Growth of an integrated economy of humans andrenewable biological resources

Liaila Tajibaeva

c Liaila Tajibaeva 2008

Department of Economics, Ryerson University, Toronto, M5B 2K3, Canada; Tele-phone: 416-979-5000 ext 7724; Fax: 416-598-5916; E-mail ltajibaeva@ryerson.ca

Abstract : This paper analyzes development of a small low-income economy thatis endowed with an open access renewable natural resource. To carry out the

analysis the paper expands the neoclassical growth model to include a renewableresource and a home production sector that specializes in harvest of the resource.There exist two saddle path stable steady states. The interior steady statepreserves the natural resource despite its open access nature while the cornersteady state depletes the natural resource stock. The initial endowment of

natural resource and capital assets determines which steady state the economyconverges to. If the economy starts with higher level of assets capital deepeningoccurs relatively quickly and labor productivity in the formal sector growsrapidly. This results in higher wages, which creates an incentive to pull laboraway from the informal harvest and preserve the natural resource. On the

contrary, if the economy starts with lower levels of both assets by the time thecapital stock accumulates to draw labor into the formal sector the natural

resource has already been depleted. In addition, if the intrinsic growth rate of thenatural resource is low or if the size of the economy relative to the biomass of theresource is high then the economy always depletes the natural resource stock andregardless of the initial endowment of assets converges to the corner steady state.

Key words: renewable natural resources, economic growth, open access re-sources, and home production

JEL classi�cation: Q2, O1

1 Introduction

The relationship between economic growth and natural resources is a topic of

continuous great interest. Are plentiful natural resource endowments an aid or an

impediment to economic growth? Does economic growth lead to environmental

improvement or degradation? Keeping these questions in mind and focusing on

low-income economies we �nd that most of them are primarily agrarian. Low-

income countries often depend on renewable natural or biological resources for

subsistence of their residents or as Partha Dasgupta puts it "Poor countries are for

the most part biomass-based subsistence economies, in that their rural folk eke out

a living from products obtained directly from plants and animals." ([14] p. 273).

Households engage in home production and harvest of natural resources al-

locating their labor away from the market activities. In such economies home

production is a signi�cant part of output, consumption, and welfare. For example,

in Tanzania over 90% of all businesses operate outside the legal system and there is

"a disconnection between government e¤orts and the vast majority of Tanzanians

who are excluded from participating in a market economy" (The Economist [1]).

Another attribute that is often observed in low-income economies is the lack of

well-de�ned and enforced property rights. This implies that renewable natural or

biological resources that are exploited by households are often open access in nature

or at least in practice. Establishing a relationship between economic growth and

natural resource endowment in a developing economy entails taking into account

market imperfections pertaining to the use of these recourses. The rights to harvest

1

the renewable natural resources are not sold in the market or protected by law.

In addition, it is often the case that the �nal product of such harvest does not

enter a formal market but instead is directly consumed by households. Market

imperfections are common in developing countries and a¤ect natural resource use

and economic growth. This paper is motivated by such low-income economies that

are dependent on their open access renewable natural resources and allocate much

of their labor to home production and self employment.

The paper analyzes the interaction between economic growth and an open

access renewable natural resource in a small developing economy. It integrates

an open access renewable natural resource in a convex growth model (along the

lines of the Ramsey, Cass, and Koopmans model). To carry out the analysis the

paper expands the neoclassical growth model in three important ways. First, it

introduces a renewable natural resource endowment and dynamics as a new factor

input. Second, it introduces market imperfections associated with the use of this

resource. Third, it introduces a home-production sector, which contributes to labor

allocation decisions, output, consumption, and ultimately welfare.

The economy has two sectors. The �rst sector is a wage employment or a

formal sector that uses labor and capital to produce a composite consumption

good that can be either consumed or invested. The second sector is a harvest or

home-production sector that uses labor and renewable natural resource to produce

a harvest good. This harvest good does not enter the market and is directly pro-

duced by households for own consumption. Every time period individuals choose

how to allocate labor between the harvest sector and wage employment in the

composite good sector. The harvest by each individual household from an open

access natural resource does not account for the negative e¤ect that it has on the

2

future productivity of that stock and it exerts negative externalities on all other

current and future users of the resource.

Analysis of equilibrium for a small closed economy, both of steady state and of

transition dynamics, shows that if an economy starts with low natural resource and

capital stocks then initially it experiences low wages. In addition, since both stocks

are low the households cannot increase their consumption through a composite

good. This combined with low opportunity cost of lost wages leads to a large

proportion of labor being allocated to harvest and the resource stock being depleted

early on. After the resource is depleted the economy starts to employ all the labor

in the composite good sector, accumulates capital, and converges to a steady state

where the natural resource is depleted. However, if an economy starts with higher

natural resource and capital stocks then initially and over time it experiences high

returns to labor in employment and converges to a steady state without depleting

the open access natural resource. Over time the opportunity cost of lost wages

is high enough so that it is not worthwhile for any household to allocate much of

their labor to harvesting the natural resource. Instead labor is primarily allocated

to employment and this leads to growth without resource depletion.

A recent book "Natural Resources and Economic Development" by Edward

Barbier [2] summarizes the existing research and further explores the contribution

of natural resources to economic development in low-income countries by recog-

nizing the fact that "the environment is not a "luxury" for economic development

but contains natural "capital" fundamental to growth and development in poorer

economies". A large body of literature has been devoted to study the relation-

ship between non-renewable and so-called �point�resources and economic growth

to the e¤ect of demonstrating both negative and positive relationship (for exam-

3

ples demonstrating negative relationship see Rodriguez and Sachs [35], Gylfason

and Zoega [19], Sachs and Warner [37], and for examples demonstrating positive

relationship see Wright and Czelusta [47], Mehlum, Moene, and Torvik [28]).

Some of the recent studies examine the relationship between renewable natural

resources and economic growth. In the existing literature, the papers by Eliasson

and Turnovsky [16] and Lopez, Anriquez, and Gulati [25] are the closest to the

current paper in a very general sense of considering a relationship between renew-

able natural resources and economic growth. However, they di¤er substantially

in the kind of economy that is analyzed, how the natural resource is used, and

the type of ownership over the resource. Eliasson and Turnovsky [16] show that a

resource sector can coexist with constant growth. To demonstrate this result the

authors study the equilibrium endogenous growth rate of a small open economy

endowed with a renewable resource. They use a continuous time approach and

unlike the current paper do not solve for the entire transition path and use instead

a linear approximation around the steady state. The resource endowment creates

a comparative advantage in trading with other countries. The renewable resource

is used for purchasing an imported consumption good. All the resource harvest

goes abroad. The consumers enjoy a higher diversity in consumption that would

not have been possible without the resource endowment. However this variety in

consumption results in a lower equilibrium growth rate. Private agents allocate

too much labor to the resource sector reducing the equilibrium growth rate below

its social optimum. The authors examine the case with perfect property rights

over the resource and brie�y mention that if there were open access to the natural

resources then problems would arise. The current paper contributes by analyzing

renewable resources in the open access framework and imperfect markets.

4

Lopez, Anriquez, and Gulati [25] examine a continuous-time optimal planner�s

problem with two �nal-good sectors: clean and dirty. The resource is a factor in-

put speci�c to the dirty sector, which represents polluting and/or natural capital

intensive sector of the economy. The economy can invest in three assets: natural re-

source, capital, and human capital. Using this model the authors examine whether

having well-de�ned property rights over the natural resource is su¢ cient for sus-

tainable growth of a distortion free economy. The authors also consider whether

it is possible to achieve sustainable economic growth when property rights over

the natural resource are ill-de�ned. They show that even if environmental policy

is fully absent or property rights are ill-de�ned with no investment in the nat-

ural asset, provided that all other assets have well-de�ned property rights and all

other markets are perfectly competitive, environmental sustainability with positive

economic growth is still feasible. The authors brie�y mention the case when con-

vergence is not possible and the resource stock is depleted. The current research

contributes by analyzing the possibility of multiple steady states with renewable

natural resource preserved or depleted.

To my knowledge, there is no existing literature that introduces a home-production

sector to study a relationship between renewable natural resources and economic

growth in a general equilibrium framework. In my opinion home production is

a substantial component in how the renewable resources are used in low-income

economies. The current paper contributes by introducing and analyzing home-

production in this framework. Home-production has been studied in economic

development literature but not in relation to renewable natural resources. For

example, Parente , Rogerson, and Wright [31] incorporate Becker�s [4] notion of

household production into their analysis of the consequences of including home

5

production for developing economies when accounting for international income

di¤erences and assessing the impact of distortionary policies. They �nd that the

key implication is that individuals spend less time working in the market in poorer

countries than in richer countries. The authors conclude that their �ndings and

existing evidence constitute "support for explicitly incorporating household pro-

duction into models of economic development". Another example, is a paper by

Gollin, Parente, and Rogerson [16] where the authors introduce home production

into the growth model and show that it accounts better for sectorial aspects of the

cross-country data than a straightforward agricultural extension of the neoclassical

growth model.

There is also a large body of literature where the sole focus is on the renewable

natural resources and these are not studied in an economic growth framework.

Brock and Xepapadeas [7] develop an approach to unify equilibrium price theory

with ecological models in which there is species competition for resources to prove

the existence of a price equilibrium for a stochastic discrete choice model. Pascual

and Hilborn [32] and Barrett and Arcese [3] conduct more applied research model-

ing more elaborate biological resource equations but treating economics harvesting

decisions as exogenously given. Pascual and Hilborn [32] focus more on the e¤ects

of alternative harvesting strategies on the resource population within a Bayesian

decision setting. While Barrett and Arcese [3] use the resource population dynam-

ics model developed by Pascual and Hilborn [32], and build onto it to explore the

interactions of wildlife populations and human consumption behavior when labor

and product markets are imperfect. This paper di¤ers substantially form that lit-

erature in the sense that it analyses natural resources as one of the components in

the contexts of economic growth in a general equilibrium framework.

6

Loibooki et al [24] investigate the relationship between illegal harvest and in-

come. An open access form of ownership creates incentives to overuse a resource

and each additional individual using it creates a negative externality on all other

users of this resource (Dasgupta and Maler, [15]). This in turn can create a cycle,

in which overexploited agricultural soils, pastures, �sheries, forests, and water re-

sources result in even smaller economic gains. However, households may not imple-

ment sustainable resource management even when the resource is privately owned.

Reardon and Vosti [44] examine the ability and willingness of rural households

to implement sustainable natural resource management. They denote by "welfare

poverty" the inability to meet basic human food, shelter, and clothing needs, while

they denote by "investment poverty" the inability to carry out sustainable nat-

ural resource management even when there is adequate wealth to prevent welfare

poverty. They also note that even though the capacity for capital-led investment is

necessary for households to invest in sustainable natural resource management, it

is not su¢ cient, because imperfect markets may prevent conversion of assets from

one form to another (Swinton and Escobar, [43]).

The rest of the paper has the following structure. The next section develops

the integrated model that incorporates economic growth and renewable natural

resource theory. Section 3 de�nes and characterizes equilibrium for that model,

solves for and analyses steady states. Section 4 parameterizes the model and solves

for and analyzes the entire transitional dynamics. Section 5 concludes the paper

and identi�es the next steps for this research.

7

2 An integrated model

This section combines the Ramsey [34], Cass [9], and Koopmans [23] convex

model of economic growth with open access renewable resources model to represent

an integrated economy of humans and biological resources.

This economy has an in�nite horizon over discrete time periods. The economy

consists of households; two production sectors, a composite good that can be used

for consumption and investment, and a harvest good that can be used for consump-

tion; and three factors of production, labor, capital stock, and renewable natural

or biological resource stock. In the following I go over each of these components

in detail.

2.1 Households�preferences and endowments

Consider a small economy with I in�nitely lived households. All households

are identical in their preferences and initial factor endowments. De�ne ct 2 <+

as consumption at time t 2 [0; 1; 2; :::;1) by a representative household. The

representative household has logarithmic utility function:

1Xt=0

�t ln (ct) , (1)

where the discount factor � satis�es 0 < � < 1. Overall consumption, ct, consists of

a composite good xt 2 <+ and a harvest good ht 2 <+. The household cares only

about the level of overall consumption and does not care about its composition

in terms of the shares of composite and harvest goods treating them as perfect

8

substitutes1:

ct = xt + ht. (2)

Each household has an endowment of time, �l, for each period. Each time period

t a household decides how to allocate its time endowment between spending lt

amount of time working in a composite good sector and earning the wage rate

wt, and spending the remaining, �l � lt, amount of labor time in self-employment

or home production harvesting the biological resource for own consumption. All

of the time endowment is spent on labor, there is no leisure in this model. In

addition to labor endowment, each household owns initial capital, k0, which can

be augmented through investment. Each household does capital accumulation.

Additions to the capital stock are made through investment minus depreciation

kt+1 � kt = it � �kt (3)

where it is period t investment by a household and 0 < � � 1 is a capital depre-

ciation rate. The composite good can be invested, it, or consumed, xt, while the

harvest good, ht, is perishable and cannot be stored from one period to another.

The household rents its capital, kt, to the �rms at the rental rate rt.

1Given the nature of consumption in poor countries, it is likley that home-produced andmarket-produced goods are close substitues. For example, in Parente, Rogerson, and Wright [31]ct = (�x�t + (1� �)h�t)

1=� with � = 0:6 such that home and market goods are close substitues(elasticity of substitution 1

1�� = 2:5). The authors report that even for the U.S. economy � isestimated to be between 0:4 (elasticity of substitution 1

1�� = 1:67) and 0:45 both using microdata (Rupert, Rogerson, and Wright [36]) and macro data (McGrattan, Rogerson, and Wright[27]).

9

A representative household�s budget constraint is

1Xt=0

pt (xt + it) �1Xt=0

pt (wtlt + rtkt) , (4)

where pt is period t price of the composite good. The harvest good, ht, does not

enter into the budget equation. Harvest of the biological resource is home-produced

by a household and is consumed directly by the household without entering the

market. Equivalently, the Arrow-Debreu budget constraint (equation 4) can be

restated as the following sequential budget constraint:

pt (xt + it) � wtlt + rtkt. (5)

Both speci�cations yield the same results. See Appendix A for the derivation of

the results using the sequential budget constraint (equation 5).

2.2 Biological resource and harvest good sector

The economy is endowed with an initial stock of a renewable biological re-

source, B0. An example of a resource to keep in mind can be wildlife. Additions to

the resource stock are made through the natural biological growth of the resource

minus the harvest of the resource. In the absence of harvesting, the resource dy-

namics are given by Bt+1�Bt = G (Bt), where G (Bt) describes the natural growth

of the resource that accounts for the natural birth and mortality rates. The func-

tion of the biological growth is given by a logistic function2 G (Bt) = sBt�1� Bt

�B

�,

2The logistic growth function has been widely used in modeling biological populations. Ingeneral, logistic growth starts at a zero, rises, peaks, falls, and reaches zero at a �nite environ-mental carrying capacity. It was �rst proposed in the nineteenth century and since then had hadempirical success (Conrad and Clarck [12]).

10

where s is the intrinsic growth rate of the biological resource and �B is the envi-

ronmental carrying capacity or the maximum biological resource stock that can be

sustained by the environment. In the absence of harvest, the biological resource

stock, Bt, converges to its maximum carrying capacity, �B.

In this economy there are no established ownership rights over the biological

resource. Any household has a free access to harvest the resource without pay-

ing a direct fee for it. Each household is endowed with a harvesting technology,

H��l � lt; Bt

�, which transforms labor, �l� lt, and biological resource stock, Bt, into

harvest by an individual household. The harvest technology takes the following

constant returns to scale formH��l � lt; Bt

�=��l � lt

�1��B�t , where harvest is mea-

sured in the same units as the resource stock, Bt, and the biological resource share

in the harvest technology � satis�es 0 < � < 1. The harvest function H��l � lt; Bt

�is increasing in the amount of labor that is allocated to harvesting, �l� lt, and the

total biological resource stock Bt. Harvesting productivity is determined by the

stock level at the start of each period. The higher the biological resource stock,

the easier it is to harvest it. As the resource stock declines harvest becomes more

labor intensive. When there is harvest it is a home-produced good all of which is

consumed by a household and is equal to ht, the consumption of the resource good

by a household in time period t:

ht =��l � lt

�1��B�t . (6)

The harvest by each individual household from an open access biological resource

stock does not account for the negative e¤ect that it has on the future productivity

of that stock and exerts negative externality on all other current and future users

11

of the resource. Total resource harvest in period t is the sum of harvest by all

individual households, I, and since all households are identical it is equal to Iht.

With harvest, the biological resource stock dynamics are given by:

Bt+1 �Bt = sBt�1� Bt�B

�� Iht. (7)

2.3 Composite good sector

There are many perfectly competitive �rms in the composite good sector.

Each �rm has access to a constant returns to scale technology F (Lt; Kt) in labor,

Lt 2 <+, and capital, Kt 2 <+: F (Lt; Kt) = AL1��t K�

t , where A is total factor

productivity and � is capital share, which satis�es 0 < � < 1. The �rm maximizes

its revenues from the composite good sales minus its labor and capital factor costs:

maxLt;Kt�0

pt�AL1��t K�

t � wtLt � rtKt

�. (8)

The following sections analyze this economy.

3 Competitive equilibrium

In this section I �rst de�ne a competitive equilibrium for the above economy

and then characterize and analyze it.

3.1 De�nition

Given the initial capital stock, Ik0, and initial biological resource stock, B0, al-

locationnfct; xt; it; ht; lt; kt+1gIj=1 ; Lt; Kt; Bt+1

o1t=0and a price system fpt; wt; rtg1t=0

12

constitute an equilibrium if:

1. Given prices and endowment vector��l; k0

�, allocation fct; xt; it; ht; lt; kt+1g1t=0

maximizes representative household�s objective function (equation 1) subject to

constraints (equations 2, 3, 4, and 6) and the non-negativity conditions 0 � lt � �l,

and ct; xt; ht; kt+1 � 0.

2. Given prices, allocation fLt; Ktg1t=0 maximizes �rm�s pro�ts (equation 8).

3. The natural resource stock fBtg1t=0 changes over time (equation 7).

4. Markets clear

I (xt + it) = AL1��t K�t for all t, (9a)

Ilt = Lt for all t, (9b)

Ikt = Kt for all t, (9c)

Iht � Bt + sBt

�1� Bt�B

�for all t. (9d)

The market clearing condition (9a) states that the total production of the com-

posite good equals its consumption and investment by all households. Condition

(9b) states that the amount of labor supplied by all households to the production

sector equals the amount of labor employed by the �rms. Condition (9c) states that

the amount of capital supplied by all households to the production sector equals

the amount of capital rented by the �rms. Condition (9d) states that harvest by

all households cannot exceed the biological resource stock.

3.2 Characterization of equilibrium

The representative household�s problem can be simpli�ed by substituting for ct

13

from equation (2) and for ht from equation (6) into the objective function (equation

1) and by substituting for it from equation (3) into the budget constraint (equation

4). With these simpli�cations the household�s problem can be stated as follows:

maxfxt;lt;kt+1g1t=0

1Xt=0

�t ln�xt +

��l � lt

�1��B�t

�(10a)

such that1Xt=0

pt (xt + kt+1 � (1� �) kt) �1Xt=0

pt (wtlt + rtkt) (10b)

xt; kt+1 � 0

0 � lt � �l

k0 > 0 given

where equation (10a) is the household�s objective function and equation (10b) is

the household�s budget constraint. The Lagrangian for the household�s constrained

utility maximization problem is

L =1Xt=0

�t ln�xt +

��l � lt

�1��B�t

�+�

( 1Xt=0

pt (wtlt + rtkt � xt � kt+1 + (1� �) kt))

Without ruling out a priori a possibility of a corner solution consider the following

Kuhn-Tucker conditions which are both necessary and su¢ cient, and a transver-

14

sality condition for capital stock:

@L@xt

=�t

x�t +��l � l�t

�1��B�t

� �pt � 0 with equality if x�t > 0 (11a)

@L@lt

= ��t (1� �)

��l � l�t

���B�t

x�t +��l � l�t

�1��B�t

+ �ptwt � 0 with equality if l�t > 0 (11b)

@L@kt+1

= �� (�pt + pt+1 (rt+1 + 1� �)) � 0 with equality if k�t+1 > 0 (11c)

@L@�

=

1Xt=0

pt�wtl

�t + rtk

�t � x�t � k�t+1 + (1� �) k�t

�� 0 (11d)

with equality if �� > 0

limt!1

�t

x�t +��l � l�t

�1��B�t(rt+1 + 1� �) k�t ! 0. (11e)

Equation (11c) states the intertemporal price condition and holds with equality

for capital stock k�t+1 > 0:

rt+1 + 1� � =ptpt+1

. (12)

Taking the ratio of equation (11a) in period t+ 1 over period t we get:

��xt +

��l � lt

�1��B�t

�xt+1 +

��l � lt+1

�1��B�t+1

=pt+1pt, (13)

which also holds with equality for a composite good x�t , x�t+1 > 0. Substitute for

the price ratio from equation (12) into equation (13) and rearrange to get:

(rt+1 + 1� �)�xt +

��l � lt

�1��B�t

�xt+1 +

��l � lt+1

�1��B�t+1

=1

�. (14)

15

Take the ratio of equations (11a) and (11b) to get:

(1� �)

Bt��l � lt

�!� � wt with equality if l�t > 0. (15)

Equations (11d), (11e), (12) through (15) constitute a representative household�s

competitive equilibrium conditions.

The �rm�s pro�t maximization problem with the constant returns to scale tech-

nology establishes labor wage and capital rental rate as:

wt = (1� �)A�Kt

Lt

��rt = �A

�LtKt

�1��.

Given that all households are identical in their preferences, market clearing con-

ditions (9b) and (9c), and CRS technology, then Kt = Ikt and Lt = Ilt, and the

above wage and capital rental rate can be restated as follows:

wt = (1� �)A�ktlt

��(16)

rt = �A

�ltkt

�1��. (17)

Substitute for wage and rental rate equations (16 and 17) into the household

conditions. Combine the representative household�s competitive equilibrium con-

ditions, �rm�s labor wage and capital rental rate conditions, and market clearing

16

conditions to derive the �ve equations that characterize an equilibrium.

��A�lt+1kt+1

�1��+ 1� �

��xt +

��l � lt

�1��B�t

�xt+1 +

��l � lt+1

�1��B�t+1

=1

�(18a)

(1� �)

Bt��l � lt

�!� � (1� �)A�ktlt

��with equality if l�t > 0 (18b)

xt + kt+1 � (1� �) kt = Al1��t k�t (18c)

Bt+1 �Bt = sBt�1� Bt�B

�� I

��l � lt

�1��B�t (18d)

I��l � lt

�1��B�t � Bt + sBt

�1� Bt�B

�(18e)

Equation (18a) is the Euler Equation. Equation (18b) requires equilibrium mar-

ginal product of labor be equal in both composite and harvest good sectors when

both sectors are active. The inequality in this condition comes from an endogenous

labor allocation decision and an endogenous possibility for one of the sectors be-

coming inactive over time. Equation (18c) is the composite good sector feasibility

condition stating that the consumption and investment is equal to production of

the composite good. Equation (18d) is the biological resource di¤erence equation.

Equation (18e) is the biological resource feasibility condition stating that the total

harvest cannot exceed the resource stock. Equations (18) characterize an equilib-

rium de�ned in section 3.1. To analyze the economy I consider its steady states

and transitional dynamics.

4 Steady State

In steady state xt+1 = xt = xss, lt+1 = lt = lss, kt+1 = kt = kss, Bt+1 =

17

Bt = Bss, that is there is no change in consumption of a composite good, labor,

capital, and resource stocks. In steady state equations (18) that characterize an

equilibrium can be restated as follows:

�A

�lsskss

�1��+ 1� � = 1

�(19a)

(1� �) (Bss)� = (1� �)A�ksslss

�� ��l � lss

��(19b)

xss + �kss = Al1��ss k�ss (19c)

sBss

�1� Bss�B

�= I

��l � lss

�1��B�ss (19d)

I��l � lss

�1��B�ss � Bss + sBss

�1� Bss�B

�(19e)

Let us examine these conditions for all possible steady states. Suppose wage labor

were equal to zero, lss = 0. Then feasibility condition (19c) implies that xss+�kss =

0 and since capital stock and consumption of a composite good can only be greater

than or equal to zero the only values of these two variables that satisfy this equation

are zeros, xss = 0 and kss = 0. This means that household consumption would be

derived solely from the natural resource stock and all available labor time of all

households in this economy would be devoted to harvest an open access resource

with zero opportunity cost of labor, which is not sustainable in the long-run steady

state. Now suppose that instead capital stock is equal to zero, kss = 0, which

further implies that investment is equal to zero. Then feasibility condition (19c)

implies that consumption of a composite good xss = 0. Since there is no composite

good sector all labor must be allocated to harvesting the open access natural

resource, lss = 0, and we are back to the contradiction of the previous case. Now

suppose that natural resource is equal to its carrying capacity, Bss = �B. This

18

implies that there is no harvest and by equation (19d) all labor time is allocated

to the composite good sector, lss = �l. Then by equation (19b) (1� �) (Bss)� = 0

or Bss = 0, which is a contradiction to the assumption that Bss = �B. Now

suppose that natural resource stock is depleted, Bss = 0. Then there is no harvest

and by equation (19b) all labor is devoted to the composite good sector, lss = �l.

This satis�es the resource feasibility constraint (19e). Then equation (19a) can be

solved for capital stock, kss = �l�

1��1+��A

�� 11��

, and equation (19c) can be solved for

consumption of a composite good, xss = �l�A'�1� �

'1

�, where '1 =

�1��1+��A

� 11��

.

Now suppose that xss > 0, kss > 0, 0 < lss > �l, and 0 < Bss < �B. In this case

the system of equations (19) can be solved for a unique interior solution. It is also

technically possible for all variables to be zero in steady state and satisfy equations

(19). Table ?? summarizes the non-zero steady states, where '1 =�

1��1+��A

� 11��

and '2 =

(1��)(1��)A

�1��1+��A

� �1��! 1

�

are constants.

Table 1: Steady StatesVariables Steady state 1 Steady state 2

(Interior) (Corner)

xss

��l � '2

��B � I �B

s('2)

1�����

A'�1� �

'1

��l�A'�1� �

'1

�lss �l � '2

��B � I �B

s('2)

1���

�l

kss�l�'2( �B� I �B

s('2)

1��)'1

�l'1

Bss �B � I �Bs('2)

1�� 0

In steady state 1 all choice and state variables are interior. Labor is allocated

between working in a composite sector for a wage and harvesting the natural

19

resource for own consumption. In this steady state the natural resource stock is

positive. In steady state 2, all labor is allocated to the wage sector and the open

access resource stock is depleted.

Lemma 1 For � 2 (0; 1) and � 2 (0; 1), then 1�> 1 � �, so that '1 > 0 and

'2 > 0.

If Bss > 0 then s > I'1��2 . In steady state 1 even though a natural resource is

an open access resource it is not depleted. A household does not pay a direct fee to

harvest the resource but it pays in opportunity cost, which is the cost of lost wages.

Every time period a household makes a labor allocation decision by weighing the

wages in the composite good sector versus labor productivity in harvesting. When

wages are high enough the opportunity cost of harvesting an open access resource

becomes so high that households choose to allocate most of their time working for

a wage. In steady state 1 the natural resource stock is greater than in steady state

2. In sections 4.3 and 4.4 of this paper I address the conditions of when the interior

and when the corner solutions occur.

Proposition 2 Assume that � 2 (0; 1), � 2 (0; 1), � 2 (0; 1), and � 2 (0; 1), then

capital stock and consumption of a composite good is always less in steady state

1, with a positive natural resource stock, than in steady state 2, with a depleted

natural resource stock, kss1 < kss2 and xss1 < xss2, respectively. In addition if

� < '3�, capital stock share in a composite good sector is much less than natural

resource stock share in a harvest good sector, then an overall consumption level,

that is composed both of the composite and harvest goods, is greater in steady state

1, with a positive natural resource stock, than in steady state 2, with a depleted

natural resource stock, css1 > css2, where '3 =�

1��1+�

1��1+��

�. Otherwise css1 < css2.

20

Proof. The proof consists of three parts for each of the respective claims. Part

I: kss1 < kss2 if and only if�l�'2 �B(1� I

s('2)

1��)'1

<�l'1. By lemma 1 '1 > 0 and

'2 > 0. Also in steady state 1 natural resource stock is positive, which means

that s > I'1��2 , thus the inequality holds. Part II: xss1 < xss2 if and only

if��l � '2

��B � I �B

s('2)

1�����

A'�1� �

'1

�< �l

�A'�1� �

'1

�. Since household�s pref-

erences satisfy the Inada condition, limc!0du(c)dc

! 1, then c > 0, in other

words an overall consumption level will be greater than zero. In steady state

2 the natural resource stock is depleted so there is no harvest. That means

that the overall consumption consists only of a composite good x. This im-

plies that xss2 > 0, which in turn implies that A'�1

> �'1. Then it remains to

show that xss1 < xss2 if and only if '2 �B�1� I

s('2)

1���> 0, which was just

shown in the �rst part of the proof, thus the inequality holds. Part III: har-

vest is always greater in steady state 1, hss1 = '1��2

��B � I �B

s('2)

1���, than in

steady state 2, hss2 = 0. Then the overall consumption in steady state 1 is

css1 = xss1+hss1 =��l � '2

��B � I �B

s('2)

1�����

A'�1� �

'1

�+'1��2

��B � I �B

s('2)

1���

and it is css2 = xss2 + hss2 = �l�A'�1� �

'1

�in steady state 2 . css1 > css2 if and only

if��l � '2

��B � I �B

s('2)

1�����

A'�1� �

'1

�+ '1��2

��B � I �B

s('2)

1���> �l

�A'�1� �

'1

�.

Rearranging this inequality and substituting for the '1 and '2 constants yields

� <

1�� 1 + �

1�� 1 + ��

!�. (20)

� 2 (0; 1) implies that 1�� 1 > 0 and � 2 (0; 1), � 2 (0; 1) imply that � > ��. This

means that 1�� 1 + � > 1

�� 1 + �� > 0 and inequality (20) holds.

Next I address stability of the two steady states.

21

5 Eigenvalues

To �nd eigenvalues I start with the system of four equations (18) in consump-

tion, x, labor, l, capital stock, k, and natural resource stock, B. When l�t > 0 equa-

tion (18b) holds with equality. Solving this equation forBt =�1��1��A

� 1�

�ktlt

��� ��l � lt�,

forwarding it by a period, and substituting it for the resource stock, Bt and Bt+1,

into equations (18a and 18d) reduces the system to three equations in one state

variable k and two choice variables x and l. This system in three equations is

��A�lt+1kt+1

�1��+ 1� �

��xt +

��l � lt

�1�� �1��1��A

� �ktlt

�� ��l � lt

���xt+1 +

��l � lt+1

�1�� �1��1��A

� �kt+1lt+1

�� ��l � lt+1

�� (21a)

=1

�,

xt + kt+1 � (1� �) kt = Al1��t k�t , (21b)

�kt+1lt+1

��� ��l � lt+1

�= (21c)

��l � lt

�0BB@�ktlt

���+ s

�ktlt

���

1� (

1��1�� A)

1��ktlt

��� (�l�lt)

�B

!�I�1��1��A

�1� 1�

�ktlt

��1CCA .

The state variable is also known as a predetermined variable, which is a func-

tion only of variables know up to time t. For example, in equation (21b) kt+1

is a function of variables at time t. The choice variables are also known as non-

predetermined variables, which can be a function of any variable up to time t+ 1.

For example, in equation (21a) xt+1 is a function of variables at times t and t+ 1

22

and so is lt+1 in equation (21c).

I linearize the above system in three equations (21) around its steady states by

taking the Taylor series approximation. The resulting linearized system is

266664~xt+1

~lt+1

~kt+1

377775 = A266664~xt

~lt

~kt

377775 ,

where ~xt = xt � xss, ~lt = lt � lss, ~kt = kt � kss, and similarly ~xt+1 = xt+1 � xss,~lt+1 = lt+1 � lss, ~kt+1 = kt+1 � kss, A is a (3� 3) matrix such that A = M�1N ,

the elements of these matrices are listed below3. A is transformed into Jordan3Elements of the M matrix are:m11 = 1,

m12 = �h(1� � (1� �)) (1� �) csslss + �

hsslss+ 1��

1��A�ksslss

��i,

m13 = (1� � (1� �)) (1� �) csskss + �hsskss;

m21 = 0,

m22 = �'���

(�l�lss)lss

+ 1

�,

m23 = '��

(�l�lss)kss

;m31 = 0,m32 = 0,m33 = 1;

where hss =��l � lss

�1��B�ss, css = xss + hss, and ' =

�1��1��A

� 1��ksslss

���

.

Elements of the N matrix are:n11 = 1,

n12 = �h�hsslss +

1��1��A

�ksslss

��i,

n13 = �hsskss;

n21 = 0,

n22 = �

2664 (1 + s)'

���

(�l�lss)lss

+ 1

�� s

�'2

�2��

(�l�lss)2

lss+ 2

��l � lss

���I 1��1��A

�ksslss

����(�l�lss)lss

+ 1

�3775,

n23 = (1 + s)'��

(�l�lss)kss

� s�'

2 2��

(�l�lss)2

kss� I�hsskss

;n31 = �1,n32 = wss,n33 = rss + 1� �;

23

canonical form:

A = V �1�V ,

where V is a (3� 3) matrix whose rows are eigenvectors of A. � is a diagonal

matrix whose diagonal elements are the characteristic roots of A. � is further

decomposed as

� =

2664 �1(m1�m1)

0

0 �2(m2�m2)

3775 ,where all eigenvalues of �1 are on or inside the unit circle ("stable" roots) and all

eigenvalues of �2 are outside the unit circle ("unstable" roots), and m1 +m2 = 3.

V and V �1 are decomposed accordingly.

For saddlepoint stability, if the number of eigenvalues of A on or inside the

unit circle is equal to the number of predetermined variables, that is if m1 = 1,

then there exists a unique solution (Blanchard and Khan [6]). If the number of

eigenvalues ofA on or inside the unit circle is less than the number of predetermined

variables, that is if m1 < 1, then there is no solution (Blanchard and Khan [6]).

If the number of eigenvalues of A on or inside the unit circle is greater than the

number of predetermined variables, that is if m1 > 1, then a unique solution is

nevertheless guaranteed through suitable linear restrictions on either the initial

conditions or the initial date and a �nite terminal date conditions (Buiter [8]).

Since here an in�nite horizon model is considered only the case of suitable linear

restrictions on the initial conditions is applicable. These linear restrictions on the

initial conditions are of the k0 = �k0 and F1l0+F2k0+F3x0 = f general form, where

�k0 is given and F1, F2, F3, and f are constants.

where wss = (1� �)A�ksslss

��, and rss = �A

�lsskss

�1��.

24

In the following section I calibrate parameter values to solve for the entire

transition paths for this economy using equations (18).

6 Numerical experiments

In this section I parameterize the above economy and use the system of equa-

tions (18) that characterizes an equilibrium de�ned in section 3.1 to solve for the

transition paths and steady states and follow up with sensitivity analysis. For

a numerical experiment consider parameters that replicate a developing economy

that is richly endowed with renewable natural resources. Currently there is a con-

siderable lack of data for countries that make potentially interesting case studies

for such application.

I would also like to point out that there are two aspects of the model that make a

calibration procedure non-standard. The �rst aspect is the biological resource stock

and its intrinsic growth rate. The biological resource stock includes populations of

animal species. If we consider a relatively small area, for example, a given national

park, then it is possible to �nd the data on the intrinsic growth rates and carrying

capacity from biological and ecological publications. However, the model in this

paper considers an economy of an entire country that can have numerous national

parks and nature conservancies each with distinct traits in its biological resource

stocks, carrying capacities, and intrinsic growth rates that are not all available for

a measurement. The second aspect is home production, which is not measured

in the national accounts but is essential for a low-income economy. This lack of

data and of measurement in the two aspects implies that it will not be possible to

compute all the parameter values solely based on data. Assumptions will have to

25

be made on some parameter values and they will be based on the existing literature,

and only where allowed by the data availability parameters will be calibrated to a

speci�c country. Because of these assumptions I will conduct sensitivity analysis

to examine how the results respond to a change the parameter values. Gollin,

Parente, and Rogerson [18] in their "Farm work, home work and international

productivity di¤erences" paper provide a similar example where the presence of

home production among other aspects of an economy preclude the authors from

basing the parameter values strictly on data and make parameterization of their

paper "somewhat exploratory in nature".

The data used in this section are from the country of Tanzania. Tanzania is

among the poorest countries in the world. In 2001 it had an estimated 36 percent

of population living below the basic needs poverty line and 18.7 percent below the

food poverty line (World Bank [48]). In the same year gross domestic product per

capita was $272, which is much lower than Sub-Saharan Africa�s average income

of $514 and low income countries�average income of $401, all measured in con-

stant 2002 USD (World Bank [49]). Like most low-income countries Tanzania is

primarily an agrarian country where agriculture is predominantly smallholder and

subsistence in nature (World Bank [50]). In recent years agriculture value added

has been about 45% of GDP and the sector employs and provides livelihood to

about 90% of the total labor force, as well as provides more than a half of the

country�s exports (World Bank [50]).

Tanzania is rich in biodiversity and has set aside a quarter of its land area

as nationally protected wildlife sanctuaries, while the conventional target is 10

percent of land area (World Bank [49]). For example, the Serengeti-Mara ecosys-

tem on the border of Tanzania and Kenya was one of the �rst areas nominated

26

as a World Heritage Site, and together with the Ngorongoro Conservation Area,

forms one of the world�s largest Biosphere Reserves and is one of the great natural

wonders of the world (Sinclair and Arcese [40]). Among Tanzania�s other natural

attractions are Mount Kilimanjaro, the Selous Reserve, Lake Victoria, and the Is-

lands of Zanzibar. These wildlife resources and spectacular landscape and scenery

attract tourists to the country and tourism sector in Tanzania is considered to

have a great economic growth potential. International tourism constitutes 47% of

Tanzania�s total exports (World Bank [49]). The tourism sector�s annual growth

rate has averaged 22 percent in the past few years, and the country aims to raise

the tourism sector�s contribution to GDP to more than 25 percent by year 2010

(World Bank [50]). However, with 18.7 percent of population living below the food

poverty line it is not surprising that the wildlife in the protected areas and around

them are poached and hunted by local communities. Current and most pressing

environment issues in Tanzania are soil degradation, deforestation, deserti�cation,

wildlife threatened by illegal hunting and trade, destruction of coral reefs, threat-

ened marine habitats, and marginal agriculture a¤ected by recent droughts (CIA

[10]). In addition to these resources, Tanzania also has a large hydropower poten-

tial, and a range of mineral deposits such as gold, diamonds, tin, iron ore, uranium,

phosphates, coal, gemstones, nickel, and natural gas deposits (World Bank [50]).

These factors make the country of Tanzania an engaging case study.

The numerical experiments section is arranged as follows. In the following sub-

section I compute the parameters for the households, �rms, and natural resource.

Using these parameters I compute the steady states and corresponding dynamics

for the entire transitional path. Then I consider a wide range of combinations of

initial conditions and follow with the sensitivity analysis.

27

6.1 Parameters

In this section I parameterize the model. Where the data are available the

parameters are calibrated and other parameters are based on the existing literature.

The data used in this section are summarized in the data appendix.

The value of the discount factor, �, is set so that the real interest rate implied by

the model asymptotically equals the average annual real interest rate of 9 percent

in Tanzania over the period from 1993 to 2005. I use the real interest rate data

from the World Bank World Development Indicators [49]. Recall equation (19a)

from the model where the capital rental rate is equal to the marginal product of

capital of the Cobb-Douglas production function. This equation can be restated

as r + 1 � � = 1�. The real interest rate is equal to r � �. The discount factor is

equal to � = 1r��+1 =

10:09+1

= 0:917.

The annual depreciation rate, �, is set to be equal to 5 percent, which is based

on the existing literature (e.g. Bergoeing, R., P. Kehoe, T. Kehoe, and Soto [5] use

� = 0:05 for Mexico and for Chile; e.g. � = 0:048 using the U.S. National Income

and Product Accounts in Cooley and Prescott [13]).

Labor share, 1 � �, is set to match the share of compensation of employees

in GDP adjusted to employment and workforce participation. Initially I compute

labor share of national income based on the share of employee compensation in

GDP. Then the returns to capital are equal to the residual. I use the data from

the U.N. National Accounts Statistics: Main Aggregates and Detailed Tables [45],

[46], and from the National Bureau of Statistics of Tanzania [30] for a range of

years from 1983 to 2003. I begin by computing total product, Yt, as GDP less net

indirect taxes (indirect taxes less subsidies). This computation assumes that net

28

indirect taxes are borne proportionally by labor and capital shares of total output.

Wages times total labor, wtLt, is equal to compensation of employees. Then labor

share,1��, is equal to wtLtYt. Using this method labor share is equal to 0:103, which

is unrealistically low.

In most countries labor share stays relatively constant over time. This stability

over long periods of time of capital and labor shares of national income within

a country appears to hold for most low-income and high-income countries, and

is often referred to as one of the "stylized facts" of growth (Kaldor [22]). The

international data appear to show wide di¤erences in labor share across countries.

However, Gollin [17], in his paper "Getting Income Shares Right", demonstrates

that employee compensation is often a poor measure of labor shares, especially for

low-income countries, because of sectorial composition of output and the struc-

ture of employment. For example, employee compensation does not include labor

income of self-employed people who in some countries constitute a large part of

the labor force. In this case compensation of employees would greatly understate

labor income. For example, in Bangladesh, Ghana, and Nigeria 75 to 80 percent

of manufacturing workers were self-employed while in the United States these were

less than 2 percent (International Labor Organization 1993 in Gollin[17]). He also

argues that the labor income of the self-employed is treated incorrectly as capi-

tal income. Gollin suggests three alternative adjustments to computing the labor

share that address these problems.

The �rst two adjustments are based on the operating surplus of private unincor-

porated enterprises (OSPUE) because most of the income of the self-employed is

allocated to this category. Since in low-income countries almost no self-employed

people will be legally incorporated this means that all their enterprise income,

29

including labor and capital income, will be reported as OSPUE. The OSPUE

is reported as part of the operating surplus together with corporate and quasi-

corporate enterprises. The third adjustment imputes employee compensation for

self-employed based on the number of employees relative to the total workforce.

Gollin [17] reports computed adjustments, for most countries, with few exceptions,

all three adjustments result in the labor shares that are relatively close to each

other. Also each adjustment results in the labor shares that are approximately

constant across most countries and fall in the range of 0:65 to 0:80.

In Tanzania compensation of employees represents only a small proportion of

total labor compensation because most of economically active and employed labor

force does not earn wages. Table 2 shows national pro�le employment statistics

for Tanzania in 2002. Only 8 percent of economically active population is paid for

work. This underestimates compensation to employees and contributes to the low

labor share.

In this paper I use the third adjustment. The reason is the availability of

employment data and the lack of OSPUE data for Tanzania. Adjusted labor

share is equal to ((employee compensation�number of employees)�total work

force)�GDP. Using this method4 I compute the adjusted labor share for Tanzania

to be 0:68.

Labor share in home production, 1��, is set equal to 0:89 and is higher than in

the formal sector. This is based on labor share in home production of 0:89 derived

for the U.S. data in Gollin et al. [18] where it is "consistent with stock of household

durables and market work outside of agriculture". In sensitivity analysis I vary

4The adjusted labor share is computed using compensation of employees and GDP in constant1990 prices for 2002; number of employees with paid work and total economically active workforcein 2002.

30

Table 2: Employment and Workforce, Tanzania, 2002Age Economically active Economically Do

Employed Unem- inactive notPaid Unpaid Work for ployed knowwork work own bene�t

Total 1,277,716 879,363 13,350,473 443,931 11,694,334 186,5375-9 5,712 165,624 231,614 24,746 4,405,095 135,70310-14 13,846 135,445 424,547 32,261 3,738,843 20,24215-19 101,310 133,028 1,462,360 112,106 1,556,369 9,33720-24 171,916 105,062 2,072,404 117,930 505,546 5,55325-29 218,674 82,053 2,019,459 65,660 296,690 3,60230-34 194,428 63,121 1,662,752 34,469 181,433 2,53035-39 158,006 46,396 1,271,144 20,079 119,268 1,55540-44 148,892 36,728 1,016,058 12,353 85,575 1,47545-49 113,378 26,370 744,035 7,887 62,353 92150-55 79,433 25,680 705,252 6,203 73,585 1,00155-59 37,234 16,678 478,309 3,781 56,500 71060-64 17,445 15,404 468,163 2,725 96,181 90065-69 9,110 11,332 326,307 1,734 88,766 57170-74 4,637 8,291 237,995 1,167 130,461 85475-79 2,137 4,461 126,006 381 92,642 53280+ 1,562 3,692 104,068 448 205,023 1,049

Source: National Bureau of Statistics of Tanzania [30]

the value of this parameter and analyze the e¤ects of this change. For example,

Parente, Rogerson, and Wright [31] vary the labor share in home production from

0:99, 0:95, 0:9, 0:8, to 0:67 concluding that when using home durables as a measure

of home capital the value of 0:95 is reasonable in the context of development. The

value of labor share in this research is di¤erent mainly because here the home

production uses natural resource stock as factor input instead of capital stock that

is used in the above examples.

The estimate of intrinsic growth rate is based on biological studies of the ungu-

late populations in the Serengeti-Mara ecosystem5. Sinclair, Dublin, and Borner

5The Serengeti-Mara ecosystem in Tanzania hosts the world�s largest ungulate herds (Sinclair

31

[42] �nd that for Serengeti wildebeest6 there exists an empirical relationship be-

tween dry season mortality rate, density, and food supply. Pascual and Hilborn [32]

use these results to further construct a wildebeest population dynamics. Account-

ing for the dry and wet seasons for the rainfall and grass availability they specify

a linear growth function and predict per capita recruitment rate for wildebeest to

be between 0:1 and 0:4, mostly centered around 0:2. Johannesen and Skonhoft

[20] use a logistic natural growth function to model the wildebeest population in

the Serengeti-Mara ecosystem. They set the intrinsic growth rate equal to 0:3.

Following the estimates of these studies the intrinsic growth rate of the renewable

natural resource, s, is initially set equal to 0:3 and then varied from 0:2 to 0:5.

The nonlinear di¤erence equation of the logistic model of the renewable re-

source is capable of producing complex behavior. More speci�cally, when the

intrinsic growth rate s 2 (0; 1] the resource population steadily approaches the

environmental carrying capacity without overshooting it (Conrad and Clark [12]).

If s 2 (1; 2] the resource overshoots but then gradually approaches its carrying ca-

pacity with oscillations decreasing over time. If s 2 (2; 2:449] the resource settles

at a two-point cycle. If s 2 (2:449; 2:570] the resource settles at a stable cycle, and

if s > 2:570 then the resource varies in a completely irregular, non-periodic way,

this behavior is referred to in mathematics as dynamic "chaos" (Conrad and Clark

[12]).

Human and natural resource populations are varied to analyze how their rel-

ative sizes a¤ect the results of the model. It is interesting to analyze these two

and Norton-Gri¢ ths [39]).6The structure and the function of the Serengeti-Mara system is de�ned by the wildebeest�s

migratory range and a collapse of their populations will result in the collapse of the entire system(Sinclair and Arcese [40]).

32

populations relative to each other because, for example, if the human population is

large relative to biomass population then there is a lot of pressure on the natural

resource and it might need more protection. While on the contrary if the bio-

mass population is large relative to human population the natural resource might

need less protection and the resources previously allocated to the natural resource

protection can be reallocated and used elsewhere in the economy.

Table 3 summarizes the parameters and initial conditions of the model.

Table 3: Parameters and initial stock conditionsParametersPreferences and technologies� 0.917 discount factor� 0.05 capital depreciation rateI 1 normalized number of households�l 1 normalized household�s labor endowment� 0.11 biological resource stock production share� 0.32 capital stock production shareA 1 total factor productivityBiological resources 0.4 intrinsic growth rate�B 1 normalized environmental carrying capacityInitial conditionsk0 80% and 20% of kss initial capital stock (two levels)B0 90% and 20% of �B initial biological resource stock (two levels)

Given these parameters I evaluate the steady states, discuss their stability, trace

the entire transitional paths to the steady states, and evaluate initial conditions.

I then follow with sensitivity analysis.

6.2 Steady states and stability

Given the parameter values both steady states are computed. The steady

state values are summarized in table 4.

33

Table 4: Steady States when s=0.4Variables Description Steady state 1 Steady state 2

(Interior) (Corner)xss composite good 1:2879 1:3053lss labor 0:9867 1kss capital stock 3:3098 3:3546Bss resource stock 0:0391 0yss output 1:4533 1:4730hss harvest good 0:0150 0

yss + hss output+harvest 1:4683 1:4730css overall consumption 1:3029 1:3053

First, observe that when the intrinsic growth rate of the natural resource is

low, s = 0:4, and the labor share in home production is high, 1 � � = 0:89, the

two steady states are close to each other, with all of the values, except for the

natural resource stock and harvest, greater in the corner steady state than in the

interior steady state. One of the reasons for that is low intrinsic growth rate of the

renewable natural resource. In fact, when the intrinsic growth rate is even lower

than 0.4, for example, equal to 0.3, 0.2, or 0.1 the interior steady state does not

exist, the resource stock is depleted, and there is a unique corner steady state.

Table 5 compares steady state values for di¤erent intrinsic growth rates.

Table 5: Steady States for di¤erent values of sVariables Steady state 1 (Interior) Steady state 2 (Corner)

s = 0:5 s = 0:65 s = 0:85 s = 0:5; 0:65; 0:85xss 1:2022 1:1231 1:0611 1:3053lss 0:921 0:8604 0:8129 1kss 3:0896 2:8864 2:7270 3:3546Bss 0:2313 0:4087 0:5478 0yss 1:3567 1:2674 1:1974 1:4730hss 0:0889 0:1571 0:2106 0

yss + hss 1:4456 1:4245 1:408 1:4730css 1:2911 1:2802 1:2716 1:3053

34

An increase in the intrinsic growth rate of the natural resource results in a

substantial di¤erence in the level of the preserved resource stock. When the growth

is low at 0.3 the stock is 0, at 0.4 the stock is 4% of its carrying capacity, at 0.5

the stock is at 23% of its carrying capacity, at 0.65 it is 41%, and at 0.85 it is

55%. The higher the resource intrinsic growth rate is the higher is the intact

natural resource stock in the interior steady state. In steady state two the natural

resource stock is always depleted. As the natural resource growth at a higher

intrinsic rate it diverts more labor resources from the formal sector to the home-

production sector resulting in an increased consumption of a home-produced good,

decreased consumption of a composite good, and a decreased capital stock.

Another aspect of interest is the comparison of labor share in home-production

sector relative to labor share in the formal sector. Table 6 compares the steady

state values across di¤erent labor shares.

Table 6: Steady States for di¤erent values of theta when s=0.6Variables Steady state 1 (Interior) Steady state 2 (Corner)

� = 0:05 � = 0:25 � = 0:45 � = 0:65 � = 0:01; 0:05; 0:45; 0:65xss 1:1285 1:1821 1:2367 1:2913 1:3053lss 0:8646 0:9056 0:9475 0:9893 1kss 2:9003 3:0379 3:1784 3:3188 3:3546Bss 0:3904 0:3003 0:1990 0:0538 0yss 1:2735 1:3339 1:3957 1:4573 1:4730hss 0:1428 0:1261 0:0956 0:0306 0

yss + hss 1:4163 1:4600 1:4913 1:4879 1:4730css 1:2713 1:3081 1:3324 1:3219 1:3053

The less labor dependent the resource sector is the better o¤ the households in

an interior steady state become. Even when labor share in the resource sector is

as high as 0.75, the households have higher overall consumption than in a corner

steady state. The interior economy also has two assets the capital stock and the

35

natural resource stock at 30% of its carrying capacity. The overall income or out-

put is interesting to compare because home production does not enter the national

accounts while the households still bene�t from it. This unmeasured home produc-

tion overstates the true di¤erences in output across countries. More speci�cally,

the model implies that true consumption is greater than reported. For example,

if we compare output, yss, then it is lower in an interior economy, however if we

take into account home production and compare overall consumption, css, then it

is higher in the interior economy as � increases. "This unmeasured consumption

may explain how individuals in some countries can survive on the very low levels of

reported income. " (Parente, Rogerson, andWright [31]). The overall consumption

is higher in the interior economy even though the formal and sometimes informal

total output is lower than in the corner economy. This happens because the inte-

rior economy has two assets, capital and natural resource, and one of them does

not depreciate. Similar to Parente, Rogerson, and Wright [31] my model allows

to quantify the amount of non-market activity to measure di¤erences in output,

consumption, and welfare.

In this paper it is assumed that the only value that the households derive

from a natural resources is consumption of harvest. However, natural resources

often provide other values, such as scenic and recreational attractions. If these

additional, non-extractive values were added to this economy, we can expect the

households with a positive natural resource stock to be even better o¤than without

it. In my next paper I investigate this question by adding non-extractive services

of the natural resources to the economy.

Figure (1) illustrates application of proposition 2 for the parameter values given

in table 4. The dashed line is a 45 degree line of �, natural resource share in harvest,

36

and the solid curve is�

1��1+�

1��1+��

��. If �, capital share in production, is anywhere

in the south-east region of the solid curve then css1 > css2. If � is anywhere

in the north-west region of the solid curve then css1 < css2. Here � = 0:11 and

� = 0:32, this combination is in the north-west region of the solid curve and overall

consumption is greater in corner steady state 2 than in interior steady state 1. If

we increase � to, for example, 0:45 then this combination will lie in the south-east

region and overall consumption in interior steady state 1 would be greater than in

corner steady state 2.

Another observation is that when we start with a one-to-one relationship of

human population relative to natural resource population an interior steady state

exists and the natural resource stock is preserved. However, if we increase human

population to, for example, two-to-one relative to the natural resource population,

then the natural resource will be depleted for all parameter values and interior

steady state does not exist. In application, if we think of a country like China

with high human population, then unless the natural biomass is protected it will

be depleted from by human pressure.

Moving in the opposite direction if we increase biomass population relative

to human population we observe that even more labor is allocated to the resource

sector drawing away 28% of factor input from the formal sector (table 7 summarizes

these results). The formal sector shrinks. Consumption of the composite good

decreases and the capital stock decreases to 28% lower than in the corner steady

state. The interior economy relies more heavily on the natural resource and harvest

for its income and consumption.

In terms of stability, eigenvalues for steady state 1 for the initial set of parameter

37

Figure 1: Capital stock share, natural resource stock share, and steady state con-sumption.

38

Table 7: Steady States when B=3, s=0.6, and theta=0.25Variables Description Steady state 1 Steady state 2

(Interior) (Corner)xss composite good 0:9356 1:3053lss labor 0:7168 1kss capital stock 2:4045 3:3546Bss resource stock 0:9010 0yss output 1:0558 1:4730hss harvest good 0:3782 0

yss + hss output+harvest 1:4340 1:4730css overall consumption 1:3138 1:3053

values summarized in table 4 are

�ss1 =

2666640:876 0 0

0 0:984 0

0 0 1:261

377775 .

Eigenvalues for steady state 2 are

�ss2 =

2666640:867 0 0

0 1:016 0

0 0 1:257

377775 .

Steady state 1 has two stable roots. That is the case, as is explained in eigenvalues

section 3.3.1, when the number of stable roots exceeds the number of predeter-

mined variables and provided a linear restriction on the initial conditions a unique

solution exists. Steady state 2 has one stable root. In this case, as is explained in

eigenvalues section 3.3.1, the number of stable roots equals the number of prede-

termined variables so a unique solution exists. Both steady states are saddlepath

stable. For example, when the economy starts with capital stock below its steady

39

state value, consumption and capital stock grow at positive and decreasing rates

that asymptotically approach zero. In the next subsection I address this further

by solving for and analyzing the transition paths for all variables.

6.2.1 Transition dynamics

In this section I analyze the dynamics of the model along its transition paths.

To solve for a transition path I use the in�nite-horizon forward shooting algorithm

(Judd [21]) and adapt it to this model. The algorithm is outlined in Appendix A.

The programming of the algorithm is done in Matlab.

The initial conditions on capital and natural resource stocks determine which

steady state an economy converges to. In the following I �rst consider a case

with a combination of high initial capital and high initial natural resource stocks,

then a case where both initial stocks are low. After that I summarize all possible

combinations of initial stocks of both assets.

In the �rst case the initial capital stock is 80% of its steady state value and

the initial natural resource stock is 90% of its carrying capacity. When both

initial stocks are relatively high the economy converges to steady state 1 with an

intact natural resource stock. It takes the economy about 120 periods to reach

its steady state. The natural resource stock decreases until it reaches its steady

state value (�gure 2). An overall consumption that includes both consumption

of the composite and harvest goods is smoothed out through time. Households

can smooth their consumption the in the initial periods by consuming more of the

harvested good (�gure 2). This is contrasted with the transition path to steady

state 2 when the economy starts with low capital stock of 20% of its steady state

value and low natural resource stock of 20% of its carrying capacity. In this case

40

Figure 2: Transition path to steady state 1 when s = 0:4 and � = 0:11.

the natural resource stock is quickly depleted (�gure 5). Over the initial forty

years there is a rapid growth of consumption as households cannot smooth it out

without the natural resource stock (�gure 5).

The transition path changes if we increase the intrinsic growth rate of the nat-

ural resource and decrease labor share in harvest. Figure 3 shows these dynamics.

The natural resource stock drops initially when the households rely more heavily

on harvest for consumption. After some twenty years though the natural resource

stock picks up and then eventually levels o¤ to its steady state value that is sig-

ni�cantly higher than in the case with a low intrinsic growth rate. Consequently

there is a much larger di¤erence between consumption of a composite good and

an overall consumption, where the di¤erence is the harvest good. Consumption

of a composite good gradually increases over time and eventually levels o¤ after

reaching its steady state value in 150 years (�gure 3). Overall consumption follows

41

Figure 3: Transition path to steady state 1 when s = 0:8 and � = 0:25.

a similar pattern to a composite good consumption but stays at all times above it.

Figure 4a shows initial growth rates along the transition path in overall con-

sumption (cg1) and consumption of a composite good (xg1). Notice that due to

the contribution of the harvest good an overall consumption it is more smooth

over time. In other words, initially before the economy has had time to accumu-

late capital stock it relies more heavily on the harvest good by compensating it

in the overall level of consumption and then over time gradually substituting for

it with a composite good. The presence of the natural resource stock enables the

households to better smooth an overall consumption and keep it at a high level at

all times. This is again observed by comparing the growth rate of capital stock,

kg1, output in a composite good sector, yg1, and income that includes both sectors

((y+h)g1) in �gure 4c. Figure 4b relates growth in natural resource stock (bg1)

42

Figure 4: Transition path to steady state 1 when s = 0:4 and � = 0:11: (a)consumption growth, (b) natural resource and harvest growth, and (c) capital andincome growth.

43

Figure 5: Transition path to steady state 2 when s = 0:4 and � = 0:11.

to growth in harvest (hg1). Both move in the same direction. Initially, when the

capital stock is low, the economy harvest level is high, so the change in resource

stock is high and negative.

Figure 6 sows the growth rates along the transition path for an economy that

starts with low stocks and converges to a corner steady state by depleting the

natural resource stock. The natural resource stock is quickly depleted (b2 in �g-

ure 6b), in fact it is depleted in the �rst period and after that the economy has

only capital stock. After period one an overall consumption coincides with the

consumption from the composite good sector (x2 in �gure 6a) as harvest is zero.

Figure 6a illustrates that growth in a composite good consumption is very large in

the initial years because initially the households consume a lot of the harvest good

until the natural resource is depleted. The initial presence of the natural resource

however, allows this economy to invest more into the capital stock in the initial

44

years and thus allow for higher capital stock from then on.

Transition paths starting from the two di¤erent initial conditions of natural

resource stock and capital stock that converge to two di¤erent steady states are

compared in �gures 7, 8, and 9. The natural resource stock is at all times greater

in the �st case (b1 in �gure 7) than in the second case (b2 in �gure 7). Overall

consumption is more smooth in the economy with the intact natural resource than

in the economy that depletes the natural resource. Initially consumption is much

higher when the resource is present. Over time consumption in both economies

becomes very similar. When labor share in harvest is much higher than in the

formal sector overall consumption is somewhat higher in the corner economy with

a depleted resource, while with a labor share in harvest that is only a little higher

than in the formal sector overall consumption is higher in the interior economy with

an intact natural resource. The di¤erence in overall consumption is especially large

in the initial years when both economies are starting to accumulate capital. Figure

8 compares the growth rates of the two economies.

Figure 9 compares transitional paths for wage and capital rental rates for the

two steady states. Initially wage is much lower on the path from the second set of

initial conditions (w2) than on the path from the �rst set of initial conditions (w1)

so the opportunity cost of lost wages on the second path is low and more labor is

allocated to harvest depleting the natural resource. Over time the wages become

closer to each other and then equal in the steady states. Capital rental rate in the

second case (r2) is initially higher than in the �rst case (r1) because of the relative

capital scarcity from the low initial level of capital in the second case.

45

Figure 6: Transition path to steady state 2 when s = 0:4 and � = 0:11: (a)consumption growth, (b) natural resource growth, and (c) capital and incomegrowth.

46

Figure 7: Comparison of a transition path to steady state 1 and a transition pathto steady state 2 when s = 0:4 and � = 0:11.

47

Figure 8: Comparison of a transition path to steady state 1 and a transition pathto steady state 2 for wage and capital rental rates.

48

Figure 9: Comparison of a transition path to steady state 1 and transition pathto steady state 2 when s = 0:4 and � = 0:11: (a) consumption growth, (b) naturalresource and harvest growth, and (c) capital and income growth.

49

6.2.2 Initial conditions

After considering the two cases of high versus low initial allocations of both

stocks I now turn to evaluating other possible combinations of the initial conditions.

When initially an economy has two low stocks then it depletes the natural resource

stock and converges to steady state 2. This is very intuitive, since both stocks are

too low to achieve a higher consumption level in the earlier years the households

rely more heavily on the natural resource and as its stock is low to begin with it is

quickly depleted. When an economy starts with a relatively higher capital stock,

even if it has a low resource stock it still manages to converge to steady state 1 with

an intact natural resource stock, a higher overall consumption level, but somewhat

lower capital stock. When an economy starts with a very low capital stock but a

relatively high natural resource stock it harvests too much, its consumption level

is too high, and it can miss the stable manifold that converges to a steady state.

If population increases, in other words if I, the number of households goes up,

then more initial combinations of assets lead to a natural resource depletion. If I

is greater than two-to-one relative to �B, then all initial combinations of the assets

lead to a steady state with natural resource depletion.

If capital share in a composite good sector increases, for example to � = 0:45,

then more labor is allocated to the composite good sector so the natural resource

stock is always preserved for all combinations of the initial conditions.

7 Conclusions and implications

This research analyzes the interaction between economic growth and an open

access renewable natural resource by combining a convex growth model with open

50

access renewable natural resource modeling. To analyze a small developing coun-

try it expands the neoclassical growth model in three important ways. First, it

introduces a renewable natural resource endowment and dynamics as a new fac-

tor input. Second, it introduces market imperfections associated with the use of

this resource. Third, it introduces a home-production sector, which contributes to

labor allocation decisions, output, consumption, and ultimately welfare.

Individuals allocate labor between wage employment in a formal production

sector and harvest of the natural resource. This framework allows to analyze the

dynamics of the households�consumption and allocation of labor between harvest

of the resource and labor wages in a general equilibrium framework where equi-

librium prices and wages are endogenous. The model also allows to quantify the

amount of non-market activity and measure di¤erences in output, consumption,

and welfare taking into account this informal home-production sector. This re-

search also develops a theoretical framework necessary to analyze a low-income

economy because in such economy harvest of a natural resource is often an impor-

tant part of the households�consumption and income.

The paper de�nes and characterizes an equilibrium for this model. There exist

two distinct steady states. First steady state is interior. The natural resource

stock is strictly positive and sustained over time, labor is allocated between home-

production and formal sectors. Second steady state is corner, in the sense that

the natural resource is fully depleted, harvest is equal to zero, and all of labor

is allocated to the formal sector. The analysis of the steady states, transition

paths, and initial conditions show that if an economy has high wage employment

it can converge to a steady state without depleting its resource stock even with

no resource regulation. However, if an economy has low wage employment, then

51

it can deplete its resource stock.

The results of the analysis show that when the intrinsic growth rate of the

renewable natural resource is low or human population is high relative to biomass

population then the economy would deplete the natural resource stock for all ini-

tial levels of the two assets. As the intrinsic growth rate of the natural resource

increases an interior steady state emerges where the natural resource stock is in-

tact in steady state despite its open access nature. When an economy has higher

initial capital and natural resource stocks it converges to an interior steady state,

and only when it starts with very low natural resource and capital stocks would it

deplete the resource stock. This suggest a policy implication of adding more initial

capital to an economy, which would allow it to preserve its natural resources.

When the biomass population is high relative to human population the economy

relies heavily on the natural resource and home production for its income and

consumption. This results in redistribution of labor away from the formal sector,

lower capital stock, and a smaller economy.

The welfare is a¤ected by the labor share in harvest relative to labor share

in the formal sector. When labor share in the formal sector is small relative to

the harvest sector consumption in an economy with depleted natural resource is

greater than in an economy with positive natural resource stock. The opposite

holds, when labor share in the formal sector is large relative to the harvest sector

consumption in an economy with positive natural resource stock is greater than in

an economy with depleted natural resource. In addition, households in an economy

that converges to a positive natural resource stock can better smooth their overall

consumption over time. It is important to note that these di¤erences in welfare

between the interior and corner steady states are relatively small. This means that

52

in the cases of low initial stocks or low intrinsic growth rates of the resource an

economy would deplete the natural resource but would not be too heavily a¤ected

in its consumption. When natural resources are important only as harvest and

consumption and are not associated with any other values there is a very small

incentive for an economy to preserve its natural biomass, which is the case for poor

economies that are primarily concerned with consumption levels.

The model of the economy in this paper assumes perfect labor and capital

markets and it assumes that there is no market for resource harvest. In many

developing countries there are imperfect labor and capital markets. In addition,

there often are markets, legal or illegal, for resource harvest (for example, an illegal

market for �bush meat�). Models in which di¤erent assumptions about how well

various markets function would be worth investigating.

53

A Appendix

Appendix A derives the characteristic equations (18) of an equilibrium de�ned

in part 3.1. but for sequential markets instead of an Arrow-Debreu market, and

shows that these characteristic equations are identical for both speci�cations.

In sequential markets, the household�s problem is state as follows:

maxfxt;lt;kt+1g1t=0

1Xt=0

�t ln�xt +

��l � lt

�1��B�t

�such that

pt (xt + kt+1 � (1� �) kt) � wtlt + rtkt

xt; kt+1 � 0

0 � lt � �l

k0 > 0 given.

The Lagrangian for this problem is:

L =1Xt=0

�t ln�xt +

��l � lt

�1��B�t

�+

1Xt=0

�t (wtlt + rtkt � pt (xt + kt+1 � (1� �) kt)) .

54

The Kuhn-Tucker conditions are:

@L@xt

=�t

x�t +��l � l�t

�1��B�t

� �tpt � 0 with equality if x�t > 0 (22a)

@L@lt

= ��t (1� �)

��l � l�t

���B�t

x�t +��l � l�t

�1��B�t

+ �twt � 0 with equality if l�t > 0 (22b)

@L@kt+1

= ��tpt + �t+1rt+1 + �t+1pt+1 (1� �) � 0 (22c)

with equality if k�t+1 > 0

@L@�

= �t (wtlt + rtkt � pt (xt + kt+1 � (1� �) kt)) � 0 (22d)

with equality if ��t > 0.

limt!1

�t

x�t +��l � l�t

�1��B�t

�rt+1pt+1

+ 1� ��kt ! 0. (22e)

Equation (22c) states the intertemporal price condition and holds with equality

for capital stock k�t+1 > 0:

rt+1pt+1

+ 1� � = �tpt�t+1pt+1

. (23)

Taking the ratio of equation (22a) at period t+ 1 over period t we get:

��xt +

��l � lt

�1��B�t

�xt+1 +

��l � lt+1

�1��B�t+1

=�t+1pt+1�tpt

, (24)

which also holds with equality for a composite good x�t , x�t+1 > 0. Substitute for

the price ratio from equations (23) into equation (24) to get:

�rt+1pt+1

+ 1� ���xt +

��l � lt

�1��B�t

�xt+1 +

��l � lt+1

�1��B�t+1

=1

�. (25)

55

Take the ratio of equations and (22a) and (22b) to get:

(1� �)

Bt��l � lt

�!� � wtptwith equality if l�t > 0. (26)

Equations (23) through (26) constitute a representative household�s competitive

equilibrium conditions.

The �rm�s pro�t maximization problem with the constant returns to scale tech-

nology establishes labor wage and capital rental rate as:

wtpt

= (1� �)A�Kt

Lt

��rtpt

= �A

�LtKt

�1��.

Given that all households are identical in their preferences, market clearing condi-

tions, and CRS technology, Kt = Ikt and Lt = Ilt, and the above wage and rental

rate can be restated as follows:

wtpt

= (1� �)A�ktlt

��(27)

rtpt

= �A

�ltkt

�1��. (28)

Substitute for wage and capital rental rate equations into the household condi-

tions. Combine the representative household�s competitive equilibrium conditions,

�rm�s labor wage and capital rental rate conditions, and market clearing conditions

56

to derive the �ve equations that characterize an equilibrium.

��A�lt+1kt+1

�1��+ 1� �

��xt +

��l � lt

�1��B�t

�xt+1 +

��l � lt+1

�1��B�t+1

=1

�(29a)

(1� �)

Bt��l � lt

�!� � (1� �)A�ktlt

��with equality if l�t > 0 (29b)

xt + kt+1 � (1� �) kt = Al1��t k�t (29c)

Bt+1 �Bt = sBt�1� Bt�B

�� I

��l � lt

�1��B�t (29d)

I��l � lt

�1��B�t � Bt + sBt

�1� Bt�B

�(29e)

Equation (29a) is the Euler Equation. Equation (29b) requires equilibrium mar-

ginal product of labor be equal in both composite and harvest good sectors when

both sectors are active. The inequality in this condition comes from an endogenous

labor allocation decision and an endogenous possibility for one of the sectors be-

coming over time inactive. Equation (29c) is the composite good sector feasibility

condition stating that the consumption and investment is equal to production of

the composite good. Equation (29d) is the biological resource di¤erence equation.

Equation (29e) is the biological resource feasibility condition stating that the total

harvest cannot exceed the resource stock. Equations (29) characterize an equilib-

rium for sequential markets. These equations are identical to equations (18). Thus

the solutions are identical as well.

57

B Appendix

This section adjusts an in�nite-horizon forward shooting algorithm (Judd [21])

to compute a stable manifold for the model in this paper. The algorithm is also

adjusted to accommodate in�nite horizon.

The objective is to solve equations (18) for xt, lt, kt, and Bt paths over t 2

f0; 1; 2; :::; Tg given k0 < kss and B0 2�0; �B

�.

Step 1. Initialize consumption in the �rst period by setting x0 = (xL + xH) =2,

where xH = f (l0; k0) = Al1��0 k�0 and xL = 0. Choose a stopping criterion " > 0,

here " is set equal to 10�15.

Step 2. Solve iteratively equations (18) starting with the initial conditions x0,

k0, and B0. Stop the initial value algorithm at the �rst t when xt+1 � xt < 0 or

kt+1 � kt < 0, and denote it by T .

Step 3. If jxT � xssj < ", then stop. If xt+1 � xt < 0, then initial x0 was too

small and a consumption level was declining, in this case set xL = x0, otherwise

set xH = x0 and reiterate by going to step 2. If kt+1 � kt < 0, then initial x0 was

too large and a capital stock was declining, in this case set xH = x0 and reiterate

by going to step 2.

58

C Data Appendix

Series Source

Real interest rate (%), Tanzania WDI

Indirect taxes, net, current prices,Tanzania (millions) U.N. NAS

Consumption of �xed capital, current prices, Tanzania (millions) U.N. NAS

Compensation of employees, current prices, Tanzania (millions) U.N. NAS

Operating surplus, current prices, Tanzania (millions) U.N. NAS

GDP, current prices, Tanzania (millions) U.N. NAS

Implicit Price De�ator, Tanzania (1990=100) U.N. NAS

Economically active persons, 5-80+, Tanzania NBST

Employed persons with paid work, 5-80+, Tanzania NBST

GDP (constant 2000 US$), Tanzania WDI

GDP de�ator (1992=100), Tanzania WDI

GDP per capita (constant 2000 US$), Tanzania WDI

GDP per capita, PPP (constant 2000 international $), Tanzania WDI

Population, total, Tanzania WDI

Population ages 15-64 (% of total), Tanzania WDI

Real GDP per worker, Tanzania PWT 6.2

Notes

WDI stands for World Development Indicators [49].

U.N. NAS stands for U.N. National Accounts Statistics: Main Aggregates and

Detailed Tables [45] and [46].

NBST stands for National Bureau of Statistics of Tanzania, National Pro�le

Statistical Tables [30].

59

PWT 6.2 stands for Penn World Tables [33].

60

References

[1] Albright, Madeline. (2007). The wold in 2007. The Economist. p. 65.

[2] Barbier, Edward B. (2005). Natural resources and economic development.

Cambridge University Press.

[3] Barrett, B. and P. Arcese. (1998). Wildlife harvest in integrated conservation

and development projects: Linking harvest to household demand, agricultural

production, and environmental shocks in the Serengeti. Land Economics. 74

(4): 449-65.

[4] Becker, Gary S. (1965). A Theory of the Allocation of Time. Economic Jour-

nal. 75 (299): 493-517.

[5] Bergoeing, Raphael, Patrick J. Kehoe, Timothy J. Kehoe, Raimundo Soto.

(2001) A decade lost and found: Mexico and Chile in the 1980s. Federal

Reserve Bank of Minneapolis.

[6] Blanchard, Oliver Jean and Charles M. Khan. (1980). The solution of linear

di¤erence models under rational expectations. Econometrica. 48 (5): 1305-

1312.

[7] Brock, W. and A. Xepapadeas. (2002). Optimal ecosystem management when

species compete for limiting resources. Journal of Environmental Economics

and Management. 44, 189-220.

[8] Buiter, Willem H. (1984). Saddlepoint problems in continuous time rational

expectations models: a general method and some macroeconomic examples.

Econometrica. 52 (3): 665-680.

61

[9] Cass, D. (1965). Optimum growth in an aggregative model of capital accumu-

lation. The Review of Economics Studies. 32 (3) Jul.: 233-2450.

[10] Central Intelligence Agency. (2004). The World Factbook.

http://www.cia.gov/cia/publications/factbook/geos/tz.html#top

[11] Cohen, Daniel, Pierre Jacquet and Helmut Reisen. (2007). Loans or grants.

Center for Economic Policy Research. Discussion Paper No. 6024 in Interna-

tional Macroeconomics.

[12] Conrad, J. and C. Clark. (1987). Natural Resource Economics. Cambridge

University Press.

[13] Cooley, Thomas F. and Edward C. Prescott. (1995). Economic growth and

business cycles. In Frontiers of Business Cycle Research, edited by Thomas

F. Cooley. 1-38. Princeton University Press.

[14] Dasgupta, Partha. (1993). An inquiry into well-being and destitution. Oxford

University Press.

[15] Dasgupta, P. and K. Maler. (1995). Poverty, institutions, and the environ-

mental resource-base. Handbook of Development Economics. 3: 2371-2462.

[16] Eliasson, L. and S. J. Turnovsky. (2004). Renewable resources in an endoge-

nously growing economy: balanced growth and transitional dynamics. Journal

of Environmental Economics and Management. 48: 1018-1049.

[17] Gollin, Douglas. (2002). Getting income shares right. Journal of Political

Economy. 110 (2): 458-474.

62

[18] Gollin, Douglas, Stephen L. Parente, and Richard Rogerson. (2004). Farm

work, home work and international productivity di¤erences. Review of Eco-

nomic Dynamics. 7:827-850.

[19] Gylfason, Thorvaldur and Gyl� Zoega. (2006). Natural resources and eco-

nomic growth: the role of investment. The World Economy. 8: 1091-1115.

[20] Johannesen, Anne Borge and Anders Skonhohft. (2004). Property rights and

natural resource conservation. A bioeconomic model with numerical illustra-

tions from the Serengeti-Mara ecosystem. Environmental and Resource Eco-

nomics. 28: 469-488.

[21] Judd, Kenneth L. (1998). Numerical methods in economics. MIT Press.

[22] Kaldor, Nicholas. (1961). Capital Accumulation and Economic Growth. In

The Theory of Capital, edited by Friedrich A. Lutz and Douglas C. Hague.

New York: St. Martin�s Press.

[23] Koopmans, T. (1965). On the concept of optimal economic growth. In The

Econometric Approach to Development Planning. North Holland.

[24] Loibooki. M, H. Hofer, K. Campbell, and M. East. (2002). Environmental

Conservation. 29 (3): 391-398.

[25] Lopez, R., G. Anriquez, and S. Gulati. (2007). Structural change and sustain-

able development. Journal of Environmental Economics and Management. 53:

307-322.

[26] Magill, M. and M. Quinzii. (1996). Theory of Incomplete Markets. The MIT

Press.

63

[27] McGrattan, Ellen R., Richard Rogerson, and Randall Wright. (1997). An

equilibrium model of the business cycle with household production and �scal

policy. International Economic Review. 38: 267-90.

[28] Mehlum, Halvor, Karl Moene, and Ragnar Torvik. (2006). Cursed by resources

or institution? The World Economy. 8: 1117-1131.

[29] Meza, David de and J. R. Gould. (1992). The social e¢ ciency of private deci-

sions to enforce property rights. Journal of Plotical Economy. 100 (3): 561-580.

[30] National Bureau of Statistics of Tanzania. (2007). Population

and Housing Census 2002. National Pro�le Statistical Tables

http://www.nbs.go.tz/NationalPro�le/NationalPro�leTables.htm.

[31] Parente, Stephen L., Richard Rogerson, and Randall Wright. (2000). Home-

work in Development Economics: Household Production and the Wealth of

Nations. Journal of Political Economy. 108 (4): 680-687.

[32] Pascual, M. A. and R. Hilborn. (1995). Conservation of Harvested Populations

in Fluctuating Environments: The Case of the Serengeti Wildebeest. The

Journal of Applied Ecology 32 (3) Aug.: 468-480.

[33] Penn World Tables PWT 6.2. (2006). By Alan Heston, Robert Summers, and

Bettina Aten, Penn World Table Version 6.2, Center for International Com-

parisons of Production, Income and Prices at the University of Pennsylvania,

September 2006.

[34] Ramsey, F. (1938). A mathematical theory of savings. Economic Journal. 387

(152) Dec.: 543-559.

64

[35] Rodriguez, F. and J. D. Sachs. (1999). Why do resource-abundant economics

grow more slowly? Journal of Economic Growth. 4:277-303.

[36] Rupert, Peter, Richard Rogerson, and Randall D. Wright. (1995). Estimating

substitution elasticities in household production models. Economic Theory. 6:

179-93.

[37] Sachs, Je¤rey D. and Andrew M. Warner. (2001). Natural resources and eco-

nomic development. European Economic Review. 45: 827-838.

[38] Sinclair, A. R. E. (1979). Dynamics of the Serengeti ecosystem: process and

pattern. In Serengeti: Dynamics of an ecosystem, ed. A. R. E. Sinclair and

M. Norton-Gri¢ ths. 1-30. Chicago: The University of Chicago Press.

[39] Sinclair, A. R. E. (1979). The Serengeti environment. In Serengeti: Dynamics

of an ecosystem, ed. A. R. E. Sinclair and M. Norton-Gri¢ ths. 31-45. Chicago:

The University of Chicago Press.

[40] Sinclair, A. R. E. (1995). Serengeti past and present. In Serengeti II: Dynam-

ics, Management, and Conservation of an Ecosystem, ed. A. R. E. Sinclair

and Peter Arcese. 3-30. Chicago: The University of Chicago Press.

[41] Sinclair, A. R. E. and Peter Arcese. (1995). Serengeti in the Context of the

Worldwide Conservation E¤orts. In Serengeti II: Dynamics, Management, and

Conservation of an Ecosystem, ed. A. R. E. Sinclair and Peter Arcese. 3-30.

Chicago: The University of Chicago Press.

65

[42] Sinclair, A. R. E., H. Dublin, and Markus Borner. (1985). Population regu-

lation of Serengeti Wildebeest: a test of the food hypothesis. Oecologia. 65:

266-268.

[43] Swinton, S. and G. Escobar. (2003). Poverty and Environment in Latin Amer-

ica: Concepts, evidence and policy implications. World Development. 31:

1865-1872.

[44] Reardon, T. and S. Vosti, (1995). Links between rural poverty and the en-

vironment in developing countries: Asset categories and investment poverty.

World Development. 23 1495-1506.

[45] United Nations. (1994). National Accounts Statistics: Main Aggregates and

Detailed Tables, Parts I and II. New York: U.N. Pub. Div.

[46] United Nations. (2006). National Accounts Statistics: Main Aggregates and

Detailed Tables, Parts I and II. New York: U.N. Pub. Div.

[47] Wright, Gavin and Jesse Czelusta. (2003). Mineral resources and economic

development. Conference on sector reform in Latin America, Stanford Center

for International Development.

[48] TheWorld Bank. (2004). The United Republic of Tanzania: Poverty reduction

strategy.

[49] The World Bank. (2005). The World Development Indicators.

[50] The World Bank and Government of the United Republic of Tanzania. (2002).

Tanzania at the Turn of the Century. A World Bank Country Study.

66