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)