Thesis Proposal
“A Multisector Out-of-Equilibrium
Model of Growth”
Matteo Degasperi
CIFREM – University of Trento
30 January 2007
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1. Background and Motivation
Modern economic models can be divided into two subgroups: models based on
the Walrasian type of general economic equilibrium and those that seriously challenge
this approach. The dominant view seems to be that in which the convergence towards,
and the existence of the general economic equilibrium is postulated.
In spite of the claimed microfoundation of macroeconomics, the standard macro
models are highly aggregated and the micro level interactions are almost always
disregarded. There is a representative consumer that maximizes a neoclassical wellbehaved (intertemporal) utility function subject to the constraints of a neoclassical wellbehaved production function (consumption goods and capital goods are
homogeneous).
In this class of models, the representative consumer-producer takes decisions as
if he was a price-taker, but the ‘micro’ economic system is assumed to be such that
prices turn out to be ‘flexible’ so as to guarantee instantaneous general equilibrium.
(Sargent 1972, 1976 are representative of these classes of models)
Deviations from the above descriptions are models in which prices do not adjust
instantaneously, but the difference is in terms of the speed of adjustment with the
tendency for the system to adjust towards general equilibrium prices only occasionally
questioned (for example, see Woodford, 2003; Chiarella and others, 2005).
Starting from the late 1960s, there has being a growing tendency of requiring
‘microfoundations’ for ‘macroeconomic’ models. Most of these models assume that
markets are always on full employment equilibrium (market clearing) and that
adjustment to exogenous shocks is instantaneous. The standard assumption is General
Equilibrium of the Walrasian type. In particular, I would cite as examples of this
tendency the models of Rational Expectation Monetary Business Cycle (Lucas, 1972;
1975), the Real Business Cycles (Kydland and Prescott, 1982) and the Overlapping
Generation (Auerbach and Kotlikoff, 1987)
A typical basic structure of these models is given by some variations of the
Ramsey-Cass-Koopmans model (see, for example, Romer, 2001, chapters 1 and 2) where
a representative agent maximizes an intertemporal neoclassical well-behaved utility
function
∞
(1)
max ∫ e − ρt u (c(t ))dt
0
subject to a budget constraint defining the feasible set with a well behaved aggregated
neoclassical production function
(2)
Y (t ) = F ( K (t ), A(t ) L (t )) = w(t ) L (t ) + r (t ) K (t ) ≥ c(t )
completed with the usual transversality conditions and assumptions with respect to the
temporal evolutions of the variables L(t), labour, and A(t), knowledge or labour
productivity.
The assumed Walrasian General Equilibrium ‘guarantees’ that w(t) and r(t) are
market clearing prices: the wage rate is equal to the marginal productivity of labour and
the profit rate is equal to the marginal productivity of capital.
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A problem often disregarded is the relation existing between a highly aggregated
model like the above and a real-world situation in which there are heterogeneous goods
that can be produced in many alternative ways. As a matter of fact, during the 1960s
there was a vigorous debate between two different schools of economic thought that
went under the name of “Cambridge-Cambridge controversy”. It was designated in this
way because it involved the economists Piero Sraffa and Joan Robinson who worked in
Cambridge (UK) and the economists Robert Solow and Paul Samuelson who worked at
the MIT, in Cambridge, Massachusetts. This dispute focused on the theory of
distribution and the nature and the role of capital goods and led to the famous
Quarterly Journal of Economics Symposium, 1966. The problem was precisely that of
the tenability of the aggregate neoclassical production function, i.e., on the possibility of
finding an index-number for the aggregated capital, let us call it Kv(t), such that in the
aggregate dKv(t)/dr(t)<0 and dL(t)/dw(t)<0 hold. Obviously if there is only one good,
and only one method of producing it, the aggregation problem does not, by definition
and/or by construction, exist.
At the end of the 1960s and during the 1970s, several authors recognized the
logical possibility of the existence of the aggregation problem in the index-number
problem just specified above, but most authors (for example, see Samuelson, 1966;
Ferguson, 1969; Sato, 1976) considered that possibility as if it was just an unlikely
perversity and supported the view that “there exists a not too-small world” (Sato, 1974,
p.568)” in which the neoclassical postulates hold. Apart from these acts of faith, very
little research has been conducted in order to assess the magnitude of this likelihood.
Zambelli (2004) has provided, through the implementation of a simulation
model, the first attempt to conceptually evaluate this likelihood. In short, given an
heterogeneous production system of the von Neumann-Sraffa-Leontief type, in which
the input matrix is given by A and the labour inputs by L, while output by the diagonal
matrix B and assuming that the system is in the long-run self-replacing equilibrium,
where the rate of profits are uniform across industries, we have the following set of
equilibrium relations:
(3)
(4)
Ap (1 + r ) + Lw = Bp
η ′p = 1
[
]
−1
w(r) = η′[B − A(1+ r)] L
(5)
[
−1
]
p(r) = [(B − A(1+ r))] Lη[B − A(1+ r)] L
−1
−1
−1
Given the above equilibrium prices, a measurement of ‘aggregate output’ and
‘aggregate capital’ is given by
(6)
(7)
Yv = e′( B − A) p(r )
K v = e′Ap(r )
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Zambelli’s conclusion is that the likelihood that in a situation in which several
methods of production are available and the most ‘efficient’ methods of production are
always chosen and the long-run equilibrium holds, the production function is
neoclassical for the whole feasible domain only 40 percent of the time. Moreover, this
being the case, the fact that the value of capital measured by (7) may be an upward
sloping function of the profit rate r implies that at least for some industries (and hence
for the firms belonging to them) the equilibrium micro isoproduct curves are also non
‘well-behaved’.
However, the problem of aggregation and the implicit assumption of general
equilibrium have also been present in the old as well as in the new models of growth.
Indeed, in the 1980s, there was a renewed interest in models of growth that were
pioneered by Romer (1986) and Lucas (1988). This new generation of models
maintained some fundamental elements that characterized the first wave of growth
models (Solow, 1956; Swan, 1956; Cass, 1965; Koopmans, 1965).
Nonetheless, the new models incorporated some innovations with respect to the
older ones. In fact, they have been labeled “endogenous growth models” in opposition
to the “exogenous growth models” previously developed. The Solow’s model and its
successors were defined as “exogenous” because the rate of growth was determined
outside of the model. More specifically, growth was possible only in the presence of an
assumed rate of technological progress or in the presence of labour force growth, but
such phenomena were not explained inside the model. In order to overcome these
shortcomings, the new models endogenized technological progress. The two common
methods employed by endogenous growth theorists were to incorporate the
assumption of growth in the form of spillovers, and by the assumption of increasing
return.
Following the classification of Aghion and Howitt (1998), these efforts have taken
different directions. Romer (1990) built a model of horizontal innovation where growth
is caused by the development of new products. Other researchers created models of
vertical innovation that ensure economic progress by imposing innovation on the
existing products (Aghion and Howitt 1992). Finally, more ambitious studies tried to
model complex phenomena as the distinction between fundamental and applied
research. Fundamental research can guarantee potential gains that they will materialize
only if the new knowledge can be applied in practice (Aghion and Howitt 1998).
Notwithstanding such progress, endogenous growth models confronted by
severe drawbacks concerning adoption of a representative consumer and the use of an
aggregate production function faces empirical as well as technical difficulties. Many
authors (for example, see Pasinetti, 2000) point out that the new growth theorists have
used an aggregate production function without worrying about its practical and
mathematical justification.
Actually, one of the main issues during the “Cambridge-Cambridge
controversy” involved the problem of adopting the aggregate production function. The
aggregate production function, which is a mathematical function that relates the total
output generated to such input factors as labour, land and capital. While the
aggregation of labour and land is easily represented by the adoption of a common
measure to express their apportion by using the number of hours worked and land
productivity, the aggregation of capital is highly problematic. Capital is formed by
many heterogeneous types of machinery that can be expressed with a unique measure
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based on their value. Robinson (1956) pointed out that the value of capital changes as
the profits or the wage rates change so that the same physical capital can represent a
different value, whereas a different stock of capital goods can have the same value.
The use of a utility maximizer representative agent constitutes another serious
limitation for models of endogenous growth. Kirman (1992) pointed out that “the
reduction of the behaviour of a group of heterogeneous agents, even if they are all
themselves utility maximizers, is not simply an analytical convenience, but is both
unjustified and leads to conclusion which are usually misleading and often wrong”.
Hence, there are many reasons to be unsatisfied with the new and old models of
growth. Their foundation in the neoclassical paradigm is essential for their existence.
Guarini (1998) showed that by imposing a positive relation between wage and labour or
between rate of interest and capital on the model of Aghion and Howitt (1998), it would
provide a result at odds with respect to what the authors found. In addition, the
introduction of the hypothesis of increasing return of scales, which makes it closer to
the real-world situation, will prevent the model from reaching the equilibrium.
In the literature, it is rare to find models of growth built by assuming the
existence of two or more sectors. Uzawa (1961, 1963) developed a two-sector model of
economic growth, but his work had the objective of distinguishing between two
productive sectors: one producing consumption goods and the other producing
investment goods. Nevertheless, this model still assumes production functions that
fabricate homogeneous goods by the use of homogeneous inputs. Moreover, the model
is grounded in the neoclassical framework, where constant return-to-scale and
diminishing marginal rates of substitution are the rule.
Other interesting examples are the so called dual models, which are common in
the framework of analyzing the growth of developing countries. The economy of these
countries is usually divided into agricultural and non-agricultural sectors, each of them
with their output, inputs, and growth rates. However, the disaggregation of the
economy into two sectors has been still too simplistic for grasping the complexity of the
reality.
The aim of this research is thus to propose an alternative approach to model the
production side of the economy that moves away from the neoclassical paradigm and is
to closer the classical tradition. The idea is to follow the path traced by von Neumann
(1937) when he built a general equilibrium model where the production of each
commodity is made by the use of other commodities. The model of Von Neumann can
be seen as a precursor of the well-known essay written by Sraffa (1960), Production of
Commodities by Means of Commodities. The objective is thus to model a system where the
flows between sectors are evident and where the production of each sector utilizes
heterogeneous types of capital coming from the other sectors. An interesting example of
these studies is the work of Salvadori (1998) and obviously also the tradition of the
steady state growth models presented in studies such as Pasinetti (1981). These works
all analyze the properties of the economic system by assuming that the system moves
along a balanced path. While these authors avoid aggregating, they assume a sort of
general equilibrium to hold at each point in time.
Notwithstanding this progress, these models are not exempt from criticism. In
particular, they assume that the economic system operates in a situation of general
equilibrium. This assumption is common in mainstream macroeconomics, and
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according to some authors it represents one of its weakest points. Temple (2006, pp.313314) observes:
“one of the central conceptual weaknesses in general
equilibrium models has always been the requirement that all
trades take place without any friction, only at equilibrium
prices, at one instant in time: the useful but abstract device
of the Walrasian auctioneer. Like many commentators before
me, I suspect that macroeconomics without an auctioneer
might look very different to the world of Arrow-Debreu.”
In the same article he argues that this leads to a “controversial conclusion”:
“I think any researcher who writes down or estimates an
aggregate production function needs to be aware of how
dangerous this can be, and aware of some of the alternatives.
At the same time, if orthodox general equilibrium
macroeconomics has an Achille heel, it is perhaps more
likely to be the auctioneer than the strict conditions needed
for aggregation.”
Other criticism have came from some economists of the Department of
Economics at the University of Trento where there is a tradition of challenging some
elements of neoclassical orthodox economics (for example, see Leijonhuvfud, 1981;
Velupillai 2005(a), 2005(b); De Antoni, 2005). In particular, Leijonhvfud (1968)
questioned for the first time the truthfulness of the Walrasian auctioneer.
Finally, I would cite the tradition of out-of-equilibrium studies present in the
works of the Santa Fe Institute. Arthur is one of the main exponents of this tradition,
and in a recent article he observes (Arthur, 2005, p.2)
“It is natural to ask how agents’ behaviour might not just be
consistent with the aggregate pattern it creates, but how
actions, strategies or expectations might in general react to
the pattern they create. In other words, it is natural to ask
how economy behaves when it is not a steady state – when it
is out of equilibrium. At this more general level, we can
surmise that economic pattern might settle down over
sufficient time to a simple, homogeneous equilibrium. Or,
that they might not: they might show ever changing,
perpetually novel behaviour. We might also surmise they
might show new phenomena that do not appear in steady
state.”
This statement is particularly true in the framework of economic growth, where
the stimuli that are at the base of the behaviour of consumers and firms are the desire of
ensuring a larger slice of an increasing pie. It is thus natural to assume that the agents’
behaviour will no longer be accommodating to the steady state equilibrium growth
path, but will show continuous variations.
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2. The Multisector Out-of-Equilibrium Model of Growth
It seems that General Equilibrium Theory of the Walrasian-Arrow-Debreu type is
inadequate as a foundation for a reasonable model of an economy and at the same time
it also seems that the available disequilibrium fix-price models operate with highly
aggregated variables and few behavioural functional forms (for example, see
Woodford, 2003; Chiarella and others, 2005; Godley and Lavoie, 2007).
The first objective of this project is to specify the behaviour of a multisector outof-equilibrium model. The above equation (3) characterizes a long-run equilibrium
condition. An interesting question is: What types of additional assumptions and
functional forms have to be added in order to study the dynamics of a multisector
model in dis-equilibrium?
Equation (3) can be reformulated in the following way:
(8)
X t At pt ⊗ (1 + rt ) + X t Lt ⊗ wt = X t +1 Bt +1 pt +1
where now rt and wt are vectors and the quantities actually bought and sold are not
necessarily such that the system will reproduce itself with equilibrium proportions.
How can such a process be described without assuming market clearing conditions (like
those implicit in (3))?
Concerning the production side of the economy, the procedure for constructing
the model will be implemented in such a way that an empirical use could entail the
possibility of adapting the input-output production matrix relative to a country or a
region.
It is known that the input-output production matrices present limitations linked
to the fact that they have fixed technical coefficients, and the estimation of these
coefficients is often made through past surveys that might not be appropriate for the
present. Moreover, the matrices seem more proper for a relatively closed economy,
since they do not take into account import and export flows.
As a positive counterpart, input-output matrices overcome the problem of
aggregation of capital and they have constituted one the best attempts to deal with
complexity on the production side. As a matter of fact, they provide the opportunity for
understanding macro behaviour by looking through the interconnections between
economic sectors, and thus realizing a micro-macro relation of the economic system.
The realization of the model entails three steps.
i. The description and simulation of disequilibrium
Clearly, an out-of-equilibrium dynamic behaviour has to include the
specifications of the functional forms describing consumers as well as producers
demands and supplies and their interaction. The above equation (8) encapsulates, so to
speak, most of the elements that are fundamental for national accounts and hence
express the simple fact that if some goods and/or services are purchased by some
agents it means that they are sold by others. While in equilibrium this simple fact may
be disregarded (because the purchasing power to buy goods is precisely matched by the
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purchasing power acquired when selling them) a disequilibrium condition is by
definition a situation in which exchange can take place only if some form of credit-debt
relations emerge. This simple reflection should shed light on the fact that
disequilibrium logically requires the specifications of behavioural relations to be
defined and studied at a more disaggregated level. This behavioural description should
contain information regarding consumption, investment, and saving decisions as well
as trading rules at some lower level with respect to the macro-relations already
encapsulated in (8).
The method will be first to complicate the model gradually so as to be able to
account for disequilibrium relation, but new unnecessary elements or complication
should not be introduced. For example, allowing for (8) to be able to move from time t
to time t+1 outside the equilibrium means to be able to specify some form of decision
processes and trading rules, but does not imply that the dimension of the problem in
terms of the number of commodities, of agents considered, and of the methods of
production available have to be unnecessarily large.
In the presence of non market clearing prices there will be some sectors with
higher wages or higher profits than others. This inequality in wages and profits could
be maintained in presence of imperfect mobility for workers and owners of capital.
How does the economy behave in this condition of disequilibrium? Will the system
converge toward long-run equilibrium at the end?
Given the nature of the proposed model, it is most likely that analytical solutions
will not be available: therefore, a method of investigation like the one implemented, for
example, in Zambelli (2004) could be followed. In short, the method is to collect
statistics of results emerging from a large number of simulations where some
fundamental parameters are allowed to vary.
ii. The implication of disequilibrium for growth
In this step, the research will try to investigate whether the disequilibrium or the
equilibrium are the best condition for economic growth. So far it is not clear whether
the situation of perfect market clearing drives the economy towards a higher growth
rate than the condition of perpetual disequilibrium. A steady state where agents have
no interest in changing their behaviour could represent an ideal frame for economic
growth given the high level of certainty that it secures. At the same time, an everchanging condition where the agents modify their behaviour according to some
learning rule could boost the growth rate even more dramatically.
iii. The role of fiscal policy
Once the results of the previous steps are compiled questions of vitality,
convergence, and growth of the system should become, in the context of the model,
clear. At this level a typical question would then be that of investigating the importance
of certain policies with respect to others in terms of some objective functions. Questions
of taxation, redistribution of resources, Government expenditure and economic policy
could be addressed and new results – compared with those computed during steps i
and ii – would be collected.
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The role of the public sector will be examined in two interrelated directions. The
first looks at the influence of different public policies on growth. Assuming a certain
level of taxation and the extent of the public sector in the economy it will be possible to
determine under which policy the system can reach the highest growth rate. The second
examines the role of the public sector in order to push the economy toward equilibrium
when the system does not converge on it. Clearly, these two aspects exhibit many
linkages either between them or among the results that step ii will provide.
3. Expected Results and Possible Applications
This new type of model could be a useful starting point in order to develop a
generalized version where there are n individuals and m production functions. On
empirical grounds, this model can be tested using parameters that come from survey
and input-output production matrices. The introduction of fiscal parameters as well as
the role of the public sector ensures more credibility for the model. In order to
incorporate data inside the model it could be useful to follow the method of calibration
that characterizes the works of real business cycle theorists. The model allows a broad
range of applications that include national to regional analysis of economic growth.
Given the adoption of input-output tables for empirical testing, the predictive validity
of the model should be more powerful in a relatively closed economy.
Besides the possible usages of the model, each single step of this research is
capable of generating interesting theoretical results. The analysis of disequilibrium in a
von Neuman – Sraffa – Leontief economic context has received little attention from
economists up to now. Moreover, it is rare to find in the literature works that examine
the effect of disequilibrium on growth.
Hence, this research can constitute a useful starting point to illuminate the effect
of interactions among diverse productive agents on the economy as a whole. Finally,
the introduction of the public sector can give interesting results that in future could be
translated into public policy advice.
Appendix: Connection with Data Programming Languages and Other Projects.
The instruments of programming languages and simulation tools are essential
for implementing a theoretical out-of-equilibrium model where different agents and
firms operate. The description of the diverse agent’s behaviour through mathematical
functional forms, the description of interaction between them, and the introduction of
dynamic into the model require a high computational capacity that only a
programming language can provide. Matlab seems to be one of the most appropriate
tools for this simulation.
The above research program is highly theoretical. Nevertheless there is ground to
expect that some ‘calibrations’ of the models and some of the results could be made
using available data sets. In particular, it would be interesting to see whether it is
possible to use the OECD Input-Output Data Base and Regional Input-Output
production matrices. Indeed, the model envisages the possibility of future empirical
testing on the Province of Trento. Recently, the Statistical Office of the Province has
updated input-output coefficients, and these data could be exploited to develop a fresh
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analysis of the local economy. At the same time, Trentino as well as other Italian regions
are developing local econometric models with the cooperation of Prometeia. These
projects consist of a model, which integrates traditional simultaneous linear equations
with input-output tables, so as to estimate the effect of aggregate demand components
on the productive sectors of economy. These models follow the trend of developing
large scale models and it is situated outside the traditional construction of models in
academic context, but it is effective for providing a model capable of carrying out
impact and scenario analyses, which are linked to hypotheses regarding the basic
factors of regional economies. Furthermore, this model disaggregates the local economy
in several fields of economic activity and thus it is appropriate to analyze the impact of
tax restructuring, scheduling different tax rates in accordance to the firms’ different
fields of activities.
The model that I am proposing complements and does not substitute
econometric models. They are two alternative instruments, which follow different
methodologies. A comparison between the outputs provided by each of the models and
their structures represents a way to overcome their limitations and to make further
improvements.
Finally, there could be linkages with the line of research followed by Boero
(2005). He developed an agent-based model of the Piemonte where the behaviour of
firms is described by the combination of empirical data coming from the input-output
table and a hypothetic stochastic rule of action. This work aim to avoid any use of
stochastic rules. The ambition is to describe agents’ behaviour in a deterministic way to
get an outcome that will be the result of economic actions and constraints and not the
result of the laws of probability.
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