Theoretical and Empirical Issues in Demand Analysis

1
Theoretical and Empirical Issues in Demand
Analysis

• By
• Anna Rapoport
• under the supervision of professor
• Yakar Kannai

2
Consumers problem
Given price p and wealth w, choose consumption
bundle x from Bpw x 0 px w

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The UMP

• The consumer aims to maximise utility...

• Subject to the budget constraint

max u(x) subject to n S pi xi w i1

• Defines the UMP

• Solution to the problem

Budget set

• x

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Comparative Statics Wealth Effects

• Definition 1 For fixed prices p, the function
  of wealth x(p,w) is called the consumers Engel
  function.
• Definition 2 At any (p,w), the derivative
  ?xm(p,w)/?w is known as the wealth (income)
  effect for the m-th good.

The wealth effects in matrix notation
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Effect of a change in income
x2

• Take the basic equilibrium

• What happens if income rises?

• Equilibrium shifts from x to x

• Demand for each good does not fall if it is
  normal

• but could the opposite happen?

• x

• x

x1
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An inferior good
x2

• The same original prices, but different
  preferences...

• Again, let income rise...

• The new equilibrium

demand for inferior good 2 falls a little as
income rises

• x

• x

x1
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Normal and Inferior Goods

• Definition 1 A commodity m is normal at (p,w) if
  ?xm(p,w)/?w 0, that is demand is nondecreasing
  in wealth. If commodity ms wealth effect is
  instead negative, then it's called inferior in
  (p,w).
• Definition 2 If every commodity is normal at all
  (p,w), then we say that demand is normal.

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Comparative Statics Price Effects

• Definition 1 The derivative ?xm(p,w)/?pk is
  known as the price effect of pk , the price of
  good k, on the demand for good m.
• Definition 2 Good m is said to be Giffen good
  at (p,w) if ?xm(p,w)/?pm gt 0.

The price effects in matrix notation
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Effect of a change in price
x2

• Again take the basic equilibrium

• ...and let the price of good 1 fall

• The effect of the price fall...

• The journey from x to x can be
  (imaginarily) broken into two parts

• A substitution effect

Income effect

• An income effect

• x

• x

x
Substitution effect
x1
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Close up.

• The income effect

how do demands respond to changes in the cost of
living?

• The substitution effect

at a given utility level how do demands respond
to relative prices?

• x

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Effect of a change in price for Giffen Good
x2

• Again take the basic equilibrium

• ...and let the price of good 1 fall

• The effect of the price fall...

• A substitution effect

• An income effect

Income effect

• x

• x

x
Substitution effect
x1
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The Slutsky equation

• Gives fundamental breakdown of effects of a
  price change

• Income effect

• Substitution effect

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Slutsky Matrix and Substitution Effects
Slutsky matrix
Negative semidefinite and symmetric
where
Substitution effects
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The good can be Giffen at (p,w) only if it is
inferior!!!
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Mean market demand

• Economy consists of a continuum consumers, which
  all have the same demand function f(p,w) but
  differ by w.
• ? - the density of distribution of w with finite
  mean
• Mean market demand
• in order to shorten notation we will write F(p).

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On the Law of Demand
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The Necessary Condition for Giffen Good
0

If
then
Mean (average) income effect term
0
lt0
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Some Assumptions
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Sufficient condition for a negative mean income
effect term
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Shochu and Special Grade Sake
Rich consumers
Special grade sake
Shochu
Poor consumers
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Inferiority and Giffen Effect (intuition)
Market Prices
1Sake2Shochu3 or 0Sake12Shochu12
Poor consumer
1Sake6Shochu7 or 0Sake12Shochu12
1x 10x 11
1x 50
1x 5
Wealth 100
Wealth 100 1x 10x 11
Giffen effect

• Inferiority

Wealth 100 ? 60
Buy 12xShochu
Price of Shochu 5 ? 8
Buy 12xShochu
Wealth? Demand Shochu?
Price Shochu? Demand Shochu?
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Data on Shochu and Sake suggests

• Special grade sake is a normal good.
• Shochu is an inferior good (UM) and (ID) hold.
• Need to examine the movements of prices and
  quantities consumed of Shochu and Sake.
• Time series data.
• Supply-and-demand model ? simultaneity.

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Demand-and-Supply Model

• Demand function QtD?0?1Pt?2dec u1t
• Supply function QtS?0?1Pt?2dec u2t
• Equilibrium condition QtD QtS

Demand function QtD?0?1Pt?2dec u1t Supply
function QtS?0?1Pt?2dec ?3int
u2t Equilibrium condition QtD QtS
Q
S1
D1
In this case we cannot distinguish between demand
and supply
We need shift in supply curve in order to
determine demand - int
S2
D2
S3
S4
P
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Simultaneous Equation Model
Demand function Qt?0?1Pt?2dec u1t Supply
function Qt?0?1Pt?2dec ?3int u2t
Structural form equation
Endogenous (dependent) variables Q ? P
Exogenous (determined outside the model) variables
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Problem Why not OLS?
Pt?0?1Qt?2dec uP Qt?0?1Pt?2dec ?3int uQ
E(Qt uP)?0 E(Pt uQ)?0
Pt?0?1 ?0?1Pt?2dec ?3int uQ ?2dec
uP Qt?0?1?0?1Qt?2dec uP ?2dec ?3int uQ
That is a violation of classical regression model
(Gauss-Markov condition ) ? OLS coefficients
biased and not consistent.
Pt?0?1?0 (?1?2?2)dec ?1?3int ?1 uQ
uP/(1- ?1?1) Qt?0?1?0 (?1?2?2)dec ?3int
?1uP uQ/(1- ?1?1)
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What can we do?
Demand function Qt?0?1Pt?2dec u1t Supply
function Qt?0?1Pt?2dec ?3int u2t
Structural form equation
Using equilibrium condition obtain
We CAN estimate these equations using OLS since
all the RHS variables are exogenous
Qt?10 ?11dec ?12int v1t Pt ?20 ?21dec
?22int v2t
Reduced form equation
But Can we obtain from ?s ?s and ?s?
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Identification Problem
We could have three possible situations for the
equation

• Underidentified We cannot get the structural
  coefficients from the reduced form estimates.
• Exactly (just) identified Can get unique
  structural form coefficient estimates.
• Overidentified More than one set of structural
  coefficients could be obtained from the reduced
  form.

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Order and Rank Conditions

• The order condition (necessary)
• Let G denote the number of structural equations.
  An equation is just identified if the number of
  variables excluded from an equation is G-1.
• If more than G-1 are absent, it is
  overidentified. If less than G-1 are absent, it
  is underidentified.

Demand function Qt?0?1Pt?2dec u1t Supply
function Qt?0?1Pt?2dec ?3int u2t
The rank condition (necessary and sufficient)
Used in practice
G-11 ? just identified
G-11gt0 ? underidentified
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2SLS

• Stage 1
• Estimating the reduced-form equation for P
•  

Pt ?0 ?1dec ?2int vt

• Stage 2
• In structural equation, regress Q on P and
  exogenous variables

Qt ?0 ?1 Pt ?2dec ut
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The results of 2SLS
Shochu is a Giffen good
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Simple Regression Model

• Yt ?0 ?1Xt ut

• ?0 and ?1 are estimated using OLS
• Statistical significance
• t-test t (?i - ?i)/?(?i ) against
  Students.
• If time series are stationary the t statistic
  will falsely reject H0 ? 5 when evaluated
  against the Students t dist at p 0.05

H0 ?10
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Stationary and Nonstationary Time Series

• Definition A stochastic process Yt is
  stationary if
• EYt? is independent of t
• VarYtE(Yt - ?)2 ?Y2 is independent of t
• CovYt, Ys E(Yt - ?)(Ys - ?) is a function
  of t-s but not of t.
• Otherwise the stochastic process is called
  nonstationary.

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Examples of Stationary Time Series

• White noise utt(-?,?) such that
• Eut0
• Varut ?u2
• Covut, us 0 for all s?t.
• AR(1) process Yt? Yt-1 ut , -1 lt ? lt 1 and
  ut is a white noise

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Nonatationary Random Walk

• The condition 1lt ? lt1 was crucial for
  stationarity. If ? 1 ?
• is a nonstationary process known as a random
  walk.

Yt Yt-1 ut
Yt Y0 u1 ut
EYt EY0 Eu1 EutY0
VarYt t?u2 is increasing with t
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More Examples of Nonstationary Time Series

• Random walk with drift
• EYtY0 ?t depends on t
• Time series with time trend
• EYt ? ?t depends on t
• Random walk with drift and linear time trend

Yt ? Yt-1 ut
Yt ? ?t ut
Yt ? ?t Yt-1 ut
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Random walk with drift Yt 0.2 Yt-1 ut
Stationary process Yt 0.7 Yt-1 ut
Random walk Yt Yt-1 ut
All three series are generated with the same set
of disturbances
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Difference between RW and LTT
RWD Yt ? Yt-1 ut ?t Y0 u1 ut
LTT Yt ? ?t ut
Random walk with drift Linear time trend
The divergence from the trend line is random walk and the variance around the trend increases without limit. The deviations from the trend are short-lived. The series sticks to its trend in the long run.
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Trend-Stationarity

• Definition A trend-stationary model is one that
  can be made stationary by removing a
  deterministic trend.

Example Series with linear time trend
Yt ? ?t ut
Yt ? ?t
Stationary
Yt Yt Yt ut

• By contrast

Yt ?t Y0 u1 ut
ZtYt- ?t Y0 u1 ut
VarZt t?u2
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Difference-Stationarity

• Definition If a nonstationary process can be
  transformed into a stationary one by
  differencing, it is said to be difference-stationa
  ry.
• Example Random walk with or without drift

Yt ? Yt-1 ut
I(1)
Zt ?Yt (Yt Yt-1) ? ut
I(0)
Many economic time series are I(1).
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Spurious regression
Xt
Yt
Yt ?0 ?1Xt ut
Granger and Newbold in a Monte Carlo experiment
fitted the model where Yt and Xt were
independently-generated random walks.
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Results

• Obviously, a regression of one random walk on
  another ought not to yield significant results
  except as a matter of Type I error. The true
  slope coefficient is 0, because Y was generated
  independently of X.
• However, performing the experiment with 100 pairs
  of random walks, Granger and Newbold found that
  the null hypothesis of a 0 slope coefficient was
  rejected 77 times (5) and 70 times (1). They
  found that in this case instead of t-critical
  value2 (5) one should use t 11.2.

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Why?
Yt ?0 ?1Xt ut
H0 ?10
ut Yt - ?0

• ut has the same autocorrelation properties as Yt
  which is nonstationary (or at best highly
  autocorrelated), but ut is white noise ? standard
  t, F statistics will fail.
• Low Durbin-Watson statistic will show that
  the regression
  is misspecified.

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Unit Root Test
If H0 is true the OLS estimator is biased
downward and conventional t and F tests will
tend incorrectly to reject H0. Dickey and Fuller
revised set of critical values (based on MC)
Yt ?Yt-1 ut
H0 ?1 against H1 ? lt 1
?Yt ?Yt-1 ut
H0 ? 0 against H1 ? lt 0
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Dickey-Fuller unit root test
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Cointegration
Yt ?0 ?1Xt ut
utYt- ?0- ?1Xt
Nonstationary

• Definition If ? ?0 and ?1 such that ut is
  stationary Xt and Yt are called cointegrated
  processes.
• Thus Y and X could both be I(1), and yet, if the
  model is correctly specified, one would expect u
  to be I(0).
• A requirement for cointegration is that all the
  variables in the relationship should be subject
  to the same degree of integration.
• Example PDI (personal disposable income) PCE
  (personal consumption expenditure)

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Empirical Results

• First step analysis of all presented time series
  for stationarity and order of integration use DF.

I(1)
I(0)
I(0)
I(1)
I(0)
I(0)
I(1)
I(0)
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Empirical Results (continuation)
Model was correctly specified.
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The End
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Glossary

• Consumer preferences rationality, desirability,
  convexity, continuity.
• Utility function representation, properties,
  UMP.
• The Walrasian demand function definition,
  properties, assumptions.
• WARP compensated law of demand.

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Consumer Preferences Rationality

• Rationality a preference relation is rational
  if it is possesses
• Completeness
• Transitivity

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Consumer Preferences Desirability

• Monotonicity is monotone on X if, for x,y ?
  X, ygtgtx implies y gt x.
• Strong monotonicity is strongly monotone if
  yx and y?x imply that y gt x.
• Local nonsatiation is locally nonsatiated if
  for every x ? X and ? gt 0 ? y ? X s.t. y-x ?
  and y gt x.

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Consumer Preferences Convexity

• Convexity The preference relation is convex
  if, ?x ? X, the upper contour set y ? X y x
  is convex, that is if y x and z x, then
  ty(1-t)z x for all 0lttlt1.
• Strict convexity The preference relation is
  strictly convex if, ?x,y,z ? X, y x, z x, and
  y ? z implies ty(1-t)z gt x for all 0lttlt1.

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Consumer Preferences Continuity

• Lexicographic (strict) preference ordering L is
  not continuous.
• For all x,y ?X?L, xLy if x1gty1, or if x1y1 and
  x2gty2, or if xiyi for i 1, , k-1 lt L-1 and xk
  gtyk.

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Utility function representation

• Definition The utility function u X? ?
  represents if
• ? x,y ? X, u(x) ? u(y) ? x y.
• Theorem (Debreu, 1954) Suppose that the
  rational preference ordering on ?L is
  continuous. Then it can be represented by a
  continuous utility function u?L??.

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A utility function
Utility Function
u
u(x1,x2)
indifference curve
x2
0
x1
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Properties of Utility Function

• For any strictly increasing function f???
• v(x) f(u(x)) is a new utility function
  representing the same preferences as u
• Convex preferences ? quasiconcave utility
  function.

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The Utility Maximization problem

• We assume that the consumer has a rational,
  continuous and locally nonsatiated and u(x) is
  a continuous utility function.
• UMP

Formalization of consumers problem

• If pgtgt0 and u() continuous gt UMP has a
  solution.

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The Walrasian Demand Function

• Definition The consumers Walrasian demand
  correspondence x(p,w) assigns a set of optimal
  consumption vectors in UMP to each price-wealth
  pair (p,w)gtgt0.
• In principle this correspondence can be
  multivalued. When x(p,w) is single-valued, we
  refer to it as a demand function.

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Properties of the Demand Function

• Theorem u() is a continuous utility function
  representing a locally nonsatiated preference
  relation defined on the consumption set X ?L.
  Then the x(p,w) possesses the following
  properties
• Homogeneity of degree zero in (p,w)
• Walras' law
• Convexity / uniqueness.

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Assumptions on x(p,w)

• x(p,w) is always single-valued.
• When convenient, we assume x(p,w) to be
• continuous
• differentiable.

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The WARP and the Law of Demand
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Proof of the Theorem 2.3.1