A Reexamination of Housing Price and Household Consumption in China: The Dual Role of Housing Consumption and Housing Investment

Abstract

It is important to investigate the correlation between housing price and household consumption to gain an understanding of the behavior of the economy and effectively handle the consequences of economic development. In the last two decades, the accumulation of housing wealth by Chinese households has not been effectively transmitted to their final consumption. We discovered that the sustained increase in household wealth and housing-ownership rate in China has been accompanied by a decrease in consumption rate. We also identified a negative correlation between housing price and household consumption for both the homeowners who own one housing unit and those who own two units of housing. We investigated this phenomenon in China both theoretically and empirically by capturing the dual nature of housing as a consumption good and an investment vehicle. We found that the demand for second housing units is motivated by increasing housing consumption demand rather than pure investment needs. To explain the mechanisms that drive household-consumption behavior, we also explored the effects on household consumption of China’s educational system, marriage market and ageing society, as well as future housing-market uncertainty. The implications of government intervention in the housing market are discussed.

Appendices

Appendix 1

Derivation of the model

In our model, the utility function at time t of a household with life length T (T > 0) is U(C_t,H _t ), which is a time-separable and time-invariant utility function describing the household’s preference for housing services (H _t ) relative to non-housing goods (C _t ). We assume that no bequest motive exists, and that the household consumes all of its remaining wealth in period T. The optimal allocation of consumption and housing is given by the Bellman equation shown below, in which the household arrives at the maximum level of current utility plus the expected discounted value in period t + 1 as a function of future consumption and housing price:

$$ {\max}_{C_t,{H}_{ct},{H}_{it},{S}_t}\left\{{U}_t\left({C}_t,{H}_{ct}\right)+E{V}_{t+1}\left({W}_{t+1}\right)\right\} $$

(1)

where E is expectation and W_t + 1 is total wealth, including housing wealth, in the period t + 1. H _ct is the unit of housing consumption, H _it is the unit of housing investment and H _t denotes the total housing owned. To simplify the model, we define only one financial asset, S _t .

Meanwhile, we assume that the household’s expected housing price in the next stage will reach P _t (1 + u _h ), where P _t is housing price in period t and u _h is growth rate. The financial asset S _t receives u _p per period in profit. The period to period budget constraint incorporates differences in housing-asset value, as follows: P _t (H _t  − H_t − 1). We assume that the household can rent out separate housing investments and obtain rent to optimize household profit, as shown in R _t H _it , where R _t is the rent price per housing unit in period t (Henderson and Ioannides 1983).

Therefore, the budget constraint can be represented as follows:

$$ {R}_{t-1}{H}_{\left(t-1\right)}+{Y}_t+{P}_t{H}_{t-1}+{S}_{t-1}\left(1+{u}_p\right)={C}_t+{P}_t{H}_t+{S}_t $$

(2)

$$ {W}_{t+1}={Y}_{t+1}+{S}_t\left(1+{u}_p\right)+{P}_t\left(1+{u}_h\right){H}_t+{R}_t{H}_{it} $$

(3)

$$ {H}_t={H}_{ct}+{H}_{it} $$

(4)

where Y _t is total income and C _t is non-housing consumption.

Homeowners who can make separate decisions regarding housing consumption and investment are motivated to consume housing and invest in housing by two distinct mechanisms. These mechanisms can be incorporated into our model. We use the Lagrangian-multiplier method to obtain the first-order extreme conditions, as follows.

$$ -{U}_1+\left(1+{u}_p\right)E\left({V}_{t+1}^{\prime}\right)=0 $$

(5)

$$ -{P}_t{U}_1+{U}_2+E\left[{P}_t\left(1+{u}_h\right){V}_{t+1}^{\prime}\right]=0 $$

(6)

$$ -{P}_t{U}_1+E\left[\left({P}_t\left(1+{u}_h\right)+{R}_t\right){V}_{t+1}^{\prime}\right]=0 $$

(7)

Thus, the equation determining H _It can be written as follows.

$$ {P}_t\left(1+{u}_p\right)E\left({V}_{t+1}^{\prime}\right)=\left({P}_t+{R}_t\right)E\left({V}_{t+1}^{\prime}\right)+E\left[\left({P}_t{u}_h\right){V}_{t+1}^{\prime}\right] $$

$$ {P}_t{u}_p={R}_t+\frac{E\left[{V}_{t+1}^{\prime}\left({P}_t+{u}_h\right)\right]}{E\left({V}_{t+1}^{\prime}\right)} $$

Using the Taylor expansion, we can write ${V}_{t+1}^{\prime }$ as the following approximate equation.

$$ {V}_{t+1}^{\prime}\approx \overline{V_{t+1}^{\prime }}+\overline{V_{t+1}^{\prime \prime }}{P}_t{H}_{it}\left({u}_h-\overline{u_h}\right) $$

We can then obtain the following.

$$ {H}_{it}=\frac{E\left[{P}_t\left({u}_h-{u}_p\right)+{R}_t\right]}{P_t^2}\times \frac{1}{\mathit{\operatorname{var}}\left({u}_h\right)}\times \frac{1}{A} $$

(8)

var(u _h ) is the variance in the expected growth rate of housing price, and indicates individuals’ confidence in the price trend as well as the risks associated with housing investment. A =  − V^′′/V^′, which denotes an individual’s relative risk aversion.

To compare housing-investment decisions with housing-consumption decisions, we rewrite Eq. (7) as follows.

$$ {P}_t{U}_1=E\left[{P}_t\left(1+{u}_h\right){V}_{t+1}^{\prime}\right]+{R}_tE\left({V}_{t+1}^{\prime}\right) $$

Combining the above equation with Eqs. (5) and (6) gives the relationship between the marginal utility of housing consumption and that of other consumption.

$$ \frac{R_t{U}_1}{1+{u}_p}={U}_2 $$

The equation above indicates that the marginal utility of housing consumption and the marginal utility of non-housing consumption are related solely to user cost, assuming that the utility function of households’ consumption is written as follows,

$$ \mathrm{U}\left({c}_t,{H}_{ct}\right)=\frac{{\left({c}_t^{\omega }{H}_{ct}^{1-\omega}\right)}^{1-\gamma }}{1-\gamma } $$

where, ω measures the household’s preference for non-housing consumption relative to housing consumption, and γ is the coefficient of relative risk aversion over the entire consumption basket.

We know that the household’s consumption structure is determined by the following principle:

$$ \frac{H_{ct}}{c_t}=\frac{1-\omega }{\omega}\bullet \frac{1+{u}_p}{R_t} $$

(9)

which suggests that the household’s housing-consumption decisions differ from its housing-investment decisions. Here, ρ is the discount rate of P _t .

$$ \frac{C_t}{H_{ct}}=\frac{\omega }{1-\omega}\rho {P}_t $$

To address the case of a homeowner for whom housing investment and housing consumption are indivisible, we follow Sebastien’s (2010) general procedure. We assume that the homeowner’s mean-variance utility is as follow:

$$ {U}^{MV}\left({\mu}_w,{\sigma}_w\right)={\mu}_w-\frac{\gamma }{2}{\sigma}_w^2 $$

where, μ _w denotes expected future wealth, γ is the risk-aversion constant and σ _w is the standard deviation of future wealth.

If the movements of stock and housing are uncorrelated, we can write:

$$ {\mu}_w=r+{\alpha}_s\left({\mu}_s-r\right)+{\alpha}_H\left({\mu}_H+\rho -r\right) $$

$$ {\sigma}_w^2={\alpha}_H^2{\sigma}_H^2+{\alpha}_s^2{\sigma}_s^2 $$

where, σ _i is the standard deviation of asset i (which may be housing asset H, financial asset S or future wealth W), α _i is the portfolio share and r is the interest rate.

We further obtain that:

$$ {\mu}_w=r+{\lambda}_s\sqrt{\sigma_w^2-{\sigma}_H^2{\alpha}_H^2}+{\alpha}_H\left({\mu}_H+\rho -r\right) $$

(10)

where λ _s is the Sharpe ratio of asset S.

Maximizing the utility function under the budget constraint determined in Eq. (10), we obtain:

$$ {\sigma}_w^2={\upgamma}^{-2}{\lambda}_s^2+{\alpha}_H^2{\sigma}_H^2 $$

$$ {\mu}_w=r+{\gamma}^{-1}{\lambda}_s^2+{\alpha}_H\left({\mu}_H+\rho -r\right) $$

And further,

$$ {\mu}_w=r+{\gamma}^{-1}{\lambda}_s^2+{\lambda}_H\sqrt{\sigma_w^2-{\gamma}^{-2}{\lambda}_s^2} $$

Substituting all of these results for the homeowner’s utility gives the marginal utility of the ratio of housing.

$$ {V}^{MV}=r+{\gamma}^{-1}{\lambda}_s^2+{\alpha}_H\left({\mu}_H+\rho -r\right)-\frac{\gamma }{2}\left({\upgamma}^{-2}{\lambda}_s^2+{\alpha}_H^2{\sigma}_H^2\right) $$

It is then easy to obtain the opportunity cost, as follows:

$$ {V}_{\alpha H}^{MV}=\left({\mu}_H+\rho -r\right)-\gamma {\sigma}_H^2{\alpha}_H $$

(11)

where

$$ {\alpha}_H^{\ast }=\frac{\mu_H+\rho -r}{\gamma {\sigma}_H^2} $$

(12)

As discussed in the text, opportunity cost influences the consumption decisions of homeowners for whom the intention to invest in housing is inseparable from the intention to consume housing.

We substitute Eq. (11) for Eq. (12) to give Eq. (3.3) in the text as follows:

$$ \frac{c_t}{H_t}=\frac{\omega }{1-\omega }{P}_t\left[\rho +A\mathit{\operatorname{var}}{\left({\mu}_{h,t}\right)}^2\left({\alpha}_{h,t}-{\alpha}_{h,t}^{\ast}\right)\right] $$

Appendix 2

Key Variables in the Empirical Tests

Table 6 Description of key variables in the empirical study