<p>There are some mistakes in the steps you took although they went in right direction. The correct steps are below.</p>
<p>First we can start by expressing output in per capita terms by dividing first two equations by <span class="math-container">$L_t$</span>:</p>
<p><span class="math-container">$$ \frac{Y_t}{L_t}=\frac{K^{\alpha}_t L^{1−\alpha}_t}{L_t} \implies y_t = k_t^{\alpha}$$</span></p>
<p>and</p>
<p><span class="math-container">$$\frac{Y_t}{L_t}=\frac{C_t}{L_t}+\frac{I_t}{L_t}+\frac{G_t}{L_t} = y_t  = c_t + i_t + g_t$$</span></p>
<p>where lower case letter refer to per capita variable (e.g. <span class="math-container">$\frac{Y_t}{L_t} = y_t$</span>). Using this and the <span class="math-container">$L_{t+1}=(1+n)L_t $</span> the evolution of capital in per capita terms will be given by:</p>
<p><span class="math-container">$$ (1 + n)k_{t+1} = (1 − \delta )k_t + i_t \implies k_{t+1} \approx (1 − n -\delta )k_t + i_t  $$</span></p>
<p>Here we can use the fact that:</p>
<p><span class="math-container">$$i_t = y_t - c_t - g_t = k_t^{\alpha} - c_t - g_t$$</span>.</p>
<p>Hence:</p>
<p><span class="math-container">$$k_{t+1} =  (1 − n -\delta )k_t + k_t^{\alpha} - c_t - g_t $$</span></p>
<p>and as a consequence</p>
<p><span class="math-container">$$k_{t+1} - k_t=  -(  n + \delta )k_t + k_t^{\alpha} - c_t - g_t $$</span></p>
<p>Now since <span class="math-container">$c_t=(1−s)(y_t-g_t)$</span>, where (<span class="math-container">$-g_t$</span>) is there because consumption depends only on disposable income after taxes and <span class="math-container">$g_t = \sigma y_t$</span> and remember <span class="math-container">$y_t= k^{\delta}_t$</span> we get:</p>
<p><span class="math-container">$$k_{t+1} - k_t  =  -(  n + \delta )k_t + k_t^{\alpha} - ( (1−s)(k_t^{\alpha}-\sigma k_t^{\alpha})- \sigma k_t^{\alpha})  \\
 =  -(  n + \delta )k_t + k_t^{\alpha} - ( k_t^{\alpha} - s (1-\sigma) k_t^{\alpha} )  \\ =   s (1-\sigma) k_t^{\alpha}  -(  n + \delta )k_t $$</span></p>
<p>where the last equation is really just this models version of <span class="math-container">$sy-(n+\delta)k_t$</span>.  Now finally dividing by <span class="math-container">$k_t$</span> to get to growth rates we get:</p>
<p><span class="math-container">$$ \frac{k_{t+1} - k_t }{k_t}=  s (1-\sigma) \frac{k_t^{\alpha}}{k_t}  -(  n + \delta ) $$</span></p>
<p>Now solving the above for steady state where <span class="math-container">$\frac{k_{t+1} - k_t }{k_t}=0$</span> (by definition a steady state),   gives us:</p>
<p><span class="math-container">$$ 0 =  s (1-\sigma) \frac{k_t^{\alpha}}{k_t}  -(  n + \delta ) $$</span></p>
<p>which finally can be solved for optimal steady state per capita capital:</p>
<p><span class="math-container">$$ k^* =  \left( \frac{s(1-\sigma)}{n+\delta} \right)^{\frac{1}{1-\alpha}}$$</span></p>
<p>In addition very similar models to this one are covered by Barro &amp; Sala-i-Martin Economic Growth 2nd ed. So you can check it out for further details.</p>

CC BY-SA 4.0
2020-11-29T15:44:10.333Z
@Tony456 you are welcome if you think this answer solved your question consider accepting it
1muflon1
CC BY-SA 4.0
2020-11-29T15:45:41.680Z
How do I accept an answer exactly? I’m still new to this process. BTW, this method would also work if I changed say $C_t$ or similar yes?
Tony456
CC BY-SA 4.0
2020-11-29T15:49:04.973Z
@Tony456 you can click on check mark - I think you already figured it out (btw consider accepting also some of the older answers you got on previous q if they solved your issue - I see you have some good answers elsewhere as well). Yes the above would work also if there are changes in $C$ I expressed everything in per capita terms and using the parameters provided but exactly the same step with just different definitions would be followed if $c$ would be some other function. You can have look at some basic Solow model derivation in that book I recommended - steps are always pretty much same
1muflon1
CC BY-SA 4.0
2020-11-29T15:54:24.527Z
I will definitely accept the answers from previous questions. This is ll really difficult (at least mathematical economics and macro) fortunately I have you guys to assist me with gaps.
Tony456
CC BY-SA 4.0
2020-11-29T15:56:59.263Z
@Tony456 you are welcome also don’t worry it might seem hard but it gets easier with practice.
1muflon1
CC BY-SA 4.0
2020-11-29T15:31:27.710Z
2020-11-29T15:42:04.503Z
2020-11-29T15:42:04.503Z
1muflon1
<p>Question: Suppose <span class="math-container">$C_t=(1-s)Y_t$</span> where <span class="math-container">$s&gt;\sigma$</span> as in the basic Solow model. Solve for capital per capita in the steady state.</p>
<p><span class="math-container">$Y_t=K^{\alpha}_tL^{1-\alpha}_t$</span></p>
<p><span class="math-container">$Y_t=C_t+I_t+G_t$</span></p>
<p><span class="math-container">$K_{t+1}=I_t+(1-\delta)K_t$</span></p>
<p><span class="math-container">$L_{t+1}=(1+n)L_t$</span></p>
<p><span class="math-container">$G_t=\sigma Y_t$</span></p>
<p>Attempt:</p>
<p><span class="math-container">$sy=sky^{\alpha}$</span>: savings and investment per capita</p>
<p><span class="math-container">$(n+d)k$</span>: investment needed to keep capital per capita constant</p>
<p>n: population growth rate</p>
<p>d: depreciation</p>
<p><span class="math-container">$\dot{k}= sky{\alpha}-(n+d)k=0$</span></p>
<p>Comment(s): I am unsure if this is even correct. My problem is that I didn’t even utilize the information given about consumption above. Any suggestions that could lead me down the right path?</p>

CC BY-SA 4.0
2020-11-29T13:30:37.367Z
what exactly is the production function in this case? Is here everything assumed to be as in standard solow swan model? also how does technology enter? Can you please provide some clarity on the assumptions behind the question?
1muflon1
CC BY-SA 4.0
2020-11-29T13:35:13.720Z
Hello there. I am sorry for not providing clarity. We assume the Solow growth model with government expenditure $G_t=\sigma Y_t$
Tony456
CC BY-SA 4.0
2020-11-29T13:35:38.380Z
I hope this helps
Tony456
CC BY-SA 4.0
2020-11-29T13:37:48.247Z
could you please just write out all assumptions/set up?
1muflon1
CC BY-SA 4.0
2020-11-29T13:52:52.823Z
Sure, I will edit them into my question
Tony456
CC BY-SA 4.0
2020-11-29T13:57:02.427Z
I just finished the edits. I hope that helps
Tony456
<macroeconomics><mathematical-economics><solow>
Would someone be able to help me solve capital per capita in the steady state (check my work)
Tony456
# Would someone be able to help me solve capital per capita in the steady state (check my work)

(Asked by: username_0 on 2020-11-29)

Question: Suppose $C\_t=(1-s)Y\_t$ where $s>\sigma$ as in the basic Solow model. Solve for capital per capita in the steady state.

$Y\_t=K^{\alpha}\_tL^{1-\alpha}\_t$

$Y\_t=C\_t+I\_t+G\_t$

$K\_{t+1}=I\_t+(1-\delta)K\_t$

$L\_{t+1}=(1+n)L\_t$

$G\_t=\sigma Y\_t$

Attempt:

$sy=sky^{\alpha}$: savings and investment per capita

$(n+d)k$: investment needed to keep capital per capita constant

n: population growth rate

d: depreciation

$\dot{k}= sky{\alpha}-(n+d)k=0$

Comment(s): I am unsure if this is even correct. My problem is that I didn’t even utilize the information given about consumption above. Any suggestions that could lead me down the right path?

**Tags:** macroeconomics, mathematical-economics, solow

> **Comments:**
> **username_1**: what exactly is the production function in this case? Is here everything assumed to be as in standard solow swan model? also how does technology enter? Can you please provide some clarity on the assumptions behind the question?  
> **username_0**: Hello there. I am sorry for not providing clarity. We assume the Solow growth model with government expenditure $G_t=\sigma Y_t$  
> **username_0**: I hope this helps  
> **username_1**: could you please just write out all assumptions/set up?  
> **username_0**: Sure, I will edit them into my question  
> **username_0**: I just finished the edits. I hope that helps  

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## 1 answers

### Answer by username_1 on 2020-11-29. Score: 1 | Accepted answer

There are some mistakes in the steps you took although they went in right direction. The correct steps are below.

First we can start by expressing output in per capita terms by dividing first two equations by $L\_t$:

$$ \frac{Y\_t}{L\_t}=\frac{K^{\alpha}\_t L^{1−\alpha}\_t}{L\_t} \implies y\_t = k\_t^{\alpha}$$

and

$$\frac{Y\_t}{L\_t}=\frac{C\_t}{L\_t}+\frac{I\_t}{L\_t}+\frac{G\_t}{L\_t} = y\_t = c\_t + i\_t + g\_t$$

where lower case letter refer to per capita variable (e.g. $\frac{Y\_t}{L\_t} = y\_t$). Using this and the $L\_{t+1}=(1+n)L\_t $ the evolution of capital in per capita terms will be given by:

$$ (1 + n)k\_{t+1} = (1 − \delta )k\_t + i\_t \implies k\_{t+1} \approx (1 − n -\delta )k\_t + i\_t $$

Here we can use the fact that:

$$i\_t = y\_t - c\_t - g\_t = k\_t^{\alpha} - c\_t - g\_t$$.

Hence:

$$k\_{t+1} = (1 − n -\delta )k\_t + k\_t^{\alpha} - c\_t - g\_t $$

and as a consequence

$$k\_{t+1} - k\_t= -( n + \delta )k\_t + k\_t^{\alpha} - c\_t - g\_t $$

Now since $c\_t=(1−s)(y\_t-g\_t)$, where ($-g\_t$) is there because consumption depends only on disposable income after taxes and $g\_t = \sigma y\_t$ and remember $y\_t= k^{\delta}\_t$ we get:

$$k\_{t+1} - k\_t = -( n + \delta )k\_t + k\_t^{\alpha} - ( (1−s)(k\_t^{\alpha}-\sigma k\_t^{\alpha})- \sigma k\_t^{\alpha}) \\
= -( n + \delta )k\_t + k\_t^{\alpha} - ( k\_t^{\alpha} - s (1-\sigma) k\_t^{\alpha} ) \\ = s (1-\sigma) k\_t^{\alpha} -( n + \delta )k\_t $$

where the last equation is really just this models version of $sy-(n+\delta)k\_t$. Now finally dividing by $k\_t$ to get to growth rates we get:

$$ \frac{k\_{t+1} - k\_t }{k\_t}= s (1-\sigma) \frac{k\_t^{\alpha}}{k\_t} -( n + \delta ) $$

Now solving the above for steady state where $\frac{k\_{t+1} - k\_t }{k\_t}=0$ (by definition a steady state), gives us:

$$ 0 = s (1-\sigma) \frac{k\_t^{\alpha}}{k\_t} -( n + \delta ) $$

which finally can be solved for optimal steady state per capita capital:

$$ k^\* = \left( \frac{s(1-\sigma)}{n+\delta} \right)^{\frac{1}{1-\alpha}}$$

In addition very similar models to this one are covered by Barro & Sala-i-Martin Economic Growth 2nd ed. So you can check it out for further details.

> **Comments:**
> **username_1**: @username_0 you are welcome if you think this answer solved your question consider accepting it  
> **username_0**: How do I accept an answer exactly? I’m still new to this process. BTW, this method would also work if I changed say $C_t$ or similar yes?  
> **username_1**: @username_0 you can click on check mark - I think you already figured it out (btw consider accepting also some of the older answers you got on previous q if they solved your issue - I see you have some good answers elsewhere as well). Yes the above would work also if there are changes in $C$ I expressed everything in per capita terms and using the parameters provided but exactly the same step with just different definitions would be followed if $c$ would be some other function. You can have look at some basic Solow model derivation in that book I recommended - steps are always pretty much same  
> **username_0**: I will definitely accept the answers from previous questions. This is ll really difficult (at least mathematical economics and macro) fortunately I have you guys to assist me with gaps.  
> **username_1**: @username_0 you are welcome also don’t worry it might seem hard but it gets easier with practice.  

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