Abstract
This paper presents a model of a housing market with a fixed supply of land available for future development. Building density and the rate of land development are both endogenous. Competition amongst atomistic landowners leads to welfare-maximizing development policies. However, a monopolist landowner develops land faster, with lower building density, than a welfare-maximizing social planner. Unless demand is very high, the first effect dominates, because a monopolist landowner increases the size of the housing stock faster than a social planner. Rapid, low-density development is a commitment device. It boosts the monopolist’s development proceeds by making it more difficult to flood the market with new housing in the future.
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Notes
The model extends real-options models of commodity markets in which agents hold inventory of a commodity and increase their inventory when spot prices are low and decrease inventory when they are high. See Evans and Guthrie (2017) for such a model of a generic commodity market and Evans et al. (2013) for a model of an electricity market in which the stored commodity is water used to generate electricity. In the current paper, the “commodity” is undeveloped land.
More recent real options analyses of individual development decisions feature a richer development environment. For example, Guthrie (2020) investigates the effects of competition between developers on individual landowners’ optimal development policies. Cheng et al (2021) model the process of obtaining all required regulatory approvals from multiple (and often overlapping) regulatory jurisdictions. Lange and Teulings (2021) adopt a more realistic process for urban growth in which the growth rate itself is stochastic.
In most cases, development occurs instantaneously. Development lasts a fixed finite amount of time in Bar-Ilan and Strange (1996), but landowners still initiate all of their desired development activity at a single point in time.
In the limit as \(\varepsilon \rightarrow 0\), z(b, y) converges to the Cobb–Douglas production function \(z(b,y) = (b^{\phi _b} y^{\phi _y})^{1/(\phi _b+\phi _y)}\). As \(\varepsilon \rightarrow \infty\), it approaches the Leontief production function.
The marginal costs of the first units of structure and land therefore equal \(\kappa _b\) and \(\kappa _y\) respectively. The curvature of c determines how quickly costs climb as the rate of development increases.
Proofs of all formal results appear in Appendix A.
From this point on, subscripts denote partial derivatives.
When apartments and townhouses are included, building consents for 3,220 new dwellings were issued in the Wellington region in 2019. I focus on the market for standalone houses in this example.
For this production function, the following house–land combinations each generate one unit of housing services: \(194\text {m}^2\) of structure on \(400\text {m}^2\) of land, \(173\text {m}^2\) of structure on \(500\text {m}^2\) of land, and \(157\text {m}^2\) of structure on \(600\text {m}^2\) of land.
The policy changes will be small and the Envelope Theorem implies that small changes in the development policy will have a negligible effect on the objective function.
I confirm this result formally as part of the proof of Proposition 2 in Appendix A.5.
\(W_l\) equals the present value of the development payoff if development of this piece of land is delayed until the total stock of undeveloped land is exhausted.
For very high levels of standardized rent (and hence very high levels of demand), the green and yellow curves cross. That is, the building density is so high that the price effect dominates the commitment effect and the monopolist’s development payoff is actually less than the payoff of an investor who owned only the land being developed.
The total height of the blue region equals the opportunity cost of the land, \(W_l \, \Delta y\), but the height of the part of this region that lies above the horizontal axis equals the marginal value of the land to the monopolist, \(F_l \, \Delta y\).
The graphs show the cost of generating the marginal standardized house. As the production function is increasing in the amount of structure and the amount of land, increased use of structure must be accompanied by decreased use of land.
Glaeser and Gyourko (2018) start with an estimate of the construction cost from RS Means Company, set the land price equal to 25% of this amount on the basis of “an industry rule of thumb based on an ad hoc survey of home builders,” and then add a 17% margin for “entrepreneurial profit” for the developer.
I follow the approach in Section 4 and choose values of \(\phi _b\) and \(\phi _y\) that are \(\varepsilon\)-dependent. I impose two conditions. First, if the marginal cost of development equals $2,500/\(\text {m}^2\) of structure and $50/\(\text {m}^2\) of land, and the price of undeveloped land is $400/\(\text {m}^2\), then the implied building density equals \(173/500 = 0.346\) square metres of structure per square metre of land. Second, a house with 173\(\text {m}^2\) of structure and 500\(\text {m}^2\) of land generates one unit of housing services.
However, there would also be an indirect effect, because this change in behavior would lead to a change in the value of the monopolist’s stock of undeveloped land.
For example, if the market-clearing house price deviates from the fundamental value of a house, then calculation of the overall welfare generated by the housing stock will be much more difficult.
The grid for l spans \([0,3\times 10^6]\) (measured in m\(^2\)) and is split into 600 subintervals; the grid for r spans \([0,1.5\times 10^5]\) (measured in dollars per annum) and is split into 300 subintervals.
I use the condition \(u(l_i,0)=0\) along the bottom boundary and one-sided finite differences along the top boundary.
I use the conditions \(f(l_i,0)=0\) and \(h(l_i,0)=0\) along the bottom boundaries and one-sided finite differences along the top boundaries.
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Acknowledgements
Cameron Murray provided helpful insights on an earlier version of this paper.
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Appendices
Appendix A: Proofs
A.1 Proof of Lemma 1
The required return, \(\rho H \, dt\), from owning a standardized house equals the sum of the expected capital gain, E[dH], and the rent it generates, \(x\psi (s) \, dt\). The standardized house price, H, therefore satisfies \(\rho H \, dt = E[dH] + x\psi (s) \, dt\). Using Itô’s Lemma to evaluate E[dH] shows that H satisfies Eq. 5. Once the supply of undeveloped land is exhausted, the supply of housing services remains constant, rent evolves according to geometric Brownian motion, and the standardized house price equals \(H(0,s,x)=x\psi (s)/(\rho -\mu )\).
A.2 Proof of Lemma 2
Overall welfare, W, is the present value of the flow \(x\Psi (s)- c(\kappa _b b + \kappa _y y)\), calculated using the discount rate \(\rho\). It therefore satisfies \(\rho W \, dt = E[dW] + (x\Psi (s) -c(\kappa _b b + \kappa _y y))\, dt\). Using Itô’s Lemma to evaluate E[dW] shows that W satisfies Eq. 6. Once the supply of undeveloped land is exhausted, the supply of housing services remains constant, the flow of total surplus evolves according to geometric Brownian motion, and the present value of this flow equals \(W(0,s,x)=x\Psi (s)/(\rho -\mu )\).
A.3 Proof of Lemma 3
The market value of the land bank, denoted F, satisfies \(\rho F \, dt = E[dF] + (z(b,y)H - c(\kappa _b b + \kappa _y y)) \, dt\), where the second term on the right-hand side is the difference between the proceeds from selling the developed land, \(z(b,y)H \, dt\), and the development expenditure, \(c(\kappa _b b + \kappa _y y) \, dt\). Using Itô’s Lemma to evaluate E[dF] shows that F satisfies Eq. 7. Once the supply of undeveloped land is exhausted, the land bank is worthless: \(F(0,s,x)=0\).
A.4 Proof of Proposition 1
The planner chooses the policies (b, y) in order to maximize \(\mathcal {P}(b,y) = z(b,y)W_s - yW_l - c(\kappa _b b + \kappa _y y)\). The first-order conditions can be written as
Eliminating \(c^{\prime }(\kappa _b b + \kappa _y y)\) between these two equations implies that
where \(d=b/y\) is building density. Solving this equation for d gives the socially optimal building density. If the planner adopts the socially optimal building density then the objective function becomes
As c is increasing and convex, the objective function is maximized at \(y^*=0\) if
and is otherwise maximized where \(y^*\) satisfies the first-order condition. Equation A.1 implies that the rate of land development satisfies
That is,
The rate of building construction is
A.5 Proof of Proposition 2
I begin the proof by deriving an optimal development policy for a price-taking landowner who takes the prices implied by the planner’s policies as given. That is, I show that if the standardized house price equals \(W_s\), the market price of undeveloped land equals \(W_l\), and the market prices of building structures and developing land equal \(\kappa _b c^{\prime }(\kappa _b b^* + \kappa _y y^*)\) and \(\kappa _y c^{\prime }(\kappa _b b^* + \kappa _y y^*)\) respectively, then an optimal development policy for the price-taking owner of \(\hat{l}\) units of undeveloped land is described by the policy functions \(\hat{b}=\hat{l}b^*(l,s,x)/l\) and \(\hat{y} = \hat{l}y^*(l,s,x)/l\), where \(b^*(l,s,x)\) and \(y^*(l,s,x)\) describe the planner’s socially optimal development policy.
-
Consider the situation facing a price-taking owner of \(\hat{l}\) units of undeveloped land. If she waits then the land is worth \(\hat{l}W_l\). In contrast, if she builds yd units of structure on y units of land (so that the building density equals d), the developed land will be worth \(W_s z(yd,y) = yW_s (\phi _b d^{-\varepsilon }+\phi _y)^{-1/\varepsilon }\). As the landowner is a price-taker in the market for construction services, she will need to pay a construction firm \((\kappa _y y + \kappa _b yd) c^{\prime }(\kappa _b b^*+ \kappa _y y^*)\), so her payoff from developing the land equals \(y(W_s (\phi _b d^{-\varepsilon }+\phi _y)^{-1/\varepsilon } - (\kappa _y + \kappa _b d) c'(\kappa _b b^*+ \kappa _y y^*))\). If she decides to develop the land, she will choose d to maximize this payoff. She will therefore choose density \(\hat{d}\) satisfying the first-order condition
$$\begin{aligned} \phi _b W_s (\phi _b + \phi _y \hat{d}^{\varepsilon })^{-(1+\varepsilon )/\varepsilon } = \kappa _b c^{\prime }(\kappa _b b^*+ \kappa _y y^*), \end{aligned}$$which implies that
$$\begin{aligned} \hat{d} = \left( \frac{1}{\phi _y} \left( \frac{\phi _b W_s}{\kappa _b c^{\prime }(\kappa _b b^*+ \kappa _y y^*)}\right) ^{\varepsilon /(1+\varepsilon )} - \frac{\phi _b}{\phi _y}\right) ^{1/\varepsilon }. \end{aligned}$$That is, the landowner will choose the same building density as the social planner. Her development payoff is \(y(W_s (\phi _b (d^*)^{-\varepsilon }+\phi _y )^{-1/\varepsilon } - (\kappa _y + \kappa _b d^*) c^{\prime }(\kappa _b b^*+ \kappa _y y^*))\).
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Proposition 1 shows that in situations where \(b^*=y^*=0\),
$$\begin{aligned} z(d^*,1)W_s \le W_l + (\kappa _y + \kappa _b d^*) c^{\prime }(0), \end{aligned}$$so that the landowner’s development payoff satisfies
$$\begin{aligned} y(W_s (\phi _b (d^*)^{-\varepsilon }+\phi _y)^{-1/\varepsilon } - (\kappa _y+\kappa _b d^*) c^{\prime }(0)) \le yW_l. \end{aligned}$$Therefore, the landowner will not want to develop her land in situations where the planner would delay development; she will choose \(y=0\).
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Proposition 1 also shows that in situations where \(b^*\) and \(y^*\) are positive,
$$\begin{aligned} c^{\prime }(\kappa _b b^*+ \kappa _y y^*) = \frac{\phi _b W_s}{\kappa _b} \left( \phi _b + \phi _y (d^*)^{\varepsilon }\right) ^{-\frac{1+\varepsilon }{\varepsilon }}, \end{aligned}$$so that her development payoff satisfies
$$\begin{aligned}&y(W_s (\phi _b (d^*)^{-\varepsilon }+\phi _y)^{-1/\varepsilon } - (\kappa _y+\kappa _b d^*) c^{\prime }(\kappa _b b^*+ \kappa _y y^*)) \\= & {} \frac{yW_s}{\kappa _b} \left( \phi _y \kappa _b (d^*)^{1+\varepsilon } - \phi _b \kappa _y \right) \left( \phi _b + \phi _y (d^*)^{\varepsilon }\right) ^{-\frac{1+\varepsilon }{\varepsilon }}. \end{aligned}$$Equation 9 implies that this equals \(yW_l\). Therefore, the landowner is indifferent between all possible values of y. In particular, the landowner cannot do better than to develop the land at the rate \(\hat{y} = \hat{l}y^*(l,s,x)/l\), with building density \(d^*(l,s,x)\).
If every individual landowner adopts this policy, then the aggregate development policy equals the social planner’s optimal policy described in Proposition 1. For example, suppose there are I landowners in total, with landowner i owning \(\hat{l}_i\) units of undeveloped land. The total amount of undeveloped land is \(\sum _i \hat{l}_i=l\) and it is developed at the rate
Similarly, new buildings are constructed at the rate
The state variables (l, s, x) therefore evolve in exactly the same way as in the solution to the planner’s problem.
In the second part of the proof I show that if landowners follow these policies then the standardized house price equals \(W_s\), the market price of undeveloped land equals \(W_l\), and the prices of construction and land-development services equal \(\kappa _b c^{\prime }(\kappa _b b^* + \kappa _y y^*)\) and \(\kappa _y c^{\prime }(\kappa _b b^* + \kappa _y y^*)\).
I start by deriving some useful properties of \(W_l\) and \(W_s\). If the market adopts the welfare-maximizing development policy in Proposition 1, then W satisfies Eq. 6. In the no-development region, this reduces to
Differentiating this equation with respect to l shows that \(W_l\) satisfies the differential equation
in this region. Similarly, differentiating it with respect to s shows that \(W_s\) satisfies
in this region. When I differentiate Eq. 6 in the development region, I use the Envelope Theorem to simplify the result. Differentiating Eq. 6 with respect to l shows that \(W_l\) satisfies the differential equation
in the development region. Similarly, differentiating Eq. 6 with respect to s shows that \(W_s\) satisfies the differential equation
in the development region. It follows that \(W_s\) satisfies Eq. 5, the differential equation for the standardized house price. Differentiating the boundary condition for W with respect to s shows that \(W_s(0,s,x) = x\psi (s)/(\rho -\mu )\), which is the boundary condition for H. This completes the proof that the standardized house price equals \(W_s\).
Now I consider an investor who owns \(\hat{l}\) units of undeveloped land and who adopts the development policy described by \(\hat{b}=\hat{l}b^*/l\) and \(\hat{y} = \hat{l}y^*/l\). The continuous cash flow generated by this asset is \(W_s z(\hat{b}, \hat{y}) - (\kappa _b\hat{b}+\kappa _y \hat{y}) c'(\kappa _b b^* + \kappa _y y^*)\), where I have used the result that \(H=W_s\). The market value, G, of this asset satisfies
The form of the development policy, together with the constant returns to scale of the production technology, implies that \(G(\hat{l};l,s,x) = \hat{l}V(l,s,x)\) for some function V to be determined. Substituting this expression for G into Eq. A.2 shows that V must satisfy
In the waiting region, the differential equation for V reduces to
which is identical to the equation for \(W_l\) in this region. Now consider the development region. In this region, the social planner’s first-order conditions imply that \(z_b (b^*,y^*) W_s = \kappa _b c^{\prime }(\kappa _b b^* + \kappa _y y^*)\) and \(z_y(b^*,y^*) W_s = W_l + \kappa _y c^{\prime }(\kappa _b b^* + \kappa _y y^*)\). The form of the CES production function means that \(bz_b + yz_y = z\), which implies that in the development region
Therefore, in the development region, the differential equation for V reduces to
As the term inside the large brackets is identical to the differential equation for \(W_l\) in this region, it follows that \(V=W_l\) satisfies the differential equation for V in this region.
This leaves the terminal conditions for the differential equations for V and \(W_l\), which apply as \(l \rightarrow 0\). As undeveloped land runs out, the price of undeveloped land approaches the payoff from developing the last (infinitesimal) piece of land. Suppose the last l units of land are developed with density \(\hat{d}\), so that the housing stock increases by \(z(\hat{d}l,l) = lz(\hat{d},1)\) units. As there is no more land to develop, these units of housing stock have market value \(lz(\hat{d},1) x\psi (s)/(\rho -\mu )\). Taking a first-order Taylor series approximation shows that the cost of developing this piece of land equals
The development payoff is therefore
so that the price of land approaches
as the stock of undeveloped land runs out. In contrast, overall welfare approaches
Differentiating this expression with respect to l implies that \(W_l\) approaches
as \(l \rightarrow 0\). Thus, the differential equations for V and \(W_l\) have the same terminal conditions, completing the proof.
A.6 Proof of Proposition 3
The proof is identical to that of Proposition 1, but with \(W_l\) and \(W_s\) replaced by \(F_l\) and \(H+F_s\), respectively.
A.7 Proof of Proposition 4
The only part of the proposition remaining to be confirmed is the claim that when it is optimal to develop land, these components sum to H; when it is optimal to suspend development, their sum exceeds H. I consider the case of the planner; the case of the monopolist is similar. From Proposition 1, if development is not occurring then
On the other hand, if development is occurring then the equation for \(y^*\) implies that
Equation 9 implies that
which implies that
Adding Eqs. A.3 and A.4 together shows that
Appendix B: Calibration
I assume that the marginal expenditure function \(c'\) is linear and choose the marginal cost parameters, \(\kappa _b\) and \(\kappa _y\), such that the cost function is consistent with arbitrary levels of the marginal costs of structure and land. Specifically, I require that
where \(\hat{w}_b\) and \(\hat{w}_y\) are the marginal costs of structure and land, and \(\hat{b}\) and \(\hat{y}\) are the rates at which structures are built and land is developed. In addition, I require that the construction sector’s variable cost equals the proportion \(\gamma\) of its total revenue. That is, I require that
These requirements uniquely determine the cost of development in the model, which is given by
with
In Section 4, I set \(\hat{w}_b=2500\), \(\hat{w}_y=50\), \(\hat{b} = 1540 \times 173\), \(\hat{y} = 1540 \times 500\), and \(\gamma =0.9\).
I set the units of housing services so that the rate of increase in housing services equals the rate at which “typical” houses are added to the housing stock. That is, \(\phi _b\) and \(\phi _y\) satisfy
where \(\hat{n}\) is the rate of construction of “typical” houses. I also impose the condition that if the marginal costs of structure and land equal \(\hat{w}_b\) and \(\hat{w}_y\), and the price of undeveloped land equals \(\hat{p}_l\), then building density equals \(\hat{b}/\hat{y}\). Equations 12 and B.1 together imply that
Solving these two conditions simultaneously for \(\phi _b\) and \(\phi _y\) shows that
and
In Section 4, in addition to the parameter values above, I set \(\varepsilon =-0.0909\) and \(\hat{p}_l = 400\), which is sufficient to calculate \(\phi\).
Appendix C: Solution Algorithm
The form of the inverse demand function and the choice of geometric Brownian motion for x make it easier to solve this problem. These assumptions mean that everything can be reduced to functions of two variables, (l, r), where \(r=xe^{-\alpha s}\) is the rent generated by a standardized house. For example, I look for a solution for the house price of the form \(H(l,s,x) = h(l,xe^{-\alpha s})\). Substituting this expression into Eq. 5 shows that h(l, r) must satisfy
I look for a solution of the overall welfare function of the form \(W(l,s,x) = x\Psi (s)/(\rho -\mu ) + u(l,xe^{-\alpha s})\) for some function u. The first term, \(x\Psi (s)/(\rho -\mu )\), equals the present value of the total surplus if no more land is developed. The second term is therefore the social value of the planner’s option to develop land that is currently undeveloped. Substituting this expression into Eq. 6 shows that u(l, r) must satisfy
Similarly, I look for a solution for the value of the land bank of the form \(F(l,s,x) = f(l,xe^{-\alpha s})\). Substituting this expression into Eq. 7 shows that f(l, r) must satisfy
The corresponding boundary conditions are \(h(0,r)=r/(\rho -\mu )\), \(u(0,r)=0\), and \(f(0,r)=0\).
In order to solve for the planner’s optimal development policy, I need to find functions u(l, r), b(l, r), and y(l, r) that simultaneously satisfy Eq. C.2 and maximize the expression in Eq. 8, which becomes
in the new coordinates. Similarly, in order to solve for the monopolist’s optimal development policy, I need to find functions f(l, r), h(l, r), b(l, r), and y(l, r) that simultaneously satisfy Eqs. C.1 and C.3 and maximize the expression in Eq. 13, which becomes
in the new coordinates.
I use an iterative method for solving the planner’s problem on a grid with points \((l_i,r_j)\), where \(l_1=0\).Footnote 24 This method starts by using the initial condition to set \(u(l_1,r_j) = 0\) for all j. I then solve the problem for each \(l_i\) in turn, starting with \(l_2\). Consider the situation when I have solved the problem for \(l_{i-1}\) and want to solve it for \(l_i\). I start with an initial estimate of the first-order derivatives of the planner’s objective function, \(u_l(l_i,r_j)\) and \(u_r(l_i,r_j)\) for all j, and use the results of Proposition 1 to calculate the corresponding policy functions \(b(l_i,r_j)\) and \(y(l_i,r_j)\) for all j. I then use a finite difference method to solve Eq. C.2 for the planner’s objective function, \(u(l_i,r_j)\) for all j.Footnote 25 I use this solution to update my estimates of \(u_l(l_i,r_j)\) and \(u_r(l_i,r_j)\) for all j. If the change in these estimates is sufficiently small then I store the policies at \(l_i\) and move onto \(l_{i+1}\), otherwise I repeat the process with the updated estimates until convergence occurs.
The iterative method for solving the monopolist’s problem is similar. I impose the initial conditions \(f(l_1,r_j) = 0\) and \(h(l_1,r_j)=r_j/(\rho -\mu )\) for all j, and then solve the problem for each \(l_i\) in turn, starting with \(l_2\). Consider the situation when I have solved the problem for \(l_{i-1}\) and want to solve it for \(l_i\). I start with an initial estimate of the first-order derivatives of the monopolist’s objective function, \(f_l(l_i,r_j)\) and \(f_r(l_i,r_j)\) for all j, and use the results of Proposition 3 to calculate the corresponding policy functions \(b(l_i,r_j)\) and \(y(l_i,r_j)\) for all j. I then use finite difference methods to solve Eqs. C.3 and C.1 for \(f(l_i,r_j)\) and \(h(l_i,r_j)\), respectively, for all j.Footnote 26 I use the solution for the monopolist’s value function to update the estimates of \(f_l(l_i,r_j)\) and \(f_r(l_i,r_j)\) for all j. If the change in these estimates is sufficiently small then I store the policies at \(l_i\) and move onto \(l_{i+1}\), otherwise I repeat the process with the updated estimates until convergence occurs.
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Guthrie, G. Land Hoarding and Urban Development. J Real Estate Finan Econ 67, 753–793 (2023). https://doi.org/10.1007/s11146-021-09880-y
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DOI: https://doi.org/10.1007/s11146-021-09880-y