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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.19 No.3 pp.589-596

Dynamic Optimization of Capital Stock: An Application of Maximum Principle

Marziyeh Esfandiari*
Department of Economics, University of Sistan and Baluchestan, Zahedan, Iran
*Corresponding Author, E-mail:
September 4, 2017 October 28, 2019 February 19, 2020


Determining the optimized level of economic variables is very influential in compiling economic plans and policies. This study deals with the determination of the country’s optimized capital stock by the use of Pontryagin’s Maximum Principle in the time period 1972-2017. The product function is assumed as Cobb-Douglas, and the coefficients of product function are measured by Eviews software. After that, the achieved differential equation applying from Maximum Principle is solved by the use of numerical methods via MATLAB software. Then, under certain hypotheses about the rate of depreciation and the ratio of savings, the optimized capital stock is calculated and is compared to the real ones. Findings of the study approve that there are big gaps between the existing situation and the optimized capital stock in Iran. In order to decrease the gap, it is urgent to increase investment and take policies for provoking the private sectors.



    The importance of mathematics in economics, like in other sciences, is known to everyone. In many economic theories, a lot of mathematic principles and premises are applied, but in practice and empirically, they have received less consideration. Mathematics plays important roles in different sciences including economics. Alpha Si Chiang expresses the identity of mathematical economics in this way: unlike general economy and international commerce, mathematical economy is not a branch of economics, but it is a method of economic analysis in which the economists apply mathematical symbols to state their issues and apply identified mathematical relationships in their reasoning.

    One of the applications of mathematics in economics is dynamic optimization. Dynamic optimization is allocation of rare resources among the rival elements in the course of time. Mathematically, it is the issue of determining time-related routes for the certain variables which are called control variables. These time routes are selected from a collection called control collection. Selection and determination of time routes for control variable is one of the differential equations called motion equations. They are selected in such a way that they maximize an assumed function which is related to the state variables. There are three methods for solving the equation of dynamic optimization: measuring the changes, optimized control and the Maximum Principle.

    Measurement of changes is the first and foremost solution for equations of control. The control equation discussed in the measurement of classic changes consists of selecting a time route, from the certain starting point to the termination point in such a way that it maximizes the value of supposed function integral. This function is a function of state variable, derivative of state variable in relation to time, and time.

    Dynamic planning is one of the modern methods for solving the control equation. In this method, we place a certain control equation which is selected for solving in a wider collection of equations that are identified with certain parameters; and by the use of the “optimization principle” we achieve the recessive basic relation that relates members of this collection of equations to each other. With some appropriate extra theories, the recessive basic relation gives an equation with small basic derivatives that is called “Belman Equation.” By solving that equation, the general answer of the wider collection of equations is achieved and as a result, the answer for the target equation, too, will be at hand (Interiligator, 1971).

    The third method for solving the control equation is “Maximum Principle.” This method is many times more useful than the previous methods and compared to the measurement of changes, this method can solve the equation in the general situation where limits of control variables are present. Compared to the dynamic planning method, this method also discusses the nature of answers. Because of this, the Maximum Principle is the basic method of measuring optimized control in many mathematical questions, engineering and economics (Interiligator, 1971). This principle which is mostly attributed to the Russian mathematician, Pontryagain, is famous because of its capability to directly discuss the special limitations of control variable (Chiang, 1974).

    Economic growth, in simple words, is increase in the production of a country in one special year compared to its amount in the base year. In the massive level, increase in the gross national production (GNP1) or gross domestic production (GDP2) in the discussed year compared to its value in the base year is considered as the economic development. The reason that for measuring economic growth the prices of the base year are applied is that the calculated increase in the gross national/domestic product should be originated from increase of amount of production, and the effect of increase of prices (inflation) be omitted.

    Out of the sources of economic growth, we can mention increase of production inputs (increase of capital or labor force), increase in the optimization of production, and application of possible capabilities in economy. The capital stock or the fixed visible produced property is the collection of visible physical capital goods which are measurable and have roles in the process of production of goods and services and in formation of income. According to the definition of the national calculation system, capital stock is the net value of the collective numbers of capital formation considering their effective lifetime. Therefore, the capital stock can be defined as the collection of building, plant, machines and equipment which are used or are usable in the process of production. According to the advice of the national accounts system, the Central Bank Office of Economic Accounts uses permanent balance check for measuring the capital. Considering the existence of human forces in the country, a lot of policy- makers believe that by increase of the capital stock (investment) production and employment can be increased. In order to determine the amount of required investment, the optimized level of capital stock should be clarified. For this purpose, by the use of maximum principle in the economy of Iran under the certain conditions and hypotheses, this study deals with determination of ideal and real capital and comparison of real and ideal gross domestic product.


    Krasovski and Trasi’ev (2007) studied the issue of controlling or finding the optimized investment in the framework of Pontryagain’s maximum principle in a study titled “dynamic optimization of investment in models of economic growth” on the basis of classic economic growth and considering the production function of lineargraphic type with three elements of labor force, capital stock and useful work (per hour).For this purpose, initially the parameters of production function were estimated by the use of the US economy data in the years 1900- 2000 (data has been formed through network making until 1900); and then by algorithm designing, the routes of optimized economic growth and investment are achieved. In this study, it is observed that the optimized routes are well-accordant with the flow of time series. The result of model design and prediction of future growth scenarios show that the growth route is S-shaped and that the growth of annual gross domestic production in 2021 in the US economy will get very close to the saturation point.

    In another study titled “dynamic optimization finding, maximum principle and capital,” Fabbri and Iacopetta (2007) presented an application of the dynamic planning and maximum principle for solving optimization finding in the course of time while considering a linear production function in relation to capital. In this study, two types of capital are considered. The first type which is considered in lots of studies is the convergent capital stock, and stable and external depreciation rates. In the other type, every unit of capital has an identified generating circle. In this study, more precise results have been achieved by the use of dynamic planning in relation to maximum principle methodology.

    In an article titled “neoclassical economic growth, environment and change of technology,” Rubio et al. (2009), by applying dynamic optimization finding, concluded that optimized investment in research and development for removing pollution are U-shaped and are reversely against the income of countries. In other words, piling up of capital along with increase of pollution happens to the point that it makes investment in research and development profitable. In this point, the equation is balanced in the Saddle point type. Then, by considering hypotheses for coefficients of production function and the equation of investment, they have extracted the optimized tax coefficient.

    In another study titled “human capital, investment and development,” Mohamed and Asma (2008) have shown, by applying optimized control in the external growth model and considering consequent educational levels, that the length of education period in the primary levels and quality improvement in the advanced levels can increase economic growth. Aseev et al. (2012) studied the development of optimized control theory for different types of limitless horizon of dynamic and optimized allocation of economic resources. In this study, a completed version of Pontryagain’s maximum principle for a two-sectioned new model of conditional optimized economic growth is presented to the random increase of prices. Escribá-Pérez et al. (2018) tried to obtain a true economic measure of capital stock according to the prescriptions of the neoclassical theory. In this way, they develop an alternative method based on the equations that solve the dynamic optimization problem of the firm, yielding an economic estimation based on indicators of profitability, such as the distributed profits and the Tobin’s q ratio. The results show an economic depreciation rate that fluctuates around the statistical rate. Moreover, they get two time profiles for the economic and statistical capital stocks that differ significantly from each other. Leung et al. (2018) find that firms with greater organization capital have significantly higher stock returns. The research findings show that the positive association between organization capital and stock returns increases with labor market flexibility.


    The model focuses on the dynamic growth of GDP, there are two sectors in the model: household and firm. Firm produces goods and services through the Cobb- Douglas production function considering the three elements of labor force, capital and total factor productivity as followed:

    y ( t ) = F [ K ( t ) , L ( t ) , T F P ( t ) ]

    I denote by Y(t), K(t), L(t), and TFP(t) the respective size of output (GDP), capital, labor and total factor productivity at time t. Partial derivatives are positive to production factors and are decreasing. It is assumed the production function satisfies the Inada Limit Condition1), capital depreciation rate is μ and labor grows exponentially, L ˙ t / L t = n . Household consumes goods and services which produced by firm. As the Figure 1 shows, with the presumption of closed economy, GDP is used for consumption and investment. Therefore:

    Y ( t ) = A L α K β T F P δ

    Y ( t ) = C ( t ) + I ( t ) = ( 1 S ( t ) ) Y ( t ) + s ( t ) Y ( t )

    I denote by C(t)≥0 the consumption, I(t)≥0 the investment and by 0≤s(t)≤1, the saving ratio (the proportion of product that is saved at time t). The growth of capital fund obeys following dynamics:

    K ˙ ( t ) = S ( t ) Y ( t ) μ K ( t )

    where, μ is capital depreciation rate. The growth of capital fund per capita is as followed2):

    K ˙ ( t ) = S ( t ) y ( t ) θ K ( t )

    The lowercase stand for per capita variables and θ = μ + n. The objective functional is as the integral of the logarithmic consumption index discounted over an infinite time horizon.

    0 + [ L n f ( k ( t ) ) L n ( 1 s ( t ) ) ] e σ t d t

    where σ>0 is the constant discount parameter. In the theory, appropriateness of logarithmic function shows relative increase of consumption in the unit of time. On the other side, in the conditions of uncertainty the relative risk-avoidance is considered as fixed.

    The equation of controlling optimum investment with considering an objective function that maximizes the present value of consumption in the time period regarding the limitations of investment is shown as the following:

    max j = 0 + [ L n f ( k ( t ) ) L n ( 1 s ( t ) ) ] e σ t d t k ˙ ( t ) = S ( t ) y ( t ) θ   k ( t ) K ( 0 ) = K 0

    where, the planner begin the investment process from the

    level K0 which is a positive number. In solving the equation of optimum control, different methods are applied. Considering the fact that this is a Mayer type of equation, it can be solved through Maximum Principle. Maximum Principle includes first degree differential equations according to state variable and co-state variable. In addition, there is another condition and necessity according to which Hamiltonian should be maximized at every point of time in relation to the variable of control. Hamiltonian function is:

    H ~ ( s , k , l , tfp , t ) = ( ln f ( k ) + ln ( 1 s ) ) , e t δ + λ ( sf ( k ) θ )

    where λ is Lagrange multiplier which is the shadow cost of capital. For simplicity, suppose that: Ψ = eλ and H ~ e t σ = H

    Therefore, H ~ e t σ λ =H= ln f(k) + ln (1-s) +Ψ(s.f(k) - θk)

    In this case, the required conditions for establishing Maximum Principle are:

    a ) H S = 0 1 1 s + ψ f ( k ) = 0 S O = 1 1 ψ . f ( k )

    where SO is the optimum level of investment. Also considering the fact that 2 H S 2 = 1 ( 1 s ) 2 is negative, then Hamilton in function in relation to “s” is concave.

    According to (5) and (9):

    K 0 = S . f ( k ) θ k = ( 1 1 ψ . f ( k ) ) f ( k ) θ k = f ( k ) 1 ψ θ k

    Substituting the investment plan SO in the dynamics of shadow costs ψ which obeys ψ 0 = δ ψ H k , leads to the following equation:

    ψ 0 = δ ψ H k = δ ψ f ' k f ( k ) ψ . S . f ' k + θ ψ = δ ψ ψ f ' k + θ ψ =    ψ ( δ + θ f ' k )

    Hamiltonian system of differential equations are consisted of the dynamics of shadow costs ψ ˙ and the capital per capita k ˙ . From this follows equations (12):

    { k 0 = f ( k ) 1 ψ θ k ψ 0 = ψ ( δ + θ f ' k )

    I introduce for convenience the new variable: z= z(t), z=ψk. The time derivative of z is given by equation (13):

    z 0 = ψ 0 k + ψk 0

    and consider the Hamiltonian system (12) of equations in the variables (k, z):

    { z 0 = z ( F f k + δ F f ' k ) 1 k 0 = f θ k k z ˙

    Applying z ˙ = 0 ,    k ˙ = 0 , the steady state of the Hamiltonian system (14) calculate as followed:

    { z 0 = 0 k 0 = 0 { z ( f k + δ f ' k ) 1 = 0 f θ k k z = 0 { f ' k * = δ + θ 1 z * = f ( k * ) k * θ

    The steady state (k, z) will be determined numerically within the framework of the proposed Algorithm in section 4. Existence and uniqueness of the solution of the equation f ' k * = δ + θ follows from Inada limit condition and f ( k ˙ ) > 0 , f ( k ) " < 0 . It is necessary to evaluate the sufficient condition. According to the condition of Mengazarian, if the functions F and f are both concave functions in relation to s, k and λ(t)≥0 , then the required condition of maximum principle is the sufficient condition as well. For this purpose, Hessian matrix is formed and by the use of especial values, I evaluate whether the function is concave or convex.

    Equation (16) make the Hamiltonian system (14) linearized in the neighborhood of steady state (k, z):

    ( k ˙ z ˙ ) = A ( k k * z z * )

    A = ( a 11 a 12 a 21 a 22 ) = ( f ˙ k λ θ 1 Z *   K * ( Z * ) 2 Z * ( K * ) 2 ( f ( K * ) k * f ˙ ( k * ) + ( k * ) 2 f " ( k * ) ) f ( K * ) K * + δ f ˙ K * )

    Applying | A θ I | = 0 , the eigenvalues of the matrix A are calculated as follows:

    | A θ I | = | a 11 θ 0 a 21 a 22 θ | = 0 θ 2 T r ( A ) θ + d = 0 , I = ( 1 0 0 1 )

    “d” is determinant of Matrix “A”, and “Tr” is the Trace of “A”, “I” is the unit matrix. Calculation of the trace and determinant of the matrix A leads to the following estimates:

    T r ( A ) = 1 z * 1 z * + δ = δ > 0 , d = d e t ( A ) = k * z * f " ( k * ) < 0

    According to (18) and (19) the eigenvalues are calculated as:

    T r 2 ( A ) 4 d > T r ( A ) 2 = δ 2 , θ 1 , 2 = T r ( A ) T r ( A ) 2 4 d 2 θ 1 = T r ( A ) T r ( A ) 2 4 d 2 < 0 , θ 2 = T r ( A ) + T r ( A ) 2 4 d 2 > δ > 0

    where θ1, θ2 are the eigenvalues of matrix A. As a result, since the eigenvalues θ1 and θ2 are real and have opposite signs, the steady state ( k * , z * ) is the saddle point, which means that for the linearized system only two trajectories converge to the steady state along the direction defined by the eigenvector corresponding to the negative eigenvalue.


    As it was pointed out, Cobb-Douglas production function is considered with three factors of labor, capital and total factors productivity, and the studied period is 1972-2017. The data was taken from world Bank for 3) Gross value added at basic prices (GVA) (constant LCU), Data from database: World Development Indicators. gross domestic product, employment and capital stock.

    The model is estimated as logarithmic. It has to be mentioned that K, L, TFP, Y are stationary in logarithmic difference. The coefficients of production function are positive and significant (the results of estimating production function are shown in appendix A).

    The algorithm to construct the optimal trajectory is as followed:

    • (1) Numerical estimation of the steady state (k∗, z∗). The steady state is determined from the equation system (13) using the method of successive approximations.

    • (2) Linearization of the system in the neighborhood of the steady state. The elements of the matrix A (15) are calculated.

    • (3) Calculation of the eigenvalues (18) and eigenvectors of the linearized system.

    • (4) Integration of the linearized system of the Hamiltonian system (13) linearized in the neighborhood of (k∗, z∗)4) in the given neighborhood along the eigenvector corresponding to the negative eigenvalue, and determination of the characteristic state of the system with nonzero velocity vector.

    • (5) Integration of the nonlinear system (13) by the Runge–Kutta method via MATLAB in reverse time from the characteristic state to the initial value of the phase variable.

    • (6) Unfolding of the integrated curve in the direct time, and scaling of the time scale (Krasovskii and Taras’ev, 2012).

    In measuring Z*, K* the depreciation rate and descent rate are considered respectively as 10% and 20%.

    ( 14.2 1796.532 × 10 5 7 × 10 5 14.4 ) Z * = 0.034 , K * = 207679.1

    By the use of appropriate capital stock which was achieved through solving control equation, the gross domestic production was calculated (GDPO) and compared to the real values (GDP) in Table 1. As shown in the Figure 2, there is a considerable gap between the real gross domestic production and its optimum value. Considering the existence of unemployment in the country and the surplus offer of labor force, the difference between existing and optimum gross domestic production shows that the productivity of labor force is low, and it is required that policies be executed to increase skills, accountabilities and professional training of labor forces. On the other hand, considering the fact that in the studied model, the gap between the real and ideal capital stock has created a big gap between real and optimum gross domestic production, it is required that policies be taken to increase investment both in the quantitative (elevation of capital stock) and also in the qualitative aspect (increase of human resources and growth in the productivity of production elements). Elevation of capital stock requires formation of an appropriate atmosphere for occupation, economic stability and attraction of foreign capitals.


    In this article, The Hamiltonian systems arising in the Pontryagin maximum principle for the problems of optimal control with infinite horizon were considered in measuring the optimum capital stock, and gross domestic production correspondent to it. For solving the differential equation, the numerical method of Range-Kouta and program writing in MATLAB software were applied. Analysis of the qualitative properties of the Hamiltonian systems gave rise to the theorems of existence and uniqueness of the steady state, as well as to characterization of the properties of the eigenvalues and eigenvectors of the linearized system. This analysis underlies the proof of the saddle nature of the steady state. The model was calibrated to the Iranian macroeconomic data under certain assumptions related to the rate of depreciation, rate of descent and coefficients of production function which can be subject to changes in the program. Computational experiments demonstrated a big gap between real capital stock and optimal capital stock as well as real GDP and optimal ones. According to the findings, the ratio of capital stock to employed labor force 0.028 is calculated in the steady state which is very low. Hence, in order to increase this ratio and to achieve economic growth, attraction and investment of domestic as well as foreign capitals is highly required.

    APPENDIX A: Estimation Results of Production Function Coefficients

    Dependent Variable: DLOG(GDP)

    Method: Least Squares

    Date: 19/06/27 Time: 13:13

    Sample (adjusted): 1972 2017

    Included observations: 44 after adjusting endpoints

    APPENDIX B: The Program to Solving Control Equation %X=[K, L, TFP];

    X=[165264.5588, 7195931, 100;

    185907.9039, 7308593, 104.2;

    207679.1385, 7394693, 109.8;

    232820.218, 7499356, 112.4;

    263878.9298, 7618748, 118.2;

    302281.3021, 7705508, 127.1;

    345282.1266, 7925777, 125.2;

    397572.5099, 8124343, 128.7;

    477525.8464, 8260218, 120.8;

    586089.9769, 8799420, 123.1;

    672638.6569, 9024018, 110;

    735635.2628, 9206557, 96;

    768158.6239, 9539965, 88.3;

    785543.5737, 9684120, 73.5;

    797120.6978, 9891959, 69.1;

    778334.5373, 10175216, 77.8;

    783200.2504, 10533630, 84.9;

    817649.425, 10660111, 80.7;

    816943.7142, 10934950, 81.5;

    772894.8343, 11056330, 76;

    774075.2806, 11369713, 74.3;

    768615.1848, 11617885, 69.8;

    788027.1622, 11925921, 72.1;

    805304.583, 12546955, 79.5;

    856865.7383, 13096615, 84.5;

    903329.5233, 13261874, 84.8;

    939502.7357, 13408492, 83.7;

    963921.5268, 13688326, 82.2;

    984559.0949, 14060756, 82.7;

    1018825.251, 14571572, 84.7;

    1061427.407, 14910239, 84.2;

    1104953.946, 15259232, 83.8;

    1151649.479, 15784467, 82;

    1200046.916, 16418521, 82.7;

    1259856.897, 16955068, 81.9;

    1330121.523, 17755060, 83.7;

    1410521.565, 18334274, 85.3;

    1496051.614, 19016442, 85.2;

    1585965.123, 19691251, 85.8;

    1676548.907, 20476343, 86.8;

    1676549, 20476343, 86.8;

    1772754, 21090633, 88.4;

    1823422, 22109765, 88.9;

    1821356, 2345659, 89.3;

    1913983, 2403476, 90.5;

    2003268, 2489574, 91.6];



    Z(1,1)=input('Enter Z0: ');

    K(1,1)=input('Enter K0: ');

    for i=1:40

    Z(1,i+1)=Z(1,i)*(((.54)*(X(i,2)^(0.53))*(X(i,1)^(- 0.54))*(X(i,3)^(1.006)))+0.20)-1;

    K(1,i+1)=((.46)*(X(i,2)^(0.53))*(X(i,1)^(0.54))*(X(i,3)^( 1.006)))-(0.1*X(i,1))-(X(i,1)/Z(1,i));




    The model.


    The optimum path: GDPO.


    GDP3), GDPO

    Reference: Research Findings .


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    3. Escribá-Péreza, F. J. , Murgui-Garcíaa, M. J. , and Ruiz-Tamarit, J. R. (2018), Economic and statistical measurement of physical capital: From theory to practice, Economic Modelling, 75, 246-255.
    4. Fabbri, G. and Iacopetta, M. (2007), Dynamic Optimization, Maximum Principle and Vintage Capital, Available from: MPRA Paper No. 5115.
    5. Interiligator, M. D. (1971), Mathematical Optimization and Economic Theory, Prentice-Hall, Englewood Cliffs, NJ.
    6. Krasovskii, A. A. and Taras’ev, A. M. (2007), Dynamic optimization of investments in the economic growth models, Automation and Remote Control, 68(10), 1765-1777.
    7. Leung, W. S. , Mazouz, K. , Chen, J. , and Wood, G. (2018), Organization capital, labor market flexibility, and stock returns around the world, Journal of Banking & Finance, 89, 150-168.
    8. Mohamed, B. M. and Asma, R. (2008), Human capital, investment and growth, a dynamic education model, Journal of Business and Public Affairs, 2(1), 1-22.
    9. Rubio, S. , Garcia, J. , and Hueso, J. (2009), Neoclassical growth, environment and technological change: The environmental Kuznets curve, The Energy Journal, 30, 143-168, CITED 2020 September 6, Available from: