Growth rate of variables on a balanced growth path (BGP)

Question

Normally, in the very basic endogeneous model of growth. It is said that all variables grow at a constant rate. Let's take the basic capital accumulation ;

$$\dot{K}=Y-C$$

If I write this in following way ;

$$g_{K}=\frac{\dot{K}}{K}=\frac{Y}{K}-\frac{C}{K}$$

Normally, at balanced growth path (BGP), $g_{K}$ grows at a constant value (an arbitrary constant)

In this case, this means that variation of $g_{K}$ according to time at BGP should be equal to zero.

So ;

$$\frac{dg_{K}^{BGP}}{dt}=\left(g_{Y}-g_{K}\right)-\left(g_{C}-g_{K}\right)=0$$

which implies at BGP ;

$$g_{Y}=g_{K}=g_{C}$$

Is this reasoning correct ?

Because in fact, it seems me weird to differentiate a growth rate $g_{K}$ according to time.

Answer 1

The concept of "balanced growth path" in economics incorporates three characteristics at the same time (related to the main macroeconomic aggregates):

1) Growth rates are constant (reflecting a notion of equilibrium/stability)

2) Growth rates are strictly positive (otherwise the economy would eventually vanish)

3) Growth rates are equal to each other (hence there is "balance" and no macroeconomic aggregate becomes negligible relative to another)

To obtain the first characteristic you have to impose that the derivative of the growth rate with respect to time is zero. Nothing weird about that, just a valid mathematical operation.