<p>If $Y = C \cdot X$ where $C$ is constant and $\frac{\dot{X}}{X} = g$ then we can solve for $\frac{\dot{Y}}{Y}$ as follows:
$$ \frac{d}{dt} Y = \frac{d}{dt} C \cdot X = C \cdot \frac{dX}{dt} = C \cdot \dot{X} \Rightarrow$$
$$ \frac{\dot{Y}}{Y} = \frac{C \cdot \dot{X}}{ C \cdot X} = \frac{\dot{X}}{X} = g$$</p>

<p>Therefore, as you concluded, they both grow at rate $g$.</p>

CC BY-SA 3.0
2016-01-12T22:46:14.377Z
2016-01-12T22:46:14.377Z
BKay
<p>I have been given the following setup:</p>

<p>$$ Y=K^\theta (AL)^{1-\theta }$$</p>

<p>Where Y = Output, K = Capital, L = Labour and A = Productivity.</p>

<p>$$ \frac{\dot{L}}{L} = n $$
$$ \frac{\dot{A}}{A} = g $$</p>

<p>The Capital Accumulation Equation has also been given as:</p>

<p>$$ \dot{K} = sY - \delta K $$</p>

<p>Using this, I have calculated expressions for capital per worker and output per worker in the steady state as:</p>

<p>$$ \frac{K}{L} = A(\frac{s}{n+g+\delta })^{\frac{1}{1-\theta }} $$</p>

<p>$$ \frac{Y}{L} = A(\frac{s}{n+g+\delta })^{\frac{\theta }{1-\theta }} $$</p>

<p>The next questions asks: <strong>Calculate the growth rates of capital per worker and output per worker that will hold as the economy moves along a steady state growth path.</strong></p>

<p>This is confusing because I was under the impression that growth of capital and labour is the steady state is 0, and then therefore growth of K/L and Y/L should also be 0. However this seems wrong and I believe the answer should be g. Can anyone help please?</p>

CC BY-SA 3.0
2016-01-12T20:29:41.670Z
Hint: Everything in the RHS of the bottom two equations is constant except A.
BKay
CC BY-SA 3.0
2016-01-12T22:09:19.920Z
@BKay So because A grows at rate g, K/L and Y/L also grow at rate g?
James Baker
<economic-growth><production-function><capital-returns><steady-state>
Solow Model, Growth rate of K/L and Y/L in steady state
James Baker
# Solow Model, Growth rate of K/L and Y/L in steady state

(Asked by: username_0 on 2016-01-12)

I have been given the following setup:

$$ Y=K^\theta (AL)^{1-\theta }$$

Where Y = Output, K = Capital, L = Labour and A = Productivity.

$$ \frac{\dot{L}}{L} = n $$
$$ \frac{\dot{A}}{A} = g $$

The Capital Accumulation Equation has also been given as:

$$ \dot{K} = sY - \delta K $$

Using this, I have calculated expressions for capital per worker and output per worker in the steady state as:

$$ \frac{K}{L} = A(\frac{s}{n+g+\delta })^{\frac{1}{1-\theta }} $$

$$ \frac{Y}{L} = A(\frac{s}{n+g+\delta })^{\frac{\theta }{1-\theta }} $$

The next questions asks: **Calculate the growth rates of capital per worker and output per worker that will hold as the economy moves along a steady state growth path.**

This is confusing because I was under the impression that growth of capital and labour is the steady state is 0, and then therefore growth of K/L and Y/L should also be 0. However this seems wrong and I believe the answer should be g. Can anyone help please?

**Tags:** economic-growth, production-function, capital-returns, steady-state
> **Comments:**
> **username_1**: Hint: Everything in the RHS of the bottom two equations is constant except A.  
> **username_0**: @BKay So because A grows at rate g, K/L and Y/L also grow at rate g?  

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## 1 answers

### Answer by username_1 on 2016-01-12. Score: 3 | Accepted answer

If $Y = C \cdot X$ where $C$ is constant and $\frac{\dot{X}}{X} = g$ then we can solve for $\frac{\dot{Y}}{Y}$ as follows:
$$ \frac{d}{dt} Y = \frac{d}{dt} C \cdot X = C \cdot \frac{dX}{dt} = C \cdot \dot{X} \Rightarrow$$
$$ \frac{\dot{Y}}{Y} = \frac{C \cdot \dot{X}}{ C \cdot X} = \frac{\dot{X}}{X} = g$$

Therefore, as you concluded, they both grow at rate $g$.


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