Unifying Empirical and Theoretical Models of Housing ~ Supply

Housing supply plays an important role in the Volatility of macroeconomic cycles and the speed with wl-iich house prices respond to changes in demand, yet it is understudied in the current literature. In this paper we present and estimate a new model of the supply of residential construction that is consistent with the theoretical treatment of land deveiopment and urban growth. The model shows that new housing construction is best described as a function of changes in house prices and Costs rather than as a function of the levels of those variables. Previous researCh that uses the price levels specification has the drawback that a one-time increase in the number of households that raises the level of real house prices leads to a permanent jump in new construction and thus an infinite increase in the stock Of housing. The empirical tests of the model support our new specification, which performs better than alternative models in out-of-sample forecasts. Our estimates suggest a fairly moderate response of supply to house price changes. A 10 percent rise in real hous~ prices leads to an 0.8 percent increase in the housing stock, which is accomplished by a temporary 180 percent increase in the average number of quarterly starts, spread over four quarters. Introduction Construction of new housing plays a critical role in the economy. Housing starts are more volatile than the overall economy and tend to lead recessions and recoveries. Residential construction influences overall output both directly, as construction and manufacturing employment rises with housing starts, and indirectly, through the multiplier effect and because new home buyers tend to purchase other consumer durables contemporaneously with the purchase of their house. Changes in new" housing supply also affect the price of existing units, which in turn has a large influence on the wealth position of existing homeowners and helps determine housing affOrdability. Finally, the elasticity of supply is a key determinant of how housing prices would respond to fundar~ental tax reform. (See Capozza, Green, and Hendershott 1996.) Despite its importance in the macroeconomy, empirical research on housing supply is surprisingly rare. This dearth of work is striking when compared to the extensive literature on housing demand, a discrepancy noted in housing market overviews by Olsen (1987) and Smith, Rosen, and Fallis (1988), among others. Most of the existing empirical work treats housing like or-her types of capital. This approach fails to recognize the defining role of land in residential development and construction. In this paper we present and estimate a model of the supply of new single-family residences that is consistent with the theoretical treatment of land development and urban growth. Unlike other empirical models, our approach is also consistent with the time series characteristics of the data. After developing and testing the model,,we use out-of-sample forecasts to compare it with several alternative empirical treatments of new housing supply. Our model of new housing supply is developed from the relationship between city size 1 and land and housing prices found in Capozza and Helsley (1989). Housing has two components: the structure (capital), which is supplied at constant cost in the long run, and land which is inelastically supplied, even in the long run. Over time, differences in housing are solely determined by differences in land prices. In a given metropolitan area, vacant land is available at the fringe of a city, but this land is inferior to existing locations that are closer to the downtown or other suburban subcenters. Unlike other investment goods, the long run cost curve for land is upward sloping. A one-time increase in new conStruction leads to a permanent increase in land prices to ensure a spatial equilibrium. House prices are a stock variable that equilibrates the total quantity of housing with the total demand for residential space. Housing starts, on the other hand, are a flow" variable, representing the change in the stock of housing. Starts should be a function of other flow variables, including the change in house prices. In contrast, the standard empirical model characterizes new construction as a function of the level of house prices. This traditional specification ignores the fact that variables such as city size and the opportunity cost of new land help to determine land prices and thus housing price levels at existing locations, but have no effect on the number of new starts in the steady state. A simple example demonstrates the intuition of treating housing starts as a function Of houSe price changes. Imagine a city composed of a stable number of homogeneous households. If the city is not growing and housing units do not depreciate, then the housing market will be at its long run equilibrium, house prices will be constant, and housing starts will equal zero.~ 1If households are heterogeneous and changing, then there can be new starts to meet the changing needs of the stable existing population. Even with positive depreciation, housing starts will just equal a 2 Suppose that the city has an unexpected one-time influx of population. Demand for new residences increases, land and house prices rise, new construction occurs, and the city increases in size to accommodate the new residents. At the new equilibrium, the city is physically larger. To ensure that households are indifferent between living in houses in the newer, more distant locations and existing units, the price of developed locations must rise above the level that they were before the population inflow. In the new spatial equilibrium, population is again stable and there are no expectations of further growth, so starts are again equal to zero and prices are constant, though at a higher level. In our example, starts occur only when the city makes the transition from one equilibrium ~o another, a period identified by the increase in the price level. A mode! where starts are a function of the price level would predict a permanent increase in the number of housing starts resulting from the one-time unexpected increase in population.2 Yet starts will increase only as needed to accommodate the new residents, a one-time event. The difference between housing starts and housing price levels is also apparent in the data on U.S. housing prices and starts shown in Figure 1. (An explanation of the data is provided below in Section IV.) Between 1987 and 1994, house prices remained above the tevel of earlier periods, yet starts during this period were consistently below the number of starts recorded in the late 1970s. This figure suggests the limitations Of using price levels to explain housing starts. By contrast, Figure 2 show’s that the constant percentage of the stock that does not vary each period 2If we allow for depreciation in the model, we can obtain a positive correlation between housing prices and starts. When the population of the city is higher, the city occupies a greater land area, so that housing prices are higher, and the stock of housing is larger. With a constant removal rate, the larger city requires a greater number of housing starts to maintain its existing stock of units. Thus starts after the increase in population would be higher than before, as would house prices. 3 relationship between housing starts and the change in house prices is much more stabie. Unlike the conventional treatments of housing supply, our model generates a stable measure of the true supply elasticity, the percentage change in the housing stock from a percentage change in prices. A one-time increase in house prices leads to a one-time increase ~n the stock of housing, because of the temporary increase in new construction. In contrast, existing research uses estimates from the housing starts function to identify, a starts elasticity, the percentage change in starts caused by a percentage change in the level of house prices. With this treatment, a change in the level of house prices results in a permanent increase in new construction. Though the change in new construction is finite, the stock of housing will increase without bound as starts are higher in all future periods. Thus, a fixed starts elasticity yields an infinite supply elasticity. Our estimate of the true elasticity of supply with respect to prices is qhite small because housing starts are a small percentage of the stock annual starts are 2.2 percent of the stock. Treating starts as a function of house price changes is also consistent with the time series properties of housing stock and prices. Previous research (for example, Holland 1991, Meese and Wallace 1994, and Rosenthal 1995) shows that the real price of existing housing is not stationary in levels (I(!?)), but is instead stationary in differences (I(1)). The stock of housing is also non-stationary. Advances in time series analysis have revealed problems in estimating relationships between data series that are non-stationary. Although over short time periods or in small samples trending and stationary variables may be correlated, in the long run this correlation will disappear. Furthermore, regressions using multiple non-stationary series can lead to spurious correlations (Granger and Newbold 1974"1. If the stock of housing and real house prices