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A model of individual household temperature demand and energy-related welfare changes using satiety
A model of individual household temperature demand and energyrelated welfare changes using satiety
John E. Kushman and Joan Gray Anderson
A model ofthe utility-maximizing household is developed in which data on satiety levels for indoor temperature can be used along with budget constraint parameters and actual temperature choices to estimate parameters of the utility function. Using the utility parameters, equivalent variations in income can be estimated to measure welfare effects of policy and price changes that effect home heating. All of these operations can be performed at the individual household level and for any member of a popular class of utilityfunctionforms. The required data are easy to obtain, although they typicall_y are not collected in household surveys. Keywords: Energy demand: Households:
Economic models of household behaviour usually assume away potential satiation in any good. This is remarkable since the combination of goods at which satiation occurs should give information about preferences, and preferences are the key to predicting quantities demanded and to measuring welfare changes. We show that data on satiation temperatures can be used along with temperatures actually chosen to obtain parameters for temperature demands and welfare changes. This applies to a widely used class of John E. Kushman is with the Department of Agricultural University of California at Davis, Davis, Economics, CA95616, USA, and Joan Gray Anderson is with the Consumer Affairs Program, University of Rhode Island. The research reported here was supported
by regronal research project W159-Consequences of Energy Conservation Policies for Western Region Households: the Califorma Energy Studies Program ofthe University ofCalifornia: the Public Service Research and Dissemination Program. University of California at Davis: and the California Agricultural Experiment Station, Giannini Foundation Paper 795. Final manuscript
received
5 August
0140-9883/86/030147-08
$03.00
1985.
Q 1986 Butterworth
Welfare effects
utility functions including the quadratic, translog, and Diewert functions. The functional forms must be truncated so that they lose some oftheir flexibility, but there is a gain in flexibility in that the parameters can differ among households. The gain in flexibility among households is important because it allows detailed examination of variations in preferences, demands, and welfare effects among households. Computations with our procedure are simple. The required data are easy to obtain, although they typically are not collected in household surveys, and are somewhat unorthodox in relying on introspection and testimonial evidence. Strategic additions to surveys and a further research agenda are indicated. The methods developed and illustrated here may be significant in facilitating policy analysis of energyrelated initiatives. Many public initiatives exercised through tax provisions, regulated prices, and mandated low-interest home improvement loan programmes have the effect of changing the price of home heating. The substantive purposes of these initiatives are to encourage energy conservation and to
& Co (Publishers)
Ltd
147
redistribute income in the face of higher fuel costs. A model that permits the effects of policies on home temperature, fuel consumption, and welfare changes to be evaluated at the individual household level has in examining important potential uses, especially Willig and redistributive effects (Posner [7], Bailey [lo]).
The model Consider a household that maximizes utility during the heating season as a function of indoor home temperature. T. and all other goods, M. Let the utility function be of the class
~r=u,.l‘(T)-u,f'(T)'+u,f(M)+~l,f(M)~ +u,f(T)f‘(M) (1) where the zli are parameters and f(x) (x = T,M), is any function such that f(x) >O and f,>O (subscripts indicating derivatives in context). In particular, any of the popular family of Box-Cox member transformations.
f(s) = log x
/i=o
(2)
will do. The translog function corresponds to the case i = 0. The quadratic and Diewert functions correspond to i = 1 and i = 0.5. We truncate the functional form by deleting the last two terms. Restoring the last term is considered below. The resulting functions are less flexible as approximations to arbitrary utility functions. and deleting the last term imposes the assumption of additive separability. The truncated quadratic function requires constant marginal utility of all other goods. The truncated translog and Diewert functions impose diminishing marginal utility of all other goods. It is obvious that for each consumer (household) there exists a satiety temperature below which higher temperatures are preferred and above which lower temperatures are preferred. If 7* is the satiety temperature. in terms of the truncated function,
In order for a sign switch, Equation (4). to occur. u2 must be positive, since only the increase of f(T)with T in the second factor of Equation (3) can produce the switch. Dividing the utility function by 112gives a ne\ order-preserving utility function (5)
c=Pl.f‘(T)-.~(T)‘+P2.f‘(M)
This reduces the number of parameters to two. If the satiety temperature is known for a household, 8, can be found by setting t+(T*)=Oas in Equation (4) and rearranging, PI = I.f(T*)
(6)
The first-order conditions for utility maximization can be used to find f12 from the temperature actually chosen and the elements of the budget constraint. The household is assumed to produce indoor temperature at least cost given fixed heating-related elements of the household capital stock (insulation, type and size of structure, etc). This household production activity is embedded in the utilitymaximization problem by entering a cost function for temperature into the budget constraint. Using the production function for indoor temperature derived by Collins and Gray [3]. assuming the heating-related elements of household capital stock are fixed and the marginal and average prices of fuel are equal and constant, the short-run budget constraint for the household is
Y-PT(T-To)-PllM=O
where Y is income, P, and P, are prices of temperature and all other goods, and To is ambient temperature. Characteristics of the house and heating plant and the price of fuel are incorporated into P,’ Solving Equation (7) for M in terms of T. substituting for M in Equation (5), and optimizing subject to T>,To gives first-order conditions (8)
T-T,20
(9)
(T- To)r,=O HII,
UT=
-2~r,.l‘(T)l
(3)
II,
>
0
1r,=0
I[,<0
14x
T*>T when
T*=T T*
(4)
(10)
short-run cost function for temperature is C=P, ’ where P, is the priceoffuel.! is theaverageinsulation value for the house. and r reflects other dwelling characteristics. P,= P, 1 I-‘. For details of the derivation of the short-run cost function for temperature. see Anderson [I]. The model presented here is a short-run model and does not address the capital stock aspect of energy use decisions. For a discussion see Hausman [4]. No&-pricing for fuel could be incorporated, but it is omitted here for expositional simplicity. Blocking-pricing data were not available for the empirical work to follow.
IThe
[z(T-
and
(7)
&)I/
ENERGY
ECONOMICS
July I986
A model of household temperature demand: J. E. Kushmun tr17rl J. G. .4ndrrson
From
Equation
(8)
uT=rit
(11) where Equations (8) and (11) are equalities whenever the household heats. When T> TO,Bz can be found by calculating Equation (11). When T= TO, Equation (11) provides a ‘best guess’ lower bound on p2. In fact, /I1 can always be recovered by asking at what temperatures households currently not heating would start to heat. Parameters /?i and /I1 can be calculated for each household. With the utility parameters available, each household’s demand for temperature and (by Shepard’s lemma and the cost function for temperature) the demand for fuel are implied by the first-order conditions (8)-(10). An explicit solution for the quadratic case is
Equivalent and compensating variations also can be calculated for changes in fuel prices, home insulation levels, or other changes that will be reflected in changes in P, In econometric studies, it usually is assumed that all households (or at least all households in a given class defined by income, number, and types of persons, or other characteristics) have identical utility functions, and a single set of parameters is estimated (eg Murray [5], Scott [S] and Pindyck [6]).2 This approach loses much of the potential richness of the data. The approach presented here allows the utility parameters to vary among households. Aside from being potentially more informative and accurate about household demands for temperature and fuel and about welfare changes, this technique permits exploration of the variability of preferences and any systematic relationships of preferences with socioeconomic characteristics. It may be argued that the usual econometric approach has an advantage in that it does not require truncating the functional form, Equation (l), so that the additional parameters provide a more flexible fit to the utility function in the goods variables. In particular, objection may be raised to excluding the last term of Equation (l), thereby imposing additive separability. With the interactive term in the functional form the counterpart to Equations (8)( IO) that implicitly defines the satiety temperature is ‘The discussion is in terms of continuous choice variables. The random utility functions of discrete choice models arise in a different context (Hausman [4]).
ENERGY
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July 1986
-2u,f(T*)+u,
(M)=O
(13)
Rearranging Equation (13) shows the testable implication of additive separability that the satiety temperature is independent of consumption of all other goods, f(T*)=
(u,!2)Cu, + u&M)3
(14)
A sign switch at T* for a given level of M still requires u,>O, so that Equation (14) may be rewritten f(T*) =
( VW, + PJ-(WI
(15)
wherev=p,f(T)-f(T)‘+P2f(M)+83.f(T).f’(M).The hypothesis of additive separability can be tested most directly by asking each household whether their ideal temperature depends on their other consumption. For those households that respond affirmatively. a series of equations could elicit pairs of T* and M, and a regression of Equation (15) would provide estimates of /!It and p3. The counterpart of Equation (I 1) will give an estimate of pz as B2 B
CL6
-
2f(T)
+
BJW)l U=,,lP,N.J;iJ;,) - BJ’( T) (16)
A less direct way of testing additive separability is to obtain pairs of T* and M for each household and conduct an hypothesis test on each regression, Equation (15). Either approach allows a test for additive separability and recovery of the utility function parameters on an individual household basis. The parameter for,f(M)2 cannot be recovered and must be assumed to be zero. The loss of flexibility entailed in this truncation, however. is likely to be unimportant. The remaining terms are likely to approximate the utility surface adequately, especially since these terms can differ among households. An additional degree of flexibility among households can be added by permitting different households to have different members of the class of utility functions, Equation (1). For instance. the household could be asked what temperatures it would choose if the price of fuel doubled, and the utility function form (eg value for i. in the Box-Cox transformation) could be chosen that best predicted the response from parameters (fi) based on the satiety temperature and current temperature.
Illustrative
computations
The methods described above were applied to data from 94 households who responded to a survey of residents of California on energy use, conservation behaviours and attitudes, and to a follow up survey on
149
A model qf household
temperature
demand: J. E. Kushman
Table I. Descriptive statistics for sample distributions of parameters for truncated quadratic and translog utility functions.
Statistic Mean Standard deviation Minimum Maximum
Utility function form Quadratic
Translog
/I, 141.74 7.33 130.00 160.00
PI 8.53 0.10 8.35 8.76
“; 740,60 8 663.40 113.42 52 074.00
B2 12.17 13.74 0.03 50.34
insulation decisions. Of the survey questions suggested by the model and not ordinarily included in questionnaires, the data include only the satiety temperature. In addition, estimates of the utility parameters were obtained by regression from the pooled data for comparison. This is the usual approach of assuming the same utility function for all households except as captured by shift variables. The respondents were asked, ‘To what temperature would you heat your house if temperature were free?‘. If the truncated additively separable model is used, this question and questions on actual temperature, housing characteristics, and characteristics of the heating plant are sufficient to implement the model. For households not heating, only a lower bound on b2 would be available. The data used here were for January, and all households in the sample were heating. If it were desired to test for additive separability or to search for the function form for each household, additional questions would be needed. The satiety temperature in the present data would correspond to a consumption of other goods equal to VP,,. The sample available for the calculations is known to be biased, and no generalizations should be made from the results.3 The results do serve, however, to illustrate the procedures. The quadratic and translog functional forms were used. The quadratic offers simplicity while the translog is a popular flexible form. As noted in the previous section. the quadratic and translog require constant and diminishing marginal utility of other goods. respectively. The question arises whether the different forms and techniques will lead to substantial differences in results. Table 1 describes the parameters, pi and /3,. that were estimated for the respondent households using the method of the first part of this paper. The different functional forms require different parameters for the same household, and this is evident in the mean parameter values. The functional forms do not require “For details on the surveys and on variable construction. see Anderhon [I]. Complications in defining home temperature were dealt with only in w,eighting the diNerent temperatures maintained at different times ofthe day. Scott [X] and Scott and Capper [9] discuss alternetive measures of temperature.
150
and J. G. Anderson
any great differences in parameter values untong households. In fact, for each functional form there is relatively little dispersion among the j3, values and much more dispersion in the flzs as indicated by the standard deviations and the minimum and maximum parameter values. The relatively tight distributions of p,s arise from the relatively small variation in satiety temperatures, see Equation (6). The flZs reflect variations in the flis, temperatures actually chosen, prices, and incomes (see Equation (11)). The wide variation in p2s illustrates the flexibility of the model across households, In the usual econometric approach to estimating parameters of the utility function (or, equivalently, of the demand function) all households are assumed to have the same parameters except as captured by the coefficients of household characteristic variables included in a rather adhoc fashion. Alternatively, household characteristics may be used to stratify the sample o priori with separate parameter estimates for each group. The model proposed here allows a test es post for relationships between utility parameters and household characteristics. The test results can be used to classify households on the basis of their characteristics according to sensitivity to economic parameters or to group households for further econometric analysis. A simple test for relationships between utility parameters and household characteristics is illustrated in Table 2. For each form of the utility function considered, the j?s are regressed on potential measures of tastes for indoor temperature and other goods. Since this analysis is exploratory, a 10% standard is used for two-tailed hypothesis tests. In the regressions for p,, the constant and opinion ofthe energy problem are the only statistically significant variables regardless of the functional form. Households with a more serious opinion of energy scarcity have lower satiety temperatures, all else constant. In the regressions for pz, the results depend on the functional form. The constant, young children, and older persons are statistically significant variables with the quadratic form. For the translog form the constant, education, older persons, and proportion of time at home are significant. Since flZ is the coefficient of other goods in the utility function, results for both functional forms indicate that households with older persons have a relatively strong preference for indoor heating over other goods. Otherwise, the results differ between functional forms. The question arises whether the sensitivity to functional form carries over to questions of more direct policy relevance. Two scenarios were simulated with the respondent households, a doubling of fuel prices and a mandatory retrolit to RI9 attic insulation. For
ENERGY
ECONOMICS
July 1986
A model
Table 2.
Regressions of utility parameters
Household characteristic
Constant Household head years of education Number of children less than five years old Number of persons 55 years old
more than
qf household
temperature
demartd: J. E. Kushman
and J. G. Andersor7
on potential measures of tastes. Regression coeffiients (r-values) Utility function form and parameter Quadratic
Translog
81
/iI
PI
-
I{:
141 .Ol (14.04) - 0.07 (0.17)
19 399.00 (1.75) 327.60 (0.70)
8.50 (60.76) 0.w (0.17)
36.2h
-0.41 (0.23)
4 734.00 (2.41)
- 0.01 (0.24)
O.OiY’ (0.00)”
- 1.34 (1.41)
- 2 158.00 (2.06)
- 0.02
12.10) I.26 (1.72)
(1.42)
-3.87 12.36)
I .72 (1.12)
-1.66 (1.86)
1050.00 (1.07)
- 0.02 (1.83)
Attitude toward voluntary home energy conservation measures’
0.03 (0.14)
- 28.88 (0.12)
0.W (0.19)
-0.4x
Attitude toward mandatory conservation measure?
0.07 (0.72)
- 163.55 (1.45)
0.00” (0.68)
- 0.I4
Willingness to undertake energysaving measures at home specifically related to heating and cooling’
0.03 (0.32)
37.73 (0.37)
0.00” (0.33)
0.08 (0.49)
Proportion of time spent at home
2.62 (0.51)
- 7 084.80 (1.24)
0.04 (0.53)
- 21.62
Local mean outdoor temperature
0.06 (0.34)
- 127.41 (0.64)
0.09
0.21
Opinion of energy problemb
energy
R2
4All equivalent
ENERGY
variations
are for one month.
ECONOMICS
July 1986
(0.7X)
(2.43)
0.00” (0.36)
- 0.23 (0.74)
0.09
“Rounded to two digits. bA higher score indicates the respondent regards the energy problem as more serious. ‘Attitude toward free or low-cost actions to reduce home energy consumption like using curtains to maximize indicates a more pro-conservation attitude. dA higher score indicates a more positive attitude toward governmentally-imposed energy conservation measures highway speed limits). ‘A higher score indicates greater willingness
each scenario, the new optimal temperatures and the equivalent variation were computed with each functional form. In nearly all cases, the differences in new optimal temperatures for the quadratic and translog functions was less than one degree, a difference attributable to the algorithms used. It follows that there were only very small differences in the fuel demanded as well. The calculated equivalent variations also were very similar for the two functional forms. For the quadratic and translog functions, respectively, the mean equivalent variations were $13.89 and $14.95 (doubled fuel prices) and -$9.25 and - $9.35 (mandatory attic insulation retrofit).4 For both scenarios, the correlation coefficient between the quadratic and translog equivalent variations was 0.99. The price elasticities of demand for heating degrees were calculated, and weighted means (weights equal to
(1.26)
0.23
heat retention.
A higher score
not necessarily
in
the home (eg
heating degrees) were computed for the sample. The elasticities for heating degrees will be equal to the elasticities for fuel as will the weighted elasticities. The weighted mean price elasticities are - 0.68 and - 0.53 for the quadratic and translog functions, respectively. Unweighted means are - 1.05 and - 0.78 in the same order. Estimates of the utility parameters, welfare effects. and price elasticities were made from the pooled data for comparison with those above. The pooled estimates were obtained by procedures paralleling those used above to enhance comparability. First, /j, was estimated by specifying p, = a’X + E in Equation (6) where X is the vector of a constant and the taste variables described in Table 2 and c is an crtl koc error term. The resulting equations have been presented in the first and third columns of Table 2. Next, the [jI equations were reestimated, excluding all except the statistically significant variables to refine the pooled estimates. The resulting equations were:
151
A mode/ of household
quadratic
temperature
demand:
and J. G. Anderson
J. E. Kushnan
fi, = 146.80- 1.3322 (opinion of the energy problem) (66.48) (1.95) b, = 8.59 - 0.02 (opinion of the energy problem) (279.12) (1.89)
translog
Table 3.
Regression results for /j2 parameters
Regression coefficients (r-value) Quadratic Translog
Variable
12 665.00
Constant
Parameter p2 was estimated conditional on the values of /?r given by these equations. Equation (11) was rearranged to give [S,
-2A771 [PdPTl rahrl=Pz
(17)
with j?, replaced by 8, and pz=l~‘X+p. Again, the vector X contains the household characteristics and p is an LU/Itoc error term. This regression, _r= $X + p, is conditional on fi, as were the previous estimates of fi2. In both approaches, X is introduced through the Ps. The regression results for Equation (17) are given in Table 3. Generally, the coefficients have the same signs as those from the individual household method (Table2), and they are of the same orders of magnitude, but the statistical significance of some variables shifts between methods. As in Table 2, the statistical significance of some variables differs between the functional forms. The statistically insignificant variables were dropped from the regressions, and the equations were rerun to give quadratic
8, = 17 932.00+4 (4.58) + 1 126.6 (1.67)
049.70 children (2.63)
- 1906.50 (2.29)
Household education
head years of
Number of children 5 years old
less than
Number of persons 55 years old
more than
Opinion
of energy
309.35 (0.80)
I .09 (1.57)
I 365.50
Attitude toward mandatory energy conservation measures Willingness to undertake energysaving measures at home specifically related to heating and cooling of time spent at home temperature
42.51 (2.59)
_ 1 720.30 (1.98)
problem
Local mean outdoor
(3.37)
4281.40 (2.64)
Attitude toward voluntary home energy conservation measures
Proportion
from pooled data.
-0.19 (0.07
1.9x
(1.67)
(1.35)
- 155.43 (0.77)
-0.41 (1.13)
-0.18 (1.90)
-0.18 (1.06)
- 13.58 (0.16)
0.08 (0.56)
- 10214.00 (2.17)
- 24.80 (2.94)
-113.11 (0.68)
- 0.34 (1.15)
0.26
R’
I
-3.17 (2.04)
0.24
older Table 4. Descriptive statistics for utility parameters, demand price elasticities, and equivalent variations from estimation with pooled data.
(opinion of energy problem) Parameter
and statistic
Utility function form Quadratic Translog
PI
- 227.84 (attitude toward mandatory conservation) (2.98)
Mean Standard deviation Minimum Maximum
-9 052.70 (time at home)
Pz Mean Standard deviation Minimum Maximum
(2.15) translog
j2
= 21 .I 1 - 2.80 older - 21.93 (time at home) (2.99) (7.45) (2.06)
Having obtained pooled estimates of the utility parameters and how these parameters are associated with household characteristics, individual household utility parameters, demand price elasticities. and were computed. Table 4 equivalent variations describes the results. The utility parameters calculated from the pooled equations are similar, in terms of the sample means, to those obtained with the individual household method. however, are much less The pooled parameters, variable over the sample. The weighted mean price regardless of elasticities are of similar magnitudes
152
142.75
1.46 141.47 146.80 7 524.10 3 639.30 190.93 18 863.00
8.53 0.02 x.52 8.59 I I .09 5.02 0.88 20.19
Price elasticity for heating or fuel Weighted mean Unweighted mean
-0.56 - 1.37
-0.53 - 0.72
Mean equivalent variation Doubled fuel prices R I9 retrofit
6.18 _ 2.65
2.32 - 0.95
method or functional form. The estimated welfare effects using the pooled equations are much smaller than those estimated by the individual household method. In contrast to the case with the individual met hod, the mean equivalent variations computed
ENERGY
ECONOMICS
July 1986
A model
of ho~rseholritemperarure demand: J. E. Kuslm~~t~urd J. G. Adcrsor~
from the pooled estimates are quite sensitive to the functional form used. An important potential advantage of the individual household method is its flexibility among households. This is reflected in the narrower range of estimated utility parameter values for the pooled method. Also, the demand elasticities and equivalent variations are only poorly correlated between the pooled and individual methods over households. For instance, using the quadratic function and mandating R19 retrofit, equivalent variations over the sample are correlated at only R =0.25. The pooled equivalent variations also show little correlation (R=0.28 and 0.49) between functional forms. Comparison of the demand price elasticities calculated here with those of other studies is problematic, since the samples are different. Barnes et al ([2], p 548) have estimated a short-run elasticity for space-heating electricity using household data at -0.9269 (standard error 0.2962). This elasticity is substantially larger than any of our weighted mean elasticities, although it is closer to our unweighted means. Barnes er al [2] calculated their elasticity using pooled parameter estimates and for a ‘representative household’ characterized by the sample mean values. Using the same procedure on the present sample gives price elasticities of - 1.3943 (quadratic) and - 1.5871 (translog).
Research agenda At least two reservations may be held about the proposed technique. First, the analysis has not dealt with the statistical properties of the parameter estimates. Second, the data are based, in part, on introspection and testimonial evidence, in contrast to the behaviouralist tradition in economics. With respect to the properties of the estimators, a straightforward answer is that, regardless of their properties, they are the only fully household-specific estimates available from a cross-section. Or. by making the required ad hoc assumptions about error terms apriori, any desired properties can be obtained. A more satisfactory answer to the question of statistical properties is that the nature of any errors should be determined experimentally. This approach also would address the adequacy of testimonial data. For example, errors in estimating fii will arise from errors in reporting Tr, see Equation (6). The nature of the errors could be examined by obtaining estimates of 7* for an experimental group of households and then paying their heating bills to observe the true satiety temperatures. The resulting distribution of errors in estimating Tr would indicate the nature and magnitude of any error specification for Equation (6)
ENERGY
ECONOMICS
July 1986
and the properties of the estimator for /il. This procedure would permit the relatively inexpensive collection of testimonial evidence based on introspection and its systematic linking to a behavioural model. Determining 7* experimentally for a very large number of households is likely to be prohibitively expensive. but work with a smaller group can validate and appropriately qualify less costly data. Experimental procedures also could be used to obtain estimates of j13, although the expense would be ev’en greater. Limited experiments to relate testimonial data to experimental data would be more cost effective.
Summary and implications This paper has presented a model of household heating decisions that uses satiety in temperature and utility maximization to identify the parameters of a class of utility functions. A series of survey questions were identified that allow flexibility of the utility function across households. The class of utility functions includes slightly truncated versions of the popular quadratic, translog, and Diewert functions and, in fact, the entire class of Box-Cox transformations. With appropriate survey questions all of the parameters of the utility functions can be obtained. The flexibility of the model across households seems more important than the loss of some flexibility from truncating the functional form. Once the utility parameters have been obtained for each household, many empirical questions can be addressed. In the set of households available to this study, the association of utility parameters with household characteristics depended, to some extent. on whether the quadratic or translog utility function was chosen. The answers to questions of more direct policy relevance, however, were not sensitive to functional form. A doubling of fuel prices or mandatory attic insulation retrofit had nearly identical effects on temperature choice and welfare regardless of functional form. The price elasticities of demand for heating degrees of fuel also were insensitive to functional form. These findings presently cannot be generalized to other households or functional forms, but they illustrate the potential of the model as a research tool. Estimates of utility parameters, price elasticities of demand. and welfare measures also were made from the pooled sample. The mean values of estimate utility parameters and demand elasticities were similar for the pooled and individual household techniques. The pooled estimates, however, were less variable over the sample. Pooled estimates of welfare effects were much smaller on average than those from the individual
153
A model of household
temperature
demand:
J. E. Kushman
and J. G. Anderson
household method and more sensitive to choice of functional form. Properties of the estimators for utility parameters based on introspection and testimonial data may be determined experimentally. Testimonial data can then be used to obtain household specific estimates for policy analysis at relatively low costs with appropriate qualifications.
References Joan Gray Anderson, Indoor Temperature and Insulation Choice: Theoretical and Econometric Models for Policy Analysis, unpublished PhD dissertion,
Department of Agricultural Economics, University of California, Davis, 1984. Roberta Barnes et al, ‘The short-run residential demand for electricity’, Review of Economics and Statistics, Vol63, No 4, August 1981, pp 541-552. Robert A. Collins and Joan K. Gray, ‘Derivation of a theoretical form of a production function for home
1.54
a comment’, Energy Economics. Vol5, No 4, October 1983, pp 273-275. Jerry A. Hausman, ‘Individual discount rates and the purchase and utilization of energy-using durables’, Bell Journal ofEconomics. Vol 10. No 1, Spring 1979, pp 3354. Michael P. Murray, ‘The distribution of tenant benefits in public housing’, Econometrica, Vol43, No 4, July 1975, pp 771-788. Robert S. Pindyck, ‘International comparisons of the residential demand for energy’, European Economic Review, Vol 13, pp l-24. Richard A. Posner, ‘Taxation by regulation’, Bell Journal ofEconomics, Vol2, No 1, Spring 1971, pp2250. Alex Scott, ‘The economics of house heating’, Energy Economics, Vol2, No 3, July 1980, pp 130-141. Alex Scott and Graham Capper, ‘Production functions for house temperatures: problems and issues’, Energy Economics, Vol5, No 4, October 1983, pp 275-278. Robert D. Willig and Elizabeth E. Bailey, ‘Income distribution concerns in regulatory policymaking’. in G. Fromm, ed, Studies in Public Regulation, MIT Press, Cambridge, MA, 1983, pp 79-107. temperature:
4
5
6
7
8 9
10
ENERGY
ECONOMICS
July 1986
John E. Kushman and Joan Gray Anderson
A model ofthe utility-maximizing household is developed in which data on satiety levels for indoor temperature can be used along with budget constraint parameters and actual temperature choices to estimate parameters of the utility function. Using the utility parameters, equivalent variations in income can be estimated to measure welfare effects of policy and price changes that effect home heating. All of these operations can be performed at the individual household level and for any member of a popular class of utilityfunctionforms. The required data are easy to obtain, although they typicall_y are not collected in household surveys. Keywords: Energy demand: Households:
Economic models of household behaviour usually assume away potential satiation in any good. This is remarkable since the combination of goods at which satiation occurs should give information about preferences, and preferences are the key to predicting quantities demanded and to measuring welfare changes. We show that data on satiation temperatures can be used along with temperatures actually chosen to obtain parameters for temperature demands and welfare changes. This applies to a widely used class of John E. Kushman is with the Department of Agricultural University of California at Davis, Davis, Economics, CA95616, USA, and Joan Gray Anderson is with the Consumer Affairs Program, University of Rhode Island. The research reported here was supported
by regronal research project W159-Consequences of Energy Conservation Policies for Western Region Households: the Califorma Energy Studies Program ofthe University ofCalifornia: the Public Service Research and Dissemination Program. University of California at Davis: and the California Agricultural Experiment Station, Giannini Foundation Paper 795. Final manuscript
received
5 August
0140-9883/86/030147-08
$03.00
1985.
Q 1986 Butterworth
Welfare effects
utility functions including the quadratic, translog, and Diewert functions. The functional forms must be truncated so that they lose some oftheir flexibility, but there is a gain in flexibility in that the parameters can differ among households. The gain in flexibility among households is important because it allows detailed examination of variations in preferences, demands, and welfare effects among households. Computations with our procedure are simple. The required data are easy to obtain, although they typically are not collected in household surveys, and are somewhat unorthodox in relying on introspection and testimonial evidence. Strategic additions to surveys and a further research agenda are indicated. The methods developed and illustrated here may be significant in facilitating policy analysis of energyrelated initiatives. Many public initiatives exercised through tax provisions, regulated prices, and mandated low-interest home improvement loan programmes have the effect of changing the price of home heating. The substantive purposes of these initiatives are to encourage energy conservation and to
& Co (Publishers)
Ltd
147
redistribute income in the face of higher fuel costs. A model that permits the effects of policies on home temperature, fuel consumption, and welfare changes to be evaluated at the individual household level has in examining important potential uses, especially Willig and redistributive effects (Posner [7], Bailey [lo]).
The model Consider a household that maximizes utility during the heating season as a function of indoor home temperature. T. and all other goods, M. Let the utility function be of the class
~r=u,.l‘(T)-u,f'(T)'+u,f(M)+~l,f(M)~ +u,f(T)f‘(M) (1) where the zli are parameters and f(x) (x = T,M), is any function such that f(x) >O and f,>O (subscripts indicating derivatives in context). In particular, any of the popular family of Box-Cox member transformations.
f(s) = log x
/i=o
(2)
will do. The translog function corresponds to the case i = 0. The quadratic and Diewert functions correspond to i = 1 and i = 0.5. We truncate the functional form by deleting the last two terms. Restoring the last term is considered below. The resulting functions are less flexible as approximations to arbitrary utility functions. and deleting the last term imposes the assumption of additive separability. The truncated quadratic function requires constant marginal utility of all other goods. The truncated translog and Diewert functions impose diminishing marginal utility of all other goods. It is obvious that for each consumer (household) there exists a satiety temperature below which higher temperatures are preferred and above which lower temperatures are preferred. If 7* is the satiety temperature. in terms of the truncated function,
In order for a sign switch, Equation (4). to occur. u2 must be positive, since only the increase of f(T)with T in the second factor of Equation (3) can produce the switch. Dividing the utility function by 112gives a ne\ order-preserving utility function (5)
c=Pl.f‘(T)-.~(T)‘+P2.f‘(M)
This reduces the number of parameters to two. If the satiety temperature is known for a household, 8, can be found by setting t+(T*)=Oas in Equation (4) and rearranging, PI = I.f(T*)
(6)
The first-order conditions for utility maximization can be used to find f12 from the temperature actually chosen and the elements of the budget constraint. The household is assumed to produce indoor temperature at least cost given fixed heating-related elements of the household capital stock (insulation, type and size of structure, etc). This household production activity is embedded in the utilitymaximization problem by entering a cost function for temperature into the budget constraint. Using the production function for indoor temperature derived by Collins and Gray [3]. assuming the heating-related elements of household capital stock are fixed and the marginal and average prices of fuel are equal and constant, the short-run budget constraint for the household is
Y-PT(T-To)-PllM=O
where Y is income, P, and P, are prices of temperature and all other goods, and To is ambient temperature. Characteristics of the house and heating plant and the price of fuel are incorporated into P,’ Solving Equation (7) for M in terms of T. substituting for M in Equation (5), and optimizing subject to T>,To gives first-order conditions (8)
T-T,20
(9)
(T- To)r,=O HII,
UT=
-2~r,.l‘(T)l
(3)
II,
>
0
1r,=0
I[,<0
14x
T*>T when
T*=T T*
(4)
(10)
short-run cost function for temperature is C=P, ’ where P, is the priceoffuel.! is theaverageinsulation value for the house. and r reflects other dwelling characteristics. P,= P, 1 I-‘. For details of the derivation of the short-run cost function for temperature. see Anderson [I]. The model presented here is a short-run model and does not address the capital stock aspect of energy use decisions. For a discussion see Hausman [4]. No&-pricing for fuel could be incorporated, but it is omitted here for expositional simplicity. Blocking-pricing data were not available for the empirical work to follow.
IThe
[z(T-
and
(7)
&)I/
ENERGY
ECONOMICS
July I986
A model of household temperature demand: J. E. Kushmun tr17rl J. G. .4ndrrson
From
Equation
(8)
uT=rit
(11) where Equations (8) and (11) are equalities whenever the household heats. When T> TO,Bz can be found by calculating Equation (11). When T= TO, Equation (11) provides a ‘best guess’ lower bound on p2. In fact, /I1 can always be recovered by asking at what temperatures households currently not heating would start to heat. Parameters /?i and /I1 can be calculated for each household. With the utility parameters available, each household’s demand for temperature and (by Shepard’s lemma and the cost function for temperature) the demand for fuel are implied by the first-order conditions (8)-(10). An explicit solution for the quadratic case is
Equivalent and compensating variations also can be calculated for changes in fuel prices, home insulation levels, or other changes that will be reflected in changes in P, In econometric studies, it usually is assumed that all households (or at least all households in a given class defined by income, number, and types of persons, or other characteristics) have identical utility functions, and a single set of parameters is estimated (eg Murray [5], Scott [S] and Pindyck [6]).2 This approach loses much of the potential richness of the data. The approach presented here allows the utility parameters to vary among households. Aside from being potentially more informative and accurate about household demands for temperature and fuel and about welfare changes, this technique permits exploration of the variability of preferences and any systematic relationships of preferences with socioeconomic characteristics. It may be argued that the usual econometric approach has an advantage in that it does not require truncating the functional form, Equation (l), so that the additional parameters provide a more flexible fit to the utility function in the goods variables. In particular, objection may be raised to excluding the last term of Equation (l), thereby imposing additive separability. With the interactive term in the functional form the counterpart to Equations (8)( IO) that implicitly defines the satiety temperature is ‘The discussion is in terms of continuous choice variables. The random utility functions of discrete choice models arise in a different context (Hausman [4]).
ENERGY
ECONOMICS
July 1986
-2u,f(T*)+u,
(M)=O
(13)
Rearranging Equation (13) shows the testable implication of additive separability that the satiety temperature is independent of consumption of all other goods, f(T*)=
(u,!2)Cu, + u&M)3
(14)
A sign switch at T* for a given level of M still requires u,>O, so that Equation (14) may be rewritten f(T*) =
( VW, + PJ-(WI
(15)
wherev=p,f(T)-f(T)‘+P2f(M)+83.f(T).f’(M).The hypothesis of additive separability can be tested most directly by asking each household whether their ideal temperature depends on their other consumption. For those households that respond affirmatively. a series of equations could elicit pairs of T* and M, and a regression of Equation (15) would provide estimates of /!It and p3. The counterpart of Equation (I 1) will give an estimate of pz as B2 B
CL6
-
2f(T)
+
BJW)l U=,,lP,N.J;iJ;,) - BJ’( T) (16)
A less direct way of testing additive separability is to obtain pairs of T* and M for each household and conduct an hypothesis test on each regression, Equation (15). Either approach allows a test for additive separability and recovery of the utility function parameters on an individual household basis. The parameter for,f(M)2 cannot be recovered and must be assumed to be zero. The loss of flexibility entailed in this truncation, however. is likely to be unimportant. The remaining terms are likely to approximate the utility surface adequately, especially since these terms can differ among households. An additional degree of flexibility among households can be added by permitting different households to have different members of the class of utility functions, Equation (1). For instance. the household could be asked what temperatures it would choose if the price of fuel doubled, and the utility function form (eg value for i. in the Box-Cox transformation) could be chosen that best predicted the response from parameters (fi) based on the satiety temperature and current temperature.
Illustrative
computations
The methods described above were applied to data from 94 households who responded to a survey of residents of California on energy use, conservation behaviours and attitudes, and to a follow up survey on
149
A model qf household
temperature
demand: J. E. Kushman
Table I. Descriptive statistics for sample distributions of parameters for truncated quadratic and translog utility functions.
Statistic Mean Standard deviation Minimum Maximum
Utility function form Quadratic
Translog
/I, 141.74 7.33 130.00 160.00
PI 8.53 0.10 8.35 8.76
“; 740,60 8 663.40 113.42 52 074.00
B2 12.17 13.74 0.03 50.34
insulation decisions. Of the survey questions suggested by the model and not ordinarily included in questionnaires, the data include only the satiety temperature. In addition, estimates of the utility parameters were obtained by regression from the pooled data for comparison. This is the usual approach of assuming the same utility function for all households except as captured by shift variables. The respondents were asked, ‘To what temperature would you heat your house if temperature were free?‘. If the truncated additively separable model is used, this question and questions on actual temperature, housing characteristics, and characteristics of the heating plant are sufficient to implement the model. For households not heating, only a lower bound on b2 would be available. The data used here were for January, and all households in the sample were heating. If it were desired to test for additive separability or to search for the function form for each household, additional questions would be needed. The satiety temperature in the present data would correspond to a consumption of other goods equal to VP,,. The sample available for the calculations is known to be biased, and no generalizations should be made from the results.3 The results do serve, however, to illustrate the procedures. The quadratic and translog functional forms were used. The quadratic offers simplicity while the translog is a popular flexible form. As noted in the previous section. the quadratic and translog require constant and diminishing marginal utility of other goods. respectively. The question arises whether the different forms and techniques will lead to substantial differences in results. Table 1 describes the parameters, pi and /3,. that were estimated for the respondent households using the method of the first part of this paper. The different functional forms require different parameters for the same household, and this is evident in the mean parameter values. The functional forms do not require “For details on the surveys and on variable construction. see Anderhon [I]. Complications in defining home temperature were dealt with only in w,eighting the diNerent temperatures maintained at different times ofthe day. Scott [X] and Scott and Capper [9] discuss alternetive measures of temperature.
150
and J. G. Anderson
any great differences in parameter values untong households. In fact, for each functional form there is relatively little dispersion among the j3, values and much more dispersion in the flzs as indicated by the standard deviations and the minimum and maximum parameter values. The relatively tight distributions of p,s arise from the relatively small variation in satiety temperatures, see Equation (6). The flZs reflect variations in the flis, temperatures actually chosen, prices, and incomes (see Equation (11)). The wide variation in p2s illustrates the flexibility of the model across households, In the usual econometric approach to estimating parameters of the utility function (or, equivalently, of the demand function) all households are assumed to have the same parameters except as captured by the coefficients of household characteristic variables included in a rather adhoc fashion. Alternatively, household characteristics may be used to stratify the sample o priori with separate parameter estimates for each group. The model proposed here allows a test es post for relationships between utility parameters and household characteristics. The test results can be used to classify households on the basis of their characteristics according to sensitivity to economic parameters or to group households for further econometric analysis. A simple test for relationships between utility parameters and household characteristics is illustrated in Table 2. For each form of the utility function considered, the j?s are regressed on potential measures of tastes for indoor temperature and other goods. Since this analysis is exploratory, a 10% standard is used for two-tailed hypothesis tests. In the regressions for p,, the constant and opinion ofthe energy problem are the only statistically significant variables regardless of the functional form. Households with a more serious opinion of energy scarcity have lower satiety temperatures, all else constant. In the regressions for pz, the results depend on the functional form. The constant, young children, and older persons are statistically significant variables with the quadratic form. For the translog form the constant, education, older persons, and proportion of time at home are significant. Since flZ is the coefficient of other goods in the utility function, results for both functional forms indicate that households with older persons have a relatively strong preference for indoor heating over other goods. Otherwise, the results differ between functional forms. The question arises whether the sensitivity to functional form carries over to questions of more direct policy relevance. Two scenarios were simulated with the respondent households, a doubling of fuel prices and a mandatory retrolit to RI9 attic insulation. For
ENERGY
ECONOMICS
July 1986
A model
Table 2.
Regressions of utility parameters
Household characteristic
Constant Household head years of education Number of children less than five years old Number of persons 55 years old
more than
qf household
temperature
demartd: J. E. Kushman
and J. G. Andersor7
on potential measures of tastes. Regression coeffiients (r-values) Utility function form and parameter Quadratic
Translog
81
/iI
PI
-
I{:
141 .Ol (14.04) - 0.07 (0.17)
19 399.00 (1.75) 327.60 (0.70)
8.50 (60.76) 0.w (0.17)
36.2h
-0.41 (0.23)
4 734.00 (2.41)
- 0.01 (0.24)
O.OiY’ (0.00)”
- 1.34 (1.41)
- 2 158.00 (2.06)
- 0.02
12.10) I.26 (1.72)
(1.42)
-3.87 12.36)
I .72 (1.12)
-1.66 (1.86)
1050.00 (1.07)
- 0.02 (1.83)
Attitude toward voluntary home energy conservation measures’
0.03 (0.14)
- 28.88 (0.12)
0.W (0.19)
-0.4x
Attitude toward mandatory conservation measure?
0.07 (0.72)
- 163.55 (1.45)
0.00” (0.68)
- 0.I4
Willingness to undertake energysaving measures at home specifically related to heating and cooling’
0.03 (0.32)
37.73 (0.37)
0.00” (0.33)
0.08 (0.49)
Proportion of time spent at home
2.62 (0.51)
- 7 084.80 (1.24)
0.04 (0.53)
- 21.62
Local mean outdoor temperature
0.06 (0.34)
- 127.41 (0.64)
0.09
0.21
Opinion of energy problemb
energy
R2
4All equivalent
ENERGY
variations
are for one month.
ECONOMICS
July 1986
(0.7X)
(2.43)
0.00” (0.36)
- 0.23 (0.74)
0.09
“Rounded to two digits. bA higher score indicates the respondent regards the energy problem as more serious. ‘Attitude toward free or low-cost actions to reduce home energy consumption like using curtains to maximize indicates a more pro-conservation attitude. dA higher score indicates a more positive attitude toward governmentally-imposed energy conservation measures highway speed limits). ‘A higher score indicates greater willingness
each scenario, the new optimal temperatures and the equivalent variation were computed with each functional form. In nearly all cases, the differences in new optimal temperatures for the quadratic and translog functions was less than one degree, a difference attributable to the algorithms used. It follows that there were only very small differences in the fuel demanded as well. The calculated equivalent variations also were very similar for the two functional forms. For the quadratic and translog functions, respectively, the mean equivalent variations were $13.89 and $14.95 (doubled fuel prices) and -$9.25 and - $9.35 (mandatory attic insulation retrofit).4 For both scenarios, the correlation coefficient between the quadratic and translog equivalent variations was 0.99. The price elasticities of demand for heating degrees were calculated, and weighted means (weights equal to
(1.26)
0.23
heat retention.
A higher score
not necessarily
in
the home (eg
heating degrees) were computed for the sample. The elasticities for heating degrees will be equal to the elasticities for fuel as will the weighted elasticities. The weighted mean price elasticities are - 0.68 and - 0.53 for the quadratic and translog functions, respectively. Unweighted means are - 1.05 and - 0.78 in the same order. Estimates of the utility parameters, welfare effects. and price elasticities were made from the pooled data for comparison with those above. The pooled estimates were obtained by procedures paralleling those used above to enhance comparability. First, /j, was estimated by specifying p, = a’X + E in Equation (6) where X is the vector of a constant and the taste variables described in Table 2 and c is an crtl koc error term. The resulting equations have been presented in the first and third columns of Table 2. Next, the [jI equations were reestimated, excluding all except the statistically significant variables to refine the pooled estimates. The resulting equations were:
151
A mode/ of household
quadratic
temperature
demand:
and J. G. Anderson
J. E. Kushnan
fi, = 146.80- 1.3322 (opinion of the energy problem) (66.48) (1.95) b, = 8.59 - 0.02 (opinion of the energy problem) (279.12) (1.89)
translog
Table 3.
Regression results for /j2 parameters
Regression coefficients (r-value) Quadratic Translog
Variable
12 665.00
Constant
Parameter p2 was estimated conditional on the values of /?r given by these equations. Equation (11) was rearranged to give [S,
-2A771 [PdPTl rahrl=Pz
(17)
with j?, replaced by 8, and pz=l~‘X+p. Again, the vector X contains the household characteristics and p is an LU/Itoc error term. This regression, _r= $X + p, is conditional on fi, as were the previous estimates of fi2. In both approaches, X is introduced through the Ps. The regression results for Equation (17) are given in Table 3. Generally, the coefficients have the same signs as those from the individual household method (Table2), and they are of the same orders of magnitude, but the statistical significance of some variables shifts between methods. As in Table 2, the statistical significance of some variables differs between the functional forms. The statistically insignificant variables were dropped from the regressions, and the equations were rerun to give quadratic
8, = 17 932.00+4 (4.58) + 1 126.6 (1.67)
049.70 children (2.63)
- 1906.50 (2.29)
Household education
head years of
Number of children 5 years old
less than
Number of persons 55 years old
more than
Opinion
of energy
309.35 (0.80)
I .09 (1.57)
I 365.50
Attitude toward mandatory energy conservation measures Willingness to undertake energysaving measures at home specifically related to heating and cooling of time spent at home temperature
42.51 (2.59)
_ 1 720.30 (1.98)
problem
Local mean outdoor
(3.37)
4281.40 (2.64)
Attitude toward voluntary home energy conservation measures
Proportion
from pooled data.
-0.19 (0.07
1.9x
(1.67)
(1.35)
- 155.43 (0.77)
-0.41 (1.13)
-0.18 (1.90)
-0.18 (1.06)
- 13.58 (0.16)
0.08 (0.56)
- 10214.00 (2.17)
- 24.80 (2.94)
-113.11 (0.68)
- 0.34 (1.15)
0.26
R’
I
-3.17 (2.04)
0.24
older Table 4. Descriptive statistics for utility parameters, demand price elasticities, and equivalent variations from estimation with pooled data.
(opinion of energy problem) Parameter
and statistic
Utility function form Quadratic Translog
PI
- 227.84 (attitude toward mandatory conservation) (2.98)
Mean Standard deviation Minimum Maximum
-9 052.70 (time at home)
Pz Mean Standard deviation Minimum Maximum
(2.15) translog
j2
= 21 .I 1 - 2.80 older - 21.93 (time at home) (2.99) (7.45) (2.06)
Having obtained pooled estimates of the utility parameters and how these parameters are associated with household characteristics, individual household utility parameters, demand price elasticities. and were computed. Table 4 equivalent variations describes the results. The utility parameters calculated from the pooled equations are similar, in terms of the sample means, to those obtained with the individual household method. however, are much less The pooled parameters, variable over the sample. The weighted mean price regardless of elasticities are of similar magnitudes
152
142.75
1.46 141.47 146.80 7 524.10 3 639.30 190.93 18 863.00
8.53 0.02 x.52 8.59 I I .09 5.02 0.88 20.19
Price elasticity for heating or fuel Weighted mean Unweighted mean
-0.56 - 1.37
-0.53 - 0.72
Mean equivalent variation Doubled fuel prices R I9 retrofit
6.18 _ 2.65
2.32 - 0.95
method or functional form. The estimated welfare effects using the pooled equations are much smaller than those estimated by the individual household method. In contrast to the case with the individual met hod, the mean equivalent variations computed
ENERGY
ECONOMICS
July 1986
A model
of ho~rseholritemperarure demand: J. E. Kuslm~~t~urd J. G. Adcrsor~
from the pooled estimates are quite sensitive to the functional form used. An important potential advantage of the individual household method is its flexibility among households. This is reflected in the narrower range of estimated utility parameter values for the pooled method. Also, the demand elasticities and equivalent variations are only poorly correlated between the pooled and individual methods over households. For instance, using the quadratic function and mandating R19 retrofit, equivalent variations over the sample are correlated at only R =0.25. The pooled equivalent variations also show little correlation (R=0.28 and 0.49) between functional forms. Comparison of the demand price elasticities calculated here with those of other studies is problematic, since the samples are different. Barnes et al ([2], p 548) have estimated a short-run elasticity for space-heating electricity using household data at -0.9269 (standard error 0.2962). This elasticity is substantially larger than any of our weighted mean elasticities, although it is closer to our unweighted means. Barnes er al [2] calculated their elasticity using pooled parameter estimates and for a ‘representative household’ characterized by the sample mean values. Using the same procedure on the present sample gives price elasticities of - 1.3943 (quadratic) and - 1.5871 (translog).
Research agenda At least two reservations may be held about the proposed technique. First, the analysis has not dealt with the statistical properties of the parameter estimates. Second, the data are based, in part, on introspection and testimonial evidence, in contrast to the behaviouralist tradition in economics. With respect to the properties of the estimators, a straightforward answer is that, regardless of their properties, they are the only fully household-specific estimates available from a cross-section. Or. by making the required ad hoc assumptions about error terms apriori, any desired properties can be obtained. A more satisfactory answer to the question of statistical properties is that the nature of any errors should be determined experimentally. This approach also would address the adequacy of testimonial data. For example, errors in estimating fii will arise from errors in reporting Tr, see Equation (6). The nature of the errors could be examined by obtaining estimates of 7* for an experimental group of households and then paying their heating bills to observe the true satiety temperatures. The resulting distribution of errors in estimating Tr would indicate the nature and magnitude of any error specification for Equation (6)
ENERGY
ECONOMICS
July 1986
and the properties of the estimator for /il. This procedure would permit the relatively inexpensive collection of testimonial evidence based on introspection and its systematic linking to a behavioural model. Determining 7* experimentally for a very large number of households is likely to be prohibitively expensive. but work with a smaller group can validate and appropriately qualify less costly data. Experimental procedures also could be used to obtain estimates of j13, although the expense would be ev’en greater. Limited experiments to relate testimonial data to experimental data would be more cost effective.
Summary and implications This paper has presented a model of household heating decisions that uses satiety in temperature and utility maximization to identify the parameters of a class of utility functions. A series of survey questions were identified that allow flexibility of the utility function across households. The class of utility functions includes slightly truncated versions of the popular quadratic, translog, and Diewert functions and, in fact, the entire class of Box-Cox transformations. With appropriate survey questions all of the parameters of the utility functions can be obtained. The flexibility of the model across households seems more important than the loss of some flexibility from truncating the functional form. Once the utility parameters have been obtained for each household, many empirical questions can be addressed. In the set of households available to this study, the association of utility parameters with household characteristics depended, to some extent. on whether the quadratic or translog utility function was chosen. The answers to questions of more direct policy relevance, however, were not sensitive to functional form. A doubling of fuel prices or mandatory attic insulation retrofit had nearly identical effects on temperature choice and welfare regardless of functional form. The price elasticities of demand for heating degrees of fuel also were insensitive to functional form. These findings presently cannot be generalized to other households or functional forms, but they illustrate the potential of the model as a research tool. Estimates of utility parameters, price elasticities of demand. and welfare measures also were made from the pooled sample. The mean values of estimate utility parameters and demand elasticities were similar for the pooled and individual household techniques. The pooled estimates, however, were less variable over the sample. Pooled estimates of welfare effects were much smaller on average than those from the individual
153
A model of household
temperature
demand:
J. E. Kushman
and J. G. Anderson
household method and more sensitive to choice of functional form. Properties of the estimators for utility parameters based on introspection and testimonial data may be determined experimentally. Testimonial data can then be used to obtain household specific estimates for policy analysis at relatively low costs with appropriate qualifications.
References Joan Gray Anderson, Indoor Temperature and Insulation Choice: Theoretical and Econometric Models for Policy Analysis, unpublished PhD dissertion,
Department of Agricultural Economics, University of California, Davis, 1984. Roberta Barnes et al, ‘The short-run residential demand for electricity’, Review of Economics and Statistics, Vol63, No 4, August 1981, pp 541-552. Robert A. Collins and Joan K. Gray, ‘Derivation of a theoretical form of a production function for home
1.54
a comment’, Energy Economics. Vol5, No 4, October 1983, pp 273-275. Jerry A. Hausman, ‘Individual discount rates and the purchase and utilization of energy-using durables’, Bell Journal ofEconomics. Vol 10. No 1, Spring 1979, pp 3354. Michael P. Murray, ‘The distribution of tenant benefits in public housing’, Econometrica, Vol43, No 4, July 1975, pp 771-788. Robert S. Pindyck, ‘International comparisons of the residential demand for energy’, European Economic Review, Vol 13, pp l-24. Richard A. Posner, ‘Taxation by regulation’, Bell Journal ofEconomics, Vol2, No 1, Spring 1971, pp2250. Alex Scott, ‘The economics of house heating’, Energy Economics, Vol2, No 3, July 1980, pp 130-141. Alex Scott and Graham Capper, ‘Production functions for house temperatures: problems and issues’, Energy Economics, Vol5, No 4, October 1983, pp 275-278. Robert D. Willig and Elizabeth E. Bailey, ‘Income distribution concerns in regulatory policymaking’. in G. Fromm, ed, Studies in Public Regulation, MIT Press, Cambridge, MA, 1983, pp 79-107. temperature:
4
5
6
7
8 9
10
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ECONOMICS
July 1986