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

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.

• Why does it seems weird? The concept would be akin to acceleration in physics. – Giskard Jul 6 '16 at 18:09