Title

Welfare analysis of extraction of a nonrenewable natural resource in a small open
economyAuthor(s)Aliyev, Kamil

Citation

大阪大学経済学. 2023, 73(2・3), p. 148-161

Version Type VoR
URL

https://doi.org/10.18910/93338

rights
Note

Osaka University Knowledge Archive : OUKA
https://ir.library.osaka-u.ac.jp/
Osaka University

Vol. 73 Nos. 2・3

OSAKA ECONOMIC PAPERS

December 2023

Welfare analysis of extraction of a non-renewable natural
resource in a small open economy
Kamil Aliyev †Abstract
We develop a small open economy model with non-renewable natural resource stock, where thegovernment can improve the extraction technology by government expenditure, which is financed
by income tax. We analytically examine the welfare-maximizing income tax rate and show thefollowing results. First, the welfare-maximizing tax rate increases when the income share ofresource export rises or when the efficiency of the improvement of the extraction technology rises.
Second, it decreases when the size of the resource stock increases or the world interest rate rises.
JEL Classification: H21, O40, Q32
Keywords: non-renewable natural resource, small open economy, optimal tax

1. Introduction
Despite thoroughly studied, the existence of non-renewable natural resource in economy stillcauses many questions for different situations. Yet the possession of non-renewable natural resourcealone cannot unambiguously predict the future development of any economy. Still we see varying
degree of development of economies with substantial amounts of non-renewable natural resourceand shortages, oversupply of natural resource into world market, price fluctuations, impact of market
power of supplier or buyers even now remain important factors requiring attention. A number ofstudies has been conducted to address such issues and the role of natural resource in economic growthis one of main directions of studies in this topic. Although the events in recent past turned the interestof researchers towards different topics of economics, the events in the last few years confirmed theimportance of studying the influence of non-renewable natural resource on economy. Nevertheless,very useful and important literature about the role of natural resources on economic development
stretch back to more than 80 years. We only reviewed those focusing on issues which pose interest for
us and briefly described some of them below.
Barbier (2012) sees the cause of low economic performance of many resource rich countries in not†

Graduate Student, Graduate School of Economics, Osaka University, E-mail: kamil.aliyev@yahoo.com

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Welfare analysis of extraction of a non-renewable natural resource in a small open economy－ 149 －

exploiting the natural resource efficiently and ending up with detrimental growth and development
as the consequence of poor policies and inefficient management of natural resource. Assuming thatthe amounts of the natural resource extracted every period are directly transformed into consumption
good through international trade, C. Le Van et al. (2010) show the circumstances where a resourcerich poor country escapes the poverty trap. They define the extraction costs as an increasing functionof extraction and describe it in detail. Their model allows the natural resource earnings to be invested
into the development of capital stock inputted in the output of non-resource good. Following the oilcrisis of 1973, the scarcity of natural resource became a prominent direction in economic research andSmith (1978, 1979, 1980), Brown and Field (1978), Pindyck (1978), Johnson et.al (1980), Devarajan
and Fisher (1982), Halvorsen and Smith (1984) study the measurement of scarcity of the natural
resource. Groth and Schou (2007) has studied the impact of tax on interest income and subsidy tocapital accumulation in one-sector model with the stock of non-renewable natural resource and foundout that neither of them affect the long-run growth rate.
Our model investigates how a stock of natural resource affects the welfare in small open economy,
which also produces a non-resource good. Our main research question is to find out whether an
optimal way of improving the welfare exists in the long run in small open economy endowed with thestock of non-renewable natural resource and what might be the role of the government in maximizing
the welfare. We follow the model proposed by Nakamoto and Futagami (2016); however, the natural
resource is non-renewable and has finite stock in our model. The economy produces non-resource
good inputting labor and capital and exhausts the natural resource stock to increase the welfare. The
household behavior is different too, where in our model households do not derive direct utility fromhaving the natural resource. The approach of C. Le Van et al. (2010) also seemed interesting; however,their model focuses on economic transformation and does not study the role of the government.
Although Groth and Schou (2007) seemed appealing in terms of studying government policies, incontrast with Nakamoto and Futagami (2016) that model had only one sector and did not separate theresource and non-resource goods. Like in the case of perfect competition in Hotelling (1931), in ourmodel the natural resource stock exhausts in finite time. Households get utility from consumption,
whereas in Hotelling’s model, utility comes from price of extraction and for free competition, it isseen as the social value of the resource.
We keep our model as simple as possible, avoid technological spillovers between production
sectors, and keep exogenous factors constant only to stress the role of natural resource stock on thewelfare. However, despite the simplicity of our approach, we attempt to keep the model robust enoughto describe the real world economies with appropriate adjustments. The natural resource is extracted
inputting portion of extraction earnings and non-resource good produced using labor and capital in
our model. The extraction amount depends on extraction capabilities and the level of domestic capitalstock. The objective of the households is to maximize the utility from consumption, and the natural
resource stock is only exchanged to foreign assets and its non-renewable nature is of no account. The
extraction industry in our economy exists in competitive world market. The non-resource good isan aggregated output of all other goods in our economy. Such simplicity makes the market price of－ 150 －OSAKA ECONOMIC PAPERS

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the natural resource extraction exogenous; therefore, we assume it is constant over time. In contrast
with existing literature, we used non-linear resource extraction technology, which is described as afunction of total output of economy and its value every period is used to extract natural resource.
Our extraction technology can be interpreted as inefficiency parameter of turning the stock of natural
resource to an extraction delivered into world market.
We organized the rest of the paper as the following. Section 2 describes the basic model setup.
The optimal path of variables in dynamic model is derived in section 3 where we find out that theconsumption becomes constant over time. Section 4 derives the optimal policy of the government inbasic model and shows that an optimal tax rate to finance the extraction of natural resource exists insmall open economy that maximizes the welfare. Concluding remarks are given at the end.
2. Basic Model
This analysis is devoted to small open economies with non-renewable natural resource stock. Bothworld interest rate and population

are constant for simplicity.

denotes the amount of non-renewable natural resource stock anddenotes extraction1.

denotes the extraction technology, which is a certain known function of extraction expenditures.

We can show the depletion of resource stock as the following:(1)We assume that the initial level of resource stock is.

and can be exhausted completely. Hence,

has a known value for every period of time, . In addition, we assume that negative

extraction is impossible and, given that the stock of natural resource is sufficiently large, the extraction
rate for any period has some restriction determined by exogenous factors on highest value it can get.
We do not try to reveal such factors and just accept that2

. The extraction technology

is a continuous function with continuous first and second derivatives, where
Extraction expenditures
extraction technology,.

, is determined for every period and does not accumulate over time. The
always, meaning that, delivering

to market is unavoidably

accompanied by certain loss of the resource stock. The exact functional form of extraction technology
is undetermined and later we will assign a functional form to get practical results. Extracted natural
resource is exchanged to foreign assets at a constant relative price .
In addition to non-renewable natural resource extraction, the economy also produces non-resource
good sold in competitive market. The non-resource good is an aggregated good representing all goodsin economy except the extraction obtained from non-renewable natural resource stock. This good is anumeraire in this model and is produced under the perfect competition by inputting capital, and labor
inelastically supplied by households. No particular production function is assigned to non-resource
1

2

We have to clarify that,
does not denote the amount of mineral unearthed, but the amount of resource delivered tomarket. Viewing
from this point better explains losses of resource stock economically.
Some extraction processes can be interpreted as negative extraction; for example, in case of heterogeneous resource
stock, re-injection of natural gas coming out of reservoir back into oil bearing layers to improve more desired crude oil
extraction. Such complexities allowing negative interpretation of extraction are ignored in this model for simplicity.

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Welfare analysis of extraction of a non-renewable natural resource in a small open economy－ 151 －

good, and we denote its output just as a function of capital stock and labor force,. Firms

producing non-resource good maximize their profit with respect to both labor and capital. We denoteandper capita values of capital stock and production function as

respectively.

Substituting these to profit maximization condition reveals that capital stock and wage rate become
constant:

Extraction expenditures are generated by taxing the output of the economy,, which in turn iscomposed of non-resource good and resource extraction:
(2)The economy accumulates foreign assets, which evolves as the following:(3)The initial level of foreign assets

is given.

Defining per-capita consumption as, households derive utility from consumption,

which is assumed to be a concave function. Denoting the subjective discount rate by,

lifetime utilityof households is given as the following:(4)

3. Optimal Paths
In this model, households maximize their lifetime utility (4) with respect to extraction of nonrenewable resource and consumption. We can separate the households’ maximization problem into thefollowing two steps. In the second step, the households solve the following maximization problem.

subject to:(5)where Z denotes the discounted sum of after-tax extraction revenue, and is given by(6)In the first step, the households solve the following maximization problem.

subject to:

We can easily solve the problem starting from the first step. Assigning costate variable
define the Hamiltonian function for this problem as the following:

for

, we

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(7)Households take the extraction expenditures as given. Therefore, we skip the first-order condition
with respect to. The optimality conditions are the following:(8)As for the choice of, the terms of (7) represent the revenue and cost of extraction respectively

in this model. The Hamiltonian is linear in
immediately, since the former requires. The cases ofandto maximize and the latter means that, andvariable in this problem at all. As we will show later,

are ruled out
is not a control

would jump to the highest

possible value.
Prebisch–Singer hypothesis3 justifies the idea that once stock is available, owner will want to exploit
it completely, hardly starting from zero. Therefore, it is reasonable to assume that, given sufficiently
large stocks of natural resource compared to extraction volume, starting the extraction path fromthe vicinity of zero is highly unlikely. The geological exploration prior to extraction is essential todetermine the magnitude of deposit along with other important objectives in modern days.


(9)
(10)

From (9),is constant over time. We define that the constant value is . Then, from (10), we get:(11)From (8) and (11), at time ,From (8),. From (11), in the interval
is constant over time, and thus the government

expenditure for the improvement of extraction technology is constant also as follows:(2’)

As the extraction rate turns out to be constant, we can rewrite the non-renewable natural resource
constraint (1) as the following:(1’)

From (9) we know that the shadow value of natural resource stock is constant in this model.
Since the depletion of natural resource stock is given by an equality constraint (1’) and it is allowed
to be completely exhausted, this control problem can be interpreted as an isoperimetric problem4.
Constant shadow value means indifference whether stocks are preserved or exhausted completely.
3
4

Arezki et al (2014).
Alpha C. Chiang. Elements of Dynamic Optimization. 1992. Chapter 10.1 Constraints involving control variables,
p:280.

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Welfare analysis of extraction of a non-renewable natural resource in a small open economy

Consequently, the time path of corresponding state variable, the resource stock－ 153 －

is of no directinterest, however, considering the constant rate of depletion we can identify when the resource will be
exhausted. Thus, integrating the resource constraint (1’) over the length extraction period gives us thevalue ofas the ratio of initial resource stock to the depletion rate
(12)Then (6) is determined and now we can rewrite the maximization problem at the second step as thefollowing:

subject to:Assigning the costate variablefor, the Hamiltonian function can be written as the following:(13)The necessary conditions are the following:


(14)

(15)
Taking the natural logarithm of (14) and differentiating with respect to time gives us:

Substituting (15) into above gives:

We assume that in small open economy. In such case, the above condition equals zero whichmeans the consumption becomes constant over time:

Using the value of extraction duration , optimal values of extraction rate

, and extraction

expenditures , we can determine the contribution of non-renewable natural resource to foreign
assets over the whole course of extraction. Moreover, substituting optimal values into foreign assets
constraint given by (3) allows us to determine the level of consumption as the following:

We integrate the above equation for the entire duration of existence of the households. Therefore,
we separate the integration period of extraction expenditures to before and after of exhaustion of

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natural resource, because after the exhaustion there would be no need for tax and expenditures would
equal zero. We get the following expression of the consumption:
(16)Hence, consumption equals the sum of interest earnings on initial foreign assets and after tax output
of economy less discounted resource earnings. An important implication of (16) is the role of lengthof extraction period on consumption level; the longer the extraction period becomes, more significant
is the role of extraction earnings on consumption.
4. Income Tax Policy
In this model, the social planner only controls tax rate to finance the resource extraction.
Differentiating the expression for consumption, we can determine the effect of tax rate on
consumption:

Considering (2’) and (12), and further simplifying and rearranging the above equation gives us:
(17)We obtain the following propositions.
Proposition 1. A positive income tax rate
Proof: Optimal tax rate requires

maximizes welfare.

. In such case,

LHS∗

Figure 1. The existence of optimal tax rate

1
RHS

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Welfare analysis of extraction of a non-renewable natural resource in a small open economy－ 155 －

(18)
When

approaches zero,. We verified the slopes of both sides of (18) by taking thederivative with respect to tax rate separately which is provided in Appendix A.
As shown in the Figure 1, there is an intersection of LHS and RHS of (18) in the region.

the RHS gets negative value. Therefore, the optimal tax rate exists in the range

Moreover, at, which maximizes welfare in this economy. QED
Proposition 2. Larger the initial natural resource stock or higher the world interest rate, lower is thelevel of optimal tax; higher the share of resource export in total output or higher the elasticity of
substitution of extraction technology, higher is the level of optimal tax.
Proof:

andare only at the left-hand side,andare only on the right-hand side of (18).

Differentiating each side of (18) with respect to corresponding arguments, we can verify the sign ofeach derivative as the following:

When

or

increases, the LHS of (18) shifts upward causing intersection with RHS move leftward,

thus decreasing the optimal tax rate.

In both cases, the RHS of (18) would shift upward, causing optimal tax move to right. QED
Although, the Proposition 1 is provable theoretically, it does not say much about plausibility of
result within the reasonable values of variables. Therefore, we assign sensible numerical values andtest the equation in Proposition 1 to see whether optimal tax rates get adequate values in the following
section.
A simple function for extraction technology meeting the assumptions can be assigned to better
understand the model. It is obvious that the extraction technology hugely affects the total durationof extraction. However, the variables of our small open economy cannot influence the efficiency of
extraction. Generally, small open economies do not produce much of technology applied in miningof natural resource and most of equipment is just obtained from specialized manufacturers. This is
the reason why we take extraction technology,, as exogenous.

is a convex function, which

has imperfection of depleting the stock more than actual extraction obtained. We used the following

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Vol. 73 Nos. 2・3function to describe the extraction technology:(19)

Keeping in mind that extraction takes place in a real world small open economy, we used the values
summarized in Table 1 for 3 different cases. In all three cases we take the same output level, 5000
per capita, for example in US dollars. This is a typical value for developing countries. The stock ofnatural resource and the extraction rate are in physical measures, such as weight, volume or energy.
The differences of cases are on the ratio of resource and non-resource good in total output of economy
and the ratio of extraction rate to resource stock. We took the 2% world interest rate arbitrarily. In case(i) the higher relative resource price means that the economy is less advanced and resource dependent.
The difference between case (i) and case (ii) is the amount of resource stock. Case (iii) illustrates the
economies with less expensive natural resource. Case (iii) might also represent advanced economies
less dependent on natural resource as well.
Table 1. Parametric values for simulation to determine the welfare maximizing tax ratecase (i)

5000

2000

0.02

80

50

case (ii)

5000

4000

0.02

80

50

case (iii)

5000

4000

0.02

10

50

For simplicity, we chose a decimal value forparameters remain unchanged. Changing the value ofyielding the highest level of tax when rest of
did not result in extremely high tax rates. The

optimal tax rates obtained with the values given in Table 1 areand

0.0305 (3%),

0.0231, (2.3%)

0.0039 (0.4%) respectively.

5. Conclusions
This model grasps the idea of exploitative approach to natural resource in small open economy anddemonstrates it in the simple form. We found out that, if the economy exchanges the natural resource
to foreign assets, it exhausts its natural resource stock in finite time.
The main finding of this paper is the possibility of constant consumption levels in a small open
economy, which is affected by the availability of the natural resource. The natural resource would
have been extracted as soon as possible and exchanged to foreign assets, which bring interest while
being consumed at constant rate.
We showed that the shadow value of the natural resource depends on inefficiency parameter ofextraction technology, tax rate and discounted relative price for the duration of extraction. Our modelallows natural resource to be extracted even with negligible expenditures, which result significant
inefficiency. Such approach allows to deliver limited amount of natural resource to market, although
it would cause loss of portion of natural resource stock. This approach can describe many cases ofwasteful extraction practices. According to Khuduzade and Gasimova (2019), the depth of soil layer

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Welfare analysis of extraction of a non-renewable natural resource in a small open economy－ 157 －

contaminated with oil and oil products near Baku in Azerbaijan reaches 2.0-2.5 meters and the content
of oil products in the soil reach to 26%, whereas 6% is called ‘heavy contamination’. No data isprovided to give idea about how much of extraction loss would have caused it.
We showed that the optimal tax rate in this economy is achievable and its level is related inversely
to initial natural resource stock and world interest rate. In addition, the optimal tax would be higher
when the natural resource export has higher share in total output.
Further, this model can be extended to cover negative externalities of extraction, like pollution,other government policies like subsidization of extraction, or to employ extraction technology, which
depends on stock variable. Additionally, empirically determining extraction expenditures for any
extraction industry might simplify the analysis of optimality of tax rate.
APPENDIX A. Verification of slopes of both sides of (18).
We can verify that the left-hand side (LHS) of (18) is an upward sloping curve by taking thederivative with respect to tax rate as the following:

Similarly, by taking the derivative of right-hand side (RHS) with respect to tax rate we can show it
is downward sloping curve:

APPENDIX B. Adjustment of basic model to accommodate the price of non-resource good andpopulation growth.
In contrast with basic model, population grows at constant rate, , the price of non resource good is, and the price of extraction sold is . Prices are constant for simplicity. Denoting
and

,

,

the resource constraint, the extraction expenditures and the evolution of foreign

assets can be expressed in per-capita terms as the following respectively:
(B1)
(B2)
(B3)
Then, the lifetime utility can be given as

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(B4)In the same way as in the basic model, we can separate the households’ maximization problem into
the following two steps. In the second step, the households solve the following maximization problem.

subject to:(B5)where denotes the discounted sum of after-tax extraction revenue, and is given by(B6)In the first step, the households solve the following maximization problem.

subject to:

We can easily solve the problem starting from the first step. Assigning costate variable

for , wedefine the Hamiltonian function for this problem as the following:(B7)The optimality conditions are the following:
(B8)

(B9)
(B10)
From (B9),. Then, from (B10), we get:(B11)

From (B8) and (B11), at time ,. From (B11), in the interval,

From (B8),Then the natural resource constraint can be shown as the following:(B1’)

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Welfare analysis of extraction of a non-renewable natural resource in a small open economy

From the natural resource constraint－ 159 －

satisfies(B12)
Then (B6) is determined and now we can rewrite the maximization problem at the second step as thefollowing:

subject to:Assigning the costate variablefor, the Hamiltonian function can be written as the following:(B13)The necessary conditions are the following
(B14)
(B15)Taking the natural logarithm of (B14) and differentiating with respect to time gives us:

Substituting (B15) into above gives:

We assume that in small open economy. In such case, the above condition equals zero whichmeans the consumption becomes constant over time:(B12) determines the value of . Considering this and solving the foreign assets constraint gives us thefollowing level of consumption:
(B16)
Then, differentiating (B16) we can examine the optimality of tax rate:
(B17)
Now we take the natural logarithm of (B12) and then differentiate with respect to and get:－ 160 －OSAKA ECONOMIC PAPERS

Denoting

and rewriting forVol. 73 Nos. 2・3

gives us:Substituting the above into (B17) gives us:

From (B12) we have:

Then by substituting this into

gives the following condition for optimal tax rate:
(B18)The left-hand side is an increasing function, and the right-hand side is a decreasing function of tax
rate. Therefore, an optimal tax rate exists where the left side of (B18) equals the right-hand side asshown in Figure B1.

LHS

∗

1
RHS

Figure B1. The existence of optimal tax rate

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