Micro meets macro via aggregation

TLDR: In this article, the authors studied the fundamentals of the aggregation problem and gave an idea of under what kind of conditions the representative agent approximation is appropriate, and showed how the macro relations arise from the micro relations via aggregation.

Abstract: This paper studies the fundamentals of the aggregation problem. The concept is most central in understanding the relations between micro and macroeconomics. Though aggregation is mentioned often it is not studied explicitly in this generality - only some special cases are governed so far. The study shows how the macro relations arise from the micro relations via aggregation. In addition, we try to give areader an idea of under what kind of conditions the representative agent approximation is appropriate.

Figures

Figure 4: Results of the two Monte Carlo experiments. MB’s are drawn with filled symbols, NLE’s with triangles and HE’s with red circles. In the left figure behaviors are exponential (Aixbi) and in the right quadratic (ai +bix+ cix 2).

Figure 3: Two cases of equalization of inputs. Final states are identical and total amount of inputs remains the same. Still in the left hand side total output grows and on the right hand side it turns down. Filled symbols mark the final state.

Figure 1: Illustration of the average behavior. In the upper graph three different behaviours, fi, are drawn with solid lines and the average behaviour, f , with the broken line. In the lower graph the deviations from the average behaviour, δfi, are drawn for the different micro units.

Figure 2: Two cases of growth. Initial states and total growth of the inputs is same in both figures. Despite of that, the case on the left graph has higher growth in total output. Filled symbols mark the final state.

Full text

öMmföäflsäafaäsflassflassflas
ffffffffffffffffffffffffffffffffffff
Discussion Papers
Micro meets Macro via Aggregation
Jussi Lintunen, Olli Ropponen and Yrjö Vartia
University of Helsinki, Department of Economics
and HECER
Discussion Paper No. 259
March 2009
ISSN 1795-0562
HECER – Helsinki Center of Economic Research, P.O. Box 17 (Arkadiankatu 7), FI-00014
University of Helsinki, FINLAND, Tel +358-9-191-28780, Fax +358-9-191-28781,
E-mail info-hecer@helsinki.fi
, Internet www.hecer.fi

HECER
Discussion Paper No. 259
Micro meets Macro via Aggregation*
Abstract
This paper studies the fundamentals of the aggregation problem. The concept is most
central in understanding the relations between micro and macroeconomics. Though
aggregation is mentioned often it is not studied explicitly in this generality - only some
special cases are governed so far. The study shows how the macro relations arise from
the micro relations via aggregation. In addition, we try to give areader an idea of under
what kind of conditions the representative agent approximation is appropriate.
JEL Classification: B41, C43, C81, C82
Keywords: Aggregation, Micro Foundations, Methodology of Economics.
Jussi Lintunen Olli Ropponen
Researcher University Lecturer
Finnish Forest Research Institute (Metla) University of Helsinki
P.O. Box 18 (Jokiniemenkuja 1) Department of Economics
FI-01301 VANTAA P.O. Box 17 (Arkadiankatu 7)
FINLAND FI-00014
FINLAND
e-mail: jussi.lintunen@metla.fi
e-mail: olli.ropponen@helsinki.fi
Yrjö Vartia
Professor of Econometrics
University of Helsinki
Department of Economics,
P.O. Box 17 (Arkadiankatu 7)
University of Helsinki
FI-00014
FINLAND
e-mail: yrjo.vartia@helsinki.fi
* We thank Vesa Kanniainen and the participants of the postgraduate seminar for
supporting comments.

1 Introduction
Aggregation is mentioned in many cases and almost in every branch of eco-
nomics. The importance in studying the aggregation problem lies in the fre-
quency it is met. Who really knows what is going on in the aggregation
problem? In 1946 the aggregation problem was studied by May, Pu and
Klein. They all had a different approach to this problem, because they all
wanted to get answers to somewhat different questions. As Klein was study-
ing the aggregation problem, he wanted to know the conditions under which
the aggregate variables were enough to explain other aggregate variable. The
conditions for these are quite restrictive and exclude many of the real life
phenomena out of the scope. Unlike in Pu’s approach, the underlying dis-
tribution did not matter in Klein’s approach. Pu wanted to know whether
there exists a distribution that could be responsible for an observed macro
variable. May’s approach differ from the above two as he starts with the mi-
cro equations. The generality of his study is reduced as he has the solutions
for the micro equations that are restricted to the case in which the theo-
retical equilibrium constraints are binding. Our analysis is closest to May’s
approach. The differences appear as we relax the assumptions that May has
and we analyze aggregation explicitly, not implicitly like May. As we have
different approaches when studying the aggregation, we are faced with an
unclearly stated problem which also happens to appear in many places. This
unclearness is pointed out multiple times by Felipe and Fisher in their paper
(2003) that provides a survey on the theoretical literature on aggregation of
production functions and especially what an applied economists should know
about the aggregation. Independently of the approach, the problem of aggre-
gation is about how to fit together three things: micro equations, aggregation
rule and macro equations.
The aggregation problem seems to be rarely understood in all its glory
and for this reason we try to clarify a reader about the concept. As we
are studying this problem we are interested in how the micro and macro
systems are related with each other. We concentrate here on the core of the
aggregation problem: How does macro arise from a known micro system? By
leaving out all the randomness in this paper we emphasize aggregation being
a logical, mathematical step between micro and macro - just like calculating
1

an average from a given set of numbers is a logical step. One of the problems
in studies on the aggregation seems to be that the aggregation problem is
mixed with the other problems like the ones of modelling or estimation.
1
By
leaving the randomness out, we can isolate the aggregation problem of these
other arising problems.
In addition to already mentioned problems in aggregation, the third prob-
lem arising when studying aggregation seems to be the difficulty of the task.
According to Lewbel (1989), representative consumer models are typically
employed when one wants to ignore the complications caused by aggrega-
tion. Or as Kirman (1992) puts it: "To many macroeconomists, the aggrega-
tion problems of the sort implied by the research just described look difficult
enough that the simplification of the representative individual looks more
attractive, rather than less".
There has been a number of studies that are concerned with the aggrega-
tion at some level. The papers by Grunfeld and Griliches (1960) and Pesaran
et al. (1989) both deal with choosing between micro and macro regression
equations. They treat micro and macro as substitutes, which is very oppo-
site to what we are doing here. We want to find a bridge between the micro
system and the corresponding macro system. We want to unite these, not
separate.
Also Klein (1946a, 1946b) has a different approach than we do. There
is a wide technical literature on Klein’s approach, where macro relations
are assumed to be similar to micro relations. Leontief, Nataf and Gorman
as Classic experts of Klein-aggregation have shown, that it leads to very
restrictive separability and additivity assumptions (see their references). The
strict conditions under which his approach works exactly and the aggregates
as such are enough to describe the situation are shown by Nataf (1948)
and Gorman (1953, 1959).
2
More recent experts are Green (1964), Fisher
(1992) and Pokropp (1972, 1978). If the conditions are not met, we face
aggregation bias
3
by using Klein’s approach. This is the bias because of
1
It seems that the aggregation problem is sometimes even confused with the estimation
problem with aggregate variables.
2
It is emphasized also by Ando (1971), Gupta (1971) and Browning (1993) that the
aggregate variables are not enough to describe the situation on their own.
3
The studies by Theil (1957), Gupta (1971) and Buse (1992) give a description for this
2

not aggregating, but estimating with aggregate variables. In this case the
macro parameters are biased and according to Gupta, the existence of the
aggregation bias can sometimes completely distort, not only the magnitude,
but also the signs of the macro parameters. As Nataf and Gorman show
the exact conditions, we try to give a description of when the representative
consumer approximation is plausible (even if not exact) and thus parallel the
Klein’s approach in this sense. The general conclusion seems to be that Klein-
aggregation is practically impossible (see Felipe and Fisher (2003) chapters
3-5 for discussion).
The notion of the macro relations not being similar to the micro rela-
tions is made both in linear cases
4
(Theil (1959), Klock (1961), Ando (1971),
Gupta (1971), Pesaran et al (1989)) and in nonlinear cases (Muellbauer
(1975), Stoker (1986), Buse (1992), Vartia (2008a, 2008b, 2009)). Because
of these differences, which can be seen as the aggregation bias, the represen-
tative agent approximation is not an appropriate approximation to describe
the macro in general. Carroll (2000) thinks that in many cases the repre-
sentative agent should have no future as he states: "For many purposes, the
representative-consumer model should be abandoned in favor of a model that
matches key microeconomic facts." In our study, we show what are the causes
that separate macro, that is aggregated from micro system, from the repre-
sentative consumer approximation. This way we can give for the macro the
micro foundations and deduce the conditions under which the representative-
agent approximation is plausible.
The basic problems preceding the logical step of aggregation are pointed
out by Blundell and Stoker (2005). Because, we can define the rule for the
aggregate
5
in many ways, how to choose what the appropriate aggregate is.
After the choice of the aggregation rule we still have to choose what are the
suitable micro and macro levels to work with. In this paper, we take these
as given as also the knowledge of the micro system. Given these choices, we
are faced with the step that we call the aggregation step. In this step, we
deduce what kind of macro arises from these via aggregation and for what
very central concept appearing in the context of aggregation.
4
The linearity is a special case and this makes aggregation much simpler than in the
case of non-linearities. This was noticed by De Wolff already in 1941.
5
For example averages and index numbers constitute sets of aggregates.
3

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Micro meets macro via aggregation" ?

This paper studies the fundamentals of the aggregation problem. Though aggregation is mentioned often it is not studied explicitly in this generality only some special cases are governed so far. The study shows how the macro relations arise from the micro relations via aggregation. In addition, the authors try to give areader an idea of under what kind of conditions the representative agent approximation is appropriate. 

Q2. How many Euros does a perturbation cost?

In addition, let the standard deviation of the perturbation be σ = Dε = 100 Euro, that is about 50% of the standard deviation of the original variable. 

Q3. What is the simplest way to explain the macro dependencies in economics?

Macro dependencies in Economics are context dependent and vary slowly in time, but once the circumstances are specified they are very stable. 

Q4. What is the aggregation of micro level outputs?

Micro level outputs can be collected together into the micro systemyt = f(xt) = A(f, xt).3◦ Macro output Y t is the total of yt(a), that is Y t = ∑Ht y t(a). 

Q5. how can i decompose the change of the output?

To ease the aggregation phase one can decompose the change of the output a bit further by adding and subtracting fi ′(x(i))dxdy(i) = fi ′(x)dx + [fi ′(x(i))− fi ′(x)] dx + fi ′(x(i)) [dx(i)− dx] . (19)The change in macro behavior can be calculated normally as a weighted averagedy = n∑i=1widy(i). 

Q6. What are the papers by Grunfeld and Griliches?

The papers by Grunfeld and Griliches (1960) and Pesaran et al. (1989) both deal with choosing between micro and macro regression equations. 

Q7. What is the definition of a macro behavior?

If changes in inputs are correlated with the behavior at the margin, then changes of the representative behavior are not adequate approximations of the macro behavior. 

Q8. what is the difference between the representative consumer approximation and the true aggregate?

From here the authors see that the the representative consumer approximation differs from the true aggregate and this is due to non-linearity of the cross-terms.