<p>You can use the following results provided below to try and answer the problem yourself.</p>
<ol>
<li><p>Consider a Constant Returns to Scale(CRS) production function <span class="math-container">$F:\mathbb{R}_+^2\to \mathbb{R}$</span><br />
Let <span class="math-container">$Y=F(K,L)$</span> denote the level of output produced using <span class="math-container">$K$</span> units of capital and <span class="math-container">$L$</span> units of labor given the production function <span class="math-container">$F$</span>.</p>
<p>By CRS, <span class="math-container">$F$</span> must satisfy <span class="math-container">$: \quad F(\lambda K, \lambda L)=\lambda F(K,L)$</span> where <span class="math-container">$\lambda \geq 0$</span> is some constant</p>
<p>Since the above holds for any positive <span class="math-container">$\lambda$</span>, we can set <span class="math-container">$\lambda=\frac{1}{L}$</span> and let <span class="math-container">$k=\frac{K}{L}$</span> and <span class="math-container">$y=\frac{Y}{L}$</span>.</p>
<p>This gives us:
<span class="math-container">$$\begin{eqnarray}
&amp; F\left(\frac{K}{L},1\right)=\frac{F(K,L)}{L} \\
&amp; F(k,1)=\frac{Y}{L} \\
&amp; y=f(k) &amp;&amp; \text{where }F(k,1)\equiv f(k) \tag{*}
\end{eqnarray}$$</span></p>
<p>The expression <span class="math-container">$\boxed{y=f(k)}$</span> is known as the intensive form (per-worker production) function (which is what you are asked to find in part a)</p>
</li>
<li><p>Consider a Solowian Economy with the following features:<br />
<span class="math-container">$\bullet$</span> Time is continuous and there is no distinction between the labor force and the population at any given period of time. Labor (or population) growth rate <span class="math-container">$\frac{\dot{L}}{L}=n$</span>  , where <span class="math-container">$\dot L= \frac{dL}{dt}$</span> is exogeneous<br />
<span class="math-container">$\bullet$</span> In each period a constant proportion of income <span class="math-container">$s\in(0,1)$</span> is saved and capital depreciates by <span class="math-container">$\delta$</span> in each period<br />
<span class="math-container">$\bullet$</span> There is no technological progress and output in any period is given by <span class="math-container">$Y_t=F(K_t,L_t)$</span> with the assumption that <span class="math-container">$F$</span> satisfies all the properties of a neo-classical production function</p>
<p>Denoting the growth rate of any object <span class="math-container">$x$</span> (say) as <span class="math-container">$g_x$</span> let us use the above data to solve for the steady state.</p>
<p>The law of motion of capital is given by: <span class="math-container">$\begin{eqnarray} &amp;\dot K = I-\delta K\\
\implies &amp;\dot K= sY-\delta K \tag{**} \end{eqnarray}$</span></p>
<p>we defined per-capita capital as <span class="math-container">$k=\frac{K}{L}$</span> and at steady state we must have <span class="math-container">$\frac{\dot k}{k}=g_k=0$</span> i.e., the growth rate of per-capita capital must be zero at the steady state. Let us find an expression for <span class="math-container">$\frac{\dot k}{k}$</span> or <span class="math-container">$g_k$</span> as a function of <span class="math-container">$k$</span> in order to find steady state per-capita capital.
<span class="math-container">$$\begin{align} 
k&amp;=\frac{K}{L} \\
\ln k&amp;= \ln K -\ln L &amp; \text{taking log on both sides} \\
\frac{\dot k}{k}&amp;=\frac{\dot K}{K}-\frac{\dot L}{L} &amp; \text{taking time derivative on both sides} \\
&amp;=\frac{sY-\delta K}{K}-n &amp; \because \; \frac{\dot L}{L}=n \text{ and using (**)}\\
&amp;=s\frac{Y/L}{K/L}-(n+\delta) \\
&amp;=\frac{sy}{k}-(n+\delta)=\frac{sf(k)-(n+\delta)k}{k} &amp; \text{using (*)}
\end{align}$$</span></p>
<p><span class="math-container">$$\therefore \quad g_k(k)=\frac{sf(k)-(n+\delta)k}{k}$$</span></p>
<p>The steady-state per-capita capital is given by <span class="math-container">$k^*$</span> such that <span class="math-container">$g_k(k^*)=0$</span>. Which can be obtained as follows:
<span class="math-container">$$\begin{eqnarray} &amp;g_k(k^*)\overset{set}{=}0 \\
&amp;\boxed{sf(k^*)=(n+\delta)k^*}\tag{1} \end{eqnarray}$$</span></p>
<p>A solution to <span class="math-container">$(1)$</span> will give you the steady state <span class="math-container">$k$</span> which in this case must be unique because of the assumption we made about production technology <span class="math-container">$F$</span> being neo-classical.</p>
<p>Once you have found out the steady-state capital per-capita <span class="math-container">$k^*$</span>, we can use <span class="math-container">$(^*)$</span> to find the steady-state output per worker and consumption per worker.</p>
<p>steady-state output per worker: <span class="math-container">$\boxed{y^*=f(k^*)}$</span><br />
steady-state consumption per worker: <span class="math-container">$\boxed{c^*=(1-s)f(k^*)}$</span></p>
</li>
</ol>

CC BY-SA 4.0
2023-06-01T11:34:54.557Z
2023-06-01T15:03:52.903Z
2023-06-01T15:03:52.903Z
mynameparv
<p>To find the per-worker production function we need to divide the given production function with L i.e. <span class="math-container">$\frac{F}{L}$</span>  = <span class="math-container">$$\frac{K^{0.3}L^{0.7}}{L}$$</span> <span class="math-container">$$y=\frac{K^{0.3}}{L^{0.3}} = k^{0.3}; k =\frac{K}{L}$$</span> This completes first part of the question.<br />
For the second part, we have to recall the key equation of the solow model i.e.,<br />
<span class="math-container">$$\dot{k}= sk^{\alpha}-(\delta+n+g)k$$</span>
As stated in the question growth rate of population and technology is 0,  <span class="math-container">$$\therefore n = g = 0$$</span>
Which transforms our equation into <span class="math-container">$$\dot{k}=sk^{\alpha}-{\delta}k ; {\alpha} = 0.3$$</span>
We know at steafdy state <span class="math-container">$\dot{k}=0$</span> therefore the above equation trandforms into
<span class="math-container">$$sk^{0.3}={\delta}k$$</span> <span class="math-container">$$\Rightarrow k^{0.7}=\frac{s}{\delta}$$</span> <span class="math-container">$$\Rightarrow k= (\frac{s}{\delta})^ \frac{1}{0.7}$$</span> and since output per worker i.e. y = k<sup>0.3</sup>
<span class="math-container">$$\therefore y = (\frac{s}{\delta})^{\frac{0.3}{0.7}}$$</span> and since consumption per worker <span class="math-container">$$c=(1-s)y$$</span> <span class="math-container">$$\therefore c= (1-s)(\frac{s}{\delta})^{\frac{0.3}{0.7}}$$</span></p>

CC BY-SA 4.0
2023-06-01T11:43:10.350Z
hey, it's against the homework policy to provide answers to problems that show no attempt
mynameparv
CC BY-SA 4.0
2023-06-01T10:37:38.717Z
2023-06-01T10:37:38.717Z
Aditya Agarwal
<p>Consider an economy descried by the production function:  = (, ) = <sup>0.3</sup><sup>0.7</sup>.<br />
a) Derive the per-worker production function.<br />
b)Assuming no population growth or technological progress, find the steady-state capita stock per worker, output per worker and consumption per worker as a function of the saving rate and depreciation rate.</p>

CC BY-SA 4.0
2023-05-30T11:52:48.277Z
What's your question? and what trouble are you facing with the problem?
mynameparv
<economic-growth><solow><model>
Revisiting development of Solow Model with no population growth and no technical progress
Gibson Kashindwa
# Revisiting development of Solow Model with no population growth and no technical progress

(Asked by: username_0 on 2023-05-30)

Consider an economy descried by the production function: = (, ) = 0.30.7.  
a) Derive the per-worker production function.  
b)Assuming no population growth or technological progress, find the steady-state capita stock per worker, output per worker and consumption per worker as a function of the saving rate and depreciation rate.

**Tags:** economic-growth, solow, model
> **Comments:**
> **username_1**: What's your question? and what trouble are you facing with the problem?  

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## 2 answers

### Answer by username_1 on 2023-06-01. Score: 2

You can use the following results provided below to try and answer the problem yourself.

1. Consider a Constant Returns to Scale(CRS) production function $F:\mathbb{R}\_+^2\to \mathbb{R}$  
   Let $Y=F(K,L)$ denote the level of output produced using $K$ units of capital and $L$ units of labor given the production function $F$.

   By CRS, $F$ must satisfy $: \quad F(\lambda K, \lambda L)=\lambda F(K,L)$ where $\lambda \geq 0$ is some constant

   Since the above holds for any positive $\lambda$, we can set $\lambda=\frac{1}{L}$ and let $k=\frac{K}{L}$ and $y=\frac{Y}{L}$.

   This gives us:
   $$\begin{eqnarray}
   & F\left(\frac{K}{L},1\right)=\frac{F(K,L)}{L} \\
   & F(k,1)=\frac{Y}{L} \\
   & y=f(k) && \text{where }F(k,1)\equiv f(k) \tag{\*}
   \end{eqnarray}$$

   The expression $\boxed{y=f(k)}$ is known as the intensive form (per-worker production) function (which is what you are asked to find in part a)
2. Consider a Solowian Economy with the following features:  
   $\bullet$ Time is continuous and there is no distinction between the labor force and the population at any given period of time. Labor (or population) growth rate $\frac{\dot{L}}{L}=n$ , where $\dot L= \frac{dL}{dt}$ is exogeneous  
   $\bullet$ In each period a constant proportion of income $s\in(0,1)$ is saved and capital depreciates by $\delta$ in each period  
   $\bullet$ There is no technological progress and output in any period is given by $Y\_t=F(K\_t,L\_t)$ with the assumption that $F$ satisfies all the properties of a neo-classical production function

   Denoting the growth rate of any object $x$ (say) as $g\_x$ let us use the above data to solve for the steady state.

   The law of motion of capital is given by: $\begin{eqnarray} &\dot K = I-\delta K\\
   \implies &\dot K= sY-\delta K \tag{\*\*} \end{eqnarray}$

   we defined per-capita capital as $k=\frac{K}{L}$ and at steady state we must have $\frac{\dot k}{k}=g\_k=0$ i.e., the growth rate of per-capita capital must be zero at the steady state. Let us find an expression for $\frac{\dot k}{k}$ or $g\_k$ as a function of $k$ in order to find steady state per-capita capital.
   $$\begin{align}
   k&=\frac{K}{L} \\
   \ln k&= \ln K -\ln L & \text{taking log on both sides} \\
   \frac{\dot k}{k}&=\frac{\dot K}{K}-\frac{\dot L}{L} & \text{taking time derivative on both sides} \\
   &=\frac{sY-\delta K}{K}-n & \because \; \frac{\dot L}{L}=n \text{ and using (\*\*)}\\
   &=s\frac{Y/L}{K/L}-(n+\delta) \\
   &=\frac{sy}{k}-(n+\delta)=\frac{sf(k)-(n+\delta)k}{k} & \text{using (\*)}
   \end{align}$$

   $$\therefore \quad g\_k(k)=\frac{sf(k)-(n+\delta)k}{k}$$

   The steady-state per-capita capital is given by $k^\*$ such that $g\_k(k^\*)=0$. Which can be obtained as follows:
   $$\begin{eqnarray} &g\_k(k^\*)\overset{set}{=}0 \\
   &\boxed{sf(k^\*)=(n+\delta)k^\*}\tag{1} \end{eqnarray}$$

   A solution to $(1)$ will give you the steady state $k$ which in this case must be unique because of the assumption we made about production technology $F$ being neo-classical.

   Once you have found out the steady-state capital per-capita $k^\*$, we can use $(^\*)$ to find the steady-state output per worker and consumption per worker.

   steady-state output per worker: $\boxed{y^\*=f(k^\*)}$  
   steady-state consumption per worker: $\boxed{c^\*=(1-s)f(k^\*)}$


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### Answer by username_2 on 2023-06-01. Score: 1

To find the per-worker production function we need to divide the given production function with L i.e. $\frac{F}{L}$ = $$\frac{K^{0.3}L^{0.7}}{L}$$ $$y=\frac{K^{0.3}}{L^{0.3}} = k^{0.3}; k =\frac{K}{L}$$ This completes first part of the question.  
For the second part, we have to recall the key equation of the solow model i.e.,  
$$\dot{k}= sk^{\alpha}-(\delta+n+g)k$$
As stated in the question growth rate of population and technology is 0, $$\therefore n = g = 0$$
Which transforms our equation into $$\dot{k}=sk^{\alpha}-{\delta}k ; {\alpha} = 0.3$$
We know at steafdy state $\dot{k}=0$ therefore the above equation trandforms into
$$sk^{0.3}={\delta}k$$ $$\Rightarrow k^{0.7}=\frac{s}{\delta}$$ $$\Rightarrow k= (\frac{s}{\delta})^ \frac{1}{0.7}$$ and since output per worker i.e. y = k0.3
$$\therefore y = (\frac{s}{\delta})^{\frac{0.3}{0.7}}$$ and since consumption per worker $$c=(1-s)y$$ $$\therefore c= (1-s)(\frac{s}{\delta})^{\frac{0.3}{0.7}}$$

> **Comments:**
> **username_1**: hey, it's against the homework policy to provide answers to problems that show no attempt  

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