The phase diagram for the simple RBC model

Question

I'm reading these notes in the RBC model. In page 14, the author, tries to explain the direction of the arrows, in the space $(\tilde C_t,\tilde K_t)$ (tilde refers to proportional deviations from steady state) , to explain the dynamics.

If I look at equations (47) and (49), the ones that are used to derive the isoclines (equations (50) and (51)), I do not understand how to see the directions Up/Down, Left/Right...

Any help would be appreciated.

Answer 1

There are two tricks to safely construct a phase diagram, as regards the dynamics off it.

First, solve for the "isoclines" as weak inequalities rather than as equalities. This method makes also clear the dynamics off the isocline.

Consider a standard capital accumulation equation

$$K_{t+1} = (1-\delta)K_t + F(K_t) - C_t \tag{1}$$

We want to obtain the isocline, but also the dynamics off it. Re-write as

$$K_{t+1} - K_t = -\delta K_t + F(K_t) - C_t$$

Now require that

$$K_{t+1} - K_t \geq 0 \implies -\delta K_t + F(K_t) - C_t \geq 0$$

$$\implies C_t \leq F(K_t) - \delta K_t \tag{2}$$

With equality, $(2)$ is the expression for the isocline. The inequality tells you that for capital to tend to increase ($K_{t+1} - K_t > 0$), consumption must be lower than the level indicated by the isocline. Etc

The second trick is already used in $(2)$: although this is the capital difference equation, I wrote it as though it is an expression determining consumption. In this way, one can look at the phase diagram and for both difference equations, treat the variable in the vertical axis as the "dependent" variable of the two isocline functions. This is the most natural to the eye, and helps avoid the up/down, right/left confusion that can result if one tries to rotate in its mind the phase diagram, for one of the two equations. It also makes easier to determine the slope/shape of the two isoclines.