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Uncertainty, liquidity, and housing choices
ELSEVIER Regional Science and Urban Economics 25 (1995) 223-236
Uncertainty, liquidity, and housing choices Yuming Fu Department of Economics and Finance, City Polytechnic of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
Received November 1992, final version received June 1994
Abstract
This paper analyzes the housing choices of illiquid households facing uncertain future housing prices. It shows that illiquidity in the housing consumption-investment model of Henderson and Ioannides (The American Economic Review, 1983, 73, no. 1, 98-113) causes a conflict between the housing consumption and investment motives. Consequently, (1) a higher permanent income may not make housing investment and home owning more attractive, (2) the income path affects housing choices, and (3) greater expected housing price appreciation, or lesser uncertainty as to the future price of housing, does not necessarily encourage housing investment. These results indicate the importance of controlling for household liquidity in empirical housing studies. Key words: Housing; Portfolio choice; Liquidity JEL classification: D91; R21
I. Introduction
T h e purpose of this p a p e r is to investigate the impact of household liquidity on housing choices, focusing on separate consumption and investm e n t motives when the future housing price is uncertain. It is widely recognized in the housing literature that owning a h o m e involves both a consumption choice and a portfolio decision (e.g. Ranney, 1981; Rothenberg, 1983; and Mills, 1990). Due to the absence of institutional arrangements that would allow households to own part of their home, h o m e o w n e r s must own at least as much housing as they consume. This investment 0166-0462/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSD1 0166-0462(94)02067-1
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
224
constraint for homeowners, introduced by Henderson and Ioannides (1983), provides a basis for studying housing demand and tenure choice from separate consumption and investment motives. Nevertheless, the housing investment-consumption model of Henderson and Ioannides (1983), and the existing studies based on this model (e.g. Ioannides and Rosenthal, 1994, and Brueckner, 1993), ignore the liquidity of households. This paper shows that illiquidity in the housing consumption-investment model of Henderson and Ioannides (1983) results in a conflict between the housing consumption and investment motives and thus affects the portfolio choice of households. In particular: (1) a higher permanent income may not make housing investment, and hence home owning, more attractive; (2) the income path affects both housing consumption and investment; and (3) greater expected housing price appreciation, or lesser uncertainty as to the future price of housing, does not necessarily encourage housing investment. These results indicate the importance of controlling for household liquidity in studying housing tenure choice and homeowners' housing demand. The impact of household liquidity on housing choices is stressed in Artle and Varaiya (1978), Ranney (1981) and Brueckner (1986). The conflicts between housing investment and current consumption motives are stressed in Ranney (1981) and Rothenberg (1983). The present analysis offers new insights by combining a liquidity constraint with uncertainty and endogenous investment benefits. A certainty-equivalent approach is employed to simplify the analysis.
2. Analysis Following Henderson and Ioannides (1983), let household utility depend on the current consumption of housing (h~) and a numeraire non-housing good (x) in period 1, and on future consumption in period 2. Future consumption, in turn, depends on savings (s) and an uncertain equity in housing investment (hi). The uncertainty arises from the housing price appreciation rate (0), which is assumed to be a random variable with an expected value of E{0} =0, a variance of 0 "2, and zero-valued moments above the second order. E{. } is the expectations operator. Let w 1 and w2 denote the consumption budget for periods 1 and 2, respectively. Let v~(w~) and v2(w2) denote the indirect utility functions that give the present value of utility in each period conditional on the consumption budget. It is assumed that v~ =-dvj/dwj > 0 and v~ ~-dZvj/dw 2 < 0 (j = 1, 2). The risk-averse household selects he, x, s, and hi, subject to its budget constraints, so as to maximize expected utility o -- o , ( w , )
+ E{~2(w2)} •
(1)
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
225
For a home-owning household, the choice of consumption and investment is subject to the investment constraint hc/> hi. For ease of exposition, the following analysis initially focuses on the choice of optimal housing consumption and investment without the investment constraint, denoted h,* and h*, respectively. If h* is at least as large as h'c, the investment constraint will have no impact on housing choices. If h* < h*, the homeowning household would have to distort the optimal choices in order to accommodate the investment constraint. This distortion reduces the attainable utility; therefore, the larger is the difference h* - h*~, the more costly it is to own a home and, ceteris paribus, the more likely it is that the household will choose to be a renter free from the investment constraint. In the second stage of the analysis, the comparative statics of h* and h* are derived and their implications for housing choices are discussed. 2.1. Optimal
housing
consumption
and investment
Let the household choose h c and h i optimally without taking into account the investment constraint. The household receives incomes of YI and Y2, respectively, in periods 1 and 2. Let P be the price, L be an interest-accrual mortgage loan, and R be the (imputed) rental income for period 1, all of which are per unit of housing. Finally, let r be a riskless interest rate. The budget constraints are then given by w~ = y l - s - ( P -
L - R)h i ,
w2 =Y2 +s(1 + r) + [P(1 + 0) - L(1 + r ) ] h i ,
(21)
(3)
where ( P - L - R ) h i is the net (of rental income) downpayment on housing investment in period 1, and [P(1 + 0) - L(1 + r)]h i is the equity in housing in period 2. The household also faces a liquidity constraint, s i> O,
(4)
so that borrowing against future income is not allowed. The possibility of short-selling housing is also excluded so that h i ~>0. For the purpose of graphical illustration, the consumption-investmentchoice problem is solved in two steps. In Step 1, the household chooses an optimal portfolio of hi and s, for each given level of wl, so as to maximize certainty-equivalent wealth in period 2. This results in a certainty-equivalent intertemporal budget curve, denoted ffz(wl). In Step 2, the household chooses an optimal current consumption w~ according to its intertemporal preferences.
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Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
2.1.1. Step 1 Let W=-y I +y2/(1 + r) be household wealth, and w 2 ~ E { w 2 } be the expected budget in period 2. Rearranging budget constraints (2) and (3): s + (P-
L -R)h
i =yl
(5)
- w, ,
(6)
w2 = (1 + r ) ( W - wl) + [P(0 - r) + R(1 + r)]h i .
Eq. (5) determines the investment budget. Eq. (6) determines the expected consumption budget in period 2 as a function of the current consumption and the return from housing investment. The certainty-equivalent intertemporal budget curve is defined as ~2(Wl) =- ,fmax w2 - ~'(w2, q), subject to (5) and (6)1, , [ h~ J
(7)
where Tr is a risk premium that depends on both the expected wealth in period 2 and the risk measured by q =- p2h~g2 /2. This risk premium is determined by v2(-w2 - 7r) --=E{v2) = v2(w2) + v'~(-w2)q ,
(8)
where the equality follows from a Taylor expansion of 02 around 0 =0 (Pratt, 1964). It is straightforward to verify that the partial derivatives of ~with respect to Wz and q are, respectively, 7rq = - v~(-WE)/V~(g, 2) > 0 and zr~2 = 1 - E { o 2 ) / o 2 ( w 2 ) . Furthermore, t3 -= "rrq/(1 - "try2) -- - u~(-WE)/E{v~} defines a measure of risk aversion that is smaller than the Arrow-Pratt measure of local risk aversion at w2. When the liquidity constraint is binding, the maximization problem (7) has a boundary solution in terms of h i. The boundary solution, denoted hb(wl), is determined by Eq. (5), with s = 0; that is
hb(w~)_
yI--Wl
(9)
P --£----R"
In this case the intertemporal budget curve, denoted ]~2(W1,• h ib ), depends on Yl, since h b depends on Yl. The slope of such a budget curve is given by
dlb2(Wl; hib) 01~ 2 Off' 2 - (1 - ¢r~2)-~wl + [(1 - 7r~2) Ohi dw 1
,lrpp20.2h b ] dhib(w 1)
aw 1
= - (1 - 7r~2)[1 + r + (10)
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
227
A
W2
/4
(l+r)W
y,-(P-L-R)hi*
y,
W
w 1
Fig. 1. Certainty-equivalent budget curves.
The intertemporal budget curves with s = 0 and different y l are illustrated in Fig. 1. Absent the liquidity constraint, the maximization problem (7) has an interior solution in terms of h i. The first-order condition for the interior solution, denoted h ° ( w l ) , is OW 2
Oh i = (1 - 7rw2)[P(0 - r) + R(1 + r)] - ~rqP 2o"2 h~ = 0
,
(11)
which yields
h°(Wl) =
P(O - r) + R(1 + r)
~p2o-2
(12)
This optimal housing investment absent a liquidity constraint varies in direct proportion to the expected return from housing investment and in inverse proportion to the risk-aversion measure ~ and the uncertainty in the future housing price, as reflected by the variance p2or2. A result similar to this is found in Fu (1991). The intertemporal budget curve absent the liquidity constraint, denoted I'VE(Wl, h°), is the envelope of the liquidity-constrained budget curves with different Yl- The slope of this budget curve is given by
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
228
d~'z(wl; h ° )1d w
- (1 -- "JTw2 )
OW~ lOW2-
( 1 - %2)(1 + r ) .
(13)
Due to diminishing risk aversion, this slope is always more negative than - ( 1 + r) and becomes less negative as w~ decreases. The unconstrained budget curve, ff2(Wl;h°), is also illustrated in Fig. 1. The liquidity-constrained budget curves are tangent to the unconstrained budget curve when = h°(w,).
2.1.2. Step 2 Given its intertemporal budget curve (7), the household chooses an optimal current consumption wl to maximize v, which can now be rewritten as
v = v , ( w , ) + v2(rbz(Wl) ) .
(14)
The first-order condition for w~ requires V'1
v2(~2~ -
d14, 2
dw,'
(15)
that is, the rate of intertemporal utility substitution equals the absolute value of the slope of the certainty-equivalent budget curve. Thus, depending on whether the liquidity constraint is binding or not, the optimal consumption choice requires that the intertemporal indifference curve be tangent to a liquidity- constrained budget curve v~2(w1,• h bi), or to the unconstrained budget curve w2(wl;h °) (see Fig. 2). The tangency point b in Fig. 2 gives an example of the optimal choice absent the liquidity constraint. In this particular case the household would borrow l 1 against its future income, given its current income Yr. When the liquidity constraint is binding, the optimal choice is at point b' on the liquidity-constrained budget curve associated with income Yl. A binding liquidity constraint results in both a smaller current consumption w~ and a smaller housing investment h*. The point of tangency a gives an example of the optimal choice when the liquidity constraint is not binding due to a sufficiently strong preference for future consumption. In this particular case, the household saves 12 from current income Yx. As Fig. 2 illustrates, the liquidity constraint affects the investment and consumption choices by precluding the separability of the portfolio decision from the consumption decision. Absent the liquidity constraint, the portfolio decision maximizes the consumption opportunity represented by ff2(w~; h°), independent of consumption preferences. With a binding liquidity constraint, an increase in investment forces a decrease in current consumption
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236 w2
229
i.
O+Ow
6
6.
o'
" " o - W1
y,-12 y, Fig. 2. Optimal choices.
y,+l,
w
along a budget curve ff2(w~,"hib); the conflict between the investment and consumption motives ensues. Another useful observation from Fig. 2 is that household liquidity depends not only on the incomes but also on the intertemporal preferences. 2.2. Comparative statics and housing choices The foregoing analysis shows the determination of housing demand based on separate investment and consumption motives as captured by h* and h*~. The implications for housing choices, based on the comparative statics of h and h* and the investment constraint for homeowners, are discussed below. The comparative statics of h* and h* with respect to incomes Yl and Y2, the income path T=-y I - y z / ( 1 + r) and the wealth W=-yl + y2/(1 + r) are summarized in the following proposition. Proposition. Given the assumption that both current and future consumption are normal goods and that the liquidity constraint is always binding (s - 0): (1) both investment demand (h~) and consumption demand (h~) for housing increase with period-1 income (Yt); (2) the investment demand for housing decreases and the consumption demand for housing increases with period-2 income (Y2), if the household has a constant Arrow-Pratt measure of local risk aversion; (3) with a constant Arrow-Pratt measure of local risk aversion, both the
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Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
investment and consumption demands for housing increase with the income path (T), with wealth ( W ) held constant; (4) with a constant Arrow-Pratt measure of local risk aversion, an increase in wealth ( W) homing the income path ( T) constant increases the consumption demand for housing and, if and only if the rate of intertemporal utility substitution (v']v~(~z)) decreases, increases the investment demand for housing. Proof. See the appendix. In comparison with the results in Henderson and Ioannides (1983) and Fu (1991), where, with the liquidity constraint absent, both the investment and consumption demands for housing increase with the incomes and are independent of the income path, the above proposition reveals quite different behaviour of h* and h*. The differences arise because of the conflict between the consumption and investment motives due to the inability of the household to liquidate the future gains of investment for current benefits. The conflict is reflected in the competing effects of future wealth on current consumption and investment. An increase in future wealth, on the one hand, reduces the marginal utility of future consumption, causing investment to decrease and current consumption to increase; these are the income effects. On the other hand, it reduces the future risk aversion, encouraging investment and discouraging current consumption; these are the substitution effects. The first statement in the proposition follows from the normal-good assumption as to consumption in both periods, and is illustrated in Fig. 3 by the horizontal movement of the budget curve from point T~ to T 4. The second statement says that, when risk aversion is insensitive to period-2 wealth, the income effect dominates the substitution effect and consumption in both periods increases. This statement is illustrated by the vertical movement of the budget curve from point T3 to T4 in Fig. 3. This statement can be reversed if risk aversion decreases sufficiently fast with period-2 wealth. The third statement is illustrated in Fig. 3 by the movement of the budget curve from point T~ to T2. An increase in the income path (T) means that current income increases at the expense of future income with wealth unaffected. When this happens, according to statements (1) and (2), investment demand increases. Statement (3) says that the increase in investment would not totally offset the increase in current income, so that current consumption also increases. The first half of the last statement follows from statements (1) and (2). The second half says that for investment demand to increase with wealth requires either risk aversion to decrease sufficiently fast or the rate of intertemporal utility substitution to decrease (stronger preference for future consumption). The last statement is
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
(l+r)W
231
b
~O W 1
w
Fig. 3. Comparativestatics. illustrated in Fig. 3 by the movement of the budget curve from point
T 2 to
74. Taking into account the investment constraint required of owning, some implications of the proposition for housing choices can be suggested. First, an increase in transitory income (y~) has an ambiguous effect on tenure choice. For those whose housing consumption demand is less sensitive to the current budget (w 1), the difference h* - h* would become smaller, resulting in greater incentive to own. For others, with more sensitive housing consumption demand, the difference h * - h* may become larger and the incentive to own may be reduced. A larger y~, however, would increase the housing demand of those who own. Second, for those expecting higher future income, unless their future risk aversion is sufficiently lowered, their investment-motivated housing demand would be lower and consumptionmotivated housing demand would be higher. Consequently, those households would be less likely to own their current housing. Third, an increase in the income path has effects similar to an increase in current income. Compared with the effects of Yl, however, T is likely to have a larger impact on investment and a smaller impact on current consumption, since the compensating change in future income tends to reinforce the effect of current income on investment but offset the effect on current consumption. Thus a forward tilting in the income path tends to increase both the likelihood of owning and the housing demand of homeowners. Fourth, similar to those expecting a higher future income, households
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Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
with higher permanent income (greater W) are not likely to own their current housing, unless their future risk aversion is much lower or their preference for future consumption is stronger. A final consideration is the impact of the expected rate of housing price appreciation 0 and the variance of future housing price o-2 on housing decisions. Since the decision would depend on after-tax capital gains in housing, an increase in 0 has the same effect as a decrease in the expected housing capital-gain tax rate. Similar to an increase in Y2, an increase in 0 or a decrease in cr 2 has competing effects on investment and current consumption. These effects are shown by the comparative statics of h* with respect to 0 and o-*. By Taylor expansion, the first-order condition for h* can be expressed as Ov Oh i .
( P.- L
. R)v'~+ . E{v~}[P(I+O)
y--
"~2
2~
L ( l + r ) ] + v tf 21,Wz)r o" n i
=0.
The comparative statics are
d~ - L 00
[P(1 + 0) - L(1 + r)] + E{o~}P + v'~'(ff'2)p3o-2h .2 -,
x \
Oh~/
'
(16)
dh* { aE{v~}--.. ]// 020\-1 do-----~ - ~ [t'[l + 0 ) - L(1 + r)] + v~(ff~z)P'h*iJ~- Oh---~i) , (17) (see the appendix). Notice the following: (1) the expected marginal utility E{v~} decreases with 0, since v~ is a decreasing function of 0; (2) E{v~} increases with o-2, since v 2 is a convex function of 0; and (3) 0'2'(~2) > 0 for a non-increasing Arrow-Pratt measure of local risk aversion. In Eq. (16), a higher 0, on the one hand, makes housing investment less attractive (hence current consumption more attractive) by reducing the expected future marginal utility E{v2}. On the other hand, it makes housing investment m_ore attractive by raising the expected return of housing investment, P(1 + 0 ) - L ( 1 +r), and reducing the risk premium -- v~(-w2)p2o'2h *. In Eq. (17), a larger o-2, on the one hand, makes housing investment more attractive by raising the expected marginal utility E{v~). On the other hand, it makes housing investment less attractive by increasing where c 9 2 v / O h ~ < O
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
233
the risk premium. The net impact of a change in 0 or tr 2 on h,* and h* cannot be determined.
3. Conclusion This paper examines the housing choices of illiquid households, extending the analysis in Henderson and Ioannides (1983) and Fu (1991). The analysis recognizes that housing choices involve both consumption and investment motives and that housing investment of homeowners must be at least as great as their housing consumption. The conflicts between the consumption and investment motives due to the inability of the household to liquidate the future gains of investment for current benefits are illustrated. For an illiquid household, the analysis shows that the effects on housing choices of current and future incomes, as well as those of expected housing price appreciation and uncertainty as to the future price of housing, depend on the relative magnitude of the income effect and substitution effect of future wealth, which, in turn, depend on the behaviour of future risk aversion and the rate of intertemporal utility substitution. A weakness of the analysis is the omission of other risky assets in a household's portfolio. Brueckner (1993) provides a succinct analysis of the portfolio choices of homeowners in the absence of borrowing constraints, taking into account the correlations among housing investment and other risky assets as well as the housing investment constraint for homeowners. The present analysis complements his, in the sense that it considers portfolio choice under a borrowing constraint, with restricted investment alternatives. The basic insight of the analysis may nevertheless be extended to portfolio choices involving both housing and other risky assets, where, in addition to the housing investment constraint, the liquidity constraint would also contribute to the conflict between the consumption and portfolio- diversification motives. The absence of other risky assets from the portfolio of illiquid households may not be a serious omission, since these households are unlikely to be able to afford portfolio diversification. Finally, though the risky asset in the model is interpreted as housing investment, the analysis applies to the choice problems of illiquid households involving any indivisible asset that generates consumption benefits (an expensive car, for example).
Acknowledgements I wish to thank Robert Helsley, Lawrence Jones, Vernon Henderson, Jan Brueckner, and Ira Horowitz for helpful comments. Any errors remaining are mine.
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Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
Appendix P r o o f o f proposition. When s ~-0, the first-order conditions for w~ and h*
are equivalent. To simplify the notation, let a =- (P - L - R) and/3 -= P(1 + 0 ) - L ( 1 + r). Differentiating v in (1) with respect to hi, subject to the budget constraints (2) and (3), we have the first-order condition: Ov Oh i
av', + E{v;/3} = 0.
(A1)
The second-order condition is 020
ohf
E{v~/3 2} < 0 .
-- Ol2Vtt I +
(A2)
The signs of the comparative statics of w~ apply to those of h*c, as housing is a normal good and h* increases with w~. Statement 1: dh*
,,/02u~ -I
~yl-aOlkoh~]
dw~
_
(A3)
>0,
dh* _
1
E{v2/3
_
dy 1 - 1 - a dy 1 -
>0
J\Oh~/
(14) "
Statement 2: Let p =- ( - v ~ / v ; ) be the constant Arrow-Pratt measure of local
risk aversion: dh*
,,
dy 2 -
/02v\-i
,
02v
l
E{v2/3}~i2)=pE{v2/3}(~i2)<0,
(15)
where the inequality follows from (A1). Since Yl does not change, w~ changes in the opposite direction to h*. Statement 3: yl = ( W + T ) / 2 and Y2 = ( W - T)(1 + r)/2. dh* _ 1 dh* dT 2 dy I
1 dh*(l+r)>0, 2 dy 2
(16)
which follows from Statements 1 and 2.
(0v)
do)T 2 -1 dT = (E{-v~/32} + (1 + r ) a E { v 2 f l } ) 2 ~ , 2 \
=pE{vzfl(fl-(l
( 2 0 2 v ] -1 + r ) a ) } \ Oh~/
Oh i /
(A7)
Let A =- E { v ; f l } / E { v ' 2 } and B ~- E{v;(fl - (1 + r ) a ) } / E { v ; } , a > 0 follows
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
235
from (A1) and B I>0 follows from (A1) and the fact that the liquidity constraint is binding so that Ov ~S -- -- v'l -1- (1 + r)E{v;} ~ O. Now the expression in (A7) is positive because E{v;/3(/3 - (1 + r)a)} E{v~}
E{o;(/3 - A)(/3 - (1 + r)a - B)} +A'B E{v~}
> 0,
(A8)
where the first part on the right-hand side of the equation is the correlation between /3 and / 3 - (1 + r)a based on a scaled distribution of 0, and is positive. Statement 4:
dw*~ OW
1__ --_ 1 dw~ 1 +r)>O, 2 d Y l + 2 dY2 "
(A9)
which follows from Statements 1 and 2. dh*dw
-
-
+(1
/~)"Vl av,' = ~o;
( 021 ~ -1
I)~(1~2)0;(1~2)(1 + r)E{v;/3} ) \[2 Ozv] Oh~/-1 ,
(AIO)
where the second equation follows from the assumption of constant local risk aversion. Since
O (O'I(W1) ~
--
O';O;--V'lO~(l+r) ?(-~
,
(All)
it follows from (A1) and ( A l l ) that dh*i/dW>O if and only if O(v'l/v')/ OW
References Artle, R. and P. Varaiya, 1978, Life cycle consumption and home ownership, Journal of Economic Theory 10, 35-58. Brueckner, J.K,, 1993, Consumption and investment motives and the portfolio choices of homeowners, Working paper, University of Illinois at Urbana-Champaign. Brueckner, J.K., 1986, The downpayment constraint and housing tenure choice, Regional Science and Urban Economics 16, 519-204. Fu, Y., 1991, A model of housing tenure choice: Comment, The American Economic Review 81, no. 1, 381-383.
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Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
Henderson, J.V. and Y.M. Ioannides, 1983, A model of housing tenure choice, The American Economic Review 73, no. 1, 98-113. Ioannides, Y.M. and S.S. Rosenthal, 1994, Estimating the consumption and investment demands for housing and their effect on housing tenure status, Review of Economics and Statistics, 76, no. 1, 127-141. Mills, E.S., 1990, Housing tenure choice, Journal of Real Estate Finance and Economics, 3, no. 4, 323-331. Pratt, J.W., 1964, Risk aversion in the small and in the large, Econometrica 32, no. 1,122-137. Ranney, S.I., 1981, Inflation expectations and the demand for housing, American Economic Review 72, no. 1, 143-153. Rothenberg, J., 1983, Housing investment, housing consumption, and tenure choice, in: R.E. Grienson, ed., The urban economy and housing (Lexington Books, Lexington, KY) 29-55.
Uncertainty, liquidity, and housing choices Yuming Fu Department of Economics and Finance, City Polytechnic of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
Received November 1992, final version received June 1994
Abstract
This paper analyzes the housing choices of illiquid households facing uncertain future housing prices. It shows that illiquidity in the housing consumption-investment model of Henderson and Ioannides (The American Economic Review, 1983, 73, no. 1, 98-113) causes a conflict between the housing consumption and investment motives. Consequently, (1) a higher permanent income may not make housing investment and home owning more attractive, (2) the income path affects housing choices, and (3) greater expected housing price appreciation, or lesser uncertainty as to the future price of housing, does not necessarily encourage housing investment. These results indicate the importance of controlling for household liquidity in empirical housing studies. Key words: Housing; Portfolio choice; Liquidity JEL classification: D91; R21
I. Introduction
T h e purpose of this p a p e r is to investigate the impact of household liquidity on housing choices, focusing on separate consumption and investm e n t motives when the future housing price is uncertain. It is widely recognized in the housing literature that owning a h o m e involves both a consumption choice and a portfolio decision (e.g. Ranney, 1981; Rothenberg, 1983; and Mills, 1990). Due to the absence of institutional arrangements that would allow households to own part of their home, h o m e o w n e r s must own at least as much housing as they consume. This investment 0166-0462/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSD1 0166-0462(94)02067-1
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
224
constraint for homeowners, introduced by Henderson and Ioannides (1983), provides a basis for studying housing demand and tenure choice from separate consumption and investment motives. Nevertheless, the housing investment-consumption model of Henderson and Ioannides (1983), and the existing studies based on this model (e.g. Ioannides and Rosenthal, 1994, and Brueckner, 1993), ignore the liquidity of households. This paper shows that illiquidity in the housing consumption-investment model of Henderson and Ioannides (1983) results in a conflict between the housing consumption and investment motives and thus affects the portfolio choice of households. In particular: (1) a higher permanent income may not make housing investment, and hence home owning, more attractive; (2) the income path affects both housing consumption and investment; and (3) greater expected housing price appreciation, or lesser uncertainty as to the future price of housing, does not necessarily encourage housing investment. These results indicate the importance of controlling for household liquidity in studying housing tenure choice and homeowners' housing demand. The impact of household liquidity on housing choices is stressed in Artle and Varaiya (1978), Ranney (1981) and Brueckner (1986). The conflicts between housing investment and current consumption motives are stressed in Ranney (1981) and Rothenberg (1983). The present analysis offers new insights by combining a liquidity constraint with uncertainty and endogenous investment benefits. A certainty-equivalent approach is employed to simplify the analysis.
2. Analysis Following Henderson and Ioannides (1983), let household utility depend on the current consumption of housing (h~) and a numeraire non-housing good (x) in period 1, and on future consumption in period 2. Future consumption, in turn, depends on savings (s) and an uncertain equity in housing investment (hi). The uncertainty arises from the housing price appreciation rate (0), which is assumed to be a random variable with an expected value of E{0} =0, a variance of 0 "2, and zero-valued moments above the second order. E{. } is the expectations operator. Let w 1 and w2 denote the consumption budget for periods 1 and 2, respectively. Let v~(w~) and v2(w2) denote the indirect utility functions that give the present value of utility in each period conditional on the consumption budget. It is assumed that v~ =-dvj/dwj > 0 and v~ ~-dZvj/dw 2 < 0 (j = 1, 2). The risk-averse household selects he, x, s, and hi, subject to its budget constraints, so as to maximize expected utility o -- o , ( w , )
+ E{~2(w2)} •
(1)
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
225
For a home-owning household, the choice of consumption and investment is subject to the investment constraint hc/> hi. For ease of exposition, the following analysis initially focuses on the choice of optimal housing consumption and investment without the investment constraint, denoted h,* and h*, respectively. If h* is at least as large as h'c, the investment constraint will have no impact on housing choices. If h* < h*, the homeowning household would have to distort the optimal choices in order to accommodate the investment constraint. This distortion reduces the attainable utility; therefore, the larger is the difference h* - h*~, the more costly it is to own a home and, ceteris paribus, the more likely it is that the household will choose to be a renter free from the investment constraint. In the second stage of the analysis, the comparative statics of h* and h* are derived and their implications for housing choices are discussed. 2.1. Optimal
housing
consumption
and investment
Let the household choose h c and h i optimally without taking into account the investment constraint. The household receives incomes of YI and Y2, respectively, in periods 1 and 2. Let P be the price, L be an interest-accrual mortgage loan, and R be the (imputed) rental income for period 1, all of which are per unit of housing. Finally, let r be a riskless interest rate. The budget constraints are then given by w~ = y l - s - ( P -
L - R)h i ,
w2 =Y2 +s(1 + r) + [P(1 + 0) - L(1 + r ) ] h i ,
(21)
(3)
where ( P - L - R ) h i is the net (of rental income) downpayment on housing investment in period 1, and [P(1 + 0) - L(1 + r)]h i is the equity in housing in period 2. The household also faces a liquidity constraint, s i> O,
(4)
so that borrowing against future income is not allowed. The possibility of short-selling housing is also excluded so that h i ~>0. For the purpose of graphical illustration, the consumption-investmentchoice problem is solved in two steps. In Step 1, the household chooses an optimal portfolio of hi and s, for each given level of wl, so as to maximize certainty-equivalent wealth in period 2. This results in a certainty-equivalent intertemporal budget curve, denoted ffz(wl). In Step 2, the household chooses an optimal current consumption w~ according to its intertemporal preferences.
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Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
2.1.1. Step 1 Let W=-y I +y2/(1 + r) be household wealth, and w 2 ~ E { w 2 } be the expected budget in period 2. Rearranging budget constraints (2) and (3): s + (P-
L -R)h
i =yl
(5)
- w, ,
(6)
w2 = (1 + r ) ( W - wl) + [P(0 - r) + R(1 + r)]h i .
Eq. (5) determines the investment budget. Eq. (6) determines the expected consumption budget in period 2 as a function of the current consumption and the return from housing investment. The certainty-equivalent intertemporal budget curve is defined as ~2(Wl) =- ,fmax w2 - ~'(w2, q), subject to (5) and (6)1, , [ h~ J
(7)
where Tr is a risk premium that depends on both the expected wealth in period 2 and the risk measured by q =- p2h~g2 /2. This risk premium is determined by v2(-w2 - 7r) --=E{v2) = v2(w2) + v'~(-w2)q ,
(8)
where the equality follows from a Taylor expansion of 02 around 0 =0 (Pratt, 1964). It is straightforward to verify that the partial derivatives of ~with respect to Wz and q are, respectively, 7rq = - v~(-WE)/V~(g, 2) > 0 and zr~2 = 1 - E { o 2 ) / o 2 ( w 2 ) . Furthermore, t3 -= "rrq/(1 - "try2) -- - u~(-WE)/E{v~} defines a measure of risk aversion that is smaller than the Arrow-Pratt measure of local risk aversion at w2. When the liquidity constraint is binding, the maximization problem (7) has a boundary solution in terms of h i. The boundary solution, denoted hb(wl), is determined by Eq. (5), with s = 0; that is
hb(w~)_
yI--Wl
(9)
P --£----R"
In this case the intertemporal budget curve, denoted ]~2(W1,• h ib ), depends on Yl, since h b depends on Yl. The slope of such a budget curve is given by
dlb2(Wl; hib) 01~ 2 Off' 2 - (1 - ¢r~2)-~wl + [(1 - 7r~2) Ohi dw 1
,lrpp20.2h b ] dhib(w 1)
aw 1
= - (1 - 7r~2)[1 + r + (10)
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
227
A
W2
/4
(l+r)W
y,-(P-L-R)hi*
y,
W
w 1
Fig. 1. Certainty-equivalent budget curves.
The intertemporal budget curves with s = 0 and different y l are illustrated in Fig. 1. Absent the liquidity constraint, the maximization problem (7) has an interior solution in terms of h i. The first-order condition for the interior solution, denoted h ° ( w l ) , is OW 2
Oh i = (1 - 7rw2)[P(0 - r) + R(1 + r)] - ~rqP 2o"2 h~ = 0
,
(11)
which yields
h°(Wl) =
P(O - r) + R(1 + r)
~p2o-2
(12)
This optimal housing investment absent a liquidity constraint varies in direct proportion to the expected return from housing investment and in inverse proportion to the risk-aversion measure ~ and the uncertainty in the future housing price, as reflected by the variance p2or2. A result similar to this is found in Fu (1991). The intertemporal budget curve absent the liquidity constraint, denoted I'VE(Wl, h°), is the envelope of the liquidity-constrained budget curves with different Yl- The slope of this budget curve is given by
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
228
d~'z(wl; h ° )1d w
- (1 -- "JTw2 )
OW~ lOW2-
( 1 - %2)(1 + r ) .
(13)
Due to diminishing risk aversion, this slope is always more negative than - ( 1 + r) and becomes less negative as w~ decreases. The unconstrained budget curve, ff2(Wl;h°), is also illustrated in Fig. 1. The liquidity-constrained budget curves are tangent to the unconstrained budget curve when = h°(w,).
2.1.2. Step 2 Given its intertemporal budget curve (7), the household chooses an optimal current consumption wl to maximize v, which can now be rewritten as
v = v , ( w , ) + v2(rbz(Wl) ) .
(14)
The first-order condition for w~ requires V'1
v2(~2~ -
d14, 2
dw,'
(15)
that is, the rate of intertemporal utility substitution equals the absolute value of the slope of the certainty-equivalent budget curve. Thus, depending on whether the liquidity constraint is binding or not, the optimal consumption choice requires that the intertemporal indifference curve be tangent to a liquidity- constrained budget curve v~2(w1,• h bi), or to the unconstrained budget curve w2(wl;h °) (see Fig. 2). The tangency point b in Fig. 2 gives an example of the optimal choice absent the liquidity constraint. In this particular case the household would borrow l 1 against its future income, given its current income Yr. When the liquidity constraint is binding, the optimal choice is at point b' on the liquidity-constrained budget curve associated with income Yl. A binding liquidity constraint results in both a smaller current consumption w~ and a smaller housing investment h*. The point of tangency a gives an example of the optimal choice when the liquidity constraint is not binding due to a sufficiently strong preference for future consumption. In this particular case, the household saves 12 from current income Yx. As Fig. 2 illustrates, the liquidity constraint affects the investment and consumption choices by precluding the separability of the portfolio decision from the consumption decision. Absent the liquidity constraint, the portfolio decision maximizes the consumption opportunity represented by ff2(w~; h°), independent of consumption preferences. With a binding liquidity constraint, an increase in investment forces a decrease in current consumption
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236 w2
229
i.
O+Ow
6
6.
o'
" " o - W1
y,-12 y, Fig. 2. Optimal choices.
y,+l,
w
along a budget curve ff2(w~,"hib); the conflict between the investment and consumption motives ensues. Another useful observation from Fig. 2 is that household liquidity depends not only on the incomes but also on the intertemporal preferences. 2.2. Comparative statics and housing choices The foregoing analysis shows the determination of housing demand based on separate investment and consumption motives as captured by h* and h*~. The implications for housing choices, based on the comparative statics of h and h* and the investment constraint for homeowners, are discussed below. The comparative statics of h* and h* with respect to incomes Yl and Y2, the income path T=-y I - y z / ( 1 + r) and the wealth W=-yl + y2/(1 + r) are summarized in the following proposition. Proposition. Given the assumption that both current and future consumption are normal goods and that the liquidity constraint is always binding (s - 0): (1) both investment demand (h~) and consumption demand (h~) for housing increase with period-1 income (Yt); (2) the investment demand for housing decreases and the consumption demand for housing increases with period-2 income (Y2), if the household has a constant Arrow-Pratt measure of local risk aversion; (3) with a constant Arrow-Pratt measure of local risk aversion, both the
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Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
investment and consumption demands for housing increase with the income path (T), with wealth ( W ) held constant; (4) with a constant Arrow-Pratt measure of local risk aversion, an increase in wealth ( W) homing the income path ( T) constant increases the consumption demand for housing and, if and only if the rate of intertemporal utility substitution (v']v~(~z)) decreases, increases the investment demand for housing. Proof. See the appendix. In comparison with the results in Henderson and Ioannides (1983) and Fu (1991), where, with the liquidity constraint absent, both the investment and consumption demands for housing increase with the incomes and are independent of the income path, the above proposition reveals quite different behaviour of h* and h*. The differences arise because of the conflict between the consumption and investment motives due to the inability of the household to liquidate the future gains of investment for current benefits. The conflict is reflected in the competing effects of future wealth on current consumption and investment. An increase in future wealth, on the one hand, reduces the marginal utility of future consumption, causing investment to decrease and current consumption to increase; these are the income effects. On the other hand, it reduces the future risk aversion, encouraging investment and discouraging current consumption; these are the substitution effects. The first statement in the proposition follows from the normal-good assumption as to consumption in both periods, and is illustrated in Fig. 3 by the horizontal movement of the budget curve from point T~ to T 4. The second statement says that, when risk aversion is insensitive to period-2 wealth, the income effect dominates the substitution effect and consumption in both periods increases. This statement is illustrated by the vertical movement of the budget curve from point T3 to T4 in Fig. 3. This statement can be reversed if risk aversion decreases sufficiently fast with period-2 wealth. The third statement is illustrated in Fig. 3 by the movement of the budget curve from point T~ to T2. An increase in the income path (T) means that current income increases at the expense of future income with wealth unaffected. When this happens, according to statements (1) and (2), investment demand increases. Statement (3) says that the increase in investment would not totally offset the increase in current income, so that current consumption also increases. The first half of the last statement follows from statements (1) and (2). The second half says that for investment demand to increase with wealth requires either risk aversion to decrease sufficiently fast or the rate of intertemporal utility substitution to decrease (stronger preference for future consumption). The last statement is
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
(l+r)W
231
b
~O W 1
w
Fig. 3. Comparativestatics. illustrated in Fig. 3 by the movement of the budget curve from point
T 2 to
74. Taking into account the investment constraint required of owning, some implications of the proposition for housing choices can be suggested. First, an increase in transitory income (y~) has an ambiguous effect on tenure choice. For those whose housing consumption demand is less sensitive to the current budget (w 1), the difference h* - h* would become smaller, resulting in greater incentive to own. For others, with more sensitive housing consumption demand, the difference h * - h* may become larger and the incentive to own may be reduced. A larger y~, however, would increase the housing demand of those who own. Second, for those expecting higher future income, unless their future risk aversion is sufficiently lowered, their investment-motivated housing demand would be lower and consumptionmotivated housing demand would be higher. Consequently, those households would be less likely to own their current housing. Third, an increase in the income path has effects similar to an increase in current income. Compared with the effects of Yl, however, T is likely to have a larger impact on investment and a smaller impact on current consumption, since the compensating change in future income tends to reinforce the effect of current income on investment but offset the effect on current consumption. Thus a forward tilting in the income path tends to increase both the likelihood of owning and the housing demand of homeowners. Fourth, similar to those expecting a higher future income, households
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Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
with higher permanent income (greater W) are not likely to own their current housing, unless their future risk aversion is much lower or their preference for future consumption is stronger. A final consideration is the impact of the expected rate of housing price appreciation 0 and the variance of future housing price o-2 on housing decisions. Since the decision would depend on after-tax capital gains in housing, an increase in 0 has the same effect as a decrease in the expected housing capital-gain tax rate. Similar to an increase in Y2, an increase in 0 or a decrease in cr 2 has competing effects on investment and current consumption. These effects are shown by the comparative statics of h* with respect to 0 and o-*. By Taylor expansion, the first-order condition for h* can be expressed as Ov Oh i .
( P.- L
. R)v'~+ . E{v~}[P(I+O)
y--
"~2
2~
L ( l + r ) ] + v tf 21,Wz)r o" n i
=0.
The comparative statics are
d~ - L 00
[P(1 + 0) - L(1 + r)] + E{o~}P + v'~'(ff'2)p3o-2h .2 -,
x \
Oh~/
'
(16)
dh* { aE{v~}--.. ]// 020\-1 do-----~ - ~ [t'[l + 0 ) - L(1 + r)] + v~(ff~z)P'h*iJ~- Oh---~i) , (17) (see the appendix). Notice the following: (1) the expected marginal utility E{v~} decreases with 0, since v~ is a decreasing function of 0; (2) E{v~} increases with o-2, since v 2 is a convex function of 0; and (3) 0'2'(~2) > 0 for a non-increasing Arrow-Pratt measure of local risk aversion. In Eq. (16), a higher 0, on the one hand, makes housing investment less attractive (hence current consumption more attractive) by reducing the expected future marginal utility E{v2}. On the other hand, it makes housing investment m_ore attractive by raising the expected return of housing investment, P(1 + 0 ) - L ( 1 +r), and reducing the risk premium -- v~(-w2)p2o'2h *. In Eq. (17), a larger o-2, on the one hand, makes housing investment more attractive by raising the expected marginal utility E{v~). On the other hand, it makes housing investment less attractive by increasing where c 9 2 v / O h ~ < O
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
233
the risk premium. The net impact of a change in 0 or tr 2 on h,* and h* cannot be determined.
3. Conclusion This paper examines the housing choices of illiquid households, extending the analysis in Henderson and Ioannides (1983) and Fu (1991). The analysis recognizes that housing choices involve both consumption and investment motives and that housing investment of homeowners must be at least as great as their housing consumption. The conflicts between the consumption and investment motives due to the inability of the household to liquidate the future gains of investment for current benefits are illustrated. For an illiquid household, the analysis shows that the effects on housing choices of current and future incomes, as well as those of expected housing price appreciation and uncertainty as to the future price of housing, depend on the relative magnitude of the income effect and substitution effect of future wealth, which, in turn, depend on the behaviour of future risk aversion and the rate of intertemporal utility substitution. A weakness of the analysis is the omission of other risky assets in a household's portfolio. Brueckner (1993) provides a succinct analysis of the portfolio choices of homeowners in the absence of borrowing constraints, taking into account the correlations among housing investment and other risky assets as well as the housing investment constraint for homeowners. The present analysis complements his, in the sense that it considers portfolio choice under a borrowing constraint, with restricted investment alternatives. The basic insight of the analysis may nevertheless be extended to portfolio choices involving both housing and other risky assets, where, in addition to the housing investment constraint, the liquidity constraint would also contribute to the conflict between the consumption and portfolio- diversification motives. The absence of other risky assets from the portfolio of illiquid households may not be a serious omission, since these households are unlikely to be able to afford portfolio diversification. Finally, though the risky asset in the model is interpreted as housing investment, the analysis applies to the choice problems of illiquid households involving any indivisible asset that generates consumption benefits (an expensive car, for example).
Acknowledgements I wish to thank Robert Helsley, Lawrence Jones, Vernon Henderson, Jan Brueckner, and Ira Horowitz for helpful comments. Any errors remaining are mine.
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Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
Appendix P r o o f o f proposition. When s ~-0, the first-order conditions for w~ and h*
are equivalent. To simplify the notation, let a =- (P - L - R) and/3 -= P(1 + 0 ) - L ( 1 + r). Differentiating v in (1) with respect to hi, subject to the budget constraints (2) and (3), we have the first-order condition: Ov Oh i
av', + E{v;/3} = 0.
(A1)
The second-order condition is 020
ohf
E{v~/3 2} < 0 .
-- Ol2Vtt I +
(A2)
The signs of the comparative statics of w~ apply to those of h*c, as housing is a normal good and h* increases with w~. Statement 1: dh*
,,/02u~ -I
~yl-aOlkoh~]
dw~
_
(A3)
>0,
dh* _
1
E{v2/3
_
dy 1 - 1 - a dy 1 -
>0
J\Oh~/
(14) "
Statement 2: Let p =- ( - v ~ / v ; ) be the constant Arrow-Pratt measure of local
risk aversion: dh*
,,
dy 2 -
/02v\-i
,
02v
l
E{v2/3}~i2)=pE{v2/3}(~i2)<0,
(15)
where the inequality follows from (A1). Since Yl does not change, w~ changes in the opposite direction to h*. Statement 3: yl = ( W + T ) / 2 and Y2 = ( W - T)(1 + r)/2. dh* _ 1 dh* dT 2 dy I
1 dh*(l+r)>0, 2 dy 2
(16)
which follows from Statements 1 and 2.
(0v)
do)T 2 -1 dT = (E{-v~/32} + (1 + r ) a E { v 2 f l } ) 2 ~ , 2 \
=pE{vzfl(fl-(l
( 2 0 2 v ] -1 + r ) a ) } \ Oh~/
Oh i /
(A7)
Let A =- E { v ; f l } / E { v ' 2 } and B ~- E{v;(fl - (1 + r ) a ) } / E { v ; } , a > 0 follows
Y. Fu / Regional Science and Urban Economics 25 (1995) 223-236
235
from (A1) and B I>0 follows from (A1) and the fact that the liquidity constraint is binding so that Ov ~S -- -- v'l -1- (1 + r)E{v;} ~ O. Now the expression in (A7) is positive because E{v;/3(/3 - (1 + r)a)} E{v~}
E{o;(/3 - A)(/3 - (1 + r)a - B)} +A'B E{v~}
> 0,
(A8)
where the first part on the right-hand side of the equation is the correlation between /3 and / 3 - (1 + r)a based on a scaled distribution of 0, and is positive. Statement 4:
dw*~ OW
1__ --_ 1 dw~ 1 +r)>O, 2 d Y l + 2 dY2 "
(A9)
which follows from Statements 1 and 2. dh*dw
-
-
+(1
/~)"Vl av,' = ~o;
( 021 ~ -1
I)~(1~2)0;(1~2)(1 + r)E{v;/3} ) \[2 Ozv] Oh~/-1 ,
(AIO)
where the second equation follows from the assumption of constant local risk aversion. Since
O (O'I(W1) ~
--
O';O;--V'lO~(l+r) ?(-~
,
(All)
it follows from (A1) and ( A l l ) that dh*i/dW>O if and only if O(v'l/v')/ OW
References Artle, R. and P. Varaiya, 1978, Life cycle consumption and home ownership, Journal of Economic Theory 10, 35-58. Brueckner, J.K,, 1993, Consumption and investment motives and the portfolio choices of homeowners, Working paper, University of Illinois at Urbana-Champaign. Brueckner, J.K., 1986, The downpayment constraint and housing tenure choice, Regional Science and Urban Economics 16, 519-204. Fu, Y., 1991, A model of housing tenure choice: Comment, The American Economic Review 81, no. 1, 381-383.
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Henderson, J.V. and Y.M. Ioannides, 1983, A model of housing tenure choice, The American Economic Review 73, no. 1, 98-113. Ioannides, Y.M. and S.S. Rosenthal, 1994, Estimating the consumption and investment demands for housing and their effect on housing tenure status, Review of Economics and Statistics, 76, no. 1, 127-141. Mills, E.S., 1990, Housing tenure choice, Journal of Real Estate Finance and Economics, 3, no. 4, 323-331. Pratt, J.W., 1964, Risk aversion in the small and in the large, Econometrica 32, no. 1,122-137. Ranney, S.I., 1981, Inflation expectations and the demand for housing, American Economic Review 72, no. 1, 143-153. Rothenberg, J., 1983, Housing investment, housing consumption, and tenure choice, in: R.E. Grienson, ed., The urban economy and housing (Lexington Books, Lexington, KY) 29-55.