# Balanced Growth (Solow Model)

Sometime ago, I asked for the definition of steady state in the Solow Model. I came to the conclusion after reading the answer that the steady state is the sequence $$\{(K(t), L(t), Y(t), C(t), w(t), r(t)\}_{t=0}^{\infty}$$ determined by $$\dot{k}(t) = 0 \ (\forall \ t = 0,1,2, \dots)$$. (The definition does not exclude unstable steady states.)

Questions:

1. How is a balanced growth path defined under the following assumptions:
• Population and technology grow at constant rates (which may or may not be zero).
• Harrod-neutral technological progress. (I think this is required, otherwise, ignore.)
• The production function $$F(A_t,K_t,L_t)$$ is homogenous of degree $$1$$.
2. How is the balanced growth path different from the steady state? From reading Alecos' answer on this site, I suspect that the BGP is almost similar to the steady state except that the sequence is now $$\{(k(t), y(t), c(t), w(t), r(t)\}_{t=0}^{\infty}$$ and that is determined by $$\dot{\tilde{k}}(t) = 0 \ \forall \ t \geq 0$$ where $$\tilde{k} = \frac{K}{AL}$$.
1. Is the above definition correct? Or should the determining factor be rather $$\dot{k} = 0$$ (instead of $$\tilde k = 0$$) where $$k = K/L$$?
2. Is $$y/k$$ being constant (or one of the equivalent statements like $$g_y = g_k$$ (growth of $$y$$ $$=$$ growth of $$k$$) additionally required as a determinant or it follows from $$\tilde{k} = 0$$?

As the answer of Alec Papadopoulos you quoted says, there has been a change of definitions and a various use of the terms steady state and balanced growth in time, and I refer to this answer for this aspect. As a consequence, there has been no fixed, uniform use of these terms in economic growth literature.

We can see it looking at some of the most well-known references in the field of economic growth.

For example, Barrow and Sala i Martin, in Economic Growth, MIT Pres 2004, write, referring to the case of absence of technological progress:

We define steady state a situation in which the various quantities grow at a constant (perhaps zero) rate. […] Some economists use the expression balanced growth path to describe the state in which all variables grow at constant rate and use steady state to describe the particular case when the growth rate is zero. (pp. 33-34)

Romer, Advanced Macroeconomics, Mac Graw Hill, 2012, privileges the term balanced growth path:

[…] the economy converges to a balanced growth path - a situation in which each variable of the model is growing at constant rate.(p. 18)

Acemoglu, Modern Economic Growth, Princeton, 2009, adds some observations about the concept of balanced growth, term that he uses in the case of presence of technological progress, pointing out the consistency with the so-called Kaldor facts:

The standard approach is to impose discipline on the form of technological progress […] by requiring that the resulting allocations be consistent with balanced growth, as defined by the so-called Kaldor facts. […] Throughout the book, balanced growth refers to an allocation where output grows at a constant rate and capital output ratio, the interest rate and factor shares remain constants.(p. 57)

But, in substance, the formal definitions are still the same:

With balanced growth the equations describing the law of motion of the economy can be represented by differential or difference with well-defined steady state in transformed variables [variables transformed to take into account technological progress, my note](p. 58).

In short, it is a matter of definitions and terminology chosen by the authors, often steady state and balanced growth path are considered synonymous, sometime steady state is used in absence of technological progress and balanced growth path when there is technological progress. The important thing is to clarify the use of the terms each time and understand each other.

$$***$$

You wrote

[…]that the BGP is almost similar to the steady state except that the sequence is now $$\{(k(t), y(t), c(t), w(t), r(t)\}_{t=0}^{\infty}$$ and that is determined by $$\dot{\tilde{k}}(t) = 0 \ \forall \ t \geq 0$$ where $$\tilde{k} = \frac{K}{AL}$$. 1. Is the above definition correct? Or should the determining factor be rather $$\dot{k} = 0$$ (instead of $$\tilde k = 0$$) where $$k = K/L$$?

The definition is correct, even if we have to specify the definition of $$k$$ etc. in the 'sequence' (or, better, the solution path)$$^1$$.

In presence of labor augmenting technical progress we use, as you point out, the modified variable $$\hat k \equiv \frac{K}{AL}$$, instead of $$k\equiv \frac{K}{L}$$ as in absence of technical progress. At the denominator we have now $$AL$$ instead of $$L$$: $$A$$ represents technological progress and $$AL$$ is the so-called effective labor, that is labor modified to take into account technology that modifies productivity.

The most important thing, to establish the model correctly, is to identify the equation of motion (the fundamental differential or difference equation ) of the model and the variable involved, which differs according to the assumption of absence or presence of technological progress. We must ask: what is the variable under consideration in the model, in terms of which the model states the fundamental equation of motion?

In both cases, the steady state (or balanced growth path) is defined as the solution of the fundamental equation of motion such that the derivative (or the difference for the discrete case) with respect to time of the variable involved is zero (the definition of equilibrium, or fixed point in dynamical systems.)

In the case of absence of technical progress, the fundamental equation (in continuous time, in discrete time the issue is analogous) is:

$$\dot {k}(t)= sf(k(t))- (n+\delta)k(t) \tag {1},$$ where the variables have the usual meaning and $$k\equiv \frac{K}{L}$$ is the per capita capital.

The steady state (or balanced growth path) requires $$\dot {k(t)}=0$$.

Under the assumption of labor-augmenting technological progress the equation is modified this way:

$$\dot{\hat {k}}(t)= sf(\hat k(t))- (n+\delta)\hat k(t) \tag {2}$$

The equation is exactly the same, except for the fact that we have now the modified capital-labor ratio, $$\hat k= \frac{K}{AL}$$, in which we have at the denominator the effective labor $$AL$$ .

The steady state requires now that the derivative with respect to time of the transformed variable, $$\dot{\hat {k}}(t)$$, is zero.

That is, in the steady state the per capita effective labor is constant. This implies that the ‘simple’ capital-labor $$k$$ is not constant, but grows according to the technological progress.

As you can see, the model is exactly the same in both cases, absence or presence of technological progress, except for the fact that with technological progress we use the transformed variable $$\hat {k}(t)$$. $$***$$

Also the usual graph of the Solow model is exactly the same, the only difference is the use of the variable $$\hat k$$ instead of $$k$$.

Below the graph with technical progress and the corresponding phase diagram in the lower part of the picture, as suggested in the answer by Papadopoulos: remember that in a so-called unidimensional flow, as our differential equations $$(1)$$ and $$(2)$$, in a phase diagram we have the variable involved on the $$x$$ axis and its derivative on the $$y$$ axis.

The arrows represent the direction of the motion, the change in $$\hat {k}(t)$$, while the ordinate in the phase diagram, $$\dot{\hat {k}}(t)$$, the derivative with respect to time, represents the velocity of the motion: it is positive on the left of $$\hat k^*$$, negative on the right, and 0 at the balanced growth path $$\hat k^*$$.

Its magnitude is represented by the difference $$sf(k)-(n+d)$$, which reaches its maximum, $$\hat k_{Max}$$ in correspondence of $$\hat k_m$$.

$$***$$ You wrote

1. Is $$y/k$$ being constant (or one of the equivalent statements like $$g_y = g_k$$ (growth of $$y$$ $$=$$ growth of $$k$$) additionally required as a determinant or it follows from $$\tilde{k} = 0$$?

In balanced growth path we have that $$\hat k$$ is constant.

Thanks to the assumption of constant returns to scale, in the model the intensive production function $$y=f(\hat k)$$ is used, that is $$\hat k$$ is the unique determinant of $$y$$: as a consequence, if $$\hat k$$ doesn’t vary, $$y$$ is constant, and so does $$\frac {y}{\hat k}$$.

$$^1$$ Just an observation, to avoid confusion between discrete and continuous time models: you use the term sequence, and this is correct if we speak of a discrete time model and a difference equation. if we have a continuous time model, where we have a derivative as $$\dot k$$ with respect to time and therefore a differential equation, it is not correct to speak of a sequence, but we must speak of functions of time. This is not a minor language matter, but a correct use of important mathematical notions.

• Please give me a day's time (from now). I'll give it a read and respond, and/or accept your answer. Thank you so much for the enormous effort! Commented Apr 6 at 4:35
• You are welcome! Take the time you need, don't mind. Commented Apr 6 at 9:33
• Read it. Very detailed and helpful. Thank you BakerStreet. Commented Apr 8 at 6:36
• I'm happy to have been useful! Commented Apr 8 at 11:05
• Hi, in the Solow model with technological progress, do I consider labour income as $wL$ or $wAL$ (where $w, A$ determine wage and technological progress respectively)? Further, is $w = \frac{\partial F(K,AL)}{\partial L}$ or $w = \frac{\partial F(K,AL)}{\partial (AL)}$? Commented Apr 12 at 4:28