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Ragnar Arnason*
Natural Resources and Property Rights:
The Icelandic Commonwealth
Conference on
Macroeconomic Policy:
Small Open Economies in an Era of Global Integration
Reykjavik 28-29 May 1999
Revised version
for publication in the workshop proceedings
Not to be quoted without consulting the author
*Department of Economics
University of Iceland
[email protected]
The large scale Nordic settlement of Iceland seems to have taken place mainly during
the period from about 870 to about 930 AD. During this period, there was a huge
influx of families and clans primarily from Norway and the Scottish isles. After this,
net immigration seems to have continued for several decades but at a reduced rate
(Nordal and Kristinsson 1987). In the year 930 the settlers adopted a common law and
established a country-wide governing organization called the Althing (Parliament) to
apply and further develop this law. The following period, until Iceland entered a joint
kingdom with Norway in 1262, is generally referred to as the Icelandic
Commonwealth. Here we use the term somewhat more loosely to cover the whole
period from the beginning of the settlement in 870 to 1262.
During the Commonwealth period, the Icelandic economy seems to have
experienced a number of output cycles. The initial settlers came to virgin land, well
grown and fertile and, by their standards, well suited to agriculture and livestock
farming. Wild game was also plentiful; birds, fish, seals and whales. Given these
bountiful natural resources, it is perhaps not surprising that the first half century or so
seems to have been economically highly successful. This is, in fact, evidenced by the
continuing high rate of immigration and population growth. Toward the end of the
10th century, this expansionary period seems to have come to an end. Slavery
contracted and many slaves were set free by their erstwhile owners. This suggests
falling prices of the substitute commodity, namely free labour, perhaps down to the
subsistence level (Agnarsdottir and Arnason 1988). The flow of immigrants dried up
almost completely and was actually followed by a degree of population emigration to
Greenland in the early 11th century. At the same time, accounts from this period
contain evidence of growing scarcity of and increasing pressure on natural resources.
During the 12th century, the Commonwealth's political institutions came under
increasing stress culminating in a prolonged civil war during the first two-thirds of the
13th century when the various clans and warlords fought for land, taxation rights and
power (Nordal and Kristinsson 1987). Although, the evidence is mostly indirect, it
seems likely that a increasing scarcity of natural resources, growing economic
inequality and generally declining economic standards were at least contributing
factors in this development.
This paper proposes to explore the macro-economic dynamics that may have
been instrumental in the above-described historical evolution of the Icelandic society
during the Commonwealth period. The Icelandic economy during this period was a
natural resource-based farming economy. Few of these natural resources were subject
to well-defined and enforced private property rights. Initially, of course, land was
common property and could be freely used by anyone. The settlement process
gradually transformed some of the land, primarily the homelands, into private
property. This process, however, took considerable time to work itself out. Also, large
parts of the land, e.g. woodlands, grazing areas and high country pastures tended to
continue to be held in common. The same applied to most stocks of game such as
birds, fish and sea mammals. As the period progressed, these natural resources came
under increasing pressure and apparently declined drastically. For instance,
woodlands, that initially covered large parts of the country (Ari Þorgilsson frodi
1958), were greatly reduced and almost disappeared during the following centuries.
Common grazing areas also declined over the centuries, some to the point of serious
soil erosion. Inland fish and game stocks were similarly greatly reduced. Given the
nature of the economy, the interaction between the population and the abundance (or
scarcity) of natural resources appears to have given rise to declining living standards
over time accompanied by long period output cycles of considerable intensity.
It has been understood for some time (see e.g. Dasgupta and Heal, 1979) that a
lack of private property rights results in misuse of natural resources. As we have seen,
the Icelandic Commonwealth suffered from limited private property rights and a
widespread use of common property arrangements over a range of important natural
resources (Eggertsson, 1990, 1992).1 Therefore, it may be interesting to consider the
evolution of the Commonwealth economy if full private property rights had been
instituted from the outset. This is similarly explored in the paper.
Our approach is to construct a simple model of a renewable natural resource
extraction economy and then to examine its behaviour under two sets of institutional
arrangements (i) common property rights and (b) private property rights. The model
employed is quite simple. For instance, the only factors of production are labour and
natural resources. Physical capital as such is not included. On the other hand, the
model is perfectly general with respect to the type of renewable resource included. So,
the basic approach is really applicable to natural resource based economies in general
The paper is arranged broadly as follows. In the next section we delineate the
basic economic model used in the paper. The following section deals with the
evolution of the economy under the common property situation. This is followed by
an examination of the evolutionary paths under a private property rights regime.
Finally, in the last section, we summarize the results of the paper.
The Basic Model
Consider a renewable natural resource whose quantity at time t is denoted by the
variable x(t). Let the renewal process of this resource be represented by the concave
function G(x(t)) defined over all non-negative x(t). More precisely we assume the
existence of the following resource renewal (or growth) function:
x (t) = G(x(t)), x0 and Gxx0,
where x epresents the first derivative of the resource quantity with respect to time. To
make this function interesting we assume that  x0 s.t. G(x)0. Normally, also  a
resource level, x10 s.t. G(x1)=0 and G(x) 0 for x x1. In the case of living
resources, x1 is often referred to as the virgin stock equilibrium.
Note that in addition to the traditional type of lack of private property rights, the Commonwealth
society also exhibited a certain communal property in wealth via the extensive communal insurance
and welfare aspects of the society. Thus, the local communities were required by law as well as
custom to take care of poor people and the needy. This, of course, reduced the rewards to private
enterprise as well as provided for the sustenance of large population than would otherwise have
been the case.
Equation (1) represents what may be called a self-renewable resource, i.e. one
whose renewal depends on its own stock level, x(t). Living resources such as
lifestock, game, forests, grasslands etc. are generally self-renewable. Some important
non-living resources such as clean air, underground water reservoirs etc. are also, at
least to some extent, self-renewable.
In the case of living resources the resource quantity is often referred to as
biomass and the corresponding growth function, G(x(t)), is usually taken to exhibit the
following additional properties:
G(0) = 0,  x20 s.t. x1 x2 and G(x2) = 0.
Under these assumptions, the biomass growth function is dome-shaped as described in
Figure 1:
Figure 1
Biomass Growth Function
x, biomass
It is easy to verify that of the three equilibria depicted in Figure 1, namely x=0,
x1 and x2, x=0 and x1 are locally stable while x2 is not. For biomass levels between x1
and x2 biomass growth is positive as indicated by the small arrows in the diagram.
This means that extraction amounting to this growth may be extracted without
reducing the biomass. Maximum biomass growth occurs at biomass level xmsy as
indicated in Figure 1. Outside the interval [x2, x1] biomass growth is negative
suggesting that x2 is the minimum viable biomass.
Let us denote the instantaneous aggregate rate of extraction from this resource
by the variable y(t) and define the corresponding production function as:
y(t) = Y(x(t),l(t)),
where l(t) represents the use of economic inputs. It is assumed that the function Y(.,.)
is increasing and jointly concave in both its arguments. Moreover,
Y(0,l(t))=Y(x(t),0)=0. This implies that both some positive level of biomass and
economic inputs are necessary for positive production.
The economic input variable, l(t), plays an important role in what follows. In
general it may represent any kind of an input; the use of labour, the use of capital, a
composite input such as extraction effort etc. There are in fact clear theoretic
advantages in leaving the empirical meaning of l(t) unspecified. In the current context,
however, it is perhaps most illuminating to think of l(t) as population or manpower.
Biomass growth and the extraction activity are assumed to be related in the
following simple way:
x (t) = G(x(t)) - Y (x(t),l(t)).
In what follows we will assume that the natural resource extraction, y(t),
represents benefits. Since l(t) is an economic input we may take it for granted that
there will be an opportunity cost associated with its use. Let us refer to the (market
clearing ) unit prices of outputs and inputs by the vector w. Economic behaviour
generally implies some development of the use of economic inputs over time.
Presumably this depends on both biomass and the current rate of use of economic
inputs as well as the prices, w. Let us represent this relationship by the differential
l (t) = F(x(t),l(t),w).
Given initial conditions, equations (3) and (4) define the evolution of biomass
and economic inputs over time, i.e. {x} and {l}, and consequently also that of output,
{y}, and other variables of interest such as the gross domestic product, wages and
resource rents. These equations and their solutions therefore are the focus of our
investigations in the subsequent chapters.
As an illustration consider welfare maximizing paths. More precisely, let us
assume that social welfare depends on the present value of net economic benefits
forever. Welfare maximum is found by solving the problem:
Max V=  (Y ( x, l ) - cl)exp(-rt)dt
s.t. x (t) = G(x(t)) - Y(x(t),l(t)),.
where c is the opportunity cost of economic inputs and r the rate of time discount.
Note also that the price of production is arbitrarily set at unity, so valuables are
measured in term s of units of the output, y.
Necessary conditions for solving this problem include:
Yl(x,l)(1-)-c = 0
 -r = - Yx(x,l) - (Gx(x) - Yx(x,l))
x = G(x) - Y(x,l),.
where  represents the shadow value of the resource.
Combining equations (I.1) and (I.2) yields a rather formidable looking
behavioral rule corresponding to (4) as
l = [(Yl(x,l)(Yl(x,l)-c)(r-Gx(x))/c - Yx(x,l)) - Yx,l (x,l)(G(x) - Y(x,l))]/Yll (x,l)
Solving (I.3) and (I.4) for the appropriate initial and terminal conditions then provides
a path in (l,x)-space that solves problem (I).
An interesting application of these optimal dynamic paths is to compare them
with other paths for biomass and economic inputs generated alternative non-optimal
processes. This is one of the main concern of this paper.
Equations (I.3) and (I.4) in this general form are quite complicated and the
dynamic paths that they define are not readily apparent. The same may be assumed to
apply to the paths defined by alternative processes. Therefore, in order to make
headway, we will find it useful to resort to more specific functional forms for the
biomass growth and extraction equations, (1) and (2). More specifically we will
assume the following forms for these functions:
x (t) = G(x(t)) = x - x2,
y(t) = Y(x(t),l(t)) = axlb.
In order to enable us to make numerical comparisons, particular numerical values for
the parameters, , , a, b, c and r will be assumed as listed below:
Table 1
Assumed parameter values
No unit
Common Property: Boom and Bust Cycles
The Icelandic Commonwealth economy was characterized by severely limited
property rights over natural resources. Initially, all land was more or less common
property and with it woodlands, lakes, rivers and game. As the country became
settled, homelands came under private property rights while many other natural
resources such as lakes, grasslands, wildstock and woodlands were still to a large
extent public commons. With the passage of time, property rights over these resources
became more extensive, better defined and enforced. Still, even to this day some of
these natural resources such as interior pasture areas, some game resources etc. have
remained common property. Thus, during the first few centuries of the Icelandic
commonwealth, property rights over natural resources were limited and poorly
enforced. Inevitably this arrangement had a marked influence on the process of the
economy and, subsequently society in general.
We will now employ the model developed in the previous section to throw
light on the possible evolution of the Icelandic Commonwealth economy assuming
common property natural resources.
The first step is to specify the dynamics of economic inputs, namely equation
(4) above. For this purpose let us regard the variable l(t) as population. Now,
population growth generally depends on two factors; natural growth, i.e., the excess of
births over deaths and net immigration, i.e.. the number of people immigrating less
the number emigrating. During the Icelandic Commonwealth, especially the early part
of the period, population movements between countries and regions were unrestricted.
Consequently, for most of this period, it is reasonable that population growth was
mainly determined by the latter factor.2
Under these circumstances, households (individuals) may be imagined to be
faced with the problem of determining their path of residency in Iceland or elsewhere,
that maximizes their income less the opportunity cost of moving in terms of foregone
income elsewhere and the direct moving costs and subject to the natural resource
constraint in Iceland.
Thus, consider household i. Let w1 represent the wage in Iceland and w0 the
corresponding opportunity wage in another relevant country where the household may
be currently residing. Obviously, w1= Yl (x,l), where as discussed above Y(x,l) is the
production function in Iceland. w0, on the other hand, may not unreasonably be taken
to be the subsistence wage. Also let ( l ), where l is total instantaneous emigration,
be the unit cost of moving residence. Finally, let li denote the decision by the i-th
household to emigrate. Moreover let li [0,1] where li =1 means emigration of the
whole household and li=0 no emigration whatsoever.
So, the i-th household will try to solve the following problem:
It is important to realize that this does not imply that immigration/emigration was solely responsible
for population growth but rather that population growth evolved as if it was solely determined by
Max V=  (Yl(x,l)li + w0(1-li)-li)exp(-rt)dt
x (t) = G(x(t)) - iY(x(t),l(t)),.
Among the necessary conditions for solving (II) we find:
Yl (x,l)(1-i)- w0 - ( l ) > 0  li=1, i.e., emigrate to Iceland
Yl (x,l)(1-i)- w0 - ( l ) < 0  li=0, i.e., emigrate to Iceland
where i represents household's i evaluation of the shadow value of the natural
resource. Given that household i is just one among many, this presumably is very
small. More precisely, it is possible to show that for identical households in
equilibrium the relationship between the social (i.e. true) shadow value of the resource
and household i's evaluation of this shadow value is3 :
i = */N,
where N is the total number of relevant households (i.e. those either in Iceland or
contemplating going there) and * is the social shadow value of the resource. Now, N
is generally a very high number. Thus, i is generally only a tiny fraction of *. For
this reason, the emigration decision (or more generally the population growth
decision) of the i-th household will be wrong. Too many people will move to Iceland.
Actually, i is small to the point of being negligible. Therefore, in the interest of
simplicity, we will, in what follows, take i=0.
Now, the cost of transport, presumably depends positively on the total number
of people moving at each point of time. A simple variant of this function is
b0+b1 l .Thus, provided immigration/emigration actually takes place, i.e. there is an
equilibrium in the immi/emigration market, (4) may be rewritten as:
l = (Yl (x,l) - c),
where c = w0+ b0 and = 1/ b1.
Note that although (9) is derived for immi/emigration, a similar function may
be assumed to apply for endogenous population growth. This of course holds in
particular if c is interpreted as the subsistence wage.
Now, given an initial position, equations (3) and (9) yield the evolution of the
economy over time. To explore this let's resort to the specific functional forms
specified in section 1 and the parameter values in Table 1. In addition, we take =1.
On these assumptions, the path of the economy in (x,l)-space from a virgin land initial
point may be calculated as follows:
A rigorous proof is found in Arnason 1990.
Figure 2
Lack of property rights over natural resources: Evolution of the economy
Labour ( n )
Land ( n )
Figure 2 illustrates the path of the natural resource, referred to as land, and
population, referred to as labour, over time. The initial point, is one of virgin stock
natural resource and virtually no labour, i.e. (x,l)=(1,0). As is apparent from this
figure, the economy evolves in a cyclical manner with an initial expansionary phase
based on non-sustainable natural resource extraction followed by contraction and a
spiral adjustment path to an equilibrium. The cycles are caused by the interaction of
natural resource and population adjustments. Given the resource dynamics, the faster
the adjustment in population the more pronounced are the cycles. The eventual
equilibrium is characterized by a substantially reduced resource level and a population
level that is considerably below the maximum attained one. More precisely, the
equilibrium is found at approximately (x,l)=(0.3,40.4), i.e. the resource has been
reduced to 30% of its initial size and the population has found an equilibrium at just
over 40.000 people compared to the maximum of about 90.000 people.4
The cyclical evolution of the economy is brought out more clearly by the
following diagrams of the paths of population and gross domestic product (GDP) over
time. In this simple model, the GDP = Y(x,l) = axlb.
Note that we have normalized the resource to range between 0 and 1 and scaled the population to
approximate the historical evidence.
Figure 3
Lack of property rights over natural resources: Evolution of
Labour ( n )
Figure 4
Lack of property rights over natural resources: Evolution of GDP
GDP( n )
As illustrated in Figure 4, there is rapid GNP growth for about the first half
century of the Commonwealth era. This, however, is based entirely on unsustainable
natural resource extraction and increased labour input. As increased labour input can
no longer be attracted to counteract dwindling natural resources, the expansionary
phase comes to an end and is followed by a severe contraction fuelled by continuing
falling resources and population decline as labour no longer earns its reservation wage
(See Figure 6). An equilibrium is reached several centuries down the road at a GDP
level less than half of what occurred during the initial boom years.
GDP per capita is probably a more helpful measure of welfare than GDP. The
evolution of this measure is even more depressing than the GDP itself as illustrated in
Figure 5
Figure 5
Lack of property rights over natural resources: GDP per capita
GDPperCapita ( n )
The population evolves as specified by equation (9) according to the
difference between the marginal product of labour and the opportunity cost of labour.
The former may be regarded as the actual wage rate and the latter either as alternative
wages abroad or the subsistence wage. The time path of the actual wage rate relative
to the subsistence wage (labelled "c") is illustrated in Figure 6.
Figure 6
Lack of property rights over natural resources: Evolution of wages
Wage ( n )
So, as illustrated in Figure 6, wages decline substantially from the bonanza
reaped by the initial settlers. After some time, wages dip below the subsistence level
and the population growth is reversed. The subsequent evolution consists of a cyclical
adjustment to equilibrium.
Private Property rights: Growth and Stability?
Let us now consider the same economic situation with the exception that private
property rights in the natural resources are now imposed. Under these circumstances
market forces will induce an opportunity charge for the use of natural resources that is
equal to the (optimal) shadow value of the resource. The potential immigrants to
Iceland will, of course, take this cost into account. Hence, the tragedy of the commons
experienced in our model of the Icelandic Commonwealth above, will be avoided and
the evolution of the economy will be materially different.
More formally: Let s represent the market (rental) price of resources under the
property rights regime. Under those circumstances, the net return to the households of
producing from these natural resources will be:
Y(x,l) - sY(x,l),
where of course, sY(x,li) is the extraction charge. And the marginal return to labour
Hence, a household i contemplating the move to Iceland and, consequently,
the use of these natural resources will be faced with the following maximization
Max V=  (Yl(x,l)(1-s)li + w0(1-li)-li)exp(-rt)dt
x (t) = G(x(t)) - iY(x(t),l(t)),.
Clearly, the necessary conditions for an interior solution include:
Yl (x,l)(1-s -i)- w0 - ( l ) > 0  li=1, i.e., emigrate to Iceland,
Yl (x,l)(1-s-i)- w0 - ( l ) < 0  li=0, i.e., emigrate to Iceland.
Assuming as before that i is negligible and that s equals the optimal shadow value of
the resource, *, these conditions are obviously socially optimal (i.e. welfare
maximizing) for the immi/emigration decision.
So, on this basis we may take it for granted that the path of the economy under
private property rights is essentially the optimal path. A numerical approximation to
the optimal path of biomass and labour from the same initial point as before is
described in the following diagram.5 For comparative purposes we also include in the
graph the corresponding path under the common property arrangement.
Figure 7
Resource-population evolution: Property rights (solid) vs. common property
Labopt( n )
Labour( n )
Landopt( n )  Land( n )
The solid curve indicates the approximately optimal path, the one presumably
applying under complete property rights, to equilibrium. As indicated, this converges
in an non-cyclical fashion to the long run optimal equilibrium. The other path
representing the common property arrangement is indicated by the dashed cyclical
path discussed in the previous section. In addition to being cyclical, this path clearly
terminates in a higher population and lower resource level than the optimal path. A
numerical comparison between the equilibrium values with and without property
rights is contained in Table 2.
Table 2
Equilibrium values of key variables under (a) common property rights and (b)
private property rights
Resources (Initially=1)
Annual rents (profits)
Marginal product of labour (wage rates)
Shadow value of resource
property rights
property rights
This and the other calculations in this paper were carried out with the help of the program package
According to Table 2, the common property economy results in a long term
resource level less than half and population almost double that of the private property
rights economy. The GDP is higher in the private property rights economy and, more
importantly, the GDP per capita is more than two times higher. Thus, clearly, the
private property rights economy produces an economically superior result on most
accounts, even from a global perspective.6
Comparison between the time paths for labour, natural resource (land) and
GDP for the common property and private property economies are illustrated in the
following three diagrams. In all cases the optimal paths are drawn solid and the other
Figure 8
Population paths: Property rights (solid) vs. common property (dashed)
Labopt( n )
Labour( n )
This is obvious because in the private property rights economy, the GDP is higher and the
population excluded presumably earns its reservation wage somepalce else.
Figure 9
Resource paths: Property rights (solid) vs. common property (dashed)
Landopt( n )
Land( n )
Figure 10
GDP paths: Property rights (solid) vs. common property (dashed)
GDPopt( n )
GDP ( n )
The relatively small difference between the path of GDP under the two
property rights regimes evidenced in Figure 10 and Table 2 may appear surprising.
This, however, is readily understandable. What is being maximized under the private
property rights regime is not gross production, i.e. GDP (remember all costs are
labour costs that are included in the GDP), but net economic rents, i.e. the difference
between gross production and the opportunity cost of labour as measured by its wage
rate. Given this objective, it is coincidental whether GDP under the private property
economy is higher or lower than under the common property economy. This is
illustrated in the following diagram, Figure 11, that gives the sustainable yield or
GDP from the natural resources (solid curve) as well as the costs measured at the
subsistence wage level of extracting that yield (dashed line). Also drawn in the
diagram are the optimal long term population level under the private property
arrangement, denoted by Lpriv, and the equilibrium population level under the
common property arrangement, denoted by Lcom.. As indicated in the diagram, the
optimal level is economically greatly superior to the other while the difference in
gross production is relatively small.
Figure 11
Sustainable GDP (solid), subsistence wage (dashed), optimal private
property population level (Lpriv), common property population level (Lcom)
s_yield (l)
15 Lpriv
The economic superiority of the private property economy becomes even more
obvious when the two paths of GDP per capita and economic rents are compared.
Economic rents are here synonymous with profits, i.e. Y(l,x)-Yl(l,x)l. The GDP paths
are presented in Figure 12, the rents in Figure 13, where 'Ropt' represents the
maximum rents attainable and 'Rents' represent the rents under the common property
regime. Rents
Figure 12
GDP per capita: Property rights (solid) vs. common property (dashed)
GDPoptperCapita( n )
GDPperCapita( n )
Figure 13
Economic rents: Property rights (solid) vs. common property (dashed)
Ropt( n )
Rents( n )
The simple model of a natural resource-based, common property economy developed
in this paper is able to generate macroeconomic cycles that seem qualitatively in
accordance with the known evolution of the Icelandic Commonwealth economy
during its first couple of centuries. This, of course, is not supposed to imply that he
model is correct. In fact, its extreme simplicity suggests that it is almost certainly
wrong. Nevertheless, it may capture some of the crucial aspects of the
macroeconomic dynamics of the Commonwealth and thus throw some useful light on
its actual evolution.
According to the model, the economy passed through a strong expansionary
phase during its first 40-50 years. This was followed by economic stagnation and then
contraction culminating in a severe depression about 100 years after the initial
settlement. These basic aspects (although not necessarily the severity of the
depression) seem confirmed by historical accounts from this period.
During the initial expansionary phase wages are predicted to have declined
continuously. At a certain stage of this development, especially when wages declined
below the subsistence level, slavery should have became uneconomical and withered
away. Indeed, according to the historical evidence, slavery began declining in the
early part of the 10th century and had mostly disappeared by the year 1000.
During the economic downturn, population growth is reversed according to
the model. In Iceland there is evidence of substantial emigration to Greenland about a
century after the initial settlement. Again this fits nicely with the economic cycles
generated by the model.
Christianity was adopted in the Icelandic Commonwealth at around the year
1000, about 125 years after the initial settlement. There are certain indications that
this new faith and the institution of the Catholic church rendered population control
more difficult.7 If true, this would be equivalent to a drop in the opportunity or
subsistence wage, making subsequent crises correspondingly deeper.
Economic conditions are undoubtedly among the main determinants of the
social and political developments. The question thus arises whether the economic
cycles generated by the model of this paper can be used to explain major aspects of
the socio-political developments during the Icelandic Commonwealth. We have
already seen how the model can be used to explain the decline and abolition of the
institution of slavery during the latter part of the 10th century. The model also predicts
a certain social stress due to the (predicted) economic depression during mid and
latter part of the 10th century. This may or may not provide an explanation for the
adoption of Christianity in the year 1000. At the same time, the model predicts a
prolonged upswing and diminishing cycles and consequently relatively unstressed
society during the early part of the 11th century. There is some historical evidence
supporting this.
More respect for human life, less warfare and blood revenge, infanticide outlawed.
What the model, in its current form, cannot explain are the severe social
stresses that led to the civil war toward the end of the Commonwealth in the 13th
century. According to the current simple version of the model, the economic situation
should actually have more or less stabilized by this time (at least relative to the severe
cycles of the 10th century) albeit at an economically distressed level. It is, of course,
quite possible, that some non-economic factors were primarily responsible for the
civil war. But it is also possible that factors such as declining opportunity wages8,
adverse environmental conditions etc. not included in the above modelling, lead to an
even further depression of living standards thus adding fuel to the civil war. Once
started, the civil war also reduced already depressed living standards even further.
During the course of the Commonwealth, private property rights over natural
resources seem to have been both expanded and increasingly enforced. In fact, it is
quite conceivable that the civil war was partly a consequence of the dispossession this
process implied. Improvement and expansion of property rights, of course, invalidates
the model of this paper to a corresponding degree. The process of improved property
rights, however, was apparently a slow one. By the time private property over the
countries natural resources had become the dominant institution (which actually may
not have happened for centuries), it may have been too late to reverse the situation
created by the previous common property arrangement. Given a sufficient
deterioration of the natural resource base, the only way to reverse the process (barring
a substantial technological progress) was by a sustained population reduction. This
was difficult, however, because (i) in the 11th and 12th century emigration seems to
have become infeasible (perhaps the poor people couldn't afford to emigrate), (ii)
population control was contrary to the Christian morality and selective starvation also
ran counter to the existing Nordic culture of communal help and assistance. Under
these circumstances, serious attempts at a reversal of the natural resource decline was
virtually impossible and mostly left to the sporadic intervention of natural disasters
and pestilence of which bubonic plague and black death later in the Middle Ages
seem to have been most influential.
Due to the Christian influence mentioned above and Church taxation.
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This appendix briefly examines the dynamics defined by the basic differential
equation system of the paper. Under the parameter specifications of Table 1, this
system is given by the equations:
x (t) =x - x2 - xlb,
(A.2) l = (bxlb-1 - c),
where, it may be recalled, x represents land and l labour. The rest of the variables are
parameters. To avoid unnecessarily tedious algebra we will, in what follows, assume
that the parameter b equals unity
The global dynamics of this system may be clarified by examining the
corresponding phase diagram.
Obviously, provided x is positive,
x  0  l = - x.
l  0  x= c.
These two equilibrium equations divide the positive quadrant in (x,l)-space into four
sectors or phases as illustrated in Figure A.1
Figure A.1
Phase diagram
Differentiation of equations (A.1) and (A.2) at their equilibrium levels yields:
x / l x 0  a  x <0,
l /  x l0    a >0.
So, l is growing everywhere to the right of the line l  0 and vice versa, and x is
growing everywhere above the line x  0 . Hence the direction arrows in the figure.
The direction arrows in Figure A.1 indicate that the pair (x,l) follows a cyclical
motion through time. This indication of cyclical movement receives further support
from noticing that (x,l) can only reach equilibrium from sectors II and IV in the
In the neighbourhood of equilibrium the dynamics of the system may be
approximated by linearization of equation (A.1) and (A.2). Equilibrium is given by:
xe = c
le =  - c
Taylor expansion of the dynamic system, (A.1) and (A.2) around equilibrium
 x      c  c   x  x e 
    
0   l  l e 
l  
The eigenroots of the Jacobian matrix of this system are given by:
i  (    c   2  c 2  4    c ) / 2 , i=1,2.
Clearly, if  and c are positive which they are by assumption, the real parts of
these eigenvalues are negative. Therefore, the equilibrium is locally stable.
Morover, if the the quantity 2c - 4 <0 the eigenroots will be complex
and the dynamic paths around equilibrium will be cyclical. clearly this can happen for
a wide range of the parameters involved. Note that the higher the speed of economic
adjustment, , the lower the subsistence wage, c, and the lower the  (which is ratio
between the biomass’s intrinsic growth rate, , and its carrying capacity) the more
likely is it that cycles will occur.
Under the parameter specifications in the paper the eigenroots are
1 = -0.28 + 1.82i,
2 = -0.28 - 1.82i,
which have negative real parts and are complex.