Reykjavik 28-29 May 1999
*Department of Economics
University of Iceland
ragnara@hi.is

1
0. Introduction
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

2 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.
1.
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:
(1) x ( t ) = G ( x ( t )),
 x

0 and G xx

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, x
1

0 s.t. G ( x
1
)=0 and G ( x )

0 for
 x

x
1
. In the case of living resources, x
1
is often referred to as the virgin stock equilibrium.
1 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.

3
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,
 x
2

0 s.t. x
1

x
2
and G ( x
2
) = 0.
Under these assumptions, the biomass growth function is dome-shaped as described in
Figure 1:
Figure 1
Biomass Growth Function
G
x
x
2 x msy x
1 x
It is easy to verify that of the three equilibria depicted in Figure 1, namely x =0, x
1 and x
2
, x =0 and x
1
are locally stable while x
2
is not. For biomass levels between x
1 and x
2
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 x msy
as indicated in Figure 1. Outside the interval [ x
2
, x
1
] biomass growth is negative suggesting that x
2
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:
(2) y ( t ) = Y ( x ( t ), l ( t )),

4 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:
(3) 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 function:
(4) 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:
(I) Max V =

0

( Y ( x , l ) - c
 l )
 exp (r
 t ) dt
{ l } s.t. x ( t ) = G ( x ( t )) Y ( x ( t ), l ( t )),. l ( t )

0, 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:
(I.1) Y l
( x , l )

(1-

)c = 0

5
(I.2)
 
r

= Y x
( x , l ) -

( G x
( x ) Y x
( x , l ))
(I.3) 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
(I.4) l

= [( Y l
( x , l )

( Y l
( x , l )c )

( r G x
( x ))/ c - Y x
( x , l )) - Y x,l
( x , l )

( G ( x ) Y ( x , l ))]/ Y ll
( 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:
(6) x ( t ) = G ( x ( t )) =
 x -
 x
2
,
(7) y ( t ) = Y ( x ( t ), l ( t )) = a
 x
 l b
.
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
Parameter

 a
Value
1

1 b 0.9 c 0.85 r 0.03
Units
1/time
1/resource
 time output/resource
 labour
No unit output/labour
1/time

6
2. Common Property: Boom and Bust Cycl es
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 w
1
represent the wage in Iceland and w
0
the corresponding opportunity wage in another relevant country where the household may be currently residing. Obviously, w
1
= Y l
( x , l ), where as discussed above Y ( x , l ) is the production function in Iceland. w
0
, 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 household to emigrate. Moreover let l i l i
denote the decision by the i-th

[0,1] where l i
=1 means emigration of the whole household and l i
=0 no emigration whatsoever.
So, the i-th household will try to solve the following problem:
2 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 immi/emigration.

7
(II) Max V =
 
0
( Y l
( x , l )
 l i
+ w
0

(1l i
)-
 l i
)
 exp (r
 t ) dt
{ l i
} s.t. x ( t ) = G ( x ( t )) -

1
 l i
( t )

0. i
Y ( x ( t ), l ( t )),.
Among the necessary conditions for solving (II) we find:
Y l
( x , l )(1-
 i ) w
0
-

( l

) > 0
 l i =1, i.e., emigrate to Iceland
Y l
( x , l )(1-
 i
) w
0
-

( l

) < 0
 l i
=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 is
3
:
(8)
 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 b
0
+ b
1
 l

.Thus, provided immigration/emigration actually takes place, i.e. there is an equilibrium in the immi/emigration market, (4) may be rewritten as:
(9) l

=

( Y l
( x , l ) - c ), where c = w
0
+ b
0
and

= 1/ b
1
.
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:
3 A rigorous proof is found in Arnason 1990.

Figure 2
Lack of property rights over natural resources: Evolution of the economy
93.659
100
8
50
0.62
0
0
0.143
0.5
1
1
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
 l b
.
4 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 population/labour
93.659
100
50
9
0.62
0
0
0 200 n
400
400
Figure 4
Lack of property rights over natural resources: Evolution of GDP
3
2.324
2
1
0.048
0
0
0
200 n
400
400
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.

10
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
6
4.755
4
2
0.435
0
0 200 400
0 n 400
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
4
4
2 c
0.392
0
0
0 200 n
400
400
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

11 and the population growth is reversed. The subsequent evolution consists of a cyclical adjustment to equilibrium.
3. 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 , l i ) is the extraction charge. And the marginal return to labour is:
Y l
( x , l )

(1s ).
Hence, a household i contemplating the move to Iceland and, consequently, the use of these natural resources will be faced with the following maximization problem:
(III) Max V =
 
0
( Y l
( x , l )

(1-s)
 l i
+ w
0

(1l i
)-
 l i
)
 exp (r
 t ) dt
{ l i
} s.t. x ( t ) = G ( x ( t )) -

1
 l i
( t )

0. i
Y ( x ( t ), l ( t )),.
Clearly, the necessary conditions for an interior solution include:
Y l
( x , l )(1s -
 i ) w
0
-

( l

) > 0
 l i =1, i.e., emigrate to Iceland,
Y l
( x , l )(1s -
 i ) w
0
-

( l

) < 0
 l i =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

12 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
(dashed)
93.659
100
50
0.62
0
0 0.5
0.143
Landopt n

( )
1
1
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
Variables
Common property rights
Private property rights
Population
Resources (Initially=1)
GDP
GDP/capita
Annual rents (profits)
Marginal product of labour (wage rates)
Shadow value of resource
40.416
0.303
0.619
0.95
0.065
0.855
*
20.386
0.633
0.697
2.121
0.418
1.909
0.555
5 This and the other calculations in this paper were carried out with the help of the program package
MathCad.

13
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 dashed.
Figure 8
Population paths: Property rights (solid) vs. common property (dashed)
93.659
100
50
0.62
0
0
0 200 n
400
400
6 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)
1
1
14
0.5
0.143
0
0 200 400
0 n 400
Figure 10
GDP paths: Property rights (solid) vs. common property (dashed)
3
2.324
2
1
0.048
0
0 200 400
0 n 400
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 L priv
, and the equilibrium population level under the common property arrangement, denoted by L com
.. 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 (L priv
), common property population level (L com
)
1
15
0.5
0
0 15 L priv
30 L com
60 l
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 )Y l
( 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)
6
4.755
4
16
2
0.435
0
0 200 400
0 n 400
Figure 13
Economic rents: Property rights (solid) vs. common property (dashed)
1.441
2
1
0
0.44
1
0
0 200 n
400
400

17
4. Discussion
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 10 th
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 10 th 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 11 th
century. There is some historical evidence supporting this.
7 More respect for human life, less warfare and blood revenge, infanticide outlawed.

18
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 13 th 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 10 th
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 wages
8
, 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 11 th and 12 th 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.
8 Due to the Christian influence mentioned above and Church taxation.

19
References
Agnarsdottir, A and R. Arnason. 1983. Slavery in the Saga Period, (In Icelandic),
Saga .
Anonymous. 1998. Mathcad 8: Users Gude . Mathsoft. Cambridge. USA.
Ari Þorgilsson, frodi. 1958. The Book of Settlements , (In Icelandic). Jakob
Benediktsson (ed.) Hið íslenzka fornritafélag. Reykjavík.
Arnason, R. 1990. Minimum Information Management in Fisheries. Canadian
Journal of Economics
Dasgupta, P.S. and G.M. Heal. 1979. Economic Theory and Exhaustible Resources .
Cambridge University Press
Eggertsson, Th. 1990. Economic Behavior and Institutions . Cambridge University
Press. Cambridge.
Eggertsson, Th. 1992. Analysing institutional successes and failures: A millennium of common mountain pastures in Iceland. International Review of Law and
Economics
Nordal, J. and V. Kristinsson (eds.). 1987. Iceland 1986 . The Central bank of Iceland,
Reykjavík.

20
Appendix
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:
(A.1) x ( t ) =
 x -
 x
2
- x
 l b
,
(A.2) l

=

( b
 x
 l b-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,

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
 
0
 l

/
 x l
 
0
  a
 x <0,
   a >0.

21
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 diagram.
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: x e
= c l e
=

-
 c
Taylor expansion of the dynamic system, (A.1) and (A.2) around equilibrium yields:

 x  l






 

 c

0 c




 x l

 l x e e

 .
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

2
= -0.28 + 1.82
 i,
= -0.28 - 1.82
 i, which have negative real parts and are complex.