Journal of Monetary Economics 46 (2000) 385}396

Comovement, excess volatility, and home production夽 Yongsung Chang* Department of Economics, University of Pennsylvania, Philadelphia, PA 19104, USA Received 4 June 1998; received in revised form 6 December 1999; accepted 21 December 1999

Abstract Two investment anomalies in aggregate home-production models are investigated: excess volatility and comovement. Adjustment cost in capital accumulation reduces both volatility and the negative correlation in investments on capital goods in the market and at home. Investments comove to the extent that durable goods and time are good substitutes in consumption activities. Consumers substitute durable goods for time at home when the opportunity cost of time is high during booms. Based on the Consumer Expenditure Survey, I show that households' expenditure shares on durable goods are negatively associated with leisure, indicating that durable goods are relatively good substitutes for time.  2000 Elsevier Science B.V. All rights reserved. JEL classixcation: E32; J22 Keywords: Home production; Comovement; Volatility

1. Introduction Economic development has led to a large secular decline in the workweek. The allocation of non-working time may now rival that of working time in 夽

I would like to thank Mark Bils, Hae-shin Hwang, Victor Rios-Rull, Richard Rogerson, Nicholas Souleles, Randy Wright, the Editor, Bob King, and an anonymous referee for comments and suggestions. All remaining errors are mine. * Corresponding author. Tel.: #1-215-898-6691; fax: #1-215-573-2057. E-mail address: [email protected] (Y. Chang). 0304-3932/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 3 2 ( 0 0 ) 0 0 0 2 6 - X

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importance to economic welfare. Since Becker (1965) and Mincer (1962), the value of non-market activity has been explicitly incorporated into economic analysis in terms of forgone earnings. Recently, home production has been incorporated into equilibrium business models to address a variety of issues. (See Greenwood et al., 1995, for a survey.) In particular, home production improves the quantitative performance of the labor market by providing intratemporal substitution of time in addition to intertemporal substitution of leisure. However, it tends to generate anomalies in the movement of investments. Investments in durable goods in the market and at home are extremely volatile and strongly negatively correlated; they move together in the data with a correlation of 0.47. Greenwood and Hercowitz (1991) "rst generated comovement in investments based on (i) highly correlated technology shocks between the market and the home sector, (ii) complementarity between durable goods and time in home production, and (iii) the labor-augmenting productivity shock in home production. However, the merits of home production disappear if the shocks are highly correlated between the sectors. Furthermore, assumption (ii) seems to be inconsistent with the data } durable goods appear to be good substitutes for time in consumption (e.g., Abbot and Ashenfelter, 1976; Barnett, 1979). Based on the Consumer Expenditure Survey, I estimate the cross-sectional elasticity of household's expenditure share with respect to leisure for six categories of goods, including durable and non-durable goods. This measure tells us how consumers substitute goods for time in consumption activities when their work-hours increase. Durable goods seem to be a relatively good substitute for time. A 1% (cross-sectional) increase in household leisure is associated with 0.13% and 0.47% decreases in the expenditure share of durable goods for married and non-married households, respectively. This paper suggests a simple way to resolve the volatility and comovement issues in aggregate home-production models. The resolution requires (i) a small adjustment cost in capital accumulation and (ii) substitutability between durable goods and time in consumption. Unlike in a one-sector model, in a multi-sector model investment in one sector can increase enormously at the price of investment in the other sector, without a!ecting consumption signi"cantly. This implies that in response to shifts in relative productivity, investments are extremely volatile and strongly negatively correlated. As was shown by Baxter (1996), introduction of a small adjustment cost in capital accumulation reduces not only volatility but also the negative correlation of investments. Following Becker (1965), I interpret home production as the "nal stage of consumption activities. Any consumption activity requires both goods and time as input to produce e!ective consumption. For example, cleaning a house requires tools and labor, and watching a play in a theater requires a ticket and time to watch. The full costs of these activities equal the sum of market goods' prices and the forgone value of time. A high opportunity cost of time implies that

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labor is a relatively expensive input in consumption. Procyclical demand for consumer durables is consistent with procyclical labor productivity, investment, and hours in the market sector if durable goods and time are good substitutes (e.g., a dishwasher, microwave, or refrigerator). Simulation of the model shows that both volatility and comovement of investments match those in the data under the substitution elasticities available from the empirical literature on consumer demand. Other studies on comovement include Hornstein and Praschnik (1997), Fisher (1997), and Einarsson and Marquis (1997). Hornstein and Praschnik (1997) developed an explanation based on the input}output structure. Fisher (1997) achieved comovement under the complementarity of investments in resource constraint. Complementarity in investments can be interpreted as an adjustment cost because it generates a concave production-possibility frontier. Einarsson and Marquis (1997) achieved comovement of investments and hours at the same time by introducing human capital accumulation but complementarity in home durable goods and time is still required. The paper is organized as follows. Section 2 presents the model. Section 3 calibrates and simulates the models under various speci"cations. In Section 4, based on the U.S. Consumer Expenditure Survey, the cross-sectional pattern of leisure and expenditure is investigated. This analysis shows that durable goods are a better substitute for time than are non-durable goods and services. Section 5 is the conclusion.

2. The model At time t, the representative household maximizes the utility de"ned over various types of consumption activities:  Z\N!1 ;"E bO\R O , R 1!p OR ( Z " “ ZIH k '0, HR H R H

( k "1, H H

Z "((1!a )X\CH #a (Q ¸ )\CH )CH CH \ for j"1,2, J. HR H HR H HR HR

(1)

(2) (3)

E denotes expectation at time t and b is a discount factor. Z is the utility from R HR consumption activity j that requires goods X and time devoted to it ¸ . HR HR Q represents the technological progress in home production. There are many HR di!erent ways, in terms of goods}labor intensities, in which the household can achieve the same e!ective consumption. For example, suppose Z is the utility HR

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from clean dishes. Households can use their own hands (a labor-intensive way) or use a dishwasher (a goods-intensive way). The representative consumer faces two constraints, time and money income: ( ¸ #N "1, HR R H

(4)

( C #I "(1!q )= N #(1!q )R K . HR R , R R ) R R H

(5)

N is labor supply in the market, C is purchase of good j, and I is investment in R HR R market capital stock K at time t. = and R denote the wage rate of labor and R R R rental rate of capital at time t. The tax rates on labor income and capital income are q and q , respectively. Accumulation of stocks of capital and consumer , ) durable goods is subject to an adjustment cost: K " (I /K )K #(1!d )K , R> R R R ) R X " (C /X )X #(1!d )X , HR> HR HR HR H HR

(6) j"1,2, J,

(7)

where is a concave function ( ( )'0, ( )40), X is the stock of good j at HR time t, and d and d are depreciation rates of capital in the market and goods j, ) H respectively. For non-durable goods, d "1 and consumption equals current H purchases: X "C . HR HR The fact that goods have di!erent substitutability with time has an interesting implication on the cyclical behavior of labor supply and demand for goods. In a traditional labor}leisure approach, an increase in real wages induces more labor and less leisure unless the income e!ect dominates the substitution e!ect. Here, the same result can be obtained via substitution of more goods-intensive consumption activity, that is, higher input of X 's and lower input of L 's. A rise H H in the cost of time relative to goods (an increase in real wages) would induce a reduction in the amount of time and an increase in the amount of goods used per unit of utility in consumption activity. This approach interprets the procyclical behavior of the labor supply as a change of consumption pattern over the business cycles, from a time-intensive consumption pattern to a goods-intensive consumption pattern. Therefore, a surge of demand for consumer durable goods during expansions will be consistent with procyclical labor productivity provided durable goods and time are good substitutes. The representative "rm produces output according to a Cobb}Douglas production function in labor and capital, > "A K\?(Q N )?, where A R R R R R R represents temporal total factor productivity (TFP) shift and Q is the  deterministic technological progress. The government purchases G each R period: G "q = N #q R K . The goods-market equilibrium is R , R R ) R R ( C #I #G ">. H HR R R

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389

3. Calibration and simulation This section calibrates and simulates the model economy. Consumption activities are aggregated into two categories: one requires durable goods and the other requires non-durable goods and services, Z "ZI Z\I, where subscript R 1R "R D denotes durable goods and S denotes non-durables and services. The model is solved numerically using a log-linear approximation of the system of "rst-order conditions and constraints of the economy around the steady state as in King et al. (1988). Parameter values are chosen as follows. The steady-state market labor supply is , and the relative risk aversion p is 1. Labor share in output a is . The annual   depreciation rate is 10% for both market and home capital. The tax rate for both labor and capital income is 20%. In steady state, the model implies a per capita annual growth of 1.6% and an annual interest rate of 4%. As a benchmark case, substitution elasticity between goods and time is 1 for consumption activity related to both durable goods and non-durable goods: e "e "1. For " 1 the labor share of durable-goods consumption, the average of those in the literature is used: a "0.81. With no information for non-durable goods, I use " the same value as the market-production function: a ". The average expendi1  ture ratio of durable goods to non-durable goods and services is 0.124 for 1954 : I}1996 : IV in the U.S., and this leads to the value for k"0.103 C (1!a )(1!k) " as " " in the steady state. The stochastic process of TFP d (1!a )k C 1 " 1 follows AR(1) in logs: ln A "(1!0.95) ln A#0.95 ln A #e , and the R R\ R standard deviation of e is 0.0076 as in the real-business-cycle literature. R There are no average and marginal adjustment costs in steady state (i.e., (I/K)"I/K, (C /X )"C /X and "1). Finally, we need to " " " " specify the elasticity of the investment/capital ratio with respect to Tobin's q : g"![(I/K) / ]\. There is no adjustment cost in the benchmark case, g"R ( "0). Table 1 reports the selected moments of models. Column (1) is the benchmark case, i.e., no adjustment cost in capital accumulation and unitary elasticity between goods and time in consumption. As is well known from the earlier studies, investments are extremely volatile and strongly negatively correlated with a correlation of !0.83. Furthermore, spending on consumer durables

 When the labor-augmenting technological progress is at the same rate both in the market and at home (e.g., Q "Q "Q ), given the constant returns to scale production functions in Z and Z , R "R 1R " 1 there exists a balanced growth path along which time allocation is constant.  They are 0.87 in Greenwood and Hercowitz (1991); 0.92 in Benhabib et al. (1991); 0.794 in McGrattan et al. (1997); and 0.68 in Greenwood et al. (1995).

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Table 1 Moments of models

Statistics p 7 p /p !1 7 p /p !" 7 p /p ' 7 cor(C ,Y) 1 cor(C ,Y) " cor(I,Y) cor(C , I) "

(1) g"R, e "1, " e "1 1

(2) g"80, e "1, " e "1 1

(3) g"80, e "1.50, " e "0.59 1

(4) g"80, e "2.59, " e "0.34 1

1.43 0.32 19.5 5.62 0.91 !0.18 0.68 !0.83

1.40 0.36 2.19 3.14 0.93 !0.04 0.98 !0.24

1.36 0.25 2.28 3.11 0.89 0.40 0.98 0.21

1.36 0.17 2.85 3.06 0.79 0.65 0.98 0.49

Data 1.72 0.50 2.96 3.05 0.83 0.83 0.71 0.47

Both data and the generated series are H-P "ltered. Data are quarterly for 1954:1}1996:4 from the Citibase. Y: real GDP in 1987 constant dollars (GDPQ). C : non-durables and service consumption 1 (GCNQ#GCSQ). C : expenditure on consumer durables (GCDQ) I: gross "xed nonresidential " investment (GIFQ-GIRQ). N: employed man-hours (LPMHU). All series are divided by the population over age 20. p /p : standard deviation of consumption on non-durable goods and !1 7 services relative to Y. cor(C , I): correlation of C and I. " "

shows a negative correlation with output (!0.18) while it is strongly procyclical in the data (its correlation with output is 0.83). The relative volatility of investment on consumer durable goods and market capital to output are 19.5 and 5.62, respectively, while those in the data are 2.96 and 3.05. The impulse}response function clearly illustrates this investment anomaly. Fig. 1 shows the impulse response of investments and stocks of durables in the market and home to a 1% increase in TFP in the market with an autocorrelation of 0.95. The purchase of home durable goods drops enormously in the "rst period. It decreases by 30% while market investment increases by 14%. Although the two investments move together after the "rst period, initial response dominates the later movement and generates a strong negative correlation. Moments of the model with adjustment cost are reported in column (2). With no direct estimate for g, I follow Baxter (1996) in tying this value to the volatility of investments in the data: g"80. Adjustment cost not only suppresses the volatility but also reduces the negative correlation of investments by mitigating the extreme initial response. The correlation of the two investments decreases to !0.24. Still, spending on consumer durable goods does not exhibit a procyclical pattern. As the empirical studies on consumer demand suggest, suppose durable goods are relatively better substitutes for time than non-durables and services are. Abbot and Ashenfelter (1976) report three sets of estimates for the substitution

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Fig. 1. Impulse response of investment and capital to 1% increase in TFP (g"R, e "1, e "1). " 1

elasticity between goods and leisure: (i) e "1.50 e "0.59, (ii) e "1.83 " 1 " e "0.48, and (iii) e "2.59 e "0.34. The results for the model with (i) and (iii) 1 " 1 are in columns (3) and (4), respectively. Two investments comove with correlation of 0.20 and 0.49, respectively (0.47 in the data). Investment in home durable goods exhibits a procyclical behavior as in the data. At the same time, substitutability between durable goods and time increases the volatility of expenditure on durable goods and decreases that of non-durables and services. Fig. 2 shows the impulse response to a 1% increase in TFP in the market for the model with e "2.59 and e "0.34. Because of strong demand for durable " 1 goods when the price of time is expensive, expenditure on consumer durable goods increases with investment in the market from the "rst period.  For non-durable goods and services, Abbot and Ashenfelter report estimates for "ve categories (Table V in their paper), and the above numbers are the average of these categories. Yet these numbers are conservative estimates for e's in re#ecting the di!erence in substitutability with time because their estimates are elasticities between total leisure and goods. One can show that, under our speci"cation, use of total leisure is likely to underestimate e for goods that are relatively good substitutes for time and overestimate e for goods that are complementary with time.

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Fig. 2. Impulse response of investment and capital to 1% increase in TFP (g"80, e "2.59, " e "0.34). 1

Two caveats are needed at this point. First, for the comovement of investments, what matters is the relative, not the absolute, magnitude of elasticities. As long as durable goods are relatively better substitutes for time than are nondurable goods and services, the two investments comove. Second, without adjustment cost, substitutability itself does not improve the correlation of investments signi"cantly. The extreme initial response dominates the substitution e!ect.

4. Some cross-sectional evidence from the consumer expenditure survey To illustrate the empirical relevance of substitutability between durable goods and time, I investigate the cross-sectional pattern of leisure and expenditure on various goods based on the Consumer Expenditure Survey (CEX). Of primary interest is the comparison between durable goods and non-durable goods including services. Although durable goods show strong substitutability for time

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in general, the substitutability depends on the characteristics of the goods. For example, one might expect that durables such as refrigerators, microwaves, and dishwashers save time at home, while entertainment-related durable goods such as audio equipment are likely to be complementary with time. In addition, services such as baby-sitting are likely to be a good substitute for time while the expense for traveling is likely to be complementary with time. To demonstrate this, I classify the goods into several sub-categories. Among durable goods, entertainment-related durable goods are of interest. For non-durable goods and services, food at home, food outside the home, and household operation services are examined separately. Consider the following cross-sectional regression:




ln



p x KZH K GK "b #b ln l #b ln y #e ,  J G W G GH p x K GK

(8)

where expenditure on item m by household i is p x . Leisure and income of the K GK household are l and y , respectively. The left-hand side of Eq. (8) is the log of G G expenditure share of goods in category j. Our primary interest is in the coe$cient on household's leisure, b . It measures the (cross-sectional) elasticity of J household's expenditure share of goods j with respect to the leisure of the household: a percentage change of share of goods j associated with a percentage increase in leisure. For goods that are relatively good substitutes for time, b is J negative, and for those goods that are relatively complementary with time, b is J positive. The term b captures the income e!ect on goods j, the so-called Engel W curve. It is important to control for the income e!ect, as durable goods tend to be luxury goods. Following the convention in the empirical analysis of consumer demand, total consumption is used as a proxy for household income. The use of total consumption provides two advantages: it represents the permanent income of the household and avoids a serious measurement error in reported income in the survey data. To control for other household characteristics the regression includes years of schooling of the household's head, age of household head, family size, number of children under age 18, number of children under age two, a dummy variable for home ownership, and dummies for quarter and year of the interview. The pooled cross-sectional CEX data for 1990}1994 are used. CEX is a quarterly outgoing rotation data set. Each household takes part in up to four consecutive quarterly surveys. One-fourth of the sample departs and is replaced each quarter. Because annual working hours are reported in the "rst and fourth interviews only, quarterly consumption data are aggregated over the previous four quarters (from the "rst to the fourth interview) for each household to match the annual hours in the fourth interview. Leisure of household is the average annual leisure of the head and of the spouse. Annual individual leisure is calculated as (365 days;16 h) ! (number of weeks worked);(average

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Table 2 Summary statistics of CEX for 1990}1994: married households Variables

Mean

SD

Total expenditure Age of head Schooling of head Age of spouse Sex of head Weeks worked: head Hours worked: head Weeks worked: spouse Hours worked: spouse Family size Number of children of age under 18 Number of children of age under 2 House owned

28,781 42.55 13.49 40.92 0.15 42.77 43.92 33.67 37.47 3.46 1.15 0.14 0.76

18,566 11.24 2.8 11.22 0.36 17.82 11.24 22.58 11.91 1.40 1.24 0.37 0.43

Min

Max

Obs.

!16,805 18 0 14 0 0 1 0 1 2 0 0 0

293,938 65 18 83 1 52 90 52 90 15 9 4 1

10,321 10,321 10,321 10,004 10,321 10,321 9,168 10,004 7,474 10,321 10,321 10,321 10,321

The data contain only households whose heads are between 18 and 65 yr old.

workweek). Table 2 shows the summary statistics of the sample for married households. Eq. (8) is estimated for 10,321 married households by OLS. Table 3 reports the result for six categories of goods: durable goods, non-durable goods and services, entertainment-related durable goods, food at home, food outside the home, and household services. Durable goods are luxury goods in general: a 1% increase in total expenditure is associated with 0.34% increases in its share cross-sectionally. They are relatively good substitutes for time: a 1% increase in leisure is associated with a 0.13% decrease in its share. However, as we expected, entertainment-related durables (sound and video equipment such as TVs and VCRs, musical instruments, pets, bicycles, boats, campers, camping equipment, hunting and "shing equipment, and other sports equipment) do not show such substitutability for time. For non-durable goods and services, a 1% increase of leisure is associated with 0.06% increases in its share. Food consumption at home and outside the home show an interesting contrast. While outside food consumption such as purchased meals is a relatively good substitute for time, food at home is  The two data sets are constructed based on the marital status of the household head: 10,321 married households and 7212 non-married households. Since the results are similar, I report the result based on married households only.  For some categories of goods, expenditures are zero or negative values (if the household sold them) so that we cannot take logs in the left-hand side of Eq. (8). To avoid this problem, I replace negative values with zeros, and then use the percentage deviations from the sample means for the approximation of logs of expenditure share.

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Table 3 Estimation of (8) based on the CEX for 1990}1994: married households

Category Log (income) Log (leisure) Family size Log (schooling) Log (age) Sex of head No. of children: age under 18 No. of children: age under 2 House owned

Durable goods

Entertainment durables

Nondurables and service Food at consumption home

Food outside

Household operation

0.34 (36.8) !0.13 (!3.9) !0.4 (!4.9) 0.08 (3.0) !0.43 (!18.3) !0.08 (!5.3) !0.003 (!0.3) 0.03 (2.3) 0.42 (30.4)

0.32 (9.7) 0.18 (1.6) !0.9 (!3.3) !0.15 (!1.5) !0.69 (!8.2) !0.27 (!5.2) 0.17 (5.2) !0.06 (!1.0) 0.05 (1.0)

!0.15 (!36.8) 0.06 (3.9) 0.02 (4.9) !0.03 (!3.0) 0.19 (18.3) 0.03 (5.3) 0.001 (0.3) !0.02 (!2.3) !0.18 (!30.4)

!0.07 (!4.4) !0.72 (!13.6) !0.03 (!2.1) 0.52 (11.9) 0.25 (6.6) !0.02 (!0.8) !0.06 (!4.4) !0.10 (!4.1) 0.08 (3.7)

0.18 (5.4) !1.31 (!10.8) !0.25 (!8.8) 1.07 (10.8) !0.27 (!3.2) 0.03 (0.6) 0.43 (12.9) 0.48 (8.6) 0.06 (1.2)

!0.35 (!39.5) 0.08 (2.7) 0.07 (9.8) !0.12 (!4.6) 0.33 (14.5) 0.03 (2.1) 0.03 (3.1) !0.09 (!5.8) 0.02 (1.9)

t-ratios are in parentheses.

a complementary with time: a 1% increase in leisure is associated with a 0.72% decrease of expenditure share of food consumption outside and a 0.08% increase in food consumption at home. Household operation services (baby-sitting, housekeeping services, gardening and lawn-care services, and repair services) show a very strong substitutability with time. A 1% increase of leisure is associated with 0.87% decreases of its expenditure share of households.

5. Conclusion In this paper, two investment anomalies in aggregate home-production models are investigated: excess volatility and comovement. The adjustment cost  While our "nding that household's leisure is negatively correlated with its expenditure share on durables in the CEX is consistent with the previous empirical studies on substitution elasticity in consumer demand based on time series, one should note that this is a cross-sectional correlation. Evidence on substitutability between time and goods based on a dynamic panel data would be more compelling.

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of capital accumulation decreases the volatility and negative correlation as well. Two investments comove if consumer durable goods are better substitutes for time than are non-durables and services as consumers substitute durable goods for time in consumption when the opportunity cost of time is high during booms. According to the Consumer Expenditure Survey, households' expenditure shares on durable goods and leisure show statistically signi"cant negative correlation, indicating substitutability between durable goods and time in consumption.

For further reading The following reference is also of interest to the reader: Rupert et al., 1995.

References Abbot, M., Ashenfelter, O., 1976. Labor supply, commodity demand, and the allocation of time. Review of Economic Studies 43, 389}412. Barnett, W., 1979. The joint allocation of leisure and goods expenditure. Econometrica 47, 539}563. Baxter, M., 1996. Are consumer durables important for business cycles? Review of Economics and Statistics 78, 147}155. Becker, G., 1965. A theory of the allocation of time. Economic Journal 73, 493}508. Benhabib, J., Rogerson, R., Wright, R., 1991. Homework in macroeconomics: household production and aggregate #uctuations. Journal of Political Economy 6, 1166}1181. Einarsson, T., Marquis, M., 1997. Home production with endogenous growth. Journal of Monetary Economics 39, 551}570. Fisher, J., 1997. Relative prices, complementarities and comovement among companies of aggregate expenditures. Journal of Monetary Economics 39, 449}474. Greenwood, J., Hercowitz, Z., 1991. The allocation of capital and time over the business cycle. Journal of Political Economy 99, 1188}1214. Greenwood, J., Rogerson, R., Wright, R., 1995. Household production in real business cycle theory. In: Cooley, T. (Eds.), Frontiers of Business Cycle Research. Princeton University Press, Princeton, NJ. Hornstein, A., Praschnik, J., 1997. Intermediate inputs and sectoral comovement in the business cycle. Journal of Monetary Economics 40, 573}596. King, R., Plosser, C., Rebelo, S., 1988. Production, growth and business cycles: I. the basic neoclassical model. Journal of Monetary Economics 21, 195}232. McGrattan, E., Rogerson, R., Wright, R., 1997. An equilibrium model of the business cycle with household production and "scal policy. International Economic Review 38, 267}290. Rupert, P., Rogerson, R., Wright, R., 1995. Using panel data to estimate substitution elasticities in household production models. Economic Theory 6, 179}193.