Mathematics Serving Economics: A Historical Review of Mathematical Methods in Economics

Abstract

This paper offers a historical review of the evolution of mathematical methods in economics, tracing their development from the earliest attempts in the 18th century to the sophisticated models of the late 20th century. The study begins by examining the initial integration of mathematical techniques into economic thought, highlighting key milestones that shaped the field. Symmetry concepts are naturally embedded in many of these mathematical frameworks, particularly in the balance and equilibrium found in economic models. Symmetry in economics often reflects proportional relationships and equilibrium conditions that are central to both micro- and macroeconomic analyses. Then, the paper elaborates on the progression of economic growth models, including the foundational Solow–Swan model, which introduced the concept of technological progress (knowledge) as a key factor influencing growth. The review also encompasses the Lucas growth model and the Mankiw–Romer–Weil model, both of which incorporate human capital into the growth equation, highlighting its importance in driving economic development. Finally, the paper addresses the Nonneman–Vanhoudt model, which extends the analysis of growth by integrating multiple types of capital, providing a more comprehensive framework for understanding economic dynamics. By documenting these developments, the paper demonstrates the significant role that mathematical modeling has played in advancing economic theory, providing tools to quantitatively analyze complex economic phenomena and driving the discipline towards greater analytical precision and rigor. This analysis emphasizes how symmetry principles, such as balance between inputs and outputs, equilibrium in supply and demand, and proportionality in growth models, underpin many economic theories.

1. Introduction

1.1. Mathematical Methods in Economics before 1838

The dynamic development of mathematics initiated in the second half of the 17th century, when the works of Isaac Newton and Gottfried Wilhelm Leibniz spurred the mathematization of natural and social sciences. The quest for symmetry in mathematical modeling during this period also emerged, as scholars sought to uncover balanced relationships and invariant structures within the phenomena they studied. For some representatives of these fields, mathematical methods became a desirable tool for a more precise description of the phenomena under study. Other scholars, however, believed that not all issues could be described using mathematical equations.

The first attempt to apply concepts from mathematical analysis to economics was made in 1738 by Daniel Bernoulli—a Swiss mathematician and physicist [1]. His considerations were related to problems originating from game theory, but they resulted in a significant contribution to the development of economic sciences. Bernoulli’s work illustrated early efforts to introduce symmetry into economic models by emphasizing equilibrium and the balanced distribution of wealth. D. Bernoulli is considered the first researcher to observe that the satisfaction derived from an increase in wealth depends not only on the value of the wealth increase but also on the total wealth possessed [2]. This observation was related to game theory, in which Bernoulli demonstrated that the utility of a single monetary gain diminishes as the amount of already accumulated money increases. Today, this economic principle is known as the law of diminishing marginal utility, and D. Bernoulli is regarded as a pioneer of the subjective theory of value and the first to apply geometric tools to solving economic problems [3]. Particularly interesting is that the logarithmic curve, which Bernoulli used to explain this relationship, was derived based on solving a differential equation, meaning that Bernoulli applied the most advanced mathematical analysis tools available at the time in his research [2].

The second researcher often mentioned in the context of the earliest applications of mathematical analysis in economics is the Italian philosopher, economist, and writer Pietro Verri, who, in his 1771 work, published theorems concerning the law of demand and supply [4,5]. While Verri did not use explicit mathematical formulas, his verbal formulation of the law of demand and supply reflects an underlying symmetry in economic interactions—where prices adjust to balance supply and demand in the market. Although P. Verri did not use any mathematical formulas, he verbally formulated the theorem that the price of a given good is directly proportional to the number of buyers and inversely proportional to the number of sellers. Based on his observation, he concluded that the optimal situation for society would be the presence of a large number of sellers relative to buyers in the market, which would naturally result in a low price. The person who translated P. Verri’s economic theorems into the language of mathematics was the Italian mathematician Paolo Frisi. His mathematical considerations serve as a commentary on the content found in P. Verri’s book. In translating the problem of finding the minimum price into mathematical language, P. Frisi expressed the price as a function of the number of sellers and buyers, and he referred to the properties of the derivative function [2].

Another application of differential calculus to solve an economic problem appeared in 1824, in the work of the English scholar T. Perronet Thompson [6]. T. Perronet Thompson’s article was a direct response to two other publications—the 1817 article by Sir John Sinclair [7] and the 1810 book by William Huskisson [8]. In his work, J. Sinclair argued that the money supply in the English economy must be increased, or else society would face severe negative consequences. On the other hand, W. Huskisson believed the opposite, arguing that increasing the money supply would lead to its depreciation, with negative effects on all social classes in England [2]. T. Perronet Thompson critically addressed both positions, presenting a mathematical analysis of the problem of money issuance by the state. In his model, he assumed that the state could issue an amount of money in a given period equal to, but not exceeding, the tax revenues expected to flow into the budget during the same period. The issuance of such an amount would not cause currency depreciation and would therefore be imperceptible to society. However, if the government decided to issue an amount of money exceeding tax revenues in a given period, inflation would arise. To introduce an equation describing these relationships, let us assume that p denotes the amount of money the state can issue daily without inflationary effects. Let s represent the surplus in money issuance (above the amount p), and let z denote the fraction describing currency depreciation. T. Perronet Thompson proposed that the effects of money issuance over a small period $dt$ be described by the following equation:

$$dG=sdt-(s+p)zdt,$$

(1)

where $dG$ represents the gain resulting from money issuance over the period $dt$. To calculate the total gain, Equation (1) would need to be integrated within specified limits. However, a more important problem than calculating the gain is answering the question of when the government should stop excessive money issuance. T. Perronet Thompson formulated the theorem that the maximum gain from money issuance is achieved when the marginal revenue equals the marginal gain, which, using the notations from Equation (4), can be written as follows: $sdt=(s+p)zdt$. The condition can be interpreted as a reflection of symmetry in economics, which is associated with concepts such as stability and balance. In other words, the condition is equivalent to stating that $dG=0$. T. Perronet Thompson’s work is considered the first example of the application of mathematical analysis in economics by an English scholar [9].

Apart from methods borrowed from mathematical analysis, the initial phase of the mathematization of economics was also associated with the transfer of other tools and concepts into the field of economic sciences. One of the ideas that significantly enriched the methodology of economic sciences is the general concept of a function. When the analytical formula for expressing a given quantity through independent variables is unknown, one can refer to the general concept of a function and its properties to study the mathematical relationships between the analyzed functions. This approach has found application in many later economic works because, in economics, we often do not know the exact formulas that quantitatively express the relationships between variables, but we can assume which independent variables affect the quantity under study and what mathematical character this relationship has (e.g., directly proportional). The concept of a function is also inherently linked to symmetry, as symmetrical relationships between variables often simplify the analysis of how changes in one economic factor affect another.

The first application of the general concept of a function to study the properties of aggregate quantities can be found in the work of Joseph Lang from 1811 [10]. The researcher who comprehensively presented the possibilities of applying the general notation of functions to describe phenomena of various natures was Georg von Buquoy [11]. These publications show that the benefits for economics can come not only from applying specific mathematical tools in the form of theorems but also from transferring selected concepts to economics and giving them appropriate interpretation.

When discussing the initial stage of the development of mathematical economics, it is also worth mentioning that, in addition to mathematical analysis and the general concept of functions, economists were also interested in the applications of geometry to economic issues. Cesare Beccaria, an Italian jurist and writer with an interest in natural sciences and mathematics, analyzed the problem of the profitability of smuggling in his 1764 work [12]. According to C. Beccaria, algebra allows for quick and precise reasoning and can therefore be applied wherever the increase or decrease in certain quantitative measures is analyzed, including in political science [13]. This article is considered one of the first purely theoretical analyses devoted to a specific social problem [2].

In his work, C. Beccaria provided a mathematical formula describing the profitability of smuggling under the assumption that a certain portion of contraband would be seized by customs officials. This topic was further explored by another Italian scholar, Giovanni Silio Borremans, who in 1792 published a work containing a graphical solution to the equation derived in Beccaria’s study [2,14,15].

Another name that should be mentioned in the context of the early development of mathematical economics is William Whewell, a versatile English philosopher, logician, and historian of science. His contribution to the advancement of mathematical economics is subject to varied assessments. For instance, R. M. Robertson, in his 1949 article [2], refers to W. Whewell as a “translator” of David Ricardo’s concepts into mathematical language [16]. A different perspective on the significance of Whewell’s results was presented by Reghinos D. Theocharis in her monograph, where she argues against calling W. Whewell a “translator” and asserts that the use of complex mathematical procedures greatly aided the development of economic concepts [17].

W. Whewell was a polymath, engaging in research across various fields of science. His work on economics using mathematical tools comprises four articles that were later published as a book. In his first economic work, W. Whewell discovered and defined the concept of inverse demand elasticity, predating other economists such as A. A. Cournot and A. Marshall [18]. It is also noteworthy that W. Whewell initiated the development of mathematical economics in England, inspiring a group of scholars to continue research in this discipline [19].

The early attempts to apply mathematical methods in economics focused on three aspects: the use of differential calculus, the utilization of general properties of functions (without defining them analytically), and the explanation of economic phenomena through geometric curves [2]. It is widely accepted that A. A. Cournot was the pioneer in using mathematics as a method for research and reasoning in economic issues [20,21]. Nevertheless, the results achieved by the scholars preceding Cournot, although contributing significantly less to the development of the field, also deserve to be cited in works dedicated to the origins of mathematical economics.

1.2. Antoine Augustin Cournot and the Model of Oligopoly

Antoine Augustin Cournot was born on 28 August 1801, in the small French town of Gray. In 1823, he graduated with a degree in mathematics from the Sorbonne, and in 1838 he published his book entitled Recherches sur les principes mathématiques de la théorie des richesses. The English edition was published in 1897, see Ref. [22], which is still considered his greatest contribution to the development of economics. A. A. Cournot’s publications are regarded as the first truly scientific works with a mathematical–economic character, where the applied method was mathematical [3].

One of the key concepts introduced by A. A. Cournot was the law of demand (demand should be understood as effective demand). A. A. Cournot treated effective demand as a continuous function of price, assuming that other factors that could influence the level of demand remained unchanged. The demand curve was decreasing and was plotted on a coordinate system where the price was on the horizontal axis and the quantity of demand was on the vertical axis. Denoting the price of a product by p, and the quantity sold of this product by D, we assume that the relationship between these variables is a function: $D=f\left(p\right)$. Since, according to the law of demand, an increase in price results in a decrease in the quantity sold, the derivative of the function $f\left(p\right)$ must always be negative:

$$\forall p>0{\displaystyle \frac{df\left(p\right)}{dp}}<0.$$

(2)

It is easy to notice that the assumption of the existence of a time-dependent, continuous, and differentiable demand function, which is fundamental for A. A. Cournot’s further considerations, is based on (as mentioned in Section 1.1) the use of general properties of functions in mathematical economics without introducing an analytical definition.

Assuming the existence of a functional relationship between demand and price, we can analyze models describing the behavior of a seller in various market situations. Depending on the number of competing firms, A. A. Cournot described the following in his book: monopoly, duopoly, and oligopoly [22]. Since these mathematical models are considered one of the most important (or one of the most important) achievements of A. A. Cournot in the field of economics [3] and a milestone for the development of mathematical economics, they will be briefly discussed in this dissertation.

The first market model in which seller strategies can be examined is the monopoly situation, which occurs when there is no competition in the market, and the single seller seeks to set a price to maximize profit. Within the theory of monopoly, A. A. Cournot distinguished two cases of monopoly.

The first monopoly situation occurs when the entrepreneur bears no production costs. In this case, the seller’s goal is to achieve the maximum value of the revenue function denoted by $R\left(p\right)$ and defined as follows:

$$R\left(p\right):=pf\left(p\right).$$

(3)

According to Fermat’s theorem on the zeroing of the derivative of a function, a necessary condition for the existence of an extremum of the function $R\left(p\right)$ at a certain point $p_0$ is that the value of the derivative at this point is zero, i.e.,:

$${\left.{\displaystyle \frac{dR\left(p\right)}{dp}}\right|}_{p=p_0}=0\iff f\left(p_0\right)+p_0{f}^{\prime }\left(p_0\right)=0$$

(4)

Equation (4) allows us to determine the point $p_0$ (one or more) that satisfies the necessary condition for the existence of an extremum of the function, given a specific function $f\left(p\right)$. To determine whether the extremum indeed exists and its type (minimum or maximum), we must refer to the second derivative. For the function $R\left(p\right)$ to attain a maximum at point $p_0$, in addition to the condition mentioned in Equation (4), the value of the second derivative of $R\left(p\right)$ at $p_0$ must be negative, i.e.,:

$$2f^{\prime }\left(p_0\right)+p{f}^{″}\left(p_0\right)<0.$$

(5)

The conditions given in Equations (4) and (5) describe the equilibrium of a firm under monopoly conditions.

The second case of monopoly considers the existence of costs associated with the entrepreneur’s activity. Costs depend on the level of production. Assuming that there is a continuous and differentiable function $c\left(D\right)$ (also known as cost) that describes this relationship, we obtain the formula for the monopolist’s profit:

$$R\left(p\right)=pf\left(p\right)-c\left(f\right(p\left)\right).$$

(6)

The entrepreneur’s goal is to maximize profit, which mathematically means finding the extremum of the function $R\left(p\right)$. Similar to the first monopoly case, if we expect that price $p_0$ will correspond to the highest profit, then it is necessary for $p_0$ to be a zero of the derivative function ${\displaystyle \frac{dR\left(p\right)}{dp}}$. By calculating the derivative and setting its value to zero, we obtain the following equation:

$$f\left(p_0\right)+f^{\prime }\left(p_0\right)\left(p_0-{\left.{\displaystyle \frac{dc\left(D\right)}{dD}}\right|}_{p=p_0}\right)=0,$$

(7)

from which we can determine the price $p_0$ that satisfies the necessary condition for maximizing profit $R\left(p\right)$. When calculating the derivative of $R\left(p\right)$, it is important to note that the function $c\left(D\right)$ is a composite function, which means:

$${\displaystyle \frac{dc\left(D\right)}{dp}}={\displaystyle \frac{dc\left(D\right)}{dD}}·{\displaystyle \frac{dD}{dp}}.$$

The function ${\displaystyle \frac{dc\left(D\right)}{dD}}$, describing the rate of change in total cost with respect to the level of production (in short: marginal cost), is also a function of price, so we can substitute ${\displaystyle \frac{dc\left(D\right)}{dD}}\equiv \psi \left(p\right)$, resulting in the following expression for $R^{\prime }\left(p\right)$:

$$R^{\prime }\left(p\right)=f\left(p\right)+f^{\prime }\left(p\right)\left(p-\psi \left(p\right)\right),$$

(8)

which in turn allows us to quickly calculate the second derivative of the profit function:

$$R^{″}\left(p\right)=f^{\prime }\left(p\right)\left(2-{\psi }^{\prime }\left(p\right)\right)+f^{″}\left(p\right)\left(p-\psi \left(p\right)\right).$$

(9)

For the profit function $R\left(p\right)$ defined in Equation (6) to attain a maximum value at the point $p_0$ determined from Equation (7), the following second condition must also be met:

$$R^{″}\left(p_0\right)<0.$$

(10)

Equation (7) along with inequality (10) allows a firm operating under monopoly conditions to determine the optimal price $p_0$, which corresponds to the highest profit.

The second market situation analyzed by A. A. Cournot was duopoly, which occurs when two suppliers offer the same product. According to the assumptions made by the model’s author, the offered products do not differ in quality and have the same price p. The sales quantity for each of the firms is denoted by $D_1$ and $D_2$, and the total demand is expressed as $D=D_1+D_2$. Neglecting the issue of production costs, the problem of a firm’s behavior in such a market reduces to the problem of maximizing the revenue function given by $R_i=p{D}_i$, where $i=1,2$. Each firm seeks to maximize its own revenue independently of the other seller.

In this part of the analysis, A. A. Cournot, referring to the concept of the inverse function, introduced the following relationship between D and p:

$$p=f^{-1}(D_1+D_2)=f^{-1}\left(D\right),$$

(11)

which allows us to express the revenue earned by each firm using the following equations:

$$R_1\left(D_1\right)=D_1{f}^{-1}\left(D\right),R_2\left(D_2\right)=D_2{f}^{-1}\left(D\right).$$

(12)

For firm number 1, the production level of the second seller is a parameter, the value of which firm number 1 cannot influence. Each firm adjusts its own production level, thus seeking to maximize its profits.

The optimal market situation occurs when both the first and second firms simultaneously achieve maximum profit. Therefore, from a mathematical perspective, it is necessary to determine such values of $D_1$ and $D_2$ that satisfy the necessary condition for an extremum of the functions $R_1\left(D_1\right)$ and $R_2\left(D_2\right)$. Thus, equilibrium in a duopoly can be described by the following system of equations:

$$\left\{\begin{array}{c}{\displaystyle \frac{d{D}_1{f}^{-1}(D_1+D_2)}{d{D}_1}}=0\hfill \\ \\ {\displaystyle \frac{d{D}_2{f}^{-1}(D_1+D_2)}{d{D}_2}}=0,\hfill \end{array}\right.$$

(13)

which, after calculating the derivatives, takes the following form:

$$\left\{\begin{array}{c}{f}^{-1}(D_1+D_2)+D_1{\left(f^{-1}\right)}^{\prime }(D_1+D_2)=0\hfill \\ \\ f^{-1}(D_1+D_2)+D_2{\left(f^{-1}\right)}^{\prime }(D_1+D_2)=0.\hfill \end{array}\right.$$

(14)

The system of Equation (14) describing equilibrium in a duopoly can be easily generalized to the case where a certain number n of sellers operate in the market. According to A. A. Cournot’s reasoning, each firm will be interested in maximizing its own profit function: $R_i\left(D_i\right):=D_i{f}^{-1}(D_1+\cdots +D_n)$ (where $i=1,2,\dots ,n$) by choosing the appropriate level of its own production $D_i$. Thus, in the case of an oligopoly, where n suppliers compete in the market, equilibrium is described by the following system of equations:

$$\left\{\begin{array}{c}{f}^{-1}(D_1+\cdots +D_n)+D_1{\left(f^{-1}\right)}^{\prime }(D_1+\cdots +D_n)=0\hfill \\ \\ f^{-1}(D_1+\cdots +D_n)+D_2{\left(f^{-1}\right)}^{\prime }(D_1+\cdots +D_n)=0\hfill \\ \\ ⋮\hfill \\ \\ f^{-1}(D_1+\cdots +D_n)+D_n{\left(f^{-1}\right)}^{\prime }(D_1+\cdots +D_n)=0.\hfill \end{array}\right.$$

(15)

Equation (15) reflects the idea of symmetry, as it assumes identical strategies and rules for companies in equilibrium, thereby simplifying the mathematical treatment of competitive dynamics.

The contemporary mathematical models of A. A. Cournot describing equilibrium in the market under conditions of monopoly, duopoly, and oligopoly are considered the most popular models of imperfect competition [23]. By utilizing the properties of functions and theorems derived from differential calculus, A. A. Cournot mathematically described in a very general way the entrepreneur’s pursuit of maximum sales income. The fact that A. A. Cournot achieved this without referring to specific functional forms has led to this model being regarded as the French scholar’s greatest contribution to the development of the science [24].

However, for a long time after the 1838 publication of Recherches sur les principes mathématiques de la théorie des richesses, A. A. Cournot’s book remained unnoticed by the academic world. In the first decade, only one scholar—Karl Heinrich Hagen, Professor of Political Science and Economics at the University of Königsberg—referenced A. A. Cournot’s book in his works. The first scholarly review wholly devoted to the results published by A. A. Cournot appeared in 1857 in the Canadian journal Canadian Journal of Industry, Science and Art, and was written by John Bradford Cherriman, who worked as a professor of mathematics at the University of Toronto from 1853 to 1875 [25]. Unfortunately, at that time Canada was still a colonial state (achieving full sovereignty only in 1931), so scientific works published in local journals were not read in major research centers worldwide. Consequently, even A. A. Cournot himself was unaware that his book had been reviewed by a Canadian mathematician and received a very positive reception. J. B. Cherriman’s work was brought to attention by Robert W. Dimand’s 1988 article, which sought to refute the widespread 20th-century view that A. A. Cournot’s book was underrated for 45 years after its publication [26]. In his article, R. W. Dimand emphasized that according to J. B. Cherriman, A. A. Cournot’s book represented the most significant contribution to the development of economics since the work of A. Smith.

There is an opinion that A. A. Cournot’s book remained on the margins for many years after its publication because it was too advanced for its time. Indeed, the methods employed were obscure to the economics of that era, and the reasoning and conclusions were so intricate and complex that only the next generation of distinguished researchers could uncover the significance of A. A. Cournot’s work [27].

The renewed interest in A. A. Cournot’s 1838 work (as well as subsequent economic books published in 1863 and 1877) only came with economists such as Léon Walras, Stanley Jevons, Francis Edgeworth, and Alfred Marshall (their contributions to the development of economics are presented in the following subsections). The methods and results obtained by A. A. Cournot faced both criticism and acclaim.

1.3. Hermann Heinrich Gossen and the Utility Laws

The second of the pioneering figures who had a significant impact on the development of modern mathematical economics is Hermann Heinrich Gossen, born on 7 September 1810, in Düren. By education, he was a lawyer, but he was passionate about science, especially mathematics. He completed his law studies at the University of Bonn and then took up work in public administration. In 1847, he experienced a period of depression and a crisis of faith, which led him to adopt the view that the purpose of life is the pursuit of maximizing happiness.

In 1854, he published his only scholarly work titled Entwicklung der Gesetz des menschlichen Verkehrs und der daraus fliessenden Regeln für menschliches Handeln. This book did not attract interest among German economists and was thus withdrawn from sale. H. H. Gossen died in 1858, and his contribution to science was not recognized until 20 years later—in 1878, his book was rediscovered by the English economist and logician William Stanley Jevons, a founder of the neoclassical school. W. S. Jevons describes the discovery and significance of H. H. Gossen’s book in the preface to the second edition of The Theory of Political Economy from 1879 [28]. Another prominent representative of the subjective-marginalist approach, Léon Walras, sought to reintroduce Gossen’s name into the scientific community by contacting Hermann Kortum—Professor of Mathematics at the University of Bonn and privately H. H. Gossen’s nephew [29]. The information that L. Walras received from H. Kortum allowed him to write the 1885 article Un Economiste Inconnu: Hermann-Henri Gossen [30].

H. H. Gossen’s discoveries are now known in economics as “Gossen’s Laws”. This term was introduced in recognition of Gossen’s contributions by two economists: Friedrich von Wieser and Wilhelm Lexis.

The first of Gossen’s laws, commonly referred to as the law of diminishing marginal utility, states that the utility (benefit) derived from consuming each additional unit of a good decreases. The increase in utility associated with an additional unit of consumption can be graphically represented by the so-called diminishing marginal utility curve [3].

The second of Gossen’s laws concerns the consumer’s achievement of equilibrium in the consumption of two goods. According to Gossen, the optimal situation occurs when the marginal utilities derived from the consumption of these goods are equal. Graphically, the equilibrium point is therefore found at the intersection of the diminishing marginal utility curves.

Thanks to the interest of two prominent economists—L. Walras and W. S. Jevons—H. H. Gossen’s work was not forgotten forever and found its place in economics textbooks.

2. Applications of Mathematics within the Neoclassical School

The neoclassical school developed in the second half of the 19th century as one of the streams of subjective–marginal economic thought. Alongside the Austrian school (also known as the psychological or praxeological school) and the mathematical school, it constituted one of the three main currents of the marginal revolution. Within the neoclassical school of economics, it was believed that quantitative economic phenomena and the relationships between them should be studied. For this reason, economists were interested in applying mathematical methods to economics. The two most prominent representatives of the neoclassical school are William Stanley Jevons (the founder of the school) and Alfred Marshall. A subsection of this paper is dedicated to each of these economists. Among other, less well-known representatives of this economic stream, Francis Ysidro Edgeworth stands out for his use of advanced differential calculus of multivariable functions. Therefore, a third subsection in this part of the paper is dedicated to him.

2.1. William Stanley Jevons and the Development of Utility Theory

William Stanley Jevons was born on 1 September 1835, in Liverpool. He had broad scientific interests. In 1852, he was admitted to University College London (UCL), where he studied chemistry, mathematics, and logic. Due to financial difficulties faced by his family, he had to go to Australia for a time and returned to London in 1859. He obtained his master’s degree from UCL in 1862 in logic, philosophy, and political economy. In the same year, he wrote an important work from the perspective of the history of mathematical economics: Notice of a General Mathematical Theory of Political Economy, which is frequently cited today but did not attract particular interest in 1862 and was only printed four years later under the title Brief Account of a General Mathematical Theory of Political Economy [31]. Jevons systematically presented his theory in 1871 with the publication of his book The Theory of Political Economy [32].

Jevons believed that the use of mathematical tools allowed for more rapid scientific progress. He was keenly interested in contemporary mathematical publications, including the works of English mathematician and logician George Boole and the articles by physicist Hermann Helmholtz on the applications of non-Euclidean geometry. Observing the development of mathematics and its rich applications led Jevons to adopt the view that employing mathematics in economics is just as valid as using it in the natural sciences [33].

Although economists had already been considering the possibility of mathematizing economics before Jevons’s publications, The Theory of Political Economy is regarded as the beginning of a new economic school where mathematics was used in a methodological and systematic way to solve economic problems [34]. The revolution brought about by Jevons is evident, among other things, in the use of differential calculus to describe the phenomenon of diminishing marginal utility. The occurrence of decreasing satisfaction from consuming each additional unit of a good was seen as a psychological phenomenon, whereas Jevons used the concept of a function to describe it. He denoted the quantity of a good by x, and the total utility derived from consuming this good by u. Since the value of utility depends on the amount consumed, u is a function of x, i.e., $u:=u\left(x\right)$. If the amount of consumption changes by $\Delta x$, and the corresponding change in utility is $\Delta u$, we can calculate the ratio ${\displaystyle \frac{\Delta u}{\Delta x}}$, which Jevons referred to as the degree of utility. By transitioning from the ratio of differences to the derivative of the function $u\left(x\right)$ with respect to x, we obtain the expression ${\displaystyle \frac{du\left(x\right)}{dx}}$, which Jevons called the final degree of utility. According to the law of diminishing marginal utility, the expression ${\displaystyle \frac{du\left(x\right)}{dx}}$ is a decreasing function of x, which in differential calculus terms means that the second derivative of $u\left(x\right)$ with respect to x must be negative, i.e., ${\displaystyle \frac{d^{2}u\left(x\right)}{d{x}^2}}<0$. The final degree of utility, or the derivative ${\displaystyle \frac{du\left(x\right)}{dx}}$, is decreasing but does not necessarily intersect the $Ox$ axis (the line $y=0$ may be a horizontal asymptote to the graph of ${\displaystyle \frac{du\left(x\right)}{dx}}$). However, for some goods, the final degree of utility may, after exceeding a certain threshold of consumption, begin to take on negative values, meaning that consuming each additional unit of the good would be associated not with satisfaction but with displeasure.

In his book The Theory of Political Economy, Jevons considered the various situations a consumer might face and, using differential calculus, determined the optimal situation from the perspective of utility. An interesting example analyzed by Jevons is the situation where a consumer has a certain amount of a good and intends to use it over a period of n days. The amount of the good allocated for consumption on the i-th day is denoted by $x_i$ ($i=1,\dots ,n$). The final degree of utility for each day is expressed by the derivatives:

$${\nu }_1={\left.{\displaystyle \frac{du\left(x\right)}{dx}}\right|}_{x=x_1},{\nu }_2={\left.{\displaystyle \frac{du\left(x\right)}{dx}}\right|}_{x=x_2},\dots ,{\nu }_n={\left.{\displaystyle \frac{du\left(x\right)}{dx}}\right|}_{x=x_n}.$$

The optimal allocation of the good is influenced by the probability of a need occurring on a given day, which the good satisfies, and the reduction in the value of future consumption due to the time perspective. Let the probability of the need being satisfied by the good on the i-th day be denoted by $p_i$, and the reduction in satisfaction related to delaying consumption be described by the set of weights: $q_1,q_2,\cdots ,q_n$. According to Jevons, the optimal (most beneficial for the consumer) allocation of the good occurs when the equality is satisfied:

$${\nu }_1{p}_1{q}_1={\nu }_2{p}_2{q}_2=\cdots ={\nu }_n{p}_n{q}_n,$$

which again reflects the idea of symmetry as an inherent concept behind optimal solutions in economics.

Jevons also sought the condition for optimal consumption in a situation where a person has several goods at their disposal and derives some utility from each. According to Jevons, a consumer reaches equilibrium when the final degree of utility for each good is the same, which can be graphically determined as the point of intersection of the curves of final degrees of utility. Interestingly, the conclusion formulated by Jevons is identical to the Second Law of Gossen, even though the author of The Theory of Political Economy was not familiar with Gossen’s book at the time he wrote his work [35].

Jevons applied his theory of utility to the labor market as well. According to the English scholar, performing work is associated with effort and accompanying displeasure. For the work performed, a person receives a wage, which can be assigned a certain utility (related to the goods it can be exchanged for). As long as the utility derived from the received wage exceeds the displeasure associated with the duties, the person will continue to work. To determine the supply of labor, one must plot the curve of diminishing utility and the curve of negative utility of labor per unit of output on the same graph. Jevons’s theory of labor supply is considered an original concept that enriched neoclassical economics [3].

2.2. Alfred Marshall and the Differential Definition of Marginal Utility

Alfred Marshall was born on 26 July 1842, in Bermondsey, near London. From 1861 to 1868, he studied at St John’s College at the University of Cambridge. After his studies, he worked for a time at his alma mater, and later at Newnham College in Cambridge, University College in Bristol, and Balliol College at Oxford University. From 1885 to 1908, he served as a Professor of Political Economy at King’s College, University of Cambridge.

In 1890, his book Principles of Economics was published, which is still considered the most important work of A. Marshall. In this book, the English scholar demonstrated his support for using geometric tools to explain economic laws, as evidenced by the inclusion of 45 diagrams. Interestingly, these diagrams were not included in the main text of the book but were placed in the footnotes. The main text has a literary character, without algebraic formulas, allowing the author to reach a broader audience with his theory. A. Marshall was a declared opponent of mathematical economics and believed that the diagrams presented in the work had nothing to do with mathematics [34]. Mathematical formulas related to the economic phenomena described in the book were collected in a separate mathematical appendix at the end of the book [36].

According to A. Marshall’s views, the fundamental economic category is the market, and the central problem of economics should be the analysis of demand and supply. To explain and describe the issue of demand, A. Marshall applied and developed the theory of marginal utility introduced by W. S. Jevons. For mathematical economics, the introduction of the definition of marginal utility based on the concept of the first-order differential is interesting. If the total utility is described by a function $u\left(x\right)$ dependent on the quantity of consumption denoted by x, then the marginal utility resulting from an increase in the consumption of the good by $\delta x$ is expressed as ${\displaystyle \frac{du\left(x\right)}{dx}}\delta x$ [36]. By referring to the first-order differential, A. Marshall expanded on the way of discussing utility introduced by W. S. Jevons, who used the concept of the derivative of the utility function with respect to the quantity of consumption: ${\displaystyle \frac{du\left(x\right)}{dx}}$, calling this mathematical expression the final degree of utility.

2.3. Francis Ysidro Edgeworth and the Mathematical Description of Indifference Curves

Francis Ysidro Edgeworth was born on 8 February 1845, in Edgeworthstown, Ireland. He completed his studies in humanities and philosophy at Balliol College, Oxford University. He also studied law at the University of London. He co-founded (along with A. Marshall) the academic journal Economic Journal and became its first editor in 1891. He held this position until his death in 1926. The role of editor of Economic Journal was taken over by J. M. Keynes.

F. Y. Edgeworth’s main contribution to mathematical economics is related to the use of differential calculus of functions of two variables to describe the total utility resulting from the consumption of two goods. Assuming that x and y represent the quantities of two goods, we can introduce a function describing total utility:

$$U:=F(x,y).$$

The effect of changes in the quantities of individual goods on total utility is described by the partial derivatives: ${\displaystyle \frac{\partial F(x,y)}{\partial x}}$ and ${\displaystyle \frac{\partial F(x,y)}{\partial y}}$. F. Y. Edgeworth postulated the existence of “indifference curves,” which are sets of different combinations of the quantities of both goods such that total utility remains unchanged. Graphically, an indifference curve is a set of points $(x,y)$ that correspond to the same level of utility. To analytically determine the set of points $(x,y)$ that form an indifference curve, one must solve the total differential equation:

$${\displaystyle \frac{\partial F(x,y)}{\partial x}}dx+{\displaystyle \frac{\partial F(x,y)}{\partial y}}dy=0.$$

A consumer, when choosing the quantities of both goods, must consider the existence of a budget constraint. If m denotes the total amount that the consumer has to allocate between the two goods, and $p_1$ and $p_2$ are the prices per unit of these goods, then the set of all points $(x,y)$ representing the quantities of goods that the consumer can purchase is described by the following equation:

$$p_{1}x+p_{2}y=m,$$

which graphically represents a line known as the budget line. The consumer achieves an optimum when they allocate their budget in such a way that maximizes their utility. Geometrically, the consumer’s optimum is the point of tangency between the budget line and the indifference curve corresponding to the highest utility attainable within the given budget.

The model of consumer equilibrium described by the introduction of indifference curves and the budget line has become a fundamental topic in economics and represents one of the main achievements of F. Y. Edgeworth [37].

3. Mathematics as a Fundamental Source of Research Methods for the Mathematical School

As previously mentioned, the marginalist revolution resulted in the emergence of three significant schools of economic thought: neoclassical, psychological, and mathematical. The previous section covered the leaders of the neoclassical school, also known as the Anglo-American school, and selected findings of their research, which are important from the perspective of contemporary mathematical economics. The psychological school, despite the significant contribution made by its main representative—Carl Menger, is not the focus of this discussion due to the lack of methods and results relevant to mathematical economics. On the other hand, the third school adopted mathematics as a source of methods for solving economic problems, which is why the third subsection is dedicated to two of the most important representatives of this trend—Léon Walras and Vilfredo Pareto. Both scholars worked at the University of Lausanne, where they held professorships. For this reason, the mathematical school is also known as the “Lausanne School in Economics”. Given that the principal concept of the mathematical school was general equilibrium theory, the economic trend initiated by Léon Walras is also referred to as the “general equilibrium school”.

3.1. Léon Walras and General Equilibrium Theory

Léon Marie Esprit Walras was born on 16 December 1834, and is listed alongside Carl Menger and William Stanley Jevons as one of the three most prominent representatives of the subjective–marginalist direction (also known as the marginalist revolution) [38]. L. M. E. Walras studied engineering at the Ecole des Mines in Paris. Walras’s economic views were shaped under the strong influence of his father—Auguste Antoine Walras, who was an economist by passion and a schoolmate of A. A. Cournot. It was the reading of A. A. Cournot’s mathematical works that had a key impact on the development of L. M. E. Walras’s academic career. In 1870, the French scholar was appointed professor at the University of Lausanne and took over the newly established chair of political economy.

In his career, he published several books, of which two are considered the most important: Éléments d’économie politique pure (eng. Elements of Pure Political Economy) published in 1874 and Théorie mathématique de la richesse sociale published in 1883 (translation: Mathematical Theory of Social Wealth).

In his theory, L. M. E. Walras divided economics into three directions: pure political economy, applied political economy, and social economy. According to the founder of the mathematical school, pure political economy should be a precise, theoretical, and abstract science. The goal of pure economics should be to uncover existing economic relationships and laws and describe them in the most general way, creating general models and ideal types. Pure economics should focus on the issues of exchangeability and exchange value, studying the relationship between things [3].

The flagship achievement of the mathematical school is the proposal of general equilibrium theory. In the static model, the standard problem of equilibrium involves finding such values of endogenous variables to ensure equilibrium for the economy. The main assumption regarding equilibrium is that equilibrium exists in the market if and only if demand equals supply, meaning there is neither a surplus nor a shortage in the market.

In the simplest case, when there is only one good in the market, it is sufficient to introduce three variables into the model: the quantity demanded of the good ($Q_d$), the quantity supplied of the good ($Q_s$), and the price of the good (P). To find the answer to when equilibrium will occur in the market, it is assumed that $Q_d$ is a decreasing linear function of price (since demand decreases with an increase in price), and $Q_s$ is an increasing linear function of price with a negative y-intercept. Mathematically, equilibrium is described by the following system of equations:

$$\left\{\begin{array}{c}{Q}_d-Q_s=0,\hfill \\ \\ Q_d=a-bP,\hfill \\ \\ Q_s=-c+dP,\hfill \end{array}\right.$$

(16)

where $a,b,c,d$ are positive parameters.

In the case of a simple market model described by the linear system of Equation (16), finding the equilibrium involves solving the system and calculating such values $(\overline{P},{\overline{Q}}_d,{\overline{Q}}_s)$ for which all three equations are satisfied.

The case of a single good in the market with the assumption of linear demand and supply functions is the simplest possible model of market equilibrium. This model can be transformed into more advanced forms; for example, by declaring demand or supply functions in a different form (e.g., a polynomial of higher degree) [39].

A more general market model might be a situation where there are two interrelated goods. Assuming that demand and supply functions are linear, such a model can be parameterized using six equations:

$$\left\{\begin{array}{c}{Q}_{d1}-Q_{s1}=0,\hfill \\ \\ Q_{d1}=a_0+a_1{P}_1+a_2{P}_2,\hfill \\ \\ Q_{s1}=b_0+b_1{P}_1+b_2{P}_2,\hfill \\ \\ Q_{d2}-Q_{s2}=0,\hfill \\ \\ Q_{d2}={\alpha }_0+{\alpha }_1{P}_1+{\alpha }_2{P}_2,\hfill \\ \\ Q_{s2}={\beta }_0+{\beta }_1{P}_1+{\beta }_2{P}_2.\hfill \end{array}\right.$$

(17)

Solving the system of linear Equation (17) allows for the determination of such prices and quantities of both goods that will ensure market equilibrium.

L. M. E. Walras, analyzing a market where two goods are sold, saw the need to connect the issue of market equilibrium with utility theory. A market participant, deciding in what proportions to purchase the two goods A and B, achieves maximum utility when the good A provides him with an intensive utility equal to the product of the utility of good B and the price of good A expressed in terms of good B. In other words, the price of good A relative to good B must equal the ratio of the utilities provided by both goods.

The founder of the mathematical school studied the problem of achieving optimal utility by two consumers making choices about the quantities of two goods A and B. Let $q_{ij}$ denote the quantity of the good, where i is the consumer number ($i=1,2$), and j denotes the good ($j=A,B$). Intensive utility will be denoted by $\varphi$. To achieve maximum utility, the i-th consumer changes the quantities of goods A and B by $x_i$ and $y_i$, respectively. The situation of the first market participant can be described by the following equations:

$${\displaystyle \frac{{\varphi }_{1,A}(q_{1,A}+x_1)}{{\varphi }_{1,B}(q_{1,B}+y_1)}}=p_{a,b},$$

(18a)

$$y_1=-p_{a,b}{x}_1,$$

(18b)

which, after transformations, give the following system of two equations:

$${\varphi }_{1,A}(q_{1,A}+x_1)=p_{a,b}{\varphi }_{1,B}(q_{1,B}-p_{a,b}{x}_1),$$

(19a)

$${\varphi }_{1,B}(q_{1,B}+y_1)=p_{b,a}{\varphi }_{1,A}(q_{1,A}-p_{b,a}{y}_1).$$

(19b)

The solutions to this system are values $x_1$ and $y_1$ expressed as a price-dependent demand function:

$$x_1=F_{1,a}\left(p_a\right),\mathrm{and}{y}_1=F_{1,b}\left(p_b\right).$$

(20)

Similar equations can be written and solved for the second consumer operating within the same market:

$$x_2=F_{2,a}\left(p_a\right),\mathrm{and}{y}_2=F_{2,b}\left(p_b\right).$$

(21)

For market equilibrium to occur, all units must simultaneously achieve maximum satisfaction from consumption. This condition will be satisfied when the equilibrium price is established in the market, expressed as the following:

$$p_{a,b}={\displaystyle \frac{r_{1,a}}{r_{1,b}}}={\displaystyle \frac{r_{2,a}}{r_{2,b}}},$$

(22)

where $r_{1,a},r_{1,b,\cdots }$ denote the intensive utility.

In the general case, L. M. E. Walras considered a market where the redistribution of m goods takes place. Let $D_{ij}$ denote the demand for good i in exchange for good j, and $p_{ij}$ denote the price of good i relative to j. Partial demand for each good depends on relative prices. We have $m(m-1)$ equations describing the demand:

$$D_{ij}=F_{ij}(p_{1i},p_{2i},\dots ,p_{mi}),i\ne j.$$

(23)

An important element of L. M. E. Walras’s analysis was the introduction of money into the market, fulfilled by one of the goods (e.g., the first). Instead of relative prices with double indices, prices relative to the first good can be used, denoted as $p_2,p_3,\cdots ,p_m$. Each consumer in the market will strive to maximize utility from the consumption of $m-1$ goods:

$${\varphi }_{2,1}(q_{2,1}+y_1)=p_2{\varphi }_{1,1}(q_{1,1}+x_1),$$

$$\begin{array}{c}{\varphi }_{3,1}(q_{3,1}+z_1)=p_3{\varphi }_{1,1}(q_{1,1}+x_1),\\ \cdots \end{array}$$

The above system of equations shows that the intensive utility of each good is a function dependent on the amount of all goods held by a given consumer. For the first unit operating in the market, we have the following expression:

$$\varphi (q_{1,1}+x_1,q_{2,1}+y_1,q_{3,1}+z_1,\dots ).$$

Each market participant has limited freedom to exchange goods. Increasing the quantity of one good requires giving up something else. L. M. E. Walras proposed the balance equation for the unit. For the first consumer, we have the following:

$$x_1+y_1{p}_2+z_1{p}_3+\cdots =0.$$

(24)

Solving the $m-1$ equations concerning intensive utility with the budget constraint (24) allows for obtaining the optimal values for each good:

$$y_1=F_{2,1}(p_2,p_3,\cdots ,p_m),$$

$$\begin{gathered} z_1=F_{2,1}(p_2,p_3,\cdots ,p_m), \\ \cdots \end{gathered}$$

Summing the quantities of a given good for all market participants, we obtain the equalities:

$$x_1+x_2+x_3+\cdots =X,$$

$$y_1+y_2+y_3+\cdots =Y,$$

$$\begin{gathered} z_1+z_2+z_3+\cdots =Z, \\ \cdots \end{gathered}$$

Next, each of these quantities can be expressed as a value of the demand function, i.e.,:

$$Y=F_b(p_2,p_3,\cdots ,p_m)=0,$$

$$\begin{gathered} Z=F_c(p_2,p_3,\cdots ,p_m)=0, \\ \cdots \end{gathered}$$

Achieving market equilibrium requires that the demand for each good equals the effective supply, so the sum of the quantities of each good exchanged among all market participants must be equal to zero.

Based on his analysis, the founder of the mathematical school formulated general laws regarding the relationship between the formation of prices in the market and the utility of goods. According to L. M. E. Walras, in market equilibrium conditions, an increase in the utility of a certain good leads to an increase in its price, while a decrease in the utility of a good implies a decrease in its price. L. M. E. Walras’s observations are consistent with the law of demand and supply.

Before formulating his general theory of equilibrium, L. M. E. Walras became interested in mathematical economics through the works of A. A. Cournot. The French scholar, a pioneer in the application of mathematical methods in economics, served as an inspiration for the founder of the mathematical school in economics. L. M. E. Walras continued the work initiated by A. A. Cournot and is compared with him not only regarding the methods of reasoning used but also in terms of personal characteristics. Both scholars never sought fame and did not expect recognition for their work. They wanted their ideas to be disseminated because they strongly believed in the validity of their reasoning and in the potential application of their results in practice to increase welfare [40].

In 1905, L. M. E. Walras was nominated for the Nobel Peace Prize in a letter sent to the Norwegian Nobel Committee. The letter was signed by three scholars—Ernest Roguin, Alexandre Maurer, and Maurice Millioud [41]. L. M. E. Walras’s main achievements in the context of the Nobel Peace Prize were highlighted, including his commitment to free international trade, which could potentially eliminate the problem of world hunger and war as a method of conflict resolution. L. M. E. Walras actively engaged in promoting his candidacy for the Nobel Peace Prize and regularly sent his books, publications, and articles by other scholars who appreciated and praised his scientific work to the Nobel Committee until the end of his life [41]. However, L. M. E. Walras never received the Nobel Peace Prize.

3.2. Vilfredo Pareto and the Study of Wealth Distribution and the Concept of Efficiency in General Equilibrium Theory

The continuation of the mathematical school at the University of Lausanne was carried out by Vilfredo Pareto, born 15 July 1848, in Paris, an Italian economist and sociologist. From 1893, he held a professorship at the University of Lausanne. V. Pareto contributed to the application of mathematical methods in economics and developed the theory of general economic equilibrium introduced by L. Walras. He also engaged with the concept of indifference curves proposed by F. Y. Edgeworth and conducted research related to the distribution of wealth.

Although V. Pareto was born in Paris (his mother was French), in 1858 he was sent to Italy, his father’s country of origin, to complete his education, which culminated in a Doctorate in Engineering in 1869. After finishing his education, V. Pareto worked in various managerial positions in industry, eventually becoming the president of the Italian Iron Works [42]. In 1893, V. Pareto succeeded L. M. E. Walras as a professor at the University of Lausanne. Due to his educational background and experience, V. Pareto possessed high-level mathematical training and also had a thorough understanding of business practice from his role as a senior executive in industry. In addition to these two attributes, his success as an economist was also influenced by his deep interest in economic policy and economics long before he assumed his professorship in 1893.

It is V. Pareto, not L. M. E. Walras, who is credited with establishing the mathematical school in economics in the full sense of the term. This school consisted of a close circle of collaborators made up of distinguished economists, a broader circle of followers including less prominent researchers, and a wide group of mainly young people aspiring to the academic world and fascinated by V. Pareto’s work. All these individuals collaborated, generating added value through synergy. They adhered to a single doctrine and recognized the leadership of one leader [42].

4. Mathematical Methods and the Development of Economic Growth Theory

One of the fundamental problems in macroeconomics is explaining the temporal and spatial variation in the wealth of nations. The first attempts to identify the determinants influencing the pace of economic development were made in the 18th and 19th centuries. The prominent economists of that period, such as Adam Smith, Thomas Malthus, and David Ricardo, tried to explain the causes of economic growth.

In the 20th century, one way to address the question of factors influencing the rate of economic growth became the construction of mathematical models of economic growth. This section will present an overview of mathematical growth models starting from 1939, when Roy Harrod’s work was published. The focus here will be on the assumptions, mathematical methods used, and the basic conclusions derived from the model. The models presented have been selected based on their historical significance and their role in shaping the fundamental understanding of economic growth. These models offer a structured approach to capturing the dynamic nature of economic development, and their mathematical formulations have provided critical insights into both short-term fluctuations and long-term growth trends. Given the extensive literature on the subject, this section on mathematical growth models is not exhaustive, but aims to synthesize knowledge in this area, with particular emphasis on the role of mathematical methods.

The first models discussed will be Keynesian growth models—the Harrod model and the Domar model. Their development was related to the fact that the theoretical system created by John Maynard Keynes was static, creating a need to introduce elements of dynamic analysis.

Next, three neoclassical growth models will be described—the Solow–Swan model, the Ramsey–Cass–Koopmans model, and the Diamond model. All of these models feature the introduction of the concept of technological level in an exogenous manner and use it to explain the rate of economic growth. A key feature of these models is the underlying symmetry they assume in economic interactions and resource allocation. Symmetry appears critical for ensuring that the mathematical models are balanced, allowing for the identification of equilibrium states where factors like capital and labor evolve consistently over time. In this context, symmetry is not merely a mathematical tool but a reflection of economic stability, enabling models to accurately capture the proportional relationship between inputs and outputs in growth dynamics.

The section will then cover two endogenous growth models—the learning-by-doing model by Romer and the Lucas model. Endogenous growth is achieved by departing from the neoclassical production function, which assumed diminishing returns to reproducible factors of production. In endogenous growth models, it is assumed that returns to such factors are at least constant.

In the final part of the section on mathematical growth models, two models from the so-called new growth theory will be discussed—the Mankiw–Romer–Weil growth model and the Nonneman–Vanhoudt model. Both of these models were developed in the 1990s and relate to neoclassical growth concepts. They can be considered contemporary generalizations (and extensions) of the Solow–Swan growth model.

The concept of symmetry also holds significance in the newer growth models, where it helps to explain how balanced development across different sectors or regions can lead to more sustainable long-term economic expansion. This intrinsic symmetry in economic structures forms the basis for many valid mathematical models that describe growth, ensuring they are adaptable to various real-world scenarios without losing their predictive accuracy.

There are many review studies in the literature on growth models. In addition to the original research papers by the model creators, several key reviews have been used to write the present study (see Refs. [39,43,44,45,46,47]). Despite the rich literature, there is no single widely accepted classification of these models. Therefore, the classification adopted in this section, into Keynesian, neoclassical, endogenous, and new growth theory models, should be considered one of the possible classifications.

4.1. Harrod’s Growth Model (1939)

The first growth model to be discussed was proposed in 1939 by Roy Harrod, a British economist associated with the University of Cambridge and the University of Oxford [48]. Harrod, a long-time collaborator of John Maynard Keynes, developed a model that aligns with the Keynesian economic framework [49]. The Harrod model served as a precursor to later, more advanced models of economic growth.

The Harrod model is based on five assumptions.

• Output (Y) is a function of capital (K):

  $$Y=f\left(K\right).$$
• The marginal product of capital is constant. This implies that the marginal productivity of capital equals the average. Mathematically, this is expressed using the derivative:

  $${\displaystyle \frac{dY}{dK}}=c=const.$$
• Capital is necessary for production:

  $$f\left(0\right)=0.$$
• Investment equals total savings (S), which can also be expressed as the product of the savings rate s and output Y:

  $$I=S=sY.$$
• Changes in capital are expressed as the difference between investment and depreciation:

  $$\Delta K=I-\delta K,$$

  where $\delta$ represents the depreciation rate of capital.

To explain the causes and factors influencing economic growth, Harrod distinguished between three types of economic growth:

• G: actual economic growth (actual growth rate)—the growth that actually occurs in the economy;
• $G_w$: warranted economic growth (warranted growth rate)—the rate of output growth at which entrepreneurs do not change the level of investment, as they are confident about meeting future consumer demand;
• $G_{\sigma }$: natural economic growth (natural growth rate)—the rate of growth at which total demand in the economy equals potential supply.

The fundamental equation of Harrod’s model pertains to the growth rate G (actual growth rate of output). The simplest form of this equation is as follows:

$$G={\displaystyle \frac{s}{C}},$$

(25)

where the symbols represent the following:

• s—savings rate, the fraction of income that individual consumers and corporations allocate to savings;
• C—the amount of capital required to produce one unit of output, known as the capital–output ratio.

Equation (25) was presented in Harrod’s 1939 work in several more general forms, incorporating other factors that might influence economic growth (e.g., a factor describing international capital flows expressed as the difference between imports and exports).

The capital–output ratio C can be expressed as the ratio of the change in capital $\Delta K=I$ to the change in output:

$$C={\displaystyle \frac{\Delta K}{\Delta Y}}.$$

Using this property and the assumption that $S=I$, Equation (25) can be easily justified by writing the following:

$$G={\displaystyle \frac{\Delta Y}{Y}}={\displaystyle \frac{\frac{S}{Y}}{\frac{I}{\Delta Y}}}={\displaystyle \frac{s}{C}}.$$

(26)

R. Harrod expressed the relationship in Equation (26) equivalently as follows:

$${\displaystyle \frac{S}{Y}}={\displaystyle \frac{\Delta Y}{Y}}{\displaystyle \frac{I}{\Delta Y}}.$$

(27)

By modifying Equation (25), a similar formula can be derived for the warranted growth rate:

$$G_w={\displaystyle \frac{s_d}{C_r}},$$

(28)

where the symbols are interpreted as follows:

• $s_d$—the portion of income that individuals and firms wish to save to ensure further development;
• $C_r$—the amount of capital required to increase output by one unit.

The characteristic feature of Equation (28) is its instability, which manifests in the fact that if output deviates from the equilibrium level, it results in changes in prices to restore equilibrium. Thus, the equilibrium is unstable.

For the natural rate of economic growth, R. Harrod formulated the following equation:

$$G_{\sigma }={\displaystyle \frac{s_{\sigma }}{C_r}},$$

(29)

where the symbol $\sigma$ is understood as “optimal”. Equation (29) was rearranged to determine the optimal savings rate:

$$s_{\sigma }=G_{\sigma }{C}_r.$$

(30)

In his works, R. Harrod emphasized the need for economic planning to align the actual rate of return with the natural (optimal) rate. Public authorities have at their disposal instruments of monetary and fiscal policy. Through these tools, they can influence the level of interest rates, thereby affecting aggregate demand.

4.2. Domar’s Growth Model (1946)

Evsay David Domar, an American economist of Polish origin (born in 1914 in Łódź), developed the Keynesian theory of economic growth. His most important contributions in this area include the article Capital Expansion, Rate of Growth, and Employment published in 1946 [50] and the book Essays in the Theory of Economic Growth [51] published in 1957. The results presented by E. D. Domar concerning economic growth theory are commonly referred to as the “Domar growth model”.

E. D. Domar focused his analysis on the relationship between investment and economic growth. He emphasized that the investment process has a dual impact on the economy—on the one hand, investments lead to a production effect by increasing the productive capacity of the economy, and on the other hand, they trigger an income effect. The production effect is achievable only through investment; thus, investments should be seen as a necessary and sufficient condition for economic development. The income effect ensures that the new production capabilities that arise in the economy are met with demand. Therefore, it can be stated that both effects caused by an increase in investment levels are complementary in the economy.

The production effect (also known as the supply effect) of investment expenditures $\sigma$ was described by E. D. Domar using the following equation:

$${\displaystyle \frac{dP}{dt}}=\sigma I,$$

(31)

where I denotes investment, and P denotes productive capacity.

The income effect (also known as the demand effect) is described by the following equation:

$${\displaystyle \frac{dY}{dt}}={\displaystyle \frac{dI}{dt}}·{\displaystyle \frac{1}{s^{\prime }}},$$

(32)

where $s^{\prime }$ denotes the marginal savings rate (E. D. Domar assumed a constant savings rate). The value of investment is treated as a function of time.

When the economy is in equilibrium, productive capacity equals total demand, i.e., $P=Y$. An increase in investment leads to an increase in productive capacity. To maintain the economy in equilibrium, the growth rate of productive capacity resulting from the supply effect must equal the growth rate of total demand:

$${\displaystyle \frac{dP}{dt}}={\displaystyle \frac{dY}{dt}}.$$

(33)

Using Equations (31) and (32), the equilibrium condition for the entire economy expressed by Formula (33) can be written as follows:

$$\sigma I={\displaystyle \frac{dI}{dt}}·{\displaystyle \frac{1}{s^{\prime }}}.$$

(34)

The equation can be easily transformed into the following form:

$${\displaystyle \frac{\dot{I}\left(t\right)}{I\left(t\right)}}=s^{\prime }\sigma ,$$

(35)

where $\dot{I}\left(t\right)={\displaystyle \frac{dI\left(t\right)}{dt}}$. For simplicity, a dot over a symbol is used to denote the time derivative. This is a common way to express the time derivative, especially in physics and engineering.

Equation (35) is a simple first-order differential equation, the solution of which is an exponential function:

$$I\left(t\right)=I\left(t_0\right)e^{s^{\prime }\sigma (t-t_0)}.$$

(36)

Equation (36) contains the main conclusion derived from the Domar model—sustainable economic growth is possible when investment growth is exponential, with a growth rate equal to the product of the average productivity of investment and the marginal savings rate. From Equations (31) and (32), it follows that production and income must also exhibit exponential growth (though with different growth rates resulting from Equations (31) and (32)).

4.3. Solow–Swan Growth Model (1956)

In the Solow–Swan growth model, a key assumption is that in addition to physical capital (such as equipment, machinery), there exists so-called human capital—a broad concept encompassing the knowledge and skills stored in the minds of society. The Solow–Swan model is the most well-known growth model that takes into account the role of knowledge. Originally introduced in two works by Robert Solow [52] and Trevor Swan [53], it was later described in numerous books such as Ref. [47] as well as generalized in [54,55].

In this model, four variables are involved in economic growth: K—capital, Y—output, L—labor (workforce), and A—knowledge. Knowledge, although understood in a very general sense, possesses certain characteristics that distinguish it. Most notably, access to knowledge is extremely free today; for example, the latest scientific publications are readily available online. Moreover, many people in different places around the world can use the same piece of knowledge simultaneously, which is not possible with resources such as machinery.

The fundamental function in the Solow–Swan model is the production function (denoted by the letter F), which, at any point in time t, allows the calculation of output $Y\left(t\right)$ (expressed in monetary terms) based on $K\left(t\right)$, $L\left(t\right)$, and $A\left(t\right)$, which can be written as follows:

$$Y\left(t\right)=F\left(K\right(t),L(t),A(t\left)\right).$$

(37)

Moreover, it is assumed that the function F does not directly depend on time, and the product of labor and knowledge constitutes a single argument, called effective labor. Another assumption concerning the function F is that it is a homogeneous function of degree one, which means

$$\forall \alpha \in \mathbb{R}\forall t\in {\mathbb{R}}_{+}F(\alpha K\left(t\right),\alpha A\left(t\right)L\left(t\right))=\alpha F(K\left(t\right),A\left(t\right)L\left(t\right)).$$

(38)

The assumption about the function F mathematically formulated in Equation (38) economically relates to the theory of production scale, specifically to the case of constant returns to scale.

The next step is to define the capital and output per unit of effective labor. These quantities are denoted by small letters:

$$k\left(t\right)\equiv {\displaystyle \frac{K\left(t\right)}{A\left(t\right)L\left(t\right)}},y\left(t\right)\equiv {\displaystyle \frac{Y\left(t\right)}{A\left(t\right)L\left(t\right)}}.$$

(39)

Substituting (39) into (37), we obtain

$$y\left(t\right)=F\left(k\right(t),1)=f\left(k\right(t\left)\right).$$

(40)

A particular example of a production function is the Cobb–Douglas function [43]:

$$F(K,AL)=K^{\alpha }{\left(AL\right)}^{1-\alpha },\mathrm{where}0<\alpha <1.$$

(41)

This function is simple to apply and satisfies the assumption of constant returns to scale. It is also easy to verify that by substituting the variables according to (39), we obtain

$$f\left(k\right)\equiv F\left({\displaystyle \frac{K}{AL}},1\right)={\left({\displaystyle \frac{K}{AL}}\right)}^{\alpha }=k^{\alpha }.$$

(42)

Since the increase in capital equals investments (savings) minus depreciation, we obtain the differential equation:

$$\dot{K}\left(t\right)=sY\left(t\right)-\delta K\left(t\right),$$

(43)

where s denotes the savings rate, $\delta$ the rate of capital depreciation, and the dot represents differentiation with respect to time.

Equation (43) can be rewritten in the variables defined in (39) and assuming that knowledge A and labor L grow at a constant rate, i.e.,:

$${\displaystyle \frac{\dot{A}\left(t\right)}{A\left(t\right)}}=g,{\displaystyle \frac{\dot{L}\left(t\right)}{L\left(t\right)}}=n,$$

(44)

where n and g are certain positive constants independent of time.

In other words, this assumption states that the growth of knowledge and labor over time is exponential. Solving the differential Equation (44), we easily obtain

$$A\left(t\right)=A\left(0\right)e^{gt},L\left(t\right)=L\left(0\right)e^{nt}.$$

(45)

The assumption of exponential knowledge growth underlies macroeconomic growth theories and was justified in the 1960s by research by D. J. de Solla Prince.

Thanks to the assumptions (39) and (44), Equation (43) can be rewritten by calculating $\dot{k}\left(t\right)$:

$$\dot{k}\left(t\right)=sf\left(k\left(t\right)\right)-(\delta +g+n)k\left(t\right).$$

(46)

The long-term equilibrium state is found at the intersection of the line $(\delta +g+n)k$ with the function $sf\left(k\right)$. This is the so-called steady state because the capital per unit of effective labor does not change ($\dot{k}\left(t\right)=0$).

It remains to check how in the long-term equilibrium state the total GDP (given by the production function $Y=F(K,AL)$) and GDP per capita ${\displaystyle \frac{Y}{L}}$ change. To do this, it is enough to differentiate, with respect to time, the equations $Y\equiv f\left(k\right)AL$ and ${\displaystyle \frac{Y}{L}}\equiv f\left(k\right)A$.

$${\displaystyle \frac{\dot{Y}}{Y}}={\displaystyle \frac{\dot{f}\left(k\right)}{f\left(k\right)}}+{\displaystyle \frac{\dot{A}}{A}}+{\displaystyle \frac{\dot{L}}{L}}\mathrm{and}{\displaystyle \frac{\dot{\left(\frac{Y}{L}\right)}}{\frac{Y}{L}}}={\displaystyle \frac{\dot{f}\left(k\right)}{f\left(k\right)}}+{\displaystyle \frac{\dot{A}}{A}},$$

(47)

which, for the equilibrium point, simplifies to

$${\displaystyle \frac{\dot{Y}}{Y}}=g+n\mathrm{and}{\displaystyle \frac{\dot{\left(\frac{Y}{L}\right)}}{\frac{Y}{L}}}=g.$$

(48)

The above analysis allows us to draw two important conclusions regarding the equilibrium state in the Solow-Swan model:

• The growth rate of GDP is equal to the sum of the growth rates of knowledge and labor.
• The growth rate of GDP per capita is equal to the growth rate of knowledge.

The Solow–Swan model, although very simplified, directly shows the relationship between GDP growth rate and knowledge growth. However, the concept of knowledge is not unambiguous. In the context of the Solow–Swan model, terms like ’technology’ or ’technological level’ are also used. Undoubtedly, there are many quantitative indicators that indirectly inform about the state of knowledge in the economy. However, there is a need to introduce a precise definition of knowledge and determine how it should be measured; for more see, for example, Ref. [56] (in Polish).

4.4. Ramsey–Cass–Koopmans Growth Model (1965)

Frank Ramsey, a British economist, published a paper on the optimal level of savings in 1928 [57]. The concepts proposed by Ramsey were further developed in 1965 in articles by David Cass [58] and Tjalling Koopmans [59].

Similar to the Solow–Swan model, it is assumed that the technological level, expressed by the function $A\left(t\right)$, and labor $L\left(t\right)$ change over time with constant growth rates of g and n, respectively. This implies that the time variation of the functions $A\left(t\right)$ and $L\left(t\right)$ is given by exponential formulas:

$$A\left(t\right)=A\left(0\right)e^{gt},L\left(t\right)=L\left(0\right)e^{nt}.$$

(49)

The production activities of firms are described by a production function analogous to that in the Solow–Swan model: $F\left(K\right(t),A(t\left)L\right(t\left)\right)$. The firm purchases labor from households at a wage rate w and capital at a cost of $\delta +r$ (where r represents the interest rate). From the firm’s perspective, the goal is to maximize profit, which is calculated as

$$I=F\left(K\right(t),A(t\left)L\right(t\left)\right)-(\delta +r)K\left(t\right)-wL\left(t\right).$$

(50)

To determine the values of $K\left(t\right)$ and $L\left(t\right)$ that maximize the firm’s profit, the expression in (50) should be treated as a function of two variables. According to the mathematical method for finding the extrema of a function of two variables, we must consider the system of two conditions:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial I}{\partial K}}=0,\hfill \\ \\ {\displaystyle \frac{\partial I}{\partial L}}=0.\hfill \end{array}\right.$$

(51)

After calculating the partial derivatives and using the fact that ${\displaystyle \frac{\partial F}{\partial K}}=f^{\prime }\left(k\right)$ (where $f\left(k\right)$ is defined according to Equation (42)), we obtain

$$\left\{\begin{array}{c}r=f^{\prime }\left(k\right)-\delta ,\hfill \\ \\ w=A\left(f\left(k\right)-k{f}^{\prime }\left(k\right)\right).\hfill \end{array}\right.$$

(52)

The primary difference between the previously discussed Solow–Swan model and the Ramsey–Cass–Koopmans model lies in the nature of the investment rate. In the earlier model, this rate was exogenous, i.e., it was considered an independent variable used to explain endogenous variables. In the Ramsey–Cass–Koopmans approach, households aim to maximize the utility derived from consumption, and the savings rate is determined endogenously.

Assuming that the economy consists of a fixed number of households (denoted by N), which increase the number of their members at a constant rate n, the utility function of a household can be expressed as follows:

$$U={\int }_{t=0}^{\infty }{e}^{-\rho t}u\left(c_{pc}\right){\displaystyle \frac{L}{N}}dt.$$

(53)

In Equation (53), integration occurs from 0 to infinity, which is associated with the simplifying assumption that human life lasts indefinitely. The other symbols used in Formula (53) represent:

• $\rho$—the rate of time preference ($\rho >0$), where a higher value of $\rho$ indicates that households place greater value on current consumption;
• $u\left(c_{pc}\right)$—the utility derived from per capita consumption;
• ${\displaystyle \frac{L}{N}}$—the average number of members in a household.

The function describing the utility of per capita consumption is increasing and follows the principle of diminishing marginal utility (mathematically: $u^{\prime }\left(c_{pc}\right)>0$ and $u^{″}\left(c_{pc}\right)<0$). Additionally, it is assumed that the function $u\left(c_{pc}\right)$ is of the CRRA type (constant relative risk aversion), i.e.,:

$$u\left(c\right)={\displaystyle \frac{c^{1-\sigma }-1}{1-\sigma }},$$

(54)

where $\sigma >0$ and $\sigma \ne 1$.

Substituting the formula for the time variation of labor: $L\left(t\right)=L\left(0\right)e^{nt}$ into Equation (53), the problem of maximizing utility by a household reduces to finding the maximum of the following integral:

$$U={\int }_0^{\infty }{e}^{(n-\rho )t}{\displaystyle \frac{c_{pc}^{1-\sigma }-1}{1-\sigma }}dt,$$

(55)

subject to the condition that the following equation is satisfied:

$${\dot{k}}_{pc}=w+r{k}_{pc}-c_{pc}-n{k}_{pc},$$

(56)

which states that the rate of change in per capita capital equals the sum of labor income and capital gains minus consumption and the factor related to population growth.

The solution to Equation (55) can be obtained using the Hamiltonian of the current value. Such an analysis leads to the following equations describing the dynamics of households in the Ramsey–Cass–Koopmans model:

$$\left\{\begin{array}{c}{\displaystyle \frac{{\dot{c}}_{pc}}{c_{pc}}}={\displaystyle \frac{r-\rho }{\sigma }},\hfill \\ \\ {\dot{k}}_{pc}=w+r{k}_{pc}-c_{pc}-n{k}_{pc},\hfill \\ \\ \underset{t\to \infty }{lim}\theta \left(0\right)k_{pc}{e}^{(n-r)t}=0,\hfill \end{array}\right.$$

(57)

where the function $\theta$ is conjugate with capital movement and describes how savings in a given period affect utility growth in the subsequent period.

The above equations describing the household side, combined with the equations expressing the dynamics of firms (Equation (52)), allow for a complete description of the economy’s dynamics in the Ramsey–Cass–Koopmans model:

$${\displaystyle \frac{\dot{c}}{c}}={\displaystyle \frac{f^{\prime }\left(k\right)-\delta -\rho -g\sigma }{\sigma }},$$

(58a)

$$\dot{k}=f\left(k\right)-c-(n+g+\delta )k,$$

(58b)

$$\underset{t\to \infty }{lim}\theta \left(0\right)A\left(0\right)k{e}^{(n-f^{\prime }\left(k\right)+\delta +g)t}=0.$$

(58c)

When addressing the question of the causes of economic growth in long-term equilibrium, the Ramsey–Cass–Koopmans model provides the same answer as the Solow–Swan model: the rate of GDP growth per capita equals the rate of technological progress (knowledge growth). Meanwhile, the rate of GDP growth equals the sum of the population growth rate and the rate of technological progress.

The fundamental difference between the Solow–Swan model and the Ramsey–Cass–Koopmans model lies in the fact that the latter is Pareto optimal. The savings rate, introduced into the model endogenously, prevents excessive capital accumulation.

4.5. Diamond Growth Model (1965)

The Diamond growth model is associated with the work of Peter A. Diamond from 1965 [60]. It is significantly closer to reality than the previously discussed Ramsey–Cass–Koopmans model, which assumes that human life lasts infinitely. The Diamond growth model features a finite time horizon and takes demographic changes into account. The lifespan of a household is divided into two periods. During the first, the members of the household work, and their income is divided between consumption and savings. In the second period, when the household members are older and no longer working, consumption is financed from savings plus accrued interest.

Since the lifespan of individuals is divided into two periods, discrete time is used to describe changes in economic indicators, unlike in the previous models where continuous time was assumed. The level of technology and labor force grow at constant rates of g and n, respectively, which can be expressed by the following equations:

$$A_t=A_{t-1}(g+1),L_t=L_{t-1}(1+n).$$

(59)

The mathematical description of entrepreneurs’ behavior in this model is identical to that in the Ramsey–Cass–Koopmans model. However, the approach to describing households is different. In the Ramsey–Cass–Koopmans model, the utility derived from lifetime consumption was expressed as an integral over time from 0 to infinity (see Equation (55)), based on the assumption of an infinite lifespan. In the current model, total utility from consumption is the sum of two components corresponding to utility in youth and maturity. If we denote the rate of time preference by $\rho >0$ (with individuals preferring consumption in the first period over the second), and let $c_{it}$ represent consumption in period t (where i denotes the life period, i.e., $i=1,2$), then the total utility from consumption in both life periods is given by

$$U_t={\displaystyle \frac{c_{1t}^{1-\sigma }-1}{1-\sigma }}+{\displaystyle \frac{1}{1+\rho }}{\displaystyle \frac{c_{2[t+1]}^{1-\sigma }-1}{1-\sigma }}.$$

(60)

We adopt the following notations:

• $w_t$ represents the income earned by a young person in period t;
• $s_t$ denotes the savings rate in period t;
• $r_t$ denotes the interest rate in period t.

According to the model’s assumptions, labor income is divided between consumption in the first period of life $c_{1t}$ and savings $s_t{w}_t$. Consumption in the second period is financed from savings plus interest, i.e.,:

$$c_{2[t+1]}=(1+r_{t+1})s_t{w}_t.$$

(61)

The equation describing the household’s budget constraint can therefore be written as

$$c_{1t}+{\displaystyle \frac{1}{1+r_{t+1}}}{c}_{2[t+1]}=w_t.$$

(62)

The household’s goal is to maximize utility from consumption in both periods of life (as described by Equation (60), subject to the budget constraint given by Equation (62)). In such a situation, the commonly used method in economic theory is to optimize using Lagrange multipliers, which allows us to find the conditional extremum of a function of several variables. For our case, we construct the Lagrangian function as follows:

$$L={\displaystyle \frac{c_{1t}^{1-\sigma }-1}{1-\sigma }}+{\displaystyle \frac{1}{1+\rho }}{\displaystyle \frac{c_{2[t+1]}^{1-\sigma }-1}{1-\sigma }}+\lambda \left(c_{1t}+{\displaystyle \frac{1}{1+r_{t+1}}}{c}_{2[t+1]}-w_t\right).$$

(63)

By writing and solving the corresponding first-order conditions, we obtain an equation that expresses the ratio of consumption in both periods of life:

$${\displaystyle \frac{c_{2[t+1]}}{c_{1t}}}={\left({\displaystyle \frac{1+r_{t+1}}{1+\rho }}\right)}^{{\displaystyle \frac{1}{\sigma }}}.$$

(64)

From Equation (64), we can infer that the level of consumption in the second period of life depends on the relationship between the interest rate $r_{t+1}$ and the rate of time preference $\rho$. Three cases can be distinguished:

• $r_{t+1}=\rho$—then consumption is the same in both periods;
• $r_{t+1}>\rho$—then consumption in the second period is higher than in the first;
• $r_{t+1}<\rho$—then consumption in the second period is lower than in the first.

By solving Equation (64) for $c_{2[t+1]}$ and recalling that $c_{1t}=(1-s_t)w_t$, we obtain the following:

$$c_{2[t+1]}=(1-s_t)w_t{\left({\displaystyle \frac{1+r_{t+1}}{1+\rho }}\right)}^{{\displaystyle \frac{1}{\sigma }}},$$

(65)

which, when substituted into the budget constraint described by Equation (62) and after performing simple algebraic transformations, allows us to derive an expression for the savings rate $s_t$ as a function of the interest rate $r_{t+1}$ in the following form:

$$s_t\left(r_{t+1}\right)={\displaystyle \frac{{(1+r_{t+1})}^{\frac{1-\sigma }{\sigma }}}{{(1+\rho )}^{\frac{1}{\rho }}+{(1+r_{t+1})}^{\frac{1-\sigma }{\sigma }}}}.$$

(66)

By calculating the derivative of the function in Equation (66) with respect to the variable $r_{t+1}$, we obtain the following expression:

$${\displaystyle \frac{d{s}_t\left(r_{t+1}\right)}{d{r}_{t+1}}}={\displaystyle \frac{\frac{1-\sigma }{\sigma }{(1+r_{t+1})}^{\frac{1-2\sigma }{\sigma }}{(1+\rho )}^{\frac{1}{\sigma }}}{{\left({(1+\rho )}^{\frac{1}{\sigma }}+{(1+r_{t+1})}^{\frac{1-\sigma }{\sigma }}\right)}^2}},$$

(67)

from which we can conclude that the monotonicity of the function $s_t\left(r_{t+1}\right)$ depends on the value of the parameter $\sigma$ describing the elasticity of the utility function. Three cases can be distinguished:

• $\sigma =1$—the derivative is 0, so the savings rate does not depend on the interest rate;
• $\sigma >1$—the savings rate function is decreasing;
• $\sigma <1$—the savings rate function is increasing.

To analyze the dynamics of the entire economy based on the Diamond growth model, we assume that in period t, the economy has $L_t$ young individuals working (in the first period of life), and $L_{t-1}$ individuals are in their second period of life. We denote total consumption by $C_t$, which can be expressed as follows:

$$C_t=c_{1t}{L}_t+c_{2t}{L}_{t-1}.$$

(68)

Investments in period t can be expressed as the sum of the increase in capital $K_{t+1}-K_t$ and depreciation expenditures $\delta K$. Furthermore, remembering that in the long run, production equals the sum of the remuneration of production factors: $Y_t=w_t{L}_t+r_t{K}_t+\delta K_t$, we obtain the following identity:

$$w_t{L}_t+r_t{K}_t+\delta K_t=c_{1t}{L}_t+c_{2t}{L}_{t-1}+K_{t+1}-K_t+\delta K.$$

(69)

Given that the consumption of the elderly must equal their savings, Equation (69) simplifies to

$$K_{t+1}=s_t{w}_t{L}_t,$$

(70)

which indicates that the capital in a given period is equal to the savings from the previous period. Considering how the labor force and technological level evolve over time in the Diamond model (see Equation (59)), we can recalculate the value of capital $K_{t+1}$ per unit of effective labor:

$$k_{t+1}={\displaystyle \frac{s_t{w}_t{L}_t}{A_t{L}_t(1+g)(1+n)}}.$$

(71)

The above equation, which describes the capital per unit of effective labor, can be transformed into a different form by considering the formula for the savings rate (66) and the fact that in the Diamond model, the same equations hold as in the earlier Ramsey–Cass–Koopmans model, i.e., Equation (52). Ultimately, we obtain the following:

$$k_{t+1}={\displaystyle \frac{1}{1+g}}{\displaystyle \frac{1}{1+n}}{\displaystyle \frac{{(1+f^{\prime }\left(k_{t+1}\right)-\sigma )}^{\frac{1-\sigma }{\sigma }}}{{(1+\rho )}^{\frac{1}{\sigma }}+{(1+f^{\prime }\left(k_{t+1}\right)-\delta )}^{\frac{1-\sigma }{\sigma }}}}\left(f\left(k_t\right)-k_t{f}^{\prime }\left(k_t\right)\right)$$

(72)

Equation (72) describes the dynamics of capital per unit of effective labor. It is a nonlinear difference equation. The Formula (72) is simplified by assuming that the production function is of the Cobb–Douglas type in the form: $f\left(k_t\right)=B{k}_t^{\alpha }$ (where $B>0$), and the utility parameter equals $\sigma =1$, which gives a logarithmic utility function. Under these assumptions, Equation (72) takes a simpler form:

$$k_{t+1}={\displaystyle \frac{1}{1+g}}{\displaystyle \frac{1}{1+n}}{\displaystyle \frac{1}{2+\rho }}B(1-\alpha )k_t^{\alpha }.$$

(73)

In the long-term equilibrium state, capital per unit of effective labor is constant, i.e., $k_t=k_{t+1}\equiv k^{*}$. By incorporating this condition into Equation (73), we obtain the following expression for the capital in the steady state:

$$k^{*}={\left({\displaystyle \frac{1}{1+g}}{\displaystyle \frac{1}{1+n}}{\displaystyle \frac{1}{2+\rho }}B(1-\alpha )\right)}^{{\displaystyle \frac{1}{1-\alpha }}}.$$

(74)

When the economy reaches a long-term equilibrium state, capital, output, and consumption per unit of effective labor are equal. It follows that the GDP growth rate is identical to the sum of the growth rate of the labor force and the rate of technological progress. In contrast, GDP per capita grows at the same rate as the available technology (knowledge). These conclusions imply that the Diamond model provides the same answer to the question of the causes of long-term GDP growth as the two previously analyzed models: Solow–Swan and Ramsey–Cass–Koopmans.

4.6. Romer’s Learning-by-Doing Model (1986)

Romer’s growth model, published in 1986 [61], fundamentally differs from the three previously discussed methods of analyzing economic growth. In the models of Solow–Swan, Ramsey–Cass–Koopmans, and Diamond, the concept of technological level (knowledge) was introduced exogenously and subsequently used as an independent variable to explain the causes of economic growth. Due to the similarities among these three models, they are often classified into one group of neoclassical growth models.

Romer’s learning-by-doing model belongs to the category of endogenous growth models, which depart from the neoclassical production function characterized by diminishing returns to factors of production. In his model, P. M. Romer assumed that knowledge exhibits increasing returns at the level of the entire economy (as the sole factor of production). This assumption is justified by the observation that new knowledge generated by investments can be utilized simultaneously by many firms without additional costs. The knowledge created can spread freely throughout the economy, leading to increasing returns from its application. Therefore, the production function for an individual firm depends on three variables: the level of knowledge within the firm $a_i$, the overall level of knowledge in the economy A, and the amount of other production factors $k_i$. It can be expressed as follows:

$$f_i=f_i(a_i,A,k_i).$$

(75)

Assuming that N firms operate in the economy, the following equality holds:

$$\sum _{i=1}^N{a}_i=A.$$

To simplify the analysis, it is assumed that the amount of other production factors remains constant over time, i.e., $k_i=const.$ Furthermore, it is assumed that all firms are identical, which implies that

$$\forall i{f}_i(a_i,A,k_i)=f(a,A,k)\mathrm{and}Na=A.$$

The production function is required to satisfy the following properties:

$$f(\lambda a,\lambda k,\lambda A)>\lambda f(a,k,A),$$

(76a)

$$f(\lambda a,\lambda k,A)=\lambda f(a,k,A),$$

(76b)

which indicate that the production function exhibits increasing returns with respect to all production factors and constant returns with respect to a and k.

Assuming that the amount of production factors other than knowledge remains unchanged (i.e., $k=const.$), the value k can be treated as a parameter rather than a variable in the function, and the notation can be simplified as follows:

• $f(a,k,A)=f(a,A)$—the production function for an individual firm;
• $F\left(a\right)$—the production function for the entire economy.

Regarding the rate of change in the amount of knowledge over time, it is assumed that the growth rate of knowledge can be expressed by the function $g\left({\displaystyle \frac{i}{a}}\right)$, where i denotes the value of investments made to generate new knowledge. This function is bounded above by a certain constant $\gamma$. Introducing an upper limit on the growth rate of knowledge is necessary to avoid the undesirable situation where consumption and utility grow indefinitely. Therefore, we can write the following:

$${\displaystyle \frac{\dot{a}}{a}}=g\left({\displaystyle \frac{i}{a}}\right)<\gamma .$$

(77)

As for utility from the perspective of the household sector, the utility function is assumed to be the same as in the Ramsey–Cass–Koopmans model:

$$U={\int }_{t=0}^{\infty }{e}^{-\rho t}u\left(c_{pc}\right)dt,$$

(78)

where $\rho >0$ denotes the rate of time preference.

Optimization in Romer’s model involves maximizing the utility function given by Equation (78), subject to the following constraints:

$${\displaystyle \frac{\dot{a}}{a}}=g\left({\displaystyle \frac{f(a,A)-c}{a}}\right)<\gamma ,$$

(79a)

$$\dot{a}\ge 0,$$

(79b)

$$a\left(0\right)\to \mathrm{given}$$

(79c)

The substitution in Equation (79a) arises from the fact that output f is divided between consumption c and investment i, hence we have the equation $i=f(a,A)-c$. The inequality in (79b) results from the fact that knowledge does not depreciate, meaning it can either grow or remain constant.

Solving the optimization problem in Romer’s model requires invoking control theory to construct the Hamiltonian of the discounted value and determine the first-order conditions. To draw specific conclusions from the calculations, it is necessary to adopt more detailed assumptions regarding the forms of the production function f, the function describing the growth rate of knowledge, and the utility function. We assume the following forms:

$$f(a,A)=a^{\alpha }{A}^{\beta }=N^{\beta }{a}^{\alpha +\beta },$$

(80a)

$$g={\displaystyle \frac{\gamma \frac{f-c}{a}}{\gamma +\frac{f-c}{a}}},$$

(80b)

$$u\left(c\right)=c.$$

(80c)

For the above forms of functions, the following equations describing the dynamics of the economy in Romer’s model can be derived:

$$c=f-\gamma a({\theta }^{{\displaystyle \frac{1}{2}}}-1),$$

(81a)

$$\dot{a}=a\gamma (1-{\theta }^{-{\displaystyle \frac{1}{2}}}),$$

(81b)

$${\displaystyle \frac{\dot{\theta }}{\theta }}=\rho -{\displaystyle \frac{\gamma \frac{i}{a}}{\gamma +\frac{i}{a}}}-{\displaystyle \frac{{\gamma }^2}{{\left(\gamma +\frac{i}{a}\right)}^2}}\left(a^{\alpha -1}{\left(Na\right)}^{\beta }(\alpha -1)+{\displaystyle \frac{c}{a}}\right),$$

(81c)

where $\theta$ denotes the variable conjugate to the capital motion equation.

The fundamental difference between Romer’s model and neoclassical growth models is that in the learning-by-doing model, there is no steady state. With an appropriate choice of initial conditions, the economy can find itself on an optimal trajectory leading to infinite economic growth. If the economy is on an optimal trajectory, the growth rate of knowledge increases, approaching (asymptotically) its upper limit $\gamma$.

The learning-by-doing model contradicts the concept of convergence between countries—the economic growth rate increases with income, which means that wealthier countries develop faster than poorer ones.

Another interesting conclusion from Romer’s model is the observation that the trajectory for a centrally planned economy will be higher on the graph than the curve for a perfectly competitive economy. This conclusion implies that state intervention is necessary to ensure optimal knowledge accumulation. Without state involvement, firms would be interested in minimizing their costs and maximizing private benefits, and for this reason, a perfectly competitive economy would accumulate too little knowledge.

4.7. Lucas Growth Model (1988)

The Lucas growth model was proposed in 1988 [62] and builds upon Romer’s learning-by-doing approach, incorporating the concept of increasing returns from human capital.

The Lucas growth model is centered around the existence of two sectors in the economy—considering both physical capital and human capital. Let H denote the amount of human capital ($H>0$) possessed by a given individual in the society. We denote by $N\left(H\right)$ the number of individuals who possess capital of magnitude H. Let $u\left(H\right)$ represent the fraction of time that an individual with capital H allocates to the accumulation of physical capital. Conversely, $1-u\left(H\right)$ will represent the fraction of time allocated to the accumulation of human capital. If N denotes the total population, L the effective labor force in the production of goods, and $H_1$ the average level of human capital in the economy, we obtain the following formulas:

$$N={\int }_0^{\infty }N\left(H\right)dH,$$

(82a)

$$L={\int }_0^{\infty }u\left(H\right)N\left(H\right)HdH,$$

(82b)

$$H_1={\displaystyle \frac{{\int }_0^{\infty }N\left(H\right)HdH}{N}}.$$

(82c)

To simplify the analysis, it is assumed that all individuals are identical, implying that everyone has the same amount of capital and allocates the same percentage of time to the accumulation of physical capital. Therefore, the integrals in the above equations reduce to multiplications, and we ultimately use the following formulas:

$$L=uNH\mathrm{and}{H}_1=H.$$

(83)

Moreover, in the Lucas model, it is assumed that knowledge (technological level), which in neoclassical models grew exponentially, is now constant. Additionally, it is assumed that there is no depreciation, i.e., $\delta =0$. The only exogenous variable in the Lucas model is the total population size. The population growth rate is constant, which means that the number of people increases exponentially, describable by the following differential equation:

$${\displaystyle \frac{\frac{dN}{dt}}{N}}=n.$$

(84)

In the Lucas model, it is assumed that the total production of goods can be calculated using the following function of three variables:

$$Y=A{K}^{\alpha }{\left(uNH\right)}^{1-\alpha }{H}_{\alpha }^{\gamma }.$$

(85)

The time dynamics of physical capital K and human capital H are described by the following differential equations:

$$\dot{K}=A{K}^{\alpha }{\left(uNH\right)}^{1-\alpha }{H}_{\alpha }^{\gamma }-Nc,$$

(86a)

$$\dot{H}=H^{\xi }G(1-u),$$

(86b)

where the function G must satisfy the following conditions:

$$\left\{\begin{array}{c}{G}^{\prime }>0,\hfill \\ G\left(0\right)=0.\hfill \end{array}\right.$$

(87)

For simplicity, it is assumed that the function G is linear and given by

$$G=\mu .$$

(88)

Furthermore, considering that the function describing the change in human capital cannot exhibit diminishing returns, it is assumed that $\xi =1$, which simplifies the differential equation describing changes in H over time:

$$\dot{H}=H^{\mu }(1-u).$$

(89)

When considering a centrally planned economy, the planner’s objective is to maximize the utility function over the lifespan of individuals within existing constraints. The function for which the maximum conditions are sought has the following form:

$$U={\int }_0^{\infty }{e}^{-\rho t}{\displaystyle \frac{c^{1-\sigma }-1}{1-\sigma }}Ndt.$$

(90)

Such an optimization problem is solved by constructing the Hamiltonian of the present value, which then leads to the first-order conditions that, in turn, yield the equations of motion for both physical and human capital.

By analyzing and comparing the dynamics of a centrally planned economy with a perfectly competitive economy, one can infer that the marginal productivity of capital is higher in the former model. This is due to the presence of externalities, which implies that a perfectly competitive economy is not Pareto optimal.

In the long-run equilibrium state, the growth rates of per capita consumption (c), physical capital (K), and human capital (H) are constant, which we express as follows:

$${\displaystyle \frac{\dot{c}}{c}}=g_c=\mathrm{const}.{\displaystyle \frac{\dot{K}}{K}}=g_K=\mathrm{const}.{\displaystyle \frac{\dot{H}}{H}}=g_H=\mathrm{const}.$$

(91)

The problem of finding the values of $g_c,g_K,g_H$ corresponding to the state in which the economy reaches equilibrium requires performing a series of complex transformations. For this reason, only the final results will be presented in this paper.

For the growth rate $g_H$ describing the rate of human capital growth, we obtain two results—one for a centrally planned economy (indexed as $CP$) and the other for a perfectly competitive economy (indexed as $PC$). The formulas are the following:

$$g_H^{CP}={\displaystyle \frac{1}{\sigma }}\left[\mu -{\displaystyle \frac{1-\alpha }{1-\alpha +\gamma }}(\rho -n)\right],$$

(92a)

$$g_H^{PC}={\displaystyle \frac{1}{\sigma (1-\alpha +\gamma )-\gamma }}(1-\alpha )(\mu +n-\rho ).$$

(92b)

Similarly, for the growth rate of physical capital per capita, we obtain the following formulas:

$$g_{K/N}^{CP}={\displaystyle \frac{1-\alpha +\gamma }{\sigma (1-\alpha )}}\left[\mu -{\displaystyle \frac{1-\alpha }{1-\alpha +\gamma }}(\rho -n)\right],$$

(93a)

$$g_{K/N}^{PC}={\displaystyle \frac{1-\alpha +\gamma }{\sigma (1-\alpha +\gamma )-\gamma }}(1-\alpha )(\mu +n-\rho ).$$

(93b)

Equations (93a) and (93b) also describe the growth rate of per capita consumption.

In the case where $\gamma =0$, the perfectly competitive economy exhibits the same growth rate as the centrally planned economy. In situations where externalities exist (i.e., $\gamma >0$), the centrally planned economy exhibits higher economic growth than the competitive economy. The higher the $\gamma$ coefficient, the greater the difference in growth rates between the two economies. A higher value of the $\gamma$ coefficient also increases the difference between the growth rate of consumption and the growth rate of human capital.

4.8. Mankiw–Romer–Weil Growth Model (1992)

The Mankiw–Romer–Weil growth model, also known as the extended Solow model, is not an endogenous growth model but represents an extension of the neoclassical theory of economic growth. Since this model was developed in the 1990s [63] and incorporates the role of human capital in economic development, the Mankiw–Romer–Weil model is classified among the so-called “new growth theory” models.

The main difference between the Mankiw–Romer–Weil model and the Solow model is the expansion of the production function to include a factor representing the value of human capital, denoted by H. In this model, three production factors are considered: physical capital—K, effective labor—$AL$, and human capital—H. The starting point for analyzing economic growth in the Mankiw–Romer–Weil model is the production function in the following form:

$$Y=K^{\alpha }{H}^{\beta }{\left(AL\right)}^{1-\alpha -\beta },$$

(94)

where the parameters $\alpha$ and $\beta$ must satisfy the following properties:

$$\left\{\begin{array}{c}\alpha >0,\hfill \\ \beta >0,\hfill \\ \alpha +\beta <1.\hfill \end{array}\right.$$

(95)

In the Mankiw–Romer–Weil model, several assumptions are made:

• The level of knowledge in the economy and the labor force grow exogenously at constant rates of a and n, respectively.
• Both types of capital are depreciated at the same rate, denoted by $\delta$.
• The fraction of income allocated to the accumulation of physical capital is denoted by $s_K$, while $s_H$ represents the fraction of income allocated to the accumulation of human capital.

Based on these assumptions, we can write the following equations characterizing the dynamics of physical and human capital:

$$\dot{K}=s_{K}Y-\delta K,$$

(96a)

$$\dot{H}=s_{H}Y-\delta H.$$

(96b)

In the model, a change in variables is introduced, analyzing the dynamics of economic growth based on physical capital, human capital, and production per unit of effective labor. These quantities are denoted by k, h, and y, respectively, and are defined as follows:

$$k:={\displaystyle \frac{K}{AL}},$$

(97a)

$$h:={\displaystyle \frac{H}{AL}},$$

(97b)

$$y:={\displaystyle \frac{K^{\alpha }{H}^{\beta }{\left(AL\right)}^{1-\alpha -\beta }}{AL}}=k^{\alpha }{h}^{\beta }.$$

(97c)

Taking the definition of k and differentiating it, we use Equation (96a) to obtain the following formula:

$$\dot{k}=s_{K}y-(n+a+\delta )k=s_K{k}^{\alpha }{h}^{\beta }-(n+a+\delta )k.$$

(98)

Similarly, by differentiating the definition of h and substituting Equation (96b), we obtain an analogous formula:

$$\dot{h}=s_{H}y-(n+a+\delta )h=s_H{k}^{\alpha }{h}^{\beta }-(n+a+\delta )h.$$

(99)

Equations (98) and (99) generalize Equation (46), which appeared in the Solow–Swan model.

At the optimal state, the values of both types of capital per unit of effective labor are constant. Mathematically, this means that the derivatives $\dot{k}$ and $\dot{h}$ are equal to zero. By setting Equations (98) and (99) to zero, we obtain formulas for the values of both types of capital per unit of effective labor in the equilibrium state. These values, denoted by $k^{*}$ and $h^{*}$, are

$$k^{*}={\left({\displaystyle \frac{s_K^{1-\beta }{s}_H^{\beta }}{n+a+\delta }}\right)}^{{\displaystyle \frac{1}{1-\alpha -\beta }}},$$

(100a)

$$h^{*}={\left({\displaystyle \frac{s_K^{\alpha }{s}_H^{1-\alpha }}{n+a+\delta }}\right)}^{{\displaystyle \frac{1}{1-\alpha -\beta }}}.$$

(100b)

The dynamics of the economy in the Mankiw–Romer–Weil model can be examined in the two-dimensional space $(k,h)$, where curves corresponding to the equations $\dot{k}=0$ and $\dot{h}=0$ are plotted. The point $(k^{*},h^{*})$, where the economy achieves long-term equilibrium, is located at the intersection of these curves.

At the optimal state, the growth rate of GDP equals the sum of the growth rate of knowledge and the growth rate of the population. Per capita GDP, on the other hand, grows at the rate of technological progress. This means that the Mankiw–Romer–Weil model provides the same answer to the question of the causes of sustained economic growth as the Solow–Swan model.

4.9. Nonneman–Vanhoudt Growth Model (1996)

The growth model proposed by Nonneman and Vanhoudt in 1996 [64] is a generalization of earlier growth models based on neoclassical economics—specifically, the Solow–Swan model and the Mankiw–Romer–Weil model. The more general nature of this model is reflected in the assumption that the rate of economic growth is influenced by N different types of capital. Consequently, the output in the economy is described by a generalized Cobb–Douglas production function:

$$Y=E^{1-{\sum }_{i=1}^N{\alpha }_i}\prod _{i=1}^N{K}_i^{{\alpha }_i},$$

(101)

where $K_i$ represents the quantity of the i-th type of capital ($i=1,\cdots ,N$), and E denotes effective labor.

The parameters ${\alpha }_i$ in Equation (101), which describe the elasticity of output Y with respect to each type of capital, must satisfy the following properties:

$$\left\{\begin{array}{c}\forall i=1,\dots ,N{\alpha }_i>0,\hfill \\ \sum _{i=1}^N{\alpha }_i<1.\hfill \end{array}\right.$$

(102)

Let $s_i$ denote the fraction of income allocated to the accumulation of the i-th type of capital (investment rate in that capital), and ${\delta }_i$ refer to the depreciation rate of the i-th resource. Then, for each type of capital, we have the following differential equation:

$$\forall i=1,\dots ,N{\dot{K}}_i=s_{i}Y-{\delta }_i{K}_i.$$

(103)

Similar to the Mankiw–Romer–Weil model, new variables are introduced by normalizing each type of capital per unit of effective labor. We thus have the following definitions:

$$\forall i=1,\dots ,N{k}_i={\displaystyle \frac{K_i}{E}}.$$

(104)

Similarly, the output per unit of effective labor is

$$y={\displaystyle \frac{Y}{E}}.$$

(105)

Substituting the generalized Cobb–Douglas function (see (101)) into definition (105), we obtain the formula for output per unit of effective labor:

$$y=\prod _{i=1}^N{k}_i^{{\alpha }_i}.$$

(106)

By differentiating the definition of $k_i$ (see (104)) with respect to time, we obtain

$${\dot{k}}_i={\displaystyle \frac{{\dot{K}}_{i}E-K_i\dot{E}}{E^2}}={\displaystyle \frac{{\dot{K}}_i}{E}}-k_i{\displaystyle \frac{\dot{E}}{E}},$$

(107)

which, using Equation (103) and substituting the formula for the growth rate of effective labor:

$${\displaystyle \frac{\dot{E}}{E}}=n+a,$$

(108)

(where n and a denote the growth rates of population and knowledge, respectively), leads to the following differential equations:

$$\forall i=1,\cdots ,N{\dot{k}}_i=s_{i}y-({\delta }_i+n+a)k_i.$$

(109)

Using the definition of y from Equation (106), the differential equations in the form of (109) can be written as the following system of equations:

$$\left\{\begin{array}{c}{\dot{k}}_1=s_1\prod _{i=1}^N{k}_i^{{\alpha }_i}-({\delta }_1+n+a)k_1,\hfill \\ {\dot{k}}_2=s_2\prod _{i=1}^N{k}_i^{{\alpha }_i}-({\delta }_2+n+a)k_2,\hfill \\ \cdots \hfill \\ {\dot{k}}_N=s_N\prod _{i=1}^N{k}_i^{{\alpha }_i}-({\delta }_N+n+a)k_N.\hfill \end{array}\right.$$

(110)

The system of Equation (110) is a natural generalization of Equations (98) and (99), which described the dynamics of two types of capital in the Mankiw–Romer–Weil model.

When the economy reaches a state of long-term equilibrium, the dynamics of each type of capital per unit of effective labor will be zero, i.e.,:

$$\left\{\begin{array}{c}{\dot{k}}_1=0,\hfill \\ {\dot{k}}_2=0,\hfill \\ \cdots \hfill \\ {\dot{k}}_N=0.\hfill \end{array}\right.$$

(111)

This means that the stationary point of the economy can be described by finding values $k_1^{*},k_2^{*},\cdots ,k_N^{*}$ that satisfy the system of equations:

$$\left\{\begin{array}{c}{s}_1\prod _{i=1}^N{\left(k_i^{*}\right)}^{{\alpha }_i}-({\delta }_1+n+a)k_1^{*}=0,\hfill \\ s_2\prod _{i=1}^N{\left(k_i^{*}\right)}^{{\alpha }_i}-({\delta }_2+n+a)k_2^{*}=0,\hfill \\ \cdots \hfill \\ s_N\prod _{i=1}^N{\left(k_i^{*}\right)}^{{\alpha }_i}-({\delta }_N+n+a)k_N^{*}=0.\hfill \end{array}\right.$$

(112)

The system of differential Equation (110) can equivalently be written in the following form [65]:

$$\left\{\begin{array}{c}{\left(k_1^{*}\right)}^{1-{\alpha }_1}\prod _{i=2}^N{\left(k_i^{*}\right)}^{-{\alpha }_i}={\displaystyle \frac{s_1}{{\delta }_1+n+a}},\hfill \\ {\left(k_2^{*}\right)}^{1-{\alpha }_2}\prod _{i=1\wedge i\ne 2}^N{\left(k_i^{*}\right)}^{-{\alpha }_i}={\displaystyle \frac{s_2}{{\delta }_2+n+a}},\hfill \\ \cdots \hfill \\ {\left(k_N^{*}\right)}^{1-{\alpha }_N}\prod _{i=1}^{N-1}{\left(k_i^{*}\right)}^{-{\alpha }_i}={\displaystyle \frac{s_N}{{\delta }_N+n+a}}.\hfill \end{array}\right.$$

(113)

To solve the system of Equation (113), each equation should be logarithmically transformed. This operation leads to the following equivalent form of the system (113):

$$\left\{\begin{array}{c}(1-{\alpha }_1)ln\left(k_1^{*}\right)-{\alpha }_{2}ln\left(k_2^{*}\right)-\cdots -{\alpha }_{N}ln\left(k_N^{*}\right)={\theta }_1,\hfill \\ -{\alpha }_{1}ln\left(k_1^{*}\right)+(1-{\alpha }_2)ln\left(k_2^{*}\right)-\cdots -{\alpha }_{N}ln\left(k_N^{*}\right)={\theta }_2,\hfill \\ \cdots \hfill \\ -{\alpha }_{1}ln\left(k_1^{*}\right)-{\alpha }_{2}ln\left(k_2^{*}\right)-\cdots +(1-{\alpha }_N)ln\left(k_N^{*}\right)={\theta }_N,\hfill \end{array}\right.$$

(114)

where ${\theta }_i$ denotes the following quantity:

$${\theta }_i=ln\left({\displaystyle \frac{s_i}{{\delta }_i+n+a}}\right)\forall i=1,\dots ,N.$$

(115)

After performing the appropriate transformations from the system of Equation (114), we can obtain formulas for the value of each type of capital per unit of effective labor in the long-term equilibrium state:

$$ln\left(k_i^{*}\right)={\displaystyle \frac{\left(1-{\sum }_{j=1\mathrm{and}j\ne i}^N{\alpha }_j\right){\theta }_i+{\sum }_{j=1\mathrm{and}j\ne i}^N\left({\alpha }_j{\theta }_j\right)}{1-{\sum }_{j=1}^N{\alpha }_j}}\forall i=1,\dots ,N,$$

(116)

which, by substituting the formula for ${\theta }_i$ from Equation (115), results in the following form:

$$ln\left(k_i^{*}\right)={\displaystyle \frac{\left(1-{\sum }_{j=1\mathrm{and}j\ne i}^N{\alpha }_j\right)ln\left(\frac{s_i}{{\delta }_i+n+a}\right)+{\sum }_{j=1\mathrm{and}j\ne i}^N({\alpha }_{j}ln\left(\frac{s_j}{{\delta }_j+n+a}\right))}{1-{\sum }_{j=1}^N{\alpha }_j}}\forall i=1,\dots ,N,$$

(117)

By analyzing the formulas for the value of each type of capital in the stationary state, we can draw the following conclusions—the higher the investment rate in a given type of capital, the higher the value of that type of capital per unit of effective labor in equilibrium. Similarly, the lower the depreciation rate of a given type of capital, the higher its value in the long-term equilibrium [65].

5. Usefulness of Mathematical Methods in Economics: SWOT Analysis

The application of mathematics to economics has long been seen as a way to transform the discipline into a more precise, predictive, and rigorous field [20,66,67]. The mathematization of economics allows researchers to express complex theories in a formal, logical structure and to test hypotheses empirically. This section presents a SWOT analysis [68] to evaluate the strengths, weaknesses, opportunities, and threats of using mathematical methods in economics, followed by a discussion of future research directions and open problems.

5.1. Strengths

• Precision and Rigor: One of the primary benefits of mathematical methods is the precision they bring to economic theories. Mathematical models allow economists to define assumptions clearly and to derive results logically, minimizing ambiguity. This rigor helps in establishing more universally accepted frameworks for economic analysis.
• Quantitative Analysis: Mathematics enables the quantitative treatment of economic data, facilitating the measurement of variables like growth rates, inflation, or unemployment. This approach not only helps in making predictions but also in testing theoretical models against real-world data. In this way, mathematics contributes significantly to econometrics, a subfield that is critical to policy-making.
• Abstraction and Generalization: Mathematical models can abstract complex economic phenomena, enabling the formulation of general principles that apply across different contexts. For example, game theory, optimization models, and equilibrium analysis provide versatile tools to explain diverse economic interactions, from market competition to international trade dynamics.
• Predictive Power: Well-developed mathematical models such as the Solow–Swan or Keynesian growth models have been used to predict economic trends over time. When used correctly, these models can offer insights into future economic conditions, guiding policymakers and businesses in strategic decision-making.
• Comparative Analysis: Mathematics facilitates the comparison of different economic scenarios through the use of sensitivity analysis and optimization. Economists can simulate the effects of various policy decisions and assess the trade-offs, helping them select the best course of action under given constraints.

5.2. Weaknesses

• Oversimplification of Reality: One of the most significant drawbacks of mathematical models in economics is their reliance on simplified assumptions. While abstraction is necessary, many models assume perfect competition, rational agents, and constant returns to scale, which do not always align with real-world complexities. This often leads to criticism that mathematical models can be too detached from reality.
• Dependence on Data Quality: The reliability of any mathematical model depends on the quality of the data used. Inaccurate or incomplete data can lead to erroneous conclusions, undermining the model’s predictive capabilities. Moreover, data collection in economics is often subject to limitations such as measurement errors or unobserved variables.
• Limited Applicability to Social Factors: While mathematics is highly effective in explaining financial and market-based phenomena, it struggles to account for the social, political, and psychological factors that also shape economic behavior. Human irrationality, cultural differences, and institutional influences are challenging to quantify and often require qualitative methods.
• High Entry Barrier: The complexity of mathematical methods can serve as a barrier to entry for many economists who are not familiar with advanced mathematics. This limitation may create a divide between theoretical and applied economists, with the latter being more focused on practical issues that do not easily lend themselves to formal modeling.

5.3. Opportunities

• Integration with Computational Methods: Advances in computational power and machine learning offer new avenues for expanding the use of mathematical methods in economics. Models can now handle larger datasets, more complex scenarios, and non-linear relationships, which were previously difficult to manage. Computational economics represents an exciting frontier for future research, blending mathematical rigor with practical applicability.
• Policy-Making and Business Strategy: Mathematical models are increasingly used in policy-making, especially in areas like monetary policy, fiscal planning, and climate change economics. Businesses also employ mathematical tools for risk management, market forecasting, and pricing strategies. As economic problems become more complex, the demand for robust mathematical models is likely to grow.
• Interdisciplinary Applications: Mathematics allows for a fruitful exchange of ideas between economics and other fields such as physics, biology, and engineering. Concepts like network theory and stochastic processes, borrowed from other disciplines, are now applied to understand financial markets and global economic systems. This cross-pollination of ideas enhances both the scope and depth of economic research.
• Enhancing Data-Driven Decision Making: With the proliferation of big data, mathematical models are more relevant than ever. Economic decisions—whether made by governments, corporations, or individuals—can be increasingly informed by data analytics powered by advanced mathematical models. This presents an opportunity to create more nuanced, dynamic, and accurate predictions of economic behavior.

5.4. Threats

• Over-Reliance on Formalism: There is a risk that economics, by becoming too reliant on mathematical formalism, could lose sight of the broader social and philosophical questions it originally sought to address. Critics argue that focusing too heavily on mathematics may lead to a neglect of qualitative insights that are equally valuable in understanding economic phenomena. This issue has also been discussed in the field of physics, where concerns have been raised about the dominance of mathematical elegance over empirical validation. In Lost in Math: How Beauty Leads Physics Astray, Sabine Hossenfelder critiques how the pursuit of aesthetic principles like symmetry and elegance has led to a lack of empirical progress in areas such as string theory and supersymmetry, drawing parallels to the risks that other sciences, including economics, might face when they over-prioritize formalism at the expense of real-world relevance [69]. This issue highlights the need for a balanced approach where mathematical tools are used to complement, rather than overshadow, other analytical methods.
• Ethical Concerns: The use of mathematical methods can sometimes lead to technocratic decision-making, where policies are made based on abstract models that fail to consider their broader social implications. There is a growing concern about the ethical ramifications of applying mathematical models, particularly in areas like inequality and labor markets, where the models may fail to capture the full spectrum of human experiences. Another ethical issue arises from the potential for data manipulation. While mathematical models may be perfectly valid in themselves, they rely heavily on the quality and integrity of the data fed into them. If data are deliberately skewed or manipulated, the resulting analysis—even when derived through correct methods—can produce misleading conclusions. This poses a significant risk, as decision-makers might rely on these results to shape economic policies that ultimately harm certain populations or benefit only a select group. Ensuring transparency in data collection and model assumptions is therefore crucial to maintaining ethical standards in economic research and policy-making.
• Misinterpretation of Results: Mathematical models often produce results that are open to interpretation. Policymakers or businesses may misinterpret or oversimplify these results, leading to unintended consequences. For instance, a model might suggest a policy that works under idealized conditions but fails when applied to real-world, messy environments. One common misinterpretation, particularly by journalists or non-experts, is confusing correlation with causation. Just because two variables are mathematically correlated does not mean that one causes the other, yet this distinction is often overlooked, leading to misleading conclusions. This misinterpretation can result in the implementation of policies or strategies based on faulty assumptions, which may exacerbate existing problems rather than solve them.
• Model Fragility: Many mathematical models are fragile in the sense that slight changes in assumptions or input data can drastically alter their conclusions. This fragility makes models less robust in uncertain environments, particularly during economic crises or when dealing with novel, untested phenomena such as pandemics or rapid technological shifts.

5.5. Open Problems and Future Research

While mathematics has significantly advanced the field of economics, several open problems remain. One pressing issue is how to better integrate social and behavioral factors into existing mathematical frameworks. While current models are effective in capturing financial or production-based aspects of the economy, they often fall short in modeling human irrationality, cultural influences, and institutional dynamics. Developing more holistic models that combine mathematical rigor with sociological and psychological insights remains a challenge.

Another open problem is the difficulty of modeling systemic risks and tail events—such as financial crises—that are rare but have devastating consequences. Standard models, which often assume normal distributions and equilibrium conditions, struggle to account for these extreme events. There is a need for more robust, crisis-proof models that can predict and mitigate such occurrences.

Finally, the ethical implications of mathematical modeling in areas like inequality, labor markets, and environmental sustainability deserve greater attention. Future research should focus on creating models that not only provide accurate predictions but also align with social justice and ethical considerations.

In conclusion, while mathematical methods offer tremendous value to economics by providing structure, precision, and predictive power, their limitations and potential pitfalls must be acknowledged. A balanced approach that integrates both mathematical and qualitative insights is essential for addressing the increasingly complex and interconnected challenges facing modern economies.

6. Discussion

Mathematics has long served as a foundational tool in economics, providing the rigorous framework necessary for analyzing complex economic phenomena and guiding policy decisions. Symmetry in mathematical models often emerges through the balanced relationships between variables, such as the equilibrium states in economic systems where supply equals demand or where growth rates stabilize. These symmetries reflect the underlying order within economic dynamics. The mathematical models discussed in this paper, particularly those concerning economic growth, illustrate the profound impact that mathematical techniques have on our understanding of economic systems and their dynamics.

Mathematical modeling in economics is not merely a methodological preference but a necessity for achieving precise and actionable insights into how economies function. Models like those developed by Solow–Swan, Mankiw–Romer–Weil, and Nonneman–Vanhoudt provide structured approaches to evaluating growth processes and the role of various types of capital. These models enable economists to abstract and simplify real-world complexities, making it possible to derive general principles and predict future trends. By translating qualitative economic theories into quantitative terms, mathematical models offer a means to rigorously test hypotheses and validate theoretical constructs through empirical data; see, for example, Refs. [70,71].

The selection of models for this review was based on their historical significance and their contributions to the development of economic theory. The chosen models—Solow–Swan, Mankiw–Romer–Weil, and Nonneman–Vanhoudt—are not only foundational but also representative of major advancements in mathematical methods applied to economics. These models were selected due to their influential role in shaping current economic thought and their demonstration of key principles such as the role of capital in growth and the integration of human capital into economic analysis. Other models and methods were considered, but these were chosen for their broad impact and relevance in illustrating the evolution of mathematical approaches in economics.

One of the significant benefits of mathematical modeling is its capacity to handle the complex interplay of multiple variables and their effects on economic growth. Symmetry, in this context, refers to the balanced and proportional relationships among these variables, enabling models to capture equilibrium dynamics effectively. For instance, the Nonneman–Vanhoudt model extends previous frameworks by incorporating multiple types of capital and their distinct impacts on production and growth. This allows for a more nuanced analysis of how different forms of capital—such as physical, human, and technological—contribute to economic development. The ability to model these relationships mathematically is crucial for formulating effective strategies to enhance economic performance and address disparities.

Symmetry also plays a role in the optimal allocation of resources and in ensuring balanced growth trajectories across different sectors of the economy. Symmetry-breaking phenomena, where previously stable relationships change, can indicate shifts in economic trends or the emergence of new dynamics, such as technological innovation or policy changes.

Moreover, mathematical models are instrumental in policy analysis and decision-making. They offer a systematic way to project the outcomes of different policy scenarios and their implications for sustainable development. Through sensitivity analysis and scenario planning, models help policymakers understand the potential trade-offs and consequences of their decisions. This capability is particularly important in addressing global challenges such as climate change, where the interactions between economic activities and environmental impacts are complex and far-reaching.

In addition to the historical developments in mathematical methods discussed in this paper, other contemporary models in the field of econophysics offer intriguing insights into wealth distribution and economic inequality. One such model explores a two-phase ODE dynamic for money exchange with collective debt limits, where agents either transfer or borrow money from a central bank. This leads to an equilibrium distribution and sheds light on the role of the banking system in wealth inequality, as quantified by the Gini index [72]. Another related study extends this model by incorporating interactions across multiple banks and examining money exchanges within a social network represented by a connected graph. The model demonstrates how wealth distribution converges to an asymmetric Laplace distribution in large populations, revealing how monetary flows are influenced by social and business partnerships [73]. Moreover, the broader literature on random asset exchange has been thoroughly reviewed, highlighting how stochastic agent-based models have been used to explain the emergence of economic inequality over the last 25 years. These models emphasize the need for more detailed representations of economic structures to improve their explanatory power [74]. While this paper has focused on the historical evolution of mathematical methods in economics, these developments point to significant advancements in modeling complex economic systems, particularly in the context of wealth distribution and inequality.

7. Conclusions

The iterative nature of mathematical modeling also supports ongoing refinement and adaptation of economic theories. As new data become available and as the global economic landscape evolves, models can be updated to reflect current conditions and emerging trends. This dynamic process ensures that economic analyses remain relevant and accurate, facilitating more informed and effective policy interventions.

In conclusion, the significance of mathematics in economics extends beyond theoretical exploration to practical application in policy-making and sustainable development. Mathematical models provide a vital tool for understanding economic growth, analyzing the effects of different types of capital, and evaluating the sustainability of development strategies. Symmetry is intrinsic to these models as it provides the structural balance that makes predictions and analysis reliable. Symmetry-based approaches help economists identify stable solutions and equilibrium conditions within dynamic systems. By offering a structured approach to addressing complex economic issues, these models help guide decision-making and promote a balanced and sustainable approach to economic development. As the challenges facing the global economy continue to grow in complexity, the role of mathematical modeling will only become more crucial in navigating the path towards sustainable and equitable growth.