# Transitional dynamics in Solow model

I am trying to understand the transitional dynamics of the Solow model. Let's take the case of forward movement (or when the capital-labour ratio $$k(t)$$ increases).

I have a bunch of questions regarding this:

1. How is the $$x$$-axis (and similarly, $$y$$-axis) defined? Is it over time $$t$$ (that is, a sequence of the form $$\{k(t)\}_{t \geq 0}$$ or is the time fixed at $$t$$ (that is, a sequence of the form $$\{k_t : t \text{ is fixed}\}_{0}^{\infty}$$)?
2. What is the significance of the dashed $$45^\circ$$ line?
3. What do the vertical arrows ($$\uparrow$$) represent? How exactly does the transition place?
4. Repeat (3) for the horizontal ($$\rightarrow$$) red arrows.

These may require a long answer, so you may suggest me a book too that answers these questions. It would be helpful if you could explain me (1) and (2) regardless of whether you're suggesting me a book or not.

How is the x-axis (and similarly, y-axis) defined?

$$t$$ is not fixed, you arrange data in pairs where you always pair $$k(t)$$ with $$k(t+1)$$ for every $$t$$. So for t=2001, $$k_{2001}$$ would go on x-axis and $$k_{2002}$$ on y-axis and then for t=2002, the $$k_{2002}$$ would go on x-axis and $$k_{2003}$$ on y-axis. The problem is actually continuous, but I use years to maybe better illustrate the problem.

What is the significance of the dashed 45∘ line?

Along the 45-degree line capital tomorrow is equal to capital today. When capital stock tomrrow and today are equal it means that we reached a sort of dynamic equilibrium when addition to capital is exactly equal to depreciation in capital over time.

What do the vertical arrows (↑ ) represent? How exactly does the transition place? Repeat (3) for the horizontal (→ ) red arrows.

Arrows are mathematical notation used in so called 'phase diagrams'. It is a way how to graphically represent temporal dynamics in differential equations, it shows you which way will variable evolve over time.

For example below $$k^*$$ k will be moving to the right (increasing), and it will do so iteratively in this model (up right, up right etc) until you reach point $$(k^*,k^*)$$. Arrows show you this trajectory. Above $$k^*$$ you are moving left down until you reach the equilibrium.

Regarding questions 1-2, there are lot of books that deal with growth theory. For example, Acemoglu Introduction to Modern Economic Growth is good book or Barro and Salla-i-Martin Economic Growth.

Regarding the arrows, that is just a math you should ideally know before studying growth theory, so you need some math textbook for that. Further Mathematics for Economic Analysis by Sydsaeter et al has good and economics oriented explanation of differential equations and phase diagrams. However, here any advanced calculus textbook will work even if its not focused on economics.

• I understood 1,2. I realized that it's a graph of the difference equation, so at a time (or using a single point), we can only represent times $t$ and $t+1$ rather than, say, $(t,t+7)$. As for 3,4, can you tell me what does the "up, right, up right, .." mean from an economics perspective?
– user43302
Mar 11, 2023 at 12:36
• @solowsupremacy it means that $k$ moves first from initial point at [k(0),0] to [k(0), k(1)] (literally movement up as the arrows indicate in that k(t), k(t+1) space, and then you move right from [k(0), k(1)] to that next point as indicated by arrows. From economic perspective it just tells you that stock of capital will be increasing
– 1muflon1
Mar 11, 2023 at 12:40
• As I understand, the horizontal arrows doesn't show anything happening in the economy (or with the representative firm, let's say). It's drawn as an intermediate phase to denote the movement of what actually happens in the economy: $[k_t, k_{t+1}] \to [k_{t+1}, t_{t+2}]$ (which is in turn represented by the vertical arrows). Is my understanding correct?
– user43302
Mar 11, 2023 at 12:55
• @solowsupremacy yes
– 1muflon1
Mar 11, 2023 at 12:56
• Thank you so much. This explains everything. Thanks for the books too.
– user43302
Mar 11, 2023 at 12:57