Housing and the Business Cycle

Abstract

Recent events have led to a renewed effort to understand the nature of cyclical fluctuations in the price and quantity of new investment in housing. This paper provides a brief summary of the existing literature modelling housing and the business cycle.

The boom and bust in residential investment and overall production during the first decade of the 21st century can be viewed as a continuation of patterns that are evident in post-Korean War US macroeconomic data. A few features of the data are worth highlighting. First, shown in Fig. 1, residential investment and real GDP are highly correlated at business cycle frequencies.¹ Second, residential investment is much more volatile than GDP and non-residential investment. Table 1 shows that the standard deviation of detrended residential investment is about twice as large as the standard deviation of detrended non-residential investment and more than six times greater than the standard deviation of detrended GDP. This last fact is also evident from the different scales of the axes of Fig. 1. Third, residential investment leads GDP by about one quarter, whereas investment in business capital lags GDP by about one quarter.

Housing and the Business Cycle, Fig. 1

Plot of real detrended residential investment and GDP, 1955:1–2009:3

Housing and the Business Cycle, Table 1 Properties of selected detrended US macroeconomic data, 1955:1–2009:3

Finally, house prices are contemporaneously correlated with GDP and are volatile. An older literature studied the responsiveness of housing prices and quantities to changes in incomes, construction costs and interest rates. A few examples include Alberts (1962), Fair (1972), Poterba (1984), Topel and Rosen (1988).² These papers uniformly assume interest rates are fixed, or are set outside of the model, in the sense that interest rates – the price of current consumption relative to future consumption – are not linked to changes in the marginal utility of consumption. As emphasized by Prescott (1986b), interest rates are a key price in any macroeconomic model. So, while the discussion about housing, mortgages, and so-called ‘Regulation Q’ in these older papers is interesting, they do not fit into the modern literature of business cycles.

The first business cycle models (Kydland and Prescott 1982) did not distinguish residential investment or housing from other forms of capital.³ The goal of these papers was to understand the fraction of the variability of post-war output that could be explained by a neoclassical growth model (Cass 1965; Brock and Mirman 1972) with stochastic stationary shocks to the level of multi-factor productivity around a growing trend. Fairly early on, researchers learned that, while successful along some dimensions, the standard ‘real’ business cycle model underpredicted the volatility of hours worked. In the data, the standard deviations of HPfiltered log hours worked and log GDP are roughly the same, about 1.7% (Prescott 1986a). In the first set of real business cycle models, the standard deviation of simulated hours worked was roughly equal to half of the simulated standard deviation of output.

Soon after the study of Kydland and Prescott (1982), researchers worked on adapting the standard real business cycle model such that it would correctly predict that the standard deviation of hours worked and GDP are roughly the same order of magnitude. Early papers by Hansen (1985) and Rogerson (1988) modified the Kydland and Prescott model to allow for indivisible labour supply.⁴ Soon after, researchers were augmenting the standard real business cycle model to allow for ‘home production’. In a home production model, households receive utility from market consumption, denoted c_m, and home (or non-market consumption), c_n; they accumulate capital to be rented to the market for the purposes of producing market output, k_m, and accumulate capital for the purposes of home production, k_n; and they allocate their time between work in the market, h_m, work at home, h_n, and leisure l. Both the home and market production functions are subject to shocks to productivity.⁵ For recent very good summaries of home production models, see Chang and Hornstein (2006) and Gangopadhyay and Hatchondo (2009).

The home production framework was considered an important extension of the original Kydland and Prescott (1982) model.⁶ The available data suggest that households spend about as much time engaged in working at home as they do in the market (Juster and Stafford 1991). For this reason changes to the allocation of time across the home and market sectors may be of first-order importance in accounting for the cyclical volatility of market hours. For the purposes of studying the role of housing in the business cycle, the home production models were the first papers to explicitly specify a different purpose for residential investment than investment in market capital (such as spending on equipment and software and on non-residential structures).

Researchers have had a number of challenges in calibrating a basic home production real business cycle model, in part because the inputs into the home production process are not all observed. In sum, researchers have had to take a stand on (a) the elasticity of substitution between home and market consumption in utility; (b) the statistical process characterizing shocks to productivity in the home sector and the correlation of home and market productivity shocks; and (c) what (in the data) should be considered as home capital. Taking each of the points in order: Benhabib et al. (1991) use data on hours worked at home, hours worked in the market, and data on wages from the Panel Study of Income Dynamics to estimate the elasticity of substitution between home and market consumption. They find an elasticity of substitution greater than one, i.e. with preferences of the form

$$ u\left({c}_n,{c}_m,l\right)=\frac{{\overline{c}}^{1-\sigma }}{1-\sigma}\upsilon (l)\kern2.5em \mathrm{with}\kern0.24em \overline{c}={\left[\alpha {c}_m^{\rho }+\left(1-\alpha \right){c}_n^{\rho}\right]}^{1/\rho }, $$

(1)

they estimate ρ = 0.8. McGrattan et al. (1997) estimate the process for shocks to home and market productivity using a structural estimation approach that takes advantage of the set of first-order conditions of the model. The authors show that home shocks are ‘relatively insignificant’, in the sense that ‘the result that home production matters does not depend critically on the presence of home technology shocks’ (p. 282). Finally, and importantly, when matching model statistics to data, all papers in the home production literature define the stock of home capital in the data as the sum of the stock of housing structures and the stock of consumer durables.

Generally speaking, the home production models have been challenged in matching two features of the data related to investment in the home sector. First, contrary to the data, the models tend to predict that investment in business capital is more volatile than investment in home capital (Gomme et al. 2001). Second, without adjustment costs, the home production real business cycle model predicts that investment in market and home capital are negatively correlated (Fisher 1997). In response to a positive shock to market productivity, households add to market capital first, since market capital is required to make more of everything. Later on, households increase their stock of home capital. As mentioned earlier, the data suggest that investment in home capital leads investment in market capital by about two quarters. Both of these points are returned to below.

Davis and Heathcote (2005) argue that home production models are somewhat ill suited to studying the business cycle properties of housing specifically. They make two related points. First, in home production models it is assumed that home capital (the sum of housing and durable goods) is produced using the same technology as all other output.⁷ This implies that the real price of housing is constant over time, except for fluctuations due to the presence of adjustment costs. This is clearly at odds with the data. As mentioned earlier, the detrended real price of housing is volatile. But, as shown in Fig. 2, the real price of housing also has an upward trend: After averaging through booms and busts, the trend rate of growth of real house prices has been about 0.5% per year since 1975.⁸

Housing and the Business Cycle, Fig. 2

Plot of real log house prices and trend line, 1975:1–2009:3

Second, when calibrating home production models, researchers treat the stock of housing and the stock of consumer durable goods (hereafter called ‘durable goods’) as equivalent. But housing and durable goods have quite different properties. To start, housing is a much longer-lived asset than durable goods. The depreciation rate on the housing stock is 1.6% per year whereas it is 21.4% per year for other durable goods (Davis and Heathcote 2005). Second (and related), investment in housing is much more volatile than investment in other durable goods: Table 1 shows that the the standard deviation of residential investment is about twice that of consumer durables. Third, residential investment leads GDP by one quarter but consumer durables do not: the highest correlation of detrended real expenditures on consumer durables and GDP is at period t, cell f6. Fourth, house prices are about four times more volatile than the price of durable goods (cells e1 and g1). Finally, house prices are positively correlated with GDP (and might even lead GDP; cells e5 and e6), whereas durable goods prices are negatively correlated with GDP.

Davis and Heathcote (2005; hereafter DH) specify and simulate a model that is viewed by some as the first paper that explicitly studies the business cycle properties of housing. The DH model is a frictionless, representative agent, neoclassical growth model that is a relatively straightforward extension of an otherwise standard home production model. The key extension is that DH specify that housing is produced using a different technology from other goods, allowing it to have a nontrivial relative price. The point of the DH paper is to quantify the extent to which a wellcalibrated model can match the fluctuations in residential investment and house prices observed in the data. Any significant model failures in matching the data could then point to a meaningful role for frictions and/or incomplete markets.

The household side of the DH model borrows heavily from the home production literature. DH assume that households receive flow utility of

$$ U\left({c}_m,h,l\right)=\frac{{\left(\overline{c}{l}^{\mu}\right)}^{1-\sigma }}{1-\sigma}\kern0.24em \mathrm{with}\kern0.24em \overline{c}={c}_m^{\alpha }{h}^{1-\alpha }, $$

(2)

where c_m and l are market consumption and leisure, as before, and h is the stock of housing, not the quantity of home production as in equation (1). As shown by Greenwood et al. (1995), equation (1) reduces to equation (2) when (a) households have log-separable preferences over leisure, market consumption and home consumption, (b) the home produced good is produced using a Cobb-Douglas technology from home capital and labour, and (c) ρ = 1. DH argue, contrary to the results of McGrattan et al. (1997), that available data support the assumption of a unitary elasticity of substitution between consumption and housing.⁹ DH calibrate utility function parameters to match the average share of time that households spend working and the average ratio of the value of the stock of residential structures relative to GDP.

As noted earlier, the production side of the DH model represents the most significant departure from the home production literature, and many recent macroeconomic models that generate nontrivial house prices borrow aspects of this production structure.¹⁰ DH specify three types of firm in the economy. The first set of firms use capital and labour to make one of three intermediate goods called ‘construction’, ‘manufacturing’ and ‘services’. Output of intermediate good i in period t, denoted x_it, for i equal to b (construction), m (manufacturing) and s (services), is specified as

$$ {x}_{it}={k}_{it}^{\theta_i}{\left({z}_{it}{n}_{it}\right)}^{1-{\theta}_i}, $$

(3)

where k_it and n_it are the capital and labour employed in the production of good i and z_it is a sector-specific productivity shock. θ_i is the capital share of producers of intermediate goods i, which can vary for i = b, m, s. In contrast to the home production function in the home production models, DH show that all aspects of this production technology are directly observable with available data. DH use the Gross Domestic Product by Industry Tables, produced by the Bureau of Economic Analysis (BEA), to identify the capital shares θ_i; and, given a value of θ_i, DH use data from the Gross Domestic Product by Industry tables and the Fixed Asset tables, also produced by the BEA, to uncover time series data for k_it, n_it, and z_it.¹¹

A second set of firms uses a Cobb-Douglas technology to combine the intermediate goods into two ‘final’ goods. The first final good can be costlessly split into consumption and investment in business capital; the second final good is residential investment. DH specify output of final good j in period t as y_jt for j = c (consumption and business investment) and j = d (residential investment) to equal

$$ {y}_{jt}={b}_{jt}^{B_j}{m}_{jt}^{M_j}{s}_{jt}^{S_j}, $$

(4)

where B_j, M_j and S_j are the value-added shares of construction, manufacturing and services in the production of final good j. DH show that these shares are identifiable using data from the Input–Output tables, also produced by the BEA.¹² DH show that residential investment is much more construction intensive than the other final good, which turns out to be important in explaining the relative volatility of residential investment.

A final set of firms in the DH model combine new residential investment with new land (made available by the government each period) to create new housing units. The specific production function for the quantity of new housing built in period t, y_ht, is

$$ {y}_{ht}={x}_{lt}^{\varphi }{x}_{dt}^{1-\varphi }, $$

(5)

where x_lt is the amount of newly developable land and x_dt is residential investment (produced according to equation 4). DH identify the parameter φ based on results about the share of the value of new housing attributable to raw land costs from an internal memo of the US Census Bureau.

Thus the DH model has three ingredients that allow for potentially interesting time-series variation in house prices. First, the statistical process (mean growth rate, variance, and autocorrelation) for z_it is allowed to vary across the construction, manufacturing and services sectors. Second, firms that produce residential investment use different combinations of these three intermediate goods than do firms that produce the other final good. The price of housing has a long-term upward trend according to the DH model for these two reasons: DH show that z_bt has zero trend growth, and construction accounts for about 50% of the value-added in residential investment (compared to 3% of the value-added of the other final good). Finally, new housing requires both new land and new residential investment, and new land is in fixed supply. The scarcity of land affects both the trend and the variance of house prices in the model.

Some key second moments from the data and from simulations of the DH model are reported in Table 2. The information in this table is copied directly from Table 10 of DH.¹³ Rows (a) and (b) of Table 2 show that the DH model under-predicts the volatility of consumption and of hours worked. In this regard, the results of DH are similar to previous models. However, the DH model has great success in replicating key facts about residential investment, namely that residential investment is about twice as volatile as business investment (rows c and d) and that residential and business investment are positively contemporaneously correlated (row f). DH show that the low depreciation rate on structures and the relatively high labour share of the construction sector are largely responsible for replicating the relative volatilities of residential and business investment. With a low depreciation rate, it is possible for households to ‘concentrate residential investment in periods of high productivity’ (p. 774); and, with a high labour share of the construction sector, ‘it is easier to expand output rapidly the more important is labour in production, since holding capital constant, the marginal product of labour declines more slowly’ (p. 774). The positive correlation of residential and business investment is attributable to the fact that new housing needs new land as an input in production, and new land is in fixed supply. In this regard, land in the DH model acts analogously to adjustment costs in the home production models.

Housing and the Business Cycle, Table 2 Business cycle properties of the Davis and Heathcote (2005) model^a

Although the DH model replicates some key features of housing investment, it does not match some key features of the housing data. The DH model cannot generate that residential investment leads GDP and business investment lags GDP (not shown).¹⁴ Second, the DH model cannot replicate two important features of house prices. Shown in row (e) of Table 2, the DH model under-predicts the volatility of house prices by about a factor of three. The DH model also predicts that residential investment and house prices are negatively contemporaneously correlated, whereas in the data they are positively correlated (row g). Future researchers are actively focusing on reconciling these issues.

See Also

• Household Production and Public Goods

• Housing Supply

• Housing Wealth

• Urban Housing Demand