We will first assume that no property taxation is imposed, i.e. so as to comparea b 0 resource allocation under the decentralized economy with that under the centralized economy.
Comparing equation (28) with equation (28’), and equation (29) with equation (29’) yields
d c
y
y
andQ
d Q
c if both N1 and . The more interesting case, however, isa b 0 where there is more than one landowner in which case we obtain the result stated below:Proposition 3:
When the real estate market is not monopolized, a landowner will develop property later, but more densely, than a central planner who fully internalizes the consumption-production externality.
Proof: Proposition 3 will follow given that
M
d M
c forN 2
, and .a b 0We use Figure 3 to explain the intuition behind Proposition 3. The same line
TT
depicts the dependence of the optimal date of development on the optimal density defined in both equation (25) given and equation (25’a b 0 ). This is because without any taxation, the existence of consumption-production externalities will be irrelevant to the choice of development timing.This line has a positive slope, thus indicating that, as property in the real estate market becomes more densely developed, both a landowner and the central planner will be less eager to develop property since the rent per unit of developed land will then be lower, and thus waiting will be more valuable. On the other hand, the lines
D D
d d andD D
c c depict the dependence of the optimal density on the optimal date of development defined in equation (27) given anda b 0 equation (27’), respectively. Both lines have a positive slope, which indicates that, at a better state, both a landowner and the central planner will develop property more densely.Given that
N 2
and , linea b 0D D
c c will lie to the left of lineD D .
d d This indicates that, given the same timing of development, a central planner who internalizes the externality will develop property less densely than a landowner who ignores the same externality.The central planner will thus develop earlier than the landowner, as indicated by the positive slope of the line
TT
. This is shown by Point C that denotes the equilibrium for a central planner, which is where the linesTT
andD D
c c intersect, for which the optimal density isQ
c and the optimal date of development isy .
c By contrast, Point A denotes the equilibrium for a landowner, which is where the linesTT
andD D
d d intersect, and where the optimal density isQ
1d and the optimal date of development isy .
1d Proposition 2 then follows becauseQ
d1 Q
c andy
1d .y
cDc Dd
Figure 3: Difference between the centralized and the decentralized economy.
Proposition 3 indicates that the market outcome is inefficient for
N 2
. Therefore, a social planner, who fully perceives the negative externality, can design a property taxation policy to correct this outcome. As mentioned before, our model presents a hierarchical game. At the lower level, a landowner competes with the other landowners, and chooses both a date of development equal toy
d and a density level equal toQ
d in a Cournot-Nash environment. At the upper level, the regulator acts as a leader and the landowner acts as a follower. The regulator anticipates that both the timing and density chosen by the landowner will be above the socially optimal level,y
c andQ , respectively.
c Consequently, the regulator needs to impose property taxation both before and after development so as to align the timing and density chosen by the landowner with those chosen by the central planner. Equatingy
d in equation (28) withy
c in equation (28’) and equatingQ
d in equation (29) withQ
c in equation (29’) yields the optimal tax rates after and before development, and*a , respectively, as given by*b*
17
The design of taxation before and after development can be explained by using Figure 3. As mentioned before, points A and C represent the equilibrium points for the decentralized and the centralized economy, respectively. As indicated by Proposition 1(a), when the regulator imposes a tax rate on property before development, a landowner will be induced to move downward*b along
D D
d d until point B, where he develops property at a date equal toy
d2 and a density equal toQ .
d2 As indicated by Proposition 1(b), when the regulator further imposes a tax rate on*a property after development, the landowner will be induced to move upward from point B to point C, the equilibrium point for the centralized economy. Note that if the central planner only imposes taxation before development, a landowner may be induced to develop on the same scale as the centralized economy, but earlier than will be socially optimal. For example, in Figure 3, a landowner will develop property atpointB’,wherethe development density is equal toQ , while
cthe development timing is equal to
y
d3 (y
c).Table 1 shows how changes in several exogenous variables affect optimal taxation before and after development, which we state in the following proposition:
Proposition 4: (a) The regulator should increase property taxation both before () and after (*b )*a
development if there are more landowners in the real estate market ( N is larger).
(b) Theregulator should increase property taxation after development, but may increase, reduce, or leave unchanged property taxation before development if (i) the negative externality becomes more significant ( a is larger), or (ii) demand for developed property is expected to grow more slowly
( is smaller). (c) The regulator should raise property taxation before development, while2leaving taxation after development unchanged if (i) the development costs are expected to grow more rapidly (
is larger), and (ii) uncertainty is reduced (1 is lower).Proof: See Appendix E.
We can use Figure 4 to explain the reason for Proposition 4(a). Suppose that the initial equilibrium is depicted by point A, which is the intersection of lines
TT
andDD
, both of which are the common lines of the centralized economy and the decentralized economy with appropriate property taxation to correct market inefficiencies. When the scale of development is fixed, an increase in the number of landowners will reduce the expected development time by the same magnitude for both the centralized and the decentralized economy, as indicated by equation (E11)'
(This is shown by a downward shift from line
TT
toT T ' '
). Furthermore, when the choice of development timing is fixed, an increase in the number of landowners will induce a landowner to increase the development density by a magnitude that outweighs the increase in the development density for a central planner because the landowner ignores the negative externality. This is indicated by equations (E12) and (E13), and is shown by an outward shift from lineDD
to lines' '
d d
D D
andD D
c' c' for the decentralized and the centralized economy, respectively. As indicated by Proposition 1, the regulator needs to raise taxation both before and after development because the new equilibrium for the centralized economy, point B, is on the south-western side of the new equilibrium for the decentralized economy, point C. The other results stated in Proposition 3 can also be derived using similar arguments.Figure 4: An increase in N.
The results of Proposition 3 (or Table 1) accord well with intuition. First, as the negative externality becomes more severe either by itself (
a is larger) or results from an increase in the
number of landowners (N
is larger), the regulator should raise taxation both before and after development in response.11 Second, as future demand for developed property is expected to grow more slowly ( is lower), a developer will be induced to develop property earlier, but on a2 smaller scale as compared to a social planner. The regulator thus needs to raise taxation after development, as indicated by Proposition 1(b) (The impact on taxation before development, however, is indefinite). Third, as the costs of development are expected to grow more rapidly, the optimal condition for the choice of density of development for both a landowner (equation (25)) and a social planner (equation (25’)) will not be affected. Consequently, the regulator only needs11 Note that an increase in the magnitude of the externality has a positive impact on taxation before development, *b, only in the region where *a*b.
19
to raise taxation before development so as to encourage a landowner to develop earlier, but on a smaller scale, while leaving taxation after development unchanged. Finally, the total instantaneous volatility () will be greater as
r
12 is lower, i.e., asx t
1( ) andx t
2( ) move in the opposite directions. That is, uncertainty will be greater if more (less) advantageous supply conditions are associated with more (less) prospective demand conditions in the real estate market.The impact of greater uncertainty resembles that of a lower expected growth rate in relation to the development costs, and thus the regulator only needs to reduce taxation before development.
We assume that developed property exhibits a negative externality on urban residents, while vacant land does not exhibit any externality. To correct this externality, however, it is uncertain whether the rate of property taxation after development should be higher than that before*a
development . We, however, can compare the order of*b and*a around the region where*b
* *
a b
, as stated in the Proposition below:
Table 1: Comparative-Statics Results
of the rent of developed land, 2
the number of landowners increases; (b) the costs of development are expected to grow less rapidly, or the total instantaneous volatility is greater.
Proof: See Appendix F.
Proposition 5(a) follows because as the number of landowners increases, both and*a *b will then be raised (as stated in Proposition 4(a)), but the latter will be raised by a smaller magnitudes than the former as indicated by equation (F1). Proposition 5(b) follows because as the costs of development are expected to grow less rapidly, or the total instantaneous volatility is greater (as stated in Proposition 4(c)), will then be lower, while*b will remain unchanged.*a
VI. Conclusion
This article investigates the design of property taxation both before and after development in a real options framework where a fixed number of landowners irreversibly develop property in an uncertain environment. We assume that densely developed properties reduce open space, and thereby harms urban residents. However, landowners will ignore this negative externality, and will thus develop properties more densely than is socially optimal. The regulator can correct this tendency by imposing taxation on property both before and after development. We then find that the tax on the former should be increased if the real estate market consists more landowners, the costs of development are expected to grow more rapidly, and uncertainty is less significant. In addition, taxation on the latter should be increased if the real estate market consists more landowners, the externality is more significant, or if demand for property after development is expected to grow less rapidly. Future studies may empirically test these theoretical predictions.
This article can be extended to investigate the issue discussed in Henry George’s seminal book Progress and Poverty (1897), i.e., taxation on vacant land should be higher than taxation on land development. Brueckner (1986) has investigated how a shift to a graded tax system (where the tax rate is lowered and the land tax rate it raised) affects the level of development, the value of land and the price of housing in the long-run. Anderson (1993b) has extended the Brueckner’s analysis by employing a perfect foresight model. His focus is how this shift affects choices regarding the timing and density of development. If we replace taxation on property after development by taxation on development, we not only can investigate the issue discussed in Anderson (1993b), but can also investigate whether there exists a tradeoff between land value taxation and land development taxation when these two instruments are employed to correct the externality associated with land development.
21
Appendix A:
Totally differentiating equation (25) with respect to
Q , and using equations (28) -(30) yields
d12
Totally differentiating equation (27) with respect to
y , and using equations (28)-(30) yields
d21
The Jacobian condition also requires that
11 22 12 21 0.
(A7)
We depict the impact of