Soc 663
October 2, 2003
Malthus and Malthusian regimes: randomness and homeostasis
Note: the concept of homeostasis is frequently used in biology; it is equivalent with the concept
of density dependency or with population regimes with feedback.
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Preamble
The lessons to be learned from the Irish example (and there are others similar to it) are two. The
fist is that during the preindustrial period, human populations could not grow beyond certain
points without triggering mechanisms that would produce more breathing room between the
absolute numbers and some carrying capacity. Growth did occur. And it did so in response to a
number of conditions. most of them related increased demand for labor. Thus, ths introduction of
a new crop, the adoption of a new cultivation technique or even the occupation of marginal
lands, could produce some additional room for the population to growth further.
The second is that the filtering mechanism through which exogenous changes eventually led to
population growth or decay were complex and involved the economic system ( property systems,
inheritance rules, financing and credit, technology etc...) and societal rules and institutions,
particularly those regulating marriage and cohabitation as well as fertility within and outside
marriage.
Mortality mattered as well. And indeed increases or decreases of mortality were reactions that a
population could experience to accommodate to carrying capacity. But the Irish example is
extreme and here mortality increases and migration did play a role. In other cases their role was
more peripheral.
The mystery of whether the increase in population before 1841 and the population collapse
thereafter was due the introduction of the potato and to the population pressure it unleashed, is
still being debated.
Text from Zuckerman, pp 228
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A. Motivation
1. The appearance of a graph displaying birth rates and death rates (or rates of growth) for the
period 1200-1800 in Europe (where we do have good data, say in Colyton from Wrigley’s book)
suggests the existence of random forces behind the population trajectory of population. There is,
to be sure, a trend towards increased life expectancy (starting in 1750 approximately) and
decreasing fertility (starting in 1900 approximately). But these are points past the so called preindustrial population regimes.
Furthermore, if one could measure the 'randomness' of the series of vital statistics over time (say
by the area of the graph covered by fluctuations) one would notice that these tend to disappear
after 1750-1800. Was population dynamic really random before this period? Did nonrandomness enter as soon as industrialization occurred?
Remark: similar fluctuations in population numbers and births and deaths can be observed in
population of animals and insects. It is speculated that species have evolved through selection
that favors those developing mechanisms to adjust population density to the constraints of the
environment.
I would like to show you that, as in animal populations, there was randomness in the preindustrial population regime, but that this is not the entire story. In particular, we can show that
the population dynamic of the period obeyed some precise laws but that these tended to make
themselves felt over the long run, not the short run. Over the short run things behaved very much
as a random system. Much as it occurs in chaotic dynamics in some physical systems, the end
product looks random but obeys to very specific laws. Some of them are part of the so called
Malthusian theory.
I now need to prepare the ground and to introduce a few concepts.
B. Malthus and homeostasis: main principles
B.0 Historical context of Malthusian theory
-Malthus work develops during the years1785-1830 against a backdrop conformed by the
aftermath of the French revolution and the idealistic and optimistic theories of Condorcet in
France and Godwin in England. Essentially these theories preached unlimited progress, much as
today overconfidence in technology.
-England was witnessing a challenge to her self sufficiency in grains, the passage pf poor laws to
protect the destitute and migration to America. Plus, Malthus was a witness of the rapid Irish
population growth following the introduction of the potato.
-Malthus theory is developed as a response to idealistic theories of progress as well as an
indictment of the Poor Laws.
-The population essay written by Malthus is in fact a number of different essays that progressed
over time. The earliest version contain a stark picture, one where population density is
continually clashing against higher levels of mortality. The last version of the essay is more
optimistic in the sense that Malthus views how social institutions can, under certain conditions,
anticipate and prevent the reaction of higher mortality
Note: graphs referred to below are in the power point presentation corresponding to this class
B.1. Linear, Geometric, Exponential and logistic growth
-If a population grows linearly, it adds the same amount every time interval. Thus, if we start at
time t=0 with P(0) individuals and if we add (subtract) r individuals per year we will have the
following yearly sequence:
(P(0)+ r) after one year; (P(0)+ 2r) after two years etc...
P(t)=P(0)+r*t
-Geometric growth refers to a process where the amounts added is proportional to the size of the
population. For example, we can add r individuals per year per person. The yearly sequence will
now be:
(P(0) +r*P(0)) after one year; (P(0) + r*(P(0)+r*P(0)) after two years, etc...
P(t)= P(0) *(1+r)t
-Exponential growth refers to growth with a constant rate of growth; it is equivalent to a
geometric growth but where the addition of new individuals rather than taking place only once a
year (geometric) it takes place continuously during the year. Mathematically it is a trajectory that
is the limit of the following (geometric growth or increase). If r is the yearly rate of growth and
we divide the year in d very small intervals of width 1/d we will have
P(t)= P(0)*(1+r/d)d
and when the quantity d tends to infinity we get:
P(t)= P(0)*exp(r*t)
Logistic growth refers to a trajectory where the rate of growth is constant for a while but as
population begins to increase the rate of growth decreases until it reaches 0. And the population
stops growing.
{graph #1 for logistic and exponential}
Logistic growth is a model applicable to animal populations who develop in a fixed environment
and with no possibilities of migration. In animal population there are checks to ensure that
populations remain below a certain threshold
B.1. The concept of a production function and elasticity
Suppose that O is total output and that only two inputs can contribute to it, land and labor.
Assume for the time being that we cannot change the amount of land cultivated and that over the
short run it cannot shrink either. So it is a fixed quantity. In theory then one could say that given
a level of technology you will produce output in proportion to the amount of labor you have
available. These ideas can be formalized by defining a production function as follows:
O= A*L
Where O is the amount of output and A is a constant that reflects the effects of land and
technology. In this system, the output per capita is a constant, O/L =A. Thus, the addition of one
unit of labor will end up adding the same amount all the time, namely A. But this is probably
unrealistic. Given that land and technology are fixed, the addition of too many workers may lead
to each of them producing less than what was being produced before the addition of more
workers. Or, you could think that, on the contrary, the addition of more workers adds to the
productivity of each one in which case the addition of an extra worker will lead to increase in the
product per capita. To reflect these possibilities we transform the expression above as follows:
O=A*Lb
Where b is a positive real number. When it is below 1, the product per worker goes down with
the addition of a new worker. When it is above one, the product per worker goes up with the
addition of one worker. The parameter b is the elasticity of the product relative to labor or the
proportionate change in O associated with a unit proportionate change in L. Thus, if b=.5 it
means that a 10 % change in L leads to a 5% change in O. If b= .05 then a 1% change in L leads
to a .5% change in O.
In theory, you can have elasticities simply as measures of the strength of the relation between
any two quantities. For example, if income is related to the birth rate we can speak of the
elasticity of births relative to income. Similarly, if changes in temperature in the stratosphere are
related to population we can speak of the elasticity of global warming relative to population size.
An elasticity is simple a measure of the change in the logarithmic value in one quantity
associated with the change in the logarithmic value in another. It is short hand for the strength of
a response to a stimulus.
(See Graph 2 (lower panel) for examples of relations between
Wages and labor inputs. Remember that wages are defined as
Output per worker and Labor is here made equivalent to
Population}
Remarks:
-you can increase the output by increasing L but also by shifting technology or expanding to
marginal lands. These changes could be captured by a new value of A.
-One can get more complicated and think of quantity of land (say Q) as another variable input.
In this case we could have the following production function:
O=A’ * Qa*Lb
where A’ is a new constant and a is now the elasticity of O relative to Q (land).
B.2. Other important concepts
-There is a minimum subsistence level below which starvation and death will follow;
-All societies have a carrying capacity
-Means of subsistence grow arithmetically;
-Population grows exponentially;
Remarks:
-the idea that there is minimum subsistence level is nothing new and probably reflects a physical
reality, but economists have in mind levels that may vary with the social and cultural contexts;
the idea of 'minimum' needs to be satisfied is relative. Malthus, however, wavers on this and
moves from the idea of 'minimum' as physical minimum (below which starvation or acute
malnutrition sets in) and a social-cultural minimum.
-Analogous comments apply to the concept of carrying capacity. This refers to level of
population supportable at some given standard of living for a given level of technology and
resources. It is a concept akin to demand for labor. It does not refer to MAXIMUM
SUSTAINABLE POPULATION
-the idea of exponential growth is very accurate for the period and, in addition, makes sense
biologically. What Malthus left out was the idea of ceilings which is better represented in a
logistic formulation.
-The main idea that means of subsistence grow only arithmetically as opposed to geometrically
the Malthusian establishes principle No 1: diminishing returns to labor This idea, central to
economic theory of the time, simply states that the product per capita (and hence wages and
income) has a tendency to decrease as the number of workers increase. We will use income,
wages and product per capita interchangeably. It is important to note how crucial is the
unmentioned assumption on which this idea rest: that there are no technological innovations able
to increase productivity of the labor force.
{Graph # 2 real wages/income as a function of population}
-The second principle on which the Malthusian theory rests, Malthusian principle No 2 is that
vital rates are responsive to economic conditions. In particular, that income is inversely
related to income and positively related to fertility.
{Graph # 3 vital rates and natural rate of growth vs income}
-Finally, from the first and second principles we get a corollary that identifies the feedback
mechanism, namely, the rate of growth must be responsive to population size. A homeostatic
regime depends on how tight this relation is: the stronger it is, the stronger is homeostasis. The
weaker it is, the more the system will be at the mercy of random forces.
{Graph # 4 natural rate of growth as a function of population size}
The text from Malthus’s:
i. On diminishing returns .........pp. 242
ii.On population growth and diminishing returns....pp.238
Remarks:
-the notion of population equilibrium is one that is somewhat capricious but in the absence of
technological innovation that increase productivity, the equilibrium rate of increase must be 0.
Any other value would—as a result of feedback-- turn the system back to a rate of increase equal
to 0
-the functioning of the system would be more or less as follows: suppose that there is an excess
of population at time t. This will lead to a reduction in income and this in turn to a reduction in
vital rates (r).
-Malthus envisioned the existence of two types of mechanisms that would bring population into a
balance or a return to equilibrium. These are the mediating mechanisms that make the system
work. He referred to them as the population checks:
-positive check (increased mortality/ wars/ famines/pestilence)
-preventive check (decreased fertility through decreased marriage
There are important differences in the emphasis placed on these two mechanisms by Malthus. In
the first two editions of the Essay he thought that positive checks would do most of the work.
But in the last editions and in the Summary he leans heavily toward the idea that social
institutions, particularly those promoting late marriage, would do most of the work. He was
naturally disinclined to believe that marital fertility control was viable at least morally.
The texts from Malthus:
iii. On nature of checks.....pp.249-50
iv. On what determines a check....pp252..
C. Malthusian dynamics
How does a Malthusian system work? Suppose the system is in equilibrium and that due to some
technological improvement (say a new crop), the curve of demand for labor is displaced to the
right. That is, keeping elasticities constant the output per worker rises (and so does income per
capita or wages). What this means is that the same amount of labor (or P) will generate a higher
wage or income. Under the Malthusian theory, this leads to a higher CBR. And here comes the
feedback: unless there is a similar increase in the CDR, there will be a positive rate of natural
increase and total population will grow. But an increase of population (indicating that the rate of
increase is no longer 0) implies a new equilibrium wage which is below the previous one.
Graph # 5: wages vs Labor; DR vs BR
An important remark:
That the new population size P (hat) will be the new equilibrium population assumes that the
surplus population can really be absorbed at the going wage rate. If not or if the population
increase overshoots the target population size (P (hat)) then, the malthusian framework would
have us believe that either positive checks come into play or preventive checks are activated
until the quantity of labor supplied is consistent with the carrying capacity or demand for labor.
D. Malthus: the corollaries of the theory.
Under a system governed by a Malthusian dynamics, we should expect:
D.1 Short-term fluctuations of birth rates and death rates (and rates of increase);
If average economic well-being declines below some threshold, then marriages should decrease
and lead to a short term lowering of the birth rates. But how can this happen? It actually did
happen in Europe and the main social institution that forced marriages to adjust was property:
land and access to wage labor. There were strict societal rules that defined age at marriage as a
function of ability to maintain a household.
D.2.A decrease in economic well-being should increase mortality;
This did happen also but to a lesser extent. Starvation is difficult to achieve even under wretched
conditions. More frequent epidemics due to lowered resistance and very high infant mortality
seem to have been responsible for increased mortality in bad times.
3. Other short-term fluctuations of birth rates and deaths rates are to be expected;
The process here works slightly differently: changes in the rate of increase lead to changes in the
age distribution which will translate into either higher or lower pressure on labor markets and
hence on real wages. Roughly, it would work as this:
An increase in economic well-being at time T translates into a temporary increase of the birth
rate. At time T+25 approximately there will be a relative inflation of labor demand, more people
will be in the labor force seeking jobs, due to the inflated size of the corresponding birth cohorts.
This will translate into more pressure over wages and thus on lower real wages. Eventually this
will lead to the preventive check and birth rates will be lower. At time T+50 we will have a
shortage of labor and therefore optimal conditions for wage workers. This will increase the
marriage rate and thus the birth rate thus leading to inflated birth cohorts...and so on...
Of course shortage of labor can be the result of mortality crises or wars or diseases or a
technological breakthrough . Similarly, overabundance of labor can be a result of excellent
health status produced by a mild winter, peace treaties and disappearance of epidemics
Remark:
-The last mechanism is a mechanism behind oscillations of birth rates in the US; it is referred to
as the Easterlin hypothesis.
E. Malthus and the historical evidence
1. Econometric analyses of historical data show that there is a tight relation between wages and
population. The estimate is that a 1 percent increase in pop size brings a 1.6 percent decrease in
real wages; this is evidence for diminishing returns.
{Graph # 6 of log wags and log of population}
2. Econometric analysis of historical data shows that there is a relation between wages and
fertility .
{Graph # 7 of TFR and wages}
The relation is weak though as the elasticity is .20: a 1 percent increase in wages leads to a .2
percent increase in TFR; to get the elasticity of TFR relative to population simply multiply this
number (.20) by –1.6 to get -.32. This is evidence for density dependence
3. The evidence is less strong for the association between mortality and economic well-being.
But, though weak, the link appears to have existed. The elasticity is about .16: a 1 percent
increase in wages leads to a .1 percent decrease in mortality rates; to get the relation between
mortality and population simply multiply this number by –1.6 to get -.26; this is evidence for the
Malthusian positive check
{Graph # 8 of deaths and prices (wages)}
Is this evidence compelling? Think of the possibility that the relation between wages and
mortality could be confounded because of the possibility of spurious association caused by a
third factor, such as wars/epidemics)
4. How did the system works? Malthus in numbers.
-A random event increases population size by a certain proportion, say K.
(The event must be totally random)
-It follows that wages will go down by a fraction H = K * (-1.6); (diminishing returns)
-It then follows that GRR will go down by a fraction I = H *.20 (density dependence)
-Finally, survival will go down by a fraction J= H*.10 (positive check)
From this we can calculate the net result on NRR:
Remember that
NRR=GRR*S(Am)
Suppose that Am~25, GRR~2.6 and S(25)~.40 so that NRR=1.04 and r~.0016
where GRR is gross reproduction rate and Am is mean age at childbearing.
New NRR=New GRR*New S(Am)
~(old GRR)*(1+I)*((old S(A))*(1+J))
~(old NRR)*(1+.20*K*(-1.6))*(1+.10*K(-1.6)
~(old NRR)*(1-.032-.016)
If K=.10,
~(old NRR)*(.96)=.9984
Since old NRR was 1.04 the relative change in NRR is about 4%. That is, a ten percent increase
in population would decrease NRR by about 4%. And from this we can calculate the final impact
on r, the rate of natural increase. Use the following expression
NRR=exp(r*Tg)~exp(r*Am)
new r=ln (new NRR)/ 25
= -.0001
Remark:
-if one used elementary calculus we could conclude that elasticity of NRR relative to population
is about .25. Thus, an increase of .10 in population size leads to a decrease of .025 in NRR (a bit
less than calculated above) and to an r~.0006 or something less than calculated before.
F. Homeostasis vs randomness
What are the implications of a system with a feedback mechanism such as the one postulated
before? What difference did it make?
The answer is: it made all the difference between extinction and survival. This is somewhat
exaggerated as it turns out but it reflects well the situation.
Suppose that there are random events such as epidemics, wars, climatic disasters and the like.
Some of these force a decrease of the population. Under a system with homeostasis, the feedback
mechanism will lead to a period of recovery. Since the elasticity of NRR relative to population
density is weak, the period of recovery will be long, say 100 years. Similarly, when the
population went over the carrying capacity, the feedback mechanism would kick in and
equilibrium would be slow to return, 100 years.
When homeostasis is weak, what matters is the recent past, not the distant past. When
randomness prevails (elasticity of NRR relative to population density is 0), everything matters a
lot, including events that happened 1000 years ago. This can be illustrated by thinking of a
population that suffered three crises in succession after reaching a steady state following each
one. In this case, what matters for the population before it returns for the third time to
equilibrium has nothing to do with what happened during the first and second crises
An elasticity of .25 is very small so homeostasis is very weak. Indeed, the implied R squared of
the natural rate of increase relative to population density is only about .005. Just about nothing.
Almost all short run variation in r is density independent. But the key thing is this: despite being
weak density dependence makes itself felt. And, despite being weak, it comes to dominate over
the long run. It may not be important over the short run but it is over the long run.
To show this one can simulated populations with and without weak homeostasis. And the results
are striking:
–weak homeostasis leads to over the long run to greater stability
-randomness leads to great instability and with values close to those experienced in preindustrial
England to extinction.
{Graph No 9 from Ron Lee’s article: pp.453}
G. Malthus becomes irrelevant?
1. Evidence for density dependence diminishes even further shortly after 1750
=population fluctuations are reduced sharply (population crises disappear)
=life expectancy increases steadily from levels of about 35-45 to 75 and over;
the elasticity of mortality relative to income crashes.
=fertility begins a secular decline in developed countries, a trend inconsistent with increasing
real wages. Fertility declines does not begin to occur until the late sixties and seventies in
developing countries. The elasticity of fertility relative to income turns negative: a perverse
result in Malthus framework. And all this has happened because the cost of fertility or children is
proportionately higher for those who are most productive
=Technological progress and continuous productivity gains erode the so called iron law of wages
(diminishing returns)
All these changes lead to explosive population growth:
2. Is Malthus dead or has he temporarily fainted?
=possibility that “Boserupian” mechanism took over briefly enhancing the capacity of population
to growth; these mechanisms, goes the theory, lead from population density to higher levels of
technology and higher levels of productivity
=in doing so we have affected the environment-a phenomenon that did not pose stresses prior to
industrialization (Is this true?). Could it be that we are due for a population crash of some sort to
resume growth at or near stationarity?
Main mechanisms for this new feedback mechanism are:
O Poverty (technology grows at the expense of inequalities)-> Malthus goes local: O
Wars, old diseases that return
O Environmental stress -> new diseases, catastrophic events->Malthus goes global
O Drastic reduction of fertility
-Can technological advancement keep pace with increased population and consumption and with
the impact on environment that these cause?
-Are we seeing the first signs of forces that will drive population growth back to a level close to
ZERO?