Housing Appreciation (Depreciation) and Owners’ Welfare: An Alternative View

Abstract

Based on a lifetime consideration, previous research has extended Frank’s graphical model with borrowing, property taxes and moving costs to analyse the welfare effects of housing appreciation and depreciation, and has shown that appreciation can make homeowners worse off, but homeowners’ welfare cannot decrease with depreciation. The latter is counter-intuitive, especially for owners with a high portion of mortgage in a housing slump. Alternatively, this paper treats property tax as an annual taxation and measures mortgage loan as a portion of house value when it was purchased. It is shown that if a house owner adjusts his/her house size in response to a price change, then with a low mortgage loan (or no mortgage), the owner will be better off only when there is a large appreciation or depreciation, while he/she cannot be better off under depreciation if he/she bought that house largely via a mortgage loan.

1. Introduction

In his microeconomics textbook, Frank (2006, pp. 166–167) shows that, starting from a fixed bundle of housing and a composite good, either an increase in the price of housing (appreciation) or a decrease in the price of housing (depreciation) makes an owner better off.¹ The intuition is as follows: a house owner can sell his/her house at a higher price after housing appreciation and use this revenue to purchase more composite goods with a constant price and thus raise his/her utility. On the other hand, after housing depreciation an owner can also sell his/her house and repurchase a larger house at a cheaper price. His/her utility is increased in the sense that he/she replaces the composite good with a cheaper good (housing). However, Frank’s (2006) model is too simple to capture some real-world factors such as property tax, mortgage loan and moving cost.

In a parallel study on this issue, Lai, McDonald and Merriman (2010) (LMM model, hereafter)² use a diagrammatic analysis³ and a lifetime span consideration to verify the correction of Frank’s (2006) model with moving cost and mortgage loan. In the case of housing appreciation, their conclusion is close to that of Frank (2006). In other words, an owner will be better off if this appreciation is large enough to compensate for the fixed cost (moving cost or mortgage loan), while the owner will be worse off after a small appreciation. In the case of housing depreciation, the result is quite different. First, if there is no mortgage loan, the owner’s utility is definitely increased, no matter whether the owner moves to another house or not. This is because the property tax liabilities fall. Secondly, if an owner still has some remaining mortgage balance, he/she will be better off by moving to another house with the optimal size if the depreciation is large. Even if the depreciation is small, by choosing to stay in the same house, the owner cannot be worse off due to the fall of property taxes liabilities. Based on these two points, the owner can never be worse off after housing depreciation. Is this right for the LMM model in reality?

The world economy encountered a financial crisis in 2008, which was initially caused by the collapse of the housing bubble in the US and many financial institutions holding mortgage or mortgage-backed securities being driven to near-bankruptcy. From July to December 2008, housing prices fell 27 per cent and 2.3 million foreclosures occurred (Mankiw, 2010). It is obvious that many house owners felt worse off during this financial crisis. Even now, the housing market has not returned to its normal track.

The LMM model is restricted by the diagrammatic analysis and some assumptions should be modified. For example, the lifetime property tax assumption in the LMM model can simplify the mathematical process, but it may not fit reality. In the real world, the property tax is an annual tax and is proportional to the housing value, and thus the tax payment can change from year to year due to housing price changes. Therefore, the composite good consumption is also affected by the property tax change. A crucial difference between this paper and the LMM model is that the LMM model considers the property tax to be included based on a lifetime framework, while this paper defines the wealth coming from a house as the total money that the owner can receive when the house is sold, which does not include tax. It is obvious that tax is paid to the government and cannot be counted as the owner’s wealth. Consequently, only when the price change is larger than the tax payment will an owner’s utility be raised.⁴

Moreover, in the LMM model, the remaining mortgage loan is assumed to be a constant, something like a constant moving cost, to accommodate to their analysis framework. However, in the real world, a mortgage loan is, in general, a proportion of the original housing value. In a housing slump, the actual housing value is evaluated at the current price, but the monthly repayment is still based on the original price. The owner therefore may have great difficulty changing to another house, because he/she needs to pay back the loan, but his/her house value is no longer as high. Based on the observation in the real world, this paper develops a mathematical model with more reasonable assumptions such that the property tax is counted on an annual basis and the mortgage loan is a portion of the original housing value. It will be shown that house owners with a high portion of mortgage will be worse off during a housing slump, instead of being better off as Frank (2006) and LMM (2010) claim.

The rest of the paper is organised as follows. Section 2 is the model; section 2.1 recapitulates Frank’s (2006) model; section 2.2 revises the model to add property taxes; section 2.3 is the general case which includes property taxes and mortgage loans; section 2.4 discusses the choice of staying in the same house; section 2.5 is an extension; and section 2.6 provides two numerical examples. Some concluding remarks are discussed in section 3.

2. The Model

Suppose there is a city where each resident should have one house in this city. In order to ease our analysis, a Cobb–Douglas utility function is employed hereafter as a standard example to depict the current model mathematically. Suppose each agent has a utility function $u = h^{α} \cdot c^{(1 - α)}$ , where h is the housing unit; c is the volume of a composite good (price normalised as one),⁵ and α is seen as the housing expenditure proportion and α < 1. The agent’s budget constraint is $w_{0} = p (1 + t - β) \cdot h + c$ , where w₀ represents the initial wealth; p is the per unit house price; t is the property tax rate; and β is the mortgage percentage of the housing value. Using a monotonic transformation of the utility function, such that $\hat{u} = \ln u = α \ln (h) + (1 - α) \ln (c)$ , simplifies the solving process.

Solving for the Lagrangian $L = \hat{u} + λ (w_{0} - p (1 + t - β) h - c)$ yields

\begin{matrix} h_{0} = \frac{α w_{0}}{p (1 + t - β)} \\ c_{0} = (1 - α) w_{0} \end{matrix}

(1)

The equilibrium indirect utility is then

\begin{matrix} v_{0} = h_{0}^{α} \cdot c_{0}^{(1 - α)} = α^{α} \cdot {(1 - α)}^{(1 - α)} \cdot w_{0} \\ \cdot {(\frac{1}{p (1 + t - β)})}^{α} \end{matrix}

(2)

Suppose the housing price now changes to p′. The new wealth is then $w' = p' h_{0} - β p h_{0} + c_{0}$ .⁶

This is because his/her current housing value is now p′h₀, while his/her mortgage is still βph₀. If the agent decides to change the house size, then he/she must first pay off the old mortgage loan and apply for a new mortgage loan.⁷ The Lagrangian is thus

L' = \hat{u} + λ (w' - p' (1 + t - β) h - c)

(3)

Solving for the Lagrangian yields

\begin{matrix} h' = \frac{α w'}{p' (1 + t - β)} \\ c' = (1 - α) w' \end{matrix}

(4)

The indirect utility function is therefore

\begin{matrix} v' = {h'}^{α} \cdot {c'}^{(1 - α)} = α^{α} \cdot {(1 - α)}^{(1 - α)} \cdot w' \\ \cdot {(\frac{1}{p' (1 + t - β)})}^{α} \end{matrix}

(5)

We should notice that the major difference between LMM’s assumptions and those in the current paper is the definition of wealth when housing prices change. Precisely speaking, in their model, the new wealth after housing prices change from p₀ to p′ is $w' = p' (1 + t) h_{0} + c_{0}$ , while in the current paper it is $w' = p' h_{0} + c_{0}$ . The reason for the current setting is because an owner expends $p_{0} (1 + t) h_{0}$ to buy his/her house, while he/she can only receive p′h₀ when the house is sold. In other words, the property tax is paid by the buyer, but never received by the seller. Due to this setting, Figures 2 and 4 in their paper show that when $p' / p_{0} = 1$ , the (indirect) utility ratio $v' / v_{0} = 1$ , while in our results, $v' / v_{0} < 1$ whenever $p' / p_{0} = 1$ . Another crucial difference between these two papers is that they treat the remaining mortgage loan as a constant cost (similar to a constant moving cost), while the current paper assumes that the mortgage loan is a percentage of the house value. Therefore, in their paper if the depreciation is large enough, then the owner is still better off. In the current paper, depreciation makes the owner with a high portion of mortgage loan worse off if he/she pays off the loan and then buys a new house, because the loan is evaluated at price p₀, while when he/she tries to sell a house, the house is evaluated at price p′ which is much less than p₀. His/her utility thus falls when the housing size is adjusted.

Note that the agent has another choice, which is staying in his/her current house. If he/she stays in the original house, it can be considered that he/she repurchases his/her house with the current conditions each year.⁸ The new wealth in this case is

w_{A} = c_{0} + p' h_{0} - β p h_{0}

(6)

where, the subscript A represents the case where the owners stay in their original houses. The Lagrangian in this case is thus

L_{A} = \hat{u} + λ (w_{A} - p' (1 + t - β) h_{0} - c)

(7)

where, $p' (1 + t - β) h_{0}$ means that the house owner repurchases his/her original house with the current conditions. Solving for the Lagrangian yields

c_{A} = c_{0} - tp' h_{0} + β h_{0} (p' - p)

(8)

The indirect utility function is then

\begin{matrix} v_{A} = h_{0}^{α} \cdot c_{A}^{(1 - α)} = {(\frac{α w_{0}}{p (1 + t - β)})}^{α} \\ {(c_{0} + (\frac{α w_{0}}{p (1 + t - β)}) (- tp' + β p' - β p))}^{1 - α} \end{matrix}

(9)

The purpose of our analysis is to compare the relative utility of a house owner before and after housing price change. Dividing equation (5) by equation (2) yields

\frac{v'}{v_{0}} = \frac{w'}{w_{0}} {(\frac{p}{p'})}^{α} = (\frac{α (p' - p (1 + t))}{p (1 + t - β)} + 1) {(\frac{p}{p'})}^{α}

(10)

In order to ease our mathematical derivation, we define a new function g as

g \equiv \frac{v'}{v_{0}} - 1

(11)

g = \frac{{(\frac{p}{p'})}^{α} (α p' + p (1 + t - α - α t - β)) - p (1 + t - β)}{p (1 + t - β)}

(12)

It is clear that the change in the price of housing makes a house owner better off if g > 0, in the sense that v′ > v₀. Conversely, a house owner feels worse off after housing price change if g<0, in the sense that v′ < v₀. Since the denominator in equation (12) is unambiguously positive, the sign of function g depends only on the numerator in equation (12). If the numerator in equation (12) is positive, then g > 0 and v′ > v₀. If the numerator in equation (12) is negative, then g < 0 and v′ < v₀. The following discussions will be separated into three parts.

2.1 When t = 0 and β = 0 (Frank’s case)

In Frank’s (2006) case, there are no property taxes (t = 0) and no mortgage loans (β = 0). Substituting t = 0 and β = 0 into equation (12) yields

\begin{matrix} g_{F} \equiv g (t = 0, β = 0) \\ = \frac{{(\frac{p}{p'})}^{α} (α p' + p (1 - α)) - p}{p} \end{matrix}

(13)

Substituting p′ = p into equation (13) yields

g_{F} (p' = p) = 0

Differentiating equation (13) with respect to p′ yields

\frac{\partial g_{F}}{\partial p'} = \frac{{(\frac{p}{p'})}^{α} α (1 - α) (p' - p)}{p' p}

(14)

It is clear that $\partial g_{F} / \partial p' > (<) 0$ if $p' > (<) p$ and $\partial g_{F} / \partial p' = 0$ if p′ = p. The second-order derivative of the function g_F is calculated as

\frac{\partial^{2} g_{F}}{\partial {(p')}^{2}} = \frac{{(\frac{p}{p'})}^{α} α (1 - α) (p (1 + α) - α p')}{{(p')}^{2} p}

It follows that the function g_F is concave in p′ if $p' \geq p (1 + α) / α$ and the function g_F is convex in p′ if $p' \leq p (1 + α) / α$ . When there are no property taxes (t = 0) and no mortgage loans (β = 0), the function g_F can be depicted as Figure 1. Note that g_F>0 if p′ ≠ p and g_F = 0 if p′ = p, thus yielding the following proposition

Proposition 1. When t = 0 and β = 0, then v′ > v₀ if p′ ≠ p and v′ = v₀ if p′ = p. In other words, Frank’s (2006) result is correct when there are no taxes and no mortgage loans. A new finding is that g_F is concave (convex) when $p' \geq (\leq) p (1 + α) / α$ . In other words, the utility will increase rapidly starting from p′ > p and will be eventually diminished when $p' \geq p (1 + α) / α$ .

Figure 1.

The function g when t = 0 and β = 0 (g_F).

2.2 When t > 0 and β = 0

When a property tax is introduced and house owners do not use any mortgage loans, then substituting β = 0 into equation (12) yields

\begin{matrix} g_{0} \equiv g (β = 0) \\ = (\frac{α (p' - p (1 + t))}{p (1 + t)} + 1) {(\frac{p}{p'})}^{α} - 1 \\ = \frac{{(\frac{p}{p'})}^{α} (α p' + p (1 + t - α - α t)) - p (1 + t)}{p (1 + t)} \end{matrix}

(15)

Differentiating equation (15) with respect to p′ yields

\frac{\partial g_{0}}{\partial p'} = \frac{{(\frac{p}{p'})}^{α} α (α - 1) (p (1 + t) - p')}{p (1 + t) \cdot p'}

(16)

From equation (16), the sign of $\partial g_{0} / \partial p'$ depends on whether $p' > p (1 + t)$ or $p' < p (1 + t)$ . If $p' > p (1 + t)$ , then $\partial g_{0} / \partial p' > 0$ . If $p' < p (1 + t)$ , then $\partial g_{0} / \partial p' < 0$ . Substituting $p' = p (1 + t)$ into equation (15) yields

g_{0} (p' = p (1 + t)) = {(\frac{1}{1 + t})}^{α} - 1 < 0

(17)

Similarly, the sign of function g₀ depends only on the numerator in equation (15). The following equation (18) indicates that the value of the function g₀ is positive when p′ is approaching zero.⁹

\begin{matrix} lim_{p' \to 0} {(\frac{p}{p'})}^{α} \\ \underset{(+)}{\underset{︸}{(α p' + p (1 + t - α - α t))}} - p (1 + t) > 0 \end{matrix}

(18)

The second-order derivative of the function g₀ is calculated as

\begin{matrix} \frac{\partial^{2} g_{0}}{\partial {(p')}^{2}} \\ = \frac{{(\frac{p}{p'})}^{α} α (1 - α) (p (1 + α) (1 + t) - α p')}{{(p')}^{2} p (1 + t)} \end{matrix}

It follows that the function g₀ is concave in p′ if $p' \geq p (1 + α) (1 + t) / α \equiv \tilde{p}$ and the function g₀ is convex in p′ if $p' \leq \tilde{p}$ .

Combining equations (16), (17) and (18), the function g₀ can be depicted as Figure 2. Note that g₀ > 0 if $p' < \bar{p}$ or if $p' > \bar{\bar{p}}$ , and g₀ < 0 if $\bar{p} < p' < \bar{\bar{p}}$ . This result can be depicted by the following proposition.

Proposition 2. When t > 0 and β = 0, then v′ > v₀ if $p' < \bar{p}$ or if $p' > \bar{\bar{p}}$ , and $v' < v_{0}$ for $\bar{p} < p' < \bar{\bar{p}}$ . Moreover, g₀ is convex when $p' \leq \tilde{p}$ , while it is concave when $p' \geq \tilde{p}$ .

Figure 2.

The function g when t >0 and β = 0 (g₀).

The implication of Proposition 2 is that, when there is no mortgage, v′ (the utility of changing to the house with an optimal size) is higher than v₀ (the original utility level) when housing price appreciation (depreciation) is large. If this price change is not sufficiently large, then changing to a new house (with optimal size) makes the owner worse off, because the benefit of appreciation (depreciation) is less than the property tax payment.¹⁰

2.3 When t > 0 and β >0 (the general case)

The general case¹¹ of our model is more complicated. To reach this general case, several steps will be employed. First, the first-order derivative of function g with respect to p′ is calculated as

\begin{matrix} \frac{\partial g}{\partial p'} \\ = \frac{{(\frac{p}{p'})}^{α} α (p' (1 - α) - p ((1 - α) (1 + t) - β))}{p' p (1 + t - β)} \end{matrix}

(19)

Solving $\partial g / \partial p' = 0$ from equation (19) yields

p' = \frac{p ((1 - α) (1 + t) - β)}{1 - α} \equiv p'_{a}

(20)

Note that p′_a is the critical value of p′. From equation (19), the sign of $\partial g / \partial p'$ depends on the sign of $p' (1 - α) - p ((1 - α) (1 + t) - β)$ . If $p^{'} (1 - α) - p ((1 - α) (1 + t) - β) \begin{matrix} > \\ = \\ < \end{matrix} 0$ , then $\partial g / \partial p^{'} \begin{matrix} > \\ = \\ < \end{matrix} 0$ . Therefore,

\frac{\partial g}{\partial p^{'}} \begin{matrix} > \\ = \\ < \end{matrix} 0 iff p^{'} \begin{matrix} > \\ = \\ < \end{matrix} \frac{p ((1 - α) (1 + t) - β)}{1 - α} = {p^{'}}_{a}

(21)

Secondly, we check the sign of function g when p′ is evaluated at the critical value (i.e. the stationary value of function g). Subtracting p from p′_a yields

p'_{a} - p = \frac{p (t (1 - α) - β)}{1 - α}

(22)

Substituting p′ = p into equation (12) yields

g (p' = p) = \frac{- α pt}{p (1 + t - β)} < 0

(23)

If the numerator of equation (22) is positive, then p′_a > p. In this case, $g (p' = p) < 0$ and $\partial g / \partial p' < 0$ for p′ < p′_a, thus $g (p' = p'_{a}) < 0$ . If the numerator of equation (22) is negative, then p′_a < p. In this case, $g (p' = p) < 0$ and $\partial g / \partial p' > 0$ for p′ > p′_a, thus $g (p' = p'_{a}) < 0$ . Therefore, it is concluded that in either case $g (p' = p'_{a}) < 0$ .

Finally, we then check the sign of function g when p′ approaches zero.

lim_{p' \to 0} \frac{w'}{w_{0}} = \frac{(1 - α) (1 + t) - β}{1 + t - β}

(24)

The sign of equation (24) depends on the numerator of equation (24). If $(1 - α) (1 + t) > β$ , then $lim_{p' \to 0} \frac{w'}{w_{0}} > 0$ and $lim_{p' \to 0} g = lim_{p' \to 0} \frac{w'}{w_{0}} {(\frac{p}{p'})}^{α} - 1 > 0$ . Moreover, we also check the concavity and convexity of function g. The second-order derivative of the function g is calculated as

\begin{matrix} \frac{\partial^{2} g}{\partial {(p')}^{2}} = \\ \frac{{(\frac{p}{p'})}^{α} α [p (1 + α) ((1 + t) (1 - α) - β) - p' α (1 - α)]}{{(p')}^{2} p (1 + t - β)} \end{matrix}

It follows that the function g is concave in p′ if $p^{'} \geq p (1 + α) ((1 + t) (1 - α) - β) / (α (1 - α)) \equiv \tilde{\tilde{p}}$ and the function g is convex in p′ if $p^{'} \leq \tilde{\tilde{p}}$ . The function g when $(1 - α) (1 + t) > β$ can be depicted as shown in Figure 3. It is clear that g > 0 if $p' < \hat{p}$ or if $p^{'} > \hat{\hat{p}}$ , and g < 0 if $\hat{p} < p^{'} < \hat{\hat{p}}$ . On the other hand, if $(1 - α) (1 + t) < β$ , then p′_a < 0, $\tilde{\tilde{p}} < 0$ and $\partial g / \partial p' > 0$ for all p′ > 0 In this case, $lim_{p' \to 0} \frac{w'}{w_{0}} < 0$ and $lim_{p' \to 0} g = lim_{p' \to 0} \frac{w'}{w_{0}} {(\frac{p}{p'})}^{α} - 1 < 0$ and the function g is concave in p′ > 0. The function g when $(1 - α) (1 + t) < β$ can be depicted as shown in Figure 4. This means that g > 0 if $p^{'} > \hat{\hat{p}}$ , and g < 0 if $p^{'} < \hat{\hat{p}}$ . Therefore, in the general case, it can be summarised in the following proposition.

Proposition 3. Suppose t > 0 and β > 0. When $(1 - α) (1 + t) > β$ , then v′ > v₀ for $p' < \hat{p}$ or $p^{'} > \hat{\hat{p}}$ and v′ < v₀ for $\hat{p} < p^{'} < \hat{\hat{p}}$ . When $(1 - α) (1 + t) < β$ , then v′ > v₀ for $p^{'} > \hat{\hat{p}}$ and v′ < v₀ for $p^{'} > \hat{\hat{p}}$ .

Figure 3.

The function g when t > 0 β > 0, and $(1 - α) (1 + t) > β$ .

Figure 4.

The function g when t > 0, β > 0 and $(1 - α) (1 + t) < β$ .

The intuition of Proposition 3 is as follows. For a housing price change, a house owner can sell his/her house at a higher price after housing appreciation and use this revenue to purchase more composite goods to raise his/her utility. In contrast, an owner can also sell his/her house in a depreciation case and repurchase a larger house with a cheaper price and raise his/her utility. However, if a property tax is considered, an owner is better off only when the appreciation or depreciation is large enough to compensate for the tax payment. Furthermore, if a house is financed through a mortgage loan, two situations should be clarified. In a housing appreciation, an owner is obviously better off, because the housing value is increased, while the mortgage loan remains constant (based on the original price). In a housing depreciation, an owner is unambiguously worse off, because the housing value is decreased, while the mortgage loan remains constant (based on the original price). Therefore, when the percentage of a mortgage loan over the total housing price is relatively low ( $(1 - α) (1 + t) > β$ ), then the situation is similar to Proposition 2, such that the owner is better off when the housing price appreciation or depreciation is large enough. However, when the mortgage loan covers a high proportion of the housing purchase ( $(1 - α) (1 + t) < β$ ), then when a slump occurs, moving to a new house indeed makes the owner worse off. This is because the gain from a price change is dominated by the negative effect with a high percentage of a mortgage loan. This result is the major difference between this paper and the LMM model. The utility change which results from the housing expenditure share (α) is similar to the explanation of β. The higher the proportion of a housing expenditure ratio is, the more likely a house owner is to be worse off in moving to a new house when a housing slump occurs.

2.4 The Choice of Staying in the Same House

A house owner always has the option of staying in the original house when the housing price changes. The utility of this situation is denoted by v_A in equation (9). There should be no difference between the analysis of changing to a new house and the analysis of staying in the same house except that an agent needs to incur a moving cost if he/she chooses to move.

If a fixed moving cost is incorporated into the model, the previous conclusions in the paper are basically held, and those equations are omitted here to save space. The conclusions of the general model with moving cost are as follows.¹² First, if the percentage of a mortgage loan over the total housing price is relatively low, then the owner is better off (worse off) under a large (small) appreciation or depreciation. Furthermore, an owner changes to a new house only if the housing price appreciation or depreciation is large enough. Secondly, if people buy their houses largely relying on a mortgage loan, it is impossible for them to be better off when the housing price falls. An owner stays in the same house under small depreciation, but changes to a new house under large depreciation. Our results can be used to explain the observation that some people who had bought their houses with a high portion of mortgage felt worse off under a housing market slump during the financial crisis in 2008, instead of being better off as predicted in Frank (2006) and LMM (2010) models.

2.5 Extension

In the housing depreciation case, the owners may have another choice, which is very common in reality. That is, he/she can hand over the key to the bank and move to another house.¹³ In this case, the critical point for handing over the key to the bank is p′ = βp. In other words, when p′ < βp, the current value of his/her house is p′h₀, which is less than his/her mortgage loan βph₀. Therefore, if he/she chooses to return the house, his/her new wealth is c₀ instead of w′. The Lagrangian is thus

L_{K} = \hat{u} + λ (c_{0} - p' (1 + t - β) h - c)

(25)

where the subscript K represents the case of handing over the keys to the bank. Solving for the Lagrangian yields

h_{K} = \frac{α (1 - α) w_{0}}{p' (1 + t - β)}, c_{K} = w_{0} {(1 - α)}^{2}

(26)

The indirect utility function is therefore

v_{K} = {h_{K}}^{α} \cdot {c_{K}}^{(1 - α)}

(27)

Comparing equations (25) and (3), we find that the two optimisation problems are the same, except that w′ is replaced by a larger c₀ when p′ < βp. It is unambiguous that $v_{K} > v'$ when p′ < βp.

However, in the real world, when a house owner hands over the key to the bank, he/she may not be able to get another mortgage from the bank. Therefore, β = 0 in equation (25) and the critical point of handing over the key is less than βp, which is the critical point when β > 0 . In the next section, we will use numerical examples to demonstrate how the owner decides whether to hand over the key or pay back the mortgage and move to another house.

2.6 Numerical Examples

Two numerical examples (low mortgage loan, high mortgage loan) will be demonstrated in this section. First, suppose that w₀ = 100, t = 0.02, α = 0.3,¹⁴ β = 0.3, z = 2 and p = 10.¹⁵ This is shown as a general case with a low mortgage loan. As seen in Figure 5, v′ is greater than v₀ whenever p′ < 2.484 or p′ > 12.080. Note that v_A is greater than v′ when the change in housing price is small (5.490 < p′ < 15.975) and that owners are better off either under large appreciation or large depreciation. Moreover, consider the appreciation case; note that when housing appreciation is very small (10 < p′ < 10.714), an owner will stay in the same house and his/her utility is lower than v₀. When appreciation is larger (10.714 < p′ < 15.975), the owner will choose to stay in the same house and his/her utility is higher than v₀. When appreciation is even larger (p′ > 15.975), the owner will move to another house with an optimal size and his/her utility is higher than v₀. Consider the depreciation case: with a small depreciation (5.490 < p′ < 10), an owner will stay in the same house and his/her utility is lower than v₀. When depreciation is larger (2.484 < p′ < 5.490), the owner will change to another house and his/her utility is still lower than v₀; and when depreciation is even larger (p′ < 2.484), the owner will move to another house with an optimal size and his/her utility is higher than v₀.

Figure 5.

The general case when β is low.

Secondly, if t = 0.02, and w₀ = 100, α = 0.3, β = 0.85, z = 2 and p = 10, then v₀, v′ and v_A can be drawn as in Figure 6. Note that the v′ curve intersects the v₀ curve once (p′ = 10.377), meaning that if the house owner changes to a new house with optimal size, only the appreciation case can make the owner better off, while the depreciation case never enables the owner to be better off. Moreover, v_A is greater than v′ when the change in housing price is small (when 7.408 < p′ < 15.868). This means that with a large appreciation or depreciation, the owner is worse off staying in the same house than moving to a new house with optimal size. Finally, we also draw two lines $v_{K} (β > 0)$ and $v_{K} (β = 0)$ , which represent the utility of handing over the key to the bank and moving to another house.¹⁶ It is unambiguous that $v_{K} (β > 0) > v'$ when p′ < βp = 8.5. If this owner can no longer get another mortgage then the critical point falls to p′ = 6.898. The intuition is straightforward: if he/she hands over the key, he/she will lose the chance to get another loan, then he/she will tolerate a larger depreciation not to hand over the key.

Figure 6.

The general case when β is high.

3. Concluding remarks

Frank (2006) provides a paradoxical example which demonstrates that housing appreciation and depreciation can both benefit owners. The LMM (2010) model embodies factors (borrowing, property taxation and moving costs) into Frank’s (2006) model from a lifetime span and shows that appreciation can make house owners worse off, but can never make them worse off under depreciation if the house owner chooses to stay in the same house. The current paper constructs a general model similar to the LMM model, but under an annual basis, stressing that a person needs to pay an annual property tax, and the mortgage loan is a percentage of the house value. Thus, it is shown that Frank’s conclusion is valid only for those cases with no property tax, no mortgage loans and no moving costs. A new finding on Frank’s (2006) model is that the function of utility difference is first convex and then concave in the appreciation case. With a property tax, a small change in the housing price never raises the owners’ utility. When people buy their houses largely relying on a mortgage loan, it is impossible to be better off when the housing price falls, which is very different from the conclusions recommended in the LMM model.

Specifically, housing appreciation is not always better for owners who intend to adjust their house sizes when a property tax is considered, especially for a small appreciation. If the gain coming from appreciation does not cover the property tax payment, then the owner is still worse off. For the depreciation case, changing to a new house may not be the best choice for owners when property tax and moving costs are considered. More importantly, if people buy their houses largely relying on a mortgage loan it is impossible for them to be better off when the housing price falls, no matter whether they stay in the same house or not. This result is close to the reality during the financial crisis of 2008, especially for house owners financed by sub-prime and related loans. When a housing market slumps, people who bought their houses with a high mortgage loan will be worse off, instead of being better off as predicted in the Frank (2006) and LMM (2010) models. This is because an owner cannot pay off his/her mortgage loan due to the house value no longer being so high.

Since our model is fully analytical in mathematics, our model can be a benchmark and could be extended to other related issues for future study. It is worth delving deeper into policy implications. For instance, in the US, starting from the mid 1990s, but accelerating after 2000, the east and west coasts and some metropolitan areas inland have experienced housing bubbles (see Schwartz, 2010, for details). A government may enact a mortgage restriction on the total outstanding loan or a ceiling of the loan-to-value ratio in order to prevent overbanking and the consequent housing bubbles, especially for low-income people, because some of them may be forced to hand over their houses to the bank and encounter some difficulties in a housing slump.

Moreover, housing is one kind of long-run asset, instead of a short-run speculative asset. A government may enact a variable property tax rate to dampen the extraordinary fluctuation of housing prices. Therefore, an optimal property tax scheme based on long-run considerations is also a possible future research direction. Finally, since our analysis is based on the demand side, a proper policy evaluation should also consider the supply side of a housing market, therefore the social welfare implications of a property tax or housing subsidy on the efficiency of the housing market could also motivate further study.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

Notes

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