Discussion Papers In Economics
And Business

Optimal Government Expenditures on Production and Extraction Technologies
in a Small Open Economy with a Non-renewable Natural Resource

Kamil Aliyev

Discussion Paper 24-01

January 2024

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANOptimal Government Expenditures on Production and Extraction Technologies
in a Small Open Economy with a Non-renewable Natural Resource 
Kamil Aliyev†

Abstract
This paper sets a small open economy model in which the government can raise the efficiencies of production andextraction of a non-renewable natural resource by spending the budget raised through income tax. Using this model,
we examine the optimal government expenditure on the production technology and that on the extraction
technology. The results are as follows. If revenue of natural resource is high, the optimal expenditure on theextraction technology is high, while the optimal expenditure on production technology is independent of the natural
resource revenue. Moreover, if the ratio of the optimal expenditure for improvement of extraction technology to
the resource revenue is higher (lower) than the optimal tax rate under the depletion of the natural resource, theoptimal tax rate before the depletion is higher (lower) than that after the depletion.

Keywords: non-renewable natural resource, labor productivity, optimal tax, small open economy
JEL classification: H21, H54, O41, Q32

 I would like to extend my sincere thanks to Tatsuro Iwaisako, and Kazuhiro Yamamoto for their valuable comments andrecommendations. All remaining errors are of course my own.
†
Graduate School of Economics, Osaka University, E-mail: kamil.aliyev@yahoo.com

1. Introduction
The obvious characteristics of non-renewable natural resource extraction is limited capability to support
income over extended periods; once the resource is exhausted the economy loses one of the growth factors. Overthe years, a lot of models theorized to reflect the mechanisms of incorporation of natural resources to achieve longterm development goals. Groth (2007) summarizing the role of non-renewable natural resource on endogenous
growth theory emphasizes declining nature of growth with exogenous or endogenous technical change andsuggests with hesitation that, endogenous technical change might bring about the technological basis for a rising
per capita consumption in the long run or at least non-decreasing per capita consumption. Developing a model
similar to Stiglitz (1974) and Groth and Schou (2002), Sasaki (2021) considers positive or negative population
growth and increasing or decreasing returns to both capital and labor and shows 4 cases where positive per capita
output growth rate is achievable by positive or negative set of parameters. Such mixed outcome of theories is also
accompanied by empirical findings. The literature survey by Mousavi and Clark (2021) concludes that majority ofliterature studying the effects of natural resource abundance or dependence on human capital accumulation, a key
factor in development and economic growth, find adverse effects of natural resource abundance or dependence on
education and health, but that a small to sizeable minority find mixed or beneficial effects. Even if we consider theliterature covering past few years on the role of natural resource extraction on labor productivity, still there is noconsensus and results are mixed. S. Rahim, M. Murshed, S. Umarbeyli et al. (2021) conduct an empirical study on
so called Next Eleven countries, consisted of Bangladesh, Egypt, Indonesia, Iran, Korea, Mexico, Nigeria, Pakistan,Philippines, Turkey, and Vietnam over the period of 1990 – 2019. The results reveal a significant negative effectof natural resource rents on the economic growth and concluded that the resource curse hypothesis exists for thesenations, however, the human capital is found to positively influence the growth through a joint impact with natural
resources. Ramli et.al (2022) investigate the relationship between public investment in human capital (educationand health) and economic growth in Algeria and by using an ARDL approach, conclude that public investment in
human capital and growth are not cointegrated in the long run suggesting policy failure. The empirical study by
Le Clech et.al. (2023) investigate the effect of non-renewable natural resources on the accumulation of humancapital for the period of 1995 – 2018 in a sample of eighteen Latin American and Caribbean countries and conclude
that, non-renewable natural resources have a positive impact on human capital accumulation in the long term,
although, estimates demonstrate relatively low impact. However, they identify physical capital, institutionalframework and the economic openness as the determinants of the human capital and estimate that increasing thephysical capital by 1% would increase the human capital by approximately another 1%. Institutions and economic
openness also promote human capital accumulation. As pointed out in Pritchett (2000), it is difficult to confirm
that all of the public investments creates economically valuable capital. Thus, this argument along with omitted
peculiar factors generally represented as characteristics of a given group of economies or regions can explaininconclusive outcome of empirical literature on the efficiency of the investment into labor productivity on
economic growth.
1The research question of this paper is to determine whether the government can facilitate welfare maximization
through its policies optimizing natural resource extraction and non-resource output. In a simplified approach where
the factors of production in economy reduced to physical assets, labor and natural resources, the only option to
improve welfare after the exhaustion of natural resource stock is to enhance the output of either both of physicalassets and labor or at least any of them. In this paper we are going to study welfare maximization perspectives ofresource exporting country through improvement of labor productivity. Thus we introduce a theoretical frameworkof a small open economy where the government plays a central role in organizing the extraction of natural resourceand supporting the productivity improvement in non-resource output, since concentrating the efforts on narrowgroup of economies with more common features might provide better insights. We incorporate the basic model
introduced in Aliyev (2023) and modify the non-resource output to reflect the government support of productivity
improvement. Our model is distinct from Sun et al. (2018) in that, the natural resource extraction inputs neither
physical nor human capital.
This paper is organized as the following. The Section 2 describes the model and variables introduced. Section
3 describes the optimizing behavior of firms and households in this economy. Section 4 focuses on welfare
maximizing behavior of the government deriving optimal values of extraction expenditure, investment into
productivity improvement, optimal rate of income tax before and after the depletion of non-renewable natural
resource. Section 5 summarizes the analysis with concluding remarks.

2. The ModelIn this model the firms maximize their profit producing output inputting labor and capital, the natural resourceis extracted from non-renewable stock and exchanged to foreign assets and the households derive utility fromconsumption. For this analysis we decided to use the same approach as we did in Aliyev (2023) for some of thecomponents of the model. It is a small open economy with non-renewable natural resource stock. The natural
resource is extracted using similar technology and exchanged to foreign assets 𝑏𝑡 at a price 𝑝 relative to nonresource output. The labor, world interest rate and subjective discount rate are assumed to be constant as well.
Particularly, the preference of the households remains same, to derive utility form consumption only given as,∞𝑢[0] = ∫ 𝑒 −𝜌𝑡 𝑢(𝑐𝑡 )𝑑𝑡 ,(1)0

where 𝜌 denotes the subjective discount rate.
We describe the non-resource output in a popular Solow-Swan style, well covered in economic literature, where
the output is labor augmented. Firms produce non-resource output 𝑌𝑡 inputting capital 𝐾𝑡 , labor 𝐿 and the labor issubject to improvements through another variable. That is, generally referred as knowledge or productivity
parameter1, the labor augmenting variable in turn is a function of investment. The government invests some 𝑔𝑡

1 See Barro, Robert J., Sala-i-Martin, Xavier (2004) Economic Growth, second edition, The MIT Press, chapter 1.1, pp 23-26

or Aghion, Philippe, Howitt, Peter (2009) The Economics of Growth, The MIT Press, chapter 1.2.2, pp 27-29.

2

into labor productivity, however, this investment is exogenous to firms’ profit maximization problem and yields
diminishing returns to scale on productivity. The production function in a general form can be given as𝜂

𝑌𝑡 = 𝐹(𝐾𝑡 , 𝑔𝑡 𝐿), 𝜂 ≤ 1,
and the profit maximization conditions would be 𝐹𝐾 = 𝑟, 𝐹𝐿 = 𝑤 where 𝑟 and 𝑤 denote the interest rate and thewage rate respectively.
The stock of natural resource 𝑆𝑡 diminishes at higher rate than the extraction 𝑋𝑡 obtained. We can show theextraction technology as𝑆𝑡̇ = −Γ(ℎ𝑡 )𝑋𝑡 ,(2)

where 𝑆𝑡 ∈ [𝑆0 , 0], Γ(ℎ𝑡 )>1 denotes the (in)efficiency of extraction which is a function of expenditures ℎ𝑡 incurred
to extract. The expenditure does not accumulate over time. Γ ′ (ℎ𝑡 ) < 0 and Γ ′′ (ℎ𝑡 ) > 0 show that the efficiencyof extraction increases with diminishing return as expenditures increase.
The government runs an intertemporal budget collecting taxes to finance the natural resource extraction andimprovement of labor productivity in non-resource output technology. The whole output of the economy will betaxed. Hence, the budget constraint of the government is given as𝑇𝑇∫ 𝑒 −𝑟𝑡 𝜏[𝑓(𝑘𝑡 , 𝑔𝑡 ) + 𝑝𝑋𝑡 ]𝑑𝑡 = ∫ 𝑒 −𝑟𝑡 (𝑔𝑡 + ℎ𝑡 )𝑑𝑡 .
0

(3)

0

3. Optimal Paths
3.1. Firms
We specify the following Cobb-Douglas production function for non-resource output:
𝜂

𝜂

1−𝛼

𝐹(𝐾𝑡 , 𝑔𝑡 𝐿) = 𝐴𝐾𝑡𝛼 (𝑔𝑡 𝐿),

where 𝐴 is constant parameter of the technology. Then, using the profit maximization condition of firms withrespect to capital we obtain
𝐹𝐾 = 𝑟:where 𝑘 ≡𝐾𝐿

𝑘𝑡
𝛼𝐴 ( 𝜂 )
𝑔𝑡

𝛼−1

= 𝑟,

denotes capital per capita. After rearranging this condition, we can show that1

𝛼𝐴 1−𝛼 𝜂
𝑘=( )
𝑔 .
𝑟

(4)

Since the amount of investment into labor productivity is exogenous into households’ problem, the level ofcapital becomes constant and in order (2) to hold, implying that the investment rate becomes constant too.
Then we determine the per capita output of non-resource sector as𝑦≡

𝑌 𝐹(𝐾, 𝑔𝜂 𝐿)== 𝐹(𝑘, 𝑔𝜂 ) = 𝐴𝑘 𝛼 𝑔𝜂(1−𝛼) ,𝐿𝐿and substituting the value of capital stock per capita from (4) and rearranging the above, express the non-resource
output in terms of technology, government investment and interest rate:
3

𝛼
1
𝛼 1−𝛼
𝐴̅ ≡ 𝐴1−𝛼 ( ).

𝑦 = 𝐴̅𝑔 ,
𝜂

(5)

𝑟

3.2. Households
Households maximize their discounted utility subject to natural resource constraint and foreign assets
constraint. The problem can be separated into two steps, where in the first step the natural resource earnings
maximized with respect to natural resource constraint and in the second step the discounted utility is maximized
with respect to foreign assets constraint. Since the extraction 𝑋𝑡 is a choice variable and its power is one, theHamiltonian function of this problem in the first step is linear in extraction rate. This means that the rate ofextraction is constant over time and has the highest possible finite value. Moreover, the stock of natural resource
does not appear in Hamiltonian function, thus making the shadow value of natural resource constant, which in turntogether with constant extraction rate mean that the economy deliberately will exhaust the stock of non-renewable
natural resource. Consequently, the extraction expenditure becomes constant over time. This part of our model isderived exactly the same way as in Aliyev (2023), hence we skip it here2. After determining the optimal paths ofresource stock, its shadow value, extraction rate and its expenditure, we can show the next step of households’
problem as the following:∞max ∫ 𝑒 −𝜌𝑡 𝑢(𝑐𝑡 )𝑑𝑡
0

subject to:𝑏𝑡̇ = 𝑟𝑏𝑡 + (1 − 𝜏)[𝑓(𝑘, 𝑔) + 𝑝𝑋̅] − 𝑐𝑡 for 𝑡 ∈ [0, 𝑇],𝑏𝑡̇ = 𝑟𝑏𝑡 + (1 − 𝜏̅)𝑓(𝑘, 𝑔̅ ) − 𝑐𝑡

for 𝑡 ∈ (𝑇, ∞)(6)

The intertemporal budget constraint of household is obtained from foreign assets constraint as the following:∞∫ 𝑒 −𝑟𝑡 𝑐𝑡 𝑑𝑡 = 𝑏0 +0

1 − 𝑒 −𝑟𝑇
𝑒 −𝑟𝑇
𝑍+
𝑍̅,
𝑟
𝑟

𝑍 ≡ (1 − 𝜏)[𝑓(𝑘, 𝑔) + 𝑝𝑋̅],𝑍̅ ≡ (1 − 𝜏̅)𝑓(𝑘, 𝑔̅ ).

(7)

The solution and the result to this step of households’ maximization problem is also similar to Aliyev (2023)3,
hence by skipping it we only show the level of consumption for households, which we obtain from (7):𝑐̅ = 𝑟𝑏0 + (1 − 𝑒 −𝑟𝑇 )(1 − 𝜏)[𝑓(𝑘, 𝑔) + 𝑝𝑋̅)] + 𝑒 −𝑟𝑇 (1 − 𝜏̅)[𝑓(𝑘, 𝑔̅ )].

(8)

Hence, the consumption becomes constant and equals the sum of interest earnings from initial foreign assets,
and discounted after-tax earnings from natural extraction and non-resource output over the duration of resource
extraction.

4. Government
The process of natural resource extraction is financed through the government and performed at highest
possible rate of extraction. As turns out, the expenditures are constant over time too. Thus the level of expenditure

2 See Aliyev (2023), pages 151-152 to derive these results.
3 See Aliyev (2023), pages 153-154.

4

affects the duration of extraction until exhaustion of the natural resource stock affecting the (in)efficiency of
extraction technology. Denoting the duration by 𝑇 we can express it in terms of initial natural resource stock,
extraction rate and technology4:𝑇=𝑆0.
Γ(ℎ)𝑋̅

(9)

Hence, tax rate affects the value of T.
Using the optimal values from households’ and firms’ maximization problems the intertemporal budget of thegovernment can be given as𝑇∫ 𝑒 −𝑟𝑡 {𝜏[𝑓(𝑘, 𝑔) + 𝑝𝑋̅] − (𝑔 + ℎ)}𝑑𝑡 = 0,0⟹

1 − 𝑒 −𝑟𝑇
{𝜏[𝑓(𝑘, 𝑔) + 𝑝𝑋̅] − (𝑔 + ℎ)} = 0𝑟∴ 𝜏[𝑓(𝑘, 𝑔) + 𝑝𝑋̅] = 𝑔 + ℎ for 𝑡 ∈ [0, 𝑇].

(10)

After the exhaustion of the natural resource the government supports the labor productivity with investment by
imposing tax on the output of non-resource industry and runs the following balanced budget:𝜏̅𝑓(𝑘, 𝑔̅ ) = 𝑔̅ 𝑓𝑜𝑟 𝑡 ∈ (𝑇, ∞).
The government maximizes the welfare of households by determining the optimal levels of tax rate, extraction
expenditures and investment into labor productivity improvement. Deriving the optimality condition for 𝑍̅ is easier,so we give it first.
Since the extraction is not financed anymore after the depletion of the natural resource, substituting (5) for
output in above we can show that1𝑔̅ = (𝜏̅𝐴̅)1−𝜂 ,(11)and substituting (11) into 𝑍̅ in (7) gives us:𝜂1𝑍̅ = (1 − 𝜏̅)𝜏̅ 1−𝜂 𝐴̅ 1−𝜂 .

(12)𝜏̅ is determined so as maximize 𝑍̅. Hence, the optimal tax rate can be given as
1−2𝜂

−
1 𝜏̅ 1−𝜂 (𝜏̅ − 𝜂)𝜕𝑍̅
= 0: − 𝐴̅ 1−𝜂
= 0,
𝜕𝜏̅
1−𝜂

⟹ 𝜏̅ ∗ = 𝜂.

(13)

We applied the following approach to obtain the solution to maximization of 𝑍 by the government.
Substituting (5) into (10) turns the budget constraint into the following:(𝑔 + ℎ) = 𝜏[𝐴̅𝑔𝜂 + 𝑝𝑋̅],(14)

which determines 𝑔. We let 𝜙(𝜏, ℎ) denote the implicit function of 𝑔. Then the maximization problem of thegovernment can be given from 𝑍 at (7) as

4

See Aliyev (2023) pages 152-153 to derive this result.

5(1 − 𝜏)[𝐴̅𝑔𝜂 + 𝑝𝑋̅] = (1 − 𝜏)[𝐴̅𝜙(𝜏, ℎ)𝜂 + 𝑝𝑋̅],
where the government chooses 𝜏 to maximize. The first-order condition with respect to 𝜏 is:(1 − 𝜏)𝐴̅𝜂𝜙(𝜏, ℎ)𝜂−1 𝜙𝜏 = 𝐴̅𝜙(𝜏, ℎ)𝜂 + 𝑝𝑋̅.

(15)

Since 𝜙(𝜏, ℎ) satisfies (14), substituting for 𝑔 to get:𝜙 + ℎ = 𝜏[𝐴̅𝜙 𝜂 + 𝑝𝑋̅].
Further, differentiating both sides with respect to 𝜏 yields:𝜙𝜏 = [𝐴̅𝜙 𝜂 + 𝑝𝑋̅] + 𝜏𝐴̅𝜂 𝜙 𝜂−1 𝜙𝜏 ,which after rearranging gives us:𝜙𝜏 =𝐴̅𝜙 𝜂 + 𝑝𝑋̅
.
1 − 𝜏𝜂𝐴̅𝜙 𝜂−1(16)

Substituting this into (15) and simplifying gives us an expression which represent the optimal level of investment,𝑔∗ as the following:1

𝜙 = (𝜂𝐴̅)1−𝜂 ≡ 𝑔∗ .

(17)

From (11) and (13), this is the same as the optimal expenditure after the depletion of the natural resource. From
the definition of 𝐴̅ in (15), we obtain the following proposition.

Proposition 1. If the productivity of the non-resource good production 𝐴 is higher, the optimal expenditure for
production technology 𝑔∗ is higher. If the world interest rate 𝑟 is higher, the optimal expenditure for production
technology 𝑔∗ is lower.

Using the optimal values obtained in (14) we can show that:ℎ = 𝜏[𝐴̅𝑔∗𝜂 + 𝑝𝑋̅] − 𝑔∗ .

(18)

Next, the government’s maximization problem can be reduced to:max𝜏

1 − 𝑒 −𝑟𝑇
𝑒 −𝑟𝑇
(1 − 𝜏) [𝐴̅𝑔∗𝜂 + 𝑝𝑋̅] +𝑍̅
𝑟𝑟

subject to𝑇=

𝑆0,Γ(ℎ)𝑋̅

ℎ = 𝜏[𝐴̅𝑔∗𝜂 + 𝑝𝑋̅] − 𝑔∗ .
The first-order condition with respect to 𝜏 is obtained as the following:𝑒 −𝑟𝑇 {(1 − 𝜏)[𝐴̅𝑔∗𝜂 + 𝑝𝑋̅] − 𝑍̅}𝜕𝑇 𝜕ℎ 1 − 𝑒 −𝑟𝑇[𝐴̅𝑔∗𝜂 + 𝑝𝑋̅],
=
𝜕ℎ 𝜕𝜏
𝑟

(19)

𝜕𝑇
𝑆0
=−
Γ ′ (ℎ),𝜕ℎΓ(ℎ)2 𝑋̅𝜕ℎ
= [𝐴̅𝑔∗𝜂 + 𝑝𝑋̅] > 0.
𝜕𝜏
6

(20)

Then we simplify (19) by defining −Γ′ (ℎ)ℎΓ(ℎ)≡ 𝜀 and using (9), its derivative with respect to ℎ and (20) to obtain:(1 − 𝜏)[𝐴̅𝑔∗𝜂 + 𝑝𝑋̅] − 𝑍̅𝑒 𝑟𝑇 − 1
𝜀=
.
ℎ
𝑟𝑇

(21)

Proposition 2. There is a single value of extraction expenditures which maximizes income earnings of thehouseholds.

Proof: Refer to (21) and substitute (18) into it. At the left-hand side h increases as 𝜏 increases, however, the lefthand side is a decreasing function of 𝜏 overall.
(𝐴̅𝑔∗𝜂 + 𝑝𝑋̅) − 𝜏(𝐴̅𝑔∗𝜂 + 𝑝𝑋̅) − 𝑍̅,ℎ∴(𝐴̅𝑔∗𝜂 + 𝑝𝑋̅) − 𝑔∗ − 𝑍̅ − ℎ𝑒 𝑟𝑇 − 1𝜀=.
ℎ𝑟𝑇From 𝑔∗ = ̅̅̅𝑔∗ and the definition of 𝑍̅, 𝐴̅𝑔∗𝜂 − 𝑔∗ = 𝑍̅ . Therefore, we obtain𝑝𝑋̅ − ℎ𝑒 𝑟𝑇 − 1
𝜀=
.
ℎ
𝑟𝑇

(22)

The right-hand side is an increasing function of 𝑇, whereas 𝑇 is an increasing function of ℎ. Hence, ℎ is anincreasing function of 𝜏. As shown in Figure 2, there is a single intersection of the left-hand side and the righthand side of (22), which constitute the optimal value. QED𝑅𝐻𝑆ℎ∗ 𝑝𝑋̅ℎ𝐿𝐻𝑆−𝜀Figure 1. Determination of optimal level of extraction expenditures

Proposition 3. If 𝑝𝑋̅ and 𝜀 are higher, the optimal expenditure for extraction technology ℎ is higher, and thus theduration of natural resource extraction is longer.

7

Increasing the values of 𝑝𝑋̅ and 𝜀 would shift the left-hand side of (22) upward shifting the intersection point ℎ∗shown in Figure 1 rightward. Since 𝑇 is an increasing function of ℎ the duration of natural resource extraction
would be longer as a result of better extraction efficiency. QED

Finally, we compare the optimal tax rate before the depletion of the natural resource and that after the depletion.
Letting ℎ∗ the optimal expenditure on productivity of extraction, from (18), the optimal income tax rate is givenby 𝜏 ∗ =𝑔∗ +ℎ∗.
𝐴̅𝑔∗𝜂 +𝑝𝑋̅𝜏 ∗ − 𝜏̅∗ =𝑔∗ + ℎ ∗𝑔∗ + ℎ∗ − 𝜏̅∗ (𝐴̅𝑔∗𝜂 + 𝑝𝑋̅ )
̅∗ =
−
𝜏
.
𝐴̅𝑔∗𝜂 + 𝑝𝑋̅
𝐴̅𝑔∗𝜂 + 𝑝𝑋̅

From (13) and (17), 𝑔∗ = 𝜏̅∗ 𝐴̅𝑔∗𝜂 . Substituting this into the equation, we obtain𝜏 ∗ − 𝜏̅∗ =Thus, ifℎ∗ − 𝜏̅∗ 𝑝𝑋̅
.
𝐴̅𝑔∗𝜂 + 𝑝𝑋̅

(23)

ℎ∗
> (<)𝜏̅∗ ,then 𝜏 ∗ > (<)𝜏̅∗ .
𝑝𝑋̅

5. Conclusion
In this research we considered a situation where government can support the labor productivity. In our model,
government imposes income tax to invest into labor productivity and support the natural resource extraction. Whilethe economy gains from the extraction of non-renewable natural resource along with the output produced inputting
labor and capital, the optimal amount of expenditure for natural resource extraction depends on the relative shareof resource earnings, the (in)efficiency of extraction technology as well as the parameters of non-resource output
technology. The higher the world interest rate, the lower the value of optimal investment is.
Our key finding is that, when the extracted natural resource is not used as an input for another output in theeconomy, the optimization of extraction and productivity improvement in remaining aggregate output are separate
problems for the government. That is, if revenue of natural resource is high, the optimal expenditure on theextraction technology is high, while the optimal expenditure on production technology is independent of the natural
resource revenue.
The optimal tax rate is determined to provide the optimal level of expenditures while the natural resource isbeing extracted. However, after depletion of natural resource, the tax rate to support labor productivity equals to
output elasticity of investment into labor productivity when the whole output is produced labor and capital.
Comparison of tax rates before and after the depletion shows that if the ratio of the optimal extraction expenditure
to resource revenue is higher (lower) than the optimal tax rate after the depletion of the natural resource, the optimal
tax rate before the depletion is higher (lower) than that after the depletion.
An empirical study might test our key theoretical finding as the possible cause of the “resource curse” from thecontext of growth theory.
8

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