Download Instability in a Macroeconomic Agent Based Model, filling the gap

Working paper
Instability in a Macroeconomic Agent Based Model,
filling the gap between micro and macro theories
Florian Botte, Lille 1 University
First draft : October 2014
Abstract
The aim of this paper is twofold: first, it deals with the macroeconomic analysis of instability inside a Macroeconomic Agent-Based
Model (MABM); second, it explains why Agent-Based Modeling can
be a useful tool for theoretical investigation in economics regarding
the debates on the microfoundations of macroeconomics. We implement Kaleckians’ hypothesis which should avoid instability in a simple
MABM and show that instability keeps emerging. In the model, full
employment is a ceiling for upward instability and can reverse the dynamic to downward instability. The system can produce endogenous
cycle if aggregate demand is increased by an autonomous expenditure.
We also clarify our position in the debates on the so-called microfoundations of macroeconomics and about the place of MABM in economic
modeling.
1
Introduction
The question of macroeconomic instability has been around for a long time
since Harrod (1939) extension of Keynes’ ideas to the long run. There is a
lot of reasons, theoretical, practical, empirical, to not take it into account in
long-run analysis. The main one is probably about the empirical relevance
of Harrodian instability : economies don’t explode or implode, the Harrod’s
knife edge does not seem relevant at all.
Despite the fact that stability is generally considered as an imperative
for the real-world relevance of the model, there is a recurrent critic of this
implicit assumption (Dallery, 2010; Skott, 2012). Among them, there is
propositions for alternative models which consider Harrodian instability seriously (Cordonnier & Van de Velde, 2009; Ferri et al., 2011; Skott, 2012;
1
Fazzari et al., 2012). In a recent article, Fazzari et al. (2012) develop a
model subject to Harrodian instability and explain why it can play a major
role in long-run dynamic. On the other side, the Kaleckian responses can
look quiet convincing when they argue that richer versions of the Kaleckian model can cop with instability and preserve their main properties (Hein
et al., 2012).
To enrich the reflection, we propose a new framework to investigate
the question of stability. Considering the fact that the main arguments
brought by Kaleckians concentrate on the firms behaviours and particularly
the investment decisions, we choose to give up the representative agents
and aggregate functions to focus on the behaviour of individual firms in a
Macroeconomic Agent Based Model (MABM) in order to investigate the
question of instability through simulations.
We propose implentation of some of the hypothesis presented by Kaleckians as more realistic and which should avoid instability such as : range of
values rather than single value, non-systematic reaction to stress in capacity
utilisation or endogenous expectations based on past events1 .
Our point is to explain why this question should be central into macro
dynamic analysis despite it doesn’t seem attractive regarding the empirical
facts. We argue that instability is a fundamental characteristic of monetary
production economies subject to radical uncertainty.
The rest of the paper is organized as follows : the second section presents
the demand-driven Macroeconomic Agent Based Model, potentially subject
to Harrodian instability. Section 3 discuses the results of the simulations.
We show that full employment can be a ceiling for upward instability and can
reverse the dynamic to downward instability. The downward instability can
be followed by a recovery if aggregate demand is raised with an autonomous
expenditure. The system can produce endogenous cycles. In section 4,
we precise our position in the debate about micro-foundation. Section 5
concludes.
2
The Model
2.1
Underlying hypothesis
The underlying hypothesis of the model are :
i) historical time : “in a historical model, causal relations have to be
specified. Today is a break in time between an unknown future and an
1
See Hein et al. (2012) for a survey around stability issues in Kaleckian models.
2
irrevocable past. What happen next will result from the interactions of the
behaviour of human beings within the economy. Movement can only be
forward” (Robinson, 1962).
ii) radical uncertainty : “In a complex economy, since the consequences
of individual choices depend on what all the others are autonomously doing, people take actions into an environment characterized by radical or
endogenous uncertainty.” (Gatti et al., 2011).
iii) procedural rationality : “Having shown that fundamental uncertainty
precludes the existence of a best solution, and that the limited computational
abilities of the human mind preclude the search for the best solution, we are
thus left with the notion that procedural rationality is the process that leads
to finding good solutions.” (Lavoie, 1992).
For the sake of simplicity, there is a fixed-coefficient production function,
the availability of external finance does not constrain investment, investment
is irreversible, wages and productivity are fixed.
2.2
Firms
Because firms have to cope with radical uncertainty, they behave following rules of thumb rather than maximisation principles. “Simple adaptive
heuristics are the most plausible assumption ex-ante, i.e. from the empirical
evidence (...).” (Assenza et al., 2014)
2.2.1
Desired level of production
The firm i determine its desired level of production Qdi,t by choosing the level
of desired capacity utilisation udi,t related to its production capacity P Ci,t .
Qdi,t = udi,t · P Ci,t
(1)
The firm keep a stock of finished products ηi,t in order to respond to an
unexpected increase in demand, it plays a role of buffer in short run. The
level of stock considered as normal is equal to the normal level of production
of one period (un · P Ci,t ). If the level of stocks is considered as normal, the
firm determines it desired capacity utilisation udi,t according to its exoected
e and its production capacity P C .
sales Si,t
i,t
udi,t =

1
e
Si,t

P Ci,t
e ≥ PC
si Si,t
i,t
else
3
(2)
The firm take into account a variation in its stock only if it exceed a
certain percentage τ of the normal level of production. According to this
rule, the firms corrects its cumulated forecasting errors by a change of factor
κ of its production based on expected sales.
udi,t =

a
(1 − κ) · Si,t
P Ci,t
if ηi,t ≥ un · P Ci,t · (1 + τ ),
e
(1 + κ) · Si,t
P Ci,t
if ηi,t ≤ un · P Ci,t · (1 − τ )
(3)
The expectations of sales are endogenously determined by an adaptive
rule considering the sales the firm experienced in the past. The expectation
e is a weighed average of current and past sales with exponentially
of sales Si,t
decaying weights (Assenza et al., 2014).
e
e
Si,t
= σSi,t−1
+ (1 − σ)(Si,t−1 )
(4)
The anticipations of sales implies inertia in the decision of production,
that is to say if σ is not null, the firm keeps in memory informations about
its previous levels of sales because the firm considers that a change for one
period does not necessarily mean this change will remain and “The fact that
adjustments entail costs (and large adjustments entail large costs) suggest
that firms will try to avoid frequent drastic adjustments and oscillatory
over-shooting.” (Wood, 1975).
The equation 5 is a synthetic representation of the firms’ utilization
decision :
udi,t =

Sa

(1 − κ) · P Ci,ti,t













if ηi,t ≥ un · P Ci,t · (1 + τ ),
else if

Se


(1 + κ) · P Ci,ti,t


if ηi,t ≤ un · P Ci,t · (1 − τ ),
else if





1
e
Si,t

P Ci,t
e ≥ PC
si Si,t
i,t
else
(5)
2.2.2
Labour requirement
When the firm knows its desired level of production, she computes the numd according to their productivity α and post vaber of worker required Ni,t
cancies. There is a possibility that the firm is not able to employ the desired
number of workers, is case of full employment for example.
d
Ni,t
= Qdi,t /α
4
(6)
2.2.3
Actual level of production
The actual level of production Qi,t equal the desired level of production Qdi,t
when firms employ all the workers needed, if not, the production depends
on the actual number of workers employed Ni,t .
Qi,t = αNi,t
2.2.4
(7)
Investment
In this model, investment is an investment of pure capacity 2 . Investment
targets a normal capacity utilisation un .
The firm decides to invest or not each λi periods (λi ∈ (λmin , λmax )),
by means of a difference between its expected capacity utilisation uei,t =
e /P C
Vi,t
i,t and the reaction capacity utilisation ur . The firm chooses the
level of investment considering the difference between the expected capacity
utilisation uei,t and the normal capacity utilisation un . Because there is only
one good in the model we have to specify that when firms invest, they buy
ω goods to increase their production capacity of one unit.
(
Ii,t =
P Ci,t−1 · (uei,t − un )
0
if uei,t > ur ,
else
(8)
So the production capacity of the firm evolves depending on the investment and the depreciation of capital δ.
(
P Ci,t =
2.2.5
P Ci,t−1 · (1 + (uei,t − un ) − δ)
P Ci,t−1 · (1 − δ)
if uei,t > ur ,
else
(9)
Financing the production
Firms request for a bank loan according to their production plan and their
liquid resources, i.e. if there is a financing gap between their liquidity and
their requirement for wage bill and investment. For simplicity we assume
their is no credit rationing in the model, the bank always accommodate the
demand for credit.
Firms pay off their loan according to a simple rule : at each step the firm
i will pay to the bank an instalment corresponding to a fraction θ ∈ (0, 1)
of its total debt.
2
See Cordonnier & Van de Velde (2009) for further details.
5
Ri,t = θLi,t
2.2.6
(10)
Pricing
The pricing function is an oversimplified version of Wood (1975) in which
the margin only serve the growth strategy of the firm. The firm modifies its
price each period.
pi,t =
2.2.7
w
Ri,t
+
α un · P Ci,t
(11)
Micro behaviours and stability
Hein et al. (2012) survey the reasons why the Kaleckian models are not
necessarily subject to Harrodian instability in more elaborated version. Despite the fact their paper focuses on a macro model, they bring up arguments
about the behaviours of individuals firms. We try to implement this kind of
behaviour in the model because we consider they are quiet consistent with
individuals trying to deal with the real world uncertainty.
First, we implement an adaptive rule of expectation that is reminiscent
of the argument developed by Wood (1975).
Second, the firms behave considering an acceptable range of values for
their capacity utilization and their inventories of finished products rather
than focussing on a single value. This kind of rule of thumb has a buffer effect
consistent with the fact that agents in the real world don’t change entirely
their plans because of a small change in their environment. If they are still
in a range they consider comfortable, they keep their strategy constant.
Third, “Given real-world uncertainty and the fact that capital decisions
are irreversible to a large extent, firms may be very prudent, so that the
Harrodian instability may not be a true concern in actual economies, at least
within a range of utilisation rate values.” (Hein et al., 2012). In the model,
firms are really careful and don’t engage in investment strategy without a
significant and durable stress on their utilization capacity.
In addition, firms only adjust their productive capacity considering the
difference between their effective rate and their desired (or normal) rate of
utilization. So in the best case, accumulation will never be up to 1 − un in
proportion of their actual productive capacity. For example, if ever a firm
i fully utilize its productive capacity uei,t = 1 since an infinite number of
periods and has a normal rate of utilization un = 0.85, the accumulation
potential of the firm is 15% every λi periods, this without considering any
6
constraint on labour or capital goods supply. This case is an extreme limit
for an individual firm and is very unlikely, let alone the possibility of all the
firms being in this situation.
We wanted to see if instability can arise in even in models built with
assumptions that are used to prevent instability.
2.2.8
Bankruptcy
At the end of each period, firms are declared bankrupt if their net worth
is negative, i.e. if the sum of their liquidity, their capital and their stock
vauates at the period selling price is under the total of their debt.
We use a simple ad hoc rule to maintain the number of firms constant:
when a firm goes bankrupt, it is replaced by a new one with the productive
capacity to employ only one household.
2.3
Households
The household j computes its expectated income considering their actual
earnings ej,t - that can be w if it is employed or 0 if they are not - their past
income, and a memory parameter φ ∈ [0, 1] :
eej,t = φ · eej,t−1 + (1 − φ)ej,t
(12)
The household j define its budget Bj,t dedicated to consumption given
its propensity to consume, its expectation of income and a fraction of its
deposits Dj,t :
Bj,t = c · eej,t + · Dj,t
2.4
(13)
Bank
The bank have a very passive role in the model. It keeps the liquid assets
of the other agents and offer loans when firms need it. There is no risk
management implemented and interest rate is fixed.
2.5
Competition
The model lies on imperfect competition characterized by the fact that consumers cannot visit all sellers and only know information about price and
the availability of goods of the firms he visited.
7
3
Some scenarios of dynamic instability
The following figures represent the aggregate outcomes of the model : u
correspond to the weighted average of capacity utilization by productive
capacity of firms, N correspond to the rate of employment, P C correspond
to the sum of productive capacity of firms, Real GDP or Q correspond to
the sum of output of firms in real terms.
The model shows that its dynamic don’t allow a durable full employment,
if it ever attains it. The reason of that lies in the fundamental hypothesis
we implement - i.e. historical time and bounded rationality. The following
sections illustrate this point.
3.1
The baseline scenario : instable growth and collapse
The first simulation on the figure 1 shows a fast expansion until the model
reach the ceiling of full employment and then collapse. The upward instability is driven by the combination of the multiplier-acceleration effect
(Harrod, 1939) : tension on capacity utilisation increases investment and
investment increases global demand that induce stress on capacity utilisation. This circular causation is interrupted when the system reach full employment because this constraint on supply stop the acceleration principle.
Delayed firms’ reactions make them continue to invest for a certain time,
this over-accumulation of productive capacity make the utilization rate fall
and reverse the system to downward instability.
8
3e+04
1
2,5e+04
0,8
0,6
u, N
PC, real GDP
2e+04
1,5e+04
0,4
1e+04
0,2
5 000
0
0
100
200
300
step
400
500
600
700
0
Figure 1: Scenario 1, time series
1
N
0,8
0,6
0,4
0,2
0
0,2
0,4
u
0,6
0,8
Figure 2: Scenario 1, utilization and employment
9
1
3.2
Endogenous cycles, how the model endogenously shifts
from downward to upward instability
To cope with downward instability, we feed aggregate demand with an autonomous expenditure set proportionally to potential GDP. If we set the
autonomous expenditure 0.5%, the system produce endogenous cycle, see
the figure 3. The upward instability and the reverse to downward instability in this second scenario is similar to the first one. The difference lies in
the exogenous expenditure that allow the model to shift from downward to
upward instability.
On the figures 7 and 5 we decompose the observable cycle in time series
on figure 8.
The autonomous expenditure led phase From step 600 to 850, the
productive capacity decreases while the employment and the utilization rate
go up. This is the recovery phase, where the system is led by the autonomous
expenditures.
The upward instability phase Between step 850 and 930, this is the
accelerator - multiplier phase : employment and utilization continue to grow,
but firms invest to increase their productive capacity.
Full Employment as a ceiling, the compression phase This upward
dynamic is stopped when the system attains full employment at the step
930 approximately, accelerator effect stops. Full employment is maintained
until around step 1000, during this phase, accumulation continue because of
the lagged reactions of firms, and logically utilization rate goes down.
The downard instability phase The next phase is characterized by
downward instability where utilization rate, productive capacity and employment collapse. There is no possibility to reverse this dynamic unless an
autonomous expenditure is set up. It is only because their is an autonomous
component of demand that the system can get back to the next phase of the
cycle.
10
4e+04
1
0,9
0,8
0,7
2e+04
0,6
0,5
0,4
0
0,3
0
500
1000
1500
2000
2500
3000
step
Figure 3: Scenario 2, endogenous cycles driven by harrodian instability
1
0,9
u
0,8
0,7
0,6
0,5
0,4
1,5e+04
2e+04
2,5e+04
3e+04
PC
Figure 4: Scenario 2, utilization-accumulation cycle
11
step 850-930
step 930-1000
1
0,9
0,9
0,8
0,8
u
1
0,7
u
0,7
0,6
0,6
0,5
0,5
0,4
1,5e+04
2e+04
0,4
1,5e+04
2,5e+04
2e+04
2,5e+04
PC
PC
step 600-850
step 1000-1250
0,9
0,9
0,8
0,8
u
1
u
1
0,7
0,7
0,6
0,6
0,5
0,5
0,4
1,5e+04
2e+04
0,4
1,5e+04
2,5e+04
2e+04
PC
2,5e+04
PC
Figure 5: Scenario 2, phases of the utilization-accumulation cycle
1
0,9
0,8
N
0,7
0,6
0,5
0,4
0,3
0,4
0,5
0,6
0,7
0,8
0,9
u
Figure 6: Scenario 2, utilization-employment cycle
12
1
step 850-930
step 930-1000
0,9
0,9
0,8
0,8
0,7
0,7
u
1
u
1
0,6
0,6
0,5
0,5
0,4
0,4
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0,3
0,4
0,5
0,6
N
step 600-850
0,8
0,9
1
0,8
0,9
1
step 1000-1250
1
1
0,9
0,9
0,8
0,8
u
u
0,7
N
0,7
0,7
0,6
0,6
0,5
0,5
0,4
0,4
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0,3
0,4
0,5
0,6
N
0,7
N
Figure 7: Scenario 2, phases of the utilization-employment cycle
1
0,9
0,8
0,7
0,6
0,5
0,4
0
600
700
800
900
1000
1100
1200
0,3
1300
step
Figure 8: Scenario 2, a single cycle time series
3.3
Durable quasi-full employment, not even a challenge in
this simplistic economy
With an autonomous expenditure set to 1%, the system is able to maintain
a quasi full employment during the entire simulation.
13
4
The question of microfoundation
The Agent Based methodology is a branch of computational economics
which focuses on the behaviours of individuals by definition. This behaviours
are the staring point of the elaboration of the model, MABM is generally
qualified as Macroeconomics from the bottom-up. However this individual
behaviours are not necessarily at the origin of the reflection around economic
theory that led to the model.
4.1
Post-Keynesian economics and microfoundation
There is at least two elements that make this question tricky for PostKeynesians. First, the fact is what mainstream call “rigorous microfoundation” clearly entails non-macro theories3 . Second, the recurrent disqualification by the “microfoundation dogma”(King, 2012) of heterodox models reveal the institutional domination of their authors by mainstream economists.
Since “there is virtually no discussion among economists of a need for macrofoundation for microeconomics (. . . ). In contrast, the demonstration of the
existence of microfoundations for macrotheories is considered essential by
many lending economists.” (Boland, 2003).
Facing this state of affairs, we identify three kinds of reactions. The first
one is rejection mainly based on the argument of “fallacy of composition”,
“[it] entails that an entire economy may behave in ways that cannot be
inferred from the behaviours of its individuals agents” (King, 2012). This
argument has a clear holistic connotation : the whole is more than the sum
of its parts. Consequently, the following argument is macroeconomics has
to be an autonomous intellectual discipline.
The second reaction consist in requiring the exact opposite argument
i.e. it is not macroeconomics that has to be microfounded, but microeconomics that have to be macrofounded. This implies that the institutional
framework, the socio-economic history and the macro-dynamic have to be
understood if man wants to develop micro theories. In other words, it is
basically a reversal of the argument i.e. a critic of the mainstream critic by
the same kind of arguments.
Finally, the third reaction lies in the claim where Post-Keynesian macroeconomics is already microfounded. The fact is a lot of Post-Keynesian work
on microeconomics and examine the interrelations between micro and macro
3
King (2012) explains that the Representative Agent With Rational Expectations is the
necessary condition for microfoundation considered as rigorous by mainstream economists.
14
theories4 . This is our position.
4.2
MABM and microfoundation
The microfoundation induced by MABM are quiet different from the rigorous microfoundation described by King (2012) as reductionist. Gatti et al.
(2010) posited that “the ultimate reason to discard modern mainstream
macroeconomics lies in the pitfalls hidden in its methodological background,
namely equilibrium microfoundations, since it does not allow the macroeconomist to recognize the essence of macroeconomics ; the emergence of
aggregate outcomes and structures as aggregate unintended and unplanned
consequences of individual human actions and dispersed interactions.”
Furthermore, MABMs have no dependence on the methodological individualist requirement of neoclassical economics. As the ways individuals
interacts is an essential part of the elaboration of the model – the procedure
of imperfect competition for example - and the emergent macro dynamic
has significant feedbacks on individual behaviours, the instability dynamic
is the emergent result of this micro-macro feedbacks.
The focus of ABM literature on emergent phenomenons is representative
of its fully different approach from the keeper of the rigorous microfoundation dogma. By definition, emergent phenomenons are aggregate outcomes
that are not exhibited by agents. This emergent properties are not allowed
by rigorous microfoudations. In fact, the macroeconomic paradoxes are similar to emergent phenomenons. The well known paradox of thrift is a good
illustration of an emergent phenomenon. A higher propensity to save at
the individual level leads to a higher saving rate ceteris paribus. But the
aggregate outcome of an increase of the collective propensity to save will be
lower overall savings.
Our position is that the aim of MABM is not to propose a faithful representation of the real world economy but intend to illustrate a specific
problem of an economic system. According to Radzicki (2010), “Properly
undertaken, system dynamics modeling is a problem-based, rather than a
system-based process. That is, instead of modeling systems, system dynamicists identify and model problems from a systems perspective. Experience
has shown that attempting to model systems rather than problems typically
results in excessively large models that are difficult to understand and that
do not yield insights into the fundamental causes of poor system behavior.”
4
A lot of books and articles supports this view, number of them are quoted in Lavoie
(1992), especially Adrian Wood, Gardiner Means, Alfred Eichner, Frederick Lee, etc
15
MABM are not intended to replace aggregate modeling. In the same
way, stock flow consistent modeling did not replace Kaleckian models of accumulation and repartition. In practice, these approaches complement one
another : what we gain in realism, complexity, or scalability with MABM,
we lose in tractability. It’s the dark side of complex dynamic systems that
can be fascinating but yet very obscure. As highlighted by Radzicki (2010),
“Linear systems can be solved either numerically (i.e., via simulation) or analytically (i.e., in closed form), while nonlinear systems can only be solved
numerically. The main differences between analytical and numerical solutions are that analytical solutions are exact, global, and nonrecursive, while
numerical solutions are approximate, local, and recursive.”
Finally, we believe macroeconomic agent-based modeling can be a useful
tool for theoretical investigation. It allows us to study the relations between
micro and macro levels, the effects of heterogeneity, dynamic adjustments
rather than equilibrium. It focuses on processes, causal relationships, feedbacks and cumulative causations and allow us to gain insights monetary
production economies work.
5
Conclusion
In this paper we show that a simplistic agent based model of a capitalist
system driven by demand is fundamentally unstable. This instability can
be tamed by an autonomous expenditure (i.e. government deficits for exemple), but there is no spontaneous mechanism that bring and maintain the
system to full employment. Therefore the policy prescription gets back to
basics, government expenditure appears to be the simplest way to reverse
the downward instability.
The model presented does not consider finance nor foreign exchange. It
focuses on the basic intuition of Harrodian instability and has no ambition
to fully explain the recent events or to be a realistic representation of an
actual developed economy. Nevertheless, it can be an interesting framework
to gain insight about the relations between micro behaviours and macro
dynamics.
16
References
Assenza, T., Gatti, D. D., & Grazzini, J. (2014). Emergent Dynamics of
a Macroeconomic Agent Based Model with Capital and Credit. CESifo
Working Paper Series 4765, CESifo Group Munich.
Boland, L. (2003). Foundations of Economic Method: A Popperian Perspective. Taylor & Francis.
Cordonnier, L. & Van de Velde, F. (2009). Le capitalisme est-il in trinsèquement instable ? formes de capitalisme et dynamique économique.
Dallery, T. (2010). Le divorce rentabilité/croissance dans le capitalisme financiarisé. Changements de régimes, équilibres, instabilités et conflits.
PhD thesis, Lille 1.
Fazzari, S. M., Ferri, P. E., Greenberg, E. G., & Maria, A. (2012). Aggregate
Demand, Instability, and Growth. INET Research Notes 2, Institute for
New Economic Thinking (INET).
Ferri, P., Fazzari, S., Greenberg, E., & Variato, A. (2011). Aggregate demand, harrod’s instability and fluctuations. Computational Economics,
38(3), 209–220.
Gatti, D., Gaffeo, E., & Gallegati, M. (2010). Complex agent-based macroeconomics: a manifesto for a new paradigm. Journal of Economic Interaction and Coordination, 5(2), 111–135.
Gatti, D. D., Desiderio, S., Gaffeo, E., Cirillo, P., & Gallegati, M. (2011).
Conclusions. In Macroeconomics from the Bottom-up, number 1 in New
Economic Windows (pp. 101–110). Springer Milan.
Harrod, R. F. (1939). An essay in dynamic theory. The Economic Journal,
49(193), 14.
Hein, E., Lavoie, M., & Treeck, T. v. (2012). Harrodian instability and the
‘normal rate’ of capacity utilization in kaleckian models of distribution
and growth—a survey. Metroeconomica, 63(1), 139–169.
King, J. E. (2012). The Microfoundations Delusion: Metaphor and Dogma
in the History of Macroeconomics. Edward Elgar Publishing.
Lavoie, M. (1992). Foundations of Post-Keynesian Economic Analysis. Edward Elgar Pub.
17
Radzicki, M. (2010). Was alfred eichner a system dynamicist? In Money and
Macrodynamics: Alfred Eichner and Post-Keynesian Economics. M.E.
Sharpe.
Robinson, J. (1962). Economic Philosophy. Doubleday & Company.
Skott, P. (2012). Theoretical and empirical shortcomings of the kaleckian
investment function. Metroeconomica, 63(1), 109–138.
Wood, A. (1975). A Theory of Profits. CUP Archive.
18