Exercise on Matlab

1. UNIVERSITY OF TRENTO
School of Social Sciences
January, 2014
Assignment nr. 2
Metzler (1941):
Insertion of modifications in the Metzler-model(s)
to account for nonlinearities. Dynamic IS-LM
Macroeconomics - PhD - School of Social Sciences
Instructor: Prof. Stefano Zambelli
PhD Student : Boccardo Serena

2. METZLER’S BUSINESS CYCLE THEORY
In Metzler’s view, production and employment adapt to cyclical changes of some parametres.
Given values and oscillations of these parametres, we have unstable business cycles that
would otherwise be stable. Starting from the general function of production in Metzler :
YP(t) = CP(t) + IP(t)
how a change in parametres can affect the above equation?... The following equation shows
how they affects production : Yp changes according to the “law of motion” of the system set up
by the producer as well as his desired level of investment in inventories and the relation
between the level of non-induced investment (I0) and the production function:
YP(t) = Et-1(CD(t)) + s*(t) + I0
First model
The model "Simple system with passive expectations and inventory adjustments" represents a
business cycle based uniquely on the initial level of inventories which thus decrease passively
exactly of the same amount of sales per each period. Because of the continuos reduction of
inventories, the income equilibrium varies according to the initial level of non-induced
investments. That is, an equilibrium level of income is assumed to correspond to any given
level of non-induced investment and the system is assumed to move toward this equilibrium.
Second model
"The Pure Inventory Cycle" model assumes the emergence of cycles as a consequence of an
additional assumption in the production function: the producer’s desire to keep the level of
investment stocks constant over time. In order to do that, he will produce or deplete a certain
amount of production goods in addition to sells at time t. The level of additional stocks to be
produced can be either positive or negative, basing on the “law of motion” of the expected
sales for that period. This generates a certain oscillation in production (which ultimately will
reach the equilibrium) and in the amount of inventories over time.

3. Third model: “Passive inventory adjustments and positive expectations”
Since expectations about future sales are realistically based not only on previous sales but
also on the direction of change of those sales, Metzler introduces a coefficient of expectations
(eta) defined as the ratio between the expected change of sales between period t and t-1 and
the observed change of sales between periods t-1 and t-2 (when the coefficient is zero, no
change is expected, as in the Lundberg model. If the coefficient is 0.5 sales are expected to
increase by 50% between t-1 and t). So a positive coefficient creates expectations of a further
rise in sales and viceversa. In the positive case, income will approach a higher equilibrium
comparing to before, given the role of the multiplier assuming passive inventory adjustments .
Obviously, the marginal propensity to consume plays also a role in changing our results,
because it implies different savings which ultimately act as stabilizer. So whether the system
approach the first or the second equilibrium in the below graphs depends, indeed, on the
relation between eta and c, given the role of savings.
Metzler with eta=0.8 e c=0.6 Metzler with eta=0.1 and c=0.9

4. U=unstable S=stable equilibrium
Fourth model: “Inventories at the constant level attempt and expectations”
The above table showing the relation between Eta and Beta changes significantly when the
producer attempts to maintain the inventories at the constant level. Meaning that parametres
play a key role as well as the initial conditions set by the producer: in this fourth model, a high
coefficient of expectation may create an unstable situation even with a marginal propensity to
consume less than unity.
1.stable case
With eta and c into the positive area (nothing change with eta negative for istance eta=-0.5
and c=0.6)
2.unstable case

5. Model 5 : “Inventory accelerator model”
By introducing an accelerator we set the desired inventory level of firms as proportional to
expected sales; firms may not correctly predict consumer demand and thus if the firms have
been too optimistic (pessimistic) with respect to consumer demand, the inventory stock is
higher (lower) than the desired inventory stock.
I=Y t − b(2 + k)Y t −1 + b(1+ k)Y t −2
And so setting Y = Yt = Yt− 1 = Yt− 2
The above is satisfied for Y =
!
!!!
∗ I, which is identical to the traditional Keynesian
multiplier solution.
The below figure represents this model with c=0.6 and eta=0.5, which reaches the equilibrium
at a slower convergence rate: the model generates business cycles with decreasing amplitude
when some conditions are satisfied, otherwise it explodes. Differences between the regions of
explosion of the business cycles have to be attributed to different assumptions on the nature
of inventories (economic and technological factors versus others).
The critical conditions for the equilibrium are:
b =
!
!!!
and b <
!(!!!)
!!! !
In this case, both quantities fluctuate around their equilibrium values and eventually
approach them as time proceeds.
However, if only the first condition holds, the model generates oscillations with increasing
amplitude, as from the second graph where only the value of k has been changed.
The right-hand panel shows that the inventory stock quickly takes on negative values, which,
in reality, is not possible.
Thus, introducing an accelerator imposes severe limitations upon stability of the system: for

6. low values of and k , the model thus always generates cycles with decreasing amplitude. If one
or both parameters increase sufficiently enough, the b =
!
!!!
gets violated and we observe
cycles with increasing amplitude.
General Metzler’s model
The substantial amplitude of cycles when we make even a small change to parametres
suggests that even when small changes production can become unstable, leading to major
consequences into the economy. The consequences are exploding cycles and irregular flow of
inventories, as we can observe in the following graphs:

7. Introducing pseudo-random shocks:
Ceilings and floors:
Since in reality it is not possible for inventory stocks to assume negative values, in order to
add more realism to the values of the production function, we could set a ceiling and floor to
the values the function can assume (Hicks, 1950): this additional hypothesis shows the
importance of inventory adjustments for the emergence of endogenous business cycles.
The above models can be considered as special cases of the general one where, in order to be
stable, a business cycle should have parametres ranging around certain values, in particular
they have to satisfy :
1 + 𝛼 2 + 𝛼 𝜂𝛽!
− 1 + 𝛼 1 + 2𝜂 𝛽 + 1 > 0
and 3 − 𝛽(2𝛼 + 3) > 0

8. NB k=alfa, b=marg.prop.consume in this graph
Description:
Parametres which lead to business cycles stability are in the red area.
In light gray, parametres values which lead to explosion.
The dark gray area is associated with dynamics where production is positive and negative.
The parameter space between the red and the dark gray area generates dynamics where
production is always positive.
The white area stands for cycles with a period length larger than 42 or quasi-periodic motion.
The other colors indicate lower cycle lengths: in this parameter space, the simple model of
Metzler, buffeted with an inventory floor, yields interesting endogenous business cycle
dynamics. This may in particular be relevant when the marginal propensity to consume is
relatively high.
Main points:
1) when 𝑏 <
!
!!!
, 𝑌𝑝 =
!
!!!
∗ 𝐼(𝑡) oscillates.
3) when alfa=0.15 (and other parameters have not been changed) the model generates
oscillations with increasing amplitude;
4) the larger the marginal propensity to consume, the lower the critical value of alfa which
ensures local stability of the fixed point.
5) for low values of b and alfa , the model thus always generates cycles with decreasing
amplitude, whereas if one or both parameters increase sufficiently enough, 𝑏 <
!
!!!
gets
violated and we observe cycles with increasing amplitude.
CONCLUSIONS:
The inventory model of Metzler may produce dampened fluctuations in economic activity and
thus contributes to our understanding of business cycle dynamics. For some parameter
combinations, however, the model generates oscillations with increasing amplitude, implying
that the inventory stock of the firms eventually turns negative.
Hicks (1950) thus suggested adding boundaries for some economic variables to such models.
In particular, his focus was on the investment part of the multiplier-accelerator model for
which he introduced a floor and a ceiling.

9. Metzler assumes that the producers desire to keep inventory proportional to expected sales of
consumption goods. This may have some important consequences. Suppose, for instance, that
the economy enters a recession so that consumer demand shrinks.
Besides reducing production of consumption goods, the firms may further decide to cut their
inventory stock, i.e. they produce even less than the consumers demand. Obviously, this may
deepen the recession. The opposite occurs in an upswing where the firms produce more than
the consumers demand. Adjustment of the inventory stock may thus amplify business cycles.
If the firms have been too optimistic (pessimistic) with respect to consumer demand, the
inventory stock is higher (lower) than the desired inventory stock.
FROM METZLER TO IS-LM in KEYNES
The IS-LM model is an Hicks’ interpretation of the fundamental relationships present in the
General Theory: in Keynes IS-LM model we can see an orbitating area of equilibrium rather
than a single point because money-income is not considered as fixed and given. As a
consequence, the IS and LM curves are not linear. Money demand is indeed fluctuating
because of expectations, uncertainty, propensity to consume, state of confidence concerning
the prospective yelds of capital-assets,etc (the so colled “animal spirits”) and this affects in
turn the level of the interest rate, which in turn, affects the marginal efficiency of capital. The
rate of new investments, thus, will be affected (and in turn, aggregate production). So one of
the main difference between Keynes and the neoclassicals arises: it is income which pushes
production, not supply which creates its own demand (Say’s law). It is the demand of money,
which is driven by many factors, to be the driver of the economic system. The “one point-
equilibrium” IS-LM exists only in an unrealistic world where the level of money-income is
given: by assuming that income is given, the neoclassicals are implicitly assuming that the
monetary policy is such as to maintain the rate of interest at that level which is compatible
with full employment.
Taxation=Government expenditures Hypothesis
With government expenditure equal to tax and eta=0 we have convergence to the equilibrium.
(Meaning the system is stable under some unrealistic assumptions)

10. Changes in the values of eta, tax and alfa lead to damped oscillations in production and
demand and in a not orbitating equilibrium but rather an irregular trend of IS-LM:
Government debt hypothesis
If we introduce some more realistic assumptions, such as : Government exp>Taxation with a
major public debt value (i.e. 20,000) , an increased tax rate from 0.3 to 0.6 in case of arising
public debt, a certain eta=0.3, debt owned by households and so affecting their income, a non-
negative production function, we can see that our aggregate income and production
function,after some years of decreasing trend, reach a cycle with regular oscillations of the
same amplitude of production and demand. (Nevertheless, changing the values of initial
parametres produces damped and irregular oscillations).

11. Government debt hypothesis with ceilings and floors
We can observe a totally different trend when, keeping a high value of gamma, low
expectations (0.3) and the same level of high public debt, we insert reasonable assumptions
about the values of production, inventories, consumption demand and even interest rate
(which can be limited in order to not to fall in any liquidity trap or produce inflationary
effects). The result is not converging demand and production trends, and their increasing
magnitude. The IS-LM is definitely not orbitating around any equilibrium:
No initial Government debt, Government debt hypothesis, ceilings and floors
We can observe meaningful differences in production and demand trends over the long run
when we erasing the initial situation of a high public debt:
Extention to the model: Basic Income Hypothesis
Under this simplified model, we can imagine to introduce a policy change in the direction of a
basic income for all citizen. How would this affect the business cycle?... Two alternative trends
can be expected: either a higher level of taxation/increased public debt undermines the
business cycle, since households will save more than before keeping consumption constant;
or an increasing disposable income pushes expectations about future consumption demand
up since producers will expect a higher propensity to consume, given a higher aggregate
disposable income (consequently increasing investments and so boosting production).

12. In my view, the way this policy will affect the business cycle depends on producers’
expectations about the potential effects of this policy on the demand function, in accordance
with Keynes’ theory. Thus, consumption demand function should increase (raising
expectations about future consumers’ demand) and this, in turn, should stimulate the
aggregate production function.
A simulation could be run on Matlab in order to check for this trend: I simulated that both in
the case of a high initial public debt and in the case of a null initial public debt. The outcomes
are significantly different, as we can see from the graphs below:
Case with significant initial public debt:
Blue: Yp
Red: Yd
Case with zero initial public debt:

13. Their trends appear definitely opposite in the short run: where high public debt is assumed,
both the demand function (red) and the production function (blue) have an initial pick,
viceversa where the initial public debt is zero. Moreover, in the latter case they almost
coincide in the first period. Nevertheless, in the long run they both stabilize on different
values with the demand always higher than production.
Nevertheless, the above trends might be done to wrong parametres modification on the
model (since I exogenously modified the marginal propensity to consume, for instance). In
order to run the above models, the modifications I added to the original IS-LM model are:
- increased government expenditures rate (gov) and constant in the for loop, regardless of the
trend of the public debt: I found reasonable to increase the tax rate in both the cases (zero and
consistent public debt). I did so just in order to give the model a more realistic feature,
nevertheless I made different hypothesis on public debt, so this modification was not
necessary.
- increased propensity to consume : this might be consequential to an increase in ydisp.
Should it be exogenously added to the matlab equations?.....
- higher floor for minimum demand function (as above)
The main question concerning this modification of the model is, in my view: is the effect of an
increased disposable income (aggregate) positive or negative for the business cycle, given the
fact that it can occur at the cost of an increase in public debt/taxes?.....
Serena Boccardo