Household Housing Demand: Empirical Analysis and Theoretical Reconciliation

Abstract

Since owner-occupied housing is partly a financial asset, expectations of capital gain or loss play a role in housing demand. In recent years, some “hot” housing markets have exhibited an increase in demand when housing prices rise and a decrease when they fall, suggesting the presence of capital gains forces that outweigh the traditional neoclassical demand response associated with the standard consumer good. To explore whether this behavior is systematic, we estimate individual household housing demand equations for two large and geographically diverse metropolitan areas, San Francisco and Atlanta. The data base consists of forty nine Public Use Microdata Area samples. The econometric results indicate that own-housing demand is downward sloping in one market but upward sloping in the other. These disparate results are reconciled by showing that they are consistent with two different and explicit special case predictions of the same theoretical model of housing demand and reflect the differing relative strengths of a standard consumption good demand response and of an asset based capital gains effect.

Technical Appendix

In general, the consumer’s utility function is given by

$$ U = U\left( {{x_1},{ }h_1^\circ, { }{x_2},{ }h_2^r} \right) $$

(A1)

and his expectations about the uncertain second period prices are depicted by the conditional subjective probability distribution $F\left( {{p_2}|{p_1}} \right)$.²¹ In the light of the consumer’s utility function and expectation formulations, expected utility is defined as

$$ {V_1}\left( {{x_1},h_1^\circ, {p_1}} \right) = \mathop\int \nolimits_{{p_2}} U\left( {\phi \left( {{x_1},h_1^\circ, .} \right)} \right){\hbox{d}}F\left( {.|{p_1}} \right) $$

(A2)

where V ₁(.) is the von Neumann-Morgenstern expected utility function of the action $\left( {{x_1},h_1^\circ } \right)$ if p ₁ is quoted in period 1. It is assumed that the consumer maximizes expected utility, so that the optimization problem becomes maximizing (A2) subject to the first period budget constraint.²² In the usual manner, the first order conditions can be solved for the consumption good demand functions and the demand function for owner-occupied housing, and the properties of the demand functions can be determined from the respective Slutsky equations that are deriveable from a total perturbation of the first order conditions.

The Slutsky equations for the own-demands are given by

$$ \frac{{\partial {q_i}}}{{\partial {p_i}}} = - {q_i}\frac{{\partial {q_i}}}{{\partial Y}} + \lambda \frac{{{D_{i,i}}}}{D} - \left[ {\mathop \sum \limits_{j = 1}^n {V_{{q_j}{p_i}}}\left( . \right)\frac{{{D_{j,i}}}}{D} + {V_{{h^\circ }{p_i}}}\left( . \right)\frac{{{D_{n + 1,i}}}}{D}} \right],\,\,\forall i = 1,...,n + 1 $$

(A3)

where $i = 1,...,n$ refers to n consumer goods and $i = n + 1$ refers to owner-occupied housing services, ${V_{{q_j}{p_i}}}\left( . \right) = \frac{{{\partial^2}V\left( . \right)}}{{\partial {q_j}\partial {p_i}}}$, ${V_{{h^\circ }{p_i}}}\left( . \right) = \frac{{{\partial^2}V\left( . \right)}}{{\partial {h^o}\partial {p_i}}}$, D is the determinant of the Jacobian of the first order conditions, D _j,i is the cofactor of the element in row j, column i, λ is the Lagrange multiplier in the maximization of (A2) subject to the first period budget constraint. For convenience, we have dropped the time subscript with the understanding that all variables are for period 1. The left-hand side of (A3) is the uncompensated (i.e., the Marshalian) price effect. The first and the second terms on the right-hand side of (A3) consist of the weighted income effect and the substitution effect, respectively, common to neoclassical theory. They represent the effect of a change in own-price on the optimal value of own-demand through the first period budget constraint. The term in squared brackets on the right-hand side of (A3) is the effect of a change in own-price on the optimal value of own-demand by altering the objective function (i.e., the expected utility function in (A2)), which we designate as the expectation effect.

We now modify the model for the purpose of establishing a clear empirical distinction between housing demand and that of the standard consumer goods. We seek to reinstate the traditional demand properties for the standard goods while preserving the possibility that the housing own-demand response to changing housing prices could be positive or negative. The key modifications embody the assumptions that the underlying utility function is intertemporally separable, a restriction common to many applied fields, and that the consumer employs a general forecasting scheme with additive error term. Starting with the von Neumann-Morgenstern expected utility function in (A2), recalling that ${x_2} = {x_2}\left( {h_1^o,{p_2}} \right)$ and $h_2^r = h_2^r\left( {h_1^o,{p_2}} \right)$ and introducing the two simplifications, the expected utility function becomes

$$ {U^1}\left( {{x_1},h_1^o} \right) + \mathop \int\nolimits_{{ \in_1}} \mathcal{W}\left( {h_1^o,g\left( {{p_1}} \right) + { \in_1}} \right)f\left( {{ \in_1}} \right){\hbox{d}}{ \in_1} $$

(A4)

where ϵ ₁ is the random variable. We next use Taylor’s theorem and expand $\mathcal{W}\left( . \right)$ around g(p ₁) regarding p ₂ as a unique random variable while treating $h_1^o$ as exogenous and substitute the resulting expression in (A4). This yields

$$ {U^1}\left( {{x_1},h_1^o} \right) + \mathcal{W}\left( {h_1^o,g\left( {{p_1}} \right)} \right) + \mathop \sum \limits_{i = 1}^{n - 1} \frac{{{\mathcal{W}^i}\left( {h_1^o,g\left( {{p_1}} \right)} \right)}}{{i!}}M_{{ \in_1}}^i + \frac{{{\mathcal{W}^n}\left( {h_1^o,{\delta_1}} \right)}}{{n!}}M_{{ \in_1}}^n $$

(A5)

where $\mathop \int \nolimits_{{ \in_1}} \in_1^if\left( {{ \in_1}} \right){\hbox{d}}{ \in_1} = M_{{ \in_1}}^i$ is the ith moment of the random variable ϵ ₁. Noting that δ ₁ is some constant between p ₂ and g(p ₁) and that all moments are some constants, we have

$$ {U^1}\left( {{x_1},h_1^o} \right) + Z\left( {h_1^o,{p_1}} \right) + \Gamma \left( {h_1^o} \right). $$

(A6)

We now allow for a partially separable real wealth effect, which additively separates consumer good prices from the middle term, to arrive at

$$ {U^1}\left( {{x_1},h_1^o} \right) + g\left( {h_1^o,{p_{ \circ 1}},{p_{r1}}} \right) + h\left( {{p_{c1}}} \right) + \Gamma \left( {h_1^o} \right). $$

(A7)

As a consequence of the specification in (A7), the Slutsky expressions for the consumption good own-demand curves and the housing own-demand curve given in (A3) reduce to

$$ \frac{{\partial {x_i}}}{{\partial {p_i}}} = - {x_i}\frac{{\partial {x_i}}}{{\partial Y}} + \lambda \frac{{{D_{i,i}}}}{D},{ }\forall i = 1,...,n $$

(A8)

and

$$ \frac{{\partial {h^o}}}{{\partial {p_\circ }}} = - {h^o}\frac{{\partial {h^o}}}{{\partial Y}} + \left( {\lambda - {V_{{h^o}{p_o}}}} \right)\frac{{{D_{n + 1,n + 1}}}}{D}, $$

(A9)

respectively.