<p>I am trying to prove the existence of at least one stable steady state in a situation where I am given properties of the production function but not an explicit functional form. I have the following (implicit) first order difference equation for the evolution of capital (comes from an overlapping generations model with no uncertainty): </p>

<p><span class="math-container">$$ (1+n)k_{t+1}(k_t)=s(w_t,R_{t+1}) $$</span>
Where <span class="math-container">$k$</span> is per capita capital, <span class="math-container">$n$</span> is population growth, <span class="math-container">$s$</span> is savings, <span class="math-container">$w$</span> is wages (the problem assumes that labor is inelastically supplied), and <span class="math-container">$R$</span> is rental rate on capital, with the appropriate time subscripts. From the firm's problem, and the assumption that the production function is CRS, this can be re-written as: 
<span class="math-container">$$ (1+n)k_{t+1}(k_t)=s(f(k_t)-k_tf'(k_t), f'(k_t)-\delta)$$</span> 
Where <span class="math-container">$\delta$</span> is depreciation. We are given standard assumptions on <span class="math-container">$f(k)$</span>: <span class="math-container">$f(0)=0, f'(k)&gt;0, f''(k)&lt;0$</span>, Inada conditions: <span class="math-container">$ \lim\limits_{k \to \infty}f'(k)=0,\lim\limits_{k \to 0}f'(k)=\infty$</span></p>

<p>I have been able to prove that </p>

<p>1) <span class="math-container">$k_{t+1}(0)=0$</span> </p>

<p>2) <span class="math-container">$\lim\limits_{k \to \infty} \frac{k_{t+1}}{k_t}=0 $</span>, implying that at some point, <span class="math-container">$k_{t+1}$</span> lies below the <span class="math-container">$45^\text{o}$</span> line</p>

<p>3) Solving for the derivative of <span class="math-container">$k_{t+1}$</span> with respect to <span class="math-container">$k_t$</span>: 
<span class="math-container">$$ k_{t+1}^{'}(k_t)= \frac{-\frac{\partial s}{\partial w_t}k_tf''(k_t)}{1+n-\frac{\partial s}{\partial R_{t+1}}f''(k_{t+1})} $$</span></p>

<p>Making some additional assumptions on the derivatives of savings with respect to wages and interest (that aren't relevant to my question), I have that this is positive. </p>

<p>However, in order to show that there exists at least one stable steady state, I need to show that <span class="math-container">$k_{t+1}$</span> crosses the <span class="math-container">$45^\text{o}$</span> line from above. This is where I am stuck. My professor says that this follows from the Inada conditions on <span class="math-container">$f(k)$</span>, but I'm not seeing it. If I can show that <span class="math-container">$\lim\limits_{k \to 0}k_{t+1}^{'}(k_t)=\infty$</span>, then I think I'm set. And this will hold if <span class="math-container">$\lim\limits_{k \to 0}k_tf''(k_t)=\infty$</span> (I think). But the Inada conditions are on <span class="math-container">$f'(k_t)$</span>, not <span class="math-container">$f''(k_t)$</span>, so I don't know how to get what I need. </p>

<p>Thanks very much in advance for the help!! </p>

CC BY-SA 4.0
2019-07-17T16:58:05.647Z
Hi Julia, wouldn't your postulation (2) that $\lim\limits_{k \to \infty} \frac{k_{t+1}}{k_t}=0$ imply that it reaches a steady state? Isn't $\frac{k_{t+1}}{k_{t}}=0$ the growth rate of capital being zero and hence stable? I am not familiar with OLG as I solved a similar problem but in terms of differentials. I interpret that as the growth rate approaching a steady state. You could probably back that argument up using the Inada conditions to state that the graph of $k_{t}$ with respect to time starts vertically and becomes horizontal, where it necessarily starts above in the first instance.
Brennan
CC BY-SA 4.0
2019-07-19T19:59:07.430Z
Hi @Brennan 2, Thanks for responding. The interpretation of $ \frac{k_t+1}{k_t} $ is a bit different than how you've written it. Suppose t=3 and $ k_3 $ is very large (approaching the limit). Then $ k_4 $ is quite small in relation to $ k_3 $ (hence the fraction approaches zero) so the difference between the two is large, and the growth rate = $ \frac {k_4-k_3} {k_3} $ is large (and negative) - sort of the opposite of a steady state!
Julia B
CC BY-SA 4.0
2019-07-19T20:00:01.550Z
Postulation (2) implies that, if $ k_{t+1} $ starts out steep enough (i.e. with a slope greater than one), then at some point it will cross the 45 degree line, i.e. where $ k_{t+1} = k_t $, implying a stable steady state, but I don't have (yet) that that slope is high enough.  I hope that helps. Thank you again!
Julia B
<macroeconomics><steady-state>
Proving the existence of a steady state
Julia B
# Proving the existence of a steady state

(Asked by: username_0 on 2019-07-17)

I am trying to prove the existence of at least one stable steady state in a situation where I am given properties of the production function but not an explicit functional form. I have the following (implicit) first order difference equation for the evolution of capital (comes from an overlapping generations model with no uncertainty):

$$ (1+n)k\_{t+1}(k\_t)=s(w\_t,R\_{t+1}) $$
Where $k$ is per capita capital, $n$ is population growth, $s$ is savings, $w$ is wages (the problem assumes that labor is inelastically supplied), and $R$ is rental rate on capital, with the appropriate time subscripts. From the firm's problem, and the assumption that the production function is CRS, this can be re-written as:
$$ (1+n)k\_{t+1}(k\_t)=s(f(k\_t)-k\_tf'(k\_t), f'(k\_t)-\delta)$$
Where $\delta$ is depreciation. We are given standard assumptions on $f(k)$: $f(0)=0, f'(k)>0, f''(k)<0$, Inada conditions: $ \lim\limits\_{k \to \infty}f'(k)=0,\lim\limits\_{k \to 0}f'(k)=\infty$

I have been able to prove that

1) $k\_{t+1}(0)=0$

2) $\lim\limits\_{k \to \infty} \frac{k\_{t+1}}{k\_t}=0 $, implying that at some point, $k\_{t+1}$ lies below the $45^\text{o}$ line

3) Solving for the derivative of $k\_{t+1}$ with respect to $k\_t$:
$$ k\_{t+1}^{'}(k\_t)= \frac{-\frac{\partial s}{\partial w\_t}k\_tf''(k\_t)}{1+n-\frac{\partial s}{\partial R\_{t+1}}f''(k\_{t+1})} $$

Making some additional assumptions on the derivatives of savings with respect to wages and interest (that aren't relevant to my question), I have that this is positive.

However, in order to show that there exists at least one stable steady state, I need to show that $k\_{t+1}$ crosses the $45^\text{o}$ line from above. This is where I am stuck. My professor says that this follows from the Inada conditions on $f(k)$, but I'm not seeing it. If I can show that $\lim\limits\_{k \to 0}k\_{t+1}^{'}(k\_t)=\infty$, then I think I'm set. And this will hold if $\lim\limits\_{k \to 0}k\_tf''(k\_t)=\infty$ (I think). But the Inada conditions are on $f'(k\_t)$, not $f''(k\_t)$, so I don't know how to get what I need.

Thanks very much in advance for the help!!

**Tags:** macroeconomics, steady-state

> **Comments:**
> **username_1**: Hi username_0, wouldn't your postulation (2) that $\lim\limits_{k \to \infty} \frac{k_{t+1}}{k_t}=0$ imply that it reaches a steady state? Isn't $\frac{k_{t+1}}{k_{t}}=0$ the growth rate of capital being zero and hence stable? I am not familiar with OLG as I solved a similar problem but in terms of differentials. I interpret that as the growth rate approaching a steady state. You could probably back that argument up using the Inada conditions to state that the graph of $k_{t}$ with respect to time starts vertically and becomes horizontal, where it necessarily starts above in the first instance.  
> **username_0**: Hi @username_1 2, Thanks for responding. The interpretation of $ \frac{k_t+1}{k_t} $ is a bit different than how you've written it. Suppose t=3 and $ k_3 $ is very large (approaching the limit). Then $ k_4 $ is quite small in relation to $ k_3 $ (hence the fraction approaches zero) so the difference between the two is large, and the growth rate = $ \frac {k_4-k_3} {k_3} $ is large (and negative) - sort of the opposite of a steady state!  
> **username_0**: Postulation (2) implies that, if $ k_{t+1} $ starts out steep enough (i.e. with a slope greater than one), then at some point it will cross the 45 degree line, i.e. where $ k_{t+1} = k_t $, implying a stable steady state, but I don't have (yet) that that slope is high enough.  I hope that helps. Thank you again!  

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## 0 answers

