On the stability of the steady-state of a general model of endogenous growth with two $CES$ production functions

This paper investigates the steady-state properties of a general Bond-type endogenous growth model featuring two distinct CES production functions and demonstrates that such a system does not necessarily exhibit saddle-path stability.

The Gist

Imagine the economy as a giant, self-sustaining factory that never stops growing. This factory has two main departments:

1. The Goods Department: This builds physical things like machines, buildings, and tools (Physical Capital).
2. The Education Department: This builds knowledge, skills, and expertise (Human Capital).

For this factory to keep growing forever without running out of steam, it needs to balance how much time and resources it spends on building machines versus teaching people.

The Old Story (Bond et al., 1996)

In 1996, a group of economists named Bond and colleagues wrote a famous paper about this factory. They said:

"We have found a perfect, stable way for this factory to run. If you start anywhere, the factory will naturally find its way to this perfect path, like a train on a track that always leads to the station. We call this 'saddle-path stability.' It means the system is predictable and safe."

They assumed the factory's production rules were simple and flexible (like a standard recipe).

The New Story (Chilarescu, 2026)

The author of this paper, Constantin Chilarescu, decided to test that claim with a more complex, realistic recipe. Instead of a simple recipe, he used CES production functions.

The Analogy: The "Mixing Bowl"
Think of the production function as a recipe for a smoothie.

• Simple Recipe (Cobb-Douglas): You must use exactly 50% fruit and 50% yogurt to make the perfect smoothie. If you change the ratio, the taste changes predictably.
• Complex Recipe (CES): You can mix fruit and yogurt in almost any ratio, but the "taste" (efficiency) changes in a tricky, non-linear way depending on how much of each you use. Sometimes adding more fruit makes the yogurt taste better; sometimes it makes it worse.

Chilarescu says: "Bond assumed the factory used the simple recipe. But what if the factory uses the complex, tricky recipe (CES) where the two departments have different mixing rules?"

The Big Discovery

Chilarescu ran the math on this complex factory and found a shocking result: The "Saddle-Path" does not exist.

What does that mean in plain English?
Imagine you are trying to balance a broom on your finger.

• Bond's claim: "If you move your hand slightly off-center, the broom will naturally swing back to the center and stay there." (Stable).
• Chilarescu's finding: "If the broom is made of this weird, complex material (CES functions), and you move your hand slightly, the broom might wobble, spin, or fall over in a way we can't predict. It doesn't have a guaranteed 'track' to follow back to stability."

In technical terms, the mathematical "map" that guides the economy back to its steady growth path breaks down. The system becomes too complex to guarantee that it will settle down neatly. The "determinant" (a mathematical measure of stability) becomes zero, which is like a traffic light turning yellow—it doesn't tell you to go or stop; it tells you the situation is ambiguous and requires a much deeper look.

Why Does This Matter?

1. It's not just a math game: If the economy is like this complex factory, we can't be 100% sure that government policies or market forces will naturally guide us back to a stable growth path after a shock (like a pandemic or a financial crash).
2. The "Special Case" Exception: Chilarescu notes that if the two departments have the exact same mixing rules (same elasticity of substitution), then Bond was right, and the system is stable. But in the real world, making machines and teaching people are very different processes with different rules. So, the "unstable" scenario is likely the real one.
3. Cobb-Douglas is still okay: Interestingly, when he tested the "simple recipe" (Cobb-Douglas) again, the system was stable. This suggests that the instability only happens when we use those more complex, realistic mixing rules.

The Bottom Line

This paper is a warning to economists. It says: "Don't assume the economy is a simple, predictable machine that always finds its way back to balance. If the rules of production are complex and different between sectors, the economy might get stuck in a wobble, and we need new tools to understand how to steady it."

It's like realizing that while a car with a simple engine drives straight, a car with a complex, high-tech engine might drift if you don't have a very sophisticated steering system.

Technical Summary

1. Problem Statement

The paper addresses a specific theoretical claim made in the seminal work of Bond, Trask, and Wang (1996) regarding a general two-sector endogenous growth model (goods sector and education sector). Bond et al. claimed that their general model, regardless of the specific functional forms of the production functions, possesses a unique balanced growth path (BGP) that is saddle-path stable.

Chilarescu challenges this assertion. The paper investigates whether the saddle-path stability property holds when the two sectors are modeled using distinct Constant Elasticity of Substitution (CES) production functions, where the elasticities of substitution differ between the sectors ($\sigma_1 \neq \sigma_2$). The central problem is to determine if the system retains the stability properties claimed by Bond et al. under these more general and flexible functional forms.

2. Methodology

The author employs a rigorous dynamic optimization framework using the Hamiltonian approach.

• Model Structure:

  • Sectors: A goods sector (producing consumption and physical capital $k$) and an education sector (producing human capital $h$).
  • Production Functions: Both sectors utilize distinct CES functions.
    • Goods Sector: $Y_1 = A_1 [\alpha_1 (kv)^{\psi_1} + (1-\alpha_1)(hu)^{\psi_1}]^{1/\psi_1}$
    • Education Sector: $Y_2 = A_2 [\alpha_2 (k(1-v))^{\psi_2} + (1-\alpha_2)(h(1-u))^{\psi_2}]^{1/\psi_2}$
    • Here, $u$ and $v$ represent the allocation of human and physical capital, respectively. $\psi_i = \frac{\sigma_i-1}{\sigma_i}$ are the substitution parameters, with $\psi_1 \neq \psi_2$.
  • Objective: Maximize intertemporal utility $V_0 = \int_0^\infty \frac{c^{1-\varepsilon}-1}{1-\varepsilon} e^{-\rho t} dt$ subject to capital accumulation constraints.

• Analytical Steps:

  1. Derivation of First-Order Conditions (FOCs): The author constructs the Hamiltonian and derives the necessary conditions for optimality, including the evolution of costate variables ($\lambda, \mu$) and the allocation rules for $u$ and $v$.
  2. Transformation to Stationary Variables: To analyze the steady state, the system is transformed. Since $k$, $h$, and $c$ grow at a common rate $r^*$ on the BGP, the author defines stationary variables:
     • $z = k/h$ (ratio of physical to human capital)
     • $q = c/k$ (consumption-capital ratio)
     • $u$ and $v$ (allocation shares, which are constant on the BGP).
  3. System Reduction: The original 5-variable dynamic system is reduced to a system of differential equations describing the dynamics of $z, q, u,$ and $v$.
  4. Stability Analysis: The author evaluates the Jacobian matrix of the reduced system at the steady state to determine the eigenvalues and assess stability.

3. Key Contributions

The paper makes three primary theoretical contributions:

1. Refutation of General Saddle-Path Stability: The author proves that the claim of saddle-path stability by Bond et al. (1996) does not hold for the general case of two distinct CES production functions.
2. Identification of a Zero Determinant: A critical technical finding is that in the CES case with distinct substitution parameters, the variable $z$ (capital ratio) depends explicitly on the control variables $u$ and $v$. Consequently, the determinant of the Jacobian matrix evaluated at the steady state is zero.
3. Distinction from Cobb-Douglas Case: The paper contrasts the CES case with the Cobb-Douglas (CD) case. It shows that while the CD case (where $\psi_1 = \psi_2 = 0$) allows for a reduction to a 3-variable system that can be saddle-path stable (confirmed via numerical simulation), the general CES case leads to a degenerate system where standard linear stability analysis is insufficient to claim saddle-path stability.

4. Results

A. Existence and Uniqueness of the Steady State

The paper establishes that a unique balanced growth path exists under specific parameter restrictions (Proposition 1).

• Growth Rate: The common growth rate $r^*$ is determined by the parameters of the economy and the unique solution $w^*$ to the equation $P(w)=0$.
• Conditions: The existence requires specific bounds on the rate of time preference ($\rho$) relative to depreciation and productivity parameters.
• Transversality: The solution satisfies transversality conditions provided the inverse elasticity of intertemporal substitution $\varepsilon > 1$.

B. Stability Analysis (The Main Result)

• The Degeneracy: In the reduced system of stationary variables $(z, q, u, v)$, the relationship between $z$ and the controls $(u, v)$ creates a dependency that causes the Jacobian determinant to be zero.
• Implication: A zero determinant implies that the system has at least one zero eigenvalue. Therefore, the system is not hyperbolic, and one cannot claim saddle-path stability based on the standard eigenvalue criteria used in the Bond et al. paper. The system may exhibit bifurcation or other complex behaviors rather than converging along a unique stable manifold.

C. Comparison with Cobb-Douglas and Numerical Evidence

• Cobb-Douglas Case: When the production functions are restricted to Cobb-Douglas (a special case of CES), the dependency structure changes. The system reduces effectively to three independent variables.
• Numerical Simulations: The author performs numerical simulations for the Cobb-Douglas case with two different parameter sets. In both cases, the eigenvalues of the Jacobian show one negative and two positive real parts (e.g., $EV = [0.16; 0.89; -0.73]$). This confirms that the Cobb-Douglas version is indeed saddle-path stable, validating the Bond et al. claim only for that specific functional form, but not for the general CES case.

5. Significance

• Theoretical Correction: The paper corrects a widely accepted assumption in endogenous growth literature. It demonstrates that the stability properties of general equilibrium models are highly sensitive to the functional form of production technologies.
• Methodological Rigor: It highlights the dangers of assuming saddle-path stability in general models without verifying the rank and determinant of the Jacobian matrix, particularly when the number of state and control variables creates dependencies.
• Policy Implications: Since the stability of the growth path determines whether an economy converges to a specific trajectory or diverges, the finding that general CES models may lack saddle-path stability suggests that policy interventions in economies with heterogeneous substitution elasticities could have unpredictable long-term effects compared to the standard Cobb-Douglas predictions.

In conclusion, Chilarescu proves that while a unique steady state exists for the general two-sector CES model, the system fails to be saddle-path stable due to a zero determinant in the Jacobian matrix, thereby invalidating the general stability claim made by Bond et al. (1996) for this class of models.