Analyzing Economic Growth Dynamics in Emerging Economies Using Contraction Fixed Point Approach

Abstract

This paper applies contraction fixed point theory to analyze the stability and convergence properties of traditional economic growth models in the context of Nigeria's economy. The results show that both models satisfy the contraction condition, ensuring global asymptotic stability and convergence to a unique steady state capital stock and equilibrium growth path, respectively. The simulation results illustrate the state dynamics and convergence behavior of the models, highlighting the stabilizing role of policy and institutional parameters. However, when the contraction condition is violated, the models exhibit divergence or persistent instability, emphasizing the importance of policy discipline in maintaining macroeconomic stability. The research provides insights for policymakers and economic planners in emerging economies, suggesting that policy coordination and institutional strengthening are crucial for ensuring sustainable economic growth.

Keywords

Introduction

Economic growth is characterized by a rise in the production of economic goods and services over a given period, typically measured using gross domestic product (GDP) or gross national product (GNP) [1], [2]. This growth is often driven by technological advancements, investments in human capital, and increases in capital goods production, leading to a rise in national income. Nigeria, with a population of approximately 223 million, is the most populous black nation in the world [3]. Despite steady economic growth, the country faces significant challenges, including uneven distribution of wealth and vulnerability to internal and external shocks. Traditional economic growth models, such as the Solow-Swan and endogenous growth models, assume equilibrium and stability without fully examining the mathematical conditions required for stability under real-world conditions [1], [4], [5]. These models fall short in accounting for critical growth-influencing factors in emerging economies like Nigeria, including exchange rate volatility, oil price fluctuations, and technological advancements. The Contraction Fixed Point Theorem, also known as Banach's Fixed Point Theorem, is a fundamental concept in mathematics that deals with the existence and uniqueness of fixed points for certain mappings in complete metric spaces. Recent studies have extended and generalized this theorem in various ways, including the introduction of new contraction mappings and the application of fixed-point theory to different types of spaces. Researchers have investigated fixed point theorems for T-Hardy-Rogers contraction mappings in complete cone b-metric spaces [6], strict set contractions in Banach spaces [7], and Krasnoselskii type mappings [8]. Others have explored fixed point theorems in complex-valued metric spaces [9], complex partial b-metric spaces [10], and F-metric spaces [11]. Contraction fixed point theory, specifically the Banach Contraction Principle, offers a powerful tool for assessing stability and convergence properties of dynamic systems [12]. This theory can be applied to economic growth modeling to develop a more robust framework for understanding how economies like Nigeria's respond to shocks, adapt to policy changes, and sustain growth under complex conditions. The application of contraction fixed point theory addresses a fundamental challenge in economic growth modeling, accurately analyzing and predicting stability and convergence under variable conditions. By introducing this theory, the research provides a rigorous foundation for understanding economic growth dynamics and offers insights for policymakers and economic planners in emerging economies [4], [5]. The application of contraction fixed point theory, notably the Banach fixed-point theorem, provides a robust framework for examining the convergence properties of dynamic systems. Notwithstanding its potential and the utilization of contraction fixed point theory in the context of economic growth models remains underinvestigated, particularly in relation to emerging economies characterized by distinctive challenges. This study aims to bridge the gap by employing contraction fixed point approach to analyze conventional economic growth models.

METHOD

Contraction Mapping Condition and Theorem

Lemma 1 [13]

Let f(x, t) be a vector value function. It is said to satisfy a Lipschitz condition in a region ℜ in space (x, t), if, for some constant say L such that

fx, t-f(y,t)≤Lx-y                (1)

where L is known as Lipschitz constant and and whenever x, t∈R,(y,t)∈R1

Theorem 1 (Conventional Mean Value Theorem)

Let f:a,b→R be a real value function, such that:

1. f is continuous on [a, b]and
2. f is differentiable on (a, b) 

Then, there exists at least one point ψ ∈ (a, b), such that the following holds:

f'ψ=fb-fab-a                                  (2)

Theorem 2 [13]

Let X be a metric space and let T : X → X be a mapping, then there exist 0 < L < 12 , for all x, y ∈ X, such that

dT x, T y≤ Ldx, T x+ dy, T y          (3)

Then T has a unique fixed a ∈ X and for every x ∈ X, the sequences xn iterates Tnx by xn+1=Txn

Corrollary 1 [14]

Let T be a contraction mapping on a complete metric space X and β be a contraction constant with a fixed point a0. Then, for every  x0∈ X, with T- iterates {xn}, the following estimate holds:

dxn,a0≤βn1-βdx0,Tx0      

dxn,a0≤βdxn-1,a0            

dxn,a0≤βn1-βdxn-1,a0        (4)

Contraction Conditions and Theorems for Economic Growth Model

Theorem 3 (Contraction Mapping)

Let (X, d) be a complete metric space, where X represents the set of possible state variables defined the economic growth model (capital stock, output per-capita, or a vector of economic variables such as oil prices, exchange rates, policy indices). A mapping φ : X → X, representing the economic growth transformation over a period of time, t is said to be a contraction if there exists a constant 0 ≤ δ < 1 such that:

dφx, φy≤ δ dx, y∀ x, y ∈ X        (5)

Where d(·, ·) measures the difference between two possible economic state variables. The contraction condition means that, after each iteration (for example yearly or quarterly growth rate), the variables move closer to one another, ensuring that shocks or deviations diminish over time rather than amplify.

Theorem 4 (Contraction Mean Value Theorem)

Let fρ= ρα, 0 < α < 1 such that f(ρ2) - f(ρ1) for some δ ∈ (ρ1,ρ2), the following holds:

ρ2α-ρ1α≤αδα-1ρ2-ρ1           (6)

Where ρ2α-ρ1α=f'(δ)(ρ2-ρ1) and f'δ=αδα-1

Theorem 5 (Contraction Fixed-Point Theorem for Convergence)

If (X, d)

 is a complete metric space and φ : X → X

 is a contraction mapping with contraction constant δ ∈ [0, 1) , then there exists a unique fixed-point x*∈ X  such that:

φx* = x*                (7)

Moreover, for any initial state x0∈ X , the sequence defined by:

xn+1 = φxn           (8)

converges to x*  as n → ∞, with the error bound:

dxn,x*≤δn1-δdx1,x0        (9)

Remark: This theorem ensures that the growth model converges to a unique steady state (equilibrium) regardless of the initial economic conditions, provided the contraction property holds.

Theorem 6 (Contraction Fixed-Point Theorem for Stability)

If the mapping φ

 describing the economic growth process is a contraction, then its unique fixed point x* is globally asymptotically stable. That is:

limn→∞d(xn, x* ) = 0          (10)

for any starting point  x0∈ X

Remark: Stability here means that even if the economy is perturbed by shocks (for example, sudden changes in oil prices, taxation policy, or exchange rates), as long as the contraction condition is maintained, the economy will return to its steady-state growth over time.

Solow-Swan Growth Model (Exogenous Growth Model)

The Solow-Swan growth model in [4] is an exogenous growth model that focused on long-term economic growth by considering capital accumulation, labor or population growth, and technological progress as key drivers of economic output. It assumes diminishing returns to capital and labor and predicts convergence to a steady-state growth based on exogenously determined technological progress. The Solow-Swan model is the Cobb-Douglas production function, given by:

Y t= AtKtαLt1-α            (11)

where:

Y (t) =

output (GDP)  at a given time A(t) = capital stock at a given time K(t) =  labor input (population or workforce) at a given time L(t) =  level of technology at a given α = output elasticity of capital (α ∈ [0, 1])

Growth Model [1]

The author considers economic growth model based on international trade with the following variables: oil exports, non-oil exports, oil imports, and non-oil imports to reflect the nature of the Nigerian trade framework. The model control variables were gross fixed capital formation, inflation rate, and exchange rate. The model equations are given by;

RDDP=fNONEXP,OIEXP,NONIMP,OIMP,GFCF,EXR LRDP=β0+β1NONEXP+β2OIEXP+β3NONIMP+β4OIMP+β5GFCF+β6IR+β7LEXP+?   (12)

where

LRGDP  is Log of Real Gross Domestic Product GrowthNONEXP

is Non-oil ExportsOIEXP  is Oil Exports NONIMP

 is Non-oil Imports

OIMP  is Oil Imports GFCF  is Gross Fixed Capital Formation IR is Inflation Rate LEXR is Log of Exchange Rateβ0  is Constant Term βi (1, 2, ..., 7) are Parameters

RESULTS

Contraction Fixed-Point Analysis of the Solow–Swan Growth Model

In this section, we apply the contraction mapping conditions and fixed-point theorems stated in chapter three to analyze the convergence and stability properties of the Solow-Swan economic growth model.

The Solow-Swan growth model (11) is given by the Cobb-Douglas production function as

Y t= AtKtαLt1-α,   0 < α < 1