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ANALYTICAL AND NUMERICAL EXPRESSIONS OF THE GOLDEN RULE
OF CAPITAL ACCUMULATION
Doctoral Student ORCID: 0000-0002-4431-3151 e-mail: ahmadxonazibayev@ gmail.com
Namangan State University
Azibaev Akhmadkhon Gulomjon ugli
Abstract. This article examines the importance of consumption and savings rate for economic growth and improvement of living standards. It explores the concept of the golden rule of savings, which represents the optimal savings rate for sustainable economic development. The paper discusses the relationship between consumption, savings and investment. In order to illustrate the relationship between the savings rate and sustainable consumption, the Solow - Swan model is studied. Using the Solow model, analytical and numerical solutions are presented to determine the sustainable savings rate and consumption rate by the golden rule of savings rate. The article emphasizes the importance of finding a balance between current consumption and future investment for long-term economic growth and well-being of present and future generations.
Keywords: consumption, savings, investment, economic growth, standard of living, golden rule of savings rate, Solow - Swan model, sustainable consumption, intertemporal budget constraint, capital accumulation, population growth, per capita income, Cobb - Douglas production function, empirical analysis.
KAPITAL JAMG'ARISHNING OLTIN QOIDASI BO'YICHA ANALITIK VA SONLI YECHIMLAR
Azibayev Ahmadxon G'ulomjon o'g'li
doktorant Annotatsiya. Ushbu maqolada iqtisodiy o'sish hamda turmush darajasini oshirishda
iste'mol va jamg'arma stavkalarining ahamiyati ko'rib chiqilgan. Unda iqtisodiyot-Namangan davlat universiteti ning barqaror rivojlanishi uchun jamg'armalarning optimal darajasini ifodalovchi
"Oltin qoida" jamg'arma stavkasi tushunchasi o'rganilgan. Maqolada iste'mol, jamg'arma va investitsiyalar o'rtasidagi o'zaro bog'liqlik muhokama qilingan. Jamg'arma stavkalari va barqaror iste'mol o'rtasidagi bog'liqlikni ko'rsatish uchun Solou - Swan modeli tahlil qilingan. Solou modeli yordamida barqaror holatdagi iste'mol ifodasi bilan birga "Oltin qoida"ning barqaror holatdagi jamg'arma stavkasining analitik va sonli yechimlari taqdim etilgan. Maqolada uzoq muddatli iqtisodiy o'sish hamda hozirgi va kelajak avlodlar farovonligi uchun joriy iste'mol va kelajakdagi investitsiyalar o'rtasidagi muvozanatni topish muhimligi ta'kidlangan. Kalitso'zlar: iste'mol, jamg'arma, sarmoya, iqtisodiy o'sish, turmush darajasi, "Oltin qoida" jamg'arma darajasi, Solou - Swan modeli, barqaror iste'mol, vaqtlararo byudjet cheklovi, kapital jamg'arish, aholi o'sishi, jon boshiga daromad, Kobb -Duglas ishlab chiqarish funksiyasi, empirik tahlil.
АНАЛИТИЧЕСКИЕ И ЧИСЛЕННЫЕ ВЫРАЖЕНИЯ ЗОЛОТОГО ПРАВИЛА НАКОПЛЕНИЯ КАПИТАЛА
Азибаев Ахмадхон Гуломжон угли
докторант
Наманганский
государственный
университет
Аннотация. В данной статье рассматривается важность нормы потребления и сбережений для экономического роста и повышения уровня жизни. В ней исследуется понятие золотого правила накопления, представляющее оптимальный уровень сбережений для устойчивого развития экономики. В статье обсуждается взаимосвязь между потреблением, сбережениями и инвестициями. С целью иллюстрации взаимосвязи между нормой сбережений
H^TH6ocnHK^HTHpoBaHHe/citation: Azibaev, A. G. (2024). Analytical and numerical expressions of the golden rule of capital accumulation. Science and Innovative Development, 7 (4), 15-26. 15
Kelib tushgan/ Получено/ Received: 18.07.2024
Qabul qilingan/Принято/ Accepted: 09.08.2024
Nashr etilgan/
Опубликовано/Published:
26.08.2024
иустойчивым потреблением изучена модель Солоу - Свона. С помощью модели Солоу представлены аналитические и численные решения по определению устойчивых нормы сбережений и уровня потребления по золотому правилу нормы сбережений. В статье подчёркивается важность поиска баланса между текущим потреблением и будущими инвестициями для долгосрочного экономического роста и благополучия нынешнего и будущих поколений. Ключевые слова: потребление, сбережения, инвестиции, экономический рост, уровень жизни, золотое правило нормы сбережений, модель Солоу - Свона, устойчивое потребление, межвременное бюджетное ограничение, накопление капитала, рост населения, доход на душу населения, производственная функция Кобба - Дугласа, эмпирический анализ.
Introduction
In the pursuit of improving the living standards, countries employ various methods to enhance well-being of their population. Among these methods, consumption plays a pivotal role in determining a nation's living standards. As individuals, we naturally strive to increase our consumption throughout our lives. However, both at the national and personal level, it is essential to strike a balance between consumption and saving or investment.
National accounts reveal that neither a country nor an individual can consume every unit of output they produce. This is because saving or investment serves as a foundation for future periods of output. If everything is consumed, there will be insufficient resources to continue producing goods or services. Consider a farmer who plants seeds in spring and harvests crops in autumn. While the farmer consumes or sells a significant portion of the harvest, they also reserve some seeds for the following year. This approach ensures sustainability of their farming activism.
Societies and countries appreciate the prudent behavior of farmers, as they consume and sell a portion of their output while ensuring future productivity. But how do we determine the appropriate saving rate? Returning to our previous example, how much of the harvest should the farmer save for a succeeding year? If the farmer consumes all of the produce, there will be nothing left to sow for an upcoming season. On the other hand, if everything is saved, what will the farmer consume in the current year? Therefore, it becomes crucial to find the optimal saving rate that strikes a balance between present consumption and future investment.
Previously known as a Golden Rule saving rate, this concept signifies the range between 0 and 1 that represents an ideal level of saving. It is a crucial consideration in determining sustainable economic growth and ensuring the well-being of both individuals and nations alike. In this article, we will delve deeper into the significance of consumption and saving rates, exploring their impact on economic development and the quest for better living standards.
In the field of economics, the concept of the Golden Rule savings rate holds significant importance as it determines an optimal rate of savings that maximizes a steady-state growth of consumption. This notion finds its roots in the influential works of John von Neumann and Maurice Allais, although it is often attributed to Edmund Phelps, who introduced the term in 1961. Phelps proposed that the Golden Rule, a principle rooted in the idea of treating others as we wish to be treated, can be applied inter-generationally within economic models to establish an "optimum" or ideal savings rate for the well-being of future generations.
In the Solow-Swan model, a savings rate of 100% implies that all income is directed towards investment capital, ensuring the creation of capital for future production. However, these results in a steady-state consumption level of zero, as all resources are devoted to building of capital. Conversely, a savings rate of 0% signifies that no new investment capital is being generated, leading to the depreciation of existing capital without any replacement.
Consequently, a steady state becomes unsustainable unless output is reduced to zero, once again resulting in a consumption level of zero.
The Golden Rule savings rate lies between these extremes, representing the savings propensity that enables per-capita consumption to reach its maximum constant value. In other words, it identifies the level of savings that allows for the highest sustainable level of permanent consumption. This concept serves as a guide for policymakers and economists seeking to strike a balance between present consumption and future investment, ensuring sustainable economic growth and well-being of current and future generations.
Literature review
Grzywinska-Rqpca & Olejarz (2021) conducted a comprehensive analysis of G7 household savings rates over the period from 2000 to 2018. To uncover the underlying dynamics of the data, they employed econometric tools for time series decomposition and component identification. The automatic seasonal adjustment procedures, TRAMO-SEATS and ARIMA-X-12, were used in the analysis. Furthermore, the derived models were subjected to empirical verification. The decomposition of the G7 household savings rate time series shed light on the factors influencing this phenomenon. By employing the Tramo-Seats procedure, Grzywinska-Rqpca and Olejarz discovered that the savings rates of the United States, Canada, and France exhibited no discernible seasonal fluctuations. Rather, the observed fluctuations were attributed to moving average processes and outliers. In contrast, Japan, Germany, Italy, and Great Britain displayed seasonal variations in their savings rates. The nature and magnitude of these seasonal deviations differed across countries, with Germany and Italy exhibiting the largest deviations.
Asia from 1966 to 2007, as analyzed by Horioka & Terada-Hagiwara (2012). The authors identify the aged dependency ratio, income levels, and level of financial development as the main factors influencing saving rates in the region. The study also projects future trends in domestic saving rates in developing Asia from 2011 to 2030, based on the estimation results. The authors find that aging of population will be the primary determinant of future saving rate trends in developing Asia. However, they note that the decline in saving rates may not be uniform across the region, as there will be significant variations among countries in the speed and timing of population aging.
In Misztal's (2011) study, the objective was to analyze the cause-and-effect relationship between economic growth and savings in advanced economies and emerging and developing countries. The research employed macroeconomic and international finance studies and econometric methods such as co-integration models and Granger's causality test. Data from the International Monetary Fund's World Economic Outlook database was utilized. The findings revealed a one-way causal relationship between gross domestic savings and gross domestic product in developed and developing countries. Interestingly, no causal relationship was observed between gross domestic product and gross domestic savings across all analyzed economies.
Ribaj & Mexhuani (2021) examined the correlation between savings and economic growth, focusing on the case of Kosovo. The study employed both qualitative and quantitative research methodologies using data from 2010 to 2017. Augmented Dickey - Fuller tests, Johansen co-integration tests, and Granger causality tests were conducted for analysis. The study confirmed stationarity of the data and demonstrated that deposits have a significant positive impact on Kosovo's economic growth. The results indicated that savings stimulate investment, production, employment, and sustainable economic growth. Additionally, loans and remittances were found to contribute to Kosovo's economy through their direct impact on investment. The study highlighted that countries with high national savings rates are less dependent on volatile foreign direct investment, reducing associated risks.
Mei & Qing-Ping (2022) explored the impact of saving rates on economic growth in Asian countries, considering the success of East Asian economies and China's rapid development. Panel data analysis was made using a dataset comprising 46 Asian countries and regions from 1969 to 2021. The study found significant relationships between the gross domestic saving rate, GDP per capita, urban population growth rate, and annual GDP growth rates in Asian countries. Notably, the positive effect of the gross saving rate was significant during the period 1960-1990, but became insignificant in 1991-2021. The urban population growth rate had a more significant positive impact in the later period. GDP per capita exhibited a significant negative effect in both periods. Furthermore, the study demonstrated that a high saving rate was a critical factor for rapid economic growth in developing countries, while urban population growth and GDP per capita also exerted significant influences on economic growth.
Materials and Methods
Analytical solution of the Golden Rule steady state savings rate
The Cobb - Douglas production function, represented by an equation (Cottrell, 2019):
is a widely used model in economics to describe the relationship between inputs and output in a production process. In this equation, Yt represents the output at time t, Kt represents the capital input, Lt represents the labor input, B is a constant factor, and a is the capital's share of the output.
The equation shows that the output Yt is a function of the capital input Kt raised to the power of a and the labor input Lt raised to the power of (1 - a). This functional form reflects the assumption that output is determined by the combined contributions of capital and labor, where a represents the importance or weight given to capital in the production process usually called capital share, and 0<a<1; (for most countries a « 0.3) (Jones, 2003). Taking the partial derivative concerning Kand L we get real factor prices: ;
where: r - rental price; wt - real wage rate.
The intertemporal budget constraint is a concept in economics that captures the relationship between capital accumulation and saving over different periods. It can be expressed as (Dar et al., 2011):
where: Y - represents thecapital stockinthefutureperiod; Kt - represents the capital stock in the current period; St - represents the level ofsavingin thecurrentperiod; S - represents the depreciation rateofcapital.
The equation states that the change in capital stock from one period to the next is equal to the difference between saving and the depreciation of capital. In this context, the saving value St is determined by the product ofthe saving rate s and the incomeoroutput Yt in the current period, represented as:
Yt = ад = B(Kt )а (Lt )
(1)
(2)
(3)
St = Y (5)
Here, s is a saving rate, which is a value between 0 and 1, indicating the proportion of income that households save rather than consume.
Additionally, the "biological" behavior of households is described by an exogenous growth rate n of the population or, more specifically, the labor force. This growth rate can be represented as (Peterson, 2017):
lt+1 = (1 + n) Le (6)
where: L r - epresents the size of the labor force in the next period.
Lt - represents the size of the labor force in the current period.
n - represents the population growth rate, which is greater than (-1).
By dividing both sides of equation (1) by the labor force L, we can express the equation in terms of output per worker yt and capital per worker kt:
yt=Y/L.
kt = K/L.
Now, let's rewrite equation (1) using these new variables:
L t+1 = (1+ n) L t.
Dividing both sides by L, we get:
Lt+1 /Lt = (1 + n).
The left-hand side of the equation represents the growth rate of the labor force (population growth rate plus 1), denoted as (L /Lt). The right-hand side represents the sum of 1 and the population growth rate (n).
In the Solow model, we focus on output per worker yt and capital per worker kt because GDP itself does not directly measure the standard of living or productivity of a nation. By analyzing output per worker and capital per worker, we can gain insights into the efficiency and productivity of the economy.
The equation states that the labor force in the succeeding period is equal to the current labor force multiplied by the sum of 1 and the population growth rate. These equations capture dynamics of capital accumulation, saving behavior, and population growth, providing insights into the interplay between these factors in an intertemporal context.
For prosperity of a nation, it is GDP per worker or per capita, not GDP itself, that is important. In the Solow model, we are therefore interested in output per worker, yt = Y/L . Define similarly capital per worker in the period t, also called the capital-labour ratio, or the capital intensity: kt = K/L . From the first equation of the Solow model, (1) above, it follows from dividing on both sides by L that:
yt=Bka (7)
We can analyze the Solow model directly in terms of the variables we are interested in, the per capita (per worker) magnitudes, kt and y. First, insert the savings behavior (7) into the capital accumulation equation:
K+i = sYt + a - S) K. (8)
Then divide this equation on both sides by Lt , on —he right-hand side using that
L+i = (1+n)Lt+1.
Using also the definitions of kt andy^ one gets:
1 r n kt+i = f~;~" ^ m fi - ^J.
1 -I- n
Finally, from the per capita productios function, we can subside Bkt foryt to arrive at :
fct+n =-M^? + (1 - (9)
1+n
There is another illustrative way to itate the economy's law of motion, where one expresses the change in the capital intensity as a SuncSion of the current capital intensity. Subtracting k from bstS sMes in (9) gives the so-cslled Solow equation:
kt+1 = = 70S - trf + 5)/)t] (10)
The steady state capital-labour raho ns given ait the uniciu) (constant solution, k = k = k, to (9) or (10) above. From the loster, such a k must fulfill Bkt - (n + S) kt = 0, or k~a = sB/ln + S1. Hence:
O
k* r^O-l^p. (11)
V n/SZ
Given the assumption n + S > 0, this is a meaningful expression. Output per worker in steady state is found by inserting the particulat vai/t k* for kt in the percapita production function, y = Bkt:
y ' = B=" (12)
Conaumption per worker is c = (1- s)yt in any period t, so:
^h^to-sor-ipr-5 (13)
Nova, lets try to identify tpe maximumvalue of theconsumption.
To do this, we'll take derivativefromconsumptionfunction c* respect to savingsrate s:
(cay = fi^Tf1 - s)' (^p^ m h°=i(i - ( ^(¡srPg-sP)^))'
( a a -, ( S \n—a a 20/0/ 1 \1—a fCa)' = Sl-o (-(-- ) g f1 -SV-S 0—a (--)
v y \ Vnm6>- ) 01-a °,n g 5/
Mathematically to find maximum of theconsumptionwe shouldsolvethisequation:(c*)' = 0. This means,
- (—+ (1 - s) — (—= 0
Vn+S/ v y1-a Vn+S/
a a
a. a 201-1 ( 1 \i-a ( s \i-a
— S)-S i-« (-) =-)
J 1—a Vn+S/ WS/
« or
(-)1-a ^ 0 that's why, w- divide both sides of equation to this term and get:
Vn+S/
(7 2a-l a
(1 — s)-S 1-a = S1"«
1 — a
No w, solve this equation with re s pect to s:
a 2«~c as 2a—1 a S 1-я--SC-я = SC-a
1 — a 1 — a
a 2a-c a
-Sl-a--S1 -a = SC
1 — a 1 — a
_2_/ a 2 a —~c
SC_a (--+ 1 I = -S C_a
V1 — a J 1 — a
1 - a 2a-i
-si-a =-s i-a if we multiply both sides of the equation by 1 - a we'll get:
1-a 1-«
a 2a-i c ..i ■ « _ 2«-i
si-a = as i-a rrom s si-a i-« = a => s c- a
Eventually, we determine that the Golden Rule Steady - State Savings rate is equal to capital share a. This means, it'll be the best option to choose the savings rate in the rate of capitalshareto maximize consumptio- and pr-vide economic growth.
Numerical solutionof theGoIden Rule steady state savings rate
The relationship between consumption and various economic factors can be quantified using the consumptionfunction.In this case, the consumption function is represented by equation:
a
C* = s-vo - 5) (^)i-a. (13)
Here, B is a constant variable. Consequently, consumption is dependent on the savings rate (s), capital share (a), depreciation rate (5), and population growth rate (n). We can express this relationship as c~c(s,n,5).
To provide numerical evidence, let's consider specific values for the variables. For instance, we'll assume n = 0.02, 5 = 0.2, and a is equal to the world average (a = 0.3).
By substituting these values into the consumption function, we can determine corresponding consumption levels (table).
This result suggests an optimal balance between savings and consumption, where a savings rate of 0.3 leads to the highest level of consumption. It implies that allocating a portion of income to savings is beneficial for long-term economic growth and maximizing consumption levels.
By studying the relationship between consumption and key economic variables, such as savings rate, capital share, depreciation rate, and population growth rate, policymakers and individuals can gain insights into how different factors impact consumption patterns and make informed decisions to promote economic well-being.
Table
Numerical values of consum
îtion respect to savings rate
s n s Consumption
0,02 0,01 0,2 0,35774496
0,04 0,01 0,2 0,4716627
0,06 0,01 0,2 0,54948499
0,08 0,01 см50,2 0,60836046
0,1 0,01 0,2 0,65486009
0,12 0,01 0,2 0,69234651
0,14 0,01 0,2 0,72282101
0,16 0,01 0,2 0,74759312
0,18 0,01 0,2 0,76757774
0,2 0,01 0,2 0,78344562
0,22 0,01 0,2 0,79570703
0,24 0,01 0,2 0,80476167
0,26 0,01 0,2 0,81093017
0,28 0,01 0,2 0,81447483
0,3 0,01 0,2 0,81561385
0,32 0,01 0,2 0,8145313
0,34 0,01 0,2 0,81138436
0,36 0,01 0,2 0,8063087
0,38 0,01 0,2 0,79942254
0,4 0,01 0,2 0,79082971
0,42 0,01 0,2 0,78062211
0,44 0,01 0,2 0,76888161
0,46 0,01 0,2 0,7556816
0,48 0,01 0,2 0,74108819
0,5 0,01 0,2 0,72516126
0,52 0,01 0,2 0,70795529
0,54 0,01 0,2 0,68952003
0,56 0,01 0,2 0,66990112
0,58 0,01 0,2 0,64914054
0,6 0,01 0,2 0,62727705
0,62 0,01 0,2 0,60434656
0,64 0,01 0,2 0,58038238
0,66 0,01 0,2 0,55541557
0,68 0,01 0,2 0,52947509
0,7 0,01 0,2 0,50258804
0,72 0,01 0,2 0,47477983
0,74 0,01 0,2 0,44607434
0,76 0,01 0,2 0,41649405
0,78 0,01 0,2 0,38606013
0,8 0,01 0,2 0,35479262
0,82 0,01 0,2 0,32271045
0,84 0,01 0,2 0,28983157
0,86 0,01 0,2 0,25617302
0,88 0,01 0,2 0,22175098
0,9 0,01 0,2 0,18658086
0,92 0,01 0,2 0,15067733
0,94 0,01 0,2 0,1140544
0,96 0,01 0,2 0,07672544
0,98 0,01 0,2 0,03870323
1 0,01 0,2 0
Understanding the dynamics of the consumption function and its relationship with various economic factors is crucial for formulating effective economic policies and strategies to ensure sustainable growth and prosperity.
Research results
The analysis of consumption and saving rates in relation to economic growth and living standards reveals the significance of finding the right balance between present consumption and future investment. The Golden Rule saving rate, which represents the optimal level of savings for sustainable economic development, plays a crucial role in determining the well-being of individuals and nations alike.
The Solow - Swan model provides insights into the relationship between savings rates and steady-state consumption. A savings rate of 100% directs all income towards investment capital, ensuring capital accumulation for future production. However, these results in zero steady-state consumption, as all resources are allocated to building capital. Conversely, a savings rate of 0% leads to the depreciation of existing capital without any replacement, rendering a sustainable steady state unattainable. The Golden Rule saving rate lies between these extremes and represents the savings propensity that maximizes sustainable level of per capita consumption.
The Graph below shows the consumption function and highlighted point (0.3; 0.81561385) determines the maximum value of consumption according to table above.
Consumption function
Savings rate Fig. Consumption function
The analytical solution of the Golden Rule steady-state savings rate reveals that it depends on various factors. The capital share, represented by a, determines the weight given to capital in the production process. A typical value for a is around 0.3 for most countries. The steady-state capital-labor ratio, k*, is determined by the saving -, population growth - and
depreciation rates, and the productivity factor B. The steady-state output per worker, y*, and consumption per worker, c*, are also derived based on these factors.
To determine the optimal saving rate that maximizes consumption, we analyze the consumption function and take the derivative with respect to the savings rate. By solving the equation (c*)' = 0, we find the critical point that represents the maximum consumption. This critical point is where the marginal benefit of saving equals the marginal cost of saving. The specific value of the savings rate at the maximum consumption depends on the population growth rate, depreciation rate, and capital share.
Analysis of research results
Countries often aim to optimize their savings rates to align with economic growth targets. A high savings rate, like Uzbekistan's 37% of GDP, can indicate strong investment potential but might also suggest under-consumption, which could hinder short-term economic activity. Ideally, the savings rate should balance between supporting investment and maintaining healthy consumption levels. For many economies, this "optimum" level is around 25-30% of GDP. Germany's savings rate, nearly equal to this optimum, reflects a balanced approach where savings effectively fuel investment without overly restricting consumer spending.
The findings from this study emphasize the importance of consumption and saving rates in achieving economic growth and better living standards. National and individual levels of consumption must strike a balance with saving and investment to ensure sustainable economic development. Consuming all output without saving or investing leads to insufficient resources for future production, while excessive saving with minimal consumption hampers current economic activity.
The concept of the Golden Rule saving rate provides guidance for policymakers and economists in determining the optimal savings level. By considering the interplay between present consumption and future investment, it allows for the highest sustainable level of per capita consumption. This principle reflects the idea of treating future generations as well as we wish to be treated and highlights the importance of long-term economic growth and well-being of current and future generations.
Empirical analyses have been made to understand the dynamics of saving rates in different countries and regions. Studies have utilized econometric tools and time series decomposition techniques to uncover the factors influencing saving rates. Factors such as income levels, financial development, and population aging have been identified as significant determinants of saving rates. These empirical findings contribute to deeper understanding of the complex relationship between consumption, saving, and economic growth.
Conclusion
In conclusion, achieving sustained economic growth and improving the living standards depend critically on careful management of consumption and saving rates. The Golden Rule saving rate stands as a key principle in this endeavor, guiding the optimal balance between current consumption and future investment to maximize long-term economic outcomes.
The Solow - Swan model, along with the analytical solutions presented, provides a robust framework for understanding the intricate relationship between savings rates and steady-state consumption levels. Through this analysis, it becomes clear how varying savings rates can influence an economy's capacity to maintain growth and elevate living standards over time.
Empirical analyses are equally important as they anchor these theoretical insights in the context of real-world data. By examining saving behaviors across different countries and regions, we gain a deeper understanding of the factors that drive economic performance and the potential for growth in diverse economic environments.
The quest to determine the optimal saving rate is not merely a theoretical exercise. It is a crucial element of sustainable economic development. Policymakers must carefully weigh the trade-offs between present consumption and future investment to ensure that economic policies foster not only immediate growth but also the long-term well-being of people.
In essence, finding the right balance between saving and consumption is essential for promoting the continuous and inclusive economic growth. By adhering to the principles of the Golden Rule saving rate, economists and policymakers can better design strategies that support both current prosperity and future economic sustainability, ensuring that the benefits of growth are shared across generations.
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