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Life-cycle theory, inflation, and the demand for housing
JOURNAL
OF URBAN
Life-Cycle
ECONOMICS
l&161-179
Theory,
(1985)
Inflation, and the Demand Housing’
WILLIAM
for
C. WHEATON
Deportments of Economics and Urban Studies and Planning. Massachwetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Received June 21,1983; revised November 29,1983
There has been considerable interest lately in the influence of economywide inflation on the demand for housing. Under suitable equilibrium conditions, such inflation creates three effects, each of which may influence housing decisions. First, inflation raises initial mortgage payments through higher interest rates. Second, it causes such payments to fall rapidly over time in real terms, and third, it creates a real growth in housing equity, as borrowed debt is leveraged against the inflating value of homes. Two empirical studies, one by Kearl [6], and more recently by Follain [3], have concluded that the sum of these impacts is adverse, that is, housing demand is reduced by greater economy-wide inflation. Each suggests that credit constraints may explain this result. In an attempt to understand better the microeconomic foundations of this empirical result, Schwab [9], and Dougherty and Van Order [2] have developed Fisher-type models of intertemporal utility maximization, which include a housing consumption choice. Schwab concludes that with a perfect capital market, inflation should induce no distortion in consumer decisions about housing consumption. Dougherty and Van Order, building on some unpublished work by Poterba [8], argue that with interest deductibility from income taxes, inflation reduces the “after tax cost of housing capital,” and thereby increases housing demand. These conclusions suggest that capital market imperfection must exist, to explain the empirical results, but just what kind? Schwab’s paper addresses this question by introducing a borrowing constraint into a two-period model [9]. The constraint is absolute, although Schwab acknowledges that allowing collateralized borrowing is more realistic. Schwab’s conclusions are two. First, he argues that if the constraint is nonbinding then the perfect capital market results prevail, and second, he demonstrates that if the constraints are binding, the effect of inflation on ‘Support for this research was provided by the Ford Foundation. 161 0094-1190/85 $3.00 Copyright D lYX5 by Academic Press. Inc All rights of reproduction in any form resewed.
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WILLIAM
C. WHEATON
housing demand can be of either sign, is complicated, and depends on many circumstances and parameters. Against this background, the objectives of this paper are to extend and clarify the results of Schwab [9]. In particular, it is useful to rephrase the Fisher model into a continuous life-cycle model, and examine two kinds of credit market constraints; those requiring liquidity and those prohibiting all forms of borrowing (besides an initial housing mortgage). An analysis of these constraints suggestsseveral conclusions. (1) Liquidity constraints prohibit borrowing against future earnings, but allow borrowing against accumulating housing equity (through refinancing, second mortgages, etc.). In this case, greater inflation increases housing demand, as in a perfect capital market, by an amount which is related to the after tax cost of capital. (2) When absolute borrowing constraints are present and binding, inflation reduces housing demand if the real (pretax) interest rate is less than the consumers’ rate of time preference. If it is not, then inflation can increase housing demand-even with binding borrowing constraints. (3) When absolute borrowing constraints are present, but nonbinding, the Schwab assertion that perfect capital market results hold-is not true. The reason is that the equity in housing is not usable. Even if consumers are saving, to tilt their consumption stream forward, borrowing constraints allow them only to dissolve their accumulated nonhousing wealth, not their housing equity. In this situation, the impact of inflation on housing demand becomes ambiguous-at least over moderate inflation levels. It would seem, therefore, that life-cycle theory in the presence of imperfect capital markets, does not automatically explain the Kearl-Follain [3,6] empirical results. Liquidity constrained borrowing, which Tobin [ll] suggests is most realistic on a priori grounds, still produces a positive relationship between housing demand and inflation. Even with absolute borrowing constraints, a negative effect emergesonly when the constraints are binding, and only when real interest rates are less than consumer discount rates. Section II presents the liquidity constrained model and its conclusions, while Section III develops the borrowing constrained model. Some conclusions are offered in Section IV. II. Life-cycle theory was first developed as a way of analyzing intertemporal resource allocation in the presence of only a single lifetime constraint on all forms of wealth. Since the original formulation, Tobin [ll] and others have argued that this perfect capital market never exists. They have suggested
163
DEMAND FOR HOUSING
instead, that even the best organized capital markets operate with liquidity constraints. This means that while individuals may save any amount of their income (and earn the competitive rate of return), at any time they can go into debt only up to the value of their real (easily liquidated) assets. In other words, they cannot borrow against future earnings-loans must be collateralized with a real asset. Within the context of the issues posed in this paper, a liquidity-constrained capital market allows homeowners to save from income in addition to building up housing equity, or alternatively to borrow against that wealth. It is important to remember that since the cost of debt and the return to savings are identical, individuals desiring to save do not set up savings accounts until they have first paid off their housing debt. Were the effective rates of return on assets substantially different, this convenient symmetry would be lost. Given that individuals operate in a capital market which is fluid enough to permit borrowing against real assets,it also makes senseto assume that individuals can adjust their housing consumption over time. The ability to expand or contract housing debt encourages such house trading, although there may be substantial transactions costs associated with selling and buying a house or moving. Rather than trying to model such costs realistically, the theory here considers two extreme alternative cases: that where consumers can instantly and costlessly adjust the level of housing services over time, and that where adjustment costs are sufficient to restrict them to a single choice. Given this discussion, one can imagine a household with fixed real income, y, which begins its life-cycle plan with no wealth, and which goes into debt at time t = 0 to cover the full value of its first house h(0). Over time, it can allow that debt to fall in real terms, and thus as its nominal income grows, its real nonhousing consumption will as well. Alternatively, it can pay off the debt faster, and perhaps build up substantial wealth to use later. Finally, it may increase its debt over time, up to the value of the house. Of course, if it adjusts its housing over time-such adjustments set limits on the changes that are possible in its debt. At the end of its life, if we assume no bequest motive, the rational household uses up its wealth. This means that both initially and at time T, it must be in debt up to the value of its housing. The following differential equation describes these possibilities. d(t) = y - c(t) - h(t)[(l
a(O)= a(T)> u(r) 2 0 all t
0
- +)r - +i] + u(t)[(l
- +)r - +i]
(1)
(2) (3)
164
WILLIAM
C. WHFATON
where a(t) y c(t) h (t ) i, r, +
= = = = =
total (housing and other) wealth in real terms, real income, real nonhousing consumption, housing consumption (in real dollars), inflation rate, real interest rate, tax rate.
In (l)-(3), if a(t) = 0, the full value of housing is encumbered with debt, whose annual cost is (1 - +)r - (pi. This expression is analogous to the real, after tax user cost of capital discussedby Dougherty and Van Order [2], as well as others. Conversely, when a(t) = h(t), there is no debt or additional income earning assets. Finally, if a(t) > h(t), the household has accumulated assetsbeyond the value of its house-yielding an after-tax real rate of return (1 - +)r - +i. Equation (1) says that the change in the households’ asset position equals its residual income minus consumption minus (plus) the cost (income) of debt (net wealth). The liquidity constraint in (3) requires that one’s overall net wealth always be positive. The equation of motion (1) is similar to that in a model developed by Artle and Varaiya [l] to investigate tenure choice by households over their life cycle. The current formulation differs primarily through the inclusion of taxes, inflation, and the level of housing services h. The objective of consumers in this model, when faced with the intertemporal budget constraint (l)-(3) is to select the time path of consumption c(t) and housing services h(t), which maximize the present value of utility (4), given a subjective discount rate m.
subject to (l)-(3). The optimal control problem in (4) is easily analyzed by applying the maximum principle. Denoting p(t) as the costate variable, and h(t) as the continuous Lagrangian applying to the nomegative restriction on u(t), the Hamiltonian of the problem is H = u(h(t),c(t))e-“’ +p(t)[y
- c(t) +((I
- @Jr - +i)(u(t)
- h(t))1 + A(t)u(t). (5)
D!ZMAND FOR HOUSING
165
The first-order conditions along any optimal c, h trajectory are (6)-(10) below: &A ace
--mf
= p(t)
(6)
au - mf = p(t)[(l dhe
- +)r -
$4
ffJ(t> = - [PM1 - (P>r - (Pi)+ WI A( = 0, A(t)2 0, a(t) 2 0 p(T)=O=a(T).
(8)
all 0 I t I T
(9) 00)
The nature of this solution is most easily analyzed by first considering the case where the constraint is nonbinding over all t. In this situation a(t) > 0, and A(t) = 0, and from (6) to (8) the following three equations can be used to solve for c, h, and p at each point in time. -[(l
p(t) = Poe
au x
g/g
-#)r-$i]?
[m-(1-.#a)r+bilt =
Poe
(14
= (1 - +)r - +i.
In the case where the constraint is binding, a(t) = 0, k(t) = 0, and A(t) > 0. Combining (1) with the marginal conditions (6) and (7), yields the following two equations to be solved for the two variables c and h. c = y - h [(l - (p)r - +i] -:a/$
(14)
= (1 - +)r - +i.
It is clear that the two solution cases differ substantially. When the constraint is binding, no asset accumulation or dissolution occurs, and the consumption of housing and other goods is constant over time. When the constraint is nonbinding, however, (12) and (13) suggestthat both c and h either increase or decrease over time as m is less or greater than (1 - $)r - +i. It is important, therefore, to establish the circumstances under which the constraint is likely to be binding. PROPOSITION 1. If m < (1 - +)r - $i, then an optimum solution requires that A(t) = 0 and u(t) > 0 over aN 0 I t I T.
This proposition can be demonstrated most easily by contradiction. If the constraints are binding over some subinterval from 0 to T, then over that
166
WILLIAM
C. WHEATON
interval there must exist a A(t) > 0 such that
a% .
see
-ml
_ & -qm - ace
-(l
- +)r + +i] -h.
This equation follows from differentiating (6) and applying (8). However, since a2u/ac2 < 0 by convexity, and m - (1 - +)r + Cpi< 0 by assumption, k must be positive. This is impossible if the constraints are binding. The above condition shows that only when m 2 (1 - #)r - cpi can there exist a X(r) 2 0 such that ? = 0 (the solution when the constraint is binding). This suggeststhe next proposition. PROPOSITION 2. If m 2 (1 - $)r - +i then an optimum solution requires that A(t) > 0 and a(t) = 0, over all 0 I t I T. Again this proposition can be shown by contradiction. Suppose m 2 (1 - +)r - (pi, and over some subinterval t, I t I t,, the constraint is nonbinding. By assumption until t1 the constraint is binding and so a(tl) = 0. In order to be nonbinding after t,, ir(tl) > 0. If m > (1 - $)r - +i, then (12) and (13) imply t -C0 and ir < 0, and from (13) this implies ii(t) > 0 for t, < t I t,. In other words, asset accumulation must grow continuously and at an increasing rate over the interval since c and h are decreasing all the time. At the end of the interval, t,, a large amount of wealth will be left which cannot be disposed of (if t, = T) or which must be instantly disposed of (if t, < T and the constraint binds from t, to T). These outcomes are not optimal and violate the terminal constraint. It should be pointed out that Proposition 2 holds over all t only when a(T) must equal zero. If the individual must (or desires to) leave some wealth, then when m > (1 - +)r - +i, the constraint binds until some t, after which c and h fall, and assetsrise just sufficiently so that at T, the required positive assetposition is obtained. In summary, the magnitude of the consumer’s discount rate m relative to the real after-tax cost of capital (1 - +)r - +i determines whether the household wants to increase or decreaseutility over time. In the former case, it accumulates assetsinitially and then later consumes them, in order to tilt its consumption forward. In the latter case,it wants first to go into debt and then to pay it off, but the liquidity constraint prevents this. Its next best alternative is to operate along the constraint with constant levels of c and h -no matter how much m exceeds(1 - +)r - +i. If, on the other hand, m is less than (1 - +)r - +i, then the magnitude of the difference is important. From (11) to (13), the more the term (1 - cp)r - (pi exceedsm, the greater the increase in c and h over time. Such a slant in c and h can be
DEMAND
FOR HOUSING
167
accomplished only through a more rapid accumulation and then dissolution of assets. The discussion so far has ignored what is perhaps the most interesting feature of the model-the choice of housing consumption relative to other goods. In both the constrained and unconstrained solutions, the same marginal condition appears, in (13) and (15). This condition establishes that the relative shadow price of housing is (1 - +)r - +i. Particularly interesting is that this term can become negative for reasonable values of cp, r, and i. This is the case, for example, if the real interest rate is 4%, the tax rate 25%, and inflation greater than 12%. Alternatively if the tax rate is 501, the expression becomes negative whenever inflation exceeds the real interest rate. In either of these examples, as (1 - +)r - +i approaches zero, the only way that the marginal condition can be satisfied (with a regular indifference surface) is for housing consumption to become infinite. While the relative consumption of c and h depends exclusively on the term (1 - (p)r - (pi, the absolute levels of consumption are also determined by the degree of tilt in the time paths of c and h. As discussed above, this depends on the magnitude of (1 - c#I)~- +i in comparison to m. Thus as the term (1 - +)r - +i decreases,two changes happen in the time paths of c and h. First, both paths become more nearly level over time, and second, the path of h is raised relative to that for c. PROPOSITION 3. If rp or i increases, or r decreases,then at any t h rises relative to c, while the gradients of each rise less steeply over time-yielding constant consumptionpaths if (1 - +)r - +i I m.
Given any change in the term (1 - (p)r - +i, then, the impact on h at any time depends on the sum of a gradient effect and a relative price effect. If the term decreases,for example, because C#Ior i increase, then at t = 0, both the gradient and relative price effect imply a higher h(0). At t = T, however, the gradient and relative price effects operate in opposite directions. Thus the net impact on h(T) is difficult to determine without specifying the utility function. It is perhaps useful at this point to introduce a liquidity-constrained model, in which the level of housing consumption is held fixed over time rather than varying continuously. This assumption might be more reasonable, for example, if there were high transactions costs associated with purchasing a house and moving. The consumption of other goods, however, and the accumulation of assetsvary as before. The objective function and equation of motion remain as in (3)-(4); only h is no longer treated as a continuous control variable. Rather, it now becomes a static, optimizable constant. In this case, the solution conditions (7) and (9)-(11) still hold, but the marginal condition for optimal housing consumption (8) is changed to
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WILLIAM
C. WHEATON
the static optimization condition,
JoT~e-“‘dr
= ir&)[(l
- +)r - +i] dr.
(16)
If the constraint is nonbinding (h(t) = 0, a(t) > 0), then (11) still defines the solution equation for the costate variable and when this is combined with (16), the marginal efficiency condition for housing consumption becomes
-“‘dr=po(l_,-T((l-+)r-+‘pi))07) Recalling from (12) that at t = 0, au/& utility is separable, (17) reduces to
= pa, and assuming further that
08)
Thus in the unconstrained case,(ll), (12), and now (17) are used to solve for p(t), c(t), and h. If the constraint is binding then the solution in the case of a constant h is the same as when h is variable-that is (14) and The impacts of a change in up,r, or i in this version of the model are also similar-with the exception that only c, and not h, is varied over time. As (1 - +)r - (pi decreases(because C$and i increase, or r decreases)then (18) implies that h must rise relative to c(0). Since (12) implies that c(t) becomes less slanted with time, c(0) must increase absolutely and therefore h must as well. As (1 - +) - +i decreases to less than the consumers’ discount rate m, the constraint becomes binding and the levels of c and h increase through (14) and (15). For illustration, a set of simulations may be undertaken, based on the log linear utility function: u = cylogh + /3loge. Using this utility function, (ll)-(13) are solved for c, h, and p as functions of time and the constant pa. These are then substituted into the equation of motion and p0 is determined so that from the starting point of n(O) = 0, a(T) winds up equaling zero. The model in this form is easy to solve, given specific utility parameters, for both the case in which h can vary over time and with h constrained to be constant. The ratio of a//3 in both simulations was set at around 0.5 under the observation that roughly one-third of household
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DEMAND FOR HOUSING
TABLE 1 Simulated Consumption: Liquidity-Constrained Capital Market (Consumption in $1000’~) Simulation 1
2
3
4
5
r T
0.04 2 0.04
‘D = (1 - +)r - +i
Any 0.04
-m+D
10
0.04 t .04 0.5 0.02 0.01 SO
0.04 0.02 0 Any 0.04 0.02
0.04 0.01 0.25 0.04 0.02 0.01
0.16 0.02 0.5 0.04 0.06 0.04
13.4 13.4 165.0 165.0 0
13.4 13.4 660.0 660.0 0
10.3 20.4 127.0 251.0 60.0
11.6 16.4 287.0 403.8 28.0
8.2 32.0 67.3 262.3 136.0
11.6 16.4 145.0 204.0 32.5
13.4 13.4 165.0 0 Bind
13.4 13.4 660.0 0 Bind
10.3 20.4 165.0 38.0 N.Bind
11.6 16.4 331.0 17.0 N.Bind
8.2 32.0 110.0 88.0 N.Bind
11.6 16.4 165.0 21.5 N.Bind
Parametersa
-
6 0.08 0.03 0.25 0.08 0.04 0.01
Solution for variable h 44
c(T) h(O) h(T) max a(r)* Solution for tixed h 40)
47’) h
max a(t) Constraint
ar = real capital costs, m = real discount rate, i = inflation rate, + = tax rate, y = $20,000, utility = a log h + fi log c where a = 0.33, j? = 0.67. *Constraints require a(O) = a(T) = 0, where T = 35 years.
resources are spent for housing services.The results of these simulations are in Table 1. In Table 1, the first two simulations involve parameter values which lead to a binding constraint. In keeping with a log linear utility function the consumption of other goods is always a constant 67% of income, while the product of the discount rate (D = (1 - cp)r - cpi) and housing consumption equals the residual 33% of income. The contrast between the first two simulations, in terms of housing consumption, is dramatic. The introduction of tax deductability with a 50% tax rate, when there is a modest 2% inflation, causes housing consumption to quadruple. This results because as stated in Propositions 1-3, the constrained solutions are shaped solely by the magnitude of the term (1 - +)r - (pi. With the given parameters, tax deductability reduces this term by 75%. These results stand in sharp contrast
170
WILLIAM
C. WHEATON
to the view that inflation, by raising nominal interest rates, reduces housing demand. In the model at hand inflation never decreasesdemand; rather it most often exerts a very powerful positive effect on housing consumption. In the third through sixth simulations, the parameters have been chosen so that the constraint is nonbinding. In all of these cases,then, either both c and h increase over time (when the latter is variable), or just c increases (when h is constrained to be constant). This slant in consumption is achieved by having households save for approximately the first 20 years, reaching the wealth level described in the line labeled max a(t). After that, wealth is consumed up to the 35th (or terminal) year. In simulation 5, this tilt in consumption is greatest, as is the accumulated assetposition, since the difference between R and m is the largest among the six simulations. In keeping with Proposition 3, comparison of simulations 4 and 6 shows that when both m and D increase by identical amounts, the consumption gradient remains the same, but average housing consumption falls. Similarly, as between simulations 6 and 3, if D is fixed and m decreases(i.e., -m + D increases) then the growth in consumption over time increases, but the average level of h remains about the same. If i or + individually increase, or r decreases,then both D and -m + D are reduced. This creates both a reduction in the slope of consumption over time and an increase in the average level of h consumption. For example, if r = 0.04, m = 0.01, (p = 0.25, and inflation is 0.04, then D = 0.02 and - m + D = 0.01, and the results would be exactly as shown in simulation 4. If all other parameters remained constant, but inflation increased from 0.04 to 0.08, then the results would be exactly as in simulation 2. The consumption gradient would be flat and the new constant level of housing consumption would be double the average level prevailing at the lower inflation rate. Thus between simulations 4 and 2, an increase from 4% in inflation to 8% would cause housing consumption to jump by 180% at t = 0 while at 1 = T, it would increase by 65%. From the propositions and simulations, then, it is clear that liquidity-constrained borrowing is not very constraining-at least in how it affects housing consumption. The strong positiue impact of inflation on housing demand, which results in this model, is almost identical to that hypothesized by earlier authors in a perfect capital market with no constraints. The reasons why liquidity constraints appear to matter very little are two. First, when individuals desire to slant their consumption forward, the constraints are simply nonbinding. Second, and more importantly, when the constraints are binding, they permit a constant share of real income to be spent on housing-insulating the consumer from the normal tilt in real mortgage payments which accompanies inflation. With a fixed payment mortgage, nominal housing debt is constant, but equity grows ammally by ih. If the individual can borrow against this each year, and use the funds to consume
DEMAND
FOR HOUSING
171
other goods, then in the absence of a tax subsidy, his budget constraint is c = y - h(i + r) + ih = y - rh. The effective cost of housing is therefore only r, and economy-wide inflation induces no distortion in behavior. Once tax deductability is introduced, the annual cost of debt service drops to h(i + r)(l - c#J),but the individual can still borrow the untaxed amount ih. Thus when the constraint is binding, the level of c consumption possible becomes y - h(i + r)(l - $) + ih = y - h[(l - +)r - @iI. As C#I or i increase sufficiently, the individual finds that if he increaseshis housing consumption, the rise in his debt costs is less than the extra money he can borrow annually against his mounting equity. In this case he does not have to sacrifice h to consume more c, but instead can consume more c by consuming more h ! III. It is clear from the previous discussion that more stringent constraints than liquidity must be placed on consumer borrowing if any negative relationship between housing consumption and inflation is to be obtained from a life-cycle model. In this section, it is assumed that while consumers may save any amount, and therefore slant their consumption forward, they cannot engage in any borrowing beyond an initial housing mortgage. Given a falling real value of fixed mortgage payments, then, the consumption of other goods, and therefore utility, is automatically tilted toward the future. By additional saving, a consumer may increase this tilt, but he cannot reduce or offset it by borrowing against his accumulating housing equity. Without the ability to borrow against this wealth, the consumer must end his life in a positive assetposition. Given that refinancing or additional borrowing is not possible on one’s house, it makes little senseto allow housing consumption to vary over time. Surely if one can freely move and adjust his housing consumption, it is primarily because additional credit is available. A credit constraint on one’s existing house, but not on a new purchase, would be difficult to enforce. Thus, the models in this section assume that the consumption of housing services is fixed over time. In this sense,they can be compared to (16) and (17) in the previous section. Because housing wealth is inaccessible in this model, it will be treated separately from other assetswhich the consumer might accumulate through savings. Thus while in Section II, the term a(t) referred to total assets, here it will refer to only nonhousing assets. With the exception of this new definition for nonhousing assets,all definitions from Section II remain the same, and the consumer’s borrowing-constrained optimization problem is (19)
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WILLIAM
C. WHEATON
subject to : ci =y - c - h(i + r)(l - $)e-”
+ a[(1 - +)r - Cpi]
u(r) 2 0 a(0) = u(T) = 0.
(20)
The constraint in (20) reflects the fact that the real value of the initial nominal mortgage payment h(l - $)( i + r) declines over time with the rate of inflation (P). The constraint also assumes a real after tax return on savings of only (1 - +)r - $ai.Thus unlike the liquidity-constrained model, the return to investment and the cost of housing services are different here, once borrowing constraints are introduced. As with the liquidity constrained model, the solution to (19)-(20) is different, depending on whether the borrowing constraint is binding or not. In either case, the first- and second-order conditions for the optimization of housing consumption, h, are dw z=Z=
Tau o ae -“‘dr -(l J
- +)(i + r)~‘$e-(‘+“)‘dt
d2w dZ -=,,
= 0 (21)
If the borrowing constraint is binding over all T I t I 0, then ci = a = 0, and is determined from the equation of motion, c(t) = y - h(1 - +)(i -I- r)e-”
if a(t) = 0.
(24
If the borrowing constraint is nonbinding over the entire interval from 0 to T, then the optimum level of c is determined with the following pair of conditions-derived by forming the Hamiltonian for the optimizing problem (19)-(20).
au --m'= p(f) ac p(t) = -f(l)[(l
-e
- 4+- - @I
if a(t) > 0.
(24)
In (23)-(24) p(t) is the dynamic Lagrangian associated with (20). Integrating (24) yields
au =
_
ac
poeIm-(l-$)r+w
if a(t) ) 0.
(25)
Thus (21) and (22) can together be used to solve for h and c(t) when the borrowing constraint is binding, while (21) and (25) are used to determine
DEMAND
FOR HOUSING
173
the solution when the constraint is nonbinding. It should be mentioned that in the latter case, the constant of integration, p,,, is determined with the equation of motion so that starting at a(0) = 0, a(t) rises and then falls to a(T) = 0. Taking (21) and (22), the impact of greater inflation on housing demand, when the borrowing constraint is binding, can be determined by total differentiation. This yields dh z=
dZ dZ -- dZ di x”OasiPO. I
(26)
PROPOSITION4. When borrowing constraints are present and binding, greater injlation reduceshousing demand if the internal discount rate is not less than the real interest rate (m 2 r). If m < r, the sign of dh/di is indeterminant. For the sake of brevity, the lengthy proof of this proposition is contained in Appendix A. It is important to point out that the proposition only establishes a sufficient condition for dh/di < 0. When m 2 r, consumers desire to shift utility toward the present, and so the opposite tilt in consumption, introduced by inflation together with fixed housing payments, should discourage housing consumption. When m < r, on the other hand, it seemspossible for the tilt in mortgage payments to reinforce the consumer’s preference for greater future consumption. Of course, if the consumer wishes to shift utility toward the future, he might also do it through direct savings-in which case the constraints are nonbinding. When the constraint on a(t) is nonbinding, (21) can be simplified to (27) by incorporating (25). -"I&
= ,-,[l
-
e-(l-+')(i+r)T]
(27)
If it is further recalled that at t = 0, &/Lk must equal p,,, and if in addition, the utility function is separable, then (27) above can be further simplified to
This equation is similar to (18) in the liquidity-constrained model, except that the exponent contains the term (1 - cp)(i + r) rather than (1 - +)r +i. The difference is important. In the borrowing constrained model, as inflation increases, so will the right-hand side of (28), implying that the ratio
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WILLIAM
C. WHEATON
of h to c at time zero must fall. By contrast, in (18), inflation increased the ratio of h to c(0). Equally important, however, is the fact that as i increases, (25) implies that future consumption is reduced and that c(0) rises. In the liquidity-constrained model, this rise in c(O), together with a rise in the ratio of h to c(0) guarantees that h increases in response to higher inflation. In the borrowing constrained model, however, the effect is ambiguous. As c(0) increases, but the ratio of h to c(0) falls, the net effect on h is indeterminant. Thus when borrowing constraints are present, but nonbinding, the results are not at all as in a perfect or even liquidity-constrained capital market. PROPOSITION 5. When borrowing constraints are present, but nonbinding over a consumer’s lifetime, greater inflation has an ambiguous eflect on the demand for housing.
An interesting implication of borrowing constraints, which does not result when there are only liquidity constraints, is that under a range of circumstances, mixed solutions are possible. That is, over some subinterval of time, the borrowing constraint can be binding, while afterward, it may not. The reason is that even if a(t) equals zero, there is an irremovable growth in consumption over time through the fixed level mortgage payment-and this varies over time. Equation (22) shows that the growth in consumption is greatest in the early years of a mortgage payment and less later on. In fact, as inflation increases, the growth in consumption over some initial period increases, but then decreases over a later period. This suggests that if individuals save some income to slant their consumption further forward, they do so when that consumption is less slanted from the mortgage effect-or later in their life. To see this formally, it is clear from (25) that a necessary condition for the borrowing constraint to be nonbinding, is that m - (1 - $)r + Qi be negative. Just being negative is not enough, however, for it must be sufficiently negative to overcome the percentage reduction in au/& over time-due to the mortgage tilt. Only in this case will additional savings be induced. Differentiating (25) then, it must be true that when the borrowing constraint is nonbinding: m -(1 - +)r + +i = CJj$ > (ih(1 - +)(i + r)eVif)g.
(29)
At t = 0, the right-hand side of (29) can be quite large, necessitating an unrealistically large difference between (1 - +)r - +i and m in order to induce savings. Further along in the mortgage term, however, the right-hand side of (29) will become quite small, generating the possibility of savings.
DEMAND
175
FOR HOUSING
TABLE 2 Simulated Housing Consumption with Borrowing Constraints Housing consumption ($1000~) when inflation rate = (constraint: B, NB, M)’
Parameters” Simulation
r
m
+
0
4
8
(1)
0.04 0.04 0.04 0.04 0.08
0.04 0.08 0.04 0.08 0.04
0 0 0.5 0.5 0.5
165(B) 165(B) 33qB) 330(B) 165(B)
129(B) 118(B) 258(B) 237(B) 172(B)
112(B) 99(B) 224(B) 198(B) 168(B)
78(B) 180(B) 157(B) 150(B)
lll(NB) 2868(l) WW 4864(l)
116(NB) 425(l) 81(NB) 2142(l)
124(B)
lWB1
(2) (3) (4) (5)
(6) (7)
0.08 0.04 0.25 (annual savings beginning at r =) 0.08 0.04 0 (annual savings beginning at t =)
0 83(M) 2414(9)
16
WBI
0 74( M: 3561(10)
“All simulations use the utility function u = nlog h + j? log c, where (I = 0.33, /3 = 0.67, and ri income = 20,000, and T = 35. ‘B = binding constraint over all t = 0,35; M = nonbinding initially, binding later: NB nonbinding over all t = 0,35.
Thus the most likely behavior finds the borrowing constraints binding throughout one’s life if m > (1 - +)r - +i. On the other hand if m < (1 +)r - (pi, then additional savings may occur but only after the growth in consumption, due to fixed mortgage payments, begins to wear off. Proposition 7 summarizes these various effects of parameter values on both savings behavior and housing consumption. PROPOSITION 6. When m > r, the borrowing constraint will be binding and inflation decreasesthe demand for housing. When r > m > (1 - +)r - +i, the borrowing constraints continue to be binding, but infition has an ambiguous impact on housing demand. When (1 - +)r - C#I~ > m, the borrowing constraint can becomenonbinding, particularly with the passage of time, and the eflect of infition on housing demand remains ambiguous.
To illustrate all of these results, a set of simulations was undertaken, again using a logarithmic utility function, with the housing expenditure share set at 0.33. Simulations were undertaken with the consumer’s discount rate m equaling 0.04 and 0.08, while the real cost of capital was inversely 0.08 and 0.04. Tax rates were varied between 0 and 0.5, while inflation ranged from 0 to 16%. The results are presented in Table 2. The first five simulations in Table 2 all involve parameters in which the borrowing constraint is binding throughout a consumer’s life-under all of
176
WILLIAM
C. WHEATON
the tested levels of inflation (i.e., m 2 (1 - +)r - $i). The results show that when m is greater than or equal to r, inflation reduces housing demand, but when r > m > (1 - $)r - +i, greater inflation can initially increase housing demand, before eventually decreasing it (simulation 5). In the sixth simulation, at low levels of inflation, (1 - $)r - c#G is sufficiently greater than m to induce savings throughout one’s lifetime. In other words, consumption growth due to fixed mortgage payments is not enough to satisfy the preference for future consumption. As inflation increases, however, the mortgage effect becomes more than enough, and savings activity ceases.The sixth simulation also creates a housing pattern in which demand initially increasessignificantly with inflation, from $111,000 at OS to $124,000 at 8% before eventually declining as inflation reaches higher levels. The seventh simulation shows how inflation can have little effect on housing consumption, but alter savings behavior quite dramatically. When r substantially exceedsm, and there is no income tax deductibility, consumers generally wish to postpone consumption through savings. As higher inflation pushes consumption into the future, through the mortgage effect, however, such saving becomes unnecessary and is either reduced or postponed. Propositions 4-6, then, together with these simulation results, demonstrate that the presence of true borrowing constraints-even if they are not binding-results in different behavior from that in perfect or liquidity-constrained models. Housing consumption can initially rise with greater inflation before eventually falling as inflation reacheshigher levels. This can even occur when the constraints are binding. In fact only when they are both binding, and m I r, will greater inflation unambiguously reduce the demand for housing. IV. The results of this paper have suggestedthat only under a very particular set of circumstances, will life-cycle theory yield the conclusion that housing demand is reduced by higher economy-wide inflation-which both increases mortgage rates and causes housing price appreciation. The assertion by Kearl [6], and the conclusions by Schwab [9] that it is credit constraints which cause this effect, must be highly qualified. First of all, if the constraints on credit merely require that any loan be collateralized with a real asset, rather than future human capital, then greater inflation creates an unambiguous increase in housing demand-by magnitudes which simulations suggest are quite high. If in addition credit constraints are extended to prohibit all kinds of borrowing, beyond an initial mortgage debt, then housing demand can be shown to decreasewith high inflation, but onb if the consumer’s discount rate exceeds the real, pretax, return to capital.
DEMAND
FOR HOUSING
177
When the discount rate is less than the real, pretax cost of capital, the impact of inflation on housing demand is very mixed, and generally small-at least in the range of inflation experienced by most western countries (O-16%). It is important to realize that the determining factor is not whether the borrowing constraints are binding or not, because this depends on more complicated considerations than whether the discount rate is larger or less than the cost of capital. What is somewhat puzzling about these results is that the conditions necessary to produce negative inflationary impacts would seemto be overly restrictive. At least in the United States, consumers have generally had the ability to either refinance their homes or take out second or third consumer loans against their equity. This clearly creates an opportunity set which most closely resembles the liquidity-constrained model. When or whether consumers choose to exercise this option is irrelevant, what matters only is that it is available. To produce the empirical results estimated by Kearl [6] and Follain [3], absolute prohibitions on borrowing must exist-together with a real pretax return to capital that is less than household rates of time preference. APPENDIX A (1) From (22) and (23), with a little manipulation z
= ir[-1
+(i + r)t]e-(‘+m)‘[u,-
UC&e-“(i + r)(l
- $)I(1 - (p)dt
where Z,(t)=[-l+(i+r)t]e-(i+m)r>O
ift>:
CO Z,(t)=
iftc-
1 1+r
1 i+r
[24,-u,,he-“(i+r)(l-+)](l-+)>O
and
dz, dtco
by convexity.
all t
178
WILLIAM
C. WHEATON
Now; p,(t)z,(t)dt
=~‘*z,(t)z,(t)dt
+ /Tz&)Z*(t)dt
0
0 s
I’
/‘*Z,(t)Z,(t*)
dr + /Zl(t)Z,(t*) 1+
0
zz Z,(t*fz,(t)
dt
dt
0
where t* = - 1 i+r
since dZ 2<0 dt
and
Ut*)
t < t*
< z&L (z&*)
’ z,(t)7
t > t*).
Thus if joTZl(t) dt -c 0 so will be dZ/di. Now ii,(t)
dt =i’-
e-(i+m)rdt + /‘t(i+
r)e-(i+m)fdt
(when the latter is integrated by parts),
ifm2r, if m < r.
Q.E.D. REFERENCES
1. R. Artle and P. Varaiya, Life cycle consumption and homeownership, J. .&on. Theory, 18, 38-58 (1978). 2. A. Dougherty and R. Van Order, Inflation, housing costs and the consumer price index, AER, 72 (l), 154-165 (1982). 3. J. Follain, Does inflation affect real behavior: The case of housing, Southern Ron. J., January (1982). 4. P. Hendershott, Real user costs and the demand for single family housing, Bookings Papers on Economic Activity, No. 2, 1980.
DEMAND
FOR HOUSING
179
5. D. W. Jorgenson, The theory of investment behavior, in “The Determinants of Investment Behavior” (Ferber, Ed.), NBER, New York (1967). 6. J. R. Kearl, Inflation, mortgages and housing, J. PO/. Econ., 87 (5), 1115-1138 (1979). 7. F. Modighani and D. Lessard, @is.), “New Mortgage Designs for Stable Housing in an Inflationary Environment,” Federal Reserve Bank of Boston (1976). 8. J. Poterba, Inflation, income taxes, and owner occupied housing, NBER, Working Paper 553, 1980. 9. R. M. Schwab, Inflationary expectations and the demand for housing, AER, 72 (l), 154-165 (1982). 10. L. C. Thurow, The optimum lifetime distribution of consumption expenditures, AER, 63, 344-353 (1972). 11. J. Tobin, Wealth, liquidity and the propensity to consume, in “Human Behavior in Economic Affairs”, Strumpel, Morgan and Zahn (Eds.), Elsevier, New York (1972).
OF URBAN
Life-Cycle
ECONOMICS
l&161-179
Theory,
(1985)
Inflation, and the Demand Housing’
WILLIAM
for
C. WHEATON
Deportments of Economics and Urban Studies and Planning. Massachwetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Received June 21,1983; revised November 29,1983
There has been considerable interest lately in the influence of economywide inflation on the demand for housing. Under suitable equilibrium conditions, such inflation creates three effects, each of which may influence housing decisions. First, inflation raises initial mortgage payments through higher interest rates. Second, it causes such payments to fall rapidly over time in real terms, and third, it creates a real growth in housing equity, as borrowed debt is leveraged against the inflating value of homes. Two empirical studies, one by Kearl [6], and more recently by Follain [3], have concluded that the sum of these impacts is adverse, that is, housing demand is reduced by greater economy-wide inflation. Each suggests that credit constraints may explain this result. In an attempt to understand better the microeconomic foundations of this empirical result, Schwab [9], and Dougherty and Van Order [2] have developed Fisher-type models of intertemporal utility maximization, which include a housing consumption choice. Schwab concludes that with a perfect capital market, inflation should induce no distortion in consumer decisions about housing consumption. Dougherty and Van Order, building on some unpublished work by Poterba [8], argue that with interest deductibility from income taxes, inflation reduces the “after tax cost of housing capital,” and thereby increases housing demand. These conclusions suggest that capital market imperfection must exist, to explain the empirical results, but just what kind? Schwab’s paper addresses this question by introducing a borrowing constraint into a two-period model [9]. The constraint is absolute, although Schwab acknowledges that allowing collateralized borrowing is more realistic. Schwab’s conclusions are two. First, he argues that if the constraint is nonbinding then the perfect capital market results prevail, and second, he demonstrates that if the constraints are binding, the effect of inflation on ‘Support for this research was provided by the Ford Foundation. 161 0094-1190/85 $3.00 Copyright D lYX5 by Academic Press. Inc All rights of reproduction in any form resewed.
162
WILLIAM
C. WHEATON
housing demand can be of either sign, is complicated, and depends on many circumstances and parameters. Against this background, the objectives of this paper are to extend and clarify the results of Schwab [9]. In particular, it is useful to rephrase the Fisher model into a continuous life-cycle model, and examine two kinds of credit market constraints; those requiring liquidity and those prohibiting all forms of borrowing (besides an initial housing mortgage). An analysis of these constraints suggestsseveral conclusions. (1) Liquidity constraints prohibit borrowing against future earnings, but allow borrowing against accumulating housing equity (through refinancing, second mortgages, etc.). In this case, greater inflation increases housing demand, as in a perfect capital market, by an amount which is related to the after tax cost of capital. (2) When absolute borrowing constraints are present and binding, inflation reduces housing demand if the real (pretax) interest rate is less than the consumers’ rate of time preference. If it is not, then inflation can increase housing demand-even with binding borrowing constraints. (3) When absolute borrowing constraints are present, but nonbinding, the Schwab assertion that perfect capital market results hold-is not true. The reason is that the equity in housing is not usable. Even if consumers are saving, to tilt their consumption stream forward, borrowing constraints allow them only to dissolve their accumulated nonhousing wealth, not their housing equity. In this situation, the impact of inflation on housing demand becomes ambiguous-at least over moderate inflation levels. It would seem, therefore, that life-cycle theory in the presence of imperfect capital markets, does not automatically explain the Kearl-Follain [3,6] empirical results. Liquidity constrained borrowing, which Tobin [ll] suggests is most realistic on a priori grounds, still produces a positive relationship between housing demand and inflation. Even with absolute borrowing constraints, a negative effect emergesonly when the constraints are binding, and only when real interest rates are less than consumer discount rates. Section II presents the liquidity constrained model and its conclusions, while Section III develops the borrowing constrained model. Some conclusions are offered in Section IV. II. Life-cycle theory was first developed as a way of analyzing intertemporal resource allocation in the presence of only a single lifetime constraint on all forms of wealth. Since the original formulation, Tobin [ll] and others have argued that this perfect capital market never exists. They have suggested
163
DEMAND FOR HOUSING
instead, that even the best organized capital markets operate with liquidity constraints. This means that while individuals may save any amount of their income (and earn the competitive rate of return), at any time they can go into debt only up to the value of their real (easily liquidated) assets. In other words, they cannot borrow against future earnings-loans must be collateralized with a real asset. Within the context of the issues posed in this paper, a liquidity-constrained capital market allows homeowners to save from income in addition to building up housing equity, or alternatively to borrow against that wealth. It is important to remember that since the cost of debt and the return to savings are identical, individuals desiring to save do not set up savings accounts until they have first paid off their housing debt. Were the effective rates of return on assets substantially different, this convenient symmetry would be lost. Given that individuals operate in a capital market which is fluid enough to permit borrowing against real assets,it also makes senseto assume that individuals can adjust their housing consumption over time. The ability to expand or contract housing debt encourages such house trading, although there may be substantial transactions costs associated with selling and buying a house or moving. Rather than trying to model such costs realistically, the theory here considers two extreme alternative cases: that where consumers can instantly and costlessly adjust the level of housing services over time, and that where adjustment costs are sufficient to restrict them to a single choice. Given this discussion, one can imagine a household with fixed real income, y, which begins its life-cycle plan with no wealth, and which goes into debt at time t = 0 to cover the full value of its first house h(0). Over time, it can allow that debt to fall in real terms, and thus as its nominal income grows, its real nonhousing consumption will as well. Alternatively, it can pay off the debt faster, and perhaps build up substantial wealth to use later. Finally, it may increase its debt over time, up to the value of the house. Of course, if it adjusts its housing over time-such adjustments set limits on the changes that are possible in its debt. At the end of its life, if we assume no bequest motive, the rational household uses up its wealth. This means that both initially and at time T, it must be in debt up to the value of its housing. The following differential equation describes these possibilities. d(t) = y - c(t) - h(t)[(l
a(O)= a(T)> u(r) 2 0 all t
0
- +)r - +i] + u(t)[(l
- +)r - +i]
(1)
(2) (3)
164
WILLIAM
C. WHFATON
where a(t) y c(t) h (t ) i, r, +
= = = = =
total (housing and other) wealth in real terms, real income, real nonhousing consumption, housing consumption (in real dollars), inflation rate, real interest rate, tax rate.
In (l)-(3), if a(t) = 0, the full value of housing is encumbered with debt, whose annual cost is (1 - +)r - (pi. This expression is analogous to the real, after tax user cost of capital discussedby Dougherty and Van Order [2], as well as others. Conversely, when a(t) = h(t), there is no debt or additional income earning assets. Finally, if a(t) > h(t), the household has accumulated assetsbeyond the value of its house-yielding an after-tax real rate of return (1 - +)r - +i. Equation (1) says that the change in the households’ asset position equals its residual income minus consumption minus (plus) the cost (income) of debt (net wealth). The liquidity constraint in (3) requires that one’s overall net wealth always be positive. The equation of motion (1) is similar to that in a model developed by Artle and Varaiya [l] to investigate tenure choice by households over their life cycle. The current formulation differs primarily through the inclusion of taxes, inflation, and the level of housing services h. The objective of consumers in this model, when faced with the intertemporal budget constraint (l)-(3) is to select the time path of consumption c(t) and housing services h(t), which maximize the present value of utility (4), given a subjective discount rate m.
subject to (l)-(3). The optimal control problem in (4) is easily analyzed by applying the maximum principle. Denoting p(t) as the costate variable, and h(t) as the continuous Lagrangian applying to the nomegative restriction on u(t), the Hamiltonian of the problem is H = u(h(t),c(t))e-“’ +p(t)[y
- c(t) +((I
- @Jr - +i)(u(t)
- h(t))1 + A(t)u(t). (5)
D!ZMAND FOR HOUSING
165
The first-order conditions along any optimal c, h trajectory are (6)-(10) below: &A ace
--mf
= p(t)
(6)
au - mf = p(t)[(l dhe
- +)r -
$4
ffJ(t> = - [PM1 - (P>r - (Pi)+ WI A( = 0, A(t)2 0, a(t) 2 0 p(T)=O=a(T).
(8)
all 0 I t I T
(9) 00)
The nature of this solution is most easily analyzed by first considering the case where the constraint is nonbinding over all t. In this situation a(t) > 0, and A(t) = 0, and from (6) to (8) the following three equations can be used to solve for c, h, and p at each point in time. -[(l
p(t) = Poe
au x
g/g
-#)r-$i]?
[m-(1-.#a)r+bilt =
Poe
(14
= (1 - +)r - +i.
In the case where the constraint is binding, a(t) = 0, k(t) = 0, and A(t) > 0. Combining (1) with the marginal conditions (6) and (7), yields the following two equations to be solved for the two variables c and h. c = y - h [(l - (p)r - +i] -:a/$
(14)
= (1 - +)r - +i.
It is clear that the two solution cases differ substantially. When the constraint is binding, no asset accumulation or dissolution occurs, and the consumption of housing and other goods is constant over time. When the constraint is nonbinding, however, (12) and (13) suggestthat both c and h either increase or decrease over time as m is less or greater than (1 - $)r - +i. It is important, therefore, to establish the circumstances under which the constraint is likely to be binding. PROPOSITION 1. If m < (1 - +)r - $i, then an optimum solution requires that A(t) = 0 and u(t) > 0 over aN 0 I t I T.
This proposition can be demonstrated most easily by contradiction. If the constraints are binding over some subinterval from 0 to T, then over that
166
WILLIAM
C. WHEATON
interval there must exist a A(t) > 0 such that
a% .
see
-ml
_ & -qm - ace
-(l
- +)r + +i] -h.
This equation follows from differentiating (6) and applying (8). However, since a2u/ac2 < 0 by convexity, and m - (1 - +)r + Cpi< 0 by assumption, k must be positive. This is impossible if the constraints are binding. The above condition shows that only when m 2 (1 - #)r - cpi can there exist a X(r) 2 0 such that ? = 0 (the solution when the constraint is binding). This suggeststhe next proposition. PROPOSITION 2. If m 2 (1 - $)r - +i then an optimum solution requires that A(t) > 0 and a(t) = 0, over all 0 I t I T. Again this proposition can be shown by contradiction. Suppose m 2 (1 - +)r - (pi, and over some subinterval t, I t I t,, the constraint is nonbinding. By assumption until t1 the constraint is binding and so a(tl) = 0. In order to be nonbinding after t,, ir(tl) > 0. If m > (1 - $)r - +i, then (12) and (13) imply t -C0 and ir < 0, and from (13) this implies ii(t) > 0 for t, < t I t,. In other words, asset accumulation must grow continuously and at an increasing rate over the interval since c and h are decreasing all the time. At the end of the interval, t,, a large amount of wealth will be left which cannot be disposed of (if t, = T) or which must be instantly disposed of (if t, < T and the constraint binds from t, to T). These outcomes are not optimal and violate the terminal constraint. It should be pointed out that Proposition 2 holds over all t only when a(T) must equal zero. If the individual must (or desires to) leave some wealth, then when m > (1 - +)r - +i, the constraint binds until some t, after which c and h fall, and assetsrise just sufficiently so that at T, the required positive assetposition is obtained. In summary, the magnitude of the consumer’s discount rate m relative to the real after-tax cost of capital (1 - +)r - +i determines whether the household wants to increase or decreaseutility over time. In the former case, it accumulates assetsinitially and then later consumes them, in order to tilt its consumption forward. In the latter case,it wants first to go into debt and then to pay it off, but the liquidity constraint prevents this. Its next best alternative is to operate along the constraint with constant levels of c and h -no matter how much m exceeds(1 - +)r - +i. If, on the other hand, m is less than (1 - +)r - +i, then the magnitude of the difference is important. From (11) to (13), the more the term (1 - cp)r - (pi exceedsm, the greater the increase in c and h over time. Such a slant in c and h can be
DEMAND
FOR HOUSING
167
accomplished only through a more rapid accumulation and then dissolution of assets. The discussion so far has ignored what is perhaps the most interesting feature of the model-the choice of housing consumption relative to other goods. In both the constrained and unconstrained solutions, the same marginal condition appears, in (13) and (15). This condition establishes that the relative shadow price of housing is (1 - +)r - +i. Particularly interesting is that this term can become negative for reasonable values of cp, r, and i. This is the case, for example, if the real interest rate is 4%, the tax rate 25%, and inflation greater than 12%. Alternatively if the tax rate is 501, the expression becomes negative whenever inflation exceeds the real interest rate. In either of these examples, as (1 - +)r - +i approaches zero, the only way that the marginal condition can be satisfied (with a regular indifference surface) is for housing consumption to become infinite. While the relative consumption of c and h depends exclusively on the term (1 - (p)r - (pi, the absolute levels of consumption are also determined by the degree of tilt in the time paths of c and h. As discussed above, this depends on the magnitude of (1 - c#I)~- +i in comparison to m. Thus as the term (1 - +)r - +i decreases,two changes happen in the time paths of c and h. First, both paths become more nearly level over time, and second, the path of h is raised relative to that for c. PROPOSITION 3. If rp or i increases, or r decreases,then at any t h rises relative to c, while the gradients of each rise less steeply over time-yielding constant consumptionpaths if (1 - +)r - +i I m.
Given any change in the term (1 - (p)r - +i, then, the impact on h at any time depends on the sum of a gradient effect and a relative price effect. If the term decreases,for example, because C#Ior i increase, then at t = 0, both the gradient and relative price effect imply a higher h(0). At t = T, however, the gradient and relative price effects operate in opposite directions. Thus the net impact on h(T) is difficult to determine without specifying the utility function. It is perhaps useful at this point to introduce a liquidity-constrained model, in which the level of housing consumption is held fixed over time rather than varying continuously. This assumption might be more reasonable, for example, if there were high transactions costs associated with purchasing a house and moving. The consumption of other goods, however, and the accumulation of assetsvary as before. The objective function and equation of motion remain as in (3)-(4); only h is no longer treated as a continuous control variable. Rather, it now becomes a static, optimizable constant. In this case, the solution conditions (7) and (9)-(11) still hold, but the marginal condition for optimal housing consumption (8) is changed to
168
WILLIAM
C. WHEATON
the static optimization condition,
JoT~e-“‘dr
= ir&)[(l
- +)r - +i] dr.
(16)
If the constraint is nonbinding (h(t) = 0, a(t) > 0), then (11) still defines the solution equation for the costate variable and when this is combined with (16), the marginal efficiency condition for housing consumption becomes
-“‘dr=po(l_,-T((l-+)r-+‘pi))07) Recalling from (12) that at t = 0, au/& utility is separable, (17) reduces to
= pa, and assuming further that
08)
Thus in the unconstrained case,(ll), (12), and now (17) are used to solve for p(t), c(t), and h. If the constraint is binding then the solution in the case of a constant h is the same as when h is variable-that is (14) and The impacts of a change in up,r, or i in this version of the model are also similar-with the exception that only c, and not h, is varied over time. As (1 - +)r - (pi decreases(because C$and i increase, or r decreases)then (18) implies that h must rise relative to c(0). Since (12) implies that c(t) becomes less slanted with time, c(0) must increase absolutely and therefore h must as well. As (1 - +) - +i decreases to less than the consumers’ discount rate m, the constraint becomes binding and the levels of c and h increase through (14) and (15). For illustration, a set of simulations may be undertaken, based on the log linear utility function: u = cylogh + /3loge. Using this utility function, (ll)-(13) are solved for c, h, and p as functions of time and the constant pa. These are then substituted into the equation of motion and p0 is determined so that from the starting point of n(O) = 0, a(T) winds up equaling zero. The model in this form is easy to solve, given specific utility parameters, for both the case in which h can vary over time and with h constrained to be constant. The ratio of a//3 in both simulations was set at around 0.5 under the observation that roughly one-third of household
169
DEMAND FOR HOUSING
TABLE 1 Simulated Consumption: Liquidity-Constrained Capital Market (Consumption in $1000’~) Simulation 1
2
3
4
5
r T
0.04 2 0.04
‘D = (1 - +)r - +i
Any 0.04
-m+D
10
0.04 t .04 0.5 0.02 0.01 SO
0.04 0.02 0 Any 0.04 0.02
0.04 0.01 0.25 0.04 0.02 0.01
0.16 0.02 0.5 0.04 0.06 0.04
13.4 13.4 165.0 165.0 0
13.4 13.4 660.0 660.0 0
10.3 20.4 127.0 251.0 60.0
11.6 16.4 287.0 403.8 28.0
8.2 32.0 67.3 262.3 136.0
11.6 16.4 145.0 204.0 32.5
13.4 13.4 165.0 0 Bind
13.4 13.4 660.0 0 Bind
10.3 20.4 165.0 38.0 N.Bind
11.6 16.4 331.0 17.0 N.Bind
8.2 32.0 110.0 88.0 N.Bind
11.6 16.4 165.0 21.5 N.Bind
Parametersa
-
6 0.08 0.03 0.25 0.08 0.04 0.01
Solution for variable h 44
c(T) h(O) h(T) max a(r)* Solution for tixed h 40)
47’) h
max a(t) Constraint
ar = real capital costs, m = real discount rate, i = inflation rate, + = tax rate, y = $20,000, utility = a log h + fi log c where a = 0.33, j? = 0.67. *Constraints require a(O) = a(T) = 0, where T = 35 years.
resources are spent for housing services.The results of these simulations are in Table 1. In Table 1, the first two simulations involve parameter values which lead to a binding constraint. In keeping with a log linear utility function the consumption of other goods is always a constant 67% of income, while the product of the discount rate (D = (1 - cp)r - cpi) and housing consumption equals the residual 33% of income. The contrast between the first two simulations, in terms of housing consumption, is dramatic. The introduction of tax deductability with a 50% tax rate, when there is a modest 2% inflation, causes housing consumption to quadruple. This results because as stated in Propositions 1-3, the constrained solutions are shaped solely by the magnitude of the term (1 - +)r - (pi. With the given parameters, tax deductability reduces this term by 75%. These results stand in sharp contrast
170
WILLIAM
C. WHEATON
to the view that inflation, by raising nominal interest rates, reduces housing demand. In the model at hand inflation never decreasesdemand; rather it most often exerts a very powerful positive effect on housing consumption. In the third through sixth simulations, the parameters have been chosen so that the constraint is nonbinding. In all of these cases,then, either both c and h increase over time (when the latter is variable), or just c increases (when h is constrained to be constant). This slant in consumption is achieved by having households save for approximately the first 20 years, reaching the wealth level described in the line labeled max a(t). After that, wealth is consumed up to the 35th (or terminal) year. In simulation 5, this tilt in consumption is greatest, as is the accumulated assetposition, since the difference between R and m is the largest among the six simulations. In keeping with Proposition 3, comparison of simulations 4 and 6 shows that when both m and D increase by identical amounts, the consumption gradient remains the same, but average housing consumption falls. Similarly, as between simulations 6 and 3, if D is fixed and m decreases(i.e., -m + D increases) then the growth in consumption over time increases, but the average level of h remains about the same. If i or + individually increase, or r decreases,then both D and -m + D are reduced. This creates both a reduction in the slope of consumption over time and an increase in the average level of h consumption. For example, if r = 0.04, m = 0.01, (p = 0.25, and inflation is 0.04, then D = 0.02 and - m + D = 0.01, and the results would be exactly as shown in simulation 4. If all other parameters remained constant, but inflation increased from 0.04 to 0.08, then the results would be exactly as in simulation 2. The consumption gradient would be flat and the new constant level of housing consumption would be double the average level prevailing at the lower inflation rate. Thus between simulations 4 and 2, an increase from 4% in inflation to 8% would cause housing consumption to jump by 180% at t = 0 while at 1 = T, it would increase by 65%. From the propositions and simulations, then, it is clear that liquidity-constrained borrowing is not very constraining-at least in how it affects housing consumption. The strong positiue impact of inflation on housing demand, which results in this model, is almost identical to that hypothesized by earlier authors in a perfect capital market with no constraints. The reasons why liquidity constraints appear to matter very little are two. First, when individuals desire to slant their consumption forward, the constraints are simply nonbinding. Second, and more importantly, when the constraints are binding, they permit a constant share of real income to be spent on housing-insulating the consumer from the normal tilt in real mortgage payments which accompanies inflation. With a fixed payment mortgage, nominal housing debt is constant, but equity grows ammally by ih. If the individual can borrow against this each year, and use the funds to consume
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other goods, then in the absence of a tax subsidy, his budget constraint is c = y - h(i + r) + ih = y - rh. The effective cost of housing is therefore only r, and economy-wide inflation induces no distortion in behavior. Once tax deductability is introduced, the annual cost of debt service drops to h(i + r)(l - c#J),but the individual can still borrow the untaxed amount ih. Thus when the constraint is binding, the level of c consumption possible becomes y - h(i + r)(l - $) + ih = y - h[(l - +)r - @iI. As C#I or i increase sufficiently, the individual finds that if he increaseshis housing consumption, the rise in his debt costs is less than the extra money he can borrow annually against his mounting equity. In this case he does not have to sacrifice h to consume more c, but instead can consume more c by consuming more h ! III. It is clear from the previous discussion that more stringent constraints than liquidity must be placed on consumer borrowing if any negative relationship between housing consumption and inflation is to be obtained from a life-cycle model. In this section, it is assumed that while consumers may save any amount, and therefore slant their consumption forward, they cannot engage in any borrowing beyond an initial housing mortgage. Given a falling real value of fixed mortgage payments, then, the consumption of other goods, and therefore utility, is automatically tilted toward the future. By additional saving, a consumer may increase this tilt, but he cannot reduce or offset it by borrowing against his accumulating housing equity. Without the ability to borrow against this wealth, the consumer must end his life in a positive assetposition. Given that refinancing or additional borrowing is not possible on one’s house, it makes little senseto allow housing consumption to vary over time. Surely if one can freely move and adjust his housing consumption, it is primarily because additional credit is available. A credit constraint on one’s existing house, but not on a new purchase, would be difficult to enforce. Thus, the models in this section assume that the consumption of housing services is fixed over time. In this sense,they can be compared to (16) and (17) in the previous section. Because housing wealth is inaccessible in this model, it will be treated separately from other assetswhich the consumer might accumulate through savings. Thus while in Section II, the term a(t) referred to total assets, here it will refer to only nonhousing assets. With the exception of this new definition for nonhousing assets,all definitions from Section II remain the same, and the consumer’s borrowing-constrained optimization problem is (19)
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subject to : ci =y - c - h(i + r)(l - $)e-”
+ a[(1 - +)r - Cpi]
u(r) 2 0 a(0) = u(T) = 0.
(20)
The constraint in (20) reflects the fact that the real value of the initial nominal mortgage payment h(l - $)( i + r) declines over time with the rate of inflation (P). The constraint also assumes a real after tax return on savings of only (1 - +)r - $ai.Thus unlike the liquidity-constrained model, the return to investment and the cost of housing services are different here, once borrowing constraints are introduced. As with the liquidity constrained model, the solution to (19)-(20) is different, depending on whether the borrowing constraint is binding or not. In either case, the first- and second-order conditions for the optimization of housing consumption, h, are dw z=Z=
Tau o ae -“‘dr -(l J
- +)(i + r)~‘$e-(‘+“)‘dt
d2w dZ -=,,
= 0 (21)
If the borrowing constraint is binding over all T I t I 0, then ci = a = 0, and is determined from the equation of motion, c(t) = y - h(1 - +)(i -I- r)e-”
if a(t) = 0.
(24
If the borrowing constraint is nonbinding over the entire interval from 0 to T, then the optimum level of c is determined with the following pair of conditions-derived by forming the Hamiltonian for the optimizing problem (19)-(20).
au --m'= p(f) ac p(t) = -f(l)[(l
-e
- 4+- - @I
if a(t) > 0.
(24)
In (23)-(24) p(t) is the dynamic Lagrangian associated with (20). Integrating (24) yields
au =
_
ac
poeIm-(l-$)r+w
if a(t) ) 0.
(25)
Thus (21) and (22) can together be used to solve for h and c(t) when the borrowing constraint is binding, while (21) and (25) are used to determine
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the solution when the constraint is nonbinding. It should be mentioned that in the latter case, the constant of integration, p,,, is determined with the equation of motion so that starting at a(0) = 0, a(t) rises and then falls to a(T) = 0. Taking (21) and (22), the impact of greater inflation on housing demand, when the borrowing constraint is binding, can be determined by total differentiation. This yields dh z=
dZ dZ -- dZ di x”OasiPO. I
(26)
PROPOSITION4. When borrowing constraints are present and binding, greater injlation reduceshousing demand if the internal discount rate is not less than the real interest rate (m 2 r). If m < r, the sign of dh/di is indeterminant. For the sake of brevity, the lengthy proof of this proposition is contained in Appendix A. It is important to point out that the proposition only establishes a sufficient condition for dh/di < 0. When m 2 r, consumers desire to shift utility toward the present, and so the opposite tilt in consumption, introduced by inflation together with fixed housing payments, should discourage housing consumption. When m < r, on the other hand, it seemspossible for the tilt in mortgage payments to reinforce the consumer’s preference for greater future consumption. Of course, if the consumer wishes to shift utility toward the future, he might also do it through direct savings-in which case the constraints are nonbinding. When the constraint on a(t) is nonbinding, (21) can be simplified to (27) by incorporating (25). -"I&
= ,-,[l
-
e-(l-+')(i+r)T]
(27)
If it is further recalled that at t = 0, &/Lk must equal p,,, and if in addition, the utility function is separable, then (27) above can be further simplified to
This equation is similar to (18) in the liquidity-constrained model, except that the exponent contains the term (1 - cp)(i + r) rather than (1 - +)r +i. The difference is important. In the borrowing constrained model, as inflation increases, so will the right-hand side of (28), implying that the ratio
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of h to c at time zero must fall. By contrast, in (18), inflation increased the ratio of h to c(0). Equally important, however, is the fact that as i increases, (25) implies that future consumption is reduced and that c(0) rises. In the liquidity-constrained model, this rise in c(O), together with a rise in the ratio of h to c(0) guarantees that h increases in response to higher inflation. In the borrowing constrained model, however, the effect is ambiguous. As c(0) increases, but the ratio of h to c(0) falls, the net effect on h is indeterminant. Thus when borrowing constraints are present, but nonbinding, the results are not at all as in a perfect or even liquidity-constrained capital market. PROPOSITION 5. When borrowing constraints are present, but nonbinding over a consumer’s lifetime, greater inflation has an ambiguous eflect on the demand for housing.
An interesting implication of borrowing constraints, which does not result when there are only liquidity constraints, is that under a range of circumstances, mixed solutions are possible. That is, over some subinterval of time, the borrowing constraint can be binding, while afterward, it may not. The reason is that even if a(t) equals zero, there is an irremovable growth in consumption over time through the fixed level mortgage payment-and this varies over time. Equation (22) shows that the growth in consumption is greatest in the early years of a mortgage payment and less later on. In fact, as inflation increases, the growth in consumption over some initial period increases, but then decreases over a later period. This suggests that if individuals save some income to slant their consumption further forward, they do so when that consumption is less slanted from the mortgage effect-or later in their life. To see this formally, it is clear from (25) that a necessary condition for the borrowing constraint to be nonbinding, is that m - (1 - $)r + Qi be negative. Just being negative is not enough, however, for it must be sufficiently negative to overcome the percentage reduction in au/& over time-due to the mortgage tilt. Only in this case will additional savings be induced. Differentiating (25) then, it must be true that when the borrowing constraint is nonbinding: m -(1 - +)r + +i = CJj$ > (ih(1 - +)(i + r)eVif)g.
(29)
At t = 0, the right-hand side of (29) can be quite large, necessitating an unrealistically large difference between (1 - +)r - +i and m in order to induce savings. Further along in the mortgage term, however, the right-hand side of (29) will become quite small, generating the possibility of savings.
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FOR HOUSING
TABLE 2 Simulated Housing Consumption with Borrowing Constraints Housing consumption ($1000~) when inflation rate = (constraint: B, NB, M)’
Parameters” Simulation
r
m
+
0
4
8
(1)
0.04 0.04 0.04 0.04 0.08
0.04 0.08 0.04 0.08 0.04
0 0 0.5 0.5 0.5
165(B) 165(B) 33qB) 330(B) 165(B)
129(B) 118(B) 258(B) 237(B) 172(B)
112(B) 99(B) 224(B) 198(B) 168(B)
78(B) 180(B) 157(B) 150(B)
lll(NB) 2868(l) WW 4864(l)
116(NB) 425(l) 81(NB) 2142(l)
124(B)
lWB1
(2) (3) (4) (5)
(6) (7)
0.08 0.04 0.25 (annual savings beginning at r =) 0.08 0.04 0 (annual savings beginning at t =)
0 83(M) 2414(9)
16
WBI
0 74( M: 3561(10)
“All simulations use the utility function u = nlog h + j? log c, where (I = 0.33, /3 = 0.67, and ri income = 20,000, and T = 35. ‘B = binding constraint over all t = 0,35; M = nonbinding initially, binding later: NB nonbinding over all t = 0,35.
Thus the most likely behavior finds the borrowing constraints binding throughout one’s life if m > (1 - +)r - +i. On the other hand if m < (1 +)r - (pi, then additional savings may occur but only after the growth in consumption, due to fixed mortgage payments, begins to wear off. Proposition 7 summarizes these various effects of parameter values on both savings behavior and housing consumption. PROPOSITION 6. When m > r, the borrowing constraint will be binding and inflation decreasesthe demand for housing. When r > m > (1 - +)r - +i, the borrowing constraints continue to be binding, but infition has an ambiguous impact on housing demand. When (1 - +)r - C#I~ > m, the borrowing constraint can becomenonbinding, particularly with the passage of time, and the eflect of infition on housing demand remains ambiguous.
To illustrate all of these results, a set of simulations was undertaken, again using a logarithmic utility function, with the housing expenditure share set at 0.33. Simulations were undertaken with the consumer’s discount rate m equaling 0.04 and 0.08, while the real cost of capital was inversely 0.08 and 0.04. Tax rates were varied between 0 and 0.5, while inflation ranged from 0 to 16%. The results are presented in Table 2. The first five simulations in Table 2 all involve parameters in which the borrowing constraint is binding throughout a consumer’s life-under all of
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the tested levels of inflation (i.e., m 2 (1 - +)r - $i). The results show that when m is greater than or equal to r, inflation reduces housing demand, but when r > m > (1 - $)r - +i, greater inflation can initially increase housing demand, before eventually decreasing it (simulation 5). In the sixth simulation, at low levels of inflation, (1 - $)r - c#G is sufficiently greater than m to induce savings throughout one’s lifetime. In other words, consumption growth due to fixed mortgage payments is not enough to satisfy the preference for future consumption. As inflation increases, however, the mortgage effect becomes more than enough, and savings activity ceases.The sixth simulation also creates a housing pattern in which demand initially increasessignificantly with inflation, from $111,000 at OS to $124,000 at 8% before eventually declining as inflation reaches higher levels. The seventh simulation shows how inflation can have little effect on housing consumption, but alter savings behavior quite dramatically. When r substantially exceedsm, and there is no income tax deductibility, consumers generally wish to postpone consumption through savings. As higher inflation pushes consumption into the future, through the mortgage effect, however, such saving becomes unnecessary and is either reduced or postponed. Propositions 4-6, then, together with these simulation results, demonstrate that the presence of true borrowing constraints-even if they are not binding-results in different behavior from that in perfect or liquidity-constrained models. Housing consumption can initially rise with greater inflation before eventually falling as inflation reacheshigher levels. This can even occur when the constraints are binding. In fact only when they are both binding, and m I r, will greater inflation unambiguously reduce the demand for housing. IV. The results of this paper have suggestedthat only under a very particular set of circumstances, will life-cycle theory yield the conclusion that housing demand is reduced by higher economy-wide inflation-which both increases mortgage rates and causes housing price appreciation. The assertion by Kearl [6], and the conclusions by Schwab [9] that it is credit constraints which cause this effect, must be highly qualified. First of all, if the constraints on credit merely require that any loan be collateralized with a real asset, rather than future human capital, then greater inflation creates an unambiguous increase in housing demand-by magnitudes which simulations suggest are quite high. If in addition credit constraints are extended to prohibit all kinds of borrowing, beyond an initial mortgage debt, then housing demand can be shown to decreasewith high inflation, but onb if the consumer’s discount rate exceeds the real, pretax, return to capital.
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177
When the discount rate is less than the real, pretax cost of capital, the impact of inflation on housing demand is very mixed, and generally small-at least in the range of inflation experienced by most western countries (O-16%). It is important to realize that the determining factor is not whether the borrowing constraints are binding or not, because this depends on more complicated considerations than whether the discount rate is larger or less than the cost of capital. What is somewhat puzzling about these results is that the conditions necessary to produce negative inflationary impacts would seemto be overly restrictive. At least in the United States, consumers have generally had the ability to either refinance their homes or take out second or third consumer loans against their equity. This clearly creates an opportunity set which most closely resembles the liquidity-constrained model. When or whether consumers choose to exercise this option is irrelevant, what matters only is that it is available. To produce the empirical results estimated by Kearl [6] and Follain [3], absolute prohibitions on borrowing must exist-together with a real pretax return to capital that is less than household rates of time preference. APPENDIX A (1) From (22) and (23), with a little manipulation z
= ir[-1
+(i + r)t]e-(‘+m)‘[u,-
UC&e-“(i + r)(l
- $)I(1 - (p)dt
where Z,(t)=[-l+(i+r)t]e-(i+m)r>O
ift>:
CO Z,(t)=
iftc-
1 1+r
1 i+r
[24,-u,,he-“(i+r)(l-+)](l-+)>O
and
dz, dtco
by convexity.
all t
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WILLIAM
C. WHEATON
Now; p,(t)z,(t)dt
=~‘*z,(t)z,(t)dt
+ /Tz&)Z*(t)dt
0
0 s
I’
/‘*Z,(t)Z,(t*)
dr + /Zl(t)Z,(t*) 1+
0
zz Z,(t*fz,(t)
dt
dt
0
where t* = - 1 i+r
since dZ 2<0 dt
and
Ut*)
t < t*
< z&L (z&*)
’ z,(t)7
t > t*).
Thus if joTZl(t) dt -c 0 so will be dZ/di. Now ii,(t)
dt =i’-
e-(i+m)rdt + /‘t(i+
r)e-(i+m)fdt
(when the latter is integrated by parts),
ifm2r, if m < r.
Q.E.D. REFERENCES
1. R. Artle and P. Varaiya, Life cycle consumption and homeownership, J. .&on. Theory, 18, 38-58 (1978). 2. A. Dougherty and R. Van Order, Inflation, housing costs and the consumer price index, AER, 72 (l), 154-165 (1982). 3. J. Follain, Does inflation affect real behavior: The case of housing, Southern Ron. J., January (1982). 4. P. Hendershott, Real user costs and the demand for single family housing, Bookings Papers on Economic Activity, No. 2, 1980.
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5. D. W. Jorgenson, The theory of investment behavior, in “The Determinants of Investment Behavior” (Ferber, Ed.), NBER, New York (1967). 6. J. R. Kearl, Inflation, mortgages and housing, J. PO/. Econ., 87 (5), 1115-1138 (1979). 7. F. Modighani and D. Lessard, @is.), “New Mortgage Designs for Stable Housing in an Inflationary Environment,” Federal Reserve Bank of Boston (1976). 8. J. Poterba, Inflation, income taxes, and owner occupied housing, NBER, Working Paper 553, 1980. 9. R. M. Schwab, Inflationary expectations and the demand for housing, AER, 72 (l), 154-165 (1982). 10. L. C. Thurow, The optimum lifetime distribution of consumption expenditures, AER, 63, 344-353 (1972). 11. J. Tobin, Wealth, liquidity and the propensity to consume, in “Human Behavior in Economic Affairs”, Strumpel, Morgan and Zahn (Eds.), Elsevier, New York (1972).