Testing stochastic rationality and normality of

demand with unobserved heterogeneity and

endogenous expenditures: the case of two goods

Karim Nchare

Penn State Departement of Economics

July 19, 2018
Motivation

I Normal good A good is normal if its consumption increases

with income, keeping prices fixed.
Motivation

I Normal good A good is normal if its consumption increases

with income, keeping prices fixed.

I Normality of goods imposes restrictions on how demand

changes when budgets (prices and income) change, which

implies specific testable implications. (Law of demand
Motivation

I Normal good A good is normal if its consumption increases

with income, keeping prices fixed.

I Normality of goods imposes restrictions on how demand

changes when budgets (prices and income) change, which

implies specific testable implications. (Law of demand

I Why do we care? The existing procedures always test multiple

joint hypotheses regarding the functional representation of

preferences/demand and the nature of heterogeneity across

different consumers. (Engel curve)
Motivation

I Normal good A good is normal if its consumption increases

with income, keeping prices fixed.

I Normality of goods imposes restrictions on how demand

changes when budgets (prices and income) change, which

implies specific testable implications. (Law of demand

I Why do we care? The existing procedures always test multiple

joint hypotheses regarding the functional representation of

preferences/demand and the nature of heterogeneity across

different consumers. (Engel curve)

I Measuring welfare effects associated with changing prices

and/or incomes requires predicting demand in counterfactual

price-income regimes.
Motivation

I Normal good A good is normal if its consumption increases

with income, keeping prices fixed.

I Normality of goods imposes restrictions on how demand

changes when budgets (prices and income) change, which

implies specific testable implications. (Law of demand

I Why do we care? The existing procedures always test multiple

joint hypotheses regarding the functional representation of

preferences/demand and the nature of heterogeneity across

different consumers. (Engel curve)

I Measuring welfare effects associated with changing prices

and/or incomes requires predicting demand in counterfactual

price-income regimes.

I Normality of goods improve predicting power of fully

nonparametric estimation approach.
What do we do?

I Use a nonparametric revealed preference approach to test

implications for observe consumption patterns to be consistent

with rational demand behavior and normality of goods.
What do we do?

I Use a nonparametric revealed preference approach to test

implications for observe consumption patterns to be consistent

with rational demand behavior and normality of goods.

I Two goods case. If there are only two goods, a relative price

increase of one good necessarily implies a relative price

decrease of the other good.
What do we do?

I Use a nonparametric revealed preference approach to test

implications for observe consumption patterns to be consistent

with rational demand behavior and normality of goods.

I Two goods case. If there are only two goods, a relative price

increase of one good necessarily implies a relative price

decrease of the other good.

I Not restrictive. Multi-goods setting can be reduced to a

two-goods setting using Hicksian aggregation or weakly

separability for dimensionality reduction.
What do we do?

I Use a nonparametric revealed preference approach to test

implications for observe consumption patterns to be consistent

with rational demand behavior and normality of goods.

I Two goods case. If there are only two goods, a relative price

increase of one good necessarily implies a relative price

decrease of the other good.

I Not restrictive. Multi-goods setting can be reduced to a

two-goods setting using Hicksian aggregation or weakly

separability for dimensionality reduction.

I An empirical illustration to data drawn from the U.S.

Consumer Expenditure Survey (1994-2007).
Revealed Preferences

I Two goods setting with associated quantities respectively

q ∈ R+ and z ∈ R+.
Revealed Preferences

I Two goods setting with associated quantities respectively

q ∈ R+ and z ∈ R+.

I Price of good z is normalized to 1 and the price of good q is

denoted by p ∈ R++.
Revealed Preferences

I Two goods setting with associated quantities respectively

q ∈ R+ and z ∈ R+.

I Price of good z is normalized to 1 and the price of good q is

denoted by p ∈ R++.

I A consumer j with the level of total expenditures xj

is

endowed with a demand function qj(p, xj).
Revealed Preferences

I Two goods setting with associated quantities respectively

q ∈ R+ and z ∈ R+.

I Price of good z is normalized to 1 and the price of good q is

denoted by p ∈ R++.

I A consumer j with the level of total expenditures xj

is

endowed with a demand function qj(p, xj).

I It is assumed that consumers exhaust their budgets so that

zj(p, xj) = xj − pqj(p, xj) with qj(p, xj) ∈ [0,

xj

p

].
Revealed Preferences

I Two goods setting with associated quantities respectively

q ∈ R+ and z ∈ R+.

I Price of good z is normalized to 1 and the price of good q is

denoted by p ∈ R++.

I A consumer j with the level of total expenditures xj

is

endowed with a demand function qj(p, xj).

I It is assumed that consumers exhaust their budgets so that

zj(p, xj) = xj − pqj(p, xj) with qj(p, xj) ∈ [0,

xj

p

].

I Cherchye, De Rock and Demuynck (2018): Normality of

demand in a two-goods setting. Dataset S = {pt, qt, xt}t∈T ,

is consistent with weak normality of goods q and z if and only

if for all t,s ∈ T:

if pt < ps and qt ≤ qs then zt ≤ zs
Testing Results

Normality of Goods q and z: If the cross-sectional joint

distribution of {qt,j, zt,j, xt,j}t∈T on budgets {pt}t∈T are

stochastically consistent with normality of both goods q and z,

then for every t,s ∈ T such that pt < ps, it must be that

Pr[qt,j ≤ qs,j and zt,j ≥ zs,j] = 0.

Then, for any a, b ∈ R+,

Pr[qt,j ≤ a and zt,j ≥ b] + Pr[a ≤ qs,j and b ≥ zs,j] ≤ 1.
Prediction results

Bounds on Pr[qj(p0, x0) ≤ q0] assuming goods q and z are both

normal

max

pt<p0

Pr[qt,j ≤ q0 and zt,j ≥ z0] ≤ Pr[qj(p0, x0) ≤ q0]

Pr[qj(p0, x0) ≤ q0] ≥ min

pt>p0

Pr[qt,j ≤ q0 or zt,j ≥ z0
Application

I Data Cross-sectional household consumption data for U.S.

households from 1994 until 2007 (14 years), which are

obtained from the diary survey contained in the Consumer

Expenditure Survey.
Application

I Data Cross-sectional household consumption data for U.S.

households from 1994 until 2007 (14 years), which are

obtained from the diary survey contained in the Consumer

Expenditure Survey.

I Two goods, food (good q) and other nondurable goods and

services (good z).
Application

I Data Cross-sectional household consumption data for U.S.

households from 1994 until 2007 (14 years), which are

obtained from the diary survey contained in the Consumer

Expenditure Survey.

I Two goods, food (good q) and other nondurable goods and

services (good z).

I One possible use of our method is to anticipate demand

effects of rising or falling food prices (e.g. because of political

intervention change of the indirect taxes on food)
Application

I Data Cross-sectional household consumption data for U.S.

households from 1994 until 2007 (14 years), which are

obtained from the diary survey contained in the Consumer

Expenditure Survey.

I Two goods, food (good q) and other nondurable goods and

services (good z).

I One possible use of our method is to anticipate demand

effects of rising or falling food prices (e.g. because of political

intervention change of the indirect taxes on food)

I I deflate total expenditures by OECD equivalence scales to

correct for household size.
Application

I Data Cross-sectional household consumption data for U.S.

households from 1994 until 2007 (14 years), which are

obtained from the diary survey contained in the Consumer

Expenditure Survey.

I Two goods, food (good q) and other nondurable goods and

services (good z).

I One possible use of our method is to anticipate demand

effects of rising or falling food prices (e.g. because of political

intervention change of the indirect taxes on food)

I I deflate total expenditures by OECD equivalence scales to

correct for household size.

I By conditioning on housing status and vehicle ownership, the

impact of the implicit assumption of separability between

durable and non-durable consumption is mitigated.
Application
Application
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