1 INTRODUCTION

Various models of economic development have drawn the attention of both economists and mathematicians (e.g., see [1, 2]). On the one hand, this interest is caused by practical necessity since in many cases mathematical modeling is the only available means of studying complex economic systems. On the other hand, modeling economic systems often yields new meaningful mathematical problems. Studying nonlinear models of economic dynamics that can generate cyclical movements, which, in particular, can be interpreted as recurring economic crises are of special interest. One such model, in which cyclical dynamics are possible is N. Kaldor’s business cycle model proposed in the 1940s (see [3, 4]).

The business cycle model proposed by N. Kaldor (the Kaldor model) describes the dynamics of national income and fixed capital assets (capital) in an idealized market economy under specified functions of investment and savings. The main idea of the model is that the observed periods of economic growth followed by recessions (crises) may have an internal nature associated with the non-linearities inherent in real economic processes. The goal of the Kaldor model is to study the conditions under which the combined effect of investment and the demand for the dynamics of the economic system inevitably lead to a cycle (see [4]). This model has been studied by many authors, mainly from the viewpoint of its bifurcation analysis, identification of the conditions under which a limiting cycle arises among the trajectories of the model, and also for studying the effect produced on the model’s dynamics by random perturbations and delays (see [2, 5–7]. The issue of stabilization of the dynamics in the Kaldor model was considered in [8]. For a detailed discussion of the Kaldor model in the context of the theory of business cycles, see [9].

The version of the Kaldor model discussed in this paper is slightly different from the original. In addition, the model’s dynamics were expanded by introducing a control parameter, which characterizes consumer demand stimulation by a hypothetical central planner (state). By varying this parameter, the central planner can stimulate demand and thereby affect the performance of the system. In the absence of demand stimulation, the system follows trajectories corresponding to the original uncontrolled model. In particular, it can be cyclical movements that can be interpreted as the periodically repeated ups and downs of economic activity. In the case of a controlled model, the quality of the control procedure at each time instant is estimated by the value of the national income taken with allowance for the costs of demand stimulation. The existence of the optimal stationary regime has been proved in the respective optimization problem and the conditions guaranteeing its uniqueness have been presented. It is proved that the optimization of the stationary regime always leads to a simultaneous increase in the values of the instantaneous utility and consumption against the stationary states of the uncontrolled model. Using numerical simulations, it was shown that in some situations, the trajectories of an uncontrolled system perform cyclical movements corresponding to recurring crises, while with the optimal choice of the control parameters it only has a stable optimal equilibrium state, which is economically preferable to uncontrolled cyclic movements.

It should be noted that the character of stationary states often completely determines the long-term dynamics of the considered system. If, however, its limit set consists of a single stable equilibrium state, then all trajectories of the controlled system, regardless of the initial state, asymptotically tend to it. In connection with this circumstance, the optimal stationary states play an important role in studying the processes of economic growth [10]. Note that the well-known golden rule of capital accumulation in the Solow–Swan neoclassical model of economic growth corresponds to the equilibrium state of the system, optimal in terms of the consumption value (see, for example, [1, Chapter 2], [11, Chapter 1]).

The further presentation is constructed as follows. Section 2 describes the modified Kaldor model. In Section 3, its controlled version is constructed and the instantaneous utility function, which characterizes the quality of the control procedure at each time point \(t \geqslant 0\), is determined. Then the general properties of the constructed control system are considered, the existence of the best stationary regime in terms of the selected instantaneous utility function is proved, and its properties are studied. Section 4 presents the results of the numerical simulation.

2 MODIFIED KALDOR MODEL

We consider the following version of the Kaldor model (see, e.g., [2–4, 6]):

$$\begin{gathered} \dot {Y}(t) = \alpha \left[ {I\left( {Y(t), K(t)} \right) - S\left( {Y(t)} \right)} \right], \hfill \\ \dot {K}(t) = I\left( {Y(t), K(t)} \right) - \delta K(t). \hfill \\ \end{gathered} $$
((1))

Here, \(Y(t)\) and \(K(t)\) are the values of the national income and fixed capital assets (capital) at time \(t \geqslant 0,\)\(\alpha > 0\) is the adjustment factor characterizing the reaction rate of the system, and \(\delta > 0\) is the fixed asset depreciation rate.

It will be assumed that the functions of investment \(I(Y, K),\)\(Y \geqslant 0,\)\(K \geqslant 0,\) and savings \(S(Y),\)\(Y \geqslant 0,\) are in the following form:

$$I(Y, K) = \left\{ \begin{gathered} I(Y) - \beta K {\text{ for }} K \leqslant I(Y){\text{/}}\beta , \hfill \\ 0 \,\,\,\,{\text{for }} K > I(Y){\text{/}}\beta , \hfill \\ \end{gathered} \right. \,\,\,\,S(Y) = \gamma Y,$$
((2))

where \(\beta > 0,\)\(0 < \gamma < 1,\) and the function \(I: [0; \infty ) \mapsto {{\mathbb{R}}^{1}}\) has a logistic form; i.e., \(I(Y)\) is a positive twice continuously differentiable function such that \(I(0) = {{I}_{0}} > 0,\)\({{\lim }_{{Y \to \infty }}}I(Y) = {{I}_{\infty }} < \infty ,\)\(I{\kern 1pt} '(Y) > 0\), and there is such \(\hat {Y} > 0,\) that \(I{\kern 1pt} ''(Y) > 0\) if \(Y < \hat {Y}\) and \(I{\kern 1pt} ''(Y) < 0\) if \(Y > \hat {Y}.\)

It should be noted that under the functions of investment \(I(Y, K)\) and savings \(S(Y),\) defined by Eqs. (2), system (1) is different from the original Kaldor model [4]. Here, in particular, the investment function \(I(Y,K)\) is non-negative and the savings function \(S(Y)\) is independent of the phase variable \(K.\) The non-negativity of the investment function means that in the considered economy the withdrawal of capital (i.e., the case of negative investments) is not possible. Parameter \(\gamma \) in the second equality of (2) characterizes the amount of savings \(S(t)\) as the fixed part \(\gamma Y(t)\) of the national income \(Y(t)\) at every time instant \(t \geqslant 0.\)

It is easily seen that, by the assumptions made above, the right-hand part of the system (1) is Lipschitzian. Therefore, according to the standard existence and uniqueness theorems (see, e.g., [12]), for any initial state \(({{Y}_{0}}, {{K}_{0}})\) and \({{Y}_{0}}\, \geqslant \,0,\)\({{K}_{0}} \geqslant 0,\) system (1) has a unique solution \(\left( {Y(t),} \right.\)\(\left. {K(t)} \right),\) defined on some time interval \([0, T),\)\(T > 0,\) and that satisfies the initial conditions

$$Y(0) = {{Y}_{0}},\,\,\,\,K(0) = {{K}_{0}}.$$
((3))

It follows from the positivity of the function \(I(Y)\) and Eqs. (2) that this solution lies in the non-negative orthant \(\mathbb{R}_{ + }^{2} = \left\{ {(Y,K): Y \geqslant 0, K \geqslant 0} \right\}.\)

Moreover, the following statement is true.

Theorem 1. Let functions of investment\(I(Y,K)\)and savings\(S(Y)\)be defined byEqs. (2). Then

(1) For any\({{Y}_{0}} \geqslant 0\) and \({{K}_{0}} \geqslant 0\)and any values of the parameters\(\alpha > 0,\)\(\beta > 0\)and\(0 < \gamma < 1\)the rectangle

$$G = \left\{ {(Y,K): 0 \leqslant Y \leqslant \max \left\{ {{{Y}_{0}},{\text{ }}{{I}_{\infty }}{\text{/}}\gamma } \right\}, 0 \leqslant K \leqslant \max \left\{ {{{K}_{0}}, {{I}_{\infty }}{\text{/}}\beta } \right\}} \right\}$$
((4))

is an invariant set of system (1).

(2) For any arbitrarily small\(\varepsilon > 0\), there is a Jordan curve (a homeomorphic image of a circle)\({{\Gamma }_{\varepsilon }},\)lying in the rectangle\(G\)and spaced away from its boundary at most by\(\varepsilon \) (in the Hausdorf metric boundary) such that on the curve\({{\Gamma }_{\varepsilon }}\)the vector field of system (1) is directed strictly inside the set bounded by this curve.

Proof. We prove statement (1). To do this, we define the curve \(\Gamma \subset G\) by the equality

$$\Gamma = \left\{ {(Y,K): Y \in \left[ {0,\max \left\{ {{{Y}_{0}},{{I}_{\infty }}{\text{/}}\gamma } \right\}} \right], K = I(Y){\text{/}}\beta } \right\}$$

and denote by \({{\Gamma }_{1}} = \left\{ {(Y,K) \in G:K = 0} \right\},\)\({{\Gamma }_{2}} = \left\{ {(Y,K) \in G:Y = \max \left\{ {{{Y}_{0}},{{{{I}_{\infty }}} \mathord{\left/ {\vphantom {{{{I}_{\infty }}} \gamma }} \right. \kern-0em} \gamma }} \right\}} \right\},\)\({{\Gamma }_{3}} = \left\{ {(Y,K) \in } \right.,\)\(\left. {G:K = \max \left\{ {{{K}_{0}},{{{{I}_{\infty }}} \mathord{\left/ {\vphantom {{{{I}_{\infty }}} \beta }} \right. \kern-0em} \beta }} \right\}} \right\},\) and \({{\Gamma }_{4}}\, = \,\left\{ {(Y, K) \in G: Y\, = \,0} \right\}\), the bottom, right, top, and left sides of the rectangle, respectively (see Fig. 1). By Eqs. (2), at each point \((Y,K)\, \in \,G,\) lying on the curve \(\Gamma \) or above it, the vector field of system (1) has the form \(f(Y,K) = ( - \alpha \gamma Y, - \delta K).\)

Fig. 1.
figure 1

Vector field of system (1) in G.

Full size image

Therefore, in the rectangle \(G\) above the curve \(\Gamma \) at \(Y > 0\), the trajectories of system (1) move along the integral curves of the form \(K = C{{Y}^{{{\delta \mathord{\left/ {\vphantom {\delta {\alpha \gamma }}} \right. \kern-0em} {\alpha \gamma }}}}},\) where \(C > 0\) is constant in the direction of the coordinates’ origin. If the trajectory \(\left( {Y(t), K(t)} \right)\) at some instant \(\tau \geqslant 0\) is on the side \({{\Gamma }_{4}}\) of the rectangle \(G\) above the curve \(\Gamma \) (i.e., \(Y(\tau ) = 0,K(\tau ) > {{{{I}_{0}}} \mathord{\left/ {\vphantom {{{{I}_{0}}} \beta }} \right. \kern-0em} \beta }),\) then on some finite segment \([\tau ,{{\tau }_{1}}],\)\({{\tau }_{1}} > \tau ,\) this trajectory moves down the side \({{\Gamma }_{4}}\) until at time \({{\tau }_{1}}\) it strikes the curve \(\Gamma .\)

At the points \((Y,K) \in G,\) lying below the curve \(\Gamma ,\) the vector field of system (1) has the form \(f(Y,K) = \left( {\alpha (I(Y) - \beta K - \gamma Y),I(Y) - (\beta + \delta )K} \right)\) (see Fig.1). Hence it follows that at all points \((Y,K) \in {{\Gamma }_{4}},\) lying on the curve \(\Gamma ,\) or below it and at all points \((Y,K),\) lying on the sides \({{\Gamma }_{1}},\)\({{\Gamma }_{2}}\), and\({{\Gamma }_{3}}\) of the rectangle \(G,\) the vector field of system (1) is directed inside rectangle \(G.\)

Thus, system (1) has no stationary points on the boundary of rectangle \(G\) and no solution \(\left( {Y(t),K(t)} \right)\) of system (1) satisfying the initial condition (3) can leave rectangle \(G.\)

Let us prove statement (2). At the point \((0,{{{{I}_{0}}} \mathord{\left/ {\vphantom {{{{I}_{0}}} \beta }} \right. \kern-0em} \beta }) \in {{\Gamma }_{4}} \cap \Gamma \), the right-hand side of system (1) is a vector \(f(0,{{{{I}_{0}}} \mathord{\left/ {\vphantom {{{{I}_{0}}} \beta }} \right. \kern-0em} \beta }) = (0, - \delta {{{{I}_{0}}} \mathord{\left/ {\vphantom {{{{I}_{0}}} \beta }} \right. \kern-0em} \beta }) \ne 0.\) Consequently, by the continuity of the vector field of system (1) in \(G,\) for any arbitrarily small \({{\varepsilon }_{1}}\, > \,0\), there can exist a point \({{A}_{0}}\, \in \,{{\Gamma }_{4}},\) lying below curve \(\Gamma ,\) and points \({{A}_{1}}\, \in \,\Gamma ,\)\({{A}_{2}}\, \in \,\Gamma ,\) and \({{A}_{1}}\, \ne \,{{A}_{2}},\) with point \({{A}_{1}}\) lying between \((0,{{{{I}_{0}}} \mathord{\left/ {\vphantom {{{{I}_{0}}} \beta }} \right. \kern-0em} \beta })\) and \({{A}_{2}},\) such that points \({{A}_{0}},\)\({{A}_{1}}\), and \({{A}_{2}}\) are spaced away from the point \((0,{{{{I}_{0}}} \mathord{\left/ {\vphantom {{{{I}_{0}}} \beta }} \right. \kern-0em} \beta })\) by at most \({{\varepsilon }_{1}}\) and the segment \([{{A}_{0}},{{A}_{1}}]\) is the transversal of system (1). Let \({{\Gamma }_{5}}\) and \({{\Gamma }_{6}}\) be integral curves of the trajectories of system (1) entering points \({{A}_{1}} \in \Gamma \) and \({{A}_{2}} \in \Gamma \) from a part of rectangle \(G\) above curve \(\Gamma \) (see Fig. 1). Obviously, for any sufficiently small \({{\varepsilon }_{1}} > 0\), by the definition of the system above curve \(\Gamma \) (see (2)), curves \({{\Gamma }_{5}}\) and \({{\Gamma }_{6}}\) are above \(\Gamma .\) Let \({{A}_{3}}\) be the intersection point of curve \({{\Gamma }_{6}}\) with the side \({{\Gamma }_{3}}\) of the rectangle \(G\) (see Fig. 1). Using the explicit form of integral curves of system (1) in rectangle \(G\) above curve \(\Gamma \), it can easily be shown that there exists a smooth curve \(\tilde {\Gamma },\) connecting points \({{A}_{1}}\) and \({{A}_{3}}\) and lying in \(G\) between curves \({{\Gamma }_{5}}\) and \({{\Gamma }_{6}},\) such that the vector field of system (1) is transversal to this curve. Then, by construction, curve \({{G}_{{{{\varepsilon }_{1}}}}},\) formed by sides \({{\Gamma }_{1}}\) and \({{\Gamma }_{2}}\) of rectangle \(G\) and (dependent on \({{\varepsilon }_{1}}\)) parts of sides \({{\Gamma }_{3}}\) (from \({{\Gamma }_{2}}\) to \({{A}_{3}}\)) and \({{\Gamma }_{4}}\) (from \(0\) to\({{A}_{1}}),\) fragment \([{{A}_{1}},{{A}_{2}}]\) and curve \(\tilde {\Gamma }\)connecting \({{A}_{1}}\) and \({{A}_{3}}\) is Jordanian and the vector field of system (1) is directed strictly inside the set bounded by this curve. By construction, for arbitrarily small \(\varepsilon > 0\), we can choose a small \({{\varepsilon }_{1}} > 0,\) such that curve \({{\Gamma }_{{{{\varepsilon }_{1}}}}}\) is spaced away from the boundary of rectangle \(G\) by at most \(\varepsilon \). In this case, curve \({{\Gamma }_{{{{\varepsilon }_{1}}}}}\) satisfies all conditions of statement (2). Statement (2) has been proved.

From theorem 1 the following properties of system (1) immediately follow.

Corollary 1.For any\({{Y}_{0}} \geqslant 0\) and \({{K}_{0}} \geqslant 0\)and any values of parameters\(\alpha > 0,\)\(\beta > 0\), and\(0 < \gamma < 1\), thesolution\(\left( {Y(t),K(t)} \right)\)of the Cauchy problem (1) and (3) isdefined on the entire infinite interval\([0, \infty )\)andliesin the corresponding rectangle\(G\) (see (4)).

Indeed, by the definition of rectangle \(G,\) we have \(({{Y}_{0}},{{K}_{0}}) \in G.\) By statement (1) of Theorem 1, the solution \(\left( {Y(t),K(t)} \right)\)of the Cauchy problem (1) and (3) over the entire interval of its definition \([0,T)\) lies in \(G.\) Since set \(G\) is limited, the solution \(\left( {Y(t),K(t)} \right)\) is defined over the entire infinite interval \([0, \infty ).\)

Corollary 2.For any\({{Y}_{0}} \geqslant 0\)and\({{K}_{0}} \geqslant 0\)and any values of the parameters\(\alpha > 0,\)\(\beta > 0\), and\(0 < \gamma < 1\), the open set\({{G}_{0}} = \operatorname{int} G\) (the interior of rectangle\(G)\)is invariant with respect to system (1).

Indeed, by statement (1) of Theorem 1 for any initial condition \(({{Y}_{0}},{{K}_{0}}) \in {{G}_{0}}\) there are \(\varepsilon > 0\) and the Jordan curve \({{\Gamma }_{\varepsilon }}\) in \(G\) such that the point \(({{Y}_{0}},{{K}_{0}})\) belongs to an open set bounded by the curve \({{\Gamma }_{\varepsilon }}\), and this set is invariant with respect to system (1). Consequently, the open set \({{G}_{0}}\) is invariant with respect to system (1).

Corollary 3.If system (1) has a single stationary point\((\tilde {Y},\tilde {K}) \in {{G}_{0}},\)while the real parts of the eigenvalues of the Jacobi matrix of the system at this point are negative, then for system (1) there is a closed periodic trajectory in\({{G}_{0}}\).

Indeed, by the Poincaré–Bendixson theorem [12], this corollary follows from statement (2) of Theorem 1.

For the point \((\tilde {Y},\tilde {K})\) to be stationary it is necessary and sufficient to satisfy the equalities

$$\tilde {Y} = \frac{\delta }{{(\beta + \delta )\gamma }}I(\tilde {Y}),\,\,\,\,\tilde {K} = \frac{{I(\tilde {Y})}}{{\beta + \delta }}.$$
((5))

From the logistic nature of the function, \(I(Y)\) it follows that the first equation in (5) has at least one but not more than three roots. Consequently, system (1) has at least one but not more than three stationary points \((\tilde {Y},\tilde {K}) \in G.\) Since by (2) all stationary points of system (1) are located strictly below the curve \(\Gamma ,\) in some vicinity of every stationary point, the right-hand side of system (1) is twice continuously differentiable. Therefore, the character of the behavior of system (1) near its non-degenerate stationary points can be studied using the corresponding linear approximations [12, Chapter VIII].

The Jacobi matrix \({\text{J}}(\tilde {Y},\tilde {K})\) of system (1) at the stationary point \((\tilde {Y},\tilde {K}) \in G\) has the form

$${\text{J}}(\tilde {Y},\tilde {K}) = \left( {\begin{array}{*{20}{c}} {\alpha \left( {I{\kern 1pt} '(\tilde {Y}) - \gamma } \right)}&{ - \alpha \beta } \\ {I{\kern 1pt} '(\tilde {Y})}&{ - \beta - \delta } \end{array}} \right).$$
((6))

Consequently,

$$\det {\text{J}}(\tilde {Y},\tilde {K}) = - \alpha \delta I{\kern 1pt} '(\tilde {Y}) + \alpha \gamma (\beta + \delta )$$
((7))

and

$${\text{tr J}}(\tilde {Y}, \tilde {K}) = \alpha I{\kern 1pt} '(\tilde {Y}) - (\alpha \gamma + \beta + \delta ).$$
((8))

Accordingly, the eigenvalues of the matrix \({\text{J}}(\tilde {Y},\tilde {K})\) are as follows:

$${{\lambda }_{{1,2}}} = \frac{{{\text{tr J(}}\tilde {Y}{\text{,}}\tilde {K}{\text{)}} \pm \sqrt {{{{\left( {{\text{tr J(}}\tilde {Y}{\text{,}}\tilde {K}{\text{)}}} \right)}}^{2}} - 4\det {\text{J(}}\tilde {Y}{\text{,}}\tilde {K}{\text{)}}} }}{2}.$$

Thus, the condition \(\det {\text{J}}(\tilde {Y},\tilde {K})\, > \,0\) excludes the saddle character of the stationary point \((\tilde {Y},\tilde {K}).\) In this case, by the Hartman–Grobman theorem [12], it follows from the condition \({\text{tr J}}(\tilde {Y}, \tilde {K}) < 0\) that the stationary point \((\tilde {Y},\tilde {K})\) of system (1) is asymptotically stable, and from the condition \({\text{tr J}}(\tilde {Y}, \tilde {K}) > 0\) it follows that the stationary point \((\tilde {Y},\tilde {K})\) of system (1) is asymptotically unstable.

3 OPTIMAL STATIONARY REGIMES

In a closed economy at every moment \(t \geqslant 0\) the amount of savings \(S(t)\) is an unconsumed part of the income \(Y(t)\) [3, Chapter 3]. Now we introduce in the modified Kaldor model (1) the control parameter \(u \in [0, 1],\) characterizing the share \(uS(t)\) of savings \(S(t) = \gamma Y(t),\) redistributed to consumption \(C(t) : = Y(t) - S(t)\) at the time \(t \geqslant 0\) as a result of the demand-stimulating policy implemented by the central planner. Then each value of the parameter \(u \in [0,1]\) will correspond to the new function of savings \(S(Y,u) = \gamma Y - u\gamma Y = {{\gamma }_{u}}Y,\) where \({{\gamma }_{u}} = (1 - u)\gamma .\) If \(u = 0,\) then there is no increase in consumption, the savings function remains the same: \(S(Y,0) = \gamma Y.\) If \(u = 1,\) then the entire produced product is consumed and the amount of savings in this case, is zero, i.e., \(S(Y,1) = 0.\)

Using the new savings function we transfer to the following controlled version of the Kaldor model:

$$\begin{gathered} \dot {Y}(t) = \alpha \left[ {I\left( {Y(t),K(t)} \right) - \left( {1 - u(t)} \right)\gamma Y(t)} \right], \hfill \\ \dot {K}(t) = I\left( {Y(t),K(t)} \right) - \delta K(t). \hfill \\ \end{gathered} $$
((9))

Here, as before, the investment function \(I(Y,K)\) is assumed to be determined by the first equality in (2). All Lebesgue-measurable functions \(u: [0, \infty ) \mapsto [0, 1]\) are considered as admissible controls.

Let the initial state \(({{Y}_{0}},{{K}_{0}}),\)\({{Y}_{0}} \geqslant 0,\)\({{K}_{0}} \geqslant 0,\) be given and the admissible control \(u(t)\) be specified. Then the corresponding admissible trajectory \(\left( {Y(t),K(t)} \right)\) is an absolutely continuous solution of system (9) satisfying the initial condition (3). It is readily seen that for any initial state \(({{Y}_{0}},{{K}_{0}}),\)\({{Y}_{0}} \geqslant 0,\) and \({{K}_{0}} \geqslant 0,\) and arbitrarily admissible control \(u(t)\), the corresponding admissible trajectory \(\left( {Y(t),K(t)} \right)\) is defined over the entire infinite time interval \([0, \infty )\) and lies in the band \(\left\{ {(Y,K): Y \geqslant 0, 0 \leqslant K \leqslant \max \left\{ {{{K}_{0}}, {{{{I}_{\infty }}} \mathord{\left/ {\vphantom {{{{I}_{\infty }}} {(\beta + \delta )}}} \right. \kern-0em} {(\beta + \delta )}}} \right\}} \right\}.\)

We are going to find out at what values of \(\tilde {Y} > 0\) there is continuous control \(\tilde {u}(t) \equiv u(\tilde {Y}) \in [0, 1],\)\(t \geqslant 0,\) such that under its substitution, instead of \(u(t)\), system (9) has a stationary point \(\left( {\tilde {Y},K(\tilde {Y})} \right).\) By equating the right-hand sides of system (9) to zero, by (2) we obtain the equalities

$$K(\tilde {Y}) = \frac{{I(\tilde {Y})}}{{\beta + \delta }},\,\,\,\,u(\tilde {Y}) = 1 - \frac{\delta }{{(\beta + \delta )\gamma }}\frac{{I(\tilde {Y})}}{{\tilde {Y}}}.$$
((10))

Since the admissible control \(\tilde {u}(t) \equiv u(\tilde {Y})\) must satisfy the constraint \(\tilde {u}(t) \in [0, 1],\)\(t \geqslant 0,\) for any such \(\tilde {Y} > 0\), the following inequality should be fulfilled:

$$\tilde {Y} \geqslant \frac{\delta }{{(\beta + \delta )\gamma }}I(\tilde {Y}).$$
((11))

In this case, the control \(\tilde {u}(t) \equiv u(\tilde {Y}),\)\(t \geqslant 0,\) is admissible and implements the stationary point \(\left( {\tilde {Y},K(\tilde {Y})} \right)\) (see (10)) in system (9). Further any of such points \(\left( {\tilde {Y},K(\tilde {Y}),u(\tilde {Y})} \right)\) will be referred to as the stationary regime of system (9).

It should be noted that by the properties of the function \(I(Y)\), inequality (11) holds for all sufficiently large values of \(Y > 0.\) Hence it follows that for any sufficiently large \(Y > 0\) system (9) has the corresponding stationary regime \(\left( {Y,K(Y),\,u(Y)} \right).\) Moreover, since \({{\lim }_{{Y \to \infty }}}I{\kern 1pt} '(Y){\kern 1pt} \, = \) 0, for all sufficiently large values \(Y > 0\) inequalities \(\det {\text{J}}\left( {Y,K(Y)} \right) > 0\) (see (7)) and \({\text{trJ}}\left( {Y,K(Y)} \right) < 0\) (see (8)) hold true. Thus, for all sufficiently high values of the national income \(Y > 0\) the stationary point \(\left( {Y,K(Y)} \right)\) of system (9) corresponding to it is asymptotically stable. If at the stationary point \(\left( {Y,K(Y)} \right)\) the value of the national income \(Y\) is insufficiently high, then depending on the values of the model parameters \(\alpha , \beta , \gamma \), and \(\delta ,\) this stationary point may be both stable and unstable (see (7), (8)).

This naturally suggests the problem of optimizing the stationary regime of the system (9) by stimulating the demand (consumption) value.

We obtain the cost of stimulating demand by the amount \(u\gamma Y,\)\(u\, \in \,[0,1],\) in the standard way using a quadratic function \(\varphi (Y,u)\) = \({\kern 1pt} {{\omega {{{(u\gamma Y)}}^{2}}{\kern 1pt} } \mathord{\left/ {\vphantom {{\omega {{{(u\gamma Y)}}^{2}}{\kern 1pt} } 2}} \right. \kern-0em} 2},\) where \(\omega > 0\) is some constant [13]. As the function of instantaneous utility \(\Phi (Y,u)\), we take the amount of the national income with allowance for the demand stimulation cost, i.e., assuming

$$\Phi (Y, u) = y - \varphi (Y, u) = Y - \frac{{\omega {{\gamma }^{2}}}}{2}{{(uY)}^{2}}.$$

The stationary regime \(\left( {{{Y}_{*}}, {{K}_{*}}(Y), u({{Y}_{*}})} \right)\) will be called optimal if the value \({{Y}_{*}}\) provides the largest possible value of the function

$$\Phi (Y): = \Phi \left( {Y,u(Y)} \right) = Y - \frac{{\omega {{\gamma }^{2}}}}{2}{{\left( {Y - \frac{\delta }{{(\beta + \delta )\gamma }}I(Y)} \right)}^{2}}$$
((12))

amongst all stationary modes \(\left( {Y,K(Y),u(Y)} \right)\) of system (9) (see (10), (11)).

Thus, in order to obtain the optimal stationary regime \(\left( {{{Y}_{*}},{{K}_{*}}(Y),u({{Y}_{*}})} \right)\) in the controlled Kaldor model (9), the following problem (Q) has to be solved (see (11), (12)):

$$\Phi (Y) \to \max ,\,\,\,\,Y \geqslant \frac{{\delta I(Y)}}{{(\beta + \delta )\gamma }}.$$

It should be recalled that \(\hat {Y}\) is the root of the equation \(I{\kern 1pt} ''(Y) = 0,\) where \(I(Y),\)\(Y \geqslant 0\), is the logistic function included in the definition of the investment function \(I(Y,K)\) (see (2)). Due to the properties of the function \(I(Y)\) and condition \(I(0) = {{I}_{0}} > 0\) the equation

$$Y = \frac{{\delta I(Y)}}{{(\beta + \delta )\gamma }},\,\,\,\,Y > 0,$$
((13))

has at least one but not more than three roots, each of which corresponds to the equilibrium position \(\left( {Y,K(Y)} \right)\) of the initial system (1) (see (5)). We denote by \({{Y}_{{\max }}}\) the maximum root of Eq. (13).

Theorem 2.For any value of parameters of system (9) in the problem (Q) there is a solution\({{Y}_{*}}\)and the inequality \({{Y}_{*}} > {{Y}_{{\max }}}\) is satisfied. The corresponding optimal stationary control \({{u}_{*}}({{Y}_{*}})\) is positive and defined by the equality

$${{u}_{*}}({{Y}_{*}}) = 1 - \frac{\delta }{{(\beta + \delta )\gamma }}\frac{{I({{Y}_{*}})}}{{{{Y}_{*}}}}.$$
((14))

If\(\hat {Y} \leqslant {{Y}_{{\max }}},\)then the solution\({{Y}_{*}}\)of the problem (Q)is unique.

Proof. By the logistic form of the function \(I(Y)\) and condition \(I(0) = {{I}_{0}} > 0\)Eq. (13) has the maximum root \({{Y}_{{\max }}} > 0\). Since the function \(I(Y)\) is limited, for all \(Y > {{Y}_{{\max }}}\), the inequality \(Y\, > \,{{\delta I(Y)} \mathord{\left/ {\vphantom {{\delta I(Y)} {\left( {(\beta \, + \,\delta )\gamma } \right)}}} \right. \kern-0em} {\left( {(\beta \, + \,\delta )\gamma } \right)}}\) is satisfied. Consequently, for every \(Y \geqslant {{Y}_{{\max }}}\) there corresponds a stationary regime \(\left( {Y,K(Y),u(Y)} \right)\) (see (10)). From equality (12) and the fact that \({{Y}_{{\max }}}\) is a root of Eq. (13), the inequality \(\Phi (Y) < {{Y}_{{\max }}} = \Phi ({{Y}_{{\max }}})\) follows for any \(Y < {{Y}_{{\max }}}\). Thus, the problem (Q) is equivalent to the problem of maximizing the function \(\Phi (Y)\) on the half interval \([{{Y}_{{\max }}},\infty ).\)

Let us consider the function \(\Phi (Y)\) on the half interval \([{{Y}_{{\max }}}, \infty ).\)

Since \(\Phi ({{Y}_{{\max }}}) = {{Y}_{{\max }}} > 0,\) and at \(Y \to \infty \) we have \(I(Y) \to {{I}_{\infty }},\) then by the quadratic increase in \(Y\) of the function \(\varphi \left( {Y,u(Y)} \right)\) for all sufficiently large \(Y\) the inequality \(\Phi (Y) < 0\) is satisfied. Hence the existence of \(Y > {{Y}_{{\max }}}\) such that

$$\Phi ({{Y}_{*}}) = {{\max }_{{Y \in [{{Y}_{{\max }}}, \infty )}}}\Phi (Y)$$

follows from the continuity of the function \(\Phi (Y)\) and the fact that \(\Phi {\kern 1pt} '({{Y}_{{\max }}}) = 1\).

Consequently, in the problem (Q) there is a solution \({{Y}_{*}}\) and, in addition, \({{Y}_{*}} > {{Y}_{{\max }}}.\)

Condition (10) implies Eq. (14). Since, as shown above, the inequality \({{Y}_{*}} > {{Y}_{{\max }}}\) is always satisfied, Eq. (14) implies the inequality \(u({{Y}_{*}}) > 0.\)

Now we prove the uniqueness of the optimal stationary regime in the case \(\hat {Y} \leqslant {{Y}_{{\max }}}.\) In this case, due to the properties of the function \(I(Y)\) for any \(Y > {{Y}_{{\max }}}\) we have \(I{\kern 1pt} ''(Y) < 0.\) Hence, by direct calculations for \(Y > {{Y}_{{\max }}}\), we obtain

$$\Phi {\kern 1pt} ''(Y) = - \omega {{\gamma }^{2}}{{\left( {1 - \frac{\delta }{{(\beta + \delta )\gamma }}I{\kern 1pt} '(Y)} \right)}^{2}} + \frac{{\omega \gamma \delta }}{{\beta + \delta }}\left( {Y - \frac{\delta }{{(\beta + \delta )\gamma }}I(Y)} \right)I{\kern 1pt} ''(Y) < 0.$$

Consequently, the function \(\Phi (Y)\) is strictly concave on the convex set \([{{Y}_{{\max }}},\infty ).\) Therefore, it reaches its maximum at \([{{Y}_{{\max }}}, \infty )\) at the only point \({{Y}_{*}} > {{Y}_{{\max }}}.\)

Corollary 4.If Eq. (13) has at least two different roots, then the solution\({{Y}_{*}}\)of the problem (Q) is unique.

In fact, in this case, it is easy to show that by the logistic nature of the function \(I(Y)\) the inequality \(\hat {Y} \leqslant {{Y}_{{\max }}}\) is satisfied.

4 RESULTS OF NUMERICAL SIMULATION

As shown in the previous section (see Theorem 2), for any values of the model parameters inequalities \({{Y}_{*}} > {{Y}_{{\max }}}\) and \(u({{Y}_{*}}) > 0\) are satisfied. Since \(\Phi ({{Y}_{*}}) = {{Y}_{*}}\) and \(\Phi ({{Y}_{{\max }}}) = {{Y}_{{\max }}}\) (see (12), (13)), optimization of the system’s stationary state always leads to an increase in both the value of the national income taken with allowance for the demand stimulation cost and the value of consumption against the stationary states of an uncontrolled system.

We consider the results of numerical simulation in the case where the initial uncontrolled system (1) with the functions of investment and savings determined by Eqs. (2) shows a cyclical movement.

Similarly to [5], as the function \(I:[0; \infty ) \mapsto {{\mathbb{R}}^{1}}\), we select a logistic function of the form

$$I(Y) = \frac{1}{{a + {{e}^{{ - b(Y - c)}}}}} + d,\,\,\,\,Y \geqslant 0,$$
((15))

where \(a, b, c\), and \(d\) are positive numbers. Under specified positive values \(\beta \) and \(\gamma \) the respective functions of investment \(I(Y,K)\) and savings \(S(Y)\) are determined by conditions (2).

Following [5], we choose the values of the parameters \(a = 1,\)\(b = 4.2,\)\(c = 1,\)\(d = 0.6,\)\(\alpha = 2.2,\)\(\delta = 0.5,\)\(\beta = 0.6\), and \(\gamma = 0.5.\) In this case, system (1) has in the domain \(G\) the only equilibrium position \((\tilde {Y},\tilde {K}) = (1, 1)\), and the corresponding Jacobi matrix \({\text{J}}(\tilde {Y},\tilde {K})\) (see (6)) at this point has two complex conjugate eigenvalues \({{\lambda }_{{1,2}}}\, = \,0.055 \pm 0.2279802623i.\) Thus, the equilibrium position \((\tilde {Y},\tilde {K})\) is an unsteady focus. By corollary 3, at \(G\) system (1) has a periodic trajectory \(t \geqslant 0.\) We find the point \(({{Y}_{0}},{{K}_{0}})\) lying on this trajectory and calculate its period \(T > 0.\) Obviously, at some \({{Y}_{0}} > 0\), the point \(({{Y}_{0}},1)\) lies on a periodic trajectory . By substituting the time \(t = \tau T\) we pass to the phase variables \({{Y}_{1}}(\tau ) = {{Y}_{c}}(T\tau ),\)\({{K}_{1}}(\tau ) = {{K}_{c}}(T\tau ),\)\(\tau \in [0,1],\) and consider the boundary-value problem

$$\left\{ \begin{gathered} {{{\dot {Y}}}_{1}}(\tau ) = \xi (\tau )\alpha \left( {I\left( {{{Y}_{1}}(\tau )} \right) - \beta {{K}_{1}}(\tau ) - \gamma {{Y}_{1}}(\tau )} \right),\,\,\,\,\tau \in [0, 1], \hfill \\ {{{\dot {K}}}_{1}}(\tau ) = \xi (\tau )\left( {I\left( {{{Y}_{1}}(\tau )} \right) - (\beta + \delta ){{K}_{1}}(\tau )} \right), \hfill \\ \dot {\xi }(\tau ) = 0, \hfill \\ \dot {\eta }(\tau ) = 0, \hfill \\ {{Y}_{1}}(0) = \eta (0), {{K}_{1}}(0) = 1, {{Y}_{1}}(1) = \eta (1), {{K}_{1}}(1) = 1. \hfill \\ \end{gathered} \right.$$

By solving this problem by the continuation method with respect to the parameter [14, Section 7.26], we find the period \(T = \xi (0) = 19.3477\) of the trajectory \({{Y}_{c}}(t) = {{Y}_{1}}({t \mathord{\left/ {\vphantom {t T}} \right. \kern-0em} T}),\)\({{K}_{c}}(t) = {{K}_{1}}({t \mathord{\left/ {\vphantom {t T}} \right. \kern-0em} T}),t \in [0, T],\) and the value \({{Y}_{0}} = \eta (0) = 1.0568\) for which the point \(({{Y}_{0}},1)\) lies on this periodic trajectory. Due to the form of the vector field of system (1) this trajectory is the limit cycle (see Fig. 2).

Fig. 2.
figure 2

(a) Limit cycle and vector field of system (9) before optimization; (b) vector field of system (9) and its typical trajectory after optimization of equilibrium position at \(\omega = 1.8.\)

Full size image

Let us turn to the case of a controlled system (9). We assume\(\omega = 1.8.\) Since \(\hat {Y} = {{Y}_{{\max }}} = \tilde {Y},\) the set of equilibrium states of system (9) consists of the points \(\left\{ {\left( {Y,K(Y)} \right):Y \in [\tilde {Y},\infty )} \right\}\) (see (10)). By Theorem 2 the optimal value \({{Y}_{*}}\) is the only root of the equation

$$\Phi {\kern 1pt} '(Y) = 1 - \omega {{\gamma }^{2}}\left( {Y - \frac{\delta }{{(\beta + \delta )\gamma }}I(Y)} \right)\left( {1 - \frac{\delta }{{(\beta + \delta )\gamma }}I{\kern 1pt} '(Y)} \right) = 0$$

on the interval \([\tilde {Y},\infty )\) (see (12)).

The numerical solution of this equation yields \({{Y}_{*}} = 3.6768669,\)\({{K}_{*}} = K({{Y}_{*}})\) = \(1.454533\) (see (10)) and \(u({{Y}_{*}}) = 0.60440952\) (see (14)). Since at \(u(t){\kern 1pt} \) ≡ \({\kern 1pt} u({{Y}_{*}}),\)\(t \geqslant 0,\) the Jacobi matrix \({\text{J}}(\tilde {Y},\tilde {K})\) of system (9) (see (6)) has eigenvalues \({{\lambda }_{1}} = - 0.4351377161\) and \({{\lambda }_{2}} = - 1.099890762,\) the obtained optimal equilibrium state \(({{Y}_{*}}, {{K}_{*}})\) is a stable node of system (9) taken with \({{u}_{*}}(t) \equiv u({{Y}_{*}}),\)\(t \geqslant 0.\) In this case, the trajectories of this system asymptotically tend to the given equilibrium state.

On comparing the value \(\Phi ({{Y}_{*}})\) (see (12)) with the value of the national income \(t\, \in \,[0,T],\) when the system is following the limit cycle , we obtain

The results of the numerical simulation show that, for the chosen values of the parameters, the uncontrolled system (1) has a single unstable stationary state \((\tilde {Y},\tilde {K}),\) and its limit set is a stable limit cycle. In this case, the corresponding optimal stationary state \(({{Y}_{*}},{{K}_{*}})\) is a the stable limit set of the controlled system (9), taken with \({{u}_{*}}(t) \equiv u({{Y}_{*}}),\)\(t \geqslant 0,\) and it corresponds both to a larger value \(\Phi ({{Y}_{*}})\) of the national income taken with allowance for the demand stimulation cost and a greater value of consumption than when following the periodic solution \(t \geqslant 0.\) Thus, in the considered case, regardless of the initial state, optimization of the stationary state of system (9) leads to an improvement in its long-term economic indicators compared to following the uncontrolled (market) trajectories.