l-------
c:+
l+r,+l
t is to maximize
for generation
(1) subject
to the
m’=w.
1 +r,+r >
( problem
yields the demand
functions: (7)
c: =cI(rt+r,nt+r),
(8)
mr=m(rt+l,.t+l),
(9)
and the real saving function: (10)
s’=s(r,+1,r5+1), where c2, = &Jar > 0, m, = amfdr < 0, and m, = am/h assumption. By the budget constraint, we have:4
< 0 from the normality
An increase in r will lead to an increase in c2, and demand for consumption, c1 +c,, will be increased. The government budget constraint is, for period t:
hence
the aggregate
T+[(l+n,+,)m’-mtp’]+L=G,
(11)
where
means the total revenue considering (4) we have:
crq+/la+85
the
a
=(q-ra).
1+7c
4Cle+C2e=(l--e)CZ.-(cZ-m)-(1+n-e)mp.
m,>O, c,,+c2,<0.
from
three
land
where e=l/(l+r).
taxes.
In
the
long
run,
(12)
As nr0,
Otecl,
c,,
and
T. Ihori, Economic effects of land taxes
200
Land tax revenues are equal to rental income minus net returns on land. Land taxes (T>O) imply that the price of land, a, is lower than the discounted value of future rental income, q/r. Given a sequence of fiscal actions and an initial stock of money supply, an equilibrium is a sequence in which markets clear under perfect foresight starting at t= 1. More formally, it is a sequence {a,,r,+,,~,+,},“=, that satisfies definition (2), the government budget constraint (ll), and the goods market-clearing condition: c~+c~~‘=y(r,)+K(r,)-K(r,+,)-G,
for all tzl,
(13)
where c:, c:, and m’ are given by the demand functions (7), (S), and (9) for t2_1.5 The consumption c;+l of the initial old is not given by the demand function (8) because our definition of equilibrium does not require that that generation’s expectations be f-uifilled. Unexpected movements in a, and p1 confer capital gains or losses on the initial old generation. In equilibrium the asset market-clearing condition,
s’=a,+K,+,,
(14)
holds for all t 2 1. Appendix A investigates the stability condition of economy.6 The following set of conditions will satisfy instability condition: m+(2+7c)m,>O,
clr
Ml+4
(m + nm,)cI,
O
cln=O,
>--
our inflationary the saddle-point
4x 1.
(clr +________ CZJ (m + (2II+ n)m )
The second condition implies that the elasticity of saving with respect to the interest rate is positive. The fourth condition means that m is more substitutable with c2 than with c1 and the degree of substitutability is not 5We implicitly assume that money supply is endogenous so as to meet the government budget constraint. As is well known, one of the policy instruments is given by the government budget constraint. Since we control government expenditures G, lump-sum taxes L, and the total tax revenue from land taxes T, money supply must be endogenous. ‘%alvo (1978) has shown that if the system is stable, then there exists a continuum of equilibrium paths that converge asymptotically to the stationary equilibrium without reaching it in finite time. In such a case one cannot even locally determine equilibrium prices and interest rates. As noted by Fried and Howitt (1988), in a monetary economy without productive capital, if savings is a decreasing function of the rate of interest, the system is stable. Hence, in order to rule out such a possibility, they assume that saving is an increasing function of the rate of interest.
T. Ihori, Economic effects of land taxes
201
very large. Under the saddle-point instability condition the economy is unstable except only one convergent path. Perfect foresight means that the economy always chooses this convergent path. As shown in appendix A, the dynamic paths of rt and rc, are independent of land tax parameters (a, p, 0) so long as the total revenue from land taxes is fixed. In other words, changes in the land rent tax, the land value tax, or the capital gains tax will not affect the real equilibrium of the economy so long as the total tax revenue from land taxes is constant. An increase in the land rent tax will have the same effect on the real economy as an increase in the land value tax or the capital gains tax if the resulting increase in the land tax revenue is the same.
3. Impacts of land taxes We now consider the impact of fiscal action which would change the total revenue from the land taxes. The conceptual experiment by which we investigate the effects of land taxes is the following. Suppose the economy before period 1 is in a stationary equilibrium. In period 1 the government unexpectedly raises land taxes. From period 1 on the total tax revenue from land taxes is maintained at the higher level.’ This requires the government to increase its government expenditure (the balanced budget incidence) or to reduce lump-sum taxes (the differential incidence).8 After the initial fiscal action, there will be no more surprises, so that the economy will be on a new undisturbed convergent path from period 1 forward.
3.1. An increase in government 3.1.1. Long-run The long-run
effects equilibrium
expenditure
is given by
cl@, 4 + c2(r,n) = y(r) - G,
(15)
nm(r,z)+T+L=G.
(16)
We are interested in the case where T+ L < G and hence 7~> 0. The balanced budget incidence means dT = dG > 0. As shown in appendix B, we have dr/dT < 0; an increase in the government expenditure financed by land taxes will reduce the real rate of return ‘r2 and x2 are not the same as rl and n,. The actual parameters of a, /?, and 0 may well be changed in period 2 so as to maintain the higher level of the total land tax revenue. *Our liscal action assumes M, =M,, although M, (t 22) will be changed so as to satisfy the government budget constraint.
202
T. Ihori, Economic effects of land taxes
on land. An intuitive explanation is as follows. An increase in government expenditures will lead to an excess demand for the goods. We know that the aggregate demand increases with the real rate of return on land and capital. Also, production decreases with r. In order to restore the equilibrium in the goods market, the real rate of return must be decreased. The increase in the land tax induces an increase in the capital stock. So far we have explored the effect of an increase in G on the goods market. This ‘crowding-in’ effect may also be explained in terms of the effect of an increase in T on the portfolio behavior. The arbitrage equation (4) implies that by an increase in land taxes the rate of return on land is less than the rate of return on capital for given r and n. Thus, a substitution from land to capital occurs in the portfolio of savings, and this is similar to the Tobin (1965) effect of inflation, as noted by Charnley and Wright (1987). Similarly, we have dn/dT ~0; an increase in the government expenditure financed by land taxes will reduce the long-run rate of inflation (i.e. the expansion rate of monetary growth) and hence the nominal price of land. A decrease in r means an increase in real money demand, m. Inflationary taxes would be raised at the initial rc, so that the rate of inflation must be reduced to maintain the government budget. This effect is induced by the substitution from land to money in the portfolio of savings. How is the real price of land, a, affected in the long run? Remember that s = a + K = w - ci - (1 + n)m. Considering the government budget constraint, the balanced budget incidence means d(nm) = 0. Hence, we have: da=dw-dK-dci-dm=(-K-K’)dr-(c,,dr+c,.dn) -(m,dr+m,dz).
(17)
We know m, < 0, m, CO, dr CO, and drc
effects. Let us investigate
the dynamic
9We implicitly assume that an increase in G will not affect the marginal choice of the consumption-saving behavior. This is satisfied if the utility function is additively separable between consumption and government expenditures.
203
T. lhori, Economic effects of land taxes
0
r Fig. 1
properties of this economy using a phase diagram. To analyze the behavior of rl, we find the locus of (z, v) where T,= r’t+ r. We call this locus the YYcurve. To find the behavior of ret, we investigate the locus of (n,r) where 7~,=rc,+ 1. We call this locus the rrrc curve. As shown in appendix A, both curves are downward sloping, and the (absolute) slope of the nn curve is greater than that of the rr curve. From the long-run comparative statics analysis we know of the initial that the new equilibrium point E* is to the south-west equilibrium point E, in fig. 1. Thus, in period 1 rc jumps from E, to E,, so that in period 1 the economy is on the new convergent path a’a’ toward E”. How are a, and c0 affected? In period 1 we have c: -I-ci = y, + K, -K, -G and (1 + nJm’ -m” + T = G. Since r2 and rc2 are less than the initial long-run equilibrium values, c: and m1 of generation 1 are greater than the initial G is, by assumption, raised. equilibrium values, cy and m”, respectively. Hence, c! of generation 0 must be lower than the (expected) old equilibrium value. The existing old generation will suffer from unexpected capital losses on land holdings. This aspect has also been explored in Charnley and Wright (1987) and Fried and Howitt (1988). Note that m” is defined by M,/p,, and M, is exogenously given. Changes in m” are associated with changes in pr. As rci is lower than the old equilibrium value, p1 is lower and hence m” is higher than the old equilibrium value. Intuitively, an increase in ml means an increase in inflationary taxes at the initial money holding, m”. In order to maintain the government budget, the government must ‘transfer’ these revenues to the existing older generation as a form of unexpected capital gains on money holdings. This is a new aspect, which occurs only in the monetary economy.
204
T. Ihori, Economic effects of land taxes
From (2’), when rcl is reduced, u1 must be reduced so long as 1 >fi+ 8. Thus, a, will normally be reduced. The existing old generation suffers from unexpected capital losses from land holdings, but he gets unexpected capital gains from money holdings. As for the effect on second-period consumption, the former effect is stronger than the latter effect: second-period consumption decreases unexpectedly. However, it is possible that older generations lind it beneficial to impose a land tax if they evaluate the service of money significantly.
3.2. A decrease in lump-sum 3.2.1. Long-run effects In this case the long-run
taxes
equilibrium
cl@,n, L) + cz(r, 7t, L) = y(r) TC~(~,TC,L)+T+L=G,
is given by G,
(18) (19)
where L appears in the consumption and money holdings functions explicitly. The differential incidence means dT+ dL = 0. As shown in appendix C, we can easily derive drr/dT
T. Ihori,
Economic
effects of land taxes
205
0 Fig. 2
Let us investigate the effect on the real price of land, a. If L is levied when the consumer is old, da = dw -dK -(dc, +dm), as in the previous subsection; a will normally be decreased. If a lump-sum tax is levied when the consumer is young, we have ds/dL
3.2.2. Transitionary effect We now consider the transitionary effects in fig. 2. As 7~ is reduced in the long run, the new equilibrium point E* is to the south of E,. 71~ will be reduced. As in subsection 3.1 we may derive that ct is reduced. The existing old generation will suffer from unanticipated capital losses on land holdings. The effect on m” is the same as before. 7~~ is reduced, p1 is reduced, and hence m” is increased. The existing old generation will get unexpected capital
T. Ihori, Economic effects of land taxes
206
Table Balanced r 77 a cl =2 m
_ _ _
4 m0 =I a,
budget
1
incidence
Differential
incidence
-
?” _
+ _
?b + _
+
+
_
_
+ _ _
+ _
“If
_ hr+c2,Nm+(2+W4J v sign will blf the stronger sign may
be positive. income effect due to a reduction of lump-sum taxation is than the substitution effect due to changes in r and n, this be positive.
gains from money holding. As in subsection reduced. Our results are summarized in table 1.
3.1 we may derive
that
u1 is
4. Conclusion We have examined the effects of land taxes on the price of land and the rate of inflation when factors of production are fixed in supply. We explicitly considered the general equilibrium effects of land rent taxes, land value taxes, and capital gains taxes in an inflationary situation. Land taxes may affect the real equilibrium only through changes in the total revenue from the land taxes. It is shown that land taxes will normally reduce the real rate of return on land, the real price of land, and the rate of inflation in the case of the balanced budget incidence. Land taxes may increase the long-run real price of land in the case of differential incidence if the income effect due to a reduction in lump-sum taxes on the younger generation is stronger than the substitution effects due to an increase in land taxes. Feldstein (1977) and Charnley and Wright (1987) obtained similar ‘surprising’ results. However, our dynamic incidence results are quite different from previous results in the following sense.
207
T. Ihori, Economic effects of land taxes
First, by incorporating monetary considerations, we have shown that the nominal price of land will normally be reduced by land taxes. Second, our instability condition in an inflationary economy implies that real saving is an increasing function of the real rate of return. Nevertheless, the real price of land may be raised if the income effect due to a reduction in lump-sum taxes in the younger generation is dominant. Third, it seems likely that the real price of land will initially decrease even if the long-run real price of land were to rise in the ‘surprising’ case. Furthermore, we may have the crowding-out case where the land taxes will increase the real rate of return on capital. This effect is induced by the substitution from land and capital to money in the portfolio of savings. Finally, the existing old generation will suffer from unexpected capital losses from land holding, but they get unexpected capital gains from money holding. It is possible that older generations find it beneficial to impose a land tax if they evaluate the service of money significantly. These aspects have been explored only in the three-asset inflationary economy.
Appendix A Let us investigate the stability system may be summarized by
condition
of our economy.
The dynamic
c1(rt+l~71,+1)+c2(r,,71,)= Y(r,,r,+J-G,
(A.1)
(l+~t+~Mr,+l,~,+l
(A.4
)-m(r,,n,)+T+L=G,
where Y(rt, rr+l)= y(r,) +K(r,)-K(r,+ J. We assume that G, T, and fixed. From (A.1) rt+l is given as a function of n,,,, rt, and rr,: r t+1 =R(n (+ 17rt3 d.
From
(A.3)
(A.2) x*+1 is given as a function
of r, + I, rt, and rr,:
71I+ I = Mb-,+ 1yIt, 4 Substituting
M(r f+ lr rl, 41
+ cZ(rt7 4 = W,, rt+J - G.
(A.9
(A.3) into (A.2), we have:
(1+%+1hCN~ The dynamic (A.6).
(A.4)
(A.4) into (A.l), we have:
crCrtfl, Substituting
L are
evolution
t+l,rt,7t,),n,+ll--m(r,,n,)+T+L=G. of the model
is entirely
determined
(‘4.6) by (A.5) and
208
T. Ihori, Economic effects of land taxes
The local stability of the equilibria can be analyzed mathematically. (A.5) and (A.@ we obtain at the neighbourhood of an equilibrium:
where dr = r - r*, drr = rc--71*, and expression: CllrMrO+c2rA=----------
(r*, n*)
is an equilibrium.
From
In the above
Yro
Clr+ClnMrl - r,l ’
B=
C_----_
ClnMrr+Czn clr+ClnMrl- YT’ (I+ 4mA0 - mL m+(l+7r)(m,+m,R,,)’ (1 + 7c)m,R,, -m,
DC___
m+(l where
Cir = &/dr,
+n)(m,+m,R,,)’
Gin = &Jaq
M,, = dM/&,,
M,, = aMjar,+
1, M, = aMlap,
R,,=aR/ar,+,, R,o=aR/ant, R,,=aRla71~+~ (i=1,2), Yro= Y,, =aY/&,+,. Following Charnley and Wright (1987), we disregard all the stable equilibria and concentrate on the saddle-point unstable case. Let us analyze the sign of the characteristic polynomial of the matrix R,,=aRpr,, dY/drt, and
i,G(A)=l’-(A+D)A+(AD-BC).
There exists a unique dynamic II/(-l)>O. In the long run we have:
path
near
the steady
state
if $( 1)
T. Ihori, Economic @cts
C=-
l-D= Suppose
209
of land taxes
- m,(c,, + c2J - 71172rc2, 3 c,,m+(l +~)kw,,--m,cLJ
-(l
+7C)**+Czn)+cIr(m+7TJ cl,m+(1+7t)(m,c,,--m,c,,)
for simplicity
cIn=O.
.
Then, we have:
$(l)=(l-A)(l-D)-BC
-
Y’CCl,h+ 714 -( 1+ 4w2JL
$(-l)=(l+A)(l+D)-BC
-c,~c,,rm+l(l+71)ml{-(Clr+CZ*)ClrCm+(2+n)m,l n
We know cl,.+ c2,>0, m,
O
clr
(m + 7m4cI, [ m,(l+n)
set of conditions
cln=O,
’
nh.71 1,
_ (cI,+c2r)(m+(2+4mz)
Let us investigate the dynamic properties of this economy diagram. From (A.5) the slope of the rr curve is given by
using
a phase
drrldr = (1 - A)/B. From
(A.6) the slope of the rcrc curve is given by drt/dr=C/(l-D).
From the instability condition both curves are downward sloping and the (absolute) slope of the 7~7~curve is greater than that of the rr curve. From (A.5):
T. Ihori, Economic effects of land taxes
210
which implies From (A.6):
that above
the rr curve rt < r,+ 1, and below the locus rt > rt+ 1.
an, +,/ar,= c > 0, which implies that on the right-hand side of the rrrr curve rr,+ 1 >IT,, and on the left-hand side of the rcrt curve rr,, 1
Appendix B Totally
differentiating
(15) and (16) with respect
to r and n, we have:
(B.1) Considering
dG=dT:
dr
dT
we have:
= - i (m-t 7cm,),
U3.2)
where A is the determinant of the matrix of LHD of (B.l) Let us investigate the sign of A. We know that clr+czr-y’>O. From the instability condition we have m +nm,>O and c~~+c~~=c~~>O. It is plausdr/dT
drr dT=z
1
7cm,< 0.
(B.3)
Appendix C Totally
differentiating
(18) and (19), we have:
“‘If dG =O, from (B.l) we know dn/dT
T. Ihori,
Economic
effects of land taxes
211
Considering dT + dL = 0, we have:
cc.21
CC.31
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