Theoretical background on aggregation:
Micro-macro debates
M.A. Keyzer
Lecture 2
Presentation available:www.sow.vu.nl/downloadables.htm
www.ccap.org.cn
1. Importance of the subject and approach
2. Comparable utilities, exact aggregation
3. Noncomparable utilities, optimal aggregation
4. Spatial aggregation over markets
5. Aggregation over commodities
6. Conclusions
Overview of the lecture
Question:
How to represent the behavior of many individuals by a
tractable number of agents and markets?
Principles:
1. Socio-economic environment of individuals is
described by finite number of fixed (spatial and
social) characteristics that follow a smooth joint
distribution.
2. Individuals choose optimally from options.
1. Subject and approach
Why special attention to aggregation issues?
(a) Aggregation errors are of particular importance in
China
(b) Current developments require disaggregated approach
1. Decentralisation and liberalization
2. Role of state to achieve basic economic targets
3. Increased diversity of lifestyles (migration)
4. Spatially explicit policies in agriculture and
environment
5. Increased product heterogeneity
1. Subject and approach (end)
Exact aggregation is possible and representative
agents exist.
Moreover, their response to changes in the fixed
characteristics of their environment is
smooth.
Two examples:
2.1 Farmland allocation
2.2 Transportation
2. Case 1: comparable utilities or payoffs
Index s for farmland sites
Index j for crops grown on site s
: acreage of crop j at site s (decision variables)
: revenue per acre of crop j at site s
: cost per acre of crop j at site s
: rental price of land at site s
: characteristics of farming at site s
: variation of productivity of farmers growing j at site
s
: marginal density of , from joint density
2.1 Example: Farmland allocation
s s sf ( , ) ss sf ( )
sjr
sjc
sj sj
s
sjs
Allocation by individual farmers (profitabilities ):
Allocation by all farmers jointly:
2.1 Farmland allocation (2)
sj sj
s s s s 0s s s
j0 a sj sj s 0s sj sj
( r c r ,a )
max [( r c r ) ]
sj s sj
s s s s 0s s
j0 ( ) a sj sj s 0s sj sj s s s s
( r c ,r ,a )
max [( r c r ) ( )] f ( )d
0s sjr
After adjusting the rental price to satisfy land
balance :
This allocation by all farmers jointly coincides with
the allocation by a representative farmer:
2.1 Farmland allocation (3)
sj s sj
s s s 0s s s
j0 ( ) a sj sj 0s sj sj s s s s
j sj s s s s s
( r c ,r ,a ,A )
max [( r c r ) ( )] f ( )d
subject to ( ) f ( )d A
sj
s s s 0s s s
ja 0 sj sj sj 0s s s1 sJ s
j sj s
( r c ,r ,a ,A )
max ( r c )a r G ( a ,...,a ,a )
subject to a A
sAs
The function has all the properties common in
microeconomics: it is strictly convex, non-
decreasing, differentiable, and homogeneous of
degree one.
Hence, it has major practical advantages:
- exact aggregation implementable at various scales
- continuous responses to price reforms
- estimation can be based on data generated by the
underlying density.
2.1 Farmland allocation (end)
sG
2.2 Example: Transportation
The farmland allocation model has many features that
can be used to model the transportation of goods
to the most rewarding markets and along the
cheapest route.
In example 2.1, the representative agent was a farmer
at site s allocating a given piece of land over
various crops j .
Now the representative agent will be a transport firm
at site s transporting a given quantity procured
from within the continuum of the site to some
discrete destination j .
2.2 Transportation (2)
Index s for the origin (say, production sites)
Index j for the destination (say, markets at district
centers)
: quantity shipped from site s to market j
: net revenue per ton produced at site s
: opportunity cost of shipping one ton
away from s
: characteristics of trading at site s
: cost per ton-kilometer at site s
: distance between a site in s and the
center of j
: joint density
s s sf ( , ) sj
0sr
s
s sr csja
s
2.2 Transportation (end)
The same construct as for farmland allocation applies
and leads to the following representative
‘transporter’ model.
sj
s s s 0s s s
ja 0 s s sj 0s s s1 sJ s
j sj s
( r c ,r ,a ,A )
max ( r c )a r G ( a ,...,a ,a )
subject to a A
3 Case 2: Noncomparable utilities
Overview:
3.1 Exact aggregation is no longer possible, unless
certain strict conditions are met.
3.2 Optimal aggregation is an attractive way out.
Methodologies originally developed in
mathematical statistics are increasingly made
available to economic analysis. Promising
research.
3.1 Exact aggregation
Suppose individuals in the smooth continuum maximize
utility from a discrete set of options, and subject to
a budget constraint, with given prices and
income .
A representative agent construct holds if :
a) consumers have common income characteristics,
b) the economy has a fixed income distribution, and,
c) consumers spend their last penny on a common
priced good.
m( )
k k k0 c ( ) c k k k
k k k
ˆ max ( u )c ( )
subject to p c ( ) m( )
kp
3.2 Optimal aggregation
Index i for consumers, i = 1, 2, ..., I
Index h for commodities, h = 1, 2, ..., H
: price of commodity h
: characteristics of consumer i
: individual demand by consumer i
: aggregate demand
Aggregate demand must equal sum of individual
demands:
ic( p,h,z )
iz
hp
Ii 1 i i i
1C( p,h ) w c( p,h,z ), for w
I
C( p,h )
3.2 Optimal aggregation (2)
Questions for optimal aggregation:
(a) How many income groups would be needed to
represent the underlying individual demand
functions ?
(b) How should the corresponding population weights
be determined ?
(c) How should the corresponding group demand
functions be specified ?
3.2 Optimal aggregation (3)
Answer to question (c):
The model for a group should simply be the model of
one individual of that group. This is required for an
analysis of welfare responses to policy reforms.
Questions (a) and (b):
boil down to an investigation into a choice of weights
other than with less than I groups. For this
we use kernel learning techniques from the vector-
support regression literature.
i1
wI
iw
3.2 Optimal aggregation (4)
The idea for obtaining weights for optimal
aggregation is to minimize the sum of squared
weights, subject to the aggregation constraint,
applied for all possible prices .
For given aggregate function and given feature
functions the optimal aggregation
problem reads
p
ic( p,h,z )
C( p,h )
i
2Ii 1w 0 i
2Ih i 1 i i
P
1 min w
21
subject to C( p,h ) w c( p,h,z ) dp 02
iw
3.2 Optimal aggregation (5)
The integral that appears in the constraint makes a
direct solution to this optimal aggregation problem
impossible.
Therefore approximation is required. For a series of
randomly sampled prices , and a
regularization term
that accounts for the fact that
aggregation cannot be exact, we obtain an optimal
aggregation model.
s i
2Ii 1 s0;w i s
Ii 1t i t i s
1 min w
2
subject to C( p ,h ) w c( p ,h,z ) s=t(H-1) h
s s htp ,t 1,...,T
We now write this model in matrix-vector notation. This
will clarify that it is a quadratic program that
possesses a particularly practical dual formulation.
The optimal aggregation model is rewritten as:
for , non-negative -matrix
and -vectors and with unit
elements.
3.2 Optimal aggregation (6)
T Tw; 0
1 min w w
2subject to w y ( )
S THt i[ c( p ,h,z )]
ty [ C( p ,h )]
S I
S 1
The key feature of this problem is its coincidence with the dual
formulation. For the positive semidefinite -matrix
defined as , optimal aggregation weights can also
be identified as after solving the dual model.
In the Chinese context with very large numbers of households
this pre-aggregation of information in the matrix
would seem necessary to find an optimal number of groups.
3.2 Optimal aggregation (end)
T T0
1 max y K
2subject to
TK
S S
Tw
TK
The full representation of transportation
economy is possible in a single-commodity
welfare model.
In multi-commodity welfare model either the
number of feasible flows has to be
restricted drastically, or, spatial
aggregation is required.
4 Spatial aggregation over markets
The representation of transport follows the
described under exact aggregation
Indices (s, r) for the sites (say, cells on a grid)
: quantity shipped from site s to site r
: price on the market at site s
: total availability of the good at site s
: cost associated with flow from
shipments
4.1 Transportation in welfare model
s s1 sS sC ( v ,...,v ,q )
sq
sp
srv
sq srv
The single-commodity representative trader model for site
s.
At given production and money-metric utility
one can define the corresponding welfare model.
4.1 Transportation in welfare model (end)
sr
s s
rv 0 s sr s s1 sS s
r sr s
( p ,q )
max p v C ( v ,...,v ,q )
subject to v q
sr s s s sv 0;q ,c 0 s s s s1 sS s
rs sr s s
rs rs s
max u ( c ) C ( v ,...,v ,q )
subject to c v q (p )
q v e
se s su ( c )
Spatial aggregation means application of the
welfare program at a larger scale, say
counties that will be indexed .
Now we can move from a single-commodity,
partial equilibrium to a multi-commodity
general equilibrium framework.
4.2 Spatial aggregation for transition from partial to general equilibrium
rv 0;q ,c 0 1 L
r r
r r
max u ( c ) C ( v ,...,v ,q )
subject to c v q (p )
q v e
However, the general equilibrium framework with spatial
aggregation of markets is to some extent
inconsistent. Within a location it abstracts from
price variation and allows supply to meet demand
along the cheapest route.
This problem can partly be overcome by assuming fixed
price differentials within a region:
Immediate extension of the framework is to allow for
production employing endowments as well as current
inputs .
4.2 Spatial aggregation (2)
s s sp p p
s s
gy ,g 0 s s s s
s s s s
max p y p g
subject to y f ( g ,e )
After incorporation of production and price variation within
regions, the general equilibrium welfare model reads:
4.2 Spatial aggregation (3)
j s s s s sv 0;q ;c ,g ,y 0;z ,z 0
gs s S ss s 1 L s s s s s s
s S js j
max
u ( c ) C ( v ,...,v ,q ) ( z z ) p g
subject to
c v q (p )
j s Sj s s s
s s s s s
s s s s
q v ( y z z )
c z y z (p )
y f ( g ,e )
This spatially aggregated general equilibrium model
forms the basis for the CHINAGRO welfare model.
It has many features to capture the response of Chinese
agriculture to changing prices and policy reforms.
It also has its shortcomings. It rules out changes in
routing, while the price band
may act as a price distortion rather than reflect true
cost.
Therefore, in parallel with the general equilibrium model,
a set of partial commodity-specific equilibrium models
is developed that do not require spatial aggregation.
4.2 Spatial aggregation (end)
s s sp p p
Spatial aggregation is a special case of aggregation
over commodities. It aggregates over commodities
that only differ with respect to location and can be
converted in one another through transportation.
Aggregation over commodities requires some sort of
nested hierarchy on the supply side (technology),
on the demand side (utility), or on both sides. Little
can be said in general about such aggregation.
5 Aggregation over commodities
I. On the representative agent (comparable utilities /
payoffs)
1) Nano-foundation of micro assumptions: Discrete choice
combined with smooth densities leads to strictly concave
and differentiable production and utility functions.
2) Profit maximizing farmers can be represented in a spatial
and social continuum, and yet their behavior follows
relatively standard micro models of production.
3) Likewise, the approach can deal with transportation from
a continuum to a finite number of market places.
4) Risk aversion behavior follows through aggregation, even
though the underlying choices are risk neutral.
6 Conclusions
II. On the representation of consumers:
1) Representative consumers are selected individuals,
not average individuals.
2) Exact aggregation is difficult under individual budget
constraints.
3) Kernel learning techniques can be used to determine
optimal level of aggregation.
4) It is “safer” to work with aggregate consumers with
utilities express in money metric and with an
exogenous marginal utility of income, as is done in
welfare programs.
6 Conclusions (2)
III. On spatial aggregation over markets
There is no clean solution. Hence, we operate two
models in parallel:
(a) a general equilibrium welfare model, in which intra-
regional trade is subject to fixed transportation
costs for all net purchase and net sales of the
county.
(b) a set of single commodity partial equilibrium models
on a 10 by 10 kilometer grid.
6 Conclusions (3)
IV On aggregation over commodities
Aggregation over commodities requires assuming
constant returns and a nested hierarchy in
production, in utility, or in both. Whether this is
warranted depends on the application at hand.
6 Conclusions (end)