Land Hoarding and Urban Development

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.

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

$$\begin{aligned} \phi _b W_s \left( \phi _b + \phi _y b^{\varepsilon } y^{-\varepsilon }\right) ^{-\frac{1+\varepsilon }{\varepsilon }}= & {} \kappa _b c'(\kappa _b b + \kappa _y y), \nonumber \\ \phi _y W_s \left( \phi _y + \phi _b b^{-\varepsilon } y^{\varepsilon }\right) ^{-\frac{1+\varepsilon }{\varepsilon }}= & {} W_l + \kappa _y c'(\kappa _b b + \kappa _y y). \end{aligned}$$

(A.1)

Eliminating $c^{\prime }(\kappa _b b + \kappa _y y)$ between these two equations implies that

$$\begin{aligned} \frac{\kappa _b W_l}{W_s} = \left( \phi _y \kappa _b d^{1+\varepsilon } - \phi _b \kappa _y \right) \left( \phi _y d^{\varepsilon }+\phi _b\right) ^{-\frac{1+\varepsilon }{\varepsilon }}, \end{aligned}$$

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

$$\begin{aligned} \mathcal {P}(d^*y,y) = y \left( z(d^*,1)W_s - W_l\right) - c(y (\kappa _y + \kappa _b d^*)). \end{aligned}$$

As c is increasing and convex, the objective function is maximized at $y^*=0$ if

$$\begin{aligned} z(d^*,1)W_s \le W_l + (\kappa _y + \kappa _b d^*) c'(0), \end{aligned}$$

and is otherwise maximized where $y^*$ satisfies the first-order condition. Equation A.1 implies that the rate of land development satisfies

$$\begin{aligned} c'(y(\kappa _y + \kappa _b d^*)) = \frac{\phi _b W_s}{\kappa _b} \left( \phi _b + \phi _y (d^*)^{\varepsilon }\right) ^{-\frac{1+\varepsilon }{\varepsilon }}. \end{aligned}$$

That is,

$$\begin{aligned} y = \frac{1}{\kappa _y+\kappa _b d^*} \cdot (c^{\prime })^{-1} \left( \frac{\phi _b W_s}{\kappa _b} \left( \phi _b + \phi _y (d^*)^{\varepsilon }\right) ^{-\frac{1+\varepsilon }{\varepsilon }} \right) . \end{aligned}$$

The rate of building construction is

$$\begin{aligned} b = d^*y = \frac{d^*}{\kappa _y+\kappa _b d^*} \cdot (c^{\prime })^{-1} \left( \frac{\phi _b W_s}{\kappa _b} \left( \phi _b + \phi _y (d^*)^{\varepsilon }\right) ^{-\frac{1+\varepsilon }{\varepsilon }} \right) . \end{aligned}$$

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 $y(W_s (\phi _b (d^*)^{-\varepsilon }+\phi _y )^{-1/\varepsilon } - (\kappa _y + \kappa _b d^*) c^{\prime }(\kappa _b b^*+ \kappa _y y^*))$.

• 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 $y=0$.

• 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

$$\begin{aligned} \underset{i}{\sum } y_i = \underset{i}{\sum } \frac{\hat{l}_i}{l} \cdot y^*(l,s,x) = y^*(l,s,x). \end{aligned}$$

Similarly, new buildings are constructed at the rate

$$\begin{aligned} \underset{i}{\sum } b_i = \underset{i}{\sum } \frac{\hat{l}_i}{l} \cdot b^*(l,s,x) = b^*(l,s,x). \end{aligned}$$

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

$$\begin{aligned} 0 = \frac{1}{2} \sigma ^2 x^2 W_{xx} + \mu x W_x - \rho W + x\Psi (s). \end{aligned}$$

Differentiating this equation with respect to l shows that $W_l$ satisfies the differential equation

$$\begin{aligned} 0 = \frac{1}{2}\sigma ^2 x^2 W_{l,xx} + \mu x W_{l,x} - \rho W_l \end{aligned}$$

in this region. Similarly, differentiating it with respect to s shows that $W_s$ satisfies

$$\begin{aligned} 0 = \frac{1}{2} \sigma ^2 x^2 W_{s,xx} + \mu x W_{s,x} - \rho W_s + x\psi (s) \end{aligned}$$

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

$$\begin{aligned} 0 = \frac{1}{2}\sigma ^2 x^2 W_{l,xx} - y^* W_{l,l} + z(b^*,y^*) W_{l,s} + \mu x W_{l,x} - \rho W_l \end{aligned}$$

in the development region. Similarly, differentiating Eq. 6 with respect to s shows that $W_s$ satisfies the differential equation

$$\begin{aligned} 0 = \frac{1}{2}\sigma ^2 x^2 W_{s,xx} - y^* W_{s,l} + z(b^*,y^*) W_{s,s} + \mu x W_{s,x} - \rho W_s + x\psi (s) \end{aligned}$$

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

$$\begin{aligned} 0= & {} \frac{1}{2}\sigma ^2 x^2 G_{xx} - \hat{y}G_{\hat{l}} - y^*G_l + z(b^*,y^*) G_s + \mu x G_x - \rho G\nonumber \\&+ W_s z(\hat{b}, \hat{y}) - (\kappa _b\hat{b}+\kappa _y \hat{y}) c'(\kappa _b b^* + \kappa _y y^*). \end{aligned}$$

(A.2)

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

$$\begin{aligned} 0= & {} \frac{1}{2}\sigma ^2 x^2 V_{xx} - y^*V_l + z(b^*,y^*) V_s + \mu x V_x - \rho V \\&+ \frac{1}{l} \left( W_s z(b^*,y^*) - y^*V - (\kappa _b b^* + \kappa _y y^*) c'(\kappa _b b^* + \kappa _y y^*)\right) . \end{aligned}$$

In the waiting region, the differential equation for V reduces to

$$\begin{aligned} 0 = \frac{1}{2}\sigma ^2 x^2 V_{xx} + \mu x V_x - \rho V, \end{aligned}$$

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

$$\begin{aligned}&z(b^*,y^*)W_s - (\kappa _b b^*+\kappa _y y^*) c^{\prime }(\kappa _b b^* + \kappa _y y^*) \\= & {} b^*z_b(b^*,y^*)W_s + y^*z_y(b^*,y^*)W_s - (\kappa _b b^*+\kappa _y y^*) c^{\prime }(\kappa _b b^* + \kappa _y y^*) \\= & {} y^* W_l. \end{aligned}$$

Therefore, in the development region, the differential equation for V reduces to

$$\begin{aligned} 0 = \left( \frac{1}{2}\sigma ^2 x^2 V_{xx} - y^*V_l + z(b^*,y^*) V_s + \mu x V_x - \rho V\right) + \frac{1}{l} y^*(W_l-V). \end{aligned}$$

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

$$\begin{aligned} c(\kappa _b \hat{d} l+\kappa _y l) \approx c(0) + (\kappa _b \hat{d} + \kappa _y) c^{\prime }(0)l = (\kappa _b \hat{d} + \kappa _y)l. \end{aligned}$$

The development payoff is therefore

$$\begin{aligned} l \left( \frac{z(\hat{d},1) x\psi (s)}{\rho -\mu } - (\kappa _b \hat{d} + \kappa _y) \right) , \end{aligned}$$

so that the price of land approaches

$$\begin{aligned} \frac{z(\hat{d},1) x\psi (s)}{\rho -\mu } - (\kappa _b \hat{d} + \kappa _y) \end{aligned}$$

as the stock of undeveloped land runs out. In contrast, overall welfare approaches

$$\begin{aligned} l \left( \frac{z(\hat{d},1) x\psi (s)}{\rho -\mu } - (\kappa _b \hat{d} + \kappa _y) \right) + \frac{x\Psi (s)}{\rho -\mu }. \end{aligned}$$

Differentiating this expression with respect to l implies that $W_l$ approaches

$$\begin{aligned} \frac{z(\hat{d},1) x\psi (s)}{\rho -\mu } - (\kappa _b \hat{d} + \kappa _y) \end{aligned}$$

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

$$\begin{aligned} W_s\le & {} \frac{W_l + (\kappa _y + \kappa _b d^*) c^{\prime }(0)}{z(d^*,1)} \\< & {} \frac{W_l + (\kappa _y + \kappa _b d^*) c'(\kappa _b b^* + \kappa _y y^*)}{z(d^*,1)} \\= & {} (\phi _b (d^*)^{-\varepsilon } + \phi _y)^{1/\varepsilon } W_l + (\phi _b (d^*)^{-\varepsilon } + \phi _y)^{1/\varepsilon } (\kappa _y + \kappa _b d^*) c^{\prime }(\kappa _b b^* + \kappa _y y^*). \end{aligned}$$

On the other hand, if development is occurring then the equation for $y^*$ implies that

$$\begin{aligned} \frac{\phi _b (d^*)^{-\varepsilon }}{\phi _b (d^*)^{-\varepsilon } + \phi _y} W_s = \kappa _b d^* c^{\prime }(\kappa _y y^* + \kappa _b b^*) \left( \phi _b (d^*)^{-\varepsilon } + \phi _y \right) ^{1/\varepsilon }. \end{aligned}$$

(A.3)

Equation 9 implies that

$$\begin{aligned} \kappa _b \phi _y (d^*)^{1+\varepsilon } W_s= & {} \kappa _y \phi _b W_s + (d^*)^{1+\varepsilon } \left( \phi _y+\phi _b (d^*)^{-\varepsilon } \right) ^{\frac{1+\varepsilon }{\varepsilon }} \kappa _b W_l \\= & {} \kappa _y (d^*)^{1+\varepsilon } \kappa _bc^{\prime }(\kappa _y y^* + \kappa _b b^*) \left( \phi _b (d^*)^{-\varepsilon } + \phi _y \right) ^{\frac{1+\varepsilon }{\varepsilon }} \\&+ (d^*)^{1+\varepsilon } \left( \phi _y+\phi _b (d^*)^{-\varepsilon } \right) ^{\frac{1+\varepsilon }{\varepsilon }} \kappa _b W_l, \end{aligned}$$

which implies that

$$\begin{aligned} \frac{\phi _y}{\phi _b (d^*)^{-\varepsilon } + \phi _y} W_s = \kappa _y c^{\prime }(\kappa _y y^* + \kappa _b b^*) \left( \phi _b (d^*)^{-\varepsilon } + \phi _y \right) ^{1/\varepsilon } + \left( \phi _b (d^*)^{-\varepsilon } + \phi _y \right) ^{1/\varepsilon } W_l. \end{aligned}$$

(A.4)

Adding Eqs. A.3 and A.4 together shows that

$$\begin{aligned} W_s = (\phi _b (d^*)^{-\varepsilon } + \phi _y)^{1/\varepsilon } W_l + (\phi _b (d^*)^{-\varepsilon } + \phi _y)^{1/\varepsilon } (\kappa _y + \kappa _b d^*) c^{\prime }(\kappa _b b^* + \kappa _y y^*). \end{aligned}$$

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

$$\begin{aligned} \hat{w}_b = \kappa _b c'(\hat{b} \kappa _b + \hat{y} \kappa _y) \;\;\; \text {and} \;\;\; \hat{w}_y = \kappa _y c^{\prime }(\hat{b} \kappa _b + \hat{y} \kappa _y), \end{aligned}$$

(B.1)

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

$$\begin{aligned} c(\hat{b} \kappa _b + \hat{y} \kappa _y) = \gamma \cdot (\hat{b} \kappa _b + \hat{y} \kappa _y)c^{\prime }(\hat{b} \kappa _b + \hat{y} \kappa _y). \end{aligned}$$

These requirements uniquely determine the cost of development in the model, which is given by

$$\begin{aligned} c(u) = u + \frac{1-\gamma }{(2\gamma -1)^2 (\hat{b} \hat{w}_b+ \hat{y}\hat{w}_y)} u^2, \end{aligned}$$

with

$$\begin{aligned} \kappa _b = (2\gamma -1)\hat{w}_b \;\;\; \text {and} \;\;\; \kappa _y = (2\gamma -1)\hat{w}_y. \end{aligned}$$

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

$$\begin{aligned} \hat{n} = z(\hat{b},\hat{y}) = (\phi _b \hat{b}^{-\varepsilon } + \phi _y \hat{y}^{-\varepsilon })^{-1/\varepsilon }, \end{aligned}$$

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

$$\begin{aligned} \frac{\hat{b}}{\hat{y}} = \frac{b^*}{y^*} = \left( \frac{\phi _y}{\phi _b} \cdot \frac{\kappa _b c'(\kappa _b b^* + \kappa _y y^*)}{W_l + \kappa _y c'(\kappa _b b^* + \kappa _y y^*)}\right) ^{-1/(1+\varepsilon )} = \left( \frac{\phi _y}{\phi _b} \cdot \frac{\hat{w}_b}{\hat{p}_l + \hat{w}_y}\right) ^{-1/(1+\varepsilon )}. \end{aligned}$$

Solving these two conditions simultaneously for $\phi _b$ and $\phi _y$ shows that

$$\begin{aligned} \phi _b = \frac{\hat{b}\hat{w}_b}{\hat{b}\hat{w}_b + \hat{y}(\hat{p}_l + \hat{w}_y)} \left( \frac{\hat{b}}{\hat{n}}\right) ^{\varepsilon } \end{aligned}$$

and

$$\begin{aligned} \phi _y =\frac{\hat{y}(\hat{p}_l + \hat{w}_y)}{\hat{b}\hat{w}_b + \hat{y}(\hat{p}_l + \hat{w}_y)} \left( \frac{\hat{y}}{\hat{n}}\right) ^{\varepsilon }. \end{aligned}$$

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

$$\begin{aligned} 0 = \frac{1}{2} \sigma ^2 r^2 h_{rr} - yh_l + (\mu - \alpha z(b,y)) rh_r - \rho h + r. \end{aligned}$$

(C.1)

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

$$\begin{aligned} 0 = \frac{1}{2} \sigma ^2 r^2 u_{rr} - yu_l + (\mu - \alpha z(b,y)) ru_r - \rho u + \frac{rz(b,y)}{\rho -\mu } - c(\kappa _b b + \kappa _y y). \end{aligned}$$

(C.2)

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

$$\begin{aligned} 0 = \frac{1}{2} \sigma ^2 r^2 f_{rr} - yf_l + (\mu - \alpha z(b,y)) rf_r - \rho f + z(b,y)h- c(\kappa _b b + \kappa _y y). \end{aligned}$$

(C.3)

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

$$\begin{aligned} \left( \frac{r}{\rho -\mu } - \alpha ru_r \right) z(b,y) - \left( yu_l + c(\kappa _b b + \kappa _y y)\right) \end{aligned}$$

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

$$\begin{aligned} \left( h - \alpha rf_r \right) z(b,y) - \left( yf_l + c(\kappa _b b + \kappa _y y)\right) \end{aligned}$$

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 $l_1=0$.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.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.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.