<2000年中華民國住宅學會第九屆年會論文集>
場次：1A-1Housing Production and Factor Demand:
A Cost Function Approach
by
Chu-Chia Lin[*]
Abstract
In this paper, we build up a translog function first, assuming there are only two kinds of input, namely land and capital. Applying a data set from the Land Bank of Taiwan, we estimate the translog cost function with some standard restrictions. In order to check how the production corresponds to the prices of land and capital, we then estimate the function of factor shares. Finally, the price elasticity of factor demand for land and for capital are computed by the estimated factor share functions. We find that both of the demand elasticity and cross elasticity are quite low, which shows that housing constructors in Taiwan do not have much room to substitute land and capital in housing production.
Keywords: housing production, cost function, elasticity of factor demand
1. Introduction
As a factor input of housing production, land is quite different from other factors, such as labor, machines, and construction materials. It is not only that the price of land is quite high comparing to other factor prices, but that the price of land usually has a different pattern of cycle . On the other hand, land as an input for housing production is usually constrainted by certain zoning policies, such as coverage ratio and floor-space ratio. Owing to these special features of land as an input in housing production, it will be very interesting to see the substitution of land and other factors in housing production.
There are quite a few studies on estimating the elasticity of substitution between land input and capital input. For example, Koenoker(1972), McDonald(1981), Jackson and Kaserman(1984), and Thorsnes(1997) have provided excellent empirical studies on estimating the elasticities of substitution between land and capital in housing production. Clapp(1996) showed that the measuremeant error of land and capital could result in housing production. While most studies were applying a CES production functions, Sirmans, Kau, and Lee(1979) and Sirmans and redman(1979) applied a VES production model and showed that the VES model is better than the CES model. Finally, applying a simulation method, Fare and Boon(1985) showed that elasticity of substitution of land and capital in housing production could be changed as land price changed. In other words, the VES function is more suitable in describing housing production behavior.
At the same time, there were also some articles studying the price elasticity of land input and capital input. For instance, Muth(1968) applied an Cobb-Douglas production function and found that the own-price elasticity of demand for land is -0.75; while Witte(1977) using a data set from different cities and found that the estimated elasticity is -0.70. Sirmans and Redman(1979) applied a VES production function and used a data set from 52 FHA housing market areas in the U.S. on the years of 1967, 1971, and 1975. They found that the estimated own-price elasticities of demand for land ranged from -0.35 to -0.80.
In fact, in general, the elasticity of substitution of land and capital is not only determined by the technology of housing production, but also influenced by the zoning policies. Since land input is usually one of the most important inputs in housing production, the constructors will always find some ways to escape the policy restriction. For example, Thorson (1997) argues that, though there exists a significant effect of zoning policy on housing production, the long run effect of zoning on housing production is trivial since the constructors can always find some substitution between land and capital.
In addition to zoning policy, the relative price of land with respect to capital is also crucial in determining the demand elasticity of and the cross elasticity of land demand and capital demand. For instance, at a place with a higher land price, the housing constructors will try their best to replace land input with capital input. On the other hand, if the land input is located at a place with abundant land and with a low price, then the constructors have little incentive to substitute land input with capital input.
In this study, we will apply a data set from Taiwan to answer the above questions. There are several reasons that the data set from Taiwan is perfectly fitted to answer the above questions. Firstly, both the coverage ratio and the floor-space ratio are usually binding in urban Taiwan, but not in rural area of Taiwan. Since the coverage ratio is bounded in urban area, the substitution among land and capital should be smaller. Therefore, the cross elasticity is smaller in urban area than in rural area. Secondly, after the land price had sharply increased between 1989-90, the demand elasticity of land is higher comparing to the period with lower land price, since the housing constructors try to find some ways to escape the high land price, such as choosing more capital intensive production.
The structure of this study is organized as follow: A standard translog cost function and related factor share functions will be illustrated in Section 2. Then the empirical results of the translog cost function and factor share functions are estimated in Section 3, using a data set from the Land Bank of Taiwan. Finally, the estimated own-price elasticity and the cross elasticity of factor demand on housing production are computed. We conclude this study in Section 4.
2. Cost Function and Factor Share in Housing Production
In general, there are four factors in producing housing commodity, namely land, machine, labor, and construction materials. However, we could separate the the above four factors into two groups, that is, land input (L) and capital input (K), where capital input includes machine, labor, and construction materials. According to Hicks(1946), we could group these three as a composite commodity and call it "capital" for two reasons: First, since the land price is so high in Taiwan that the average land cost is about 40% of the total construction cost. On the other hand, comparing to the different movement pattern of land price, the prices of machine, labor, and construction material are relatively stable.
Assuming that the total output of housing production is H, then the housing production function is:
Assuming that the factor prices of land and capital are PL and PK , then, according to the duality theory, the cost function could be written as:
Following Christensen, Jogensen, and Lau (1975), we assume a translog cost function as follow:
Moreover, for a technology with a homogeneous property, the following restrictions have to hold:[1]
By Shephard's lemma, the factor demand functions for land and capital could be got from the first derivative of cost function.[2] Derivating Equation (3) with respect to PL , we get
Rearranging terms in above equation, we get the demand function for land input as
The difficulty for Equation (6) for estimation is that the equation is nonlinear either for variables or for parameters. However we could rewrite the above equation in terms of factor share as follow:
Since the function form of factor share is linear in parameters, it is easy to estimate and it is commonly applied for estimation.[3]
Finally, in order to get the demand elasticity of land input, we take logarithmic form of Equation (6) and then derivate it with respect to PL and PK , and we get
Rearranging terms, the own-price elasticity of demand for land and cross elasticity of land with respect to capital price are as follow:[4]
Since the translong function is a symmetric function for land input and for capital input, the factor share function and the elasticity function for capital are similar to Equation (7) and (9), we omit them here to save space.
The estimation procedure is as follow: Firstly, we estimate the translog cost function both with restrictions and without restrictions. Secondly, we estimate the factor share functions, i.e. Equation (7), for both land share and capital share. Finally, getting the coefficients in factor share functions, we could compute own-price elasticity and cross elasticity for land input and for capital input.
3. Estimation Results
Data Description
This study applied a construction loan data set from the Land Bank of Taiwan for the period of 1981 to 1992. There are several important variables useful for this study including total construction space (TSPACE), total cost (TC), total land cost (LDCOST), total construction cost (CCOST), average land price (PL), average construction (capital) price (PK).[5] For estimation purpose, we take logarithm for all variable and put an LN in front of each variable, such as LNTC, LNTSPACE, LNPL, and LNPK, and so on.
There are 319 effective samples in the data set. We separate the data into two subsamples, i.e. urban area and rural area. The samples coming from Taipei, Kaoshiang, Keelung, Hsinchu, Taichung, Chiayi, and Tainan are classified as from urban area, and the rest is defined as from rural area. The reason that the data set is divided into urban area and rural area is that the zoning policy is different from the two areas, for example, the coverage ratio and floor-area ratio are quite restricted in urban area, but not in rural area. Moreover, since the average land price in urban area is much higher than the land price in rural area of Taiwan, the land intensity in housing production is quite different for the two areas, too.
At the same time, we also separate the total sample into three different time periods, i.e. 1981-1988, 1989-1990, and 1991-1992, since the land price has increased sharply during the time period of 1989 to 1990.[6] One interesting question we like to discuss here is to see if there is a drastic change of housing production, or if there is a significant change of demand elasticity and cross elasticity, after the dramatic land price changed.
The basic statistics of total sample and subsamples for different groups are shown on Table 1. The average total construction space (TSPACE) is 23,688 square meters, the total cost of production is NT$684.5 millions, and the average land price is NT$ 114.6 thousands per square meter. The average land input for urban area is 2,586 square meters which is smaller than rural area (5,851 square meters). In terms of land intensity, the ratio of total construction space to land input is 8.11 for urban area which is much larger than the figure for the rural area (4.85). The reason is simply because that the land price of urban area is much higher than the rural area (NT$134.8 vs 79.6 thousands per square meter).
The land intensity has a big discrepancy among different time periods, too. For instance, the ratio of total construction space to land input for the period of 1981-88, 1989-90, and 1991-92, i.e. before, during , and after the land price sharply increased, is 5.30, 5.66, and 7.09, respectively. The average land price for the three periods separately is NT$91.9, 116.5, and 127.3 thousands per square meter.
Translog Cost Function
Having an appropriate data set, now we could apply a translog form to estimate the housing cost function for housing production with some standard restrictions. Applying the ordinary least squares method (OLS) on Equation (3) and (4), the estimated results are shown on Table 2, where we also provide the results for different regions and different time periods. Overall speaking, the adjusted R2 are higher than 0.9 in all equations, which is quite satisfactory for a cross-sectional data like ours. On the other hand, almost all of the estimated coefficients are significant with correct signs. The restrictions are all significant, too, which shows that the restrictions are appropriate. Finally, most coefficients are quite stable among different equations, which implies that the translog function is an appropriate function form to describe housing constructors' cost behavior.
Table 1 ： Basic Statistics
Total 1981-88 1989-90 1991-92 Rural Urban
TSPACE 23688 20096 18884 28577 28378 20972
(m2) (34528) (39024) (24652) (35950) (41151) (29810)
TC 684.5 391.2 604.0 907.6 841.8 593.4
(NT$mill.) (973.4) (721.5) (816.8) (1126.5) (1290.6) (717.6)
LDCOST 290.5 183.0 233.9 387.6 358.0 251.5
(NT$mill.) (471.6) (352.5) (297.1) (584.4) (658.8) (311.7)
CCOST 394.0 208.2 370.1 520.0 483.9 341.9
(NT$mill.) (583.0) (392.5) (618.8) (628.6) (716.7) (483.7)
114.6 91.9 116.5 127.3 79.6 134.8
(NT$,000/m2) (130.0) (112.8) (137.6) (134.3) (109.2) (136.9)
17.7 13.0 18.4 20.1 16.7 18.2
(NT$,000/m2) (16.8) (15.9) (20.0) (14.8) (13.2) (18.6)
Number of
Observations 319 89 83 147 117 202
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Notes: The figures in parentheses are S.D.
There are several important features in the estimated translog cost function: First, we found that the coefficient of housing output (LNTSPACE) is near but less than unit. This result shows that it exhibits an increasing return to scale in housing production in Taiwan, since the coefficient of LNTSPACE is the output elasticity of total cost.[7]
Secondly, the coefficient of land price (LNPL) is much less than that of capital price (LNPK). In other words, the cost elasticity with respect to capital price is much larger than that of land price. This shows that housing constructors have a better way to deal with the change of capital price,[8] however they cannot get away from land cost much when the land price goes up.[9]
Table 2：Translog cost function with restrictions
Dependent variables：LNTC
Total 1981-88 1989-90 1991-92 Urban Rural