Stochastic Demand and Revealed Preference, November 2010

Stochastic Demand and Revealed Preference

Richard Blundell Dennis Kristensen Rosa Matzkin

UCL & IFS, Columbia and UCLA

November 2010

Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 1 / 37

Introduction

This presentation investigates the role of restrictions from economictheory in the microeconometric estimation of nonparametric modelsof consumer behaviour.

Objective is to uncover demand responses from consumer expendituresurvey data.

Inequality restrictions from revealed preference are used to improvethe performance of nonparametric estimates of demand responses.

Particular attention is given to nonseparable unobserved heterogeneityand endogeneity.

Introduction

New insights are provided about the price responsiveness of demand

especially across di¤erent income groups and across unobservedheterogeneity.

Derive welfare costs of relative price and tax changes.

Introduction

The Problem

Assume every consumer is characterised by observed and unobservedheterogeneity (h, ε) and responds to a given budget (p,x), with aunique, positive J-vector of demands

q = d(x ,p,h, ε),

where demand functions d(x ,p,h, ε) : RK++ ! RJ

++ satisfyadding-up: p0q = x for all prices and total outlays x 2 R.

ε 2 RJ�1, J � 1 vector of (non-separable) unobservable heterogeneity.Assume ε? (x ,h) for now.The environment is described by a continuous distribution of q, x andε, for discrete types h.Will typically suppress observable heterogeneity h in what follows.

The Problem

q = d(x ,p,h, ε),

ε 2 RJ�1, J � 1 vector of (non-separable) unobservable heterogeneity.

Assume ε? (x ,h) for now.The environment is described by a continuous distribution of q, x andε, for discrete types h.Will typically suppress observable heterogeneity h in what follows.

The Problem

q = d(x ,p,h, ε),

ε 2 RJ�1, J � 1 vector of (non-separable) unobservable heterogeneity.Assume ε? (x ,h) for now.

The environment is described by a continuous distribution of q, x andε, for discrete types h.Will typically suppress observable heterogeneity h in what follows.

The Problem

q = d(x ,p,h, ε),

ε 2 RJ�1, J � 1 vector of (non-separable) unobservable heterogeneity.Assume ε? (x ,h) for now.The environment is described by a continuous distribution of q, x andε, for discrete types h.

Will typically suppress observable heterogeneity h in what follows.

The Problem

q = d(x ,p,h, ε),

ε 2 RJ�1, J � 1 vector of (non-separable) unobservable heterogeneity.Assume ε? (x ,h) for now.The environment is described by a continuous distribution of q, x andε, for discrete types h.Will typically suppress observable heterogeneity h in what follows.

Non-Separable Demand

For demands q = d(x ,p, ε):

One key drawback has been the (additive) separability of ε assumedin empirical speci�cations.

We will consider the non-separable case and impose conditions onpreferences that ensure invertibility in ε (which corresponds tomonotonicity for J = 2 which is our leading case). Assume uniqueinverse structural demand functions exists - Fig 1a.Here we consider the case of a small number of price regimes and userevealed preference inequalities applied to d(x ,p, ε) to improvedemand predictions

In other related work Slutsky inequality conditions have been shownto help in �smoothing�demands for �dense�or continuously distributedprices

We will consider the non-separable case and impose conditions onpreferences that ensure invertibility in ε (which corresponds tomonotonicity for J = 2 which is our leading case). Assume uniqueinverse structural demand functions exists - Fig 1a.

Here we consider the case of a small number of price regimes and userevealed preference inequalities applied to d(x ,p, ε) to improvedemand predictions

Related Literature

Nonparametric Demand Estimation: Blundell, Browning andCrawford (2003, 2007, 2008); Blundell, Chen and Kristensen (2007);Chen and Pouzo (2009).

Nonseparable models: Chernozhukov, Imbens and Newey (2007);Imbens and Newey (2009); Matzkin (2003, 2007, 2008), Chen andLaio (2010).

Restricted Nonparametric Estimation: Blundell, Horowitz andParey (2010); Haag, Hoderlein and Pendakur (2009), Kiefer (1982),Mammen, Marron, Turlach and Wand (2001); Mammen andThomas-Agnan (1999); Wright (1981,1984)

Inequality constraints and set identi�cation: Andrews (1999,2001); Andrews and Guggenberger (2007), Andrews and Soares(2009); Bugni (2009); Chernozhukov, Hong and Tamer (2007)....

Related Literature

Revealed Preference and Expansion Paths

Suppose we have a discrete price distribution, fp (1) ,p (2) , ...p (T )g.Observe choices of large number of consumers for a small (�nite) setof prices - e.g. limited number of markets/time periods

Market de�ned by time and/or location.

Questions to address here:

How do we devise a powerful test of RP conditions in thisenvironment?How do we estimate demands for some new price point p0?

In this case Revealed Preference conditions, in general, only allowset identi�cation of demands.

How do we devise a powerful test of RP conditions in thisenvironment?

How do we estimate demands for some new price point p0?

How do we devise a powerful test of RP?

Afriat�s TheoremData (pt ,qt ) satisfy GARP if qtRqs implies psqs � psqt

� if qt is indirectly revealed preferred to qs then qs is notstrictly preferred to qt

9 a well behaved concave utility function � the data satisfy GARP

How do we devise a powerful test of RP?Afriat�s Theorem

Data (pt ,qt ) satisfy GARP if qtRqs implies psqs � psqt

Data: Observational or Experimental - Is there a best design forexperimental data?

Blundell, Browning and Crawford (Ecta, 2003) develop a method forchoosing a sequence of total expenditures that maximise the power oftests of RP (GARP).

De�ne sequential maximum power (SMP) path

fxs , xt , xu , ...xv , xw g = fp0sqt (xt ),p0tqu(xu),p0vqw (xw ), xw g

Proposition (BBC, 2003) Suppose that the sequence

fqs (xs ) ,qt (xt ) ,qu (xu) ...,qv (xv ) ,qw (xw )g

rejects RP. Then SMP path also rejects RP. (Also de�ne RevealedWorse and Revealed Best sets.)

Data: Observational or Experimental - Is there a best design forexperimental data?Blundell, Browning and Crawford (Ecta, 2003) develop a method forchoosing a sequence of total expenditures that maximise the power oftests of RP (GARP).

- great for experimental design but we have Observational Data

continuous micro-data on incomes and expenditures

�nite set of observed price and/or tax regimes (across time andmarkets)

discrete demographic di¤erences across households

use this information alone, together with revealed preference theory toassess consumer rationality and to place �tight�bounds on demandresponses and welfare measures.

So, is there a best design for observational data?

Suppose we have a discrete price distribution,fp (1) ,p (2) , ...p (T )g.Observe choices of large number of consumers for a small (�nite) setof prices - e.g. limited number of markets/time periods. Marketde�ned by time and/or location.

Given t, qt (x ; ε) = d(x ,p(t), ε) is the (quantile) expansion path ofconsumer type ε facing prices p(t).Fig 1b

Suppose we have a discrete price distribution,fp (1) ,p (2) , ...p (T )g.

Observe choices of large number of consumers for a small (�nite) setof prices - e.g. limited number of markets/time periods. Marketde�ned by time and/or location.

Given t, qt (x ; ε) = d(x ,p(t), ε) is the (quantile) expansion path ofconsumer type ε facing prices p(t).

Fig 1b

Support Sets and Bounds on Demand Responses:

Suppose we observe a set of demands fq1,q2, ...qT g which record thechoices made by a particular consumer (ε) when faced by the set ofprices fp1,p2, ...pT g .

What is the support set for a new price vector p0 with new totaloutlay x0?Varian support set for d (p0, x0, ε) is given by:

SV (p0, x0, ε) =�q0 :

p00q0 = x0, q0 � 0 andfpt ,qtgt=0...T satis�es RP

In general, support set will only deliver set identi�cation ofd(x ,p0, ε).Figure 2(a) - generating a support set: SV (p0, x0, ε) for consumerof type ε

Suppose we observe a set of demands fq1,q2, ...qT g which record thechoices made by a particular consumer (ε) when faced by the set ofprices fp1,p2, ...pT g .What is the support set for a new price vector p0 with new totaloutlay x0?

Varian support set for d (p0, x0, ε) is given by:

SV (p0, x0, ε) =�q0 :

Suppose we observe a set of demands fq1,q2, ...qT g which record thechoices made by a particular consumer (ε) when faced by the set ofprices fp1,p2, ...pT g .What is the support set for a new price vector p0 with new totaloutlay x0?Varian support set for d (p0, x0, ε) is given by:

SV (p0, x0, ε) =�q0 :

In general, support set will only deliver set identi�cation ofd(x ,p0, ε).

Figure 2(a) - generating a support set: SV (p0, x0, ε) for consumerof type ε

SV (p0, x0, ε) =�q0 :

e-Bounds on Demand Responses

Can we improve upon SV (p0, x0, ε)?

Yes! We can do better if we know the expansion pathsfpt ,qt (x , ε)gt=1,..T .For consumer ε: De�ne intersection demands eqt (ε) = qt (xt , ε) byp00qt (xt , ε) = x0Blundell, Browning and Crawford (2008): The set of points that areconsistent with observed expansion paths and revealed preference isgiven by the support set:

S (p0, x0, ε) =�q0 :

q0 � 0, p00q0 = x0fp0,pt ;q0, eqt (ε)gt=1,...,T satisfy RP

�By utilizing the information in intersection demands, S (p0, x0, ε)yields tighter bounds on demands.These are sharp in the case of 2 goods. (BBC, 2003, for RW boundsfor the many goods case).Figure 2b, c - S (p0, x0, ε) the identi�ed set of demand responses forp0, x0, ε given t = 1, ...,T .

Can we improve upon SV (p0, x0, ε)?Yes! We can do better if we know the expansion pathsfpt ,qt (x , ε)gt=1,..T .

For consumer ε: De�ne intersection demands eqt (ε) = qt (xt , ε) byp00qt (xt , ε) = x0Blundell, Browning and Crawford (2008): The set of points that areconsistent with observed expansion paths and revealed preference isgiven by the support set:

S (p0, x0, ε) =�q0 :

Can we improve upon SV (p0, x0, ε)?Yes! We can do better if we know the expansion pathsfpt ,qt (x , ε)gt=1,..T .For consumer ε: De�ne intersection demands eqt (ε) = qt (xt , ε) byp00qt (xt , ε) = x0

Blundell, Browning and Crawford (2008): The set of points that areconsistent with observed expansion paths and revealed preference isgiven by the support set:

S (p0, x0, ε) =�q0 :

Can we improve upon SV (p0, x0, ε)?Yes! We can do better if we know the expansion pathsfpt ,qt (x , ε)gt=1,..T .For consumer ε: De�ne intersection demands eqt (ε) = qt (xt , ε) byp00qt (xt , ε) = x0Blundell, Browning and Crawford (2008): The set of points that areconsistent with observed expansion paths and revealed preference isgiven by the support set:

S (p0, x0, ε) =�q0 :

By utilizing the information in intersection demands, S (p0, x0, ε)yields tighter bounds on demands.These are sharp in the case of 2 goods. (BBC, 2003, for RW boundsfor the many goods case).Figure 2b, c - S (p0, x0, ε) the identi�ed set of demand responses forp0, x0, ε given t = 1, ...,T .

S (p0, x0, ε) =�q0 :

�By utilizing the information in intersection demands, S (p0, x0, ε)yields tighter bounds on demands.These are sharp in the case of 2 goods. (BBC, 2003, for RW boundsfor the many goods case).

Figure 2b, c - S (p0, x0, ε) the identi�ed set of demand responses forp0, x0, ε given t = 1, ...,T .

S (p0, x0, ε) =�q0 :

Unrestricted Demand Estimation

Observational setting: At time t (t = 1, ...,T ), we observe a randomsample of n consumers facing prices p (t).

Observed variables (ignoring other observed characteristics ofconsumers):

p (t) = prices that all consumers face,

qi (t) = (q1,i (t) , q2,i (t)) = consumer i�s demand,

xi (t) = consumer i�s income (total budget)

Observational setting: At time t (t = 1, ...,T ), we observe a randomsample of n consumers facing prices p (t).Observed variables (ignoring other observed characteristics ofconsumers):

p (t) = prices that all consumers face,

qi (t) = (q1,i (t) , q2,i (t)) = consumer i�s demand,

xi (t) = consumer i�s income (total budget)

We �rst wish to recover demands for each of the observed priceregimes t,

q (t) = d(x(t), t, ε), t = 1, ...,T ,

where d is the demand function in price regime p (t).

We will here only discuss the case of 2 goods with 1-dimensional error:

ε 2 R,

d(x(t), t, ε) = (d1(x(t), t, ε), d2(x(t), t, ε)) .

Given t, d1(x(t), t, ε) is exactly the quantile expansion path (Engelcurve) for good 1 at prices p (t).

q (t) = d(x(t), t, ε), t = 1, ...,T ,

where d is the demand function in price regime p (t).We will here only discuss the case of 2 goods with 1-dimensional error:

ε 2 R,

q (t) = d(x(t), t, ε), t = 1, ...,T ,

where d is the demand function in price regime p (t).We will here only discuss the case of 2 goods with 1-dimensional error:

ε 2 R,

Assumption A.1: The variable x (t) has bounded support, x (t)2 X = [a, b] for �∞ < a < b < +∞, and is independent ofε � U [0, 1].

Assumption A.2: The demand function d1 (x , t, ε) is invertible in εand is continuously di¤erentiable in (x , ε).

Identi�cation Result: d1(x , t, τ) is identi�ed as the τth quantile ofq1jx(t):

d1 (x , t, τ) = F�1q1(t)jx (t) (τjx) .

Thus, we can employ standard nonparametric quantile regressiontechniques to estimate d1.

Assumption A.1: The variable x (t) has bounded support, x (t)2 X = [a, b] for �∞ < a < b < +∞, and is independent ofε � U [0, 1].Assumption A.2: The demand function d1 (x , t, ε) is invertible in εand is continuously di¤erentiable in (x , ε).

d1 (x , t, τ) = F�1q1(t)jx (t) (τjx) .

We propose to estimate d using sieve methods.

Letρτ (y) = (I fy < 0g � τ) y , τ 2 [0, 1] ,

be the check function used in quantile estimation.

The budget constraint de�nes the path for d2. We let D be the set offeasible demand functions,

D =�d � 0 : d1 2 D1, d2 (x , t, τ) =

x � p1 (t) d1 (x , t, ε (t))p2 (t)

We propose to estimate d using sieve methods.Let

ρτ (y) = (I fy < 0g � τ) y , τ 2 [0, 1] ,be the check function used in quantile estimation.

D =�d � 0 : d1 2 D1, d2 (x , t, τ) =

x � p1 (t) d1 (x , t, ε (t))p2 (t)

We propose to estimate d using sieve methods.Let

ρτ (y) = (I fy < 0g � τ) y , τ 2 [0, 1] ,be the check function used in quantile estimation.

D =�d � 0 : d1 2 D1, d2 (x , t, τ) =

x � p1 (t) d1 (x , t, ε (t))p2 (t)

Let (qi (t) , xi (t)), i = 1, ..., n, t = 1, ...,T , be i.i.d. observationsfrom a demand system, qi (t) = (q1i (t) , q2i (t))

We then estimate d (t, �, τ) by

d (�, t, τ) = arg mindn2Dn

∑i=1

ρτ (q1i (t)� d1n (xi (t))) , t = 1, ...,T ,

where Dn is a sieve space (Dn ! D as n! ∞).Let Bi (t) = (Bk (xi (t)) : k 2 Kn) 2 RjKn j denote basis functionsspanning the sieve Dn.Then d1 (x , t, τ) = ∑k2Kn πk (t, τ)Bk (x), where πk (t, τ) is astandard linear quantile regression estimator:

π (t, τ) = arg minπ2RjKn j

∑i=1

�q1i (t)� π0Bi (t)

�, t = 1, ...,T .

∑i=1

ρτ (q1i (t)� d1n (xi (t))) , t = 1, ...,T ,

where Dn is a sieve space (Dn ! D as n! ∞).

Let Bi (t) = (Bk (xi (t)) : k 2 Kn) 2 RjKn j denote basis functionsspanning the sieve Dn.Then d1 (x , t, τ) = ∑k2Kn πk (t, τ)Bk (x), where πk (t, τ) is astandard linear quantile regression estimator:

∑i=1

�q1i (t)� π0Bi (t)

�, t = 1, ...,T .

∑i=1

ρτ (q1i (t)� d1n (xi (t))) , t = 1, ...,T ,

where Dn is a sieve space (Dn ! D as n! ∞).Let Bi (t) = (Bk (xi (t)) : k 2 Kn) 2 RjKn j denote basis functionsspanning the sieve Dn.

Then d1 (x , t, τ) = ∑k2Kn πk (t, τ)Bk (x), where πk (t, τ) is astandard linear quantile regression estimator:

∑i=1

�q1i (t)� π0Bi (t)

�, t = 1, ...,T .

∑i=1

ρτ (q1i (t)� d1n (xi (t))) , t = 1, ...,T ,

where Dn is a sieve space (Dn ! D as n! ∞).Let Bi (t) = (Bk (xi (t)) : k 2 Kn) 2 RjKn j denote basis functionsspanning the sieve Dn.Then d1 (x , t, τ) = ∑k2Kn πk (t, τ)Bk (x), where πk (t, τ) is astandard linear quantile regression estimator:

∑i=1

�q1i (t)� π0Bi (t)

�, t = 1, ...,T .

Adapt results in Belloni, Chen, Chernozhukov and Liao (2010) forrates and asymptotic distribution of the linear sieve estimator:

jjd (�, t, τ)� d (�, t, τ) jj2 = OP�n�m/(2m+1)

pnΣ�1/2

n (x , τ)�d1 (x , t, τ)� d1 (x , t, τ)

�!d N (0, 1) ,

where Σn (x , τ)! ∞ is an appropriate chosen weighting matrix.

RP-restricted Demand Estimation

No reason why estimated expansion paths for a sequence of pricest = 1, ...,T should satisfy RP.

In order to impose the RP restrictions, we simply de�ne theconstrained sieve as: DTC ,n = DTn \ fdn (�, �, τ) satis�es RPg .We de�ne the constrained estimator by:

dC (�, �, τ) = arg mindn(�,�,τ)2DTC ,n

∑t=1

∑i=1

ρτ (q1,i (t)� d1,n (t, xi (t))) .

Since RP imposes restrictions across t, the above estimation problemcan no longer be split up into T individual sub problems as theunconstrained case.

In order to impose the RP restrictions, we simply de�ne theconstrained sieve as: DTC ,n = DTn \ fdn (�, �, τ) satis�es RPg .

We de�ne the constrained estimator by:

∑t=1

∑i=1

ρτ (q1,i (t)� d1,n (t, xi (t))) .

∑t=1

∑i=1

ρτ (q1,i (t)� d1,n (t, xi (t))) .

∑t=1

∑i=1

ρτ (q1,i (t)� d1,n (t, xi (t))) .

Theoretical properties of restricted estimator: In general, the RPrestrictions will be binding. This means that dC will be on theboundary of DTC ,n. So the estimator will in general have non-standarddistribution (estimation when parameter is on the boundary).

Too hard a problem for us....

Instead: We introduce DTC ,n (ε) as the set of demand functionssatisfying

x (t) � p (t)0 d (x (s) , s, τ) + ε, s < t, t = 2, ...,T ,

for some ("small") ε � 0.Rede�ne the constrained estimator to be the optimizer overDTC ,n (ε) � DTC ,n.

x (t) � p (t)0 d (x (s) , s, τ) + ε, s < t, t = 2, ...,T ,

for some ("small") ε � 0.

Rede�ne the constrained estimator to be the optimizer overDTC ,n (ε) � DTC ,n.

x (t) � p (t)0 d (x (s) , s, τ) + ε, s < t, t = 2, ...,T ,

Under assumptions A1-A3 and that d0 2 DTC , then for any ε > 0:

jjdεC (�, t, τ)� d0 (�, t, τ) jj∞ = OP (kn/

pn) +OP

�k�mn

for t = 1, ...,T . Moreover, the restricted estimator has the sameasymptotic distribution as the unrestricted estimator.

Also derive convergence rates and valid con�dence sets for thesupport sets.

In practice, use simulation methods or the modi�ed bootstrapprocedures developed in Bugni (2009, 2010) and Andrews and Soares(2010); alternatively, the subsampling procedure of CHT.

pn) +OP

�k�mn

pn) +OP

�k�mn

Demand Bounds Estimation

Simulation Study: Cobb-Douglas demand function.

95% con�dence bands of demand bounds.

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 1.020

Demand bounds, τ = 0.50

price, food

true95% conf. band

Figure: Performance of demand bound estimator.

Simulation Study: Cobb-Douglas demand function.95% con�dence bands of demand bounds.

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 1.020

price, food

true95% conf. band

Figure: Performance of demand bound estimator.

Testing for Rationality

Constrained demand and bounds estimators rely on the fundamentalassumption that consumers are rational.

We wish to test the null of consumer rationality.Let Sp0,x0 denote the set of demand sequences that are rational givenprices and income:

Sp0,x0 =

�q 2 BTp0,x0 :

9V > 0,λ � 1 :V (t)� V (s) � λ (t) p (t)0 (q (s)� q (t))

We wish to test the null of consumer rationality.

Let Sp0,x0 denote the set of demand sequences that are rational givenprices and income:

Sp0,x0 =

�q 2 BTp0,x0 :

9V > 0,λ � 1 :V (t)� V (s) � λ (t) p (t)0 (q (s)� q (t))

We wish to test the null of consumer rationality.Let Sp0,x0 denote the set of demand sequences that are rational givenprices and income:

Sp0,x0 =

�q 2 BTp0,x0 :

9V > 0,λ � 1 :V (t)� V (s) � λ (t) p (t)0 (q (s)� q (t))

Test statistic: Given the vector of unrestricted estimatedintersection demands, bq, we compute its distance from Sp0,x0 :

ρn (bq,Sp0,x0) := infq2Sp0 ,x0

kbq�qk2W testn,

where k�kW testnis a weighted Euclidean norm,

kbq�qk2W testn=

∑t=1(bq (t)� q (t))0 W test

n (t) (bq (t)� q (t)) .

Distribution under null: Using Andrews (1999,2001),

ρn (bq,Sp0,x0)!d ρ (Z ,Λp0,x0) := infλ2Λp0 ,x0

kλ� Zk2 ,

where Λp0,x0 is a cone that locally approximates Sp0,x0 andZ � N (0, IT ).

Test statistic: Given the vector of unrestricted estimatedintersection demands, bq, we compute its distance from Sp0,x0 :

ρn (bq,Sp0,x0) := infq2Sp0 ,x0

kbq�qk2W testn,

where k�kW testnis a weighted Euclidean norm,

kbq�qk2W testn=

∑t=1(bq (t)� q (t))0 W test

n (t) (bq (t)� q (t)) .Distribution under null: Using Andrews (1999,2001),

ρn (bq,Sp0,x0)!d ρ (Z ,Λp0,x0) := infλ2Λp0 ,x0

kλ� Zk2 ,

where Λp0,x0 is a cone that locally approximates Sp0,x0 andZ � N (0, IT ).

Estimating e-Bounds on Local Consumer Responses

For each household de�ned by (x , ε), the parameter of interest is theconsumer response at some new relative price p0 and income x or atsome sequence of relative prices. The later de�nes the demand curvefor (x , ε).

A typical sequence of relative prices in the UK:

Figure 4: Relative prices in the UK and a �typical�relative pricepath p0.Figure 5: Engel Curve Share DistributionFigure 6: Density of Log Expenditure.

Figure 4: Relative prices in the UK and a �typical�relative pricepath p0.

Figure 5: Engel Curve Share DistributionFigure 6: Density of Log Expenditure.

Figure 4: Relative prices in the UK and a �typical�relative pricepath p0.Figure 5: Engel Curve Share Distribution

Figure 6: Density of Log Expenditure.

Estimation

In the estimation, we use log-transforms and polynomial splines

log d1,n(log x , t, τ) =qn

∑j=0

πj (t, τ) (log x)j

∑k=1

πqn+k (t, τ) (log x � νk (t))qn+ ,

where qn � 1 is the order of the polynomial and νk , k = 1, ..., rn, arethe knots.

In the implementation of the quantile sieve estimator with a smallpenalization term was added to the objective function, as in BCK(2007).

Estimation

In the estimation, we use log-transforms and polynomial splines

log d1,n(log x , t, τ) =qn

∑j=0

πj (t, τ) (log x)j

∑k=1

πqn+k (t, τ) (log x � νk (t))qn+ ,

where qn � 1 is the order of the polynomial and νk , k = 1, ..., rn, arethe knots.

In the implementation of the quantile sieve estimator with a smallpenalization term was added to the objective function, as in BCK(2007).

Unrestricted Engel Curves

3.8 4 4.2 4.4 4.6 4.8 5 5.2

logtotal exp.

Unconstr. food demand est., t=1

τ = 0.1τ = 0.5τ = 0.995% CIs

Figure: Unconstrained demand function estimates, t = 1983.

RP Restricted Engel Curves

3.8 4 4.2 4.4 4.6 4.8 5 5.2

logtotal exp.

Constr. food demand est., t=1

τ = 0.1

τ = 0.5τ = 0.995% CIs

Figure: Constrained demand function estimates, t = 1983.Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010 29 / 37

Estimation

conditional quantile splines - 3rd order pol. spline with 5 knots

RP restrictions imposed at 100 x-points over the empirical support x .1983-1990 (T=8).Figures 9-11: Estimated e-Bounds on Demand Curve

Demand (e-)bounds (support sets) are de�ned at the quantiles of xand ε

tightest bounds given information and RP.varies with income and heterogeneity

Estimation

RP restrictions imposed at 100 x-points over the empirical support x .

1983-1990 (T=8).Figures 9-11: Estimated e-Bounds on Demand Curve

Estimation

RP restrictions imposed at 100 x-points over the empirical support x .1983-1990 (T=8).

Figures 9-11: Estimated e-Bounds on Demand Curve

Estimation

tightest bounds given information and RP.

varies with income and heterogeneity

Estimation

0.9 0.92 0.94 0.96 0.98 1 1.02 1.040

price, food

T = 4T = 6T = 8

Figure: Demand bounds at median income, τ = 0.1.

0.9 0.92 0.94 0.96 0.98 1 1.02 1.040

price, food

T = 4T = 6T = 8

0.9 0.92 0.94 0.96 0.98 1 1.02 1.040

price, food

T = 4T = 6T = 8

0.9 0.92 0.94 0.96 0.98 1 1.02 1.040

price, food

T = 4T = 6T = 8

Figure: Demand bounds at 25th percentile income, τ = 0.5.

0.9 0.92 0.94 0.96 0.98 1 1.02 1.040

price, food

T = 4T = 6T = 8

Figure: Demand bounds at 75th percentile income, τ = 0.5.

Endogenous Income

To account for the endogeneity of x we can utilize IV quantileestimators developed in Chen and Pouzo (2009) and Chernozhukov,Imbens and Newey (2007).

Chen and Pouzo (2009) apply to exactly this data using the sameinstrument as in BBC (2008).

Our basic results remain valid except that the convergence rate statedthere has to be replaced by that obtained in Chen and Pouzo (2009)or Chernozhukov, Imbens and Newey (2007).

Alternatively, the control function approach taken in Imbens andNewey (2009) can be used. Again they estimate using the exact samedata and instrument. Specify

ln x = π(z, v)

where π is monotonic in v , z are a set of instrumental variables.

Endogenous Income

ln x = π(z, v)

Endogenous Income

ln x = π(z, v)

Endogenous Income

ln x = π(z, v)

Summary

Objective to elicit demand responses from consumer expendituresurvey data.

Inequality restrictions from revealed preference used to produce tightbounds on demand responses.

Derive a powerful test of RP conditions.

Particular attention given to nonseparable unobserved heterogeneityand endogeneity.

New (empirical) insights provided about the price responsiveness ofdemand, especially across di¤erent income groups.

Derive welfare costs of relative price and tax changes across thedistribution of demands by income and taste heterogeneity.

Summary

Stochastic Demand and Revealed Preference, November 2010

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