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

Contraction fixed point theory, Economic growth models, Stability and convergence, Emerging economies, Nigeria's economy

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   

where Y (t)

denotes output, K(t) is the capital stock, L(t) is labour, and A(t) represents the level of technology. The Capital accumulation is governed by

Kt= sY t- δ + n + gKt      (13)

where s is the savings rate, δ  is the depreciation rate, n is the population growth rate, and g is the exogenous rate of technological progress. Discretizing the capital accumulation equation (13) yields the iterative mapping:

Kt+1=∅Kt        (14)

where

∅Kt=sAKtα-δ+n+gKt

Using the proposed Contraction Mapping theorem 3, let X = [0, ∞) be a complete metric space endowed with the metric: d(x, y) = |x - y|

For any K1,K2 ∈ X , we have that

∅K1-∅K2=sAK1α-K2α-δ+n+g(K1-K2)                    (15)

By mean value theorem which provides a Lipschitz bound required for a contraction condition, there exists ξ between K1 and K2 such that:

K1α-K2α≤αξα-1K1-K2        (16)

From (16) it follows from the Mean Value Theorem 4, applied to the Cobb-Douglas production function commonly used in neoclassical growth theory [15]. Hence,

∅K1-∅K2≤sAαξα-1-δ+n+g(K1-K2)      (17)

Let define the contraction constant by

δ*=sAαξα-1-δ+n+g       (18)

If 0 ≤δ*< 1 , then ∅ is a contraction mapping on X . This condition is satisfied under diminishing returns to capital and economically realistic parameter values. To establish the Convergence of the model, the proposed theorem 5 is adopted. Since (X, d) is complete and ∅ is a contraction, Theorem 5 guarantees the existence of a unique fixed point K* such that:

∅K*=K*        (19)

Solving for K* using the iterative equation (14) yields the steady-state capital stock:

K*=sAδ+n+g11-α        (20)

Moreover, for any initial capital stock K0 ∈ X , the sequence {Kt} converges to K* with the error bound:

dKt,K*≤δt1-δdK1,K0               (21)

Thus, the model convergence holds in equation (21).  For the stability condition of the Solow-Swan model (11), the unique fixed point K* is globally asymptotically stable by Theorem 6. That is:

limt→∞dKt,K*=0          (22)

for any initial condition K0 ∈ X.

Contraction Mapping Analysis of [1] Growth Model

Here, we apply the contraction mapping framework, Theorem 3 – 6, to the growth model proposed by [1], given in equation (12).

The empirical growth model formulated by [1] is given as:

LRDPt=β0+β1NONEXPt+β2OIEXPt+β3NONIMPt+β4OIMPt+β5GFCFt+β6IRt+β7LEXPt+?t          (23)

To apply contraction mapping theorems, the model relation (23) is converted into a dynamic system by collecting the relevant variables into a state vector and specifying a period-to-period mapping. Let the economic state vector be defined by:

xt=LRDPtNONEXPtOIEXPtNONIMPtOIMPtGFCFtIRtLEXPt∈R8                (24)

Assuming the time dynamic evolution of the economy is given by:

xt+1=Φxt=Bxt+c             (25)

where

B∈R8×8  is a transition matrix capturing intertemporal effects among the variables (endogenous propagation of exports, investment and exchange rate effects) and c ∈ R8  is a vector of constants (including β0  and deterministic trend components).

Let (X, d)  be a complete metric space where:

X=R8,        dx,y=x-y      

By Theorem 3, we verify the Contraction Condition as follows. For any x, y ∈ X,

d(Φ(x), Φ(y)) = ?Bx - By?           (26)

= ?B(x - y)?                      (27)

≤ ?B??x - y?                  (28)

If there exists 0 ≤ δ < 1  such that:

?B? ≤ δ < 1                      (29)

then Φ  is a contraction mapping on X .

Theorem 5, guarantees the existence of fixed point in X . Since Φ  is a contraction on a complete metric space (X, d), there exists a unique fixed point x* ∈ X  such that:

Φx*=x*                     (30)

Using equation (30) on the time dynamics given in equation (25);

xt= Bxt+c   

 we have:

I - Bx* = c            (31)

Thus, solving (31) yields:

x*=(I - B)-1c         (32)

Hence, x*  is a fixed point and is unique because (I -B)  is invertible whenever ρ(B) < 1.  Next, we determine the model convergence to the fixed point. For any initial condition x0∈ X , the following iterative sequence holds:

xt+1=Φxt                   (33)

converges to x* , with error bound:

dxt,x*≤δt1-δdx1,x0         (34)

That is, from Theorem 5, the iteration satisfies for xt+1=Φxt :

xt-x*≤Bt1-Bx1-x0         (35)

In spectral radius form for some induced norm with contraction constant δ = ?B?,

xt-x*≤Cδt                 (36)

where

C=x1-x01-δ                (37)

Hence deviations from x*  decay at least exponentially at rate δ . If the true dynamic is nonlinear, Φ  may be differentiable and satisfy a Lipschitz condition on a region  R⊂R8  :

?Φx- Φy? ≤ L?x - y?,     ∀x, y ∈ R            (38)

A sufficient (local) condition is supx∈R  ?DΦ(x)? = L < 1,  where DΦ  is the Jacobian matrix of Φ . This aligns with Lemma 1 and Theorem 3. By Theorem 6, since Φ  is a contraction mapping, the unique fixed point x*  is globally asymptotically stable:

limt→∞d(xt,x*)                                      (39)

Which implies,

limt→∞xt-x*=0                (40)

for every initial state x0 .

In Figure 2, the curve monotonically declines and linearly decay on a log scale, signifying exponential convergence with temporal uptick around the period of 2020 to 2023 reflecting Covid-19 effects and oil price shocks. The return to zero confirms global stability condition.

Figure 4: Steady-state Dynamics of Economic Growth Model (δ < 1)

In Figure 4, the results show an initial fluctuation in GDP growth and over time, the path oscillate around a stable mean. There was not sufficient divergence during the periods

Figure 5 shows how the economic growth model convergence admits a unique and global stable equilibrium. Fluctuations induced by macroeconomic shocks are damped over time. The peak spikes can be attributed to the effect exchange rate crisis, food inflation and policy changes. This validates |Φ(xt) - x* | ≤ δ|xt -x* |,   δ = 0.6 < 1

Figure 6: State Dynamics and Convergence of Growth Model (δ < 1)

Figure 6, illustrates the GDP growth index (LRGDPt) and the convergence distance to equilibrium (|LRGDPt-LRGDP*|). The decay of deviations confirms the existence of a unique globally stable equilibrium growth path for the Nigerian economy. That is, despite fluctuations due to macroeconomic shocks, deviations always return toward equilibrium.

Figure 7 shows the evolution of the growth model, LRGDPt equation (23) under the counterfactual case δ = 1.05 (δ ≥ 1). The result implies that the trajectory does not settle around a stable level, fluctuations grow larger over time, no damping of shocks and convergence to a steady state. Recall that the contraction inequality is of the form ||Φ(xt) - x* || ≤ δ||Φ(xt-1) - x*  ||. If   0 < δ < 1, deviations shrink (stability). However, if δ ≥ 1, deviations amplify or persists (instability). Figure 4.7 confirms the failure of the inequality when δ ≥ 1.

DISCUSSIONS

The study applied contraction fixed point theory to analyze the stability and convergence properties of traditional economic growth models, including the Solow-Swan model and the [1] growth model. The results show that the Solow-Swan model satisfies the contraction condition under standard neoclassical assumptions, ensuring global asymptotic stability and convergence to a unique steady state capital stock. The [1] growth model, which incorporates macroeconomic and policy related variables, also satisfies the contraction mapping condition, guaranteeing convergence to a unique equilibrium growth path. The simulation results illustrate the state dynamics and convergence behavior of the Solow-Swan model, showing exponential decay of deviations and convergence to a steady state. The [1] growth model also exhibits convergence to a unique equilibrium growth path, highlighting the stabilizing role of policy and institutional parameters. However, when the contraction condition is violated, the model exhibits divergence or persistent instability, highlighting the importance of policy discipline in maintaining macroeconomic stability.

Data Sources: CBN, IMF & NBS, Period: 1995-2024.                                          

REFERENCE

1. Adekunle EO. International trade and economic growth in an oil-dependent country: Case of Nigeria, Manag Glob Trans 2025, 23(2), 147–174.
2. The Investopedia Team. Economic growth: What it is and how it is measured, 2025. Available from: https://www.investopedia.com. Reviewed by Charles Potters.
3. Chukwuebuka EE. Nigeria economic growth and its sustainability: A time series analysis on Nigeria and Algeria [Master’s thesis]. University of East Anglia; 2024.
4. Ighodaro CA. Determinants of economic growth in Nigeria. Uni of Nig Journ of Pol Econ. 2021; 9(1), 142–157.
5. Sinha JK. An investigation into the convergence of economic growth among Indian states and the path ahead. Stat Journ of the IAOS. 2024;40, 449–460.
6. Shashi P, Kumar S. Common fixed-point theorems for t-hardy-rodgers contraction mappings in complete cone b-metric spaces with an application. Topol. Algebra Appl. 2021; 9, 105–117.
7. Mechrouk S. Fixed point theorems in the study of positive strict set-contractions. Proyecciones Jour of Math. 2021;40(6):1569–1586.
8. Abdelhamid B, Karima H, Nassima M. Fixed point index for simulation mappings and applications. Analele Stiintifice ale Universitatii Ovidius Constanta. 2023;31(3):27–45.
9. Aslam M, Bota M, Mohammad S, Guran L, Saleem N. Common fixed points technique for existence of a solution of urysohn type integral equations system in complex valued b-metric spaces. Mathematics. 2021; 9(400).
10. Arul JG, Salah MB, Gunaseelan M, Mohamed, A, Asma A. Solving integral equations by common fixed-point theorems on complex partial b-metric spaces. Jour of Funct Spaces. 2021;2021(1)
11. Hamid F, Stojan RN. Some fixed-point results for convex contraction mappings on f-metric spaces. 2007;22
12. Nawab H, Iram I. Contraction mappings and application. Adv. in Int. Eqt. Intech Open; 2019. Available from: http://dx.doi.org/10.5772/intechopen.81571
13. Adagonye O, Ayuba SA. Bi-commutative digital contraction mapping and fixed-point theorem on digital image and metric spaces. Dutse Jour of Pure and Appl Sci. 2023;9(3a).
14. Adagonye O, Bitrus BY, Isa SA. Exploring commutative functions in digital metric spaces for contraction-type fixed point in digital image processing. Intl Jour of Sci Research and Tech. 2025;7(9).
15. Barro RJ, Sala-i-Martin, X. Economic Growth (2nd ed.) Chapter 1-2: Neoclassical           Growth Models. MIT Press, Cambridge, Massachusetts London, England; 2004.