<p>The Solow-Swan has the two versions, one is in discrete time (first equation) and other in continuous time. Both give the same results, but you cannot mix them - as you did. </p>

<p>In the first equation (discrete time) you can take the first diference but never a derivative, since the variable is not continuous. The point above the variable means that it has been diferenciated with relation to time.</p>

<p>So the last equation does not make any sense because tried to mix both models.</p>

CC BY-SA 3.0
2016-01-26T04:00:55.070Z
2016-01-26T04:16:21.667Z
2016-01-26T04:16:21.667Z
Joao Luiz Pacheco
<p>I have seen the capital transition equation take two forms</p>

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

<p>Obviously I can rewrite the top one to get </p>

<p>$$K_{t+1} - K_t = sY_t - \delta K_t $$</p>

<p>So that suggests somehow </p>

<p>$$\dot{K}_t \equiv K_{t+1}-K_t$$</p>

<p>But as far as I know, </p>

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

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

<p><strong>my question</strong></p>

<p>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? </p>

CC BY-SA 3.0
2016-01-25T08:47:42.750Z
The notation $\dot{K}_t$ makes no sense to me at all because time seems to be both discreet and continuous.
Giskard
CC BY-SA 3.0
2016-01-25T09:01:44.340Z
Oh, maybe im confusing those. Not sure. That might be the problem.
Stan Shunpike
CC BY-SA 3.0
2016-01-25T10:54:07.760Z
Better to set $\Delta K$ for notational convenience. As denesp mentionned, the variables with dot are often used in continous time models.
optimal control
<macroeconomics>
Why are there two formats for the Solow capital evolution equations?
Stan Shunpike
# Why are there two formats for the Solow capital evolution equations?

(Asked by: username_0 on 2016-01-25)

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?

**Tags:** macroeconomics

> **Comments:**
> **username_1**: The notation $\dot{K}_t$ makes no sense to me at all because time seems to be both discreet and continuous.  
> **username_0**: Oh, maybe im confusing those. Not sure. That might be the problem.  
> **username_2**: Better to set $\Delta K$ for notational convenience. As denesp mentionned, the variables with dot are often used in continous time models.  

---

## 1 answers

### Answer by username_3 on 2016-01-26. Score: 2

The Solow-Swan has the two versions, one is in discrete time (first equation) and other in continuous time. Both give the same results, but you cannot mix them - as you did.

In the first equation (discrete time) you can take the first diference but never a derivative, since the variable is not continuous. The point above the variable means that it has been diferenciated with relation to time.

So the last equation does not make any sense because tried to mix both models.


---

