A Housing Supply Absorption Rate Equation

Abstract

What is the optimal rate of new housing supply? We answer this question with a simple model of new housing supply where the choice variable is the rate of new housing lot sales. This model is informed by the cost-side assumptions of the static equilibrium model but allows for demand for home-buying to vary over time. It differs from static models of housing production equilibrium by assuming that landowners hold land assets that are sold in asset markets to create new supply. Landowners maximise the present value of their balance sheet by choosing a rate of new housing lot sales, accounting for the effect on asset price growth from their sales in a housing market of finite depth. The resulting absorption rate equation has radically different parameter effects compared to the popular static housing density model. Constraints on density, for example, increase the optimal rate of supply by reducing the return to delaying development. Interest rates, land value tax rates, and demand growth, positively relate to the optimal rate of supply. The policy lessons are (1) the relationship between demand growth and the optimal supply rate limits the ability for market supply to reduce prices, and (2) increasing the cost to delaying housing development is the primary way to increase the market rate of housing supply.

Appendix: : Representative Housing Owner Absorption Rate

A more general version of this absorption rate approach assumes the representative agent not only supplies new housing lots but owns all housing once developed. In this model environment the dynamic maximisation problem is as per (14).

$$ \max\limits_{q_{t}} {\int}_{0}^{\infty} e^{-i t} \text{Return}(Q_{t},q_{t}) dt. $$

(14)

Notably, the effect of land holding taxes is removed from the discounting process as the representative agent pays these taxes with or without housing on the site and it therefore does not enter the timing decision. Here, the housing stock, Q_t, is the state variable that evolves subject to new dwelling supply, q_t, and the dwelling value, P_t, is the capitalised value of the housing rent resulting from that housing stock at a fixed capitalisation rate, i.

The two economic returns of interest for solving this maximisation problem are for (1) converting a lot into housing, and (2) retaining a housing lot undeveloped. The return from converting a lot into housing is the rent minus the interest on development cost, plus the dwelling price growth (the capital gain), minus any land rents from lower value uses on the site that must be given up. We show this in the first term of Eq. 15 which shows the problem redefined as an optimisation of the instantaneous rate of new supply according to Bellman’s maximisation principle. The economic return from delaying converting a lot into housing is the capital gain of that land, which is the dwelling price gain scaled by ω.

$$ \text{Return}_{t} =\underbrace{q_{t}(p_{t} + \dot{P}_{t}-C_{FC} i - r_{L})}_{\text{Net dwelling return}} + \underbrace{q_{t}(\omega \dot{P}_{t}(q_{t}))}_{\text{Land capital gain}} $$

(15)

To determine housing rents we borrow directly from the static model where housing rents are determined by incomes, housing preferences, and housing stock, such that p_t = αy_t/Q_t. Price growth forms part of the return to owning housing and in this setup is the capitalised time derivative of rents. The cost of development is scaled so that at the current optimal density C_FC is the per dwelling average cost, meaning that in annual flow terms the development costs is C_FCi. The forgone rents from lower value uses are r_L. The return to delay is represented by the capital gains that occur on land in the absence of development, which in this case is the capitalised change in rent (the capital gain) of the dwelling scaled by the diseconomies of density relationship ω to capture the change in optimal density with price. The complete problem setup is in Eq. 16.

$$ \text{Return}_{t} =\max\limits_{q_{t}} \left\{q_{t} \underbrace{\left( \frac{\alpha y_{t}}{Q_{t}} + \frac{\frac{\alpha \dot{y}_{t}}{Q_{t}}-\frac{\alpha y_{t} q_{t}}{{Q_{t}^{2}}}}{i}- C_{FC} i -r_{L}\right)}_{\text{Net rent plus cap. gain (now)}} + q_{t}\underbrace{\left( \omega \frac{\frac{\alpha \dot{y}_{t}}{Q_{t}}-\frac{\alpha y_{t} q_{t}}{{Q_{t}^{2}}}}{i}\right)}_{\text{Scaled cap. gain (later)}} \right\} $$

(16)

The problem is solved by taking the derivative of the return with respect to q_t and setting to zero, giving

$$ \frac{\alpha y_{t}}{Q_{t}} - \frac{2 \alpha y_{t} q_{t}}{i {Q_{t}^{2}}} - C_{FC} - r_{L} + \frac{\omega \alpha \dot{y}_{t}}{i Q_{t}} - \frac{2 \omega \alpha y_{t} q_{t}}{i {Q_{t}^{2}}}=0 $$

then solving for q_t. Second order conditions for a maximum are confirmed by the second derivative result of $-\frac {(1+\omega ) 2 \alpha y_{t}}{i {Q_{t}^{2}}}$. Solving for q_t and rearranging provides the solution for the equilibrium new housing supply in Eq. 17.

$$ q^{*}_{t} = \underbrace{Q}_{\text{Stock}} \frac{1}{2} \left( \underbrace{\frac{\dot{y}_{t}}{y_{t}}}_{\text{Income growth rate}} + \underbrace{\frac{i}{(1+\omega)}}_{\text{Intertemporal cost}} \left( \underbrace{1-\frac{Q(C_{FC}i+r_{L})}{\alpha y_{t}}}_{\text{Land rent share of gross rent}} \right) \right) $$

(17)

The first term in parentheses is the income growth rate. The second term is the product of the land rent share of gross dwelling rent (which is bound by 0 and 1 for profitable developments) and the intertemporal cost of earning that land rent which is the interest rate augmented by the efficiency of density, ω.

Substituting our optimal supply, $q_{t}^{*}$ for q provides the following housing rental growth rate (and hence price growth rate when the capitalisation rate is fixed) with the equilibrium rate new supply.

$$ \frac{\dot{r}_{t}}{r_{t}} =g^{*}_{t} = \frac{1}{2} \left( \frac{\dot{y}_{t}}{y_{t}} + \frac{i}{(1+\omega)} \left( \frac{Q(C_{FC}i+r_{L})}{\alpha y_{t}}-1\right)\right) $$

(18)

We know that $\frac {Q(C_{FC}i+r_{L})}{\alpha y_{t}}-1<0$ for any feasible development. Hence the rental growth rate will be less than half the rate of demand growth $\frac {\dot y_{t}}{y_{t}}$ in this dynamic equilibrium.

How much less depends on a number of factors. An ability to increase density cheaper, or a higher ω, will increase the price growth rate, as well a higher ratio of development costs in flow terms compared to the housing rent. The effect of interest rates depends on the relative effect on the cost of borrowing for development (in the C_FCi term) and its effect on inter-temporal optimisation (in the i/(1 + ω) term).

The main difference between the representative dwelling owner agent problem and the agent selling housing lots is the effect of the net land rent. In this formulation, the land rent from its use as housing is forgone when delaying development rather than the interest on land value. In this problem setup, reductions in the fixed cost of housing development increase net land rents from developing housing and accelerate supply. The same effect can be achieved by reducing rental income from lower value uses, or increasing holding costs in a way that reduces the net rental income while delaying housing development. In places where built-to-rent landlords dominate the housing market, these effects may be important.

Another difference is that the own-price effect from developing housing comes via the housing rental market rather than the housing asset market, i.e. α reflects the sensitivity of rent to new rental supply, while a represents the sensitivity of price to new housing lot sales. Lastly, the demand growth effect in this model comes via population and/or income growth, which translates into rental housing demand.