Ag Econ 635
Topics in Consumer Demand Analysis
State Adjustment Model
Dr. Oral Capps, Jr.
Texas A&M University
Spring 2008
Dynamics of Consumer Expenditures:
Application of Complete Demand Systems
Introduction
Non-technical description of dynamic demand systems
Dynamic models provide information on adjustment,
habit, and inventory features
Objectives:
(1) Justification of dynamic demand systems
(2) Description of methods to incorporate dynamic
structures into demand systems
(3) Description of state adjustment model and dynamic
linear expenditure model
(4) Empirical examples
Justification
Static Demand Systems:
By assumption consumer adjustment
instantaneously to new equilibria due to changes in
income or price, while all other factors remain
constant.
Dynamic Demand Systems:
Permit processes of gradual adjustment to
changes not only in the economic environment but also
to changes in tastes and preferences; account for
adjustments that occur through time due to habit
persistences or stock adjustments.
Incorporation of Dynamic Structures into Demand Systems
(1) Ad Hoc Procedures
addition of trend variables to the demand equations
introduction of trend variables into the parameters of models
state adjustment model
(2) Dynamic Utility Functions
certain parameters of the utility function depend on past consumption
Quadratic Model
Dynamic Linear Expenditure Model
examples
Translog Demand Model
Almost Ideal Demand System
}
(3) Control Theory Format
Maximization of a discounted utility function subject to wealth and
stock constraints
Phlips (1974)
Lluch (1974)
examples
Klijn (1977)
}
Popular Dynamic Models
( 1) State Adjustment Model: Houthakker and Taylor (1970)
A directly specified demand system
QIT  I  I SIT  KI M I  VI PIT
SIT = physical stock or psychological stock
αI= stock coefficient ( < 0 for physical stocks, > 0 for psychological
stocks)
θI = autonomous consumption levels
KI, VI = short-run derivatives of consumption with respect to income
and price
SI  QIT   I SIT
S I = rate of change in stock of commodity I
δI = depreciation rate
δI - αI = reaction or adjustment coefficient
(2) Dynamic Linear Expenditure System
U
  LOG(Q
I
IT
I )
I
 I  I   I SIT
SI  QIT   I SIT
Demand functions arise from maximization of utility function
βI, γI underlying structural parameters
To derive long-run conditions in both models, assume S I = 0 which
implies that the stock adjustment has reached an equilibrium state;
consequently, impacts of changes in income and/or prices can be
analyzed into short-term and long-term effects.
Empirical Examples
Green, Hassan, and Johnson (1978)
Analysis of consumer expenditures, four groups:
(1) durables
(2) semi-durables
(3) non-durables
(4) services
Structure of consumer behavior in Canada
1947-72
State adjustment model
Dynamic linear expenditure model
The Role of Inventories and Habits on Elasticities
Habit formation relative to inventory behavior decreases as the time interval
decreases
With short time periods such as a month, inventory behavior tends to dominate
demand relationships
If inventory adjustment dominates in the short-run, the elasticity of demand
generally increases as the time interval is shortened.
Pasour & Schrimper – the relative importance of inventory behavior with respect to
the length of the adjustment period is an empirical question, & its importance
can vary form commodity to commodity.
The State Adjustment Model (Houthakker and Taylor)
Key Point: past behavior is embodied in a state variable, encompassing both stocks
held by consumers and habits formed by past consumption
The model has two equations:
(1) A short-run demand function
(2) A stock depreciation equation
Example: Consumer demand for beef, pork, chicken (from Wohlgenant & Hahn,
AJAE 1982.)
(1)
(2)
qit   i   i Sit   ib Pbt   ip Ppt   ic Pct   iyYt

dSit
S it 
 qit   i Sit
dt
Stocks depreciate at a declining geometric rate over time
Si  the state of the ith commodity
Pi  the real (deflated) price of the ith commodity
y  per capita real consumer income
qi  per capita consumption of the ith meat commodity
If the inventory effect dominates Bi < 0; the larger the physical stock, the smaller
the consumer demand at t.
If habits dominate, the larger the (psychological) stock of habits and the greater
the demand at t. (Bi > 0)
The state variable in equation (1) is unobservable, but can be estimated with
equation (2). The two respective equations are formulated in continuous
time, so it is necessary for empirical implementation to approximate them by
discrete time.
qit   i   i Sit   ib Pbt   ip Ppt   ic Pct   iyYt
1
Sit  [qit   i   ib Pbt   ip Ppt   ic Pct   iyYt ]
Bi
Replace (following Winder; Houthakker and Taylor; Phlips)

Sit  Sit 1
and replace S it by Sit  Sit  Sit 1.
Sit by
2
 Sit  Sit 1 
 S it  Sit  Sit 1  qit   i 
  Sit
2



After many tedious steps of algebra,
qit 
1
[(1  1 ( Bi   i ))qit 1   i i   ib (1  1  i )Pbt
2
2
(1  1 ( Bi   i ))
2
  ib i Pbt 1   ip (1  1  i )Ppt   ip i Ppt 1   ic (1  1  i )Pct
2
2
  ic i Pct 1   iy (1  1  i )Yt   iy iYt 1 ]
2
More compactly
qit  Aio  Ai1Pbt  Ai 2 Pbt 1  Ai 3Ppt  Ai 4 Ppt 1  Ai 5 Pct
 Ai 6 Pct 1  Ai 7 Yt  Ai8Yt 1  Ai 9 qit 1  it
Let K  (1 - (1/2)(B i   i ))
Aio 
 i i
Ai1 
Ai 2 
Ai 3 
Ai 4 
K
Ai 5 
,
 ib (1  (1 / 2) i )
Ai 6 
K
Ai 7 
 ib i
K
 ip (1  (1 / 2) i )
Ai8 
 ic (1  (1 / 2) i )
K
 ic i
K
 iy (1  (1 / 2) i )
K
 iy i
K
K
 ip i
K
(1  (1 / 2)( Bi   i ))
Ai 9 
K
The structural parameter i is over-identified.
A unique estimate of i is obtained if the following restrictions hold:
Ai1  Ai 2 Ai7 / Ai8
Ai3  Ai 4 Ai7 / Ai8
Ai5  Ai6 Ai7 / Ai8
 i 
1
 Ai 7 1 


 A  2
 i8

2( Ai9  1)
 Bi   i 
1  Ai9 
Long-Run derivatives obtained by setting Sit  0
 qit  i Sit
so qit   i 
Bi qit
i
  ib Pbt   ip Ppt   ic Pct   iyYt
 iy i y
 i i
 ib i Pbt  ip i Ppt  ic i Pct
 qit 




( i  Bi ) ( i  Bi ) ( i  Bi ) ( i  Bi ) ( i  Bi )
Reaction or adjustment
coefficient
If habit persistence dominates, the long-run elasticities will be larger
than the short-run elasticities.
If inventory behavior is dominant, the short-run elasticities will be
larger.
Nerlove’s Partial Adjustment Model
qit*  i  CibPbt  Cip Ppt  Cic Pct  CiyYt
(qit  qit 1)  k (qit*  qit 1)
qit  qit 1  kqit 1  kqit*
qit
qit
1 k
*
 qit 1  qit 
 qit 1 (
)
k
k
k
qit
1 k

  it  (
)qit 1  Cib Pbt  Cip Ppt  Cic Pct  CiyYt
k
k
 qit  kit  (1  k )qit 1  kCibPbt  kCip Ppt  kCic Pct  kCiyYt
Test H0: i = 2 (partial adjustment model is same as state
adjustment model.)