Analysis of λ-Hölder Stability of Economic Equilibria and Dynamical Systems with Nonsmooth Structures

Abstract

This paper develops a mathematical approach to the analysis of the stability of economic equilibria in nonsmooth models. The $\lambda$-Hölder apparatus of subdifferentials is used, which extends the class of systems under study beyond traditional smooth optimization and linear approximations. Stability conditions are obtained for solutions to intertemporal choice problems and capital accumulation models in the presence of nonsmooth dependencies, threshold effects, and discontinuities in elasticities. For $\lambda$-Hölder production and utility functions, estimates of the sensitivity of equilibria to parameters are obtained, and indicators of the convergence rate of trajectories to the stationary state are derived for $\lambda >1$. The methodology is tested on a multisectoral model of economic growth with technological shocks and stochastic disturbances in capital dynamics. Numerical experiments confirm the theoretical results: a power-law dependence of equilibrium sensitivity on the magnitude of parametric disturbances is revealed, as well as consistency between the analytical $\lambda$-Hölder convergence rate and the results of numerical integration. Stochastic disturbances of small variance do not violate stability. The results obtained provide a rigorous mathematical foundation for the analysis of complex economic systems with nonsmooth structures, which are increasingly used in macroeconomics, decision theory, and regulation models.

1. Introduction

The problem of the stability of solutions in economic models occupies a central place in modern mathematical economics. When analyzing complex socio-economic systems, it often becomes necessary to study the behavior of solutions to nonlinear equations and dynamical systems under small perturbations of parameters. To describe such systems, functions are used that do not always possess sufficient smoothness—there are discontinuities of derivatives, nonsmooth segments, threshold effects, and saturation regions. Under these conditions, classical methods of stability analysis, based on differentiability and linearization in the neighborhood of the equilibrium state, lose their applicability.

The problem of nonsmoothness is particularly acute in economic growth and equilibrium models, where agent behavior is characterized by threshold responses—for example, minimum investment levels, technological constraints, or nonlinear costs. In such cases, the system response function $F\left(x\right)$, linking the vector of economic variables $x\in {\mathbb{R}}^n$ with the resulting indicators $y\in {\mathbb{R}}^m$, may be only partially continuous or Hölder continuous, but not differentiable. For a correct analysis of the stability of solutions to systems of the form

$$F\left(x\right)+g\left(x\right)=y$$

(1)

where $g\left(x\right)$ models external or internal disturbances, an extension of the classical analysis apparatus is required.

Modern methods of nonsmooth analysis offer solutions to this problem through the introduction of generalized concepts of differentiability and tangent spaces. In particular, the use of $\lambda$-Hölder subdifferentials allows for approximation of the behavior of nonsmooth mappings in the neighborhood of an equilibrium point, while providing the ability to assess the stability of solutions and their continuous dependence on parameters [1]. Such constructions make it possible to work not only with smooth functions, but also with mappings possessing only limited regularity, which enables the analysis of a wide class of economic models with threshold effects.

For a formal description of stability, extended versions of the implicit function theorem are used, applicable to nonsmooth equations. Let $F:{\mathbb{R}}^n\to {\mathbb{R}}^m$ be a mapping possessing a $\lambda$-Hölder subdifferential at the point $\overline{x}$, and let the regularity conditions $P\left(h\right)=0$, $\mathrm{im} P^{\prime }={\mathbb{R}}^m$ hold. Then, for small disturbances $g\left(x\right)$, there exists a solution $x(y,g)$ that depends continuously on the parameters and satisfies the approximation $x(y,g)\to \overline{x}$ as $g\to 0$. This means that the equilibrium state of the economic system is stable to small changes in technological, investment, or external factors.

The economic interpretation of this statement is that in the presence of a $\lambda$-Hölder subdifferential, one can guarantee not only the existence of an equilibrium state, but also its stability in the sense of Hölder continuity. In other words, small shocks in the system do not lead to unstable behavior, and changes in variables remain proportional in order of magnitude to the perturbation. This is especially important for macroeconomic models, where exact analytical dependencies are often replaced by piecewise-continuous or approximated functions.

Consider a typical economic growth model given by the following system:

$$\dot{K}=sF\left(K\right)-\delta K+g\left(K\right)$$

(2)

where K is aggregate capital, $F\left(K\right)$ is the production function, s is the savings rate, $\delta$ is the capital depreciation rate, and $g\left(K\right)$ describes external disturbances. If $F\left(K\right)$ is nonsmooth (for example, of the form $F\left(K\right)=A{K}^{\alpha }+\mu {\left|K\right|}^{\beta }$), standard analysis via eigenvalues of the Jacobian is impossible. However, the use of $\lambda$-Hölder subdifferentials allows for consideration of an approximating polynomial $P\left(h\right)$ of order $\lambda$ and evaluation of stability based on the asymptotic behavior of the solution.

Thus, the generalized implicit function theorem and the construction of tangent spaces to solution sets of the type $M=\{x:F(x)=y\}$ make it possible to prove the existence of a stable equilibrium even for systems with nonsmooth structure. Moreover, under the condition

$$\mathrm{dist}(x+\tau \tilde{h},M)\le C{\tau }^{1+\delta }, \delta ={\displaystyle \frac{{\gamma }_0}{{\alpha }_0}}>0,$$

it can be shown that deviations of the system from the equilibrium state decay faster than linearly, which corresponds to superstable behavior.

Recent works on nonsmooth analysis in economics show that the application of Hölder approximation methods and subdifferentials provides new opportunities for studying nonlinear and nonsmooth economic systems. In particular, similar methods have been used to analyze the stability of equilibria in general equilibrium models [2,3,4], stochastic growth models [5,6], and optimal control problems with nonsmooth constraints [7,8,9].

The issues of subdifferentiation were examined in detail by the authors of [10,11,12,13]. They noted that subdifferentials allow for the formulation of generalized stationarity conditions of the Karush–Kuhn–Tucker type for interval optimization with uncertainty, the application of an extended Pontryagin maximum principle in systems with product spoilage and shortages, and the modeling of equilibrium in supply chains through variational inequalities with nonsmooth cost functions. However, they did not account for the fact that in dynamic economic growth models, such subdifferentials can be combined with stochastic disturbances of small variance to analytically assess the superstability of equilibria, with explicit derivation of power-law convergence rates of trajectories to the stationary state when $\lambda >1$. Therefore, the need arose to study this aspect of stochastic analysis—namely, the influence of small-variance stochastic shocks on $\lambda$-Hölder stability of trajectories to the stationary state.

Thus, the goal of this paper is to develop and apply the apparatus of $\lambda$-Hölder subdifferentials for analyzing the stability of solutions in nonlinear economic growth models. This paper formulates conditions for the existence and stability of solutions, investigates the properties of tangent spaces, provides examples of economic interpretations, and presents numerical results demonstrating the effectiveness of the approach.

The rest of the paper is structured as follows. Section 2 outlines the materials and methods, formulating nonlinear equilibrium models, $\lambda$-Hölder approximations, regularity conditions for solution existence, nonlinear stability criteria, and geometric properties of equilibrium sets. Section 3 examines stability in a single-sector growth model with $\lambda$-Hölder production function, deriving approximations, perturbation estimates, stability criteria, and power-law convergence rates for capital and output trajectories under stochastic perturbations. Section 4 extends the framework to multidimensional economic systems, establishing local and global stability theorems, $\lambda$-homogeneous mappings, Hausdorff distances, tangent cones, Lyapunov functionals, second-order Hölder derivatives, and equivalence to nonsmooth optimization problems. Section 5 applies the approach to stochastic equilibrium models with perturbations, probabilistic estimates, intertemporal choice problems with $\lambda$-Hölder preferences, and generalized elasticity measures. Section 6 analyzes a multidimensional capital accumulation model with nonsmooth technologies, proving component-wise $\lambda$-regularity, local and global stability, parameter sensitivity, and optimal investment policies. Section 7 reports numerical experiments on two-sector models, including stochastic simulations, convergence rate validations, and parameter sensitivity tests. The paper concludes with a summary of key findings and implications for nonsmooth economic modeling.

2. Materials and Methods

2.1. Nonlinear Equilibrium Models and Stability of $\lambda$-Hölder Type

Consider an economic system described by the following operator equation:

$$F\left(x\right)+g\left(x\right)=y,$$

(3)

where $x\in {\mathbb{R}}^n$ is the vector of endogenous variables (e.g., capital, production capacities, investment levels), $y\in {\mathbb{R}}^m$ is the exogenous vector of target indicators (e.g., aggregate output or demand), and F and g are nonlinear operators modeling the internal structure of the economy and perturbing effects, respectively.

2.2. Main Problem Statement

The system (1) is considered in the neighborhood of the equilibrium state $\overline{x}$, satisfying $F\left(\overline{x}\right)=\overline{y}$. Let the deviations from equilibrium be small: $h=x-\overline{x}$, $\mathsf{\Delta }y=y-\overline{y}$. Then, (1) can be written in the form

$$F(\overline{x}+h)-F\left(\overline{x}\right)=\mathsf{\Delta }y-g(\overline{x}+h).$$

(4)

If F were smooth, then locally $F(\overline{x}+h)\approx F\left(\overline{x}\right)+F^{\prime }\left(\overline{x}\right)h$, and the existence of a solution $h=h(\mathsf{\Delta }y,g)$ would be determined by the non-degeneracy of the matrix $F^{\prime }\left(\overline{x}\right)$. However, in real economic models, F is often nonsmooth: investment functions have discontinuities in derivatives, production functions include saturation, and resource reallocation mechanisms are defined piecewise analytically. Therefore, the classical differential does not exist, and a generalized analysis apparatus is required.

2.3. $\lambda$-Hölder Approximation

Let the mapping F possess a $\lambda$-Hölder subdifferential at $\overline{x}$. Then, in some neighborhood of the point $\overline{x}$, the approximation holds:

$$F(\overline{x}+h)=F\left(\overline{x}\right)+P\left(h\right)+R\left(h\right),$$

(5)

where $P\left(h\right)$ is a polynomial of degree $\lambda >0$ satisfying

$$P\left(0\right)=0, \mathrm{im} P^{\prime }={\mathbb{R}}^m,$$

(6)

and the residual term $R\left(h\right)$ satisfies the condition

$$\left|R\left(h\right)\right|=O\left(\right|h{|}^{1+\lambda }), \mathrm{as} |h|\to 0.$$

(7)

In the economic context, $P\left(h\right)$ describes the local approximation of the production or equilibrium function, where nonlinearities of order $\lambda$ reflect the intensity of inelastic responses of the system to small perturbations.

Substituting (5) into (4), we obtain

$$P\left(h\right)+R\left(h\right)=\mathsf{\Delta }y-g(\overline{x}+h).$$

(8)

The goal is to show that for sufficiently small perturbations $\mathsf{\Delta }y$ and g, there exists a solution $h=h(\mathsf{\Delta }y,g)$, and to estimate its stability.

2.4. Regularity Conditions and Existence of Solution

Let the mapping F satisfy the regularity conditions:

$$\exists c_0>0:|P\left(h_1\right)-P\left(h_2\right)| \ge c_0{|h_1-h_2|}^{\lambda },$$

(9)

which is equivalent to the local invertibility of P on ${\mathbb{R}}^n$. The perturbation function g is assumed small in norm:

$$\left|g\left(x\right)\right|\le \delta |x-\overline{x}{|}^{\lambda +ϵ}, ϵ>0.$$

(10)

Then, from (12), it follows that for sufficiently small $\left|\mathsf{\Delta }y\right|$, the equation

$$P\left(h\right)=\mathsf{\Delta }y-R\left(h\right)-g(\overline{x}+h)$$

(11)

has a unique solution $h(\mathsf{\Delta }y,g)$, and there exists a constant $C>0$ such that

$$\left|h(\mathsf{\Delta }y,g)\right|\le C{\left|\mathsf{\Delta }y\right|}^{1/\lambda }.$$

(12)

This condition represents the Hölder stability of the solution: small changes in the parameters y cause changes in the endogenous variables x bounded in order by ${\left|\mathsf{\Delta }y\right|}^{1/\lambda }$.

2.5. Nonlinear Stability of Equilibrium

From (12), it follows that the equilibrium state $\overline{x}$ is stable under small perturbations of the parameters y and the function g. For dynamic economic models where the equilibrium is described by a system of ordinary differential equations,

$$\dot{x}=F\left(x\right)+g\left(x\right)-y$$

(13)

Equation (12) provides the initial approximation of trajectories to the stationary state. In particular, if F admits a $\lambda$-Hölder approximation, then from (13), the estimate follows:

$$|\dot{x}\left(t\right)|\le C_1|x\left(t\right)-\overline{x}{|}^{\lambda }+C_2\left|g\left(x\left(t\right)\right)\right|,$$

(14)

and consequently, the solution $x\left(t\right)$ tends to $\overline{x}$ as $t\to \infty$, if $\lambda >1$ or if the perturbation $g\left(x\right)$ has order higher than ${|x-\overline{x}|}^{\lambda }$.

This property can be regarded as **Hölder asymptotic stability**, where the rate of return to equilibrium is not exponential (as in the smooth case), but is determined by a power law:

$$|x\left(t\right)-\overline{x}| \le C{t}^{-{\displaystyle \frac{1}{\lambda -1}}}.$$

(15)

2.6. Geometric Properties of the Equilibrium Set

Let the set of equilibrium states be of the form

$$M=\{x\in {\mathbb{R}}^n:F\left(x\right)=y\}.$$

(16)

Then, for any $x\in M$ and any tangent vector $\tilde{h}$ constructed based on the $\lambda$-Hölder approximation, the inequality holds:

$$\mathrm{dist}(x+\tau \tilde{h},M)\le C{\tau }^{1+\delta }, \delta ={\displaystyle \frac{{\gamma }_0}{{\alpha }_0}}>0.$$

(17)

This property ensures the stability of the geometric structure of the equilibrium set: with a small shift in the state x, the system returns to M with a Hölder order of convergence rate.

In the dynamic sense, (17) means that the equilibrium is stable along trajectories—a small perturbation of the variables does not remove the system from the basin of attraction, and the magnitude of the deviation decays faster than linearly, which corresponds to superstable behavior.

2.7. Section Conclusions

The formulated relations (5)–(17) provide a rigorous foundation for the analysis of the stability of nonsmooth economic systems. The $\lambda$-Hölder subdifferential serves as a substitute for the classical differential in cases where the response function F is not smooth, and conditions (9)–(10) provide sufficient criteria for the existence and stability of equilibrium.

Further analysis will be devoted to the application of these results to a specific economic growth model, where F describes the dynamics of capital accumulation and returns on investment, and the perturbations $g\left(x\right)$ model stochastic or structural fluctuations.

3. Stability of Equilibrium States in a Growth Model with $\mathbf{\lambda }$-Hölder Structure of the Production Function

Let $K\left(t\right)$ denote the aggregate capital at time t, and $Y\left(t\right)$ the aggregate output. Consider a capital accumulation model of the form

$$\dot{K}\left(t\right)=sF\left(K\left(t\right)\right)-\delta K\left(t\right)+g\left(K\left(t\right)\right),$$

(18)

where $s\in (0,1)$ is the savings rate, $\delta >0$ is the depreciation coefficient, F is the production function admitting a $\lambda$-Hölder approximation in the neighborhood of the equilibrium value $K^{*}$, and $g\left(K\right)$ is a small structural perturbation reflecting stochastic or politico-economic deviations.

The equilibrium state $K^{*}$ is determined by the condition

$$sF\left(K^{*}\right)-\delta K^{*}=0.$$

(19)

Our goal is to show that under $\lambda$-Hölder regularity of F, the equilibrium $K^{*}$ is stable, and the nature of stability is described by a power law, rather than exponential, as in the smooth case.

The issues of economic growth were examined in detail by the authors of [14,15,16]. They noted that the trajectories of capital and output in developed and European countries are shaped by classical determinants—imports, exports, foreign direct investment, social contributions, wages, and the Cobb–Douglas production function with near-constant elasticities—as well as new drivers of sustainable transition, including digitalization, fintech, green economy, and e-commerce, with quantitative assessment of their contribution to gross domestic product, value added, and final household consumption through panel analysis with fixed and random effects on data from 38 European countries over 2009–2020, as well as combinations of ordinary and fractional differential equations with stochastic processes of Itô or Lévy type for modeling nonlinear growth dynamics under macroeconomic uncertainty and external shocks. However, these works assume smoothness of the production function, which limits applicability to real economies with threshold effects, discontinuities in elasticities, and nonsmooth dependencies. In contrast, the $\lambda$-Hölder approximation allows for analytical derivation of power-law convergence rates of capital and output trajectories to the stationary state when $\lambda >1$, even in the presence of small-variance stochastic disturbances, while preserving the superstability of the equilibrium. Therefore, the need arose to study this aspect of stochastic analysis—namely, the influence of small-variance stochastic shocks on the $\lambda$-Hölder stability of trajectories to the stationary state.

3.1. Approximation of the Production Function

Assume that $F\left(K\right)$ has a $\lambda$-Hölder subdifferential at the point $K^{*}$:

$$F(K^{*}+h)=F\left(K^{*}\right)+P\left(h\right)+R\left(h\right),$$

(20)

where $P\left(h\right)$ is a polynomial of degree $\lambda >0$, $P\left(0\right)=0$, $\mathrm{im} P^{\prime }=\mathbb{R}$, and the remainder $R\left(h\right)$ satisfies the condition

$$\left|R\left(h\right)\right|\le C_0{\left|h\right|}^{1+\lambda }.$$

(21)

Then, substituting (20) into (18) and denoting $h\left(t\right)=K\left(t\right)-K^{*}$, we obtain

$$\dot{h}\left(t\right)=s[P\left(h\left(t\right)\right)+R\left(h\left(t\right)\right)]-\delta h\left(t\right)+g(K^{*}+h\left(t\right)).$$

(22)

This differential equation describes the dynamics of capital deviations from the stationary level $K^{*}$.

Stochastic extensions of macrodynamic growth models demonstrate how random disturbances affect capital trajectories. For example, in the analysis of the Brazilian economy [17], deterministic and stochastic equations verify the interactions of fiscal/monetary policy with GDP, inflation, and interest rates, showing that shocks amplify deviations from equilibrium in developing systems. Similarly, the estimation of a competitive commodity storage model [18] using particle MCMC accounts for stochastic trends in prices, capturing nonlinear effects from seasonal and demand shocks. In the energy sector [19], a stochastic–economic framework for PV systems in the UK integrates Monte Carlo to account for uncertainty in production and costs, evaluating NPV with climate fluctuations. Finally, a review [16] for G20 combines fractional differential equations with Itô–Lévy processes, modeling nonlinear growth with memory and jumps, but primarily in smooth scenarios. In our model (18), such stochasticity $g\left(K\right)$ is complemented by the $\lambda$-Hölder approximation (20)–(22), enabling analytical capture of threshold discontinuities and power-law convergence rates under small shock variance, thereby enhancing robustness to real economic irregularities.

3.2. Hölder Linearization and Perturbation Estimation

Let $P\left(h\right)$ be of the form $P\left(h\right)=a{h}^{\lambda }$, where $a>0$ is the output sensitivity coefficient to changes in capital. Then, (22) is written as

$$\dot{h}\left(t\right)=sah{\left(t\right)}^{\lambda }-\delta h\left(t\right)+\epsilon {\left|h\left(t\right)\right|}^{\mu }sign\left(h\left(t\right)\right), \mu >\lambda ,$$

(23)

where the perturbation $g\left(K\right)$ is approximated by the expression $\epsilon |K-K^{*}{|}^{\mu }sign(K-K^{*})$ with small $\epsilon$.

To analyze stability, it suffices to consider the behavior of (23) in the neighborhood of $h=0$. For $\left|h\right|\ll 1$, the terms of order ${\left|h\right|}^{\lambda }$ dominate, and the equation takes the asymptotic form:

$$\dot{h}\left(t\right)=sah{\left(t\right)}^{\lambda }-\delta h\left(t\right)+{o\left(\right|h\left(t\right)|}^{\lambda }).$$

(24)

Consider the integral inequality following from (24):

$${\displaystyle \frac{d}{dt}}\left|h\left(t\right)\right|\le C_1{\left|h\left(t\right)\right|}^{\lambda }+C_2\left|h\left(t\right)\right|.$$

(25)

If $\lambda >1$, then for sufficiently small initial conditions $\left|h\right(0\left)\right|<\eta$, the solution satisfies

$$\left|h\left(t\right)\right|\le {\displaystyle \frac{\left|h\right(0\left)\right|}{{\left(1+C_3(\lambda -1){\left|h\left(0\right)\right|}^{\lambda -1}t\right)}^{1/(\lambda -1)}}},$$

(26)

which proves the **asymptotic stability of the equilibrium $K^{*}$**. The dependence (26) characterizes the decay rate of deviations as power law:

$$\left|h\left(t\right)\right|=O\left(t^{-1/(\lambda -1)}\right), t\to \infty .$$

(27)

Thus, when $\lambda >1$, the system returns to equilibrium more slowly than exponentially, but is guaranteed—regardless of the form of the perturbation g.

3.3. $\lambda$-Stability Criterion

Let us generalize the result (27). Let the system (18) satisfy the following conditions:

$$\left\{\begin{array}{c}|F(K^{*}+h)-F\left(K^{*}\right)-P\left(h\right)|\le C{\left|h\right|}^{1+\lambda },\hfill \\ |g(K^{*}+h){| \le \epsilon |h|}^{\mu }, \mu >\lambda .\hfill \end{array}\right.$$

(28)

Then, the equilibrium $K^{*}$ is $\lambda$-stable, that is, there exists a function $V\left(h\right)$ and a constant $c>0$ such that

$$\dot{V}\left(h\left(t\right)\right)\le -c{\left|h\left(t\right)\right|}^{1+\lambda }.$$

(29)

The Lyapunov function $V\left(h\right)={\displaystyle \frac{1}{2}}{h}^2$ is suitable as $h\to 0$, since

$$\dot{V}\left(h\right)=h\dot{h}=h[sa{h}^{\lambda }-\delta h+{o\left(\right|h|}^{\lambda }\left)\right]=-\delta h^2+sa{h}^{1+\lambda }+{o\left(\right|h|}^{1+\lambda }),$$

and the sign is negative for sufficiently small $\left|h\right|$. Consequently, the equilibrium $K^{*}$ is stable in the Lyapunov sense and asymptotically stable when $\lambda >1$.

3.4. Nonlinear Stability Assessment of Aggregate Output

For a complete analysis, consider the dynamics of aggregate output $Y\left(t\right)=F\left(K\right(t\left)\right)$. From (20) and (26), we have

$$|Y\left(t\right)-Y^{*}|=|F(K^{*}+h\left(t\right))-F\left(K^{*}\right)|\le |P\left(h\left(t\right)\right)|+|R\left(h\left(t\right)\right)|.$$

(30)

Since $\left|P\left(h\left(t\right)\right)\right|=O\left(\right|h\left(t\right){|}^{\lambda })$ and $\left|R\left(h\left(t\right)\right)\right|=O\left(\right|h\left(t\right){|}^{1+\lambda })$, taking (26) into account,

$$|Y\left(t\right)-Y^{*}|=O\left(t^{-\lambda /(\lambda -1)}\right).$$

(31)

This means that output $Y\left(t\right)$ stabilizes at a higher rate than capital: the decay exponent $\lambda /(\lambda -1)$ exceeds $1/(\lambda -1)$. Thus, the system exhibits enhanced stability of production indicators relative to capital dynamics.

3.5. Case of Limiting Smoothness ($\lambda =1$)

If $\lambda =1$, then F becomes Lipschitz, and Equation (23) takes the form

$$\dot{h}\left(t\right)=(sa-\delta )h\left(t\right)+\epsilon {\left|h\left(t\right)\right|}^{\mu }sign\left(h\left(t\right)\right).$$

(32)

Then, stability is determined by the sign of the coefficient $sa-\delta$. For $sa<\delta$, we have exponential decay:

$$\left|h\left(t\right)\right|=\left|h\left(0\right)\right|e^{-(\delta -sa)t}+O\left(\epsilon \right),$$

(33)

which corresponds to the classical Solow model case. Consequently, as $\lambda \to 1^{+}$, the power-law stability (27) smoothly transitions into exponential, and the nonsmoothness of F acts as a mechanism for slowing convergence to equilibrium.

3.6. Economic Interpretation and Calibration of the $\lambda$-Hölder Exponent

For $\lambda >1$, the local response becomes superlinear, which is typical of real-world economies exhibiting threshold effects and switching costs. Common economic sources of $\lambda$-Hölder nonsmoothness include the following:

• Minimum efficient scale of investment and fixed entry costs (productivity rises sharply only after capital exceeds a certain threshold);
• Tax and regulatory kinks (marginal tax rates or compliance costs change discontinuously at specific income or scale levels);
• Credit constraints (access to external financing opens nonlinearly once a collateral threshold is reached);
• Technology adoption and adaptation costs (S-shaped diffusion trajectories);
• Network effects and platform industries (marginal returns grow quadratically or faster).

Values $\lambda \in [1.4,1.7]$ prove most realistic for contemporary economies with significant institutional and technological thresholds. When $\lambda >1$, deviations from the steady state decay according to the following power law:

$$|K\left(t\right)-K^{*}|\sim t^{-1/(\lambda -1)},$$

which is slower than the exponential decay of the $\lambda =1$ case but still guarantees stability for sufficiently small perturbations. For $\lambda \le 1$, stability may fail, and multiple equilibria and poverty traps can emerge—a pattern characteristic of the least developed countries.

3.7. Classical Smooth Stability Versus $\lambda$-Hölder Stability

The fundamental difference between classical smooth stability ($\lambda =1$) and $\lambda$-Hölder stability lies in the nature and rate of convergence of deviations from the steady state $K^{*}$.

When $\lambda =1$, deviations decay exponentially, consistent with the standard Solow–Ramsey framework. For $\lambda >1$, convergence follows a power law $|K\left(t\right)-K^{*}|\sim t^{-1/(\lambda -1)}$, which is significantly slower yet remains guaranteed for sufficiently small perturbations. Values $\lambda \in [1.5,1.7]$ yield the most realistic trajectories for contemporary economies with pronounced institutional and technological thresholds.

Domain of Definition, Convexity, and Continuity of the Perturbation Mapping

The mapping $P:{\mathbb{R}}^n\to {\mathbb{R}}^n$ introduced in (20) and (21) is defined on the entire space ${\mathbb{R}}^n$ (or at least on ${\mathbb{R}}_{+}^n$ under the economically natural domain of the production function). The set $\{h:P(h) \mathrm{is} \mathrm{defined}\}$ is therefore the whole space and, hence, convex.

No continuity of $P\left(h\right)$ is assumed in the statements of the main theorems. The only regularity required is the $\lambda$-Hölder condition on the remainder term:

$$\left|R\left(h\right)\right|=O\left(\right|h{|}^{1+\lambda }), |h|\to 0,$$

see (21). This condition replaces pointwise differentiability of the original nonsmooth mapping F and is sufficient for all subsequent stability and convergence results. The perturbation mapping $P\left(h\right)$ itself may be discontinuous (e.g., in the presence of fixed costs, tax kinks, or credit rationing thresholds), yet the $\lambda$-Hölder estimate on $R\left(h\right)$ guarantees that small deviations are dominated by the superlinear term, ensuring power-law convergence whenever $\lambda >1$.

3.8. Section Conclusions

1

For the model (18) with a $\lambda$-Hölder production function F, the equilibrium state $K^{*}$, determined by (19), is stable when $\lambda >1$.

2

The rate of return to equilibrium is described by the power law (24), depending on the regularity parameter $\lambda$.

3

Aggregate output stabilizes faster than capital, as follows from (31).

4

When $\lambda =1$, the classical exponential stability regime is restored.

Thus, $\lambda$-Hölder analysis allows for coverage of both smooth and nonsmooth production functions, providing a unified criterion for equilibrium stability in dynamic growth models.

4. Geometric Structure of the Equilibrium Set and Tangent Spaces in $\mathbf{\lambda }$-Hölder Analysis

Consider a general equilibrium economic system given by the operator equation

$$F\left(x\right)+g\left(x\right)=y,$$

(34)

where $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$ is the vector of endogenous variables (sectoral capitals, price levels, market shares, technological parameters), $y\in {\mathbb{R}}^m$ is the vector of exogenous parameters (aggregate demand, investment norms, budget constraints), $F:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is the equilibrium relations operator, and $g\left(x\right)$ represents perturbations (shocks, structural imbalances).

The set of all equilibrium states for fixed parameters y is defined as

$$M_y=\{x\in {\mathbb{R}}^n:F\left(x\right)+g\left(x\right)=y\}.$$

(35)

The goal is to construct tangent spaces and establish the stability of $M_y$ in the sense of $\lambda$-Hölder geometry, that is, to estimate the distance from a perturbed state $x+\tau h$ to $M_y$ with accuracy $O\left({\tau }^{1+\delta }\right)$.

4.1. Local Approximation and Manifold Structure

Let the mapping F at the point $\overline{x}\in M_{\overline{y}}$ possess a $\lambda$-Hölder subdifferential, and satisfy the approximation

$$F(\overline{x}+h)=F\left(\overline{x}\right)+P\left(h\right)+R\left(h\right), |R\left(h\right){|=O(\left|h\right|}^{1+\lambda }),$$

(36)

where $P\left(h\right)$ is a $\lambda$-homogeneous mapping, $P\left(ah\right)=a^{\lambda }P\left(h\right)$, with non-degenerate linear image:

$$\mathrm{im} P^{\prime }={\mathbb{R}}^m.$$

(37)

Substituting (36) into (34), we obtain in the neighborhood of $\overline{x}$ the following:

$$P\left(h\right)+R\left(h\right)+g(\overline{x}+h)=\mathsf{\Delta }y, \mathsf{\Delta }y=y-\overline{y}.$$

(38)

The solution set of (38) for small $\mathsf{\Delta }y$ describes the local structure of the manifold $M_y$.

4.2. Generalized Tangent Space

We introduce the definition.

Definition 1.

A vector $h\in {\mathbb{R}}^n$ is called a λ-Hölder tangent direction to the set $M_{\overline{y}}$ at the point $\overline{x}$ if there exists a function $\varphi \left(\tau \right)$ such that

$$\varphi \left(0\right)=0, {\varphi }^{\prime }\left(\tau \right)=O\left({\tau }^{\lambda -1}\right),$$

and

$$F(\overline{x}+\tau h+\varphi \left(\tau \right))=\overline{y}+O\left({\tau }^{1+\lambda }\right), \tau \to 0.$$

(39)

The collection of all such directions is denoted $T_{\lambda }(M_{\overline{y}},\overline{x})$ and is called the λ-Hölder tangent space to $M_{\overline{y}}$ at the point $\overline{x}$.

4.3. Lemma on $\lambda$-Hölder Projection

Lemma 1.

Let the mapping F satisfy (36) and (37). Then, for any $h\in {\mathbb{R}}^n$, there exists $\tilde{h}=h+{O\left(\right|h|}^{1+\delta })$ such that

$$\mathrm{dist}(\overline{x}+\tau \tilde{h},M_{\overline{y}})\le C{\tau }^{1+\delta }, \delta ={\displaystyle \frac{{\gamma }_0}{{\alpha }_0}}>0.$$

(40)

Proof. 

From (36), we have

$$F(\overline{x}+\tau h)-F\left(\overline{x}\right)=P\left(\tau h\right)+R\left(\tau h\right)={\tau }^{\lambda }P\left(h\right)+O\left({\tau }^{1+\lambda }\right).$$

(41)

It is required to select a correction $\varphi \left(\tau \right)$ that eliminates the leading term $P\left(h\right)$.

If P is regular, there exists $h^{\prime }$ such that $P\left(h^{\prime }\right)=P\left(h\right)$, and $|h^{\prime }-h|=O\left({\left|h\right|}^{1+\delta }\right)$. Then,

$$F(\overline{x}+\tau h^{\prime })-F\left(\overline{x}\right)={\tau }^{\lambda }P\left(h^{\prime }\right)+O\left({\tau }^{1+\lambda }\right)={\tau }^{\lambda }P\left(h\right)+O\left({\tau }^{1+\lambda }\right),$$

which proves (40).    □

4.4. Interpretation in Economic Terms

The vector $\tilde{h}$ from (40) can be regarded as a stable direction of local shift of the economic equilibrium. For example, if $x=(K,L)$—capital and labor, then $\tilde{h}=(h_K,h_L)$ describes a coupled change in factors under which the system remains close to the equilibrium set $M_y$ with an error on the order of ${\tau }^{1+\delta }$.

Thus, $\lambda$-Hölder smoothness reflects not only the stability of a specific equilibrium, but also the stability of the geometry of the equilibrium state surface in the factor space.

4.5. Theorem on Local Stability of the Equilibrium Set

Theorem 1

($\mathbf{\lambda }$-Hölder Stability of the Manifold). Let the mapping $F:{\mathbb{R}}^n\to {\mathbb{R}}^m$ satisfy conditions (36) and (37), and let the perturbation g be small in the sense that

$$\left|g\left(x\right)\right|\le \epsilon |x-\overline{x}{|}^{1+\lambda +\eta }, \eta >0.$$

(42)

Then, for any small $\left|\mathsf{\Delta }y\right|$, there exists a unique set $M_y$, λ-Hölder close to $M_{\overline{y}}$:

$${\mathrm{dist}}_H(M_y,M_{\overline{y}})\le C{\left|\mathsf{\Delta }y\right|}^{1/\lambda },$$

(43)

where ${\mathrm{dist}}_H$ is the Hausdorff distance between sets.

Proof. 

1

From (36) and (42), we obtain that the deviation of Equation (38) from the unperturbed case is bounded by an expression of order $O\left(\right|h{|}^{1+\lambda })$.

2

Applying the $\lambda$-Hölder version of the implicit function Theorem 1, we obtain the existence of a mapping $x=\mathsf{\Psi }\left(y\right)$ with norm $|\mathsf{\Psi }\left(y\right)-\overline{x}{|=O(\left|\mathsf{\Delta }y\right|}^{1/\lambda })$.

3

Then, the set $M_y=\mathsf{\Psi }\left(y\right)$ forms a $\lambda$-smooth manifold, $\lambda$-Hölder close to $M_{\overline{y}}$.

   □

4.6. Economic Interpretation Through Multiplicity of Equilibria

In economic models, especially with nonsmooth dependencies (e.g., threshold effects, external constraints, or sectoral inelasticity), the equilibrium set may not be unique. However, Theorem 1 shows that under a $\lambda$-Hölder structure, even in nonsmooth conditions, local stability of the solution set is possible: small changes in parameters y do not lead to a rupture of the equilibrium set, but only deform it within $O\left(\right|\mathsf{\Delta }y{|}^{1/\lambda })$.

This property can be interpreted as the structural stability of the economy: even with nonsmooth resource allocation mechanisms (e.g., fixed tariffs, piecewise tax rates, or threshold technologies), equilibrium remains stable in an average geometric sense.

4.7. Tangent Cones and Variational Estimates

To analyze the system’s responses to bounded perturbations, we introduce the λ-Hölder tangent cone:

$$T_{\lambda }^{+}(M_{\overline{y}},\overline{x})=\{h\in {\mathbb{R}}^n:\exists {\tau }_k\downarrow 0, h_k\to h, \mathrm{dist}(\overline{x}+{\tau }_k{h}_k,M_{\overline{y}})=o\left({\tau }_k^{\lambda }\right)\}.$$

(44)

For any $h\in T_{\lambda }^{+}(M_{\overline{y}},\overline{x})$, the estimate holds:

$$\mathrm{dist}(F(\overline{x}+\tau h),\overline{y})\le C{\tau }^{\lambda }{\left|h\right|}^{\lambda }.$$

(45)

In the economic context, (45) means that when factors are perturbed by a small amount $\tau h$, the economic system deviates from the equilibrium state by no more than the order ${\tau }^{\lambda }$. Thus, $\lambda$ acts as an exponent of stability: the larger $\lambda$, the faster the effects of local perturbations decay.

4.8. Second-Order Stability and $\lambda$-Hölder Variation Estimates

Let $\overline{x}$ be a stationary point of the system (34). Consider a perturbation of the parameter $y\to y+\mathsf{\Delta }y$ and a solution $x=\overline{x}+h$, where h is small. The expansion (36) gives

$$F(\overline{x}+h)+g(\overline{x}+h)-(F\left(\overline{x}\right)+g\left(\overline{x}\right))=\mathsf{\Delta }y.$$

(46)

Denote $\mathsf{\Phi }\left(h\right)=P\left(h\right)+R\left(h\right)+g(\overline{x}+h)-g\left(\overline{x}\right)-\mathsf{\Delta }y$. The solution h is determined by the condition $\mathsf{\Phi }\left(h\right)=0$. Consider the functional

$$V\left(h\right)={\displaystyle \frac{1}{2}}{\left|\mathsf{\Phi }\left(h\right)\right|}^2,$$

(47)

describing the “deviation from equilibrium”. Then, the stability of the point $\overline{x}$ can be regarded as the existence of a constant $c>0$ such that

$$V\left(h\right)\ge {c\left|h\right|}^{2\lambda } \mathrm{in} \mathrm{a} \mathrm{neighborhood} \mathrm{of} h=0.$$

(48)

This inequality is analogous to the second-order strict minimum condition, but in a nonsmooth (Hölder) sense.

4.9. Variations and $\lambda$-Smooth Gradients

If F admits a $\lambda$-Hölder subdifferential ${\partial }_{\lambda }F\left(\overline{x}\right)$, then the first variation is written as

$$\delta F(\overline{x};h)=\langle p,h\rangle +{O\left(\right|h|}^{\lambda }), p\in {\partial }_{\lambda }F\left(\overline{x}\right).$$

(49)

Then,

$$V\left(h\right)={\displaystyle \frac{1}{2}}{|\langle p,h\rangle +O(\left|h\right|}^{\lambda }{\left)\right|}^2={\displaystyle \frac{1}{2}}{\left|ph\right|}^2+{O\left(\right|h|}^{1+\lambda }).$$

(50)

Consequently, if $\left|ph\right|\ge c_0{\left|h\right|}^{\lambda }$ for some $c_0>0$, then (48) holds. This provides a second-order stability criterion in terms of the $\lambda$-subdifferential.

4.10. Theorem on Second-Order $\lambda$-Hölder Stability

Theorem 2.

Let the mapping $F:{\mathbb{R}}^n\to {\mathbb{R}}^m$ satisfy conditions (36)–(38), and let its λ-Hölder subdifferential ${\partial }_{\lambda }F\left(\overline{x}\right)$ be nonempty and uniformly bounded:

$$\exists c_0>0:\left|ph\right|\ge c_0{\left|h\right|}^{\lambda }, \forall p\in {\partial }_{\lambda }F\left(\overline{x}\right).$$

(51)

Then, the point $\overline{x}$ is second-order λ-stable, that is,

$$|F\left(x\right)-F\left(\overline{x}\right)|\ge c_1{|x-\overline{x}|}^{\lambda },$$

(52)

and the equilibrium set $M_y$ satisfies the estimate

$$\mathrm{dist}(x,M_y)\le C{|F\left(x\right)-y|}^{1/\lambda }.$$

(53)

Proof. 

From (49) and (51), we have

$$|F(\overline{x}+h)-F\left(\overline{x}\right){|=|ph+O\left(\right|h|}^{1+\lambda }\left)\right|\ge c_0{\left|h\right|}^{\lambda }-C_1{\left|h\right|}^{1+\lambda }.$$

For sufficiently small $\left|h\right|$, the first term dominates, which yields (52). From inequality (52), (53) follows by the invertibility property of the $\lambda$-homogeneous mapping.   □

4.11. Application to Nonsmooth Optimization Problems

Consider the problem of minimizing the potential function of the economy:

$$\underset{x\in {\mathbb{R}}^n}{min} \mathsf{\Phi }\left(x\right)={\displaystyle \frac{1}{2}}{|F\left(x\right)-y|}^2,$$

(54)

where F has a $\lambda$-Hölder subdifferential. Then, the necessary optimality conditions take the following form:

$$0\in {\partial }_{\lambda }\mathsf{\Phi }\left(\overline{x}\right)={\left({\partial }_{31}F\left(\overline{x}\right)\right)}^{\top }(F\left(\overline{x}\right)-y),$$

(55)

which is equivalent to the equilibrium equation $F\left(\overline{x}\right)=y$. Moreover, from (52), it follows that the functional $\mathsf{\Phi }\left(x\right)$ is strictly convex in the neighborhood of $\overline{x}$ in the $\lambda$-Hölder sense, and thus $\overline{x}$ is a unique stable minimum.

Thus, equilibrium problems and minimization problems [20] in nonsmooth analysis are equivalent under condition (51), and the stability criterion (52) plays the role of a generalized strict convexity condition.

4.12. Multidimensional Dynamics and $\lambda$-Stability of Trajectories

Consider a system of differential equations of the general form

$$\dot{x}\left(t\right)=-F\left(x\left(t\right)\right)+y+g\left(x\left(t\right)\right),$$

(56)

describing the dynamic restoration of equilibrium. From (52), we have

$$\langle \dot{x},F\left(x\right)-y\rangle =-{|F\left(x\right)-y|}^2+\langle g\left(x\right),F\left(x\right)-y\rangle .$$

(57)

If $\left|g\right(x\left)\right|\le \epsilon \left|F\right(x)-y|$, then

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{1}{2}}{|F\left(x\left(t\right)\right)-y|}^2\le -(1-\epsilon ){|F\left(x\left(t\right)\right)-y|}^2.$$

(58)

Consequently,

$$|F\left(x\left(t\right)\right)-y|\le |F\left(x\left(0\right)\right)-y|e^{-(1-\epsilon )t},$$

(59)

and by inequality (52), this gives

$$|x\left(t\right)-\overline{x}{| \le C|F\left(x\left(0\right)\right)-y|}^{1/\lambda }{e}^{-{\displaystyle \frac{(1-\epsilon )t}{\lambda }}}.$$

(60)

That is, even in the nonsmooth case ($\lambda >1$), trajectories remain exponentially stable with respect to the function F, but with a weakened order of stability in $x-1/\lambda$.

Thus, the $\lambda$-Hölder structure determines the “rate of transmission of stability” from the equilibrium function to the state trajectory.

4.13. Geometric Consequence: Stability of the Equilibrium Family

Let the equilibrium set be parameterized by the vector y:

$$M=\{(\overline{x}\left(y\right),y):F\left(\overline{x}\left(y\right)\right)=y\}.$$

(61)

From (53), it follows that when the parameter y changes by $\mathsf{\Delta }y$, the point $\overline{x}(y+\mathsf{\Delta }y)$ deviates from $\overline{x}\left(y\right)$ by no more than ${C\left|\mathsf{\Delta }y\right|}^{1/\lambda }$. This property means $\lambda$-Hölder smoothness of the equilibrium mapping:

$$|\overline{x}\left(y_1\right)-\overline{x}\left(y_2\right)|\le C|y_1-y_2{|}^{1/\lambda }.$$

(62)

Consequently, the family of equilibria forms a $\lambda$-Hölder manifold in ${\mathbb{R}}^{n+m}$, stable with respect to parametric perturbations.

This is a key property for macroeconomic models with multiple external shocks—small changes in parameters (e.g., tax rate, savings norm, or exports) cause Hölder-predictable changes in the equilibrium configuration of the economy.

4.14. Section Conclusions

1

The $\lambda$-Hölder tangent space $T_{\lambda }(M_{\overline{y}},\overline{x})$ is constructed, and it is proved that under conditions (36)–(37) and (42), the equilibrium set $M_y$ is stable in the sense of Hausdorff distance.

2

A theorem on second-order $\lambda$-Hölder stability is proved, ensuring the estimate (52).

3

Equivalence is shown between equilibrium stability and strict $\lambda$-convexity in optimization problems.

4

It is established that for dynamical systems (56), trajectory stability is preserved with a weakened order proportional to $1/\lambda$.

5

The family of equilibria $\overline{x}\left(y\right)$ forms a $\lambda$-smooth manifold stable with respect to parametric perturbations, confirming the structural stability of economic systems with nonsmooth dependencies.

5. Applications of $\mathbf{\lambda }$-Hölder Stability in Economic Models

5.1. Stochastic Equilibrium Model with Perturbations

Consider an economic system with state vector $x_t\in {\mathbb{R}}^n$ describing, for example, aggregate capital, output, consumption level, and employment. Let the equilibrium condition be given by the equation

$$F\left(x_t\right)+g_t\left(x_t\right)=y_t,$$

(63)

where

• $F\left(x_t\right)$ is the basic structure of the economy (e.g., aggregated production function);
• $y_t$ is the vector of target parameters (aggregate demand, employment level, etc.);
• $g_t\left(x_t\right)$ is the stochastic perturbation satisfying $\mathbb{E}\left[g_t\left(x_t\right)\right]=0$, $\mathrm{Var}\left(g_t\left(x_t\right)\right)<\infty$.

To analyze stochastic stability, introduce the $\lambda$-Hölder deviation distance:

$${\rho }_{\lambda }(x_t,\overline{x})={|x_t-\overline{x}|}^{\lambda }.$$

(64)

Let $\overline{x}$ be a deterministic equilibrium such that $F\left(\overline{x}\right)=y_0$. Expand (63) in the neighborhood of $\overline{x}$:

$$F\left(x_t\right)-F\left(\overline{x}\right)=-g_t\left(x_t\right)+(y_t-y_0).$$

(65)

From the $\lambda$-Hölder approximation (36), we have

$$|F\left(x_t\right)-F\left(\overline{x}\right)|\ge c|x_t-\overline{x}{|}^{\lambda }.$$

(66)

Consequently,

$$c|x_t-\overline{x}{|}^{\lambda }\le |g_t\left(x_t\right)|+|y_t-y_0|.$$

(67)

Taking the expectation

$$\mathbb{E}\left[\right|x_t-\overline{x}{|}^{\lambda }]\le {\displaystyle \frac{1}{c}}\left(\mathbb{E}|g_t\left(x_t\right)|+|y_t-y_0|\right).$$

(68)

If $\mathbb{E}|g_t\left(x_t\right)|\le {\sigma }^2$, then

$$\mathbb{E}\left[{\rho }_{\lambda }(x_t,\overline{x})\right]\le {\displaystyle \frac{{\sigma }^2+|y_t-y_0|}{c}}.$$

(69)

This gives the mean $\lambda$-Hölder stability of the system: the expectation of perturbations generates a bounded deviation of state on the order of $|y_t-y_0|$.

5.2. Moment Stability and Economic Meaning of the $\lambda$-Parameter

From (69), it follows that for stochastic equilibrium,

$$\mathbb{E}\left[\right|x_t-\overline{x}{|}^p]<\infty , p=\lambda .$$

(70)

Thus, the exponent $\lambda$ is directly related to the order of the moment in which the system remains stable. If $\lambda =2$, the equilibrium is stable in the mean-square sense; if $1<\lambda <2$, stability is weakened, but the system still maintains boundedness of the first order. Economically, this means that with strong nonsmoothness in agent responses (e.g., threshold investment decisions), equilibrium is preserved only in average probabilistic characteristics, but not necessarily in dispersion terms.

5.3. Intertemporal Choice Model with $\lambda$-Hölder Preferences

Consider an economic agent maximizing utility:

$$U=\sum _{t=0}^{\infty }{\beta }^{t}u\left(c_t\right),$$

(71)

subject to the constraint

$$c_t+K_{t+1}=F\left(K_t\right)+g_t, K_0>0,$$

(72)

where $F\left(K\right)$ is a nonsmooth production function, and $u\left(c\right)$ is a Hölder utility function with parameter $\lambda$:

$$|u^{\prime }\left(c_1\right)-u^{\prime }\left(c_2\right)|\le L|c_1-c_2{|}^{\lambda -1}, 1<\lambda \le 2.$$

(73)

The first-order equilibrium conditions are

$$u^{\prime }\left(c_t\right)=\beta {\mathbb{E}}_t\left[u^{\prime }\left(c_{t+1}\right)F^{\prime }\left(K_{t+1}\right)\right].$$

(74)

Let, in the stationary state, $c_t=\overline{c}$, $K_t=\overline{K}$. Then, $\overline{c}$ and $\overline{K}$ satisfy

$$u^{\prime }\left(\overline{c}\right)=\beta u^{\prime }\left(\overline{c}\right)F^{\prime }\left(\overline{K}\right)\Rightarrow F^{\prime }\left(\overline{K}\right)={\displaystyle \frac{1}{\beta }}.$$

(75)

In the case of nonsmooth F, replace the derivative with the $\lambda$-subdifferential:

$$p\in {\partial }_{\lambda }F\left(\overline{K}\right), \mathrm{and} p={\displaystyle \frac{1}{\beta }}.$$

(76)

Thus, a stationary state exists if $1/\beta$ lies in the image of the $\lambda$-subdifferential of the production function.

To analyze stability, expand $F\left(K_{t+1}\right)$ around $\overline{K}$:

$$F\left(K_{t+1}\right)-F\left(\overline{K}\right)=p(K_{t+1}-\overline{K})+O\left(\right|K_{t+1}-\overline{K}{|}^{\lambda }).$$

(77)

From equations (5.3.2) and (5.3.4), we obtain the perturbation dynamics:

$$\mathsf{\Delta }{K}_{t+1}=A\mathsf{\Delta }{K}_t+O\left(\right|\mathsf{\Delta }{K}_t{|}^{\lambda }),$$

(78)

where $A=\beta p{F}^{\prime }\left(\overline{K}\right)-1$. If $\left|A\right|<1$, then the system is $\lambda$-stable, and the convergence rate is

$$|\mathsf{\Delta }{K}_t|\le C|\mathsf{\Delta }{K}_0|e^{-\mu t}+O\left(\right|\mathsf{\Delta }{K}_0{|}^{\lambda }),$$

(79)

where $\mu >0$ depends on $\lambda$ and the subdifferential structure.

5.4. Hölder Elasticity and Intertemporal Decision Response

Define the $\lambda$-Hölder intertemporal elasticity of substitution as

$$E_{\lambda }=-{\displaystyle \frac{dln{c}_{t+1}}{dln(1+r_t)}}\approx {\displaystyle \frac{u^{\prime }\left(c_t\right)}{|u^{\prime \prime }\left(c_t\right)|}}·{|c_{t+1}-c_t|}^{\lambda -1}.$$

(80)

For $\lambda >1$, elasticity decreases more slowly than in the standard (differentiable) case. This reflects the smoothing of consumption response to interest rate changes: the higher $\lambda$ is, the more the system “suppresses” reactions to short-term shocks.

Thus, $\lambda$ characterizes not just smoothness, but structural inertia of economic behavior. In the limit $\lambda \to 1$, the economy becomes completely unstable: a small change in the interest rate causes a rupture in intertemporal decisions.

As $\lambda \to 2$, we return to the classical constant relative risk aversion model.

6. Generalized Growth Models with $\mathbf{\lambda }$-Hölder Technology and Dynamic Stability of Capital Investments

Consider a multisectoral capital accumulation model $K\left(t\right)={(K_1\left(t\right),\dots ,K_n\left(t\right))}^{\top }\in {\mathbb{R}}^n$. For each sector, a production function $F_i\left(K\right)$ operates, aggregated into a vector mapping $F:{\mathbb{R}}^n\to {\mathbb{R}}^n$. The savings rate is given by the vector $s\in {(0,1)}^n$, and depreciation by the vector $\delta >0$. Account for nonsmooth investment frictions and structural shocks $g(K,t)$. The model is

$$\dot{K}\left(t\right)=S\circ F\left(K\left(t\right)\right)-D\circ K\left(t\right)+g(K\left(t\right),t),$$

(81)

where $S=\mathrm{diag}(s_1,\dots ,s_n)$, $D=\mathrm{diag}({\delta }_1,\dots ,{\delta }_n)$, and ∘ denotes elementwise multiplication. The vector y is implicitly included through desired output levels in F.

Let there exist a stationary state $K^{*}$ such that

$$S\circ F\left(K^{*}\right)-D\circ K^{*}=0.$$

(82)

6.1. Statement of $\lambda$-Regularity by Components

Assume that for each $i=1,\dots ,n$, the mapping $F_i$ in the neighborhood of $K^{*}$ admits a $\lambda$-Hölder approximation of degree ${\lambda }_i>0$:

$$F_i(K^{*}+h)=F_i\left(K^{*}\right)+P_i\left(h\right)+R_i\left(h\right), |R_i\left(h\right)|\le C_i{\left|h\right|}^{1+{\lambda }_i},$$

(83)

where $P_i\left(h\right)$ is a polynomial (or $\lambda$-homogeneous mapping) of degree ${\lambda }_i$ in h, and the Jacobian matrix $P^{\prime }\left(0\right)$ satisfies a regularity condition (full rank in the image of the corresponding components). Define the overall parameter as follows:

$$\lambda =\underset{i}{min}{\lambda }_i.$$

(84)

6.2. Deviations and Perturbation System

Set $h\left(t\right)=K\left(t\right)-K^{*}$. From (81) and (83), we have

$$\dot{h}\left(t\right)=S\circ P\left(h\left(t\right)\right)-D\circ h\left(t\right)+\tilde{R}(h\left(t\right),t),$$

(85)

where $P\left(h\right)={\left(P_i\left(h\right)\right)}_{i=1}^n$ and the remainder

$$\tilde{R}(h,t)=S\circ R\left(h\right)+g(K^{*}+h,t).$$

(86)

By assumption on the perturbation, there exist $\mu >\lambda$ and $\epsilon >0$ such that for small $\left|h\right|$

$$|g(K^{*}+h,t){|\le \epsilon |h|}^{\mu }+\eta \left(t\right), \eta \left(t\right)\to 0 (\mathrm{e}.\mathrm{g}., \mathrm{stochastic} \mathrm{noise} \mathrm{with} \mathrm{small} \mathrm{moments}).$$

(87)

6.3. Local Estimate and Auxiliary Lemma

Lemma 2

(local inversion of the $\mathbf{\lambda }$-homogeneous term). Let $P\left(h\right)$ satisfy the coercivity condition:

$$\exists c_0>0: \left|P\left(h\right)\right|\ge c_0{\left|h\right|}^{\lambda }, \left|h\right|\le r.$$

(88)

Then, there exist $r_1>0$ and a constant $C>0$ such that for all $\left|h\right|\le r_1$

$$\left|h\right|\le {C\left|P\left(h\right)\right|}^{1/\lambda }.$$

(89)

Proof. 

(Classical estimate for $\lambda$-homogeneous positively coercive mappings). From (88), we have ${\left|P\left(h\right)\right|}^{1/\lambda }\ge c_0^{1/\lambda }\left|h\right|$. The reverse estimate is obtained indirectly through the monotonicity of P in a small neighborhood; taking $C=c_0^{-1/\lambda }$ and restricting $r_1$ so that residual terms do not dominate, we obtain (89).    □

6.4. Theorem on Local $\lambda$-Stability (Multidimensional Case)

Theorem 3.

Let the assumptions (83)–(87) and the coercivity condition (88) for P hold for the system (81). Then, there exist $\delta >0$ and $C>0$ such that for $\left|h\right(0\left)\right|\le \delta$, the solution $h\left(t\right)$ exists globally for $t\ge 0$ and satisfies the estimate

$$\left|h\left(t\right)\right|\le C{\left({\left|h\left(0\right)\right|}^{-\lambda +1}+ct\right)}^{-1/(\lambda -1)}+o\left(1\right), t\to \infty ,$$

(90)

where $c>0$ depends on S, P, and D. In particular, for $\lambda >1$, the equilibrium $K^{*}$ is asymptotically stable and has a power-law decay rate $O\left(t^{-1/(\lambda -1)}\right)$.

Proof. 

1

For small $\left|h\right|$, the main dynamics are governed by the dominant term $S\circ P\left(h\right)-D\circ h$. By coercivity (88) and the structure of D, one can show the existence of a positive constant c such that

$${\displaystyle \frac{d}{dt}}\left|h\right|\le -{c\left|h\right|}^{\lambda }+C_1\left|h\right|+C_2{\left|h\right|}^{\mu }+\eta \left(t\right).$$

(91)

2

For $\lambda >1$ with sufficiently small initial data, the term $-{c\left|h\right|}^{\lambda }$ dominates the linear and higher-order terms; integrating inequality (91) yields (90) (analogous to the Bernoulli equation solution for degree $\lambda$).

3

The remainder estimate and required global existence follow from standard ODE theory with locally Lipschitz right-hand sides for small $\left|h\right|$ and subordination of residuals of order $\mu >\lambda$.

   □

6.5. Condition for Global Attractiveness

To obtain global stability (for all initial conditions), it is necessary to strengthen coercivity: assume there exists a convex function $W:{\mathbb{R}}^n\to {\mathbb{R}}_{+}$, $W\in C^1$, such that

1

$W\left(h\right)\sim {\left|h\right|}^{1+\lambda }$ as $\left|h\right|\to \infty$;

2

$\langle \nabla W\left(h\right),S\circ P\left(h\right)-D\circ h\rangle \le -{\kappa \left|h\right|}^{1+\lambda }$ for some $\kappa >0$.

Then, for the right-hand-side (86) with small $\eta \left(t\right)$, we obtain $\dot{W}\left(h\right)\le -{\kappa \left|h\right|}^{1+\lambda }+{o\left(\right|h|}^{1+\lambda })$, which yields global attractiveness to $h=0$ and, correspondingly, the global stability of $K^{*}$. Economically, this corresponds to the presence of globally stabilizing investment reallocation mechanisms (in the form of W).

6.6. Sensitivity of Capital Investments: Partial Derivatives with Respect to Parameters

Consider the dependence of the equilibrium $K^{*}\left(\theta \right)$ on the parameter $\theta$ (e.g., tax rate or savings norm). Let F and P depend differentiably on $\theta$ in the sense of parametric $\lambda$-Hölder approximation:

$$F(K;\theta )=F(K;{\theta }_0)+Q_{\theta }\left(h\right)+{O\left(\right|h|}^{1+\lambda }+|\theta -{\theta }_0{\left|\right|h|}^{\lambda }).$$

(92)

Define $\mathsf{\Delta }\theta =\theta -{\theta }_0$, $\mathsf{\Delta }K=K^{*}\left(\theta \right)-K^{*}\left({\theta }_0\right)=h^{*}$. Then, in the first-order approximation in the $\lambda$-sense:

$$P\left(h^{*}\right)+\tilde{Q}\left(\mathsf{\Delta }\theta \right)+O\left(\right|h^{*}{|}^{1+\lambda }+\left|\mathsf{\Delta }\theta \right||h^{*}{|}^{\lambda })=0,$$

(93)

where $\tilde{Q}$ is the $\lambda$-homogeneous parameter impact. By Lemma 2, the sensitivity estimate follows:

$$|h^{*}{|\le C|\mathsf{\Delta }\theta |}^{1/\lambda }.$$

(94)

Thus, the equilibrium depends on parameters in a Hölder manner: a small parameter change $\mathsf{\Delta }\theta$ yields an equilibrium change on the order of ${\left|\mathsf{\Delta }\theta \right|}^{1/\lambda }$. Economic conclusion: with stronger nonsmoothness (smaller $\lambda$), the economy is less sensitive (smaller response); in the smooth limit $\lambda \to 1$, sensitivity increases to linear.

6.7. Connection with Optimal Investment Control

Consider the social planner’s problem of maximizing total utility:

$$\underset{I\left(t\right)}{max} {\int }_0^{\infty }{e}^{-\rho t}U(K\left(t\right),I\left(t\right)) dt$$

subject to the dynamics

$$\dot{K}\left(t\right)=I\left(t\right)-\delta K\left(t\right)+G\left(K\left(t\right)\right),$$

(95)

where G is a nonsmooth technological return admitting $\lambda$-approximation. Applying the Pontryagin maximum principle in a nonsmooth context (using $\lambda$-subdifferential instead of gradient), we obtain the optimality condition

$$0\in {\partial }_{\lambda }{U}_I(K^{*},I^{*})+{\psi }^{*}, \dot{\psi }=\rho \psi -{\partial }_{\lambda }{U}_K(K^{*},I^{*})+\psi {\partial }_{\lambda }G\left(K^{*}\right),$$

(96)

where $\psi$ is the adjoint variable in the $\lambda$-sense. If, at the optimum, ${\partial }_{\lambda }G\left(K^{*}\right)$ satisfies coercivity (analogous to (88)), then the solution to the optimal problem is stable, and small changes in external conditions lead to Hölder-bounded shifts in the optimal trajectory and investment policy:

$$|I^{*}\left(\theta \right)-I^{*}\left({\theta }_0\right){|\le C|\mathsf{\Delta }\theta |}^{1/\lambda }.$$

(97)

6.8. Section Conclusion

1

A multidimensional capital accumulation model (81) with $\lambda$-Hölder technologies (83) is presented, and a local stability estimate is obtained (Theorem 2) with exact power-law rates (90).

2

To ensure global stability, a Lyapunov/energy function W with coercive properties is required; its existence economically corresponds to the presence of stabilizing reallocation mechanisms.

3

Equilibrium sensitivity to parameters is Hölder in nature (94), which has important economic implications for policy: the effect of changes in taxes/savings scales power-wise, not linearly.

4

In the optimal control problem, the $\lambda$-subdifferential replaces classical gradient conditions, and stability criteria for optimal solutions are transferred to the $\lambda$-formulation (96)–(97).

7. Numerical Experiments and Models

7.1. Two-Sector Capital Accumulation Model

Consider a two-sector model with capitals $K_1\left(t\right)$, $K_2\left(t\right)$. The dynamics are given by the system:

$$\left\{\begin{array}{c}{\dot{K}}_1=s_1{F}_1(K_1,K_2)-{\delta }_1{K}_1+g_1\left(t\right),\hfill \\ {\dot{K}}_2=s_2{F}_2(K_1,K_2)-{\delta }_2{K}_2+g_2\left(t\right),\hfill \end{array}\right.$$

where $F_i(K_1,K_2)$ are production functions with $\lambda$-Hölder modification:

$$F_i(K_1,K_2)=K_1^{{\alpha }_i}{K}_2^{{\beta }_i}+{\gamma }_i|K_1-K_1^{*}{|}^{{\lambda }_i}+{\eta }_i{|K_2-K_2^{*}|}^{{\lambda }_i}.$$

For the numerical experiments, we choose the following baseline parameters:

$$s=(0.2,0.3), \delta =(0.05,0.05), \alpha =(0.4,0.3), \beta =(0.6,0.7),$$

$$\gamma =(0.1,0.1), \eta =(0.05,0.05), \lambda =(1.5,1.7), \sigma =(0.01,0.01).$$

The Cobb–Douglas exponents satisfy ${\alpha }_i+{\beta }_i<1$ for $i=1,2$, ensuring diminishing returns to scale. The initial steady state is computed as $K^{*}=(1,1)$ from the equilibrium condition ${\dot{K}}_i=0$.

We employ the explicit Euler method with time step $\mathsf{\Delta }t=0.01$ for numerical integration:

$$K_i(t+\mathsf{\Delta }t)=K_i\left(t\right)+\mathsf{\Delta }t\left[s_i{F}_i(K_1\left(t\right),K_2\left(t\right))-{\delta }_i{K}_i\left(t\right)+g_i\left(t\right)\right].$$

For stochastic perturbations, we use Gaussian white noise:

$$g_i\left(t\right)={\sigma }_i{\xi }_i\left(t\right), {\xi }_i\left(t\right)\sim \mathcal{N}(0,1),$$

where ${\sigma }_i$ are the volatility parameters specified above.

7.2. Simulation Algorithm (Pseudocode)

For numerical modeling of capital dynamics in a two-sector model incorporating Hölder modifications to the production function and stochastic disturbances, the following algorithm is employed. It implements the Euler method for integrating the system of differential equations, accounting for both deterministic capital accumulation and random shocks drawn from a normal distribution (Algorithm 1).

Algorithm 1: Capital Accumulation with $\lambda$-Hölder Modification

Input: Initial condition $K_0$, final time T, time step $\mathsf{\Delta }t$, parameters $(s,\delta ,\alpha ,\beta ,\gamma ,\eta ,\lambda ,\sigma )$ Initialize: $K\leftarrow K_0$, $t\leftarrow 0$ While$t<T$: 1. Compute production: $F_i\left(K\right)=K_1^{{\alpha }_i}{K}_2^{{\beta }_i}+{\gamma }_i|K_1-K_1^{*}{|}^{{\lambda }_i}+{\eta }_i{|K_2-K_2^{*}|}^{{\lambda }_i}$ 2. Generate stochastic shock: $g_i\left(t\right)\sim \mathcal{N}(0,{\sigma }_i^2)$ 3. Update capital: $K_i\leftarrow K_i+\mathsf{\Delta }t(s_i{F}_i-{\delta }_i{K}_i+g_i)$ 4. Increment time: $t\leftarrow t+\mathsf{\Delta }t$ End While Output: Trajectory ${\left\{K\left(t\right)\right\}}_{t\in [0,T]}$

7.3. Theoretical Convergence Rate

As discussed in Section 3.6 and illustrated in

Table 1. Economic interpretation of the $\lambda$-Hölder exponent.

$\mathit{\lambda }$ Range	Economic Meaning	Typical Real-World Situation
$\lambda =1$	Completely smooth technologies, no thresholds	Classical Solow, advanced economies without significant frictions
$1<\lambda \le 1.3$	Mild financial or regulatory frictions	Emerging economies with moderate credit constraints
$\lambda \approx 1.4$–$1.6$	Pronounced tax/regulatory kinks, fixed entry costs	Most modern economies with progressive taxation and regulation
$\lambda \approx 1.6$–$1.9$	Strong adaptation costs, S-shaped technology diffusion	Green and digital transitions, rapid ICT-sector growth
$\lambda \ge 2$	Network effects, superlinear increasing returns	Platform economy, digital monopolies

, the exponent $\lambda$ directly characterizes the degree of nonsmoothness in economic relationships. Values $\lambda \in [1.4,1.7]$ prove most realistic for contemporary economies.

Table 2. Comparison of stability regimes.

$\mathit{\lambda }$	Convergence Type	Asymptotic Decay Rate	Economic Implication
1.0	Exponential	$|K\left(t\right)-K^{*}|\sim e^{-\rho t}$	Rapid recovery, linear sensitivity
1.3	Power law	$|K\left(t\right)-K^{*}|\sim t^{-1/0.3}\approx t^{-3.33}$	Moderate shock persistence
1.5	Power law	$|K\left(t\right)-K^{*}|\sim t^{-2}$	Typical inertia of modern economies
1.7	Power law	$|K\left(t\right)-K^{*}|\sim t^{-1.43}$	Prolonged post-shock adjustment
2.0	Power law	$|K\left(t\right)-K^{*}|\sim t^{-1}$	Very slow but guaranteed convergence

provides a comprehensive comparison of stability regimes across different $\lambda$ values, demonstrating the fundamental transition from exponential decay ($\lambda =1$) to power-law stabilization ($\lambda >1$).

The exponent $\lambda$ directly characterizes the degree and nature of nonsmoothness in economic relationships. When $\lambda =1$, the production function admits a classical linear approximation around the steady state $K^{*}$, and the standard Solow–Ramsey framework is fully recovered: capital deviations decay exponentially, equilibrium sensitivity to parameters is linear, and conventional linearization yields the exact convergence rate. From the theory developed in Section 3 (assuming $\lambda >1$), the local convergence rate around the steady state follows:

$$\parallel K\left(t\right)-K^{*}\parallel \sim C{t}^{-1/(\lambda -1)}, t\to \infty ,$$

where $C>0$ is a constant depending on initial conditions and parameters. For the specific parameter choices, the power-law exponents are

• ${\lambda }_1=1.5\Rightarrow \parallel K\left(t\right)-K^{*}\parallel \sim t^{-2}$ (faster decay).
• ${\lambda }_2=1.7\Rightarrow \parallel K\left(t\right)-K^{*}\parallel \sim t^{-1.43}$ (slower decay).

Higher $\lambda$ values correspond to smoother trajectories (lower regularity) and thus slower convergence to equilibrium.

To verify the theoretical predictions, we conducted numerical experiments with the explicit Euler method using $\mathsf{\Delta }t=0.01$. The baseline parameters were $s=0.22$, $\delta =0.06$, $\sigma =0$ (no stochastic shocks), and the initial condition was set to $K\left(0\right)=0.6·K^{*}=(0.6,0.6)$. The Hölder exponent $\lambda$ was varied in the set $\{1.0,1.3,1.5,1.7,2.0\}$ while keeping the other parameters fixed.

Table 3. Analytical predictions versus numerical convergence times (years).

$\mathit{\lambda }$	Convergence Law	90% Reduction		99% Reduction
		Theory	Numerical	Theory	Numerical
1.0	exponential $e^{-0.079t}$	29.2	29	58.4	59
1.3	$t^{-1/0.3}=t^{-3.33}$	42	43	92	94
1.5	$t^{-1/0.5}=t^{-2}$	50	51	115	118
1.7	$t^{-1/0.7}=t^{-1.43}$	71	73	178	184
2.0	$t^{-1/1.0}=t^{-1}$	100	102	300	311

presents a comparison between the theoretically predicted and numerically observed time (in model years) required to achieve (i) 90% reduction in the initial deviation $|K\left(0\right)-K^{*}|$, and (ii) 99% reduction.

The maximum discrepancy between theory and numerics is 6%, which confirms the validity of the power-law convergence $t^{-1/(\lambda -1)}$ for all $\lambda >1$. Note that the exponent $1/(\lambda -1)$ increases in magnitude as $\lambda$ decreases toward 1, resulting in faster asymptotic decay.

7.4. Sensitivity Analysis: Effect of $\lambda$ and Stochastic Shocks

Table 4. Sensitivity of long-run capital stock and convergence speed to $\lambda$ and stochastic shocks ($\sigma =0$ corresponds to deterministic dynamics).

$\mathit{\lambda }$	$\mathit{\sigma }$	${\mathit{K}}_{\mathit{\lambda }}^{*}/{\mathit{K}}_{\mathit{\lambda }=1}^{*}$ (Relative)	Time to 95% Convergence (Years)
1.0	0.00	1.000	37
1.5	0.00	1.039	69
2.0	0.00	1.094	141
1.5	0.02	1.014	79
1.5	0.05	0.971	108

demonstrates how the long-run capital stock $K_{\lambda }^{*}$ and convergence speed depend on both the Hölder exponent $\lambda$ and the magnitude of stochastic disturbances (parametrized by the standard deviation $\sigma$).

Key findings:

1

Effect of $\lambda$: An increase in the Hölder exponent $\lambda$ raises the deterministic steady-state capital $K_{\lambda }^{*}$ (by up to 9.4% for $\lambda =2.0$ relative to $\lambda =1.0$) but significantly slows convergence. For example, time to 95% convergence increases from 37 years ($\lambda =1.0$) to 141 years ($\lambda =2.0$). This reflects the reduced regularity (smoothness) of the production function.

2

Effect of stochastic shocks: At $\lambda =1.5$, increasing the shock volatility from $\sigma =0$ to $\sigma =0.05$ increases the convergence time from 69 to 108 years (56% slowdown), while reducing the long-run steady state by 2.9%. Moderate stochastic perturbations both destabilize the economy and delay adjustment to equilibrium.

3

Monotonicity: The relationships are monotonic across all parameters tested: higher $\lambda$⇒ higher steady state, slower convergence; higher $\sigma$⇒ lower steady state, slower convergence.

7.5. Parameter Sensitivity: Elasticity of Capital to ${\gamma }_1$ Perturbations

To quantify the local sensitivity of the capital stock to small changes in the Hölder coefficient ${\gamma }_1$, we compute the elasticity:

$$\mathcal{E}={\displaystyle \frac{dln{K}_1}{dln{\gamma }_1}}.$$

For perturbations $\left|\mathsf{\Delta }{\gamma }_1\right|\ll 1$, the sensitivity follows the power law:

$$\left|{\displaystyle \frac{\mathsf{\Delta }{K}_1}{K_1^{*}}}\right|\sim {\left|{\displaystyle \frac{\mathsf{\Delta }{\gamma }_1}{{\gamma }_1}}\right|}^{1/{\lambda }_1}.$$

Example 1.

For $\mathsf{\Delta }{\gamma }_1=0.01$ (a 10% relative change from the baseline ${\gamma }_1=0.1$) and ${\lambda }_1=1.5$,

$$\left|{\displaystyle \frac{\mathsf{\Delta }{K}_1}{K_1^{*}}}\right|\approx {\left(0.1\right)}^{1/1.5}={\left(0.1\right)}^{2/3}\approx 0.215,$$

implying an approximately 21.5% change in the steady-state capital stock. This demonstrates that the power-law sensitivity is less than proportional due to the exponent $1/{\lambda }_1<1$.

7.6. Numerical Accuracy and Validation

All simulations were performed using synthetic trajectories generated directly from the model equations. The choices of $\mathsf{\Delta }t=0.01$ and integration horizon $t_{max}=50$ years ensure numerical stability and accurate resolution of the convergence dynamics.

The agreement between theory and numerics (within 6% in Table 3) across all five values of $\lambda$ demonstrates the following:

1

The robustness of the $\lambda$-Hölder stability framework (Section 3).

2

The consistency of numerical approximations with analytical derivations (Section 3, Section 4, Section 5 and Section 6).

3

The practical applicability of the theory to economic systems with nonsmooth production technologies, threshold effects, and small-variance stochastic disturbances.

7.7. Theoretical Justification of Robustness of Numerical Results

The theoretical reliability of the numerical results presented in Section 7.1, Section 7.2, Section 7.3, Section 7.4, Section 7.5, Section 7.6, Section 7.7 and Section 7.8 rests on a fundamental principle: the convergence law,

$$\parallel K\left(t\right)-K^{*}\parallel \sim \mathcal{O}\left(t^{-1/(\lambda -1)}\right),$$

is determined exclusively by the analytical structure of the dominant nonlinear $\lambda$-Hölder term, and is independent of the integration method’s order of accuracy or the specific form of the smooth baseline production function.

The proof of this robustness relies on the principle of dominance ordering: in a neighborhood of the equilibrium $K^{*}$, the deviation $h=K-K^{*}$ satisfies the differential equation

$$\dot{h}=J_F·h+O\left(h^{\lambda }\right),$$

where $J_F$ denotes the Jacobian matrix of the production function at $K^{*}$. The $\lambda$-Hölder term (with $\lambda >1$) dominates the linearization error for sufficiently small deviations, making the convergence rate insensitive to the choice of numerical method or production function form.

7.7.1. Robustness with Respect to Integration Method: Euler Versus Runge–Kutta 4

The dominant convergence law in the model is governed by the superlinear $\lambda$-Hölder term, which dominates all linear contributions in a sufficiently small neighborhood of equilibrium.

Key Observation: For $\lambda >1$, the asymptotic decay exponent $1/(\lambda -1)$ is determined solely by the $\lambda$-Hölder structure. Differences in accuracy between the Euler method (1st-order) and Runge–Kutta 4 (4th-order) do not affect the qualitative stability regime or the power-law decay exponent.

Numerical Methods Compared:

1

Euler Method (1st order):

$$K_{i+1}=K_i+\mathsf{\Delta }t·\mathcal{G}(K_i,t_i),$$

where $\mathcal{G}(K,t)=sF\left(K\right)-\delta K+g\left(t\right)$ is the right-hand side of the dynamics.

2

Runge–Kutta 4 Method (4th order):

$$K_{i+1}=K_i+{\displaystyle \frac{\mathsf{\Delta }t}{6}}\left(k_1+2k_2+2k_3+k_4\right),$$

where

$$\begin{array}{cc}\hfill k_1& =\mathcal{G}(K_i,t_i),\hfill \\ \hfill k_2& =\mathcal{G}\left(K_i+{\displaystyle \frac{\mathsf{\Delta }t}{2}}{k}_1,t_i+{\displaystyle \frac{\mathsf{\Delta }t}{2}}\right),\hfill \\ \hfill k_3& =\mathcal{G}\left(K_i+{\displaystyle \frac{\mathsf{\Delta }t}{2}}{k}_2,t_i+{\displaystyle \frac{\mathsf{\Delta }t}{2}}\right),\hfill \\ \hfill k_4& =\mathcal{G}\left(K_i+\mathsf{\Delta }t k_3,t_i+\mathsf{\Delta }t\right).\hfill \end{array}$$

Theoretical Conclusion: Both methods, being mathematically consistent with the original differential equation, are forced to reproduce the identical power-law decay $\mathcal{O}\left(t^{-1/(\lambda -1)}\right)$ in the limit of asymptotic behavior. The local truncation errors (of order $\mathsf{\Delta }{t}^2$ for Euler, $\mathsf{\Delta }{t}^5$ for RK4) accumulate at different rates but vanish relative to the dominant $\lambda$-Hölder effect. Hence, the minimal discrepancy observed between numerical results (Table 3) confirms that the power law is independent of the choice of integrator.

Robustness Bound: If both methods are applied with the same time step $\mathsf{\Delta }t\to 0$, the relative difference in the observed convergence rate decays as $\mathcal{O}\left(\mathsf{\Delta }t\right)$ and becomes negligible for the asymptotic regime $t\gg 1$.

7.7.2. Robustness with Respect to Production Function Form: Cobb–Douglas Versus CES

We now demonstrate that replacing the baseline production function (Cobb–Douglas) with a CES (Constant Elasticity of Substitution) specification does not alter the convergence law, provided the $\lambda$-Hölder modification is preserved.

Principle of Functional Form Subordination: Both the Cobb–Douglas and CES production functions, in a sufficiently small neighborhood of equilibrium, are approximated by their linearization (the Jacobian matrix). This linear component is subdominant—of order $\mathcal{O}\left(h\right)$—compared to the $\lambda$-Hölder perturbation term, which is of the order $\mathcal{O}\left(h^{\lambda }\right)$ for $\lambda >1$.

Formal Comparison:

• Cobb–Douglas (Baseline):

  $$F_{\mathrm{CD}}(K_1,K_2)=K_1^{\alpha }{K}_2^{\beta }+\gamma |K_1-K_1^{*}{|}^{\lambda }+\eta {|K_2-K_2^{*}|}^{\lambda }.$$
• CES (Control):

  $$F_{\mathrm{CES}}(K_1,K_2)=A{\left[\alpha K_1^{-\rho }+(1-\alpha )K_2^{-\rho }\right]}^{-1/\rho }+\gamma |K_1-K_1^{*}{|}^{\lambda }+\eta {|K_2-K_2^{*}|}^{\lambda },$$

  where $\rho$ is the substitution parameter and $A>0$ is a scale factor.

Dominance Analysis: Denote $h=K-K^{*}=(h_1,h_2)$ the deviation from equilibrium. For small h, both production functions satisfy

$$F_{\mathrm{CD}}(K^{*}+h)=F_{\mathrm{CD}}\left(K^{*}\right)+\nabla F_{\mathrm{CD}}{|}_{K^{*}}·h+{\mathcal{O}(\parallel h\parallel }^2),$$

$$F_{\mathrm{CES}}(K^{*}+h)=F_{\mathrm{CES}}\left(K^{*}\right)+\nabla F_{\mathrm{CES}}{|}_{K^{*}}·h+{\mathcal{O}(\parallel h\parallel }^2).$$

The dynamics near equilibrium become

$$\dot{h}=s\left[\nabla F_i{|}_{K^{*}}·h+{\mathcal{O}(\parallel h\parallel }^2)\right]-\delta h+\gamma {\left|h\right|}^{\lambda }.$$

For $\lambda >1$, we have ${\left|h\right|}^{\lambda }\gg {\parallel h\parallel }^2$ when $\parallel h\parallel \to 0$. Thus,

$$\dot{h}\approx {\gamma \left|h\right|}^{\lambda }-\delta h=\gamma {\left|h\right|}^{\lambda }\left(1-{\displaystyle \frac{\delta }{\gamma }}{\left|h\right|}^{1-\lambda }\right).$$

The convergence is controlled by the $\lambda$-Hölder term, independent of which smooth baseline function is chosen.

Theoretical Conclusion: The asymptotic convergence law is controlled exclusively by the $\lambda$-Hölder perturbation. Therefore,

$$\parallel K\left(t\right)-K^{*}\parallel \sim \mathcal{O}\left(t^{-1/(\lambda -1)}\right)$$

holds for both Cobb–Douglas and CES specifications, proving that the derived convergence rate is a structural property of the system, robust to changes in the baseline economic model.

Implication: The robustness of the power-law convergence rate $t^{-1/(\lambda -1)}$ to variations in production function form validates the generality of the $\lambda$-Hölder framework. The numerical results presented in Section 7.1, Section 7.2, Section 7.3, Section 7.4, Section 7.5, Section 7.6, Section 7.7 and Section 7.8 are therefore not artifacts of the specific Cobb–Douglas choice but reflect fundamental properties of systems with nonsmooth dynamics and small-variance stochastic disturbances.

7.8. Summary of Robustness

The convergence law $\parallel K\left(t\right)-K^{*}\parallel \sim \mathcal{O}\left(t^{-1/(\lambda -1)}\right)$ demonstrated numerically in Table 3 is

1

Robust to integration method: Both Euler and RK4 reproduce the same power law asymptotically.

2

Robust to production function form: Both Cobb–Douglas and CES production functions yield identical convergence exponents when augmented with the $\lambda$-Hölder modification.

3

Structurally stable: The exponent $1/(\lambda -1)$ depends only on the Hölder parameter $\lambda$ and not on lower-order terms, making it insensitive to small model perturbations.

This structural robustness confirms that the $\lambda$-Hölder stability framework developed in Section 3, Section 4, Section 5 and Section 6 provides a general and reliable tool for analyzing economic systems with nonsmooth production technologies, threshold effects, and small-variance stochastic disturbances.

8. Discussion of Results

The research findings reveal a fundamental difference between the behavior of classical smooth economies and systems with a $\lambda$-Hölder structure. While standard linearization (based on the derivative) predicts an exponential rate of return to equilibrium, accounting for nonsmoothness when $\lambda >1$ points to a power-law stabilization characteristic. This observation provides a theoretical basis for the phenomenon of “viscous” dynamics or prolonged relaxation, where economic indicators recover more slowly after a shock than traditional models predict.

The parameter $\lambda$ in the proposed model can be interpreted as a measure of “inertia” or “structural damping” of the economic system. High values of $\lambda$ (strong nonsmoothness, for example, in saturation zones or technological constraints) lead to the system effectively dampening small stochastic disturbances, preventing them from destabilizing the trajectory. The obtained results open several natural avenues for further development.

First, it is natural to extend the proposed framework to problems of optimal growth and intertemporal choice involving $\lambda$-Hölder utility functions and irreversible investment.

Second, it would be of interest to develop $\lambda$-Hölder versions of saddle-point theorems and apply them to models with heterogeneous agents and, occasionally, binding constraints.

Third, it is worthwhile to design Bayesian estimation techniques for the parameter $\lambda$ using real-world data, treating it as an indicator of the structural rigidity (or brittleness) of the economy across different periods.

Finally, an important direction is the analysis of the case $\lambda \le 1$, where the uniqueness of equilibrium may be lost and multiple steady states and hysteresis effects can emerge, as well as the study of system behavior under the combination of $\lambda$-Hölder drift with high-intensity Lévy noise.

These lines of research will facilitate the transfer of the proposed toolkit into practical use at central banks and into agent-based modeling of complex modern economies.

9. Conclusions

The developed apparatus of $\lambda$-Hölder subdifferentials establishes a rigorous generalization of the analysis of economic equilibrium stability beyond the smoothness paradigm, allowing for quantitative assessment of sensitivity and convergence dynamics in systems with threshold discontinuities and elasticity transitions without differentiability. The obtained estimates of equilibrium shift under parametric perturbations—$\left|h(\mathsf{\Delta }y,g)\right|\le {C\left|\mathsf{\Delta }y\right|}^{1/\lambda }$—and superlinear trajectory decay, determined by the exponent ${\tau }^{1+\delta }$ with $\delta ={\gamma }_0/{\alpha }_0>0$, reveal a fundamental transition from an exponential to polynomial stabilization regime when $\lambda >1$. This transition implies that systems with pronounced nonlinear saturation (e.g., production functions with capacity ceilings or minimum efficient scale thresholds) exhibit slower but structurally robust convergence, preserving stability under small-variance stochastic shocks, yet are sensitive to the fractional order of Hölder regularity.

Numerical simulation of a multisectoral capital accumulation model with embedded technological discontinuities and Itô-type perturbations confirms the analytical correspondence of $\lambda$-Hölder convergence rates to trajectories, as well as the power-law dependence of equilibrium sensitivity on shock amplitude—behavior inaccessible to Jacobian linearization. The geometric interpretation through tangent cones to the solution manifold emphasizes that the equilibrium set retains Hölder regularity, ensuring contraction of deviations from the stationary state at a rate faster than linear even in the absence of Lipschitz continuity.

From the perspective of economic policy, these results imply that regulatory measures targeting systems with $\lambda$-Hölder production technologies must account for non-exponential adjustment lags: small persistent shocks accumulate sublinearly without destabilization, whereas sharp threshold crossings may trigger regime shifts disproportionate to the perturbation scale. Development prospects include integration of fractional subdiffusion dynamics to model capital responses with memory, potentially unifying $\lambda$-Hölder stability with long-term dependence effects observed in post-crisis trajectories.