# Why are there two formats for the Solow capital evolution equations?

I have seen the capital transition equation take two forms

$$K_{t+1} = sY_t +(1-\delta) K_t$$ $$\dot{K}_t = sY_t - \delta K_t$$

Obviously I can rewrite the top one to get

$$K_{t+1} - K_t = sY_t - \delta K_t$$

So that suggests somehow

$$\dot{K}_t \equiv K_{t+1}-K_t$$

But as far as I know,

$$\dot{K}_t \equiv \frac{\partial K_t}{\partial t} = \lim_{h\rightarrow 0} \frac{K_{t+h} - K_t}{h}$$

which is, as far as I know, not the same as $K_{t+1}-K_t$.

my question

Is $$\dot{K}_t \equiv K_{t+1}-K_t$$ right? If so, what does this mean and why isn't it the usual derivative?

• The notation $\dot{K}_t$ makes no sense to me at all because time seems to be both discreet and continuous. – Giskard Jan 25 '16 at 8:47
• Oh, maybe im confusing those. Not sure. That might be the problem. – Stan Shunpike Jan 25 '16 at 9:01
• Better to set $\Delta K$ for notational convenience. As denesp mentionned, the variables with dot are often used in continous time models. – optimal control Jan 25 '16 at 10:54