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Simple structural econometrics of price elasticity
Economics Letters 86 (2005) 1 – 6 www.elsevier.com/locate/econbase
Simple structural econometrics of price elasticity Catherine Cazals a, Fre´de´rique Fe`ve a, Patrick Fe`ve b,*, Jean-Pierre Florens c b
a University of Toulouse (CNRS-GREMAQ and IDEI), France GREMAQ-Universite´ de Toulouse I, manufacture des Tabacs, baˆt. F, 21 alle´e de Brienne, 31000 Toulouse, France c University of Toulouse (IUF, GREMAQ and IDEI), France
Received 15 January 2004; accepted 29 March 2004 Available online 10 November 2004
Abstract This paper delivers a methodology for the identification of the price elasticity of demand. The approach relies on a structural model of consumer behavior wherein demand and expenditure depend on a heterogeneity factor. We show that the structural parameters are not identified in the one-period case. Conversely, in a two-period case, the structural parameters, and thus the price elasticity of demand, are identified for each period. D 2004 Elsevier B.V. All rights reserved. Keywords: Structural model; Heterogeneity factor; Price elasticity JEL classification: C1; C2; D1
1. Introduction The identification of price elasticity from cross-section individual data is impossible when the price does not vary in the sample. Moreover, in the case of panel data, the slow modification of tariffs and the usual small number of periods render very hazardous identification of the price elasticity. The aim of this paper is to present a methodology for the treatment of this kind of data and to show that we can take advantages from the restrictions created by a structural model of demand. We first show that the structural parameters are not identified in the one-period case. Nevertheless, the identification can be obtained using different periods. The motivation of such an approach is that, if preferences and the distribution of tastes remain stable—or controlled—across periods, exogenous price changes allow to identify the structural parameters. We exploit this idea and we show in a series of examples that the price elasticity of demand can be identified. * Corresponding author. Tel.: +33-5-61-12-85-75; fax: +33-5-61-22-55-63. E-mail address: [email protected] (P. Fe`ve). 0165-1765/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2004.03.032
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C. Cazals et al. / Economics Letters 86 (2005) 1–6
2. A simple structural model We consider a repeated static market with a given set of producers and consumers. The quantity of a homogeneous1 good used by a consumer i in period t is denoted xi,t and the associated price—common to all consumers—is pt. The consumer i at period t seeks to maximize Uðxi;t Þ ¼
a 2 x þ xi;t hi;t pt xi;t 2 i;t
where a < 0 is the same for all consumers. The random variable hi,t is assumed to be only observed by the consumer. The first-order condition yields xi;t ¼ bðpt hi;t Þ where b = 1/a < 0. The demand xi,t is a linear function of the difference between the price pt and the random variable hi,t that represents shifts in preferences. Note that the demand function introduces a nonadditive error term hi,t. We now specify the heterogeneity factor hi,t. This random variable can be expressed as hi;t ¼ hi þ et where hi is a random variable specific to each consumer and et is an independent and serially uncorrelated aggregate random variable. The specification of hi,t can also include an exogenous trend that accounts for growth in demand independently from price variations: hi;t ¼ lt1 hi The individual demand xi,t and expenditure yi,t are xi;t ¼ at bhi;t
ð1Þ
yi;t ¼ ct at hi;t
ð2Þ
The parameter at = bpt is negative and varies over periods following a change in price. The parameter b = 1/a is negative and time-invariant and ct = bp2t is negative. Eqs. (1) and (2) are used in order to identify the structural parameters that summarize preferences. The structural model imposes cross-equations restrictions as the parameter at enters both in Eqs. (1) and (2). The idea now is to take advantages from these restrictions in order to recover the structural parameters. 1
Although excessively simplified, this structural model delivers the basis of our identification principle. See Fe`ve et al. (2003) for several extensions.
C. Cazals et al. / Economics Letters 86 (2005) 1–6
3
In what follows, we are mainly interested in the identification of the price elasticity of demand. The prices elasticities at period t of a representative consumer is given by:
et ¼
AEðxi;t Þ pt at u : Apt Eðxi;t Þ Eðxi;t Þ
ð3Þ
3. Identification in the one period case In order to illustrate the difficulties about the identification of structural parameters, we first consider a one-period case. Therefore, we discard from Eqs. (1) and (2) the time index. The demand and expenditure are thus defined by xi = a bhi and yi = c ahi. We denote E(hi) = ho and V(hi) = Vo. Moments conditions on demand and expenditure are the E(xi) = a bh0, E( yi) = c ah0, V(xi) = b2Vo, V( yi) = a2Vo and Cov(xi,yi) = baVo. The set of structural parameters to be estimated is u={a, c, b, h0, Vo}. The number of structural parameters is equal to the number of moments, i.e., the number of identifying conditions. Therefore, necessary conditions for identification appears at a first glance verified. The following proposition states that the structural parameters cannot be identified using these moment conditions. Proposition 1. The five structural parameters u={a, c, b, h0, Vo} cannot be identified from the five moment conditions. In the single-period case, Proposition 1 shows that we cannot take advantage from the cross-equation restrictions created by the structural model. However, the identification is obtained when particular assumptions about the distribution of the heterogeneity factor are formulated. We present an example where the first two moments of hi depend on a single parameter. Example 1. Let hi f fho where f is an exponential distribution of parameter ho > 0. The moments are E(hi) = ho and V(hi) = ho2. It follows that E(xi) = a bho and V(xi) = b2ho2. We obtain rxi = bho and after substitution into the mean of demand, we obtain a = E(xi) rxi. The price elasticity is thus given by q = 1 rxi/E(xi).
4. Identification in the two-period case Another—less arbitrary and more robust—way to obtain identification is to take into account for different periods. For example, Brown and Rosen (1982) and Kahn and Lang (1988) consider the hedonic price model, either with multimarket data with a single-period data or single market with a multiperiod data. The motivation of such an approach is that if preferences and the distribution of tastes remain stable—or controlled—exogenous price changes allow to identify the structural parameters of demand. Although unexplained (for a critical survey, see Ekeland et al., 2004), shifts over time in technology offer an opportunity to identify the structural parameters of demand. We exploit this idea and
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C. Cazals et al. / Economics Letters 86 (2005) 1–6
we show that the identification of demand parameters can be easily obtained. For simplicity, we consider the structural model in a two-period case.2 Example 2. Heterogeneity factor with aggregate shocks. The heterogeneity factor depends on an individual effect hi and an unobservable random aggregate variable et. Demand and expenditure are given by xi,t = at bhi bet and yi,t = ct athi atet, where E(et) = 0, V(et) = r2 and t = 1,2. The moments conditions are E(xi,1) = a1 bh0, E(xi,2) = a2 bh0 Cov(xi,1,yi,1) = ba1(Vo + r2) and Cov(xi,2,yi,2)=ba2(Vo+ r2). First, subtract the average demand over the two periods, Eðxi;2 Þ Eðxi;1 ÞuEðxi;2 xi;1 Þ ¼ a2 a1
ð4Þ
and divide the covariance of demand and expenditure over the two periods Covðxi;2 ; yi;2 Þ a2 ¼ Covðxi;1 ; yi;1 Þ a1
ð5Þ
Eqs. (4) and (5) allow to identify the two parameters a1 and a2: a1 ¼
Covðxi;1 ; yi;1 Þ Eðxi;2 ; xi;1 Þ Covðxi;2 ; yi;2 Þ Covðxi;1 ; yi;1 Þ
a2 ¼
Covðxi;2 ; yi;2 Þ Eðxi;2 ; xi;1 Þ Covðxi;2 ; yi;2 Þ Covðxi;1 ; yi;1 Þ
Using Eq. (3), we deduce the price elasticity for each period. Example 3. Heterogeneity factor with trend. The heterogeneity factor evolves exogenously over periods: hi;t ¼ lt1 hi The mean and the variance of demand now vary independently from changes in price. The parameter l represents an exogenous growth factor in preferences. The change in demand has two components: xi;tþ1 xi;t ¼ ðatþ1 at Þ bhi lt1 ðl 1Þ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} change in demand
change in price
change in preferences
When l p 1, the change in demand results in the change in preferences when the price is constant (at + 1 = at). This growth factor is normalized to 1 in the first period. The moments are E(xi,1) = a1 bh0, E(xi,2) = a2 blh0, V(xi,1) = b2Vo, V(xi,2) = b2l2Vo, Cov(xi,1,yi,1) = ba1Vo and Cov(xi,2,yi,2) = ba2Vo. Note that the parameter l enters both in the mean and the variances. The parameter l is identified from the variances of demand over the two periods, taking the square root l = rxi,2/rxi,1. Now, subtract the demand over the two periods E(xi,2) E(xi,1) = a2 a1 bho(l 1) in order to eliminate the individual 2
As we are interested in the identification of price elasticities, we only study the identification of at for t = 1,2. The remaining parameters are identified from the other moment conditions.
C. Cazals et al. / Economics Letters 86 (2005) 1–6
5
effects and divide the covariance between demand and expenditure over the two periods Cov(xi,1,yi,1)/ V(xi,1) = a1/b and Cov(xi,2,yi,2)/V(xi,2) = a2/b. It follows that a2/a1=(Cov(xi,2,yi,2)V(xi,1))/(Cov(xi,1,yi,1) V(xi,2)). Now, from the mean of demand in first period, one gets bho = E(xi,1) a1 and expected demand variation is E(xi,2) E(xi,1) = a2 a1+(E(xi,1) a1)(l 1). The values of a1 and a2 are thus given by: a1 ¼
Eðxi;2 Þ ðrxi;2 =rxi;1 ÞEðxi;1 Þ Covðxi;2 ; yi;2 Þ=V ðxi;2 Þ rxi;2 Covðxi;1 ; yi;1 Þ=V ðxi;1 Þ rxi;1
a2 ¼
Eðxi;2 Þ ðrxi;2 =rxi;1 ÞEðxi;1 Þ Covðxi;2 ; yi;2 Þ=V ðxi;2 Þ rxi;1 1 Covðxi;1 ; yi;1 Þ=V ðxi;1 Þ rxi;2
The expressions of a1 and a2 are similar to the ones of Example 2, apart from the growth correcting effect. From Eq. (3), the price elasticity can be directly deduced.
5. Concluding remarks In this paper, we present an empirical strategy that allows to identify the demand parameters of a simple structural model. Several issues may be then worth considering. First, instead of a single homogeneous product, the structural model must introduce a differentiated and/or several products that induce new conditions for identification. Second, the model delivers a linear demand. One may ask if the previous results about identification still hold in the nonlinear case. Finally, the potential of this approach with real data must be investigated. Acknowledgements We would like to thank J. Panzar for helpful comments. The traditional disclaimer applies.
Appendix A
Proof of Proposition 1. First, note that the parameters ho and c are only identified from equations the mean of demand and expenditure. So, we discard these two parameters from identification study. The identification of u˜ ={a, b, Vo} is thus based on the remained equations. The condition for identification relies on the rank of f (u˜ ) where 0
b2 Vo V ðxi Þ
B B 2 ˜ ¼B f ðuÞ B a Vo V ðyi Þ @
baVo Covðxi; yi Þ
1 C C C C A
6
C. Cazals et al. / Economics Letters 86 (2005) 1–6
Let the first derivative of f(u˜ ) with respect to u˜ 0
0
B B ˜ Af ðuÞ ¼B B 2aVo ˜ AuV @ bVo
2bVo
b2
1
0
C C a C C A
aVo
ab
2
˜ uVj ˜ = 0, i.e., f (u) ˜ is not full rank. It follows that the condition for identification is not fulfilled. 5 jAf (u)/A
References Brown, J., Rosen, H., 1982. On the estimation of structural hedonic price models. Econometrica 50, 765 – 769. Ekeland, I., Heckman, J., Nesheim, L., 2004. Identification and estimation of hedonic models. Journal of Political Economy 112, s1, S60 – S103. Fe`ve, F., Fe`ve, P., Florens, J.-P., 2003. Attribute Choice and Structural Econometrics of Price Elasticity of Demand, IDEI working paper. Kahn, S., Lang, K., 1988. Efficient estimation of structural hedonic systems. International Economic Review 29, 157 – 166.
Simple structural econometrics of price elasticity Catherine Cazals a, Fre´de´rique Fe`ve a, Patrick Fe`ve b,*, Jean-Pierre Florens c b
a University of Toulouse (CNRS-GREMAQ and IDEI), France GREMAQ-Universite´ de Toulouse I, manufacture des Tabacs, baˆt. F, 21 alle´e de Brienne, 31000 Toulouse, France c University of Toulouse (IUF, GREMAQ and IDEI), France
Received 15 January 2004; accepted 29 March 2004 Available online 10 November 2004
Abstract This paper delivers a methodology for the identification of the price elasticity of demand. The approach relies on a structural model of consumer behavior wherein demand and expenditure depend on a heterogeneity factor. We show that the structural parameters are not identified in the one-period case. Conversely, in a two-period case, the structural parameters, and thus the price elasticity of demand, are identified for each period. D 2004 Elsevier B.V. All rights reserved. Keywords: Structural model; Heterogeneity factor; Price elasticity JEL classification: C1; C2; D1
1. Introduction The identification of price elasticity from cross-section individual data is impossible when the price does not vary in the sample. Moreover, in the case of panel data, the slow modification of tariffs and the usual small number of periods render very hazardous identification of the price elasticity. The aim of this paper is to present a methodology for the treatment of this kind of data and to show that we can take advantages from the restrictions created by a structural model of demand. We first show that the structural parameters are not identified in the one-period case. Nevertheless, the identification can be obtained using different periods. The motivation of such an approach is that, if preferences and the distribution of tastes remain stable—or controlled—across periods, exogenous price changes allow to identify the structural parameters. We exploit this idea and we show in a series of examples that the price elasticity of demand can be identified. * Corresponding author. Tel.: +33-5-61-12-85-75; fax: +33-5-61-22-55-63. E-mail address: [email protected] (P. Fe`ve). 0165-1765/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2004.03.032
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C. Cazals et al. / Economics Letters 86 (2005) 1–6
2. A simple structural model We consider a repeated static market with a given set of producers and consumers. The quantity of a homogeneous1 good used by a consumer i in period t is denoted xi,t and the associated price—common to all consumers—is pt. The consumer i at period t seeks to maximize Uðxi;t Þ ¼
a 2 x þ xi;t hi;t pt xi;t 2 i;t
where a < 0 is the same for all consumers. The random variable hi,t is assumed to be only observed by the consumer. The first-order condition yields xi;t ¼ bðpt hi;t Þ where b = 1/a < 0. The demand xi,t is a linear function of the difference between the price pt and the random variable hi,t that represents shifts in preferences. Note that the demand function introduces a nonadditive error term hi,t. We now specify the heterogeneity factor hi,t. This random variable can be expressed as hi;t ¼ hi þ et where hi is a random variable specific to each consumer and et is an independent and serially uncorrelated aggregate random variable. The specification of hi,t can also include an exogenous trend that accounts for growth in demand independently from price variations: hi;t ¼ lt1 hi The individual demand xi,t and expenditure yi,t are xi;t ¼ at bhi;t
ð1Þ
yi;t ¼ ct at hi;t
ð2Þ
The parameter at = bpt is negative and varies over periods following a change in price. The parameter b = 1/a is negative and time-invariant and ct = bp2t is negative. Eqs. (1) and (2) are used in order to identify the structural parameters that summarize preferences. The structural model imposes cross-equations restrictions as the parameter at enters both in Eqs. (1) and (2). The idea now is to take advantages from these restrictions in order to recover the structural parameters. 1
Although excessively simplified, this structural model delivers the basis of our identification principle. See Fe`ve et al. (2003) for several extensions.
C. Cazals et al. / Economics Letters 86 (2005) 1–6
3
In what follows, we are mainly interested in the identification of the price elasticity of demand. The prices elasticities at period t of a representative consumer is given by:
et ¼
AEðxi;t Þ pt at u : Apt Eðxi;t Þ Eðxi;t Þ
ð3Þ
3. Identification in the one period case In order to illustrate the difficulties about the identification of structural parameters, we first consider a one-period case. Therefore, we discard from Eqs. (1) and (2) the time index. The demand and expenditure are thus defined by xi = a bhi and yi = c ahi. We denote E(hi) = ho and V(hi) = Vo. Moments conditions on demand and expenditure are the E(xi) = a bh0, E( yi) = c ah0, V(xi) = b2Vo, V( yi) = a2Vo and Cov(xi,yi) = baVo. The set of structural parameters to be estimated is u={a, c, b, h0, Vo}. The number of structural parameters is equal to the number of moments, i.e., the number of identifying conditions. Therefore, necessary conditions for identification appears at a first glance verified. The following proposition states that the structural parameters cannot be identified using these moment conditions. Proposition 1. The five structural parameters u={a, c, b, h0, Vo} cannot be identified from the five moment conditions. In the single-period case, Proposition 1 shows that we cannot take advantage from the cross-equation restrictions created by the structural model. However, the identification is obtained when particular assumptions about the distribution of the heterogeneity factor are formulated. We present an example where the first two moments of hi depend on a single parameter. Example 1. Let hi f fho where f is an exponential distribution of parameter ho > 0. The moments are E(hi) = ho and V(hi) = ho2. It follows that E(xi) = a bho and V(xi) = b2ho2. We obtain rxi = bho and after substitution into the mean of demand, we obtain a = E(xi) rxi. The price elasticity is thus given by q = 1 rxi/E(xi).
4. Identification in the two-period case Another—less arbitrary and more robust—way to obtain identification is to take into account for different periods. For example, Brown and Rosen (1982) and Kahn and Lang (1988) consider the hedonic price model, either with multimarket data with a single-period data or single market with a multiperiod data. The motivation of such an approach is that if preferences and the distribution of tastes remain stable—or controlled—exogenous price changes allow to identify the structural parameters of demand. Although unexplained (for a critical survey, see Ekeland et al., 2004), shifts over time in technology offer an opportunity to identify the structural parameters of demand. We exploit this idea and
4
C. Cazals et al. / Economics Letters 86 (2005) 1–6
we show that the identification of demand parameters can be easily obtained. For simplicity, we consider the structural model in a two-period case.2 Example 2. Heterogeneity factor with aggregate shocks. The heterogeneity factor depends on an individual effect hi and an unobservable random aggregate variable et. Demand and expenditure are given by xi,t = at bhi bet and yi,t = ct athi atet, where E(et) = 0, V(et) = r2 and t = 1,2. The moments conditions are E(xi,1) = a1 bh0, E(xi,2) = a2 bh0 Cov(xi,1,yi,1) = ba1(Vo + r2) and Cov(xi,2,yi,2)=ba2(Vo+ r2). First, subtract the average demand over the two periods, Eðxi;2 Þ Eðxi;1 ÞuEðxi;2 xi;1 Þ ¼ a2 a1
ð4Þ
and divide the covariance of demand and expenditure over the two periods Covðxi;2 ; yi;2 Þ a2 ¼ Covðxi;1 ; yi;1 Þ a1
ð5Þ
Eqs. (4) and (5) allow to identify the two parameters a1 and a2: a1 ¼
Covðxi;1 ; yi;1 Þ Eðxi;2 ; xi;1 Þ Covðxi;2 ; yi;2 Þ Covðxi;1 ; yi;1 Þ
a2 ¼
Covðxi;2 ; yi;2 Þ Eðxi;2 ; xi;1 Þ Covðxi;2 ; yi;2 Þ Covðxi;1 ; yi;1 Þ
Using Eq. (3), we deduce the price elasticity for each period. Example 3. Heterogeneity factor with trend. The heterogeneity factor evolves exogenously over periods: hi;t ¼ lt1 hi The mean and the variance of demand now vary independently from changes in price. The parameter l represents an exogenous growth factor in preferences. The change in demand has two components: xi;tþ1 xi;t ¼ ðatþ1 at Þ bhi lt1 ðl 1Þ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} change in demand
change in price
change in preferences
When l p 1, the change in demand results in the change in preferences when the price is constant (at + 1 = at). This growth factor is normalized to 1 in the first period. The moments are E(xi,1) = a1 bh0, E(xi,2) = a2 blh0, V(xi,1) = b2Vo, V(xi,2) = b2l2Vo, Cov(xi,1,yi,1) = ba1Vo and Cov(xi,2,yi,2) = ba2Vo. Note that the parameter l enters both in the mean and the variances. The parameter l is identified from the variances of demand over the two periods, taking the square root l = rxi,2/rxi,1. Now, subtract the demand over the two periods E(xi,2) E(xi,1) = a2 a1 bho(l 1) in order to eliminate the individual 2
As we are interested in the identification of price elasticities, we only study the identification of at for t = 1,2. The remaining parameters are identified from the other moment conditions.
C. Cazals et al. / Economics Letters 86 (2005) 1–6
5
effects and divide the covariance between demand and expenditure over the two periods Cov(xi,1,yi,1)/ V(xi,1) = a1/b and Cov(xi,2,yi,2)/V(xi,2) = a2/b. It follows that a2/a1=(Cov(xi,2,yi,2)V(xi,1))/(Cov(xi,1,yi,1) V(xi,2)). Now, from the mean of demand in first period, one gets bho = E(xi,1) a1 and expected demand variation is E(xi,2) E(xi,1) = a2 a1+(E(xi,1) a1)(l 1). The values of a1 and a2 are thus given by: a1 ¼
Eðxi;2 Þ ðrxi;2 =rxi;1 ÞEðxi;1 Þ Covðxi;2 ; yi;2 Þ=V ðxi;2 Þ rxi;2 Covðxi;1 ; yi;1 Þ=V ðxi;1 Þ rxi;1
a2 ¼
Eðxi;2 Þ ðrxi;2 =rxi;1 ÞEðxi;1 Þ Covðxi;2 ; yi;2 Þ=V ðxi;2 Þ rxi;1 1 Covðxi;1 ; yi;1 Þ=V ðxi;1 Þ rxi;2
The expressions of a1 and a2 are similar to the ones of Example 2, apart from the growth correcting effect. From Eq. (3), the price elasticity can be directly deduced.
5. Concluding remarks In this paper, we present an empirical strategy that allows to identify the demand parameters of a simple structural model. Several issues may be then worth considering. First, instead of a single homogeneous product, the structural model must introduce a differentiated and/or several products that induce new conditions for identification. Second, the model delivers a linear demand. One may ask if the previous results about identification still hold in the nonlinear case. Finally, the potential of this approach with real data must be investigated. Acknowledgements We would like to thank J. Panzar for helpful comments. The traditional disclaimer applies.
Appendix A
Proof of Proposition 1. First, note that the parameters ho and c are only identified from equations the mean of demand and expenditure. So, we discard these two parameters from identification study. The identification of u˜ ={a, b, Vo} is thus based on the remained equations. The condition for identification relies on the rank of f (u˜ ) where 0
b2 Vo V ðxi Þ
B B 2 ˜ ¼B f ðuÞ B a Vo V ðyi Þ @
baVo Covðxi; yi Þ
1 C C C C A
6
C. Cazals et al. / Economics Letters 86 (2005) 1–6
Let the first derivative of f(u˜ ) with respect to u˜ 0
0
B B ˜ Af ðuÞ ¼B B 2aVo ˜ AuV @ bVo
2bVo
b2
1
0
C C a C C A
aVo
ab
2
˜ uVj ˜ = 0, i.e., f (u) ˜ is not full rank. It follows that the condition for identification is not fulfilled. 5 jAf (u)/A
References Brown, J., Rosen, H., 1982. On the estimation of structural hedonic price models. Econometrica 50, 765 – 769. Ekeland, I., Heckman, J., Nesheim, L., 2004. Identification and estimation of hedonic models. Journal of Political Economy 112, s1, S60 – S103. Fe`ve, F., Fe`ve, P., Florens, J.-P., 2003. Attribute Choice and Structural Econometrics of Price Elasticity of Demand, IDEI working paper. Kahn, S., Lang, K., 1988. Efficient estimation of structural hedonic systems. International Economic Review 29, 157 – 166.