Multiple Goods, Consumer Heterogeneity and Revealed Preference - - PowerPoint PPT Presentation

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" Focus here is on identification and estimation when there are many heterogeneous consumers, a finite 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 Maxy U

• y1, ..., yG , I −

G

∑

g =1

pg yg , z, ε1, ..., εG

• 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 Maxy U

• y1, ..., yG , I −

G

∑

g =1

pg yg , z, ε1, ..., εG

• Typically dealing with a finite 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

= pg for g = 1, ..., G

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.

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

• p1, ...,

pG , 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

• p1, ...,

pG , I;

also derive results on bounds for infinitessimal 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

• p1, ...,

pG , I;

also derive results on bounds for infinitessimal 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

• p1, ...,

pG , I;

also derive results on bounds for infinitessimal 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 specification 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 ) identifies 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 ) identifies 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

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 ) identifies 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 differentiable, strictly increasing and strictly concave, and that ∂2w(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 ) identifies 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 differentiable, strictly increasing and strictly concave, and that ∂2w(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 first order conditions exists and satisfies for all p, I, ε, ∂d (p, I, ε) ∂ε = − w10 (y1, ε) v11 (y1, I − py1) − 2 v10 (y1, I − py1) p + v00 (y1, I − py1) p2 + w11 (y > 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 first order conditions exists and satisfies for all p, I, ε, ∂d (p, I, ε) ∂ε = − w10 (y1, ε) v11 (y1, I − py1) − 2 v10 (y1, I − py1) p + v00 (y1, I − py1) p2 + w11 (y > 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

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, Fε (ε) = FY |p,I (d (p, I, ε)) where Fε is the cumulative distribution of ε and FY |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, Fε (ε) = FY |p,I (d (p, I, ε)) where Fε is the cumulative distribution of ε and FY |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 Fε is strictly increasing at ε, d

• p, I , ε

− d

• p,

I, ε

• = F −1

Y |(p,I )=(p,I )

• FY |(p,I )=(

p, I) (y1)

• − y1

where y1 is the observed consumption when budget is

• p,

I

• .

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 7 / 24

Implied Restrictions on Demands

If consumer ε satisfies Revealed Preference then the inequalities:

• p1
• y

1 −

y1 + p0(y

0 −

y0) ≤ I ⇒ p

1

• y

1 −

y1 + p

0(y 0 −

y0) < I allow us to bound demand on a new budget

• p,

I

• for each consumer ε,

where y

1 = d (p, I , ε) and y 0 = (I − p 1d (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:

• p1
• y

1 −

y1 + p0(y

0 −

y0) ≤ I ⇒ p

1

• y

1 −

y1 + p

0(y 0 −

y0) < I allow us to bound demand on a new budget

• p,

I

• for each consumer ε,

where y

1 = d (p, I , ε) and y 0 = (I − p 1d (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:

• p1
• y

1 −

y1 + p0(y

0 −

y0) ≤ I ⇒ p

1

• y

1 −

y1 + p

0(y 0 −

y0) < I allow us to bound demand on a new budget

• p,

I

• for each consumer ε,

where y

1 = d (p, I , ε) and y 0 = (I − p 1d (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:

• p1
• y

1 −

y1 + p0(y

0 −

y0) ≤ I ⇒ p

1

• y

1 −

y1 + p

0(y 0 −

y0) < I allow us to bound demand on a new budget

• p,

I

• for each consumer ε,

where y

1 = d (p, I , ε) and y 0 = (I − p 1d (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:

• p1
• y

1 −

y1 + p0(y

0 −

y0) ≤ I ⇒ p

1

• y

1 −

y1 + p

0(y 0 −

y0) < I allow us to bound demand on a new budget

• p,

I

• for each consumer ε,

where y

1 = d (p, I , ε) and y 0 = (I − p 1d (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:

• p1
• y

1 −

y1 + p0(y

0 −

y0) ≤ I ⇒ p

1

• y

1 −

y1 + p

0(y 0 −

y0) < I allow us to bound demand on a new budget

• p,

I

• for each consumer ε,

where y

1 = d (p, I , ε) and y 0 = (I − p 1d (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

• f type [ε1, ..., εG ] under RP inequality restrictions

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 8 / 24

Results for infinitessimal changes in prices.

We know ∂d (p, I, ε) ∂ (p, I) = −

• ∂FY |(p,I ) (d (p, I, ε))

∂y −1 ∂FY |(p,I ) (d (p, I, ε)) ∂(p, I) (Matzkin (1999), Chesher (2003)). And since each consumer ε satisfies the Integrability Conditions ∂d(p, I, ε) ∂p ≤ − y

• ∂FY |(p,I )(y)

∂y −1 ∂FY |(p,I )(y) ∂I

• Blundell, Kristensen and Matzkin ()

Multiple Goods January 2012 9 / 24

Results for infinitessimal changes in prices.

We know ∂d (p, I, ε) ∂ (p, I) = −

• ∂FY |(p,I ) (d (p, I, ε))

∂y −1 ∂FY |(p,I ) (d (p, I, ε)) ∂(p, I) (Matzkin (1999), Chesher (2003)). And since each consumer ε satisfies the Integrability Conditions ∂d(p, I, ε) ∂p ≤ − y

• ∂FY |(p,I )(y)

∂y −1 ∂FY |(p,I )(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

• n 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

• n 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

• n 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

• n 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 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. {y0, y1, y2} 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. {y0, y1, y2} 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. {y0, y1, y2} 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, I, y2, ε1) where I 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. {y0, y1, y2} 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, I, y2, ε1) where I 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. {y0, y1, y2} 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, I, y2, ε1) where I 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]. 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

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 ε2 from c1 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 p2, and ε2, are exclusive to c2. So that we have: y1 = c1(p1, I, y2, ε1) y2 = c2(p2,

• I, y1, ε2)

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

although at the cost of strengthening assumptions on the specification of prices and/or demographics.

Likewise p2, and ε2, are exclusive to c2. So that we have: y1 = c1(p1, I, y2, ε1) y2 = c2(p2,

• I, y1, ε2)

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

although at the cost of strengthening assumptions on the specification of prices and/or demographics.

Likewise p2, and ε2, are exclusive to c2. So that we have: y1 = c1(p1, I, y2, ε1) y2 = c2(p2,

• I, y1, ε2)

Notice that the ε1 and ε2 naturally append to goods 1 and 2 and are increasing in the conditional demands for each good respectively. Extends 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

although at the cost of strengthening assumptions on the specification of prices and/or demographics.

Likewise p2, and ε2, are exclusive to c2. So that we have: y1 = c1(p1, I, y2, ε1) y2 = c2(p2,

• I, y1, ε2)

Notice that the ε1 and ε2 naturally append to goods 1 and 2 and are increasing in the conditional demands for each good respectively. Extends 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 ε2 from c1 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 p2, and ε2, are exclusive to c2. So that we have: y1 = c1(p1, I, y2, ε1) y2 = c2(p2,

• I, y1, ε2)

Notice that the ε1 and ε2 naturally append to goods 1 and 2 and are increasing in the conditional demands for each good respectively. Extends the monotonicity result to conditional demands: Permits estimation by QIV. 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 Specific 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 Specific 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 = c1(p1, I, y2, z1, ε1) y2 = c2(p2,

• I, y1, z2, ε2)

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 13 / 24

Commodity Specific 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 = c1(p1, I, y2, z1, ε1) y2 = c2(p2,

• I, y1, z2, ε2)

corresponding to standard demands y1 = d1 (p1, p2, I, z1, ε1, z2, ε2) y2 = d2 (p1, p2, I, z1, ε1, z2, ε2)

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 13 / 24

Commodity Specific 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 = c1(p1, I, y2, z1, ε1) y2 = c2(p2,

• I, y1, z2, ε2)

corresponding to standard demands y1 = d1 (p1, p2, I, z1, ε1, z2, ε2) y2 = d2 (p1, p2, I, z1, ε1, z2, ε2) We may also wish to group together the heterogeneity terms in some restricted way, for example y1 = d1 (p1, p2, I, z1 + ε1, z2 + ε2) y2 = d2 (p1, p2, I, z1 + ε1, z2 + ε2) .

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 13 / 24

Commodity Specific 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 = c1(p1, I, y2, z1, ε1) y2 = c2(p2,

• I, y1, z2, ε2)

corresponding to standard demands y1 = d1 (p1, p2, I, z1, ε1, z2, ε2) y2 = d2 (p1, p2, I, z1, ε1, z2, ε2) We may also wish to group together the heterogeneity terms in some restricted way, for example y1 = d1 (p1, p2, I, z1 + ε1, z2 + ε2) y2 = d2 (p1, p2, I, z1 + ε1, z2 + ε2) . 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 [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. 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. Note however that the MRS for y2 and y0 does depend on y1. The conditional demands then take the triangular form: y1 = c1(p1, I, z1, ε1) y2 = c2(p2,

• I, y1, z2, ε2)

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. Note however that the MRS for y2 and y0 does depend on y1. The conditional demands then take the triangular form: y1 = c1(p1, I, z1, ε1) y2 = c2(p2,

• I, y1, z2, ε2)

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. Note however that the MRS for y2 and y0 does depend on y1. The conditional demands then take the triangular form: y1 = c1(p1, I, z1, ε1) y2 = c2(p2,

• I, y1, z2, ε2)

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 − u (y2)) + y0

Blundell, Kristensen and Matzkin () Multiple Goods 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 identification (Matzkin (2010)) fY |p,I,z(y) = f ε (r (y, p, I, z))

• ∂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 (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 identification (Matzkin (2010)) fY |p,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 specific characteristics and discrete prices.

U(y, I − py) + V (y, z + ε) and fε primitive functions

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 17 / 24

An example of commodity specific characteristics and discrete prices.

U(y, I − py) + V (y, z + ε) and fε primitive functions Demands given by arg max

y

{U(y, y0) + V (y, z + ε) | py + y0 ≤ I}

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 17 / 24

An example of commodity specific characteristics and discrete prices.

U(y, I − py) + V (y, z + ε) and fε primitive functions Demands given by arg max

y

{U(y, y0) + V (y, z + ε) | py + y0 ≤ I} Assume   V1,G +1 V1,G +2 · · V1,G +G · · VG ,G +1 VG ,G +2 VG ,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 − py) + V (y, z + ε) and fε primitive functions Demands given by arg max

y

{U(y, y0) + V (y, z + ε) | py + y0 ≤ I} Assume   V1,G +1 V1,G +2 · · V1,G +G · · VG ,G +1 VG ,G +2 VG ,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 r1, ..., rG such that ε1 + z1 = r1 (y1, ..., yG , p1, ..., pK , I) · · · εG + zG = rG (y1, ..., yG , p1, ..., pK , I)

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 17 / 24

Identification

Constructive identification follows as in Matzkin (2007). Assume ∂fε(ε) ∂ε = 0 <=> ε = ε∗

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 18 / 24

Identification

Constructive identification follows as in Matzkin (2007). Assume ∂fε(ε) ∂ε = 0 <=> 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

Identification

Constructive identification follows as in Matzkin (2007). Assume ∂fε(ε) ∂ε = 0 <=> 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 |p,I,z(y) = fε (r(y, p, I) − z)

• ∂r(y, p, I)

∂y

• Blundell, Kristensen and Matzkin ()

Multiple Goods January 2012 18 / 24

Identification

Constructive identification follows as in Matzkin (2007). Assume ∂fε(ε) ∂ε = 0 <=> 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 |p,I,z(y) = fε (r(y, p, I) − z)

• ∂r(y, p, I)

∂y

• Taking derivatives with respect to z

∂fY |p,I,z(y) ∂z = ∂fε (r(y, p, I) − z) ∂ε

• ∂r(y, p, I)

∂y

• Blundell, Kristensen and Matzkin ()

Multiple Goods January 2012 18 / 24

Identification

In ∂fY |p,I,z(y) ∂z = ∂fε (r(y, p, I) − z) ∂ε

• ∂r(y, p, I)

∂y

• Blundell, Kristensen and Matzkin ()

Multiple Goods January 2012 19 / 24

Identification

In ∂fY |p,I,z(y) ∂z = ∂fε (r(y, p, I) − z) ∂ε

• ∂r(y, p, I)

∂y

• Note that

∂fY |p,I,z(y) ∂z = 0 ⇒ ∂fε (r(y, p, I) − z) ∂ε = 0

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 19 / 24

Identification

In ∂fY |p,I,z(y) ∂z = ∂fε (r(y, p, I) − z) ∂ε

• ∂r(y, p, I)

∂y

• Note that

∂fY |p,I,z(y) ∂z = 0 ⇒ ∂fε (r(y, p, I) − z) ∂ε = 0 and ∂fε (r(y, p, I) − z) ∂ε = 0 ⇒ r(y, p, I) − z = ε∗

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 19 / 24

Identification

Fix y, p, I. Find z∗ such that ∂fY |p,I,z ∗(y) ∂z = 0

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 20 / 24

Identification

Fix y, p, I. Find z∗ such that ∂fY |p,I,z ∗(y) ∂z = 0 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 ∂fY |p,I,z ∗(y) ∂z = 0 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 ∂fY |p,I,z ∗(y) ∂z = 0 Then, r(y, p, I) = ε∗ + z∗ We have then constructive identification of the function r. Identification of r ⇒ identification of h ∂fY |p,I,z ∗(y) ∂z = 0 ⇒ y = h (p, I, ε∗ + z∗)

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 20 / 24

Average derivative estimator

• ∂r(y)

∂y = ry (y) =

• TZZ (y)

−1 TZY (y)

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 21 / 24

Average derivative estimator

• ∂r(y)

∂y = ry (y) =

• TZZ (y)

−1 TZY (y) Elements of TZZ and TZY are average derivative type estimators

• Tyjzk (y)

= ∂ log fy |z (y) ∂yj ∂ log fy |z (y) ∂zk ω(z)dz

• −

∂ log fy |z (y) ∂yj ω(z)dz ∂ log fy |z (y) ∂zk ω(z)dz

• Powell, Stock, and Stoker (1989), Newey (1994)

Blundell, Kristensen and Matzkin () Multiple Goods January 2012 21 / 24

Average derivative estimator

• ∂r(y)

∂y = ry (y) =

• TZZ (y)

−1 TZY (y) Elements of TZZ and TZY are average derivative type estimators

• Tyjzk (y)

= ∂ log fy |z (y) ∂yj ∂ log fy |z (y) ∂zk ω(z)dz

• −

∂ log fy |z (y) ∂yj ω(z)dz ∂ log fy |z (y) ∂zk ω(z)dz

• 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

• bserved 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

• bserved 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

• bserved 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

• bserved 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’ 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 y

Figure 1c: The quantile expansion path y1 d(I ε) d(I,ε) increase in total budget y2 y

Figure 2a: Generating a Support Set with RP for consumer ‘ε’ y1 d (I ) d1(I1,ε) d2(I2,ε) y2

Figure 2a: Generating a Support Set with RP for consumer ‘ε’ y1 d new budget line at prices d1 new budget line at prices p0 and income x0 d2 y2

Figure 2a: Generating a Support Set with RP for consumer ‘ε’ y1 d Support set for demands d (I ) 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’

( )

1 1,I

p

( )

0 , I

Se p

1

d

2

d

( )

0,I

p

( )

2 2,I

p

Figure 2e: The best support set with many price regimes

( )

I d

( )

1 1 I

p

( )

I,ε

1

d

( )

1 1, I

p

( )

~ I d

( )

I,ε

3

d

( )

1 1 ~

I d

( )

I S p

( )

0 , I

p

(I,ε)

2

d

( )

0 , I

S p ( )

3 3 ~

I d

( )

2 2 ~

I d

( )

I p

( )

I p

( )

3 3,I

p

( )

2 2,I

p

Figure 3a. Unrestrcited Quantile Expansion Paths: Food, 1986

4 4.2 τ = 0.1 τ = 0.5 0 9 3.6 3.8 τ = 0.9 95% CIs 3.2 3.4

• od exp.

2.8 3 log-fo 2.4 2.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 2.2 log-total exp

Blundell, Matzkin and Kristensen (2011)

log total exp.

Figure 3b. RP- Restrcited Quantile Expansion Paths: Food, 1986

4 4.2 τ = 0.1 τ = 0 5 3.6 3.8 τ 0.5 τ = 0.9 95% CIs 3.2 3.4

• food exp.

2.8 3 log- 2.4 2.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 2.2 log-total exp.

Blundell, Matzkin and Kristensen (2011)

Figure 4a: Quantile (RP-Restricted) Bounds on Demand (Median Income, τ=.5 )

90 100 estimate 95% confidence interval 70 80 50 60 emand, food 30 40 d 10 20 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 )

90 100 T = 4 T = 6 T = 8 70 80 50 60 emand, food 30 40 de 10 20 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 )

90 100 T = 4 T = 6 T = 8 70 80 T 8 60 70 d, food 40 50 demand 20 30 0.92 0.94 0.96 0.98 1 1.02 10 price food

Blundell, Matzkin and Kristensen (2011)

price, food

Figure 4d: Quantile (RP-Restricted) Confidence Sets (Median Income, τ=.9 )

90 100 T = 4 T = 6 T = 8 70 80 50 60 n d , f

• d

30 40 d e m a 10 20 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 )

90 100 T = 4 T = 6 T = 8 70 80 50 60 a n d , f

• d

30 40 d e m a 20 30 0.92 0.94 0.96 0.98 1 1.02 10 price food

Blundell, Matzkin and Kristensen (2011)

price, food

Figure 4f: Quantile (RP-Restricted) Confidence Sets (75% Income, τ=.5 )

90 100 T = 4 T = 6 T = 8 70 80 50 60 d , f

• d

40 50 d e m a n 20 30 0.92 0.94 0.96 0.98 1 1.02 10 price food

Blundell, Matzkin and Kristensen (2011)

price, food

Figure 4a. Relative price data: 1975 to 1999 and price path

1.2

Figure 5. Relative price data: 1975 to 1999 and price path

1.15 1.2 1997 1998 1999 1.05 1.1 1990 1991 1992 1993 1994 1995 1996 1 1982 1983 1984 1985 1986 1987 1988 1989 0.9 0.95 1980 1981 0.85 1976 1977 1978 1979 1980 0 92 0 94 0 96 0 98 1 1 02 1 04 1 06 1 08 0.75 0.8 1975 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08

Price of food relative to nondurables

Figure 4a: Typical Joint Distribution of log food and log income

4 3.8 4 . 2 . 2 0 4 density median 3.6 0.4 . 6 3 2 3.4 . 2 0.4 0.4 0.6 . 8 0 8 1

• d exp.

3 3.2 0.2 6 0.8 1 1.2 log-foo 2.8 0.4 0.4 0.6 0.6 0.8 2.4 2.6 0.2 . 2

Blundell, Matzkin and Kristensen (2011)

3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 log-total exp.