Optimal control applied to a Ramsey model with taxes and exponential utility

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1 WSEAS RANSACIONS on MAHEMAICS Optimal control applied to a Ramsey model with taxes and exponential utility OLIVIA BUNDĂU Politehnica University of imişoara Department of Mathematics P-ţa. Victoriei, nr, 34, imişoara ROMANIA sbundau@yahoo.com Abstract: In this paper we analyze an economical growth model with taxes and exponential utility in continuous and infinite time. his economical growth model leads to an optimal control problem. he necessary and sufficient conditions for optimality are given. Using the optimality conditions we prove the existence, uniqueness and stability of the study state for a differential equations system. Also we have investigated the dependence of the steady state,c ) on the growth rate of the labor force n and the effects of fiscal policy changes on welfare. Key Words: mathematical models applied in economies, endogenous growth, optimal policy Introduction In this paper, we consider a version of the Ramsey growth model with taxes in infinite and continuous time. his economy consists of a firm, a household and a government. he firm produces goods using two factors of production, capital and labor, it rents capital at the rate of interest r and it hires labor at the wage rate w. Also, the firm sees to maximize the present value of its profits. he government imposes taxes on capital and labor income, and it provides lump sum transfer and public-consumption expenditures. In this economy, the consumer chooses at any moment in time the level of consumption so as to maximize the global utility on the infinite time taing into account the budget constraint for household and the government. he utility function is given by an exponential function. his economical growth model leads to an optimal control problem. We prove that a necessary condition for the control function to solve our optimal control problem is that it is a solution of Euler equation. Also, we give the sufficient conditions for the optimal solution of the optimal control problem. An analogous result was formulated by Ş.Cruceanu and C. Vârsan [5]. Finally we investigate the effects of the fiscal policy changes on the steady-state values of capital and consumption and on welfare. he outline of this paper is as follows. In Section, we present the model which we use in this article. In Section 3, we give the necessary and sufficient conditions for the optimal solution of the economical growth problem with public intervention. In Section 4, we determine the steady state of the optimal control problem and we show that it is a saddle point. Also, we examine the qualitative dynamic behavior of the optimal solution. In Section 5,we analyze the dependence of the steady state,c ) on the growth rate of the labor force n. In Section 6, we investigate the effects of the fiscal policy changes on welfare. Some conclusions are given in section 7. he economical growth model In this paper, based on [], [3], [] and [8], we consider an economical growth model with public intervention. he economy consists of a household, a firm and a government. he competitive equilibrium achieved in a decentralized manner through perfect competition between the firm and the household. Also, we assume the economy closed i.e. all of the stoc capital must be owned by someone in economy and the net foreign debt is zero.) he firm produces goods, pays wages for labor, and maes rental payments for capital on the competitive maret at equilibrium prices given by the net marginal products of labor and capital. Output is determined according to a constant returns to scale technology using the following production function Y = FK,L) ) where K is the capital, L is the labor force, F : IR + IR + IR + is a function of class C having the following properties FK,L) K >, FK,L) L >, K >,L > ; ISSN: Issue, Volume 8, December 9

2 FK,L) K <, FK,L) L <, K >,L > ; FλK,λL) = λfk,l) = λy, λ > ; FK,L) lim K K lim K FK,L) K = lim L FK,L) L FK,L) = lim L L = FK,) = F,L) = he firm sees to maximize the present value of profits, given by Profit=FKt),Lt)) rt) + δ)kt) wt)lt), ) taing rt) and wt) as given, r is the rental rate of capital, w is wage per worer, and δ is depreciation of capital. We define t) = yt) = Kt) the capital per worer; Lt) Y t) the output per worer. Lt) Using the homogeneity condition we have where WSEAS RANSACIONS on MAHEMAICS Y = FK,L) = L F K,) = L f), L f) = F,). hen, the output per worer is y = f) Using the properties of function F we obtain the properties of function of class C, f : f) = ;f ) >,f ) <, > ; lim f ) = ; lim f ) =. Profit for this firm can be written as P rof it = Lt)ft)) rt)+δ)t) wt)) 3) he economical condition is that on a competitive maret the maximum profit obtained by the firm is zero, and is given by f ) = r + δ f) f ) = w. 4) We assume that the government taxes labor and net capital incomes at proportional rates τ w and τ r. We also assume that the tax revenue is allocated between lump-sum transfer S and public consumption goods X. Assuming that the government must run a balanced budget at each moment in time and that tax rates, transfers, and public-consumption expenditures per unit of labor x = X L,s = S L ) are constant over time, then we have the government budget constraint =, written in terms per capita τ r f ) δ) + τ w f) f )) = x + s 5) where the left-hand side is the total tax revenue per unit of labor, and the right-hand side gives total expenditure. Given constant values of x + s and τ r, equation can be solved for the value of τ w that will preserve budget balance for each value of. In this economy the consumer chooses at any moment in time the level of consumption ct) so that to maximize the global utility, given by U = θ e n)t e θct) )dt, 6) and subject to the budget constraint for the household, given by Kt)= τ r )rt)kt)+ τ w )wt)lt)+s Ct). Where c is the consumption, > is the discount rate and θ >. he size of the household grows at rate n. he budget constraint for the household can be rewritten in per capita terms as t)= τ r )rt)t)+ τ w )wt)+s ct) nt). where ct) = Ct) Lt) is the consumption per capita. he initial stoc of capital for the household is K. hus, the initial stoc of capital per capita is. In the conditions of a competitive equilibrium and taing into account the government budget constraint, the budget constraint for household in per capita terms can be rewritten as: t) = ft)) n + δ)t) x ct). 7) 3 Determination of optimality conditions he economical problem is to choose in every moment t, the size of consumption so as to maximize the global utility taing into account the government budget constraint and the budget constraint for household and the initial stoc of capital, in the conditions ISSN: Issue, Volume 8, December 9

3 WSEAS RANSACIONS on MAHEMAICS of competitive equilibrium, leads us to the following mathematical optimization problem P) : he problem P. o determine,c ) which maximizes the following functional θ e n)t e θct) )dt 8) in the class of functions AC[, ),IR + ), and c X, where X = {c : [, ) [,A],c measurable, A < } which verifies: Proof: Let ct), t)) by an optimal trajectory of the problem P). From 9) and ), we have ct) = φt)) t). Choose [, ] such that φ t)) is continuous on [, ]. Let h by any C function on [, ] which satisfies h) = h ) =. For each real number α IR, we define a new function t) by t) = t) + αht), as in fig. t) = ft)) n + δ)t) x ct) 9) ) = ) In our problem P, is the state variable and c is the control variable. Definition A trajectory t),ct)) is called an admissible trajectory, with initial capital, for the problem P) if it verifies the relation 9)-). Definition An admissible trajectory, t),c t)), is called optimal trajectory if: θ θ e n)t e θct) )dt e n)t e θc t) )dt. for every admissible trajectory t), ct)) of the problem P). In the following theorem we prove that a solution of our optimal control problem must satisfy a certain differential equation called the Euler equation. We denote: φt)) = ft)) n + δ)t) x. ) Uct)) = θ e θct) ) ) heorem If ct), t)) is an optimal trajectory of the problem P), then it verifies the Euler-Lagrange equation: d dt e θct) n)t ) = e θct) n)t φ t)). 3) Fig. Note that if α is small, the function t) is near the function t). We define J h α)= Uφt)+αht)) t)+α ht)))e n)t dt. Because t) is an optimal trajectory, we have Uφt)) t))e n)t dt 4) Uφt) + αht)) t) + α ht)))e n)t dt for all α IR. hus, J h α) J h ) for all α IR. Hence the function J h α) has a maximum at α =, so that J h ) =. 5) Now, looing at 4), we see that to calculate J h ) we must differentiate the integral with respect to a parameter appearing in the integrand. Conversely, 5) implies U φt)) t))φ t))ht) ht))e n)t dt = 6) ISSN: Issue, Volume 8, December 9

4 WSEAS RANSACIONS on MAHEMAICS We consider ψt) = t t [, ] and where G is given by U φs)) s))φ s))e n)s ds, gt) = ψt) + G, [, ] and g is a continuous function on [, ]. Hence, we conclude gt) = U φt)) t))φ t))e n)t = = d dt U φt)) t))e n)t ). his is the Euler-Lagrange equation discovered in 744 by mathematician Euler. In next theorem, we give the sufficient conditions for the solution of our optimal control problem P. [U φt)) t))e n)t +ψt)]dt+g ) = Since we have = h) = h ) =. U φt)) t))φ t))ht)e n)t dt gt)ht)dt = hus, 6) it becomes: gt) ht)dt gt) + U φt)) t))e n)t ) ht)dt =, 7) for all functions h which are C on [, ] and which satisfy h) = h ) =. We consider ht)= t U φs)) s))e n)s +ψs))ds Gt ), t. From 7) and the definition for g and G, we obtain hus gt) + U φt)) t))e n)t ) dt =. gt) = U φt)) t))e n)t for all t [, ]. Since g is a continuous function on [, ], we see that ct) = φt)) t) is a continuous function on heorem Let t),c t)) be an admissible trajectory in problem P. If there exists an absolutely continuous function qt) such that for all t, the following conditions are satisfied qt) = qt) f t)) δ) )) 8) qt) = e θc t) 9) lim t e n)t qt) = ) then t),c t)) is an optimal trajectory in problem P. Proof: Let t),c t)) be an arbitrary admissible trajectory for P. We denote = θ e n)t e θc t) )dt θ e n)t e θct) )dt ) and we define the function of Hamilton-Pontryagin H : [, ) [,A] IR [, ) IR, given by H,c,p,t)= θ e n)t e θc )+pf) n+δ) c x) ) In the following, we simplify our notation and put t) =, c t) = c t) =,ct) = c, H,c,p,t) = H, H,c,p,t) = H. From ) we have θ e n)t e θc ) =H,c,p,t) pf) n+δ) c x) and using 9) the relation ) becomes = Since [H,c,p,t) H,c,p,t)]dt+ p H,c,p,t) = pf ) H c,c,p,t) = e n)t e θc pt) H cc,c,p,t) = θe n)t e θc, )dt. 3) ISSN: Issue, Volume 8, December 9

5 we have that H,c,p,t) is concave as a function of c and. Using a standard result on concave functions, we obtain H,c,p,t) H,c,p,t) H )+ H c c c ). 4) Using the transformation qt) = e n)t pt) in 8) we have the equation of the adjoint variable p, H = p. Using 4) and H = p in 3), we obtain that WSEAS RANSACIONS on MAHEMAICS p )+p ))dt+ H c c c)dt. 5) We differentiate ) with respect to c and using 9) we obtain the maximum condition H t),c t),pt),t) H t),c,pt),t), c [,A] i.e. H c c c), c [,A]. 6) From 6) we have that the second integral in 5) is, so or equivalently, p ) + p ))dt 7) d dt p ))dt = p ). Hence, using ) we obtain the inequality θ θ e n)t e θc t) )dt e n)t e θct) )dt, which prove that t),c t)) is an optimal trajectory in problem P. Remar 3 he optimal trajectory of the problem P) in the conditions of a competitive equilibrium is the solution of the following system t) = ft)) n + δ)t) x ct) 8) ct) = θ [ )f t)) δ)]. 9) 4 Qualitative analysis of the optimal solution Proposition 4 he system of differential equations t) = f t)) n + δ) t) x ct) 3) ct) = [ τr ) f t)) δ )] θ 3) exhibits saddle-path stability. Proof: In order to determinate the steady state of the above system we choose the stationary solutions t) =, ct) = c. From c t) = we obtain f ) = + δ. 3) Using the properties of the function f, it results that the equation 3) has a unique solution = with which we may determine c = f ) n + δ) x. 33) he point,c ) represents a steady state for the system of differential equations. In order to investigate the stability of the steady state,c ) we linearize the system 3)-3)in the steady state,c ) and we obtain t) f ) n δ ) t) ) ct) c ) ct) θ ) f ) t) ) he matrix of the linearized system is f ) ) n δ θ ) f. ) he eigenvalues are solutions of equation λ λ f ) n δ ) + θ )f ) =. 34) Because f is a concave function, it results that the determinant of the equation is positive = f ) n δ ) 4 θ )f ) > and p = θ ) f ) <. Hence, the equation 34) has two real roots for the contrary signs, = f ) n δ) ±. 35) ISSN: Issue, Volume 8, December 9

6 Using 3), it results that the relation 35) which gives the eigenvalues, can be written thus, = ± he eigenvalues of the linearized system being the real numbers with contrary signs, it results that the steady state,c ) is the saddle point. Because the steady state,c ) is saddle point, there are two manifolds passing through the steady state: a stable manifold W s and an instable manifold W i. he dynamic equilibrium follows the stable manifold heorem 3 i) he eigenvector, corresponding to the eigenvalue = WSEAS RANSACIONS on MAHEMAICS n n) τ 4 r θ τ r)f ) tangent in the steady state,c ) to the stable manifold W s is given by v = α,v ), α R n+ n) τ v = r τ 4 r θ )f ) ; ii) he eigenvector, corresponding to the eigenvalue n+ n) τ λ = r τ 4 r θ τ r)f ), tangent in the steady state,c ) to the stable manifold W i is given by ω= α,ω ), α R where n n) τ ω = r τ 4 r θ )f ). Proof: he matrix of the linearized system is : f ) n + δ) ) A= θ )f ) Its eigenvalues are the roots of the characteristic equation λ traλ + deta = where tra = f )-n+δ) = n and τ r deta = θ ) f )., So, the eigenvalues are given by n n) τ = r τ 4 r θ τ r)f ) <. and n+ n) τ λ = r τ 4 r θ τ r)f ) > Next, we shall calculate the eigenvector v = v,v ) associated to the eigenvalue. his vector is tangent in the steady state,c ) to the optimal trajectory. he eigenvector is the solution of the equation Av = v, v = v,v ) herefore, f ) ) n δ v θ ) f ) v ) = which means that f ) ) n δ v v = θ )f ) v v =. ), Normalizing v = and taing into account 3) we obtain v =,v ), where n+ n) τ v = r τ 4 r θ τ r)f ) >. he slope of the stable manifold W s in the steady state,c ) will be given by v. In the same way for the eigenvalue λ we obtain the associated eigenvector ω=,ω ), where n τ ω = r n) τ 4 r θ τ r)f ) <. he slope of the instable manifold W i in the steady state,c ) will be given by ω. 5 he dependence of the steady state, c ) on the growth rate of the labor force n Proposition 5 he function c n) is strictly decreasing with respect to n and does not depend on n. ISSN: Issue, Volume 8, December 9

7 WSEAS RANSACIONS on MAHEMAICS Proof: Differentiating with respect to n the relation we obtain consequently, f ) = + δ f ) d dn = d dn =. herefore, because d dn =, n) does not depend on n. Differentiating with respect to n the relation we obtain c = f ) n + δ) x dc dn = f ) d n + δ)d dn dn or equivalently dc dn = f ) n δ ) d dn Because d dc =, we obtain dn dn = <, therefore c n) is strictly decreasing with respect to n. Due to the fact that the parameter n represents the growth rate of labor force, the result of the above theorem from an economic point of view, would be interpreted as following: the optimum level of capital is not influenced by labor force and if the labor force would grow in time then the optimum level of consumption per capita would shrin in time as well. A reduction of labor force would lead to a rise in the optimum consumption level per capita. 6 Change in Public Policy We will study the effects of a tax change. We suppose that the economy is initially at the steady state corresponding to the given values of the different tax parameters. Also, we suppose that the government changes the tax rate on interest income from the initial value τ r to a new one τ r, eeping total expenditure per unit of labor s + x constant, and adjusting the tax rate on wage income τ w as needed to preserve budget balance at any moment in time. In the following, we will determine the effects of the fiscal policy change on the steady-state values of capital and consumption c, given by the solution of the system ct) = f ) = + δ 36) t) = c = f ) n + δ) x 37) Let,c ) be the steady state value of the capital and the consumption c. Proposition 6 he function τ r ) is strictly decreasing with respect to τ r. Proof: Due to the relation 36), verifies f ) = + δ. 38) Differentiating with respect to τ r on both members of the relation 38) we obtain Hence, we have f ) d = d = ). f ) ). his equality and the fact that the function f is strictly concave imply d <. herefore, τ r ) is strictly decreasing with respect to τ r. With x fixed, the position of the = line does not depend on the value of τ r. According to the Proposition 6, we have that a decrease in τ r increases the steady-state value of the capital from the initial value to a new one and shifts the c t) = isocline to the right, determining the new steady-state value of the consumption, as in fig.. he effect on consumption depends on the slope of the t) = isocline at the steady state,c ), given by c ) = f ) n + δ) = n 39) Because c ) >, we obtain that a decrease in τ r will grow the steady state value of consumption from the initial value c to a new one c. herefore, a decrease in τ r encourages accumulation by growing the net return on saving. In the long run, therefore, the capital is an decreasing function of τ r. ISSN: Issue, Volume 8, December 9

8 WSEAS RANSACIONS on MAHEMAICS Fig. Because the steady state utility, given by uc ) = θ e n)t e θc dt = θ n) e θc is a decreasing function of steady state consumption, decreases in τ r will grow steady state value of consumption and therefore will grow welfare. In order to calculate the net changes upon welfare, we shall evaluate the utility function with respect to the tax rate on interest income τ r. For this we define V τ r ) = θ e n)t e θctτ ) ) dt 4) which expresses the global utility of the representative household as a function of the tax rate on interest income τ r. c t τ r ) represents the consumption along the stable manifold of the system t) = ft)) n + δ)t) ct) x4) ct) = θ [ )f t)) δ)] 4) corresponding to a given value of the tax rate on interest income τ r. he stable manifold of the system 4)-4) can be looed as the graph of the function c = p,τ r ) 43) which expresses the equilibrium value of consumption c as a function of the current value of the capital and the tax rate on interest income τ r. his relation is nown in dynamic programming as a policy function: it relates the optimal value of a control variable to the state variable. Because the slope of the stable manifold at the steady state is negative, being given by the slope of the stable eigenvector corresponding to the negative eigenvalue p = v = ) n + 44) + n ) 4 θ )f ) we have that an increases of the tax rate on interest income τ r will shift the manifold upward. hus, p τr >. 45) Substituting the policy function into the evolution equation of the capital, the relation 4), we obtain the evolution equation of the capital along the stable manifold t) = ft)) n + δ)t) pt),τ r ) x = = χ t),τ r ). 46) his equation has a unique steady state that coincides with the steady state of the system 4)-4). aing into account 44) and 46), we have χ,τ r ) = p,τ r ) + 47) + f ) n δ ) = < therefore, that the steady state is stable. he solution of the differential equation 46), denote with t τ r ), us give the equilibrium value of as a function of time and the tax rate on interest income τ r. Substituting the solution of the differential equation 46) into the policy function we obtain the trajectory of consumption c as a function of time and the tax rate on interest income τ r c t τ r ) = p t τ r ),τ r ). 48) In order to analyze the effect of the tax rate on interest income τ r on the consumption we will calculate the marginal change in consumption at the moment t, so that: dc t τ r ) = p t τ r ),τ r ) d t τ r ) +p τr t τ r ),τ r ). 49) he solution of the equation 46), t τ r ), must satisfy the equation 4) t,τ r )=ft,τ r )) n+δ)t,τ r ) pt,τ r ),τ r ) x 5) ISSN: Issue, Volume 8, December 9

9 where we denote t τ r ) = t,τ r ). Differentiating the equation 5) with respect to τ r and taing into account the fact that τr t,τ r ) = d τ r dt WSEAS RANSACIONS on MAHEMAICS t,τ r ), we have: d τ r dt t,τ r) = f t,τ r )) d t,τ r ) n+δ) d t,τ r ) p t,τ r ),τ r ) d t,τ r ) p τr t,τ r ),τ r ) Substituting 55) in 49), we can calculate the derived of the consumption trajectory as function of τ r : dc t τ r ) = p,τ r ) e λt ) p τ r,τ r ) +56) λ + p τr,τ r )=p τr e λt ) p ) +. from the equation45) we obtain: d τ r dt t,τ r) = d t,τ r )f t,τ r )) n + δ) 5) p t,τ r ),τ r )) p τr t,τ r ),τ r ) dc τ r ) = p τr >. 57) he equation 5) can be rewritten at the steady state such as: d τ r dt or equivalent with t,τ r ) = d t,τ r ) p τr,τ r ) τr t) = τr t) p τr,τ r ) 5) which is a differential equation with constant coefficients he general solution of the differential equation 5) is given by: ) τ r t) = e λt τr ) + e t p τr,τ r ) 53) Because <, the steady state value of this equation is stabile and τr asymptotically converges at the steady state value, given by: τ r = p τ r,τ r ) < 54) which is the derived of the steady state value of the capital with respect to τ r. Due to the fact that τr ) is predetermined, it results that the changes in τ r does not produce changes in τ r ). Hence, the trajectory of the capital τr is given by: ) τr t) = e t p τr,τ r ) 55) From the equation 55), we note that a reduction in the tax rate on interest income τ r determines a reduction of capital, towards the new value of the steady state, at an exponential rate given by the stabile root of the system. Using 47), 39) and 45) in 56) we obtain: dc τ r ) ) ) p p + = p τr + = p τr = = p τ r f ) n δ) = 58) = p ) τ r n <. Considering the relation 57) we obtain that the immediate effects of fiscal policy consist of the growth of consumption per capita, witch determines the shift of the optimal trajectory upwards. Considering the relation 58) we obtain that the long term effects consist in the reduction of consumption per capita until the reach of the new steady state value. he relations 57)-58) show that an increase in the tax rate of the interest income τ r will discourage the accumulation due to the reduction of the net income rate, witch will consequently encourage the short-term consumption, thus, on the long-term, the capital will be reduced, a reduction witch will determine a reduction of consumption on the long-term Next, we shall analyze the effects of the change of income rate of capital τ r over social welfare. Differentiating with respect to τ r V τ r ) = θ e n)t e θctτ ) ) dt 59) evaluated in the steady state and using 56), we obtain ISSN: Issue, Volume 8, December 9

10 WSEAS RANSACIONS on MAHEMAICS V τ ) = e θc e n)tdc t τ r ) dt = 6) = e θc e n)t p τr e λt ) p ) + dt= = e θc p τr = e θc p τr p + e θc p τr p = e θc p τ r e n)t p + e t p ) dt = e n)t dt e n)t e t dt = p + n ) p = n = e θc p τr n p n) n ). Using 47) and 39) in 6), we obtain V τ r ) = e θc p τr n n) n ) = = e θc p τr n τ r. 6) Because p τr >, < and > n, we have V τ r ) <. herefore, the welfare is strictly decreasing function as function of the tax rate on interest income τ r. Hence, we have that the optimal policy is that the tax rate on interest income τ r be zero. 7 Conclusion In this paper we have analyzed an economic growth model with taxes in which the utility is given by exponential function. his economic growth model leads us to an optimal control problem. We have determined the necessary and sufficient conditions for optimality. Using these conditions, we have shown that the optimal trajectory for the optimal control problem is the solution for a differential equations system. We have proven the existence, uniqueness and stability of the study state for a differential equations system. We have shown that the optimum level of capital is not influenced by labor force and if the labor force would grow in time then the optimum level of consumption per capita would shrin in time as well. We have investigated the effects of the fiscal policy changes on the steady-state values of the capital and consumption, and on welfare. We have shown that a decrease in tax rate on interest income τ r encourages accumulation by growing the net return on saving. On the long run, therefore, the capital is a decreasing function of τ r. Also, we have shown that a decrease in tax rate on interest income τ r will grow the welfare. References [] R. J. Barro, X. Sala-i-Martin, Economic Growth, McGraw-H ill, New Yor, 995. [] O. Bundău, An economical growth model with taxes and exponential utility, Sci. Bull. of the Politehnica University of imisoara, Fasc., 8. [3] E. Burmeister, A. R. Dobell, Matematical theories of economic growth, Great Britan, 97. [4] J. Coletsos, B. Koinis, Optimization Methods for Nonlinear Eliptic Optimal Control Problems with State Constraints, Wseas ransaction on Mathematics, issue, vol. 5, pp. 69-8, 6. [5] Ş. Cruceanu and C. Vârsan, Elemente de control optimal şi Aplicaţii ˆın economie, Ed.ehnică, Bucharest, 978. [6] Z. Man, X. Yu, Modelling and Stability Analysis of Nonlinear CP Dynamics with RED, Wseas ransaction on Mathematics, issue 8, vol. 6, pp , 6. [7] A. Seierstad, K. Sydsæter, Optimal Control heory with Economic Applications, North-Holland, Amsterdam, 987. [8]. amai, Optimal fiscal policy in an endogenous growth model with public capital: a note, Journal of Economics, Vol 93, nr., pp. 8-93, 8. [9] C. Udrişte, M. Ferrara, Multitime Models of Optimal Growth, Wseas ransaction on Mathematics, issue, vol. 7, 8. ISSN: Issue, Volume 8, December 9