A Structural Model of the U.S. Housing Market: Improvement and New Construction

JOURNAL OF HOUSING ECONOMICS ARTICLE NO.

5, 166–192 (1996)

0009

A Structural Model of the U.S. Housing Market: Improvement and New Construction* CLAIRE A. MONTGOMERY Department of Forest Resources, Oregon State University, Corvallis, Oregon 97331 Received April 26, 1995

Built on foundations set in earlier structural models of housing markets, this model of the U.S. market includes endogenous household formation and household investment demand that allows different response to economics variables; household investment demand share equations for new construction and improvement; and a new construction supply equation that includes speculative supply behavior. The model predicts moderate housing price growth. Predictions of growth in total investment in housing depend heavily on assumptions about future income growth. Improvement is predicted to continue to grow in importance relative to new construction.  1996 Academic Press, Inc.

Each decade, in satisfaction of the requirements of the Forest and Rangeland Renewable Resources Planning Act of 1974, the Secretary of Agriculture prepares an assessment of the state of forest and other renewable resources in the United States, including long-run predictions of wood use and timber harvest using econometric models of wood products markets (Adams and Haynes, 1980). Residential housing is by far the most important final use for solid wood products in the United States (Haynes et al., 1995; Western Wood Products Assoc., 1993). In 1986, for instance, 67% of the softwood lumber and 62% of the structural panelboards consumed in the United States were used to increase and improve the residential housing stock. Improvement of the existing housing stock is growing in importance relative to new construction as a source of demand for wood products. For instance, the improvement share of the softwood lumber used in residential housing grew from 40 to 50% between 1980 and 1990. This paper reports a model built to produce long-run projections of housing activity in support of the Resource Planning Act assessment. It is based on previous structural models of housing markets. Its components * I gratefully acknowledge research support from the USDA Forest Service Pacific Northwest Research Station. I am also grateful for encouragement and comments from Robert A. Pollak, Richard Haynes, and anonymous reviewers. 166 1051-1377/96 $18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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include household formation, household demand for investment in housing, shares of investment demand for new construction and improvement, and new construction supply. The innovations in this model include: ● endogenous household formation and investment per household. ● demand for investment in housing as measured by the constant dollar value of construction put in place and endogenous share equations for investment via new construction or improvement (including maintenance). ● new construction supply that includes speculative behavior of developers and technological change in home construction. The model allows projection of the level of investment and the asset price of housing, along with the number of households and the level of new construction and improvement. Simulation with this model produces results that are consistent with the results of DiPasquale and Wheaton (1994) and Follain and Velz (1995). Both studies predict moderate increases in housing prices. This is in contrast to the widely publicized and criticized prediction of Mankiw and Weil (1989) that housing prices will decline dramatically over the next two decades. In Mankiw and Weil’s model, an aging population that is growing at a decreasing rate is willing to hold the housing stock only at much lower prices. In the model reported in this paper, the quantity of new construction and improvement is predicted to increase moderately, but is highly sensitive to assumptions about income growth. Because new construction supply is highly elastic, the asset price of housing is predicted to increase only moderately in response to projected increases in the quantity of new construction and to supply-induced increases in wood prices. The first section of this paper briefly summarizes past models of housing markets. The second section describes the structure of this model along with data and estimation results for each model component. The third section presents model simulations. The paper concludes with a discussion of the accomplishments and limitations of this model and suggestions for further work.

I. BACKGROUND

The 1960s and 1970s saw the development of large macroeconomic models of the United States economy. Models of the housing market were linked to the main body of these models via models of mortgage and finance markets. See, for example, the Brookings quarterly econometric model of the United States (Duesenberry et al., 1965) and the Federal Reserve–MIT– Penn Economic Model (Gramlich and Jaffee, 1972). Stand-alone models of housing and mortgage markets were also developed during this period (Smith, 1969; Jaffee and Rosen, 1979). While one of the questions posed

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to these models was ‘‘Can we house the baby boom,’’ they generally focussed on the business cycle rather than on long-run trends, attempting to predict cyclical fluctuations in housing market activity. For the most part, they modeled new construction only, although occasionally an ad hoc equation for improvement expenditure appeared. While the recession of the 1980s prompted housing economists to concentrate on questions of affordability and individual household behavior, there is renewed interest in modeling long-run trends in housing in the 1990s. The demographic structure of the population of the United States is changing. The combined effects of relatively low birth rates and increasing longevity will swell the ranks of the oldest age groups in the population relative to the age groups in which people typically form new households. Recent long-run projections of housing market activity forecast declines either in housing starts (Montgomery, 1989; Crone and Mills, 1991) or in the asset price for housing (Mankiw and Weil, 1989). These models rely on demographic variables that can be predicted with some accuracy (subject to migration)—the age–class structure of the adult population of the United States for the next 20 years. DiPasquale and Wheaton (1994), responding to the publicity surrounding Mankiw and Weil’s predictions, called for a return to the richer structural modeling approach of the 1960s and 1970s augmented by the learning of the past decade. By specifying a model of the housing market as completely as possible, the effects of exogenous variables may be isolated and the danger of attributing too much importance to any single variable (i.e., the age–class structure of the population) is avoided. DiPasquale and Wheaton explore the role of land value, gradual price adjustment, and price expectations in the determination of housing prices and housing starts. Approaches to structural modeling of the housing market can be split roughly into two groups; models of household formation and housing starts based on the work of Sherman Maisel (1963) and stock adjustment models based on the work of Richard Muth (1960). The basic identity underlying the housing market as formulated by Maisel is startst 5 (DHHt 1 Rt ) 1 (DVt 2 DIt ).

(1)

This can be viewed as a reduced form equation for housing starts. The change in the number of households, DHH, is the driving force behind demand because a household is defined by its occupancy of a discrete dwelling unit. In any period, however, DHH and housing starts differ substantially and the remainder of the identity is intended to explain the discrepancy. The net removal from the housing stock, R, includes units discarded from the housing stock or converted to or from other uses. Maisel

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took the first term, (DHH 1 R), as an exogenous determinant of demand and concentrated on the second term, (DV 2 DI), the change in inventories under construction less the change in the number of vacant units, which is a disequilibrium phenomenon related to supply that is more important for explaining the business cycle than for explaining long-run trends. While recent models that project long-run trends in housing starts were built on Maisel’s model, they emphasized change in the number of households rather than inventory adjustment. Marcin (1978), Smith (1984), Smith et al. (1984), and Montgomery (1989) estimated equations to explain the propensity of individuals to form households (headship rate) by age group as a function of per capita income and the rental price of housing. Both Marcin and Montgomery used their headship rate equations to predict DHH. Taking R and DV as given and ignoring DI, which is a short-run phenomenon, they predicted long-run trends in housing starts. Crone and Mills (1991) went one step further and linked housing starts directly to growth in the adult population, claiming that there is no discernible trend in headship rates over the last decade. In fact, over a longer time period, headship rates have exhibited an increasing trend (see Fig. 4). Trends in the number of households, and thus in the number of housing units, describe only one dimension of investment in housing. While housing starts have been relatively flat over the past four decades, housing stock per household has nearly doubled, suggesting the importance of individual household demand for predicting levels of activity in housing markets. In past studies, the relationship between the number of households and investment per household was confused, evidenced by the practice of using housing starts and the constant dollar value of new construction interchangeably as the endogenous variable. Richard Muth’s model of consumer demand for housing provides a basis for modeling individual household demand for investment in housing. A durable good such as housing stock, H, is an input into the production of a stream of services, f (H ), that are valued by the household. Utility maximizing households hold housing stock up to the point at which the marginal utility of housing stock, Uf fH , is equal to the product of the marginal utility of income, l, and the rental price of housing, r: Uf fH 5 lr R H D 5 H D (r, Y, S).

(2)

The rental price of housing is a function of the asset price of housing, its expected appreciation rate, the alternative rate of return, taxes, and depreciation of the housing stock (Hendershott and Shilling, 1981). In the context of an intertemporal utility function or subject to a wealth constraint, this yields household demand for housing stock, H D, that depends on the relative rental price of housing, permanent income and wealth, Y, and

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sociodemographic characteristics of the household, S. When faced with transaction costs for moving, households do not fully adjust holdings of housing stock to desired levels in the short run and housing markets are in disequilibrium. To account for this sluggishness, Muth proposed a partial stock adjustment model for demand for new construction i tD 5 e[H tD 2 (1 2 d)Ht21 ]

(3)

in which demand for new construction, i tD , is some portion, e, of the difference between desired housing stock and depreciated current holdings, (1 2 d)Ht21 . In recent years, economists have moved beyond the concept of disequilibrium in modeling household behavior. Instead, transaction costs are viewed as part of the household optimizing choice and are incorporated into the budget constraint, creating nonlinearities. The household chooses simultaneously which segment or kink of the budget constraint to be on (i.e., between moving and staying put) and the level of demand conditioned by that choice (Hausman and Wise, 1980; Hausman, 1985). In this context, Muth’s partial adjustment parameter, e, can be interpreted as the probability that the household chooses to move. While some portion of the increasing trend in individual holdings of housing stock can be explained by increasing size or quality of new construction (for instance the average floor space of new single family detached homes has increased from 1100 to 2100 square feet over the past four decades (Haynes et al. 1995)), households are turning more to improvement as a means to increase their holdings of housing stock without moving. In fact, in 1990 more softwood lumber was used in the United States for improvement of existing housing than for construction of new housing. Montgomery (1992) expanded the basic household model to include investment in the existing stock. In that model, households make a fourway ordered choice between moving down, doing nothing, improving, and moving up, defined by the form of the budget constraint. Improvement expenditure is conditional on the decision to improve. A conceptual restatement of the partial stock adjustment model splits demand for investment in housing into separate components for new construction and improvement as I I i tD 5 eM [H M t 2 (1 2 d)Ht21 ] 1 e [H t 2 (1 2 d)Ht21 ].

(4)

The first term on the right is the product of the probability that the household chooses to move and the conditional demand for investment given that the household moves and, likewise, the second term is the product of

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the probability that the household chooses to improve and the conditional demand for investment given that the household improves. Although most empirical studies focus on demand for housing, the asset price for housing and the quantity of investment are determined simultaneously by the equilibrium of supply and demand. In the past, to avoid simultaneity, many modelers of housing markets imposed a form of recursivity on the model structure. In the supply-based approach, the asset price of housing is set so that the existing stock is willingly held. Then builders supply more housing in response to the difference between the asset price and the cost of construction. The asset price then adjusts so that the new housing stock is willingly held and so on (Topel and Rosen, 1988). In contrast, in some long-run analyses, demand for the housing stock depends on lagged prices, presumably because of their role in the formation of expectations about future housing prices (Smith, 1984). However, only the most naive models of price expectations exclude information contained in the current period price. With the advances of the last decade in both econometric theory for estimating systems of equations and in computational technology, the temptation to make questionable assumptions in order to avoid simultaneity is reduced.

II. MODEL STRUCTURE, ESTIMATION, AND RESULTS

This section describes a model of the housing market that consists of a system of six equations that simultaneously determine: (i) the number of households, (ii) average household investment in housing and the share of household investment by new construction and by improvement, and (iii) the price of new construction. Figure 1 shows the structure of the model. The equations were estimated as a system using three stage least squares, with the set of exogenous variables in the model serving as instruments. The equations and estimation results are described individually in the following sections. In all equations, the exogenous economic variables in the empirical model are represented by Xt and the random error term by ut . Data sources and construction are described under Appendix 1. (i) The Number of Households Although neoclassical theory of consumer demand explains individual behavior, the theory is often transferred to households for empirical analysis without comment. While, for many goods the difference between individuals and households is not important, for housing it is a crucial linkage. Gary Becker set the tone for studying household formation with his pioneering work examining the behavior of individuals within households, looking at

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FIG. 1. Model structure.

household production, marriage, and child bearing as economic behavior (see, for example, Becker, 1973). In models of household formation (Marcin, 1978; Smith, 1984; Smith et al., 1984; Montgomery, 1989), individuals choose whether to remain in their current household or whether to head their own household in response to economic and sociodemographic variables. The endogenous variables are age-specific headship rates, hit , defined

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as the proportion of the population in age group i, POPit , that chooses to head a household, HHit , in time period t:

hit 5

HHit . POPit

(5)

While earlier studies each formulated the empirical model slightly differently, together they provide evidence that both income and the rental price of housing are important determinants of household formation behavior in the youngest age group and that income is important in the oldest age group. In this model, headship rate equations were estimated for three age groups: 18 to 29 years, 30 to 64 years, and over 65 years. The individual choice is dichotomous (whether to head a household) and the observed data are grouped (the proportion of a population age group that heads a household). In the logit model, individuals choose to head a household if the value of an unobservable index variable that is a linear function of exogenous variables exceeds some threshold value for that individual. If the probability of that occurrence follows a logistic cumulative density function, the natural log of the odds is equal to the index variable plus a random error (Judge et al., 1982):

ln

S D

hit 5 X 9t ai 1 uit , 1 2 hit

(6)

where ai is a vector of coefficients to be estimated. The logit model is preferable to the log-linear model used by Smith and Smith et al. because it restricts predicted values to the unit or other specified interval. The variance of the error term is [1/(POPit hit (1 2 hit ))] and the variables are transformed to correct for heteroskedasticity. Because the log of the odds are positively correlated with the probability of ‘‘success,’’ the sign of the estimated coefficients gives the sign of the effect of a change in exogenous variables on the probability of household formation. The elements of Xt include the rental price for housing and per capita disposable income. Because adjustment costs for household formation are very high (due to its reliance on the formation and dissolution of personal relationships), the partial adjustment form was used for the headship rate equations so that lagged headship rate also appears as an exogenous variable. The partial adjustment form eliminated much of the serial autocorrelation in the error term that appeared in earlier models.

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TABLE I Coefficients for Headship Rate Equations a Age group (years):

18–29

30–64

651

Constant Rental price Per capita income Lagged headship Adjusted R 2, DW statistic

20.436 (4.06) 20.325 (2.41) 12.79 (3.64) 0.728 (11.9) 0.962, 2.04

20.029 (0.97) 0.039 (0.53) 3.354 (1.43) 0.850 (12.3) 0.978, 1.71

20.083 (1.65) 20.026 (0.16) 15.07 (2.66) 0.693 (7.14) 0.950, 2.22

a

t statistics are in parentheses.

Estimation results, shown in Table I, are consistent with earlier studies. Members of the youngest age group are discouraged from household formation when the rental price for housing is high. Household formation is encouraged in both the youngest and oldest age groups by high per capita income. In the middle age group, headship rates have been relatively stable and unresponsive to fluctuations in economic variables over the past four decades. (ii) Average Household Investment Demand and Share Equations Household investment demand for housing has two components—the level and the mode of investment. In this model, these components are specified as a system of equations based on Eq. (4). The average level of housing investment per household is given by I I it 5 eM [H M t 2 (1 2 d)Ht21 ] 1 e [H t 2 (1 2 d)Ht21 ]

(7)

5 X 9t b 2 (1 2 d)eHt21 1 ut , I I M where X 9t b replaces (eMH M 1 eI) and ut is the t 1 e H t ), e replaces (e error term. The household investment demand equation was estimated in the following form derived from Eq. (7):1

Equation (8) is derived from Eq. (7) as follows. By definition, Ht21 5 (1 2 d)Ht22 1 it21 . Substitute in Eq. (7) for Ht21 to get 1

it 5 X t9b 2 e(1 2 d)[(1 2 d)Ht22 1 it21 ] 1 ut . From Eq. (7) it21 5 X t921b 2 e(1 2 d)Ht22 1 ut21 ⇒ 2e(1 2 d)Ht22 5 it21 2 X t921b 2 ut21 . Substitute for 2e(1 2 d)Ht22 and rearrange to get Eq. (8).

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it 5 [Xt 2 (1 2 d)Xt21 ]9b 1 (1 2 e)(1 2 d)it21 1 S9t l 1 vt

(8)

in which S9t l represents the effect other demand shifters and vt 5 ut 2 (1 2 d)ut21 . The elements of Xt include the rental price of housing and household disposable personal income. In quasidifferencing these variables, the gross depreciation rate, d, was assumed to be 2% (Margolis 1982). The elements of St include the average age of the housing stock and a summary variable for the age class structure of the population. For the latter, the proportion of the population in the 18- to 29-year age group fit marginally better than the average age of the adult population with which it is highly multicollinear. This age class is a particularly important one for housing demand for two reasons. It is an active age for household formation, as reflected in the headship rates described above, and also for changing housing needs as families grow. A dummy variable was included in St to account for changes in the survey used to estimate repair expenditures. It was set to one for the years 1962 to 1984, in which the Survey of Residential Alteration and Repair appears to have underestimated improvement expenditure relative to the methods used prior to 1964 and after 1983 (USDC, 1961, 1985, 1986) and zero otherwise. Estimation results for the investment demand equation are shown in Table II. Because Eq. (8) effectively imposes a structure on the error term (ut in Eq. (7)), such that it follows a first-order autoregressive process with r 5 0.98, the autocorrelation in the error term that appeared in most earlier empirical models of housing demand was largely eliminated in this model. The estimated coefficient for the lagged investment and the change in income variables are strongly significant and positive. These two variables account for most of the explanatory power of the estimated equation. The probability of either moving or improving is estimated from the coefficient on lagged investment as e 5 0.37. As a point of comparison, Muth (1960)

TABLE II Household Investment Demand Equation a Constant Household investment, lagged Rental price for housing, differenced Household income, differenced Proportion 18- to 29-year age group Survey dummy Average age of housing stock Adjusted R 2, DW statistic a

t statistics in parentheses.

21.890 (1.87) 0.626 (6.16) 1.198 (0.77) 0.232 (5.79) 3.272 (1.97) 20.042 (0.63) 0.097 (2.49) 0.694, 1.63

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estimated a partial adjustment coefficient of e 5 0.31 using per capita nonfarm housing stock as a dependent variable—so that investment included both new construction and improvement. Investment demand shows a positive and mildly significant correlation with the population age-class variable and a positive and significant correlation with average age of the housing stock. While the coefficient on the survey dummy variable is negative, as expected, it is not significantly different from zero. The rental price coefficient is not statistically different from zero. The effect of changes in the rental price on long-run housing investment demand may be very small or it simply may be impossible to detect using the noisy data that is available for long-run time series. In fact, theoretical predictions for the effect of changes in rental price on housing investment demand are ambiguous (Montgomery, 1992). An increase in the rental price that is due to an increase in the asset price of housing has two effects; the relative price of housing services increases yielding a negative price effect and the value of the household’s stock of housing increases yielding a positive wealth effect. The shares of investment by new construction and by improvement are defined by w It 5

eI [H It 2 (1 2 d)Ht21] , it

I wM t 5 1 2 wt

(9)

and estimated by w It 5 u9it 1 X 9t h 1 S9t k 1 ut .

(10)

Note that if the conditional demand for housing stock for improvers is equal to the conditional demand for housing stock for movers, so that D H It 5 H M t 5 H t from Eq. (3), then the probability of a household improving, e It , may be estimated by eIt 5 w It

S

H tD

D

it 5 w It e 2 (1 2 d)Ht21

(11)

and likewise for the probability of moving. Remember that e was estimated by Eq. (8). However, because movers and improvers face different budget constraints, H It ? H M t , and the probabilities of moving or of improving are not separately identified in this model. Estimation results for the improvement share equation are reported in

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TABLE III Improvement Share Equation a Constant Household investment Rental price Household income Proportion 18- to 29-year age group Survey dummy Average age of housing stock Wood price index Carpenters’ wage index Materials price index Adjusted R 2, DW statistic a

0.535 (1.92) 20.097 (8.55) 20.459 (3.29) 0.006 (2.26) 0.076 (0.32) 20.069 (7.28) 0.002 (0.26) 20.138 (3.20) 20.225 (4.17) 0.234 (2.00) 0.913, 1.61

t statistics in parentheses.

Table III. The new construction share is determined by the improvement share so that coefficients are equal and opposite in sign. The elements of Xt and St include the variables that appear in the investment demand equation. The elements of Xt were not quasidifferenced and lagged investment does not appear. Three cost variables for inputs to improvement are included. While new construction is purchased by the household, the household acts as a contractor for improvement and faces the costs of inputs directly. In fact, Eq. (10) can be viewed as a reduced form equation for household demand and supply for improvement based on the household production model. Since the household acts both as a consumer and a producer of improvement, demand and supply cannot be separately identified. The current level of household investment is endogenous and is a determinant of the share in Eq. (9). Its coefficient estimate is negative and highly significant. An inspection of the historical levels of total investment in housing, new construction and improvement shown in Fig. 6 suggests an explanation. While the dramatic fluctuations in total investment are mirrored in new construction, improvement is fairly stable. The negative coefficient on current investment in the improvement share equation (the positive coefficient on current investment in the new construction share equation) indicates that new construction moves more strongly with the fluctuations in total investment than improvement does. Thus, the coefficient on current investment represents the relative volatility of the two components of housing investment. Theory gives no prediction for the sign on the rental price coefficient. When the rental price for housing increases, the relative price for housing services increases for both improvers and movers. Because some households that would have improved at a lower price will choose to do nothing at

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the new price, while some households that would have moved will choose to improve at the new price, there is no basis for predicting which shift will dominate (see Montgomery, 1992). However, intuition leads one to expect the negative sign on the rental price coefficient in the improvement share equation obtained here, suggesting that demand for improvement is more elastic than demand for new construction. The reason one might expect households that are improvers to respond more strongly to rental price changes than do households that are movers is that the transaction costs for movers (search costs, monetary, and psychic moving costs) are greater than those faced by improvers (contracting costs and disruptions of the household during the construction of the improvement), so that improvers have more flexibility to respond to small price changes. The coefficient on household income is positive and significant at the 5% level of confidence suggesting that increases in income favor improvement over new construction. Theory gives no prediction for the effect of income on the relative shares of improvement and new construction because it affects the wealth of the household and, as a measure of the opportunity cost of time, it affects both the cost function for improvement and the cost of moving. The positive sign suggests that, perhaps, its role in the cost of moving dominates its role in the cost of improving. Input prices were included in the share equation because of their importance in the cost of improvement relative to new construction. While for movers the slope of the budget constraint is the rental price of housing, for improvers it is the rental price of housing plus an adjustment equal to the difference between the marginal cost of improvement and the asset price of housing (Montgomery, 1992). Thus, when the prices of inputs to improvement increase, the slope of the improvement portion of the budget constraint increases while the rest of the budget constraint stays put, resulting in a decrease in the improvement share. Three input prices were included in the model. The construction materials price index is a weighted average of a set of inputs to general construction. The weights are obtained from the benchmark input/output tables constructed from the U.S. Department of Commerce Census of Manufactures (USDC, 1991). The construction materials price index does not include labor cost, so carpenters’ wages are included in the model as a separate input price. The construction materials price index does include prices for several solid wood products. However, because the input/output weight assigned to wood in general construction is low relative to its weight in new construction and improvement,2 and because wood use is of particular importance in this research, 2 The relative importance of wood inputs relative to all physical inputs was 12% in general construction, 30% in new residential construction, and 21% in repair and improvement in 1982 as computed from weights given in the USDC input–output accounts for that year.

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a separate wood price index was included in the model. The coefficients on both the labor and the wood price variables are significant and have the expected negative sign. This suggests that wood and labor are relatively more important in the cost function for improvement than they are in the asset price of housing. In fact, data on the relative importance of wood in improvement and in new construction provides conflicting evidence. The input/output weights based on constant dollar measure of wood input indicate heavier wood use in new construction, while USDA Forest Service statistics based on physical measure of wood inputs indicate heavier wood use in improvement3 (Haynes et al., 1995). The coefficient on the materials price index is positive and not quite significant at the 5% confidence level. This result is difficult to interpret, but might arise from the use of a set of weights for aggregating material prices that is not representative for new residential construction. The survey dummy has the expected sign and is significant. The coefficients on the dwelling age and the population age variables are not significantly different from zero and omitting them does not reduce the explanatory power of the model or affect coefficients on other variables. (iii) New Construction Supply Price Equation The early empirical studies of long-run supply of new construction of Muth (1960) and of Follain (1979) found strong evidence that, in the long run, housing supply is perfectly elastic so that the asset price of housing is a function only of the cost of inputs. Topel and Rosen (1988) revisited the issue of supply elasticity of new construction with a model that incorporates short run adjustment costs for builders. They used quarterly data and estimated a long-run supply elasticity for a year that is high (3.0) but not infinite. DiPasquale and Wheaton estimated a long run elasticity of 1.0 to 1.2. Follain and Velz (1995) estimate a very high, but not perfect, supply elasticity (about 6 for housing stock supply elasticity and a large multiple of that for new construction supply elasticity). The model estimated here is the basic supply model reported in the earlier studies in which supply price is a linear function of the level of new construction and cost shifters: Pt 5 fQt 1 X 9t f 1 ut ,

(12)

where f and f are coefficients to be estimated and Qt is the total constant 3 Wood use in 1986 was 139 thousand board feet lumber and 72 thousand square feet structural panels per million dollars of new construction, and 176 thousand board feet lumber and 110 thousand square feet structural panels per million dollars of improvement.

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TABLE IV New Construction Supply Price Equation a Constant Total new construction Expected real interest rate Expected inflation Expected opportunity cost Materials price index Carpenters’ wage index Wood price index Plywood use index Waferboard use index Adjusted R 2, DW statistic a

0.535 (4.35) 1.10E-06 (4.84) 1.439 (5.61) 1.575 (9.12) 0.671 (4.86) 0.387 (3.42) 20.093 (1.54) 20.011 (0.20) 20.030 (3.63) 0.038 (0.71) 0.907, 1.76

t statistics in parentheses.

dollar value of new construction; the product of the number of households, individual household investment in housing, and the new construction share, all of which are endogenous to the model: Qt 5 w M t it

O h POP . it

it

(13)

i

The model was estimated with annual data and does not include the short run adjustment costs of Topel and Rosen. Estimation results are reported in Table IV. The elements of Xt are cost and technology variables. They include the real interest rate, expected inflation, an opportunity cost variable, prices for labor, general construction materials and wood, and technology variables representing diffusion of new wood products into residential construction. Land prices are not included, even though the building lot is important both as an input, and thus a cost, in the production of housing services and as a fixed factor in the production of improvement. Data series that include lot value are limited to shorter time series and, because the motivation for this model was its potential usefulness for explaining derived demand for wood and is thus focussed on the demand and supply of structures, I opted for the longer time series. In an earlier version of the model, a land price series constructed from the median lot value from FHA homes did not contribute to the explanatory value of the supply price equation. The real interest rate is included in new construction supply models to represent the cost of working capital for builders. The estimated coefficient on the real interest rate in this model is positive and significant as expected. However, as in past studies, the coefficient on expected inflation is also positive and significant. This empirical result suggests that builders respond

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more strongly to the nominal interest rate than theory predicts. The questions raised by this result were well articulated in Topel and Rosen and remain unresolved. An opportunity cost variable was included to represent the opportunity cost to builders of selling housing stock in the current period rather than holding and selling later. Serial autocorrelation appeared in most of the earlier supply models and was generally treated with autoregressive processes of the first or second order. In fact, the price of new construction can be fit quite well by a simple two-lag autoregressive model.4 One possible explanation for the role of past prices in the determination of current price is that they inform the speculative behavior of builders and developers. Suppose that builders and developers hold development permits, land, and even buildings under construction as assets. The opportunity cost of building and selling in the current period is the foregone earnings from building and selling in the next period. Thus, builders will postpone development, reducing current supply, as long as Pt , Pt exp( r et 2 it )

(14)

where re is the expected real rate of appreciation in the asset price of housing and i is the real interest rate. The difference in the appreciation and the interest rates represents the amount by which the rate of return on holding housing is expected to exceed rates of return on alternative investments. Inclusion of ( re 2 i) as an element in Xt , with expectations based on past prices, eliminated serial correlation as diagnosed by the Durbin–Watson statistic and improved the explanatory power of the equation. The estimated coefficient was positive and significant. Of the three input price variables, only the coefficient on the construction materials price index was statistically significant and it had the expected positive sign. However, the coefficient on the labor cost variable warrants some discussion. In an earlier model (Montgomery, 1989) and in early versions of this model, the labor cost coefficient was statistically significant and negative. In the final version, reported here, it is still marginally significant (at the 13% confidence level). Other studies, using shorter time series, yielded either the expected positive sign (Follain) or no significant difference from zero (Topel and Rosen). Over the longer time horizon of the data series used in this model, major changes in the technology of housing construction occurred. In particular, the adoption of plywood as a substitute for lumber during the 1960s and of oriented strand board (waferboard) as a substitute for both plywood and lumber during the 1980s have increased 4

Ph 5 0.15 1 1.4*Ph (21) 2 0.55*Ph (22) R 2 5 0.85 t stat: (2.3) (9.9) (3.8) DW 5 1.75.

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FIG. 2. End use factors for softwood plywood, waferboard, and carpenters’ wages.

the productivity of labor in residential construction. Figure 2 shows indexes for lumber, plywood, and waferboard use in new residential construction along with an index of carpenters’ wages. Inclusion of the plywood and waferboard indexes as technology variables resulted in lowering the significance of the negative coefficient on labor cost. The coefficient on the plywood technology variable is strongly significant and negative, which is the expected sign for a cost-saving innovation. The coefficient on the waferboard technology variable is not statistically significant and has the wrong sign. However, waferboard represents an important new product in housing construction. Dropping it from the model results in changes in the other coefficient estimates, indicating that its insignificance may be due to multicollinearity rather than unimportance. A more sophisticated model of the technology of residential construction that includes different wood products and diffusion of new technologies would be interesting to construct, but is beyond the scope of this paper. Spelter (1984, 1985) constructed innovative, but structurally weak, models representing technology change and diffusion of new products in wood-using industries, including new construction. On the question of new construction supply elasticity, this model provides further evidence that, while supply may not be perfectly elastic, it is nearly so. The coefficient on the quantity of new construction is significantly positive so that the hypothesis of perfect elasticity of supply may be rejected

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FIG. 3. Historical and projected household income growth rate.

at the 5% confidence level. However, the long-run elasticity of supply is estimated to average 9.2 over the sample period, suggesting that an assumption of perfect elasticity is not far wrong.

III. SIMULATIONS

Simulations of the model were run under two income scenarios to emphasize the importance of income projections in this model—one in which household income grows at approximately 2.3% and one in which household income growth rate continues to decline over the projection period following a trend line fit through household income growth over the past four decades (see Fig. 3). It should be noted that even in the low-income growth scenario, household income grows to levels unobserved in the historical data. Caution should always be exhibited when interpreting simulation results that use projection of exogenous variables that move outside the range of the historical data upon which the empirical model was estimated. Sources for projections of all of the exogenous variables are described under Appendix 2. Figures 4 through 7 show model simulation results for headship rates, household investment in housing, total constant dollar investment in housing for improvement and new construction, and the asset price of housing.

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FIG. 4. Historical and projected headship rate by age group.

Household income and the size and composition of the adult population are by far the most important exogenous determinants of the level of investment in housing (and thus derived demand for wood) in the estimated model. Because of the high elasticity of supply, the asset price grows moderately with the increasing level of new construction. Projected headship rates (Fig. 4) climb upward for the next two decades and grow or flatten in the last decades depending on the income growth scenario. The projected number of households ranges from 160 million at the end of the projection period with low income growth to 190 million with high income growth. In the high income growth scenario, headship rates grow to levels that some believe unlikely for sociocultural reasons. An alternative model was estimated using caps on headship rates that are thought to be consistent with observed marriage and family behavior for the different age groups. These maximums were based on ad hoc figures computed and used by Marcin (1978). With the alternative model the number of households grow to 155 to 163 million by the end of the projection period, but little else in the model differs. The rate of overall household formation is affected by both the agespecific headship rates and the age class structure of the population. Thus, although headship rates increase in both income growth scenarios, the number of households grow at a decreasing rate, largely due to the progression of the baby boom cohort through the age classes. In the first half of

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FIG. 5. Historical and projected household investment in housing.

the simulation period, they move from the youngest age class (18–29 years) to the middle age class (30–64 years), making an average increase in headship rate of over 20%. In the second half of the simulation period, they move from the middle age class to the oldest (over 65 years) class, making a corresponding increase in headship rate in the vicinity of 10%. Household investment in housing (Fig. 5) is highly sensitive to income. The low-income growth scenario shows moderate decline in household investment, while the high-income growth scenario shows household investment increasing to the levels of the mid-1950s by the end of the projection period. Total investment in housing (Fig. 6) grows at about 0.7% per year under the low-income growth scenario and at about 1.8% per year under the high-income growth scenario. The growth under the low-income growth scenario results primarily from the increasing number of households, while the growth under the high-income growth scenario is fueled by both income and demographics. In the high-income growth scenario, improvement overtakes new construction because of the role of income in the improvement share equation. As expected, the asset price of housing (Fig. 7) increases moderately under both scenarios, staying within the range of values of the past four decades under the low-income growth scenario and growing at about 0.4% per year under the high-income growth scenario. This is consistent with

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FIG. 6. Historical and projected total investment in housing by type.

the predictions of DiPasquale and Wheaton and of Follain and Velz but differs dramatically from the predictions of Mankiw and Weil. The increasing trend results, in part, from predicted increases in the cost of wood that push up the materials price index. These, in turn, result from recent dramatic restrictions on public timber supply in the Pacific Northwest for environmental reasons (Haynes et al., 1995) that are likely to be repeated in other regions where protection of habitat for old-growth dependent wildlife is a concern.

IV. CONCLUDING REMARKS

Efforts to model the U.S. housing market have spanned three decades. This model follows and builds on the traditional structural models of the past. Consequently, many of the empirical outcomes of previous models are repeated. The empirical model is fairly robust, as experiments with functional form and structure of the data variables had little effect on the key results. The important results include the following: ● Highly elastic new construction supply, leading to moderate growth in the asset price of housing in simulations. ● Responsiveness of both household formation and household investment demand to income growth, leading to very different predictions of

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FIG. 7. Historical and projected price of new construction.

total investment in housing depending on assumptions about future income growth. This model adds to the body of research into the structure of housing markets in the following ways: ● It models household formation and household investment demand separately but simultaneously, so that there is explicit recognition of the components defining the housing stock—number of dwelling units and quality of those units. Income is allowed to influence each differently through the headship rate equations and the household investment demand equation. Thus, while demographically driven models of housing starts predict declining levels of demand for housing modified only slightly by different income growth scenarios, models of household investment tell a different story—one of possible growth in total demand corresponding to the household income growth. ● It models improvement of the existing housing stock simultaneously with new construction so that households may modify their holdings of housing stock without incurring moving costs. This model gives households more flexibility to respond to economic variables. ● It includes a variable that represents the opportunity cost of building and selling now rather than later, so that the serial autocorrelation that has been pervasive in past models of construction supply is largely eliminated. DiPasquale and Wheaton’s model of gradual price adjustment resolves the

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problem, but the gradual price adjustment itself is left unexplained. One possible explanation is speculative supply behavior by builders and developers. ● It models known changes in residential construction technology with indexes rather than with a nondescriptive time trend variable. The empirical results suggest that the introduction of structural panels, such as plywood and waferboard, resulted in increased labor productivity, allowing real increases in carpenters’ wages at the same time as decreases in the asset price of housing from an overall reduction in the cost of new construction. Condensing the knowledge gained in the past two decades of micromodels of household demand for housing and models of urban growth into macromodels of housing markets that are sparse enough to be useful for generating long run predictions is a challenging task. Some of the extremely sparse models of recent years have produced implausible predictions. Others, in which the basic market structure is maintained and detail is added selectively, have generated more moderate predictions that are reasonably consistent with historical trends. This model falls in the latter category and, hopefully, provides a basis for richer future modeling efforts that might include land values and attributes of the housing stock of interest for wood use.

APPENDIX 1: DATA SOURCES AND CONSTRUCTION

The data are annual series from 1952 to 1992. Prices and income are deflated by the Consumer Price Index for all goods excluding shelter (1987 5 1.0): Number of households by age of head (thousands): from U.S. Bureau of the Census, Current Population Reports, Series P-20, Household and Family Characteristics. Investment in residential housing—constant million 1987 dollars: for new construction and construction improvements; U.S. Bureau of the Census, Construction Reports, C-30, Value of New Construction Put in Place—for maintenance; U.S. Bureau of the Census, Construction Reports, C-50, Residential Alterations and Repairs. Price of new construction—1987 5 1.0: for 1963–1992: U.S. Bureau of the Census Single Family Houses Under Construction Price Deflator. According to Pollak (1989): ‘‘there is no presumption that deflating price and expenditure by the Laspeyres index is better than deflating by a weighted average of prices whose weights were chosen with the aid of a table of random digits.’’ Use of this Paasche index is consistent with the definition of the deflated current dollar value of construction as a measure of quantity.

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For 1952–1962: Revised Deflator for New Construction from the U.S. Bureau of Economic Analysis, Survey of Current Business, Oct. 1974. Resident population by age (thousands); Disposable personal income (million $1987); Cost of capital for builders (Moody’s AAA corporate bond rate); Construction materials price (1987 5 1.0): U.S. Bureau of the Census, Statistical Abstract of the United States and Historical Statistics, Colonial times to 1970 and U.S. Bureau of Economic Analysis, Current Business Statistics. Net stock of residential capital (million $1987) and average age of net stock: U.S. Bureau of Economic Analysis, Fixed Reproducible Tangible Wealth in the United States. Expected inflation: estimated using polynomial distributed lag on actual inflation rate and the ex post real interest rate following Modigliani and Shiller (1973). Expected asset price appreciation: constructed as average growth rate in undeflated price of new construction over the past 2 years less expected inflation. User cost of capital for homeowners: [(nominal interest rate 1 property tax rate) p (1 2 marginal income tax rate) 2 expected inflation 1 depreciation rate 2 expected real asset price appreciation—the interest rate is the 10-year U.S. treasury bond rate to represent long-term interest rates; the property tax rate is constant at 1%; the marginal income tax rate is the weighted average of individual marginal income tax rates and the number of returns from the Internal Revenue Service, Statistics of Income (Barro and Sahasakul, 1983); the depreciation rate is constant at 2%; expected inflation and expected asset price appreciation are computed as above. Carpenter wages (1987 5 1.0): U.S. Bureau of Labor Statistics, Employment and Earnings. Wood price index (1987 5 1.0): constructed from producer price indexes for softwood lumber, softwood plywood, and hardwood lumber from U.S. Bureau of Labor Statistics, Producer prices and price indexes, and waferboard prices from Random Lengths Yearbook, Eugene, Oregon. Weighted average of prices using apparent U.S. consumption from Haynes et al. (1995) and USDA Forest Service Pacific Northwest Research Station, Portland, Oregon, as weights.

APPENDIX 2: PROJECTIONS OF EXOGENOUS VARIABLES

Where possible, projections of exogenous variables were those used in or produced by the most recent RPA Assessment (Haynes et al., 1995) which were based on the Wharton Econometrics long-term forecast of the

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U.S. economy using the Wharton Mini-Growth Model (WEFA, 1991) and the U.S. Census Bureau ‘‘Middle Series’’ population projection. Resident population by age (thousands): middle series from U.S. Bureau of the Census, Current Population Reports, P25-1092, Population projections of the United States by age, sex, race, and hispanic origin: 1992–2050. Disposable personal income (million $1987): from WEFA for high growth scenario—from a simple trend line through past growth for the low growth scenario.5 Average age of net stock: linked to lagged age of net stock, lagged level of stock, and twice lagged quantity of new construction.6 Expected inflation: polynomial distributed lag weights on WEFA projection of CPI. Cost of capital for builders: Moody’s AAA corporate bond rate linked to long-term corporate bond rate projection from WEFA. User cost of capital for homeowners: interest rate is from RPA Assessment; the property tax rate is constant at 1%; the marginal income tax rate is the average rate over historical period; the depreciation rate is constant at 2%. Carpenter wages (1987 5 1.0): from RPA Assessment. Wood price index (1987 5 1.0): prices and consumption from RPA Assessment. Construction materials price index (1987 5 1.0): linked to wood price index and all commodity producer price index projected by WEFA.7 Income growth rate 5 0.0447 2 0.000534 p trend (6.92) (1.91). 6 Aget 5 1.73 1 0.89 p Aget21 1 9.62E-5 p Stockt21 2 4.9E-6 p QNC,t21 1 2.85E-6 p QNC,t22 t stat: (4.6) (47.5) (4.2) (5.3) (2.9) R 2 5 0.99. 7 Pmat 5 0.068 1 0.159 p zwood 1 0.814 p Pall comm r (1) 5 0.610 R 2 5 0.96 t stat: (0.8) (6.3) (10.5) (4.7). 5

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