Multiple Goods, Consumer Heterogeneity and Revealed
Preference
Richard Blundell(UCL), Dennis Kristensen(UCL) and Rosa Matzkin(UCLA)
January 2012
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
1 / 24
This talk builds on three related papers:
Blundell, Kristensen, Matzkin (2011a) "Bounding Quantile Demand
Functions Using Revealed Preference Inequalities"
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
2 / 24
This talk builds on three related papers:
Blundell, Kristensen, Matzkin (2011a) "Bounding Quantile Demand
Functions Using Revealed Preference Inequalities"
Blundell and Matzkin (2010) "Conditions for the Existence of Control
Functions in Nonparametric Simultaneous Equations Models"
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
2 / 24
This talk builds on three related papers:
Blundell, Kristensen, Matzkin (2011a) "Bounding Quantile Demand
Functions Using Revealed Preference Inequalities"
Blundell and Matzkin (2010) "Conditions for the Existence of Control
Functions in Nonparametric Simultaneous Equations Models"
Matzkin (2010) "Estimation of Nonparametric Models with Simultaneity"
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
2 / 24
This talk builds on three related papers:
Blundell, Kristensen, Matzkin (2011a) "Bounding Quantile Demand
Functions Using Revealed Preference Inequalities"
Blundell and Matzkin (2010) "Conditions for the Existence of Control
Functions in Nonparametric Simultaneous Equations Models"
Matzkin (2010) "Estimation of Nonparametric Models with Simultaneity"
I Focus here is on identi…cation and estimation when there are many
heterogeneous consumers, a …nite number of markets (prices) and
non-additive heterogeneity.
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
2 / 24
Consumer Problem
Consider choices over a weakly separable subset of G + 1 goods, y1 , ...., yG , y0
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
3 / 24
Consumer Problem
Consider choices over a weakly separable subset of G + 1 goods, y1 , ...., yG , y0
(p1 , p2 , ..., pG , I ) prices (for goods 1, ...G ) and total budget, [p, I ]
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
3 / 24
Consumer Problem
Consider choices over a weakly separable subset of G + 1 goods, y1 , ...., yG , y0
(p1 , p2 , ..., pG , I ) prices (for goods 1, ...G ) and total budget, [p, I ]
(ε1 , ..., εG ) unobserved heterogeneity (tastes) of the consumer, [ε]
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
3 / 24
Consumer Problem
Consider choices over a weakly separable subset of G + 1 goods, y1 , ...., yG , y0
(p1 , p2 , ..., pG , I ) prices (for goods 1, ...G ) and total budget, [p, I ]
(ε1 , ..., εG ) unobserved heterogeneity (tastes) of the consumer, [ε]
(z1 , ..., zK ) observed heterogeneity, [z]
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
3 / 24
Consumer Problem
Consider choices over a weakly separable subset of G + 1 goods, y1 , ...., yG , y0
(p1 , p2 , ..., pG , I ) prices (for goods 1, ...G ) and total budget, [p, I ]
(ε1 , ..., εG ) unobserved heterogeneity (tastes) of the consumer, [ε]
(z1 , ..., zK ) observed heterogeneity, [z]
Observed demands are a solution to
G
Maxy U
y1 , ..., yG , I
∑ pg yg , z, ε1 , ..., εG
g =1
Blundell, Kristensen and Matzkin ()
Multiple Goods
!
January 2012
3 / 24
Consumer Problem
Consider choices over a weakly separable subset of G + 1 goods, y1 , ...., yG , y0
(p1 , p2 , ..., pG , I ) prices (for goods 1, ...G ) and total budget, [p, I ]
(ε1 , ..., εG ) unobserved heterogeneity (tastes) of the consumer, [ε]
(z1 , ..., zK ) observed heterogeneity, [z]
Observed demands are a solution to
G
Maxy U
y1 , ..., yG , I
∑ pg yg , z, ε1 , ..., εG
g =1
!
Typically dealing with a …nite number of markets (prices) and many
(heterogeneous) consumers.
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
3 / 24
Consumer Demand
FOC - system of simultaneous equations with nonadditive unobservables
Ug y1 , ..., yG , I
∑G
g =1 pg yg , z, ε1 , ..., εG
U0 y1 , ..., yG , I
∑G
g =1 pg yg , z, ε1 , ..., εG
Blundell, Kristensen and Matzkin ()
Multiple Goods
= pg
for g = 1, ..., G
January 2012
4 / 24
Consumer Demand
FOC - system of simultaneous equations with nonadditive unobservables
Ug y1 , ..., yG , I
∑G
g =1 pg yg , z, ε1 , ..., εG
U0 y1 , ..., yG , I
∑G
g =1 pg yg , z, ε1 , ..., εG
= pg
for g = 1, ..., G
Demand functions - reduced form system with nonadditive unobservables
Yg = dg (p1 , ..., pG , I , z, ε1 , ..., εG )
Blundell, Kristensen and Matzkin ()
Multiple Goods
for g = 1, ..., G .
January 2012
4 / 24
Consumer Demand
FOC - system of simultaneous equations with nonadditive unobservables
Ug y1 , ..., yG , I
∑G
g =1 pg yg , z, ε1 , ..., εG
U0 y1 , ..., yG , I
∑G
g =1 pg yg , z, ε1 , ..., εG
= pg
for g = 1, ..., G
Demand functions - reduced form system with nonadditive unobservables
Yg = dg (p1 , ..., pG , I , z, ε1 , ..., εG )
for g = 1, ..., G .
The aim in this research is to use the Revealed Preference inequalities to
place bounds on predicted demands for each consumer [ε, z] for any
p
e1 , ..., p
eG , eI ;
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
4 / 24
Consumer Demand
FOC - system of simultaneous equations with nonadditive unobservables
Ug y1 , ..., yG , I
∑G
g =1 pg yg , z, ε1 , ..., εG
U0 y1 , ..., yG , I
∑G
g =1 pg yg , z, ε1 , ..., εG
= pg
for g = 1, ..., G
Demand functions - reduced form system with nonadditive unobservables
Yg = dg (p1 , ..., pG , I , z, ε1 , ..., εG )
for g = 1, ..., G .
The aim in this research is to use the Revealed Preference inequalities to
place bounds on predicted demands for each consumer [ε, z] for any
p
e1 , ..., p
eG , eI ;
also derive results on bounds for in…nitessimal changes in p and I .
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
4 / 24
Consumer Demand
FOC - system of simultaneous equations with nonadditive unobservables
Ug y1 , ..., yG , I
∑G
g =1 pg yg , z, ε1 , ..., εG
U0 y1 , ..., yG , I
∑G
g =1 pg yg , z, ε1 , ..., εG
= pg
for g = 1, ..., G
Demand functions - reduced form system with nonadditive unobservables
Yg = dg (p1 , ..., pG , I , z, ε1 , ..., εG )
for g = 1, ..., G .
The aim in this research is to use the Revealed Preference inequalities to
place bounds on predicted demands for each consumer [ε, z] for any
p
e1 , ..., p
eG , eI ;
also derive results on bounds for in…nitessimal changes in p and I .
For each price regime the dg are expansion paths (or Engel curves) for each
heterogeneous consumer of type [ε, z]
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
4 / 24
Consumer Demand
FOC - system of simultaneous equations with nonadditive unobservables
Ug y1 , ..., yG , I
∑G
g =1 pg yg , z, ε1 , ..., εG
U0 y1 , ..., yG , I
∑G
g =1 pg yg , z, ε1 , ..., εG
= pg
for g = 1, ..., G
Demand functions - reduced form system with nonadditive unobservables
Yg = dg (p1 , ..., pG , I , z, ε1 , ..., εG )
for g = 1, ..., G .
The aim in this research is to use the Revealed Preference inequalities to
place bounds on predicted demands for each consumer [ε, z] for any
p
e1 , ..., p
eG , eI ;
also derive results on bounds for in…nitessimal changes in p and I .
For each price regime the dg are expansion paths (or Engel curves) for each
heterogeneous consumer of type [ε, z]
Key assumptions will pertain to the dimension and direction of unobserved
heterogeneity ε, and to the speci…cation of observed heterogeneity z.
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
4 / 24
Invertibility
The system is invertible, at (p1 , ..., pG , I , z ) if for any (Y1 , ..., YG ) , there
exists a unique value of (ε1 , ..., εG ) satisfying the system of equations.
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
5 / 24
Invertibility
The system is invertible, at (p1 , ..., pG , I , z ) if for any (Y1 , ..., YG ) , there
exists a unique value of (ε1 , ..., εG ) satisfying the system of equations.
Each unique value of (ε1 , ..., εG ) identi…es a particular consumer.
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
5 / 24
Invertibility
The system is invertible, at (p1 , ..., pG , I , z ) if for any (Y1 , ..., YG ) , there
exists a unique value of (ε1 , ..., εG ) satisfying the system of equations.
Each unique value of (ε1 , ..., εG ) identi…es a particular consumer.
Example with G + 1 = 2: (ignoring z for the time being) suppose
U ( y1 , y0 , ε ) = v ( y1 , y0 ) + w ( y1 , ε )
subject to
Blundell, Kristensen and Matzkin ()
p y1 + y0
Multiple Goods
I
January 2012
5 / 24
Invertibility
The system is invertible, at (p1 , ..., pG , I , z ) if for any (Y1 , ..., YG ) , there
exists a unique value of (ε1 , ..., εG ) satisfying the system of equations.
Each unique value of (ε1 , ..., εG ) identi…es a particular consumer.
Example with G + 1 = 2: (ignoring z for the time being) suppose
U ( y1 , y0 , ε ) = v ( y1 , y0 ) + w ( y1 , ε )
subject to
p y1 + y0
I
Assume that the functions v and w are twice continuously di¤erentiable,
strictly increasing and strictly concave, and that ∂2 w (y1 , ε)/∂y1 ∂ε > 0.
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
5 / 24
Invertibility
The system is invertible, at (p1 , ..., pG , I , z ) if for any (Y1 , ..., YG ) , there
exists a unique value of (ε1 , ..., εG ) satisfying the system of equations.
Each unique value of (ε1 , ..., εG ) identi…es a particular consumer.
Example with G + 1 = 2: (ignoring z for the time being) suppose
U ( y1 , y0 , ε ) = v ( y1 , y0 ) + w ( y1 , ε )
subject to
p y1 + y0
I
Assume that the functions v and w are twice continuously di¤erentiable,
strictly increasing and strictly concave, and that ∂2 w (y1 , ε)/∂y1 ∂ε > 0.
Then, the demand function for y1 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,
y1 = d (p1 , I , ε) that solves the …rst order conditions exists and satis…es for
all p, I , ε,
∂d (p, I , ε)
∂ε
=
v11 (y1 , I
py1 )
2 v10 (y1 , I
w10 (y1 , ε)
py1 ) p + v00 (y1 , I
py1 ) p 2 + w11 (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,
y1 = d (p1 , I , ε) that solves the …rst order conditions exists and satis…es for
all p, I , ε,
∂d (p, I , ε)
∂ε
=
v11 (y1 , I
py1 )
2 v10 (y1 , I
w10 (y1 , ε)
py1 ) p + v00 (y1 , I
py1 ) p 2 + w11 (y
> 0
and the denominator is < 0 by unique optimization.
Hence, the demand function for y1 is invertible in ε.
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
6 / 24
Identi…cation 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
Identi…cation 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,
Fε (ε) = FY jp,I (d (p, I , ε))
where Fε is the cumulative distribution of ε and FY jp,I is the cumulative
distribution of Y given (p, I ) .
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
7 / 24
Identi…cation 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,
Fε (ε) = FY jp,I (d (p, I , ε))
where Fε is the cumulative distribution of ε and FY jp,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 Fε is strictly increasing at ε,
d p0, I 0, ε
d p
e, eI , ε = FY j(1 p,I )=(p 0 ,I 0 ) FY j(p,I )=(pe,eI ) (y1 )
y1
where y1 is the observed consumption when budget is p
e, eI .
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
7 / 24
Implied Restrictions on Demands
If consumer ε satis…es Revealed Preference then the inequalities:
p
e1 y10
ye1 + p
e0 (y00
ye0 )
eI ) p10 y10
ye1 + p00 (y00
ye0 ) < I 0
allow us to bound demand on a new budget p
e, eI for each consumer ε,
where y10 = d (p 0 , I 0 , ε) and y00 = (I 0
Blundell, Kristensen and Matzkin ()
p10 d (p 0 , I 0 , ε))/p00 .
Multiple Goods
January 2012
8 / 24
Implied Restrictions on Demands
If consumer ε satis…es Revealed Preference then the inequalities:
p
e1 y10
ye1 + p
e0 (y00
ye0 )
eI ) p10 y10
ye1 + p00 (y00
ye0 ) < I 0
allow us to bound demand on a new budget p
e, eI for each consumer ε,
where y10 = d (p 0 , I 0 , ε) and y00 = (I 0 p10 d (p 0 , I 0 , ε))/p00 .
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 satis…es Revealed Preference then the inequalities:
p
e1 y10
ye1 + p
e0 (y00
ye0 )
eI ) p10 y10
ye1 + p00 (y00
ye0 ) < I 0
allow us to bound demand on a new budget p
e, eI for each consumer ε,
where y10 = d (p 0 , I 0 , ε) and y00 = (I 0 p10 d (p 0 , I 0 , ε))/p00 .
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 satis…es Revealed Preference then the inequalities:
p
e1 y10
ye1 + p
e0 (y00
ye0 )
eI ) p10 y10
ye1 + p00 (y00
ye0 ) < I 0
allow us to bound demand on a new budget p
e, eI for each consumer ε,
where y10 = d (p 0 , I 0 , ε) and y00 = (I 0 p10 d (p 0 , I 0 , ε))/p00 .
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 satis…es Revealed Preference then the inequalities:
p
e1 y10
ye1 + p
e0 (y00
ye0 )
eI ) p10 y10
ye1 + p00 (y00
ye0 ) < I 0
allow us to bound demand on a new budget p
e, eI for each consumer ε,
where y10 = d (p 0 , I 0 , ε) and y00 = (I 0 p10 d (p 0 , I 0 , ε))/p00 .
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 satis…es Revealed Preference then the inequalities:
p
e1 y10
ye1 + p
e0 (y00
ye0 )
eI ) p10 y10
ye1 + p00 (y00
ye0 ) < I 0
allow us to bound demand on a new budget p
e, eI for each consumer ε,
where y10 = d (p 0 , I 0 , ε) and y00 = (I 0 p10 d (p 0 , I 0 , ε))/p00 .
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 identi…cation results hold for each consumer
of type [ε1 , ..., εG ] under RP inequality restrictions
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
8 / 24
Results for in…nitessimal changes in prices.
We know
∂d (p, I , ε)
=
∂ (p, I )
"
∂FY j(p,I ) (d (p, I , ε))
∂y
# 1
∂FY j(p,I ) (d (p, I , ε))
∂(p, I )
(Matzkin (1999), Chesher (2003)).
And since each consumer ε satis…es the Integrability Conditions
∂d (p, I , ε)
∂p
Blundell, Kristensen and Matzkin ()
y
∂FY j(p,I ) (y )
∂y
Multiple Goods
! 1
∂FY j(p,I ) (y )
∂I
!
January 2012
9 / 24
Results for in…nitessimal changes in prices.
We know
∂d (p, I , ε)
=
∂ (p, I )
"
∂FY j(p,I ) (d (p, I , ε))
∂y
# 1
∂FY j(p,I ) (d (p, I , ε))
∂(p, I )
(Matzkin (1999), Chesher (2003)).
And since each consumer ε satis…es the Integrability Conditions
∂d (p, I , ε)
∂p
y
∂FY j(p,I ) (y )
∂y
! 1
∂FY j(p,I ) (y )
∂I
!
Which allow us to bound the e¤ect of an in…nitessimal 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 0 , I , ε) for each price regime p 0 .
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 0 , I , ε) for each price regime p 0 .
Show how a valid con…dence 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 0 , I , ε) for each price regime p 0 .
Show how a valid con…dence set can be constructed for the demand bounds
on predicted demands.
I 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 0 , I , ε) for each price regime p 0 .
Show how a valid con…dence set can be constructed for the demand bounds
on predicted demands.
I 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 .
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 0 , I , ε) for each price regime p 0 .
Show how a valid con…dence set can be constructed for the demand bounds
on predicted demands.
I 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 .
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 con…dence 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 y2 , that is not separable from y0 and y1 .
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
11 / 24
Multiple Goods and Conditional Demands
Suppose there is a good y2 , that is not separable from y0 and y1 .
fy0 , y1 , y2 g 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 y2 , that is not separable from y0 and y1 .
fy0 , y1 , y2 g 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 y2 , that is not separable from y0 and y1 .
fy0 , y1 , y2 g 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:
y1 = c1 (p1 , eI , y2 , ε1 )
where eI is the budget allocated to goods 0 and 1, as for d1 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 y2 , that is not separable from y0 and y1 .
fy0 , y1 , y2 g 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:
y1 = c1 (p1 , eI , y2 , ε1 )
where eI is the budget allocated to goods 0 and 1, as for d1 in the two good
case.
The inclusion of y2 in the conditional demand for good 1 represents the
non-separability of y2 from [y1 : y0 ].
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
11 / 24
Multiple Goods and Conditional Demands
Suppose there is a good y2 , that is not separable from y0 and y1 .
fy0 , y1 , y2 g 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:
y1 = c1 (p1 , eI , y2 , ε1 )
where eI is the budget allocated to goods 0 and 1, as for d1 in the two good
case.
The inclusion of y2 in the conditional demand for good 1 represents the
non-separability of y2 from [y1 : y0 ].
I As before we assume ε1 is scalar and c1 is strictly increasing in ε1 .
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
11 / 24
Multiple Goods and Conditional Demands
The exclusion of ε2 from c1 is a strong assumption on preferences.
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
12 / 24
Multiple Goods and Conditional Demands
The exclusion of ε2 from c1 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 ε2 from c1 is a strong assumption on preferences.
In our general framework we weaken these preference restrictions
I although at the cost of strengthening assumptions on the speci…cation of
prices and/or demographics.
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
12 / 24
Multiple Goods and Conditional Demands
The exclusion of ε2 from c1 is a strong assumption on preferences.
In our general framework we weaken these preference restrictions
I although at the cost of strengthening assumptions on the speci…cation of
prices and/or demographics.
Likewise p2 , and ε2 , are exclusive to c2 . So that we have:
y1
y2
Blundell, Kristensen and Matzkin ()
= c1 (p1 , eI , y2 , ε1 )
eI , y , ε )
= c (p , e
2
2
Multiple Goods
1
2
January 2012
12 / 24
Multiple Goods and Conditional Demands
The exclusion of ε2 from c1 is a strong assumption on preferences.
In our general framework we weaken these preference restrictions
I although at the cost of strengthening assumptions on the speci…cation of
prices and/or demographics.
Likewise p2 , and ε2 , are exclusive to c2 . So that we have:
y1
y2
= c1 (p1 , eI , y2 , ε1 )
eI , y , ε )
= c (p , e
2
2
1
2
Notice that the ε1 and ε2 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 ε2 from c1 is a strong assumption on preferences.
In our general framework we weaken these preference restrictions
I although at the cost of strengthening assumptions on the speci…cation of
prices and/or demographics.
Likewise p2 , and ε2 , are exclusive to c2 . So that we have:
y1
y2
= c1 (p1 , eI , y2 , ε1 )
eI , y , ε )
= c (p , e
2
2
1
2
Notice that the ε1 and ε2 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 ε2 from c1 is a strong assumption on preferences.
In our general framework we weaken these preference restrictions
I although at the cost of strengthening assumptions on the speci…cation of
prices and/or demographics.
Likewise p2 , and ε2 , are exclusive to c2 . So that we have:
y1
y2
= c1 (p1 , eI , y2 , ε1 )
eI , y , ε )
= c (p , e
2
2
1
2
Notice that the ε1 and ε2 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:
I Permits estimation by QIV.
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
12 / 24
Multiple Goods and Conditional Demands
The exclusion of ε2 from c1 is a strong assumption on preferences.
In our general framework we weaken these preference restrictions
I although at the cost of strengthening assumptions on the speci…cation of
prices and/or demographics.
Likewise p2 , and ε2 , are exclusive to c2 . So that we have:
y1
y2
= c1 (p1 , eI , y2 , ε1 )
eI , y , ε )
= c (p , e
2
2
1
2
Notice that the ε1 and ε2 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:
I Permits estimation by QIV.
I Implyies that the ranking of goods on the budget line [y0 : y1 ] is invariant
to y2 , (as well as to I and p) even though y2 is non-separable from [y0 : y1 ].
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
12 / 24
Commodity Speci…c Observed Heterogeneity
Mirroring the discussion of ε1 and ε2 , we also introduce exclusive observed
heterogeneity z1 and z2 .
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
13 / 24
Commodity Speci…c Observed Heterogeneity
Mirroring the discussion of ε1 and ε2 , we also introduce exclusive observed
heterogeneity z1 and z2 .
Conditional demands then take the form:
y1
y2
Blundell, Kristensen and Matzkin ()
= c1 (p1 , eI , y2 , z1 , ε1 )
eI , y , z , ε )
= c (p , e
2
2
Multiple Goods
1
2
2
January 2012
13 / 24
Commodity Speci…c Observed Heterogeneity
Mirroring the discussion of ε1 and ε2 , we also introduce exclusive observed
heterogeneity z1 and z2 .
Conditional demands then take the form:
y1
y2
= c1 (p1 , eI , y2 , z1 , ε1 )
eI , y , z , ε )
= c (p , e
2
2
1
2
2
corresponding to standard demands
y1
y2
Blundell, Kristensen and Matzkin ()
= d1 ( p1 , p2 , I , z1 , ε 1 , z2 , ε 2 )
= d2 ( p1 , p2 , I , z1 , ε 1 , z2 , ε 2 )
Multiple Goods
January 2012
13 / 24
Commodity Speci…c Observed Heterogeneity
Mirroring the discussion of ε1 and ε2 , we also introduce exclusive observed
heterogeneity z1 and z2 .
Conditional demands then take the form:
y1
y2
= c1 (p1 , eI , y2 , z1 , ε1 )
eI , y , z , ε )
= c (p , e
2
2
1
2
2
corresponding to standard demands
y1
y2
= d1 ( p1 , p2 , I , z1 , ε 1 , z2 , ε 2 )
= d2 ( p1 , p2 , I , z1 , ε 1 , z2 , ε 2 )
We may also wish to group together the heterogeneity terms in some
restricted way, for example
y1
y2
Blundell, Kristensen and Matzkin ()
= d1 ( p1 , p2 , I , z1 + ε 1 , z2 + ε 2 )
= d2 ( p1 , p2 , I , z1 + ε 1 , z2 + ε 2 ) .
Multiple Goods
January 2012
13 / 24
Commodity Speci…c Observed Heterogeneity
Mirroring the discussion of ε1 and ε2 , we also introduce exclusive observed
heterogeneity z1 and z2 .
Conditional demands then take the form:
y1
y2
= c1 (p1 , eI , y2 , z1 , ε1 )
eI , y , z , ε )
= c (p , e
2
2
1
2
2
corresponding to standard demands
y1
y2
= d1 ( p1 , p2 , I , z1 , ε 1 , z2 , ε 2 )
= d2 ( p1 , p2 , I , z1 , ε 1 , z2 , ε 2 )
We may also wish to group together the heterogeneity terms in some
restricted way, for example
y1
y2
= d1 ( p1 , p2 , I , z1 + ε 1 , z2 + ε 2 )
= d2 ( p1 , p2 , I , z1 + ε 1 , z2 + ε 2 ) .
These restricted speci…cations will be important in our discussion of
identi…cation and estimation
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
13 / 24
Triangular Demands
Suppose preferences are such that [y1 , y0 ] form a separable sub-group within
[y1 , y0 , y2 ]. In this case, utility has the recursive form
U (y0 , y1 , y2 , z1 , z2 , ε1 , ε2 ) = V (u (y0 , y1 , z1 , ε1 ), y2 , z2 , ε2 )
so that the MRS between goods y1 and y0 does not depend on y2 .
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
14 / 24
Triangular Demands
Suppose preferences are such that [y1 , y0 ] form a separable sub-group within
[y1 , y0 , y2 ]. In this case, utility has the recursive form
U (y0 , y1 , y2 , z1 , z2 , ε1 , ε2 ) = V (u (y0 , y1 , z1 , ε1 ), y2 , z2 , ε2 )
so that the MRS between goods y1 and y0 does not depend on y2 .
I Note however that the MRS for y2 and y0 does depend on y1 .
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
14 / 24
Triangular Demands
Suppose preferences are such that [y1 , y0 ] form a separable sub-group within
[y1 , y0 , y2 ]. In this case, utility has the recursive form
U (y0 , y1 , y2 , z1 , z2 , ε1 , ε2 ) = V (u (y0 , y1 , z1 , ε1 ), y2 , z2 , ε2 )
so that the MRS between goods y1 and y0 does not depend on y2 .
I Note however that the MRS for y2 and y0 does depend on y1 .
The conditional demands then take the triangular form:
y1
y2
Blundell, Kristensen and Matzkin ()
= c1 (p1 , eI , z1 , ε1 )
eI , y1 , z2 , ε2 )
= c 2 ( p2 , e
Multiple Goods
January 2012
14 / 24
Triangular Demands
Suppose preferences are such that [y1 , y0 ] form a separable sub-group within
[y1 , y0 , y2 ]. In this case, utility has the recursive form
U (y0 , y1 , y2 , z1 , z2 , ε1 , ε2 ) = V (u (y0 , y1 , z1 , ε1 ), y2 , z2 , ε2 )
so that the MRS between goods y1 and y0 does not depend on y2 .
I Note however that the MRS for y2 and y0 does depend on y1 .
The conditional demands then take the triangular form:
y1
y2
= c1 (p1 , eI , z1 , ε1 )
eI , y1 , z2 , ε2 )
= c 2 ( p2 , e
I Can relax preference assumptions to allow ε1 to enter c2 .
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
14 / 24
Triangular Demands
Suppose preferences are such that [y1 , y0 ] form a separable sub-group within
[y1 , y0 , y2 ]. In this case, utility has the recursive form
U (y0 , y1 , y2 , z1 , z2 , ε1 , ε2 ) = V (u (y0 , y1 , z1 , ε1 ), y2 , z2 , ε2 )
so that the MRS between goods y1 and y0 does not depend on y2 .
I Note however that the MRS for y2 and y0 does depend on y1 .
The conditional demands then take the triangular form:
y1
y2
= c1 (p1 , eI , z1 , ε1 )
eI , y1 , z2 , ε2 )
= c 2 ( p2 , e
I Can relax preference assumptions to allow ε1 to enter c2 .
z1 (and p1 ) is excluded from c2 and could act an instrument for y1 in the
QCF estimation of c2 , 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 (ε1 , ε2 , y2 ) + W (ε1 , y1 , y2 ) + y0
e.g.
= (ε1 + ε2 ) u (y2 ) + ε1 log (y1
Blundell, Kristensen and Matzkin ()
Multiple Goods
u (y2 )) + y0
January 2012
15 / 24
The general G+1>2 case
If demand functions are invertible in (ε1 , ..., εG ) , we can write (ε1 , ..., εG ) as
ε1
= r1 (y1 , ..., yG , p1 , ..., pG , I , z1 , ...zG )
εG
= rG (y1 , ..., yG , p1 , ..., pG , I , z1 , ...zG )
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
16 / 24
The general G+1>2 case
If demand functions are invertible in (ε1 , ..., εG ) , we can write (ε1 , ..., εG ) as
ε1
= r1 (y1 , ..., yG , p1 , ..., pG , I , z1 , ...zG )
εG
= rG (y1 , ..., yG , p1 , ..., pG , I , z1 , ...zG )
Can use the transformation of variables equation to determine identi…cation
(Matzkin (2010))
fY jp,I ,z (y ) = f ε (r (y , p, I , z ))
Blundell, Kristensen and Matzkin ()
Multiple Goods
∂r (y , p, I , z )
∂y
January 2012
16 / 24
The general G+1>2 case
If demand functions are invertible in (ε1 , ..., εG ) , we can write (ε1 , ..., εG ) as
ε1
= r1 (y1 , ..., yG , p1 , ..., pG , I , z1 , ...zG )
εG
= rG (y1 , ..., yG , p1 , ..., pG , I , z1 , ...zG )
Can use the transformation of variables equation to determine identi…cation
(Matzkin (2010))
fY jp,I ,z (y ) = f ε (r (y , p, I , z ))
∂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 speci…c characteristics and
discrete prices.
U (y , I
p 0 y ) + V (y , z + ε) and fε primitive functions
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
17 / 24
An example of commodity speci…c characteristics and
discrete prices.
U (y , I p 0 y ) + V (y , z + ε) and fε primitive functions
Demands given by
arg max fU (y , y0 ) + V (y , z + ε)
y
Blundell, Kristensen and Matzkin ()
Multiple Goods
j p 0 y + y0
Ig
January 2012
17 / 24
An example of commodity speci…c characteristics and
discrete prices.
U (y , I p 0 y ) + V (y , z + ε) and fε primitive functions
Demands given by
arg max fU (y , y0 ) + V (y , z + ε)
y
Assume
0
@
j p 0 y + y0
V1,G +1
V1,G +2
V1,G +G
VG ,G +1
VG ,G +2
VG ,G +G
Ig
1
A
is a P-matrix (e.g. positive semi-de…nite or with dominant diagonal) ...
examples..
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
17 / 24
An example of commodity speci…c characteristics and
discrete prices.
U (y , I p 0 y ) + V (y , z + ε) and fε primitive functions
Demands given by
arg max fU (y , y0 ) + V (y , z + ε)
y
Assume
0
@
j p 0 y + y0
V1,G +1
V1,G +2
V1,G +G
VG ,G +1
VG ,G +2
VG ,G +G
Ig
1
A
is a P-matrix (e.g. positive semi-de…nite or with dominant diagonal) ...
examples..
Then, by Gale and Nikaido (1965), the system is invertible: There exist
functions r 1 , ..., r G such that
Blundell, Kristensen and Matzkin ()
ε 1 + z1
= r 1 (y1 , ..., yG , p1 , ..., pK , I )
ε G + zG
= r G (y1 , ..., yG , p1 , ..., pK , I )
Multiple Goods
January 2012
17 / 24
Identi…cation
Constructive identi…cation follows as in Matzkin (2007). Assume
∂fε (ε)
=0
∂ε
Blundell, Kristensen and Matzkin ()
<=>
Multiple Goods
ε=ε
January 2012
18 / 24
Identi…cation
Constructive identi…cation follows as in Matzkin (2007). Assume
∂fε (ε)
=0
∂ε
<=>
ε=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
Identi…cation
Constructive identi…cation follows as in Matzkin (2007). Assume
∂fε (ε)
=0
∂ε
<=>
ε=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
fY jp,I ,z (y ) = fε (r (y , p, I )
Blundell, Kristensen and Matzkin ()
Multiple Goods
z)
∂r (y , p, I )
∂y
January 2012
18 / 24
Identi…cation
Constructive identi…cation follows as in Matzkin (2007). Assume
∂fε (ε)
=0
∂ε
<=>
ε=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
fY jp,I ,z (y ) = fε (r (y , p, I )
z)
∂r (y , p, I )
∂y
Taking derivatives with respect to z
∂fY jp,I ,z (y )
∂z
Blundell, Kristensen and Matzkin ()
=
∂fε (r (y , p, I )
∂ε
Multiple Goods
z)
∂r (y , p, I )
∂y
January 2012
18 / 24
Identi…cation
In
∂fY jp,I ,z (y )
∂z
Blundell, Kristensen and Matzkin ()
=
∂fε (r (y , p, I )
∂ε
Multiple Goods
z)
∂r (y , p, I )
∂y
January 2012
19 / 24
Identi…cation
In
∂fY jp,I ,z (y )
∂z
Note that
∂fY jp,I ,z (y )
∂z
Blundell, Kristensen and Matzkin ()
=
∂fε (r (y , p, I )
∂ε
=0 )
z)
∂r (y , p, I )
∂y
∂fε (r (y , p, I )
∂ε
Multiple Goods
z)
=0
January 2012
19 / 24
Identi…cation
In
∂fY jp,I ,z (y )
∂z
Note that
∂fY jp,I ,z (y )
∂z
and
∂fε (r (y , p, I )
∂ε
Blundell, Kristensen and Matzkin ()
=
∂fε (r (y , p, I )
∂ε
=0 )
z)
=0
z)
∂r (y , p, I )
∂y
∂fε (r (y , p, I )
∂ε
) r (y , p, I )
Multiple Goods
z)
=0
z=ε
January 2012
19 / 24
Identi…cation
Fix y , p, I . Find z such that
∂fY jp,I ,z (y )
∂z
Blundell, Kristensen and Matzkin ()
Multiple Goods
=0
January 2012
20 / 24
Identi…cation
Fix y , p, I . Find z such that
∂fY jp,I ,z (y )
∂z
=0
Then,
r (y , p, I ) = ε + z
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
20 / 24
Identi…cation
Fix y , p, I . Find z such that
∂fY jp,I ,z (y )
∂z
=0
Then,
r (y , p, I ) = ε + z
We have then constructive identi…cation of the function r .
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
20 / 24
Identi…cation
Fix y , p, I . Find z such that
∂fY jp,I ,z (y )
∂z
=0
Then,
r (y , p, I ) = ε + z
We have then constructive identi…cation of the function r .
Identi…cation of r ) identi…cation of h
∂fY jp,I ,z (y )
∂z
Blundell, Kristensen and Matzkin ()
=0
)
Multiple Goods
y = h (p, I , ε + z )
January 2012
20 / 24
Average derivative estimator
\
∂r (y )
bZZ (y )
=bry (y ) = T
∂y
Blundell, Kristensen and Matzkin ()
Multiple Goods
1
bZY (y )
T
January 2012
21 / 24
Average derivative estimator
\
∂r (y )
bZZ (y )
=bry (y ) = T
∂y
1
bZY (y )
T
bZZ and T
bZY are average derivative type estimators
Elements of T
!
Z ∂ log b
fy jz (y ) ∂ log b
fy jz (y )
b
Tyj zk (y ) =
ω (z )dz
∂yj
∂zk
! Z
!
Z ∂ log b
fy jz (y )
∂ log b
fy jz (y )
ω (z )dz
ω (z )dz
∂yj
∂zk
Powell, Stock, and Stoker (1989), Newey (1994)
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
21 / 24
Average derivative estimator
\
∂r (y )
bZZ (y )
=bry (y ) = T
∂y
1
bZY (y )
T
bZZ and T
bZY are average derivative type estimators
Elements of T
!
Z ∂ log b
fy jz (y ) ∂ log b
fy jz (y )
b
Tyj zk (y ) =
ω (z )dz
∂yj
∂zk
! Z
!
Z ∂ log b
fy jz (y )
∂ log b
fy jz (y )
ω (z )dz
ω (z )dz
∂yj
∂zk
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 speci…c observed heterogeneity
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
22 / 24
Empirical example for the multiple good case
Three good model with commodity speci…c observed heterogeneity
I 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 speci…c observed heterogeneity
I 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 speci…c observed heterogeneity
I 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 speci…c observed heterogeneity
I 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 speci…c observed heterogeneity
I 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.
I Figure 5....
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
22 / 24
Conclusions
I Show conditions for identi…cation 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
I Show conditions for identi…cation and estimation of individual demands in
the two good and the multiple good case with nonadditive/nonseparable
heterogeneity.
I Focus on the case of discrete prices (…nite markets) and many
heterogeneous consumers.
Blundell, Kristensen and Matzkin ()
Multiple Goods
January 2012
23 / 24
Conclusions
I Show conditions for identi…cation and estimation of individual demands in
the two good and the multiple good case with nonadditive/nonseparable
heterogeneity.
I Focus on the case of discrete prices (…nite markets) and many
heterogeneous consumers.
I 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’
y1
d(I,ε)
y2
Figure 1a: The distribution of demands across consumers indexed by ‘ε’
y1
d(I,ε)
y2
Figure 1b: Monotonicity in ‘ε’ and rank preserving on the budget constraint
y1
d(I
( ,,ε))
relative price change
y2
Figure 1c: The quantile expansion path
y1
d(I ε)
d(I,ε)
increase in total budget
y2
Figure 2a: Generating a Support Set with RP for consumer ‘ε’
y1
d1(I1,ε))
d2(I2,ε)
y2
Figure 2a: Generating a Support Set with RP for consumer ‘ε’
y1
new budget line at prices
p0 and income x0
d1
d2
y2
Figure 2a: Generating a Support Set with RP for consumer ‘ε’
y1
d1
S0(I0,ε)
Support set for demands d0(I0,ε))
for consumer ε consistent
with RP at prices p0 and income x0
d2
y2
Figure 2d. Improving the support set with e-bounds, for consumer ‘ε’
(p1 , I1 )
Se (p0 , I0 )
d1
d2
(p0 , I0 )
(p2 , I2 )
Figure 2e: The best support set with many price regimes
(p1 , I1 )
(p0 , I0 )
d1 (II,ε)
d1 (~I1 )
S (p0 , I0 )
d3 (I,ε)
d 2 (I,ε)
d3 (~I3 )
d2 (~I2 )
(p3 , I3 )
(p2 , I2 )
Figure 3a. Unrestrcited Quantile Expansion Paths: Food, 1986
4.2
τ = 0.1
τ = 0.5
4
τ = 0.9
09
95% CIs
3.8
ood exp.
log-fo
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
3.8
4
4.2
4.4
4.6
log-total
log
total exp
exp.
4.8
5
5.2
Blundell, Matzkin and Kristensen (2011)
Figure 3b. RP- Restrcited Quantile Expansion Paths: Food, 1986
4.2
τ = 0.1
4
τ = 0.5
05
τ = 0.9
95% CIs
3.8
log--food exp.
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
3.8
4
4.2
4.4
4.6
log-total exp.
4.8
5
5.2
Blundell, Matzkin and Kristensen (2011)
Figure 4a: Quantile (RP-Restricted) Bounds on Demand (Median Income, τ=.5 )
100
estimate
95% confidence interval
90
80
d emand, food
70
60
50
40
30
20
10
0
0.92
0.94
0.96
0.98
1
1.02
price, food
Blundell, Matzkin and Kristensen (2011)
Figure 4b: Quantile (RP-Restricted) Confidence Sets (Median Income, τ=.1 )
100
T=4
T=6
T=8
90
80
de
emand, food
70
60
50
40
30
20
10
0
0.92
0.94
0.96
0.98
1
1.02
price, food
Blundell, Matzkin and Kristensen (2011)
Figure 4c: Quantile (RP-Restricted) Confidence Sets (Median Income, τ=.5 )
100
T=4
T=6
T=8
90
80
d, food
demand
70
60
50
40
30
20
10
0
0.92
0.94
0.96
0.98
1
1.02
price food
price,
Blundell, Matzkin and Kristensen (2011)
Figure 4d: Quantile (RP-Restricted) Confidence Sets (Median Income, τ=.9 )
100
T=4
T=6
T=8
90
80
dema nd, food
70
60
50
40
30
20
10
0
0.92
0.94
0.96
0.98
1
1.02
price, food
Blundell, Matzkin and Kristensen (2011)
Figure 4e: Quantile (RP-Restricted) Confidence Sets (25% Income, τ=.5 )
100
T=4
T=6
T=8
90
80
dema
and, food
70
60
50
40
30
20
10
0
0.92
0.94
0.96
0.98
1
1.02
price food
price,
Blundell, Matzkin and Kristensen (2011)
Figure 4f: Quantile (RP-Restricted) Confidence Sets (75% Income, τ=.5 )
100
T=4
T=6
T=8
90
80
deman d, food
70
60
50
40
30
20
10
0
0.92
0.94
0.96
0.98
1
1.02
price food
price,
Blundell, Matzkin and Kristensen (2011)
Figure
Figure 5.
4a.Relative
Relativeprice
pricedata:
data:1975
1975toto1999
1999and
andprice
pricepath
path
1.2
1999
1.15
1998
1994
1993
1.1
1997
1995
19921996
1991
1990
1.05
1
1986
1989
1988
1983
1987
1985
1984
1982
1981
0.95
0.9
1980
1976
0.85
1979
1978
0.8
0.75
0 92
0.92
1977
1975
0 94
0.94
0 96
0.96
0 98
0.98
1
1 02
1.02
Price of food
relative to nondurables
1 04
1.04
1 06
1.06
1 08
1.08
Figure 4a: Typical Joint Distribution of log food and log income
4
3.8
0.2
density
median
0.2
0.4
0.
6
3.6
0.6
0.4
1
0.8
0.
8
32
3.2
0.
4
0.2
0.2
1.2
3
1
0.6
0.8
0.6
2.8
0.4
od exp.
log-foo
3.4
0.4
0.2
2.6
0.2
2.4
3.4
3.6
3.8
4
4.2
4.4
log-total exp.
4.6
4.8
5
5.2
Blundell, Matzkin and Kristensen (2011)