A Dynamic Model of Housing Wealth Effect: Based on the Diversity of Wealth Expectations

2.1 Improvements to the consumption function

According to the hypothesis of Hall[1] that, people always maximize their whole-life utility, we make a model as follows:

As by Hall[1], the following notation is used throughout the paper:

E = mathematical expectation conditional on all information available;

δ = rate of subjective time preference, which changes with time and different person;

r = real interest rate, which changes over time;

T = length of economic life;

u (⋅) = one-period utility function, strictly concave;

c = consumption;

w = income;

A = wealth assets apart from human capital;

In order to find the optimal solution, we build a Lagrangian equation:

Then, we take partial derivatives of c_t, c_{t + 1}, ⋯, c_T respectively to solve the Lagrangian equation and get the following equations:

Solving the first order condition, we can get:

Here we assume that πi=(1+rt+i1+δt+i)−i,thenEtu′ct+τ=πτπτ−1u′ct+τ−1, and we can easily get that

In order to get the optimal analytical solution, we borrow ideas from Hall[1], and assume that the utility function is a quadratic equation which can be expressed as uct=−12c¯−ct2, where c is the bliss level of consumption. Then we go on to solve the problem using the utility function, and get that:

Taking equation (7) and (8) into equation (6), we get equation (9):

The equation (9) means that c − c_{t + τ} = π_t (c − c_t) + ε_t.

By this time, we get the formula of c_t+τ, which is

Equation (11) is obtained when equation (10) is taken into constraint condition ∑τ=0T−t(1+rt+τ)−τct+τ−wt+τ=At:

Assuming T → ∞, we get consumers’ optimization:

Equation (12) is the new consumption function we get. We can easily find that people’s consumption depend on their current wealth and lifetime income.

The implications of our new result are presented in one conclusion and one corollary:

Conclusion 1: Coefficient of wealth in the consumption function is a variable one. Current consumption is under the influence of the expected rate of subjective time preference and expected real interest rate.

2.2 Dynamic model of housing wealth effect

Here the current wealth is defined only as the housing wealth H_t. The dynamic model of housing wealth effect is obtained:

In our equation, the level of housing wealth effect is measured by f0rte,δte. By taking logarithms of variables, we can highlight the parameters’ economic significance and better analyze the relationships between the parameters. Assuming ln c_t = cons_t, ln H_t = fpri_t, ln w_t = inco_t, we take logarithms of both sides of equation (14), and get a new equation:

Dividing both sides of equation (15) by fpri_t, we get

And then housing wealth effect frte,δte can be expressed as:

According to equation (17), we get another conclusion:

Conclusion 2: The housing price-to-income ratio (fpritincot) is a crucial factor in housing wealth effect.

2.3 Dynamic models of wealth effect for the rich and the poor

Since people with different incomes have different consumer behaviors, we clarify people into two groups according to their income. Then the housing wealth effect model (15) is transformed into two equations:

Former studies related with housing wealth effect equal to (18) plus (19), which hide many problems. For the expectation differences, addition makes lots of opposite effects cancel out. As a result, we should pay more attention to (18) minus (19). In this way, we can find some hidden problems, take economic inequality as an example.

According to the Permanent Income Hypothesis, the Life-cycle Hypothesis and our analysis, under the rationality assumption, people tend to smooth their current consumption based on their lifetime income, and changes in current income have little influence on consumption. So we assume that: g1r1te,δ1te=g2r2te,δ2te=grte,δte.

Then equation (18) and (19) can be written as:

Equation (22) is obtained by subtracting equation (21) from equation (20):

Here, Δinco_t = inco_1t − inco_2t stands for the gap of wealth, we can see clearly that gap of wealth is greatly influenced by wealth effect when we make some subtractions. Then we get conclusion 3 according to equation (22):

Conclusion 3: The relationship between the gap of wealth and housing price depends on the housing wealth effect difference, f2r2te,δ2te−f1r1te,δ1te.

3.1 Dynamic model of housing wealth effect and some extended analysis

According to consumption function equation (12)

Coefficient of wealth in the consumption function is a variable value. The unstable coefficients are decided by π_t and ∑τ=0∞1+rt+τ−τ. That is to say, the consumption coefficient of period t depends on the rate of subjective time preference of period 2t (δ_2t) and the lifetime real interest rate. In conclusion, we can say that current consumption is under the influence of the expected rate of subjective time preference and expected real interest rate.

Extending the housing wealth effect model, we get another form of the housing wealth effect equation:

1. If frte,δte>0, the housing wealth effect is positive and it increases or decreases along with the increase or decrease of the housing price-to-income ratio. So when income is a constant and the housing price is low, a proper increase in housing price is good for stimulating consumption and enlarging domestic demand. While when the housing price is high, it is unwise to enlarge domestic demand by making use of the positive impact of housing wealth effect, because although a further increase in housing price can stimulate domestic consumption, it may also bury the seeds of the housing bubble.

2. If frte,δte<0, the housing wealth effect is negative, and it increases or decreases in inverse proportion to that of the housing price-to-income ratio. So when income is a constant and the housing price is low, a proper increase in housing price is good for reducing the negative impact of housing wealth effect and may help to relieve some of the “squeeze effect” of housing price on consumption. Similarly when the housing price is high, it is not appropriate to use housing wealth effect as an index to decide whether to raise the housing price.

3.2 Influence of economic inequality on housing wealth effect

According to economic inequality formula

Assume dt=f2r2te,δ2te−f1r1te,δ1te, then equation (22) is turned into

1. If d_t > 0, housing wealth effect of the poor is larger than that of the rich and the economic inequality rises or declines in direct proportion with the rise and decline of housing price. The relative wealth change of the poor is bigger, which makes them willing to spend more to satisfy their consumption demands. As for the rich, the rise of housing price does not make much difference. The relative wealth change is rather small and they do not have the impetus to increase their consumption. Under this circumstance, with a further increase of housing price, the gap between the poor and the rich is widen.

2. If d_t < 0, housing wealth effect of the rich is larger than that of the poor and the economic inequality increases or decreases in inverse proportion with that of the housing price. At this time, the poor are cautious about the economic situation and less willing to increase their consumption. In this way, with the further increase of the housing price, the gap between the rich and the poor is diminishing.

Our analysis shows that the inequality Δinco_t changes with housing price, and the relationship between economic inequality Δinco_t and housing price fpri_t is affected by the difference of housing wealth effect between the rich and the poor d_t. If d_t > 0, economic inequality and housing price change in the same direction; if d_t < 0, economic inequality and housing price change in opposite directions.