Open access peer-reviewed chapter - ONLINE FIRST

On Some Important Ordinary Differential Equations of Dynamic Economics

By Anastasios Tsoularis

Submitted: October 4th 2020Reviewed: March 9th 2021Published: April 21st 2021

DOI: 10.5772/intechopen.97130

Downloaded: 15

Abstract

Mathematical modeling in economics became central to economic theory during the decade of the Second World War. The leading figure in that period was Paul Anthony Samuelson whose 1947 book, Foundations of Economic Analysis, formalized the problem of dynamic analysis in economics. In this brief chapter some seminal applications of differential equations in economic growth, capital and business trade cycles are outlined in deterministic setting. Chaos and bifurcations in economic dynamics are not considered. Explicit analytical solutions are presented only in relatively straightforward cases and in more complicated cases a path to the solution is outlined. Differential equations in modern dynamic economic modeling are extensions and modifications of these classical works. Finally we would like to stress that the differential equations presented in this chapter are of the “stand-alone” type in that they were solely introduced to model economic growth and trade cycles. Partial differential equations such as those which arise in related fields, like Bioeconomics and Differential Games, from optimizing the Hamiltonian of the problem, and stochastic differential equations of Finance and Macroeconomics are not considered here.

Keywords

  • Walrassian condition
  • Marshallian condition
  • homogeneous function
  • Cobb–Douglas form
  • endogenous growth

1. Introduction

Ordinary differential equations are ubiquitous in the physical sciences and are fundamental for the understanding of complex engineering systems [1]. In economics they are used to model for instance, economic growth, gross domestic product, consumption, income and investment whereas in finance stochastic differential equations are indispensable in modeling asset price dynamics and option pricing. The vast majority of the ordinary differential equations in economic are autonomous differential equations or difference equations, where time is an implicit variable, whereas the more difficult to solve delay (differential-difference) equations have received much less attention. Difference equations seem a more natural choice of modeling economic processes as key economic variables are monitored at discrete time units but they can present significant complications in their asymptotic behavior and are thus more difficult to analyse. Differential equations on the other hand, can be more amenable to asymptotic stability analysis. Partial differential equations, usually of the second order, for functions of at least two variables arise naturally in modern macroeconomics from solving an optimization problem formulated in a stochastic setting and in optimal control theory. Two books that are recommended for delving deeper into the- economic applications of differential equations are the introductory one by Gandolfo [2] and the more advanced by Brock and Malliaris [3]. Both books are excellent sources for ordinary differential equations in economic dynamics. A more recent book which requires strong mathematical background is by Acemoglu [4].

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2. Some differential equations of neoclassical growth theory and business cycles

Some of the most important differential equations developed by economists during a period spanning over sixty years are presented in this section. Most of them beginning with Solow’s development of a growth model, which was partly motivated by the works of Harrod and Domar, are models from Neoclassical Growth Theory. The main postulate of Neoclassical Growth Theory is that economic growthis driven by three elements: labour, capital, and technology. Economic growth is an important topic in economics and Solow’s growth model is the first topic taught in undergraduate economics because of its underlying simplicity and importance as argued by Acemoglu [5]. The differential equation by Samuelson is concerned with demand and supply scenarios. Phillips’ work is the earliest attempt to employ classical feedback control theory in order to steer a national economy towards a desired target. The remaining works are differential equations with time lags inherently present in production and capital accumulation. Due to space limitations, the exposition is somewhat uneven with full mathematical analyses of most models and cursory treatments of those with time lags. The choice of the differential equations presented in this chapter is a judicious one, the list is by no means exhaustive, but is meant to afford a glimpse into how the mathematical thinking of some famous economists has influenced the economic growth theory in the twentieth century.

2.1 Harrod-Domar

The Harrod-Domar model was developed independently by Roy Harrod [6] and Evsey Domar [7] to analyze business cycles originally but later was used to explain an economy’s growth rate through savings and capital productivity. Output, Y,is a function of capital stock, K, Y=FK, and the marginal productivity, dYdK=c=constant. The model postulates that the output growth rate is given by

1YdYdt=scδ,

where sis the savings rate, and δthe capital depreciation rate. The straightforward solution,

Yt=Y0escδt.

clearly demonstrates that increasing investment through savings and productivity boosts economic growth but does not take into account labour input and population size.

2.2 Samuelson

In his 1941 Paul Samuelson [8] paper employed simple differential equations to investigate the stability of equilibrium for several demand–supply scenarios. The simplest stability analysis was carried out under the Walrasian and Marshallian assumptions. In the former price increases (decreases) if excess demand is positive (negative), whereas in the latter quantity increases (decreases) if excess demand price is positive (negative). Excess demand is the difference between the quantity that buyers are willing to buy and the quantity that suppliers are willing to supply at the same price. Excess demand price is the difference between the price that buyers are willing to pay for a given quantity and the price required by the suppliers.

Let Dpαand Spdenote the demand and supply functions of price, p, respectively with αa shift parameter representing “taste”. At equilibrium, price, p, and quantity, q, are given by

q=Dpα=Sp
∂D∂α>0,∂D∂p<0.

It is the task of comparative statics to show the determination of the equilibrium values of price and quantity and their sensitivity on the “taste” parameter, α.

The dynamic formulation of the Walrasian assumption is

dpdt=fDpSp,f0=0,f0>0.

Retaining the first order term in a Taylor series expansion near the equilibrium, p, we obtain the following linear differential equation

dpdt=a0dDdpdSdpppp,

with solution for an initial price, p0

pt=p+pp0ea0tdDdpdSdpp.

The equilibrium is stable if dDdpp<dSdpp. Price must rise when demand increases.

The dynamic formulation of the Marshallian assumption is

dqdt=gpDqpSq,g0=0,g0>0.

Neglecting high order terms and using the trivial elementary calculus result, dpDdq=1dDdp,dpSdq=1dSdp, we obtain

qt=q+qq0expb0t1dDdp1dSdpq.

The equilibrium is stable if 1dDdpq<1dSdpq. Quantity supplied must rise when demand increases, while the change in price is dependent upon the algebraic sign of the supply curve’s slope.

2.3 Solow

Robert Solow [9] proposed a growth equation incorporating production, capital growth and growth in the labour force absent from the Harrod-Domar model.

  1. Production function: =FKL, the quantity of goods by Kunits of capital and Lunits of labour at time t. In a closed economy where all output is invested or consumed,

Yt=Ct+It,

where Ctand Itare the consumption and investment functions respectively.

An important assumption of the model are the Inada conditions [10]

FK>0,FL>0,2FK2<0,2FL2<0.

In the limits.

limK0FK=,limL0FL=,limKFK=0,limLFL=0.

The Inada conditions ensure that Fis strictly concave with slope decreasing from infinity to zero.

The function Fis linearly homogeneous of degree 1 in Kand L(in economic terms this is known as constant returns to scale, increasing capital and labour by a certain amount, results in a proportional rise of production) if

Y=FαKaL=αFKL,α>0.

In particular, choosing α=1Land set y=YL,k=KL, representing the output and capital per worker respectively

YL=y=FKL1=fk.

The production function is expressed in terms of a unit of labour and the capital to labour ratio. The assumption of constant returns to scale allows the simplified function, fk.

  1. Growth of Capital in Economy: The growth of the capital stock, K, is equivalent to growth in investment, I, which is used to increase capital subject to depreciation. Depreciation of capital stock will be accounted for so that Iis essentially

    investment=rate of change of capital+capital depreciation rate

or

It=dKdt+δKt,

where δis the constant capital depreciation rate.

Letting ctand itdenote the consumption and investment per labour unit

ct=CL,it=IL,
yt=ct+it=ct+1LdKdt+δk=ct+dkdt+δ+1LdLdtk.

  1. Growth of the Labour Force with full employment: The assumption in the labour market is that the labour supply is equivalent to the population. There is no unemployment and the growth of labour as function of time follows an exponential growth pattern:

L=L0ent.

The fundamental differential equation of economic growth is then

dkdt=fkδ+nkct.

The differential equations and production functions outlined in these three assumptions are the fundamental elements for Solow’s basic differential equation. In Solow’s paper, a constant fraction of income is allocated to savings, in particular, =ytct=fk1sfk=sfk, so that

dkdt=sfkδ+nk.

The equilibrium solution to the basic differential equation is found from sfk=δ+nk. A well-known function is the Cobb–Douglas production function, YKL=αKβL1β,0<β<1, where βis the elasticity of output, KYYK, with respect to capital. The use of the Cobb–Douglas production function is justified because it exhibits constant returns to scale: If capital and labour are both increased by the same factor, λ>1, output will be increased by exactly the same proportion, YKL=λαKβL1β. Also the marginal product, YK,YL, diminishes as either Kor Lincreases since 2YK2<0,2YL2<0. Introduce k=αKLβ=αkβ, so the differential equation becomes

dkdt=kβδ+nk.

From dkdt=0, k=δ+n11β. Substituting k=δ+n11βinto y=αkβ, the steady state level of per capita income is

y=a11βsδ+nβ1β.

The output per unit growth converges to n:

1YdYdt=βkdkdt+nn.

A multiplicative factor in the form of technological progress, t=A0egt, can be introduced in the production function, so that, Yt=aKtβAtLt1βand kt=KtAtLt, leading to

dkdt=sakβδ+n+gk.

The first order nonlinear differential equation has solution

kt=δ+n+g+k01βδ+n+geδ+n+g1βt11β.

This solution includes the solution to the labour growth only model, n=0. The steady state is

k=δ+n+g11β.

Differentiation of dkdt=sakβδ+n+gkwith respect to kat kgives β1δ+n+g<0, the equilibrium is stable. The steady state level of per capita income is

y=a11βsδ+n+gβ1β,

a constant, since s,δ,n,gare all constant.

Yt=αKβA0L0eg1β+nt1β=akβA0L0eg1β+nt. The output per unit growth, 1YdYdt, converges to g1β+n.

The Solow residual is the part of growth unexplained by changes in capital and labour. For Yt=aKtβAtLt1β

Yt=aβKtβ1AtLt1βdKdt+aKtβ1βAtLtβdAdtLt+dLdtAt.

The growth rate per unit output is

1YYt=βKdKdt+1β1LdLdt+1β1AdAdt,
Solow residual=1YYtβKdKdt+1β1LdLdt.

A positive Solow residual would indicate a faster output growth than that of capital and labour.

2.4 Phelps

Phelps [11] used the neoclassical growth model to address the consumption per unit of labour at equilibrium in the so-called “golden rule”. At equilibrium with labour force growth rate, n, only the consumption per unit of labour is

ct=fknk.

For a maximum consumption per unit of labour

dcdk=dfdkn=0.

Since d2fdk2<0, the turning point is a maximum given by dfdk=n. The “golden rule” concludes that the marginal output per worker must equal the growth rate of the labour force at maximum per capita consumption.

2.5 RCK

The Ramsey–Cass–Koopmans model, or RCK model, is a neoclassical model of economic growth which differs from Solow’s model in its inclusion of consumption, based primarily on the work of Ramsey [12], with later significant extensions by Cass [13] and Koopmans [14].

dkdt=fkδ+nkct.

A steady state is when ct=fkδ+nk.

There is a second equation of the RCK model, the social planner‘s problem of maximizing a social welfare function expressed by the integral

0eρtLtuctdt=0enρtuctdt,

where ρ>0is the discount rate and uctis a strictly increasing concave utility function of consumption. The objective is formally stated thus

u=maxct0enρtuctdt

subject to

dkdt=fkδ+nkct
k0=k0.

The Hamiltonian is

Hc=enρtuc+λeρntfkδ+nkct,

where λis the costate variable (Lagrange multiplier). From

Hc=enρtucλ=0,
λ=enρtuc.

Also for the costate variable

dt=Hk=λfkδ+n,

and

dt=nρλ+2uc2ucdcdtλ.

Hence

nρ+2uc2ucdcdt=fk+δ+n,

whence

dcdt=uc2uc2fk+δ+ρ.

This is a nonlinear differential equation that describes the optimal evolution of consumption, known as the Keynes-Ramsey rule. Along with the differential equation, dkdt=fkδ+nkct, form the RCK dynamical system which does not admit an analytical solution. At equilibrium,

fkk=δ+ρ,
c=fkδ+nk.

The Jacobian matrix at equilibrium,

J=ρn1uc2uc22fk2k0

has eigenvalues real and opposite in sign as its determinant is uc2uc22fk2k<0(fkand ucare both concave), therefore the equilibrium is a saddle point.

2.6 Romer

The growth in the Solow model is exogenous, the steady state depends on the exogenous parameters, ,g, which are due to outside trends. In the absence of AtLtgrowth cannot be maintained. The marginal product of capital, YK=aβAt1βLK1β=aβAt1βKL1β, is inversely proportional to the capital per labour, KL. In countries with lower capital per labour the marginal product of capital should be higher which is not the case. The disparity could be attributed to the different gvalues in At, which is treated as an exogenously given parameter in the Solow model, so an explanation is lacking.

Romer [15] proposed a mathematical theory of endogenous growth based on the following three assumptions:

  1. The production function, Y=FKALoffers increasing returns to scale, that is FλKλAλL>λFKAL.

  2. The change in capital is identical to Solow’s model, dKdt=sYδK, where sis the fraction in savings, δis the exogenous capital depreciation rate. Labour, L, is also exogenous, dLdt=nL, and is comprises labour involved in research technology, LA, and labour involved in the production of the final goods, LY,L=LA+LY.

  3. Technology is exogenous and evolves in time, dAdt=γLAθAφ,0<θ<1,φ<1.

As is evident from the three assumptions, Romer’s growth model consists of three sectors: the research sector of ideas, the intermediate goods sector which implements the ideas of the research sector and the final goods sector which produces the final output.

Let gAbe the technology growth rate, taken to be constant along the stable path,

gA=1AdAdt=γLAθAφ1,
dgAdt=γθLAθ1dLAdtAφ1+γφ1LAθAφ2dAdt=0,
θ1LAdLAdt+φ11AdAdt=0,
θn+φ1gA=0,
gA=θn1φ.

In Romer’s model, the output production function is given by

y=kβLYL1β,

and the capital dynamics is

dkdt=skβLYL1βn+gA+δk.

The respective stable equilibria are

k=LYLsn+gA+δ11β,
y=LYLsn+gA+δβ1β.

The labour involved in the production of the final goods, LY, is determined in Romer [15] by maximizing the net profit for the final goods sector and obtaining the closed form expression for LYL=rnrn+βgA, where ris the interest rate, and all parameters are exogenous except for gAwhich is derived endogenously.

A nice accessible exposition of both Solow’s and Romer’s growth models is Chu [16]. Jones [17] argued that the predicted scale effects of Romer’s theory of growth is inconsistent with the time-series evidence from industrialized economies and that long-term growth depends on exogenous parameters including the rate of population growth.

2.7 Mankiw, Romer and Weil

Mankiw, Romer and Weil [18] argued that the marginal product of capital, YK, is lower in poorer countries is due to their deficiency in human capital. Human capital is the accumulation of knowledge and skills achieved through training and education, which are essential ingredients in adding economic value. The production function is of the Cobb–Douglas type

Yt=HtαKtβAtLt1αβ=HtAtLtαKtAtLtβAtLt,
yt=YtAtLt=HtAtLtαKtAtLtβ=hαkβ,

where Htis the human capital stock which depreciates at the same rate, δ, as Kt. As in Solow’s model, a fraction of the output, sYt, is saved but in this model, it is split between human and capital stock, s=sH+sK. The evolution of the economy is determined by

dkdt=sKhαkβn+g+δk,
dhdt=sHhαkβn+g+δh.

The equilibrium is

k=n+g+δsK1αsHα1α+β1,
h=n+g+δsKβsH1β1α+β1.

In the steady state,

y=n+g+δα+βα+β1sKβα+β1sHαα+β1.

Introduce the transformations, x1=kk,x2=hh, so that the equilibrium shifts to 11.Then

dx1dt=n+g+δx1βx2αx1,
dx2dt=n+g+δx1βx2αx2.

For small deviations, ξ1,ξ2, from the equilibrium the linear system

dξ1dt=n+g+δβ1ξ1+αξ2,
dξ2dt=n+g+δβξ1+α1ξ2.

The eigenvalues of the Jacobian matrix,

n+g+δβ1αβα1,

are given by the roots of the quadratic

λ2+2αβλ+1αβ=0.

From the production function, 1αβ>0. Since the sum of the eigenvalues is α+β2<0, and the product is 1αβ>0, both roots have negative real parts and the equilibrium point is stable.

2.8 Kaldor

Kaldor [19] presented a model of the trade cycle involving non-linear investment and saving functions that shift over time in response to capital accumulation or decumulation so that the system moves from stable equilibrium to unstable equilibrium to stable equilibrium again. In Kaldor’s model investment, I, and savings, S, functions are non-linear with respect to the level of activity, X, measured in terms of employment.

Kaldor used a differential equation system with general non-linear forms. Net investment, I, and savings, S, are functions of national income, Y, and capital stock, K:

I=IYK,
S=SYK,
IY>0,IK<0,SY>0,SK<0,
IK<SK.

Also growth in capital determines investment is given by

dKdt=IYK.

Since income will rise if investment is greater than savings, the dynamics of the national income is captured by the differential equation

dYdt=αIYKSYK,α>0.

The necessary and sufficient assumptions for the generation of a perpetual cyclical movement are:

  1. For normal income levels,

    IY>SY.

  • For extreme income levels, either low or high,

    IY<SY.

  • At equilibrium, where dKdt=0, income levels are normal.

  • 2.9 Phillips

    National governments design their expenditure policies to steer the national economy towards a desired income. The theory of feedback control or servomechanisms provides the mathematical methodology of correcting deviations of the controlled variables from their target values. Feedback policies applied to economic stability were implemented by Phillips [20].

    If Yis national income and Dais the aggregate demand then for some adjustment coefficient, a>0,

    dYdt=aDaY.

    A similar differential equation holds for the actual, Dgand target government demand, Dg, with b>0, namely,

    dDgdt=bDgDg.

    Aggregate and government demand are related by

    Da=mY+Dg,

    where mis the private sector’s marginal propensity to spend.

    Eliminate Dato obtain

    dYdt=am1Y+aDg.

    Differentiate the above to obtain

    d2Ydt2=am1dYdt+abDgDg=am1dYdt+abDg+abm1YbdYdt

    or

    d2Ydt2+b+a1mdYdt+ab1mYabDg=0.

    Phillips’ model is thus described by the linear second-order differential equation where Yis the target variable and Dgis the control variable. Investigated three types of feedback policy:

    1. Proportional, Dg=kPY, where kP>0. This policy does not prevent income reduction and induces oscillations.

    2. Derivative, Dg=kDdYdt, where kD>0. This policy does not prevent income reduction but avoids oscillations.

    3. Integral, Dg=kI0tYdt, where kI>0. This policy prevents income reduction but can induce unstable movement.

    2.10 Kalecki

    Kalecki [21] was the first economist to investigate the relationship between production lags and endogenous business cycles by considering a closed economic system over a short period of time without trend. Atis the gross capital accumulation (unconsumed goods). There is a “gestation period”, θ, for any investment It. Deliveries Ltare equal to investment orders, Itθat time, tθ:

    Lt=Itθ.

    Any orders placed during the “gestation period”, tθt, remain unfulfilled, Atis equal to the average of investment orders Itallocated during the period tθt:

    At=1θtϑtIτ.

    If Ktis the capital stock, and Uits physical depreciation

    dKdt=LtU=ItθU.

    The rate of change in investment is for some constants, m>0,n>0:

    dIdt=mdAdtndKdt=mθItItθnItθU.

    Denoting the deviation of Itfrom the constant demand for restoration of the depreciated industrial equipment Uby Jt=ItU, and differentiating Jt

    dJdt=mθJtJtθnJtθ

    or

    θdJdt++mJtθmJt=0.

    During the interval tθ0Kalecki assumed that Jt=0. A standard way to solve this differential equation with delay is to assume a solution of the form, Deαt, with Dand α(where αis a complex number), to be determined. The general solution of the differential equation for some constants, c1and c2is

    Jt=ebtc1cosωt+c2sinωt.

    The sign of the real parameter, b, classifies the behavior of the model as explosive for b>0, cyclical for b=0, and damped for b<0.

    2.11 A Solow model with lags

    Zak [22] considered a version of the Solow model with delay. Capital can be used τperiods later, so at time t, the capital to be put into productive use is ktτ. If fkis the production function, s01is the constant savings rate and δ01is the constant capital depreciation rate, Zak’s model is

    dkdt=sfktτδktτ.

    At equilibrium,

    sfk=δk.

    Deviations of the form, et, from equilibrium are governed by

    dkdt=sdfdkδeτ,

    with characteristic equation

    λsdfdkδeλτ=0.

    In many cases depending on the initial conditions, the roots of the characteristic equation have real parts with opposite signs, indicating the presence of a saddle point unlike Solow’s stable model. The model exhibits endogenous cycles when the roots are purely imaginary.

    2.12 Goodwin

    Goodwin [23] presented a nonlinear model of nonlinear business cycles with time lags between decisions to invest and the corresponding outlays. Changes at time, t, in income, yt, induce investment outlays, Oit+θ, at a later time, t+θ. Therefore

    Oit+θ=φdydt=φ(y)̇.

    Hence the nonlinear delay differential equation modeling the evolution of income is

    ϵdyt+θdt+1αyt+θ=Ot+φ(y)̇,

    where Otis autonomous investment outlay and ϵ,αare constants. The derivative, (y)̇dẏ, measures the rate of growth in investment with relative to the income growth, termed as acceleration coefficient. Expanding the two leading terms in Taylor series and neglecting higher order terms, Goodwin obtained the nonlinear delay differential equation

    εθd2ydt2+1αθ+ϵdydt+1αytφ(y)̇=Ot.

    Goodwin assumed that Otis constant, Ot=O, and introduced a new variable

    zt=ytO1α,

    where O1αis the income at equilibrium. The transformed differential equation is then

    εθd2zdt2+1αθ+ϵdzdtφż+1αz=0.

    The asymptotic behavior of the transformed equilibrium, z=0, is determined by the eigenvalue solutions of the characteristic equation

    εθλ2+1αθ+ϵφ̇0λ+1α=0,

    with characteristic roots,

    λ1,2=φ̇01αθ+ϵ±1αθ+ϵφ̇024εθ1α2εθ.

    Since

    λ1λ2=1αεθ>0

    and

    λ1+λ2=φ̇01αθ+ϵεθ,

    can be either positive or negative, both eigenvalues have positive or negative real parts. So if φ̇0<1αθ+ϵthe deviations from equilibrium are damped oscillatory motions, but if φ̇0>1αθ+ϵthe system is unstable and drifts away from the locally linearized region of stability.

    2.13 A brief literature survey of current research

    We close this chapter by providing a very brief snapshot of the current state of the art in theories of economic growth. Most of the very recent works cited are predominantly mathematical in nature. There is an enormous literature, not touched upon here, which employs Econometrics methods, like for instance panel data regression to estimate economic growth based on explanatory variables such as income, investment, policy indicators, education and others over several decades.

    In a short article Zhao [24] discusses how technology was integrated into economic growth by Romer.

    Boyko et [25] use least squares linear regression to determine the values of the coefficients at which the production functions of Cobb–Douglas in Solow’s growth model provide the best fit for available statistical data. Borges et al. [26] examine the dynamics of Solow’s economic growth model assuming that the labour force growth rate function is a solution of a delay differential equation thereby avoiding the use of exponential growth, Lt=L0ent, often criticized as a rather unrealistic choice. Their approach is motivated by the fact that there are delays in entering and retiring an individual from the labour force, relative to their birth date.

    Zhang et al. [27] base their analysis of how the redistribution of emission quotas would impact short-run equilibrium in a specific market of interest and long-run growth on the Solow growth model with endogenous dynamics and exogenous technological shocks.

    Zhang [28] develops an endogenous growth model based on modifications of both Solow’s model by introducing endogenous knowledge. and Romer’s by allowing knowledge to be gained from learning as well as from research.

    The paper by Caraballo et al. [29] is devoted to analysis of the stability of the economy according to an extended version of Kaldor’s economic growth model. They consider the role of the government’s monetary and fiscal policies and we study whether or not a time delay in implementing and the fiscal policy can affect the economic stability.

    Dayal [30] considers long run historical data and uses difference equation simulation to explore the Solow growth model to assess the growth changes in the recent decade.

    Perez-Trujillo et al. [31] investigate the impact of improvement in accessing innovation and knowledge on economic growth and convergence among countries using an augmented Solow-Swan growth model on data from 138 countries.

    Turnovsky [32] discusses contemporary aspects of stabilization policy in reference to Phillips’ contributions in a lengthy paper of substantial mathematical control theory content.

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    Anastasios Tsoularis (April 21st 2021). On Some Important Ordinary Differential Equations of Dynamic Economics [Online First], IntechOpen, DOI: 10.5772/intechopen.97130. Available from:

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