dX t X t dt X t d t
, (2)where is the instantaneous expected percentage change in
X t ( )
per unit time,
is the instantaneous standard deviation per unit time, andd ( ) t
is an increment of a standard Wiener process. Such a market is thus characterized by evolving uncertainty in the state of demand for the developed property. We assume that development, which reduces open space, harms urban residents. Similar to Anderson (1993a), we assume that the external cost increases as the cash flow from developed properties increases. Consequently, from the viewpoint of a “sophisticated”social planner who takes this externality into account, the marginal value of an additional developed property is reduced to( ) (1 ) ( ), 0 1,
c d
P t
a P t
a
(1’)where
a denotes the size of the external effect.
6Landowners may add to the supply of developed properties by incurring a one-time construction cost of
K
that is proportional to the quantity of new units of developed property supplied. We assume that landowners can construct buildings instantly, and thus they earn an instantaneous profit per unit time ofP t
d( ) per unit of developed property.7 However, the construction costs, once incurred, are assumed to be irreversibly committed6 Here and in what follows, we use the subscripts “d ”and “c”to represent the decentralized economy and the centralized economy, respectively.
7 We do not consider the time-to-build problem. See Grenadier (2000) for a thorough discussion of this issue.
8
thereafter.
Like Grenadier (1995), we seek to construct a competitive equilibrium in which developers have rational expectations. Such an equilibrium involves the simultaneous determination of developers’rents and entry strategies, assuming that these developers ignore the negative externality from development. Development strategies, the rent process, and expectations are all interdependent and must be mutually consistent in equilibrium.
Developers choose strategies so as to maximize the discounted present value of profits less development costs, given their expectations concerning the probability distribution governing the rent of developed property. These strategies, together with the exogenous demand shocks, determine the actual realization of the rent and supply in the real estate industry. If expectations turn out to be rational, then the rent process the developers use to form their strategies and the actual rent process will be the same function of the current state variables.
Finally, to be a competitive equilibrium, the present discounted value of equilibrium profits at any point of entry must equal the cost of development at that time.
Such a rational expectations competitive equilibrium can be determined as the solution to a maximization problem. As in Lucas and Prescott (1971) or Dixit (1991), the equilibrium evolves as if to maximize the expected present discounted value of social welfare in the form of consumer surplus. That is, the equilibrium path of rents and quantities of developed properties can be derived from the perspective of a “naive”social planner who ignores the negative externality from development.
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The problem for the naive social planner is to choose the path of new supply of developed properties so as to maximize the value of the flow of consumer surplus. This is precisely the same as choosing the path of
Q t ( )
. The total flow rate of social surplus in the view of the naive social planner,S
d[ ( ), ( )]X t Q t , is equal to the area under the following
The sophisticated social planner, in contrast, will internalize the externality before developing properties, and thus he takes the total flow rate of social surplus to be given by
(1 ) ( ) ( )
We assume that developers are risk neutral, and thus the appropriate discount rate is the risk-free interest rate,
. This seemingly restrictive assumption can be relaxed by adjusting the drift rate, , by a risk premium in the way of Cox and Ross (1976).When property development exhibits negative externalities as shown in equation (1’), then the market outcome will be inefficient. To correct this, a regulator can adopt policies that include density ceiling control (Lee and Jou, 2007) or Pigouvian taxes such as property taxes, building fees, and entitlement fees. We focus on two kinds of property taxation, a tax on the market value of vacant land and a tax on the development costs, but abstract from the other instruments. We assume that these two policy instruments are already set before vacant land is developed. Denote by the tax rate on land value and u the tax rate on development,
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both of which are constant over time. Note that can be either positive or negative (i.e., a subsidy), while
u is always positive.
By following the literature that applies non-cooperative dynamic games to environmental management (see, e.g., Jou 2001, 2004), we model land use regulation as a hierarchical game. Developers compete to enter the real estate industry at the lower level of the game. At the upper level the game takes the form of a Stackelberg game in which a regulator acts as the leader and a developer acts as the follower. Anticipating the timing chosen by the developer, the regulator should set a tax on development and/or a negative tax on land value to induce the developer to choose the socially optimal level of development timing.
Optimal Development Timing Choices
In this section we investigate how to design optimal taxation on land value and development by sequentially analyzing the behavior of both the naive social planner and the sophisticated social planner. We show that these planners make different choices regarding the timing of development, because the former ignores the externality while the latter internalizes it.