Invertibility The system is invertible, at ( p 1 , ..., p G , I , z ) if for any ( Y 1 , ..., Y G ) , there exists a unique value of ( ε 1 , ..., ε G ) satisfying the system of equations. Each unique value of ( ε 1 , ..., ε G ) identifies a particular consumer. Example with G + 1 = 2: (ignoring z for the time being) suppose U ( y 1 , y 0 , ε ) = v ( y 1 , y 0 ) + w ( y 1 , ε ) p y 1 + y 0 ≤ I subject to Assume that the functions v and w are twice continuously differentiable, strictly increasing and strictly concave, and that ∂ 2 w ( y 1 , ε ) / ∂ y 1 ∂ε > 0 . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 5 / 24
Invertibility The system is invertible, at ( p 1 , ..., p G , I , z ) if for any ( Y 1 , ..., Y G ) , there exists a unique value of ( ε 1 , ..., ε G ) satisfying the system of equations. Each unique value of ( ε 1 , ..., ε G ) identifies a particular consumer. Example with G + 1 = 2: (ignoring z for the time being) suppose U ( y 1 , y 0 , ε ) = v ( y 1 , y 0 ) + w ( y 1 , ε ) p y 1 + y 0 ≤ I subject to Assume that the functions v and w are twice continuously differentiable, strictly increasing and strictly concave, and that ∂ 2 w ( y 1 , ε ) / ∂ y 1 ∂ε > 0 . Then, the demand function for y 1 is invertible in ε Blundell, Kristensen and Matzkin () Multiple Goods January 2012 5 / 24
Invertibility By the Implicit Function Theorem, Blundell, Kristensen and Matzkin () Multiple Goods January 2012 6 / 24
Invertibility By the Implicit Function Theorem, y 1 = d ( p 1 , I , ε ) that solves the first order conditions exists and satisfies for all p , I , ε , ∂ d ( p , I , ε ) ∂ε w 10 ( y 1 , ε ) = − v 11 ( y 1 , I − py 1 ) − 2 v 10 ( y 1 , I − py 1 ) p + v 00 ( y 1 , I − py 1 ) p 2 + w 11 ( y > 0 and the denominator is < 0 by unique optimization. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 6 / 24
Invertibility By the Implicit Function Theorem, y 1 = d ( p 1 , I , ε ) that solves the first order conditions exists and satisfies for all p , I , ε , ∂ d ( p , I , ε ) ∂ε w 10 ( y 1 , ε ) = − v 11 ( y 1 , I − py 1 ) − 2 v 10 ( y 1 , I − py 1 ) p + v 00 ( y 1 , I − py 1 ) p 2 + w 11 ( y > 0 and the denominator is < 0 by unique optimization. Hence, the demand function for y 1 is invertible in ε . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 6 / 24
Identification when G+1=2 Assume that d is strictly increasing in ε , over the support of ε , and ε is distributed independently of ( p , I ) . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 7 / 24
Identification when G+1=2 Assume that d is strictly increasing in ε , over the support of ε , and ε is distributed independently of ( p , I ) . Then, for every p , I , ε , F ε ( ε ) = F Y | p , I ( d ( p , I , ε )) where F ε is the cumulative distribution of ε and F Y | p , I is the cumulative distribution of Y given ( p , I ) . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 7 / 24
Identification when G+1=2 Assume that d is strictly increasing in ε , over the support of ε , and ε is distributed independently of ( p , I ) . Then, for every p , I , ε , F ε ( ε ) = F Y | p , I ( d ( p , I , ε )) where F ε is the cumulative distribution of ε and F Y | p , I is the cumulative distribution of Y given ( p , I ) . Assuming that ε is distributed independently of ( p , I ) , the demand function is strictly increasing in ε , and F ε is strictly increasing at ε , � � � � � � − d p � , I � , ε p , � = F − 1 � I , ε I ) ( y 1 ) − y 1 d F Y | ( p , I )= ( � p , � Y | ( p , I )=( p � , I � ) � � p , � where y 1 is the observed consumption when budget is � I . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 7 / 24
Implied Restrictions on Demands If consumer ε satisfies Revealed Preference then the inequalities: � � + � � � + p � y � p 0 ( y � y 0 ) ≤ � ⇒ p � y � 0 ( y � y 0 ) < I � � 1 − � 0 − � 1 − � 0 − � p 1 y 1 I y 1 1 � � p , � � for each consumer ε , allow us to bound demand on a new budget I 0 = ( I � − p � where y � 1 = d ( p � , I � , ε ) and y � 1 d ( p � , I � , ε )) / p � 0 . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 8 / 24
Implied Restrictions on Demands If consumer ε satisfies Revealed Preference then the inequalities: � � + � � � + p � y � p 0 ( y � y 0 ) ≤ � ⇒ p � y � 0 ( y � y 0 ) < I � � 1 − � 0 − � 1 − � 0 − � p 1 y 1 I y 1 1 � � p , � � for each consumer ε , allow us to bound demand on a new budget I 0 = ( I � − p � where y � 1 = d ( p � , I � , ε ) and y � 1 d ( p � , I � , ε )) / p � 0 . These inequalities extend naturally to J > 2 price regimes (markets). Blundell, Kristensen and Matzkin () Multiple Goods January 2012 8 / 24
Implied Restrictions on Demands If consumer ε satisfies Revealed Preference then the inequalities: � � + � � � + p � y � p 0 ( y � y 0 ) ≤ � ⇒ p � y � 0 ( y � y 0 ) < I � � 1 − � 0 − � 1 − � 0 − � p 1 y 1 I y 1 1 � � p , � � for each consumer ε , allow us to bound demand on a new budget I 0 = ( I � − p � where y � 1 = d ( p � , I � , ε ) and y � 1 d ( p � , I � , ε )) / p � 0 . These inequalities extend naturally to J > 2 price regimes (markets). BBC (2008) derive the properties of the support set for the unknown demands and show how to construct improved bounds using variation in Engel curves for additive heterogeneity. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 8 / 24
Implied Restrictions on Demands If consumer ε satisfies Revealed Preference then the inequalities: � � + � � � + p � y � p 0 ( y � y 0 ) ≤ � ⇒ p � y � 0 ( y � y 0 ) < I � � 1 − � 0 − � 1 − � 0 − � p 1 y 1 I y 1 1 � � p , � � for each consumer ε , allow us to bound demand on a new budget I 0 = ( I � − p � where y � 1 = d ( p � , I � , ε ) and y � 1 d ( p � , I � , ε )) / p � 0 . These inequalities extend naturally to J > 2 price regimes (markets). BBC (2008) derive the properties of the support set for the unknown demands and show how to construct improved bounds using variation in Engel curves for additive heterogeneity. BKM (2011a) provide inference for these bounds based on RP inequality constraints with non-separable heterogeneity and quantile Engel curves. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 8 / 24
Implied Restrictions on Demands If consumer ε satisfies Revealed Preference then the inequalities: � � + � � � + p � y � p 0 ( y � y 0 ) ≤ � ⇒ p � y � 0 ( y � y 0 ) < I � � 1 − � 0 − � 1 − � 0 − � p 1 y 1 I y 1 1 � � p , � � for each consumer ε , allow us to bound demand on a new budget I 0 = ( I � − p � where y � 1 = d ( p � , I � , ε ) and y � 1 d ( p � , I � , ε )) / p � 0 . These inequalities extend naturally to J > 2 price regimes (markets). BBC (2008) derive the properties of the support set for the unknown demands and show how to construct improved bounds using variation in Engel curves for additive heterogeneity. BKM (2011a) provide inference for these bounds based on RP inequality constraints with non-separable heterogeneity and quantile Engel curves. Figures 1 - 2 show how sharp bounds on predicted demands are constructed under invertibility/ rank invariance assumption. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 8 / 24
Implied Restrictions on Demands If consumer ε satisfies Revealed Preference then the inequalities: � � + � � � + p � y � p 0 ( y � y 0 ) ≤ � ⇒ p � y � 0 ( y � y 0 ) < I � � 1 − � 0 − � 1 − � 0 − � p 1 y 1 I y 1 1 � � p , � � for each consumer ε , allow us to bound demand on a new budget I 0 = ( I � − p � where y � 1 = d ( p � , I � , ε ) and y � 1 d ( p � , I � , ε )) / p � 0 . These inequalities extend naturally to J > 2 price regimes (markets). BBC (2008) derive the properties of the support set for the unknown demands and show how to construct improved bounds using variation in Engel curves for additive heterogeneity. BKM (2011a) provide inference for these bounds based on RP inequality constraints with non-separable heterogeneity and quantile Engel curves. Figures 1 - 2 show how sharp bounds on predicted demands are constructed under invertibility/ rank invariance assumption. In this paper we show same set identification results hold for each consumer of type [ ε 1 , ..., ε G ] under RP inequality restrictions Blundell, Kristensen and Matzkin () Multiple Goods January 2012 8 / 24
Results for infinitessimal changes in prices. We know � � − 1 ∂ F Y | ( p , I ) ( d ( p , I , ε )) ∂ F Y | ( p , I ) ( d ( p , I , ε )) ∂ d ( p , I , ε ) = − ∂ ( p , I ) ∂ y ∂ ( p , I ) (Matzkin (1999), Chesher (2003)). And since each consumer ε satisfies the Integrability Conditions � � − 1 � � ∂ F Y | ( p , I ) ( y ) ∂ F Y | ( p , I ) ( y ) ∂ d ( p , I , ε ) ≤ − y ∂ p ∂ y ∂ I Blundell, Kristensen and Matzkin () Multiple Goods January 2012 9 / 24
Results for infinitessimal changes in prices. We know � � − 1 ∂ F Y | ( p , I ) ( d ( p , I , ε )) ∂ F Y | ( p , I ) ( d ( p , I , ε )) ∂ d ( p , I , ε ) = − ∂ ( p , I ) ∂ y ∂ ( p , I ) (Matzkin (1999), Chesher (2003)). And since each consumer ε satisfies the Integrability Conditions � � − 1 � � ∂ F Y | ( p , I ) ( y ) ∂ F Y | ( p , I ) ( y ) ∂ d ( p , I , ε ) ≤ − y ∂ p ∂ y ∂ I Which allow us to bound the effect of an infinitessimal change in price. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 9 / 24
Estimated bounds on demands In BKM (2011a) we provide an empirical application for G + 1 = 2. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 10 / 24
Estimated bounds on demands In BKM (2011a) we provide an empirical application for G + 1 = 2. Derive distribution results for the unrestricted and RP restricted quantile demand curves (expansion paths) d ( p � , I , ε ) for each price regime p � . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 10 / 24
Estimated bounds on demands In BKM (2011a) we provide an empirical application for G + 1 = 2. Derive distribution results for the unrestricted and RP restricted quantile demand curves (expansion paths) d ( p � , I , ε ) for each price regime p � . Show how a valid confidence set can be constructed for the demand bounds on predicted demands. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 10 / 24
Estimated bounds on demands In BKM (2011a) we provide an empirical application for G + 1 = 2. Derive distribution results for the unrestricted and RP restricted quantile demand curves (expansion paths) d ( p � , I , ε ) for each price regime p � . Show how a valid confidence set can be constructed for the demand bounds on predicted demands. � In the estimation, use polynomial splines, 3rd order pol. spline with 5 knots, with RP restrictions imposed at 100 I -points over the empirical support I . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 10 / 24
Estimated bounds on demands In BKM (2011a) we provide an empirical application for G + 1 = 2. Derive distribution results for the unrestricted and RP restricted quantile demand curves (expansion paths) d ( p � , I , ε ) for each price regime p � . Show how a valid confidence set can be constructed for the demand bounds on predicted demands. � In the estimation, use polynomial splines, 3rd order pol. spline with 5 knots, with RP restrictions imposed at 100 I -points over the empirical support I . � Study food demand for the same sub-population of couples with two children from SE England, 1984-1991, 8 price regimes. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 10 / 24
Estimated bounds on demands In BKM (2011a) we provide an empirical application for G + 1 = 2. Derive distribution results for the unrestricted and RP restricted quantile demand curves (expansion paths) d ( p � , I , ε ) for each price regime p � . Show how a valid confidence set can be constructed for the demand bounds on predicted demands. � In the estimation, use polynomial splines, 3rd order pol. spline with 5 knots, with RP restrictions imposed at 100 I -points over the empirical support I . � Study food demand for the same sub-population of couples with two children from SE England, 1984-1991, 8 price regimes. Figures of quantile expansion paths, demand bounds and confidence sets in Figures 3 and 4. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 10 / 24
Multiple Goods and Conditional Demands Suppose there is a good y 2 , that is not separable from y 0 and y 1 . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 11 / 24
Multiple Goods and Conditional Demands Suppose there is a good y 2 , that is not separable from y 0 and y 1 . { y 0 , y 1 , y 2 } now form a non-separable subset of consumption goods Blundell, Kristensen and Matzkin () Multiple Goods January 2012 11 / 24
Multiple Goods and Conditional Demands Suppose there is a good y 2 , that is not separable from y 0 and y 1 . { y 0 , y 1 , y 2 } now form a non-separable subset of consumption goods they have to be studied together to derive predictions of demand behavior under any new price vector. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 11 / 24
Multiple Goods and Conditional Demands Suppose there is a good y 2 , that is not separable from y 0 and y 1 . { y 0 , y 1 , y 2 } now form a non-separable subset of consumption goods they have to be studied together to derive predictions of demand behavior under any new price vector. The conditional demand for good 1 , given the consumption of good 2, has the form: y 1 = c 1 ( p 1 , � I , y 2 , ε 1 ) where � I is the budget allocated to goods 0 and 1 , as for d 1 in the two good case. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 11 / 24
Multiple Goods and Conditional Demands Suppose there is a good y 2 , that is not separable from y 0 and y 1 . { y 0 , y 1 , y 2 } now form a non-separable subset of consumption goods they have to be studied together to derive predictions of demand behavior under any new price vector. The conditional demand for good 1 , given the consumption of good 2, has the form: y 1 = c 1 ( p 1 , � I , y 2 , ε 1 ) where � I is the budget allocated to goods 0 and 1 , as for d 1 in the two good case. The inclusion of y 2 in the conditional demand for good 1 represents the non-separability of y 2 from [ y 1 : y 0 ] . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 11 / 24
Multiple Goods and Conditional Demands Suppose there is a good y 2 , that is not separable from y 0 and y 1 . { y 0 , y 1 , y 2 } now form a non-separable subset of consumption goods they have to be studied together to derive predictions of demand behavior under any new price vector. The conditional demand for good 1 , given the consumption of good 2, has the form: y 1 = c 1 ( p 1 , � I , y 2 , ε 1 ) where � I is the budget allocated to goods 0 and 1 , as for d 1 in the two good case. The inclusion of y 2 in the conditional demand for good 1 represents the non-separability of y 2 from [ y 1 : y 0 ] . � As before we assume ε 1 is scalar and c 1 is strictly increasing in ε 1 . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 11 / 24
Multiple Goods and Conditional Demands The exclusion of ε 2 from c 1 is a strong assumption on preferences. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 12 / 24
Multiple Goods and Conditional Demands The exclusion of ε 2 from c 1 is a strong assumption on preferences. In our general framework we weaken these preference restrictions Blundell, Kristensen and Matzkin () Multiple Goods January 2012 12 / 24
Multiple Goods and Conditional Demands The exclusion of ε 2 from c 1 is a strong assumption on preferences. In our general framework we weaken these preference restrictions � although at the cost of strengthening assumptions on the specification of prices and/or demographics. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 12 / 24
Multiple Goods and Conditional Demands The exclusion of ε 2 from c 1 is a strong assumption on preferences. In our general framework we weaken these preference restrictions � although at the cost of strengthening assumptions on the specification of prices and/or demographics. Likewise p 2 , and ε 2 , are exclusive to c 2 . So that we have: c 1 ( p 1 , � y 1 = I , y 2 , ε 1 ) c 2 ( p 2 , � � = I , y 1 , ε 2 ) y 2 Blundell, Kristensen and Matzkin () Multiple Goods January 2012 12 / 24
Multiple Goods and Conditional Demands The exclusion of ε 2 from c 1 is a strong assumption on preferences. In our general framework we weaken these preference restrictions � although at the cost of strengthening assumptions on the specification of prices and/or demographics. Likewise p 2 , and ε 2 , are exclusive to c 2 . So that we have: c 1 ( p 1 , � y 1 = I , y 2 , ε 1 ) c 2 ( p 2 , � � = I , y 1 , ε 2 ) y 2 Notice that the ε 1 and ε 2 naturally append to goods 1 and 2 and are increasing in the conditional demands for each good respectively. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 12 / 24
Multiple Goods and Conditional Demands The exclusion of ε 2 from c 1 is a strong assumption on preferences. In our general framework we weaken these preference restrictions � although at the cost of strengthening assumptions on the specification of prices and/or demographics. Likewise p 2 , and ε 2 , are exclusive to c 2 . So that we have: c 1 ( p 1 , � y 1 = I , y 2 , ε 1 ) c 2 ( p 2 , � � = I , y 1 , ε 2 ) y 2 Notice that the ε 1 and ε 2 naturally append to goods 1 and 2 and are increasing in the conditional demands for each good respectively. E xtends the monotonicity result to conditional demands: Blundell, Kristensen and Matzkin () Multiple Goods January 2012 12 / 24
Multiple Goods and Conditional Demands The exclusion of ε 2 from c 1 is a strong assumption on preferences. In our general framework we weaken these preference restrictions � although at the cost of strengthening assumptions on the specification of prices and/or demographics. Likewise p 2 , and ε 2 , are exclusive to c 2 . So that we have: c 1 ( p 1 , � y 1 = I , y 2 , ε 1 ) c 2 ( p 2 , � � = I , y 1 , ε 2 ) y 2 Notice that the ε 1 and ε 2 naturally append to goods 1 and 2 and are increasing in the conditional demands for each good respectively. E xtends the monotonicity result to conditional demands: � Permits estimation by QIV. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 12 / 24
Multiple Goods and Conditional Demands The exclusion of ε 2 from c 1 is a strong assumption on preferences. In our general framework we weaken these preference restrictions � although at the cost of strengthening assumptions on the specification of prices and/or demographics. Likewise p 2 , and ε 2 , are exclusive to c 2 . So that we have: c 1 ( p 1 , � y 1 = I , y 2 , ε 1 ) c 2 ( p 2 , � � = I , y 1 , ε 2 ) y 2 Notice that the ε 1 and ε 2 naturally append to goods 1 and 2 and are increasing in the conditional demands for each good respectively. E xtends the monotonicity result to conditional demands: � Permits estimation by QIV. � Implyies that the ranking of goods on the budget line [ y 0 : y 1 ] is invariant to y 2 , (as well as to I and p ) even though y 2 is non-separable from [ y 0 : y 1 ] . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 12 / 24
Commodity Specific Observed Heterogeneity Mirroring the discussion of ε 1 and ε 2 , we also introduce exclusive observed heterogeneity z 1 and z 2 . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 13 / 24
Commodity Specific Observed Heterogeneity Mirroring the discussion of ε 1 and ε 2 , we also introduce exclusive observed heterogeneity z 1 and z 2 . Conditional demands then take the form: c 1 ( p 1 , � y 1 = I , y 2 , z 1 , ε 1 ) c 2 ( p 2 , � � = I , y 1 , z 2 , ε 2 ) y 2 Blundell, Kristensen and Matzkin () Multiple Goods January 2012 13 / 24
Commodity Specific Observed Heterogeneity Mirroring the discussion of ε 1 and ε 2 , we also introduce exclusive observed heterogeneity z 1 and z 2 . Conditional demands then take the form: c 1 ( p 1 , � y 1 = I , y 2 , z 1 , ε 1 ) c 2 ( p 2 , � � = I , y 1 , z 2 , ε 2 ) y 2 corresponding to standard demands y 1 = d 1 ( p 1 , p 2 , I , z 1 , ε 1 , z 2 , ε 2 ) y 2 = d 2 ( p 1 , p 2 , I , z 1 , ε 1 , z 2 , ε 2 ) Blundell, Kristensen and Matzkin () Multiple Goods January 2012 13 / 24
Commodity Specific Observed Heterogeneity Mirroring the discussion of ε 1 and ε 2 , we also introduce exclusive observed heterogeneity z 1 and z 2 . Conditional demands then take the form: c 1 ( p 1 , � y 1 = I , y 2 , z 1 , ε 1 ) c 2 ( p 2 , � � = I , y 1 , z 2 , ε 2 ) y 2 corresponding to standard demands y 1 = d 1 ( p 1 , p 2 , I , z 1 , ε 1 , z 2 , ε 2 ) y 2 = d 2 ( p 1 , p 2 , I , z 1 , ε 1 , z 2 , ε 2 ) We may also wish to group together the heterogeneity terms in some restricted way, for example y 1 = d 1 ( p 1 , p 2 , I , z 1 + ε 1 , z 2 + ε 2 ) = d 2 ( p 1 , p 2 , I , z 1 + ε 1 , z 2 + ε 2 ) . y 2 Blundell, Kristensen and Matzkin () Multiple Goods January 2012 13 / 24
Commodity Specific Observed Heterogeneity Mirroring the discussion of ε 1 and ε 2 , we also introduce exclusive observed heterogeneity z 1 and z 2 . Conditional demands then take the form: c 1 ( p 1 , � y 1 = I , y 2 , z 1 , ε 1 ) c 2 ( p 2 , � � = I , y 1 , z 2 , ε 2 ) y 2 corresponding to standard demands y 1 = d 1 ( p 1 , p 2 , I , z 1 , ε 1 , z 2 , ε 2 ) y 2 = d 2 ( p 1 , p 2 , I , z 1 , ε 1 , z 2 , ε 2 ) We may also wish to group together the heterogeneity terms in some restricted way, for example y 1 = d 1 ( p 1 , p 2 , I , z 1 + ε 1 , z 2 + ε 2 ) = d 2 ( p 1 , p 2 , I , z 1 + ε 1 , z 2 + ε 2 ) . y 2 These restricted specifications will be important in our discussion of identification and estimation Blundell, Kristensen and Matzkin () Multiple Goods January 2012 13 / 24
Triangular Demands Suppose preferences are such that [ y 1 , y 0 ] form a separable sub-group within [ y 1 , y 0 , y 2 ] . In this case, utility has the recursive form U ( y 0 , y 1 , y 2 , z 1 , z 2 , ε 1 , ε 2 ) = V ( u ( y 0 , y 1 , z 1 , ε 1 ) , y 2 , z 2 , ε 2 ) so that the MRS between goods y 1 and y 0 does not depend on y 2 . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 14 / 24
Triangular Demands Suppose preferences are such that [ y 1 , y 0 ] form a separable sub-group within [ y 1 , y 0 , y 2 ] . In this case, utility has the recursive form U ( y 0 , y 1 , y 2 , z 1 , z 2 , ε 1 , ε 2 ) = V ( u ( y 0 , y 1 , z 1 , ε 1 ) , y 2 , z 2 , ε 2 ) so that the MRS between goods y 1 and y 0 does not depend on y 2 . � Note however that the MRS for y 2 and y 0 does depend on y 1 . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 14 / 24
Triangular Demands Suppose preferences are such that [ y 1 , y 0 ] form a separable sub-group within [ y 1 , y 0 , y 2 ] . In this case, utility has the recursive form U ( y 0 , y 1 , y 2 , z 1 , z 2 , ε 1 , ε 2 ) = V ( u ( y 0 , y 1 , z 1 , ε 1 ) , y 2 , z 2 , ε 2 ) so that the MRS between goods y 1 and y 0 does not depend on y 2 . � Note however that the MRS for y 2 and y 0 does depend on y 1 . The conditional demands then take the triangular form: c 1 ( p 1 , � y 1 = I , z 1 , ε 1 ) c 2 ( p 2 , � � y 2 = I , y 1 , z 2 , ε 2 ) Blundell, Kristensen and Matzkin () Multiple Goods January 2012 14 / 24
Triangular Demands Suppose preferences are such that [ y 1 , y 0 ] form a separable sub-group within [ y 1 , y 0 , y 2 ] . In this case, utility has the recursive form U ( y 0 , y 1 , y 2 , z 1 , z 2 , ε 1 , ε 2 ) = V ( u ( y 0 , y 1 , z 1 , ε 1 ) , y 2 , z 2 , ε 2 ) so that the MRS between goods y 1 and y 0 does not depend on y 2 . � Note however that the MRS for y 2 and y 0 does depend on y 1 . The conditional demands then take the triangular form: c 1 ( p 1 , � y 1 = I , z 1 , ε 1 ) c 2 ( p 2 , � � y 2 = I , y 1 , z 2 , ε 2 ) � Can relax preference assumptions to allow ε 1 to enter c 2 . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 14 / 24
Triangular Demands Suppose preferences are such that [ y 1 , y 0 ] form a separable sub-group within [ y 1 , y 0 , y 2 ] . In this case, utility has the recursive form U ( y 0 , y 1 , y 2 , z 1 , z 2 , ε 1 , ε 2 ) = V ( u ( y 0 , y 1 , z 1 , ε 1 ) , y 2 , z 2 , ε 2 ) so that the MRS between goods y 1 and y 0 does not depend on y 2 . � Note however that the MRS for y 2 and y 0 does depend on y 1 . The conditional demands then take the triangular form: c 1 ( p 1 , � y 1 = I , z 1 , ε 1 ) c 2 ( p 2 , � � y 2 = I , y 1 , z 2 , ε 2 ) � Can relax preference assumptions to allow ε 1 to enter c 2 . z 1 (and p 1 ) is excluded from c 2 and could act an instrument for y 1 in the QCF estimation of c 2 , as in Chesher (2003) and Imbens and Newey (2009). Blundell, Kristensen and Matzkin () Multiple Goods January 2012 14 / 24
Triangular Demands Blundell and Matzkin (2010) derive the complete set of if and only if conditions for nonseparable simultaneous equations models that generate triangular systems and therefore permit estimation by the control function (QCF) approach. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 15 / 24
Triangular Demands Blundell and Matzkin (2010) derive the complete set of if and only if conditions for nonseparable simultaneous equations models that generate triangular systems and therefore permit estimation by the control function (QCF) approach. The BM conditions cover preferences that include the conditional recursive separability form above. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 15 / 24
Triangular Demands Blundell and Matzkin (2010) derive the complete set of if and only if conditions for nonseparable simultaneous equations models that generate triangular systems and therefore permit estimation by the control function (QCF) approach. The BM conditions cover preferences that include the conditional recursive separability form above. For example, V ( ε 1 , ε 2 , y 2 ) + W ( ε 1 , y 1 , y 2 ) + y 0 e.g. = ( ε 1 + ε 2 ) u ( y 2 ) + ε 1 log ( y 1 − u ( y 2 )) + y 0 Blundell, Kristensen and Matzkin () Multiple Goods January 2012 15 / 24
The general G+1 > 2 case If demand functions are invertible in ( ε 1 , ..., ε G ) , we can write ( ε 1 , ..., ε G ) as ε 1 = r 1 ( y 1 , ..., y G , p 1 , ..., p G , I , z 1 , ... z G ) · ε G = r G ( y 1 , ..., y G , p 1 , ..., p G , I , z 1 , ... z G ) Blundell, Kristensen and Matzkin () Multiple Goods January 2012 16 / 24
The general G+1 > 2 case If demand functions are invertible in ( ε 1 , ..., ε G ) , we can write ( ε 1 , ..., ε G ) as ε 1 = r 1 ( y 1 , ..., y G , p 1 , ..., p G , I , z 1 , ... z G ) · ε G = r G ( y 1 , ..., y G , p 1 , ..., p G , I , z 1 , ... z G ) Can use the transformation of variables equation to determine identification (Matzkin (2010)) � � � � ∂ r ( y , p , I , z ) � � f Y | p , I , z ( y ) = f ε ( r ( y , p , I , z )) � � ∂ y Blundell, Kristensen and Matzkin () Multiple Goods January 2012 16 / 24
The general G+1 > 2 case If demand functions are invertible in ( ε 1 , ..., ε G ) , we can write ( ε 1 , ..., ε G ) as ε 1 = r 1 ( y 1 , ..., y G , p 1 , ..., p G , I , z 1 , ... z G ) · ε G = r G ( y 1 , ..., y G , p 1 , ..., p G , I , z 1 , ... z G ) Can use the transformation of variables equation to determine identification (Matzkin (2010)) � � � � ∂ r ( y , p , I , z ) � � f Y | p , I , z ( y ) = f ε ( r ( y , p , I , z )) � � ∂ y As we show, estimation can proceed using the average derivative method of Matzkin (2010). Blundell, Kristensen and Matzkin () Multiple Goods January 2012 16 / 24
An example of commodity specific characteristics and discrete prices. U ( y , I − p � y ) + V ( y , z + ε ) and f ε primitive functions Blundell, Kristensen and Matzkin () Multiple Goods January 2012 17 / 24
An example of commodity specific characteristics and discrete prices. U ( y , I − p � y ) + V ( y , z + ε ) and f ε primitive functions Demands given by p � y + y 0 ≤ I } arg max { U ( y , y 0 ) + V ( y , z + ε ) | y Blundell, Kristensen and Matzkin () Multiple Goods January 2012 17 / 24
An example of commodity specific characteristics and discrete prices. U ( y , I − p � y ) + V ( y , z + ε ) and f ε primitive functions Demands given by p � y + y 0 ≤ I } arg max { U ( y , y 0 ) + V ( y , z + ε ) | y Assume V 1 , G + 1 V 1 , G + 2 · · V 1 , G + G · · V G , G + 1 V G , G + 2 V G , G + G is a P-matrix (e.g. positive semi-definite or with dominant diagonal) ... examples.. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 17 / 24
An example of commodity specific characteristics and discrete prices. U ( y , I − p � y ) + V ( y , z + ε ) and f ε primitive functions Demands given by p � y + y 0 ≤ I } arg max { U ( y , y 0 ) + V ( y , z + ε ) | y Assume V 1 , G + 1 V 1 , G + 2 · · V 1 , G + G · · V G , G + 1 V G , G + 2 V G , G + G is a P-matrix (e.g. positive semi-definite or with dominant diagonal) ... examples.. Then, by Gale and Nikaido (1965), the system is invertible: There exist functions r 1 , ..., r G such that r 1 ( y 1 , ..., y G , p 1 , ..., p K , I ) ε 1 + z 1 = · · · r G ( y 1 , ..., y G , p 1 , ..., p K , I ) ε G + z G = Blundell, Kristensen and Matzkin () Multiple Goods January 2012 17 / 24
Identification Constructive identification follows as in Matzkin (2007). Assume ∂ f ε ( ε ) ε = ε ∗ = 0 < = > ∂ε Blundell, Kristensen and Matzkin () Multiple Goods January 2012 18 / 24
Identification Constructive identification follows as in Matzkin (2007). Assume ∂ f ε ( ε ) ε = ε ∗ = 0 < = > ∂ε The system derived from FOC after inverting is r ( y , p , I ) = ε + z Blundell, Kristensen and Matzkin () Multiple Goods January 2012 18 / 24
Identification Constructive identification follows as in Matzkin (2007). Assume ∂ f ε ( ε ) ε = ε ∗ = 0 < = > ∂ε The system derived from FOC after inverting is r ( y , p , I ) = ε + z Transformation of variables equations for all p , I , y , z � � � � ∂ r ( y , p , I ) � � f Y | p , I , z ( y ) = f ε ( r ( y , p , I ) − z ) � � ∂ y Blundell, Kristensen and Matzkin () Multiple Goods January 2012 18 / 24
Identification Constructive identification follows as in Matzkin (2007). Assume ∂ f ε ( ε ) ε = ε ∗ = 0 < = > ∂ε The system derived from FOC after inverting is r ( y , p , I ) = ε + z Transformation of variables equations for all p , I , y , z � � � � ∂ r ( y , p , I ) � � f Y | p , I , z ( y ) = f ε ( r ( y , p , I ) − z ) � � ∂ y Taking derivatives with respect to z � � ∂ f Y | p , I , z ( y ) � � = ∂ f ε ( r ( y , p , I ) − z ) ∂ r ( y , p , I ) � � � � ∂ z ∂ε ∂ y Blundell, Kristensen and Matzkin () Multiple Goods January 2012 18 / 24
Identification In � � ∂ f Y | p , I , z ( y ) � � = ∂ f ε ( r ( y , p , I ) − z ) ∂ r ( y , p , I ) � � � � ∂ z ∂ε ∂ y Blundell, Kristensen and Matzkin () Multiple Goods January 2012 19 / 24
Identification In � � ∂ f Y | p , I , z ( y ) � � = ∂ f ε ( r ( y , p , I ) − z ) ∂ r ( y , p , I ) � � � � ∂ z ∂ε ∂ y Note that ∂ f Y | p , I , z ( y ) ∂ f ε ( r ( y , p , I ) − z ) = 0 ⇒ = 0 ∂ z ∂ε Blundell, Kristensen and Matzkin () Multiple Goods January 2012 19 / 24
Identification In � � ∂ f Y | p , I , z ( y ) � � = ∂ f ε ( r ( y , p , I ) − z ) ∂ r ( y , p , I ) � � � � ∂ z ∂ε ∂ y Note that ∂ f Y | p , I , z ( y ) ∂ f ε ( r ( y , p , I ) − z ) = 0 ⇒ = 0 ∂ z ∂ε and ∂ f ε ( r ( y , p , I ) − z ) r ( y , p , I ) − z = ε ∗ = 0 ⇒ ∂ε Blundell, Kristensen and Matzkin () Multiple Goods January 2012 19 / 24
Identification Fix y , p , I . Find z ∗ such that ∂ f Y | p , I , z ∗ ( y ) = 0 ∂ z Blundell, Kristensen and Matzkin () Multiple Goods January 2012 20 / 24
Identification Fix y , p , I . Find z ∗ such that ∂ f Y | p , I , z ∗ ( y ) = 0 ∂ z Then, r ( y , p , I ) = ε ∗ + z ∗ Blundell, Kristensen and Matzkin () Multiple Goods January 2012 20 / 24
Identification Fix y , p , I . Find z ∗ such that ∂ f Y | p , I , z ∗ ( y ) = 0 ∂ z Then, r ( y , p , I ) = ε ∗ + z ∗ We have then constructive identification of the function r . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 20 / 24
Identification Fix y , p , I . Find z ∗ such that ∂ f Y | p , I , z ∗ ( y ) = 0 ∂ z Then, r ( y , p , I ) = ε ∗ + z ∗ We have then constructive identification of the function r . Identification of r ⇒ identification of h ∂ f Y | p , I , z ∗ ( y ) y = h ( p , I , ε ∗ + z ∗ ) = 0 ⇒ ∂ z Blundell, Kristensen and Matzkin () Multiple Goods January 2012 20 / 24
Average derivative estimator � � − 1 � � ∂ r ( y ) � = � r y ( y ) = T ZZ ( y ) T ZY ( y ) ∂ y Blundell, Kristensen and Matzkin () Multiple Goods January 2012 21 / 24
Average derivative estimator � � − 1 � � ∂ r ( y ) � = � r y ( y ) = T ZZ ( y ) T ZY ( y ) ∂ y Elements of � T ZZ and � T ZY are average derivative type estimators � � ∂ log � � ∂ log � f y | z ( y ) f y | z ( y ) � T y j z k ( y ) = ω ( z ) dz ∂ y j ∂ z k � � ∂ log � � � � ∂ log � � f y | z ( y ) f y | z ( y ) − ω ( z ) dz ω ( z ) dz ∂ y j ∂ z k Powell, Stock, and Stoker (1989), Newey (1994) Blundell, Kristensen and Matzkin () Multiple Goods January 2012 21 / 24
Average derivative estimator � � − 1 � � ∂ r ( y ) � = � r y ( y ) = T ZZ ( y ) T ZY ( y ) ∂ y Elements of � T ZZ and � T ZY are average derivative type estimators � � ∂ log � � ∂ log � f y | z ( y ) f y | z ( y ) � T y j z k ( y ) = ω ( z ) dz ∂ y j ∂ z k � � ∂ log � � � � ∂ log � � f y | z ( y ) f y | z ( y ) − ω ( z ) dz ω ( z ) dz ∂ y j ∂ z k Powell, Stock, and Stoker (1989), Newey (1994) Use mode assumption on ε , to recover the level of r at some value of y . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 21 / 24
Empirical example for the multiple good case Three good model with commodity specific observed heterogeneity Blundell, Kristensen and Matzkin () Multiple Goods January 2012 22 / 24
Empirical example for the multiple good case Three good model with commodity specific observed heterogeneity � Food, services and other goods. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 22 / 24
Empirical example for the multiple good case Three good model with commodity specific observed heterogeneity � Food, services and other goods. Assume that unobserved preference for food exactly matches variation family size/age composition, and are independent conditional on income (and other observed heterogeneity). Blundell, Kristensen and Matzkin () Multiple Goods January 2012 22 / 24
Empirical example for the multiple good case Three good model with commodity specific observed heterogeneity � Food, services and other goods. Assume that unobserved preference for food exactly matches variation family size/age composition, and are independent conditional on income (and other observed heterogeneity). Similarly, assume unobserved preference for services exactly matches age/birth cohort of adults. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 22 / 24
Empirical example for the multiple good case Three good model with commodity specific observed heterogeneity � Food, services and other goods. Assume that unobserved preference for food exactly matches variation family size/age composition, and are independent conditional on income (and other observed heterogeneity). Similarly, assume unobserved preference for services exactly matches age/birth cohort of adults. Extend to an index on z . Blundell, Kristensen and Matzkin () Multiple Goods January 2012 22 / 24
Empirical example for the multiple good case Three good model with commodity specific observed heterogeneity � Food, services and other goods. Assume that unobserved preference for food exactly matches variation family size/age composition, and are independent conditional on income (and other observed heterogeneity). Similarly, assume unobserved preference for services exactly matches age/birth cohort of adults. Extend to an index on z . � Figure 5.... Blundell, Kristensen and Matzkin () Multiple Goods January 2012 22 / 24
Conclusions � Show conditions for identification and estimation of individual demands in the two good and the multiple good case with nonadditive/nonseparable heterogeneity. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 23 / 24
Conclusions � Show conditions for identification and estimation of individual demands in the two good and the multiple good case with nonadditive/nonseparable heterogeneity. � Focus on the case of discrete prices (finite markets) and many heterogeneous consumers. Blundell, Kristensen and Matzkin () Multiple Goods January 2012 23 / 24
Conclusions � Show conditions for identification and estimation of individual demands in the two good and the multiple good case with nonadditive/nonseparable heterogeneity. � Focus on the case of discrete prices (finite markets) and many heterogeneous consumers. � Show how to use restrictions implied by revealed preference / integrability to bound the distribution of predicted demand at unobserved prices (policy counterfactual). Blundell, Kristensen and Matzkin () Multiple Goods January 2012 23 / 24
Figure 1a: The distribution of demands across consumers indexed by ‘ ε ’ y 1 d(I, ε ) y 2
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