<p>1) Instead of <span class="math-container">$(9)$</span> use</p>

<p><span class="math-container">$$C = K^{\alpha}L^{1-\alpha} \tag{9*}$$</span></p>

<p>2) Equation <span class="math-container">$(10)$</span> is not correct. For constant population <span class="math-container">$N$</span> and identical agents,</p>

<p><span class="math-container">$$c^{\sigma} l^{\gamma} =  w \implies (C/N)^{\sigma} (L/N)^{\gamma} =  w \implies C^{\sigma} L^{\gamma} = N^{\sigma +\gamma}w \tag{10*}$$</span></p>

<p>Insert <span class="math-container">$(9^*)$</span> in <span class="math-container">$(11)$</span> and solve for <span class="math-container">$C$</span> as a function of <span class="math-container">$K$</span>, using <span class="math-container">$(8)$</span> for <span class="math-container">$r$</span>. Substitute back in <span class="math-container">$(9^*)$</span> and solve for <span class="math-container">$K$</span> as a function of <span class="math-container">$L$</span>. Insert into <span class="math-container">$(12)$</span> and you will get <span class="math-container">$w$</span> as a function of parameters. Etc.</p>

CC BY-SA 4.0
2019-01-26T20:41:35.630Z
Thanks @Alecos. I don’t understand what you do with equation (10). At the end, $C^{\sigma}$ is equal to what?
Tecon
CC BY-SA 4.0
2019-01-26T22:46:56.033Z
@Tecon I start with the per capita expression and then write  $c=C/N, l = L/N$, which gives a denominator of $N^{\sigma + \gamma}$ that I then move to the other side of the equation.
Alecos Papadopoulos
CC BY-SA 4.0
2019-01-28T11:02:11.543Z
Sorry for my previous stupid question, I was reading from the phone and I couldn't see the end of the equation.
I followed your steps. Given that $N=1$, equation (10*) is equal to equation (10). I substitute (9*) in (11) and i get: $C=\frac{r}{\alpha}K$. I then substitute $C$ in (9*) and obtain: $K=(\frac{\alpha}{r})^{(\frac{1}{1-\alpha})}L$. I plug this into (12) and obtain $r = \alpha (1-\alpha)^{\frac{1-\alpha}{\alpha}}w^{\frac{\alpha-1}{\alpha}}$.
Tecon
CC BY-SA 4.0
2019-01-28T11:03:26.107Z
This last expression is identical to the one obtained by "solving for the ratio", as I show [here](https://economics.stackexchange.com/questions/26580/cobb-douglas-production-function-given-w-get-r-regardless-of-input-levels). How do you interpret this?
Tecon
CC BY-SA 4.0
2019-01-26T19:10:04.843Z
2019-01-26T19:10:04.843Z
Alecos Papadopoulos
<p>There is one agent with utility function given by:</p>

<p><span class="math-container">\begin{equation}
U(c,l) = \frac{c^{1-\sigma}}{1-\sigma}-\frac{l^{1+\gamma}}{1+\gamma}\tag{1}
\end{equation}</span></p>

<p>With budget constraint:</p>

<p><span class="math-container">\begin{equation}
c_{t} + k_{t+1}   = (1+r_t)k_{t} + l_tw_t \tag{2}
\end{equation}</span></p>

<p>The technology in the economy is a standard Cobb-Douglas with CRS:</p>

<p><span class="math-container">\begin{equation}
Y = K^{\alpha}L^{1-\alpha}\tag{3}
\end{equation}</span></p>

<p>The FOC of the utility-maximization problem are:</p>

<p><span class="math-container">\begin{align}
 c_{t+1} &amp;= c_{t} [\beta (1+r_{t+1})] ^{\frac{1}{\sigma}}\tag{4}\\
 %
 c_{t}^{\sigma} l_{t}^{\gamma} &amp;= w\tag{5}\\
 %
\end{align}</span></p>

<p>While the FOCs of the profit-maximizing problem are:</p>

<p><span class="math-container">$$r = \alpha K^{\alpha-1}L^{1-\alpha} \tag{6}$$</span></p>

<p><span class="math-container">$$w = (1-\alpha) K^\alpha L^{-\alpha} \tag{7}$$</span></p>

<p>In the steady state, equation (4) implies the following</p>

<p><span class="math-container">\begin{equation}
r   = \frac{1-\beta}{\beta}\tag{8}
\end{equation}</span></p>

<p>Hence, to solve for the steady state, we need to solve for the following system of equation:</p>

<p><span class="math-container">\begin{equation}
C = rK + wL \tag{9}
\end{equation}</span></p>

<p><span class="math-container">\begin{equation}
C^{\sigma} L^{\gamma} =w\tag{10}
\end{equation}</span></p>

<p><span class="math-container">$$r = \alpha K^{\alpha-1}L^{1-\alpha} \tag{11}$$</span></p>

<p><span class="math-container">$$w = (1-\alpha) K^\alpha L^{-\alpha} \tag{12}$$</span></p>

<p>Where the unknowns are <span class="math-container">$C, K, L,w$</span> (<span class="math-container">$r$</span> is known).</p>

<p>But from here I can't solve for the exact values of these variables. What information I miss? </p>

CC BY-SA 4.0
2019-01-25T20:06:42.517Z
Is population constant? What is the assumption on capital depreciation?
Alecos Papadopoulos
CC BY-SA 4.0
2019-01-25T20:11:31.383Z
Also, in eq. 2, it appears we should have $w_tl_t$, not just $w_t$.
Alecos Papadopoulos
CC BY-SA 4.0
2019-01-25T21:32:15.533Z
@Alecos, I corrected equation 2, thanks. About depreciation, it is zero.
Tecon
CC BY-SA 4.0
2019-01-26T17:44:12.640Z
You didn't clarify whether there is population growth. Also, in equations $9$ onwards, suddenly capital and labor appear in uppercase letters (indicating perhaps aggregate magnitudes?), while consumption is still lowercase -one agent. Please clarify these two things.
Alecos Papadopoulos
CC BY-SA 4.0
2019-01-26T18:05:41.787Z
There isn’t population growth and it is constantly equal to 1. I corrected the problem with the consumption. Thanks.
Tecon
<macroeconomics><production-function><general-equilibrium>
Solve for the steady state with CRS Cobb-Douglas, problem with the system of equations
Tecon
# Solve for the steady state with CRS Cobb-Douglas, problem with the system of equations

(Asked by: username_0 on 2019-01-25)

There is one agent with utility function given by:

\begin{equation}
U(c,l) = \frac{c^{1-\sigma}}{1-\sigma}-\frac{l^{1+\gamma}}{1+\gamma}\tag{1}
\end{equation}

With budget constraint:

\begin{equation}
c\_{t} + k\_{t+1} = (1+r\_t)k\_{t} + l\_tw\_t \tag{2}
\end{equation}

The technology in the economy is a standard Cobb-Douglas with CRS:

\begin{equation}
Y = K^{\alpha}L^{1-\alpha}\tag{3}
\end{equation}

The FOC of the utility-maximization problem are:

\begin{align}
c\_{t+1} &= c\_{t} [\beta (1+r\_{t+1})] ^{\frac{1}{\sigma}}\tag{4}\\
%
c\_{t}^{\sigma} l\_{t}^{\gamma} &= w\tag{5}\\
%
\end{align}

While the FOCs of the profit-maximizing problem are:

$$r = \alpha K^{\alpha-1}L^{1-\alpha} \tag{6}$$

$$w = (1-\alpha) K^\alpha L^{-\alpha} \tag{7}$$

In the steady state, equation (4) implies the following

\begin{equation}
r = \frac{1-\beta}{\beta}\tag{8}
\end{equation}

Hence, to solve for the steady state, we need to solve for the following system of equation:

\begin{equation}
C = rK + wL \tag{9}
\end{equation}

\begin{equation}
C^{\sigma} L^{\gamma} =w\tag{10}
\end{equation}

$$r = \alpha K^{\alpha-1}L^{1-\alpha} \tag{11}$$

$$w = (1-\alpha) K^\alpha L^{-\alpha} \tag{12}$$

Where the unknowns are $C, K, L,w$ ($r$ is known).

But from here I can't solve for the exact values of these variables. What information I miss?

**Tags:** macroeconomics, production-function, general-equilibrium

> **Comments:**
> **username_1**: Is population constant? What is the assumption on capital depreciation?  
> **username_1**: Also, in eq. 2, it appears we should have $w_tl_t$, not just $w_t$.  
> **username_0**: @username_1, I corrected equation 2, thanks. About depreciation, it is zero.  
> **username_1**: You didn't clarify whether there is population growth. Also, in equations $9$ onwards, suddenly capital and labor appear in uppercase letters (indicating perhaps aggregate magnitudes?), while consumption is still lowercase -one agent. Please clarify these two things.  
> **username_0**: There isn’t population growth and it is constantly equal to 1. I corrected the problem with the consumption. Thanks.  

---

## 1 answers

### Answer by username_1 on 2019-01-26. Score: 1 | Accepted answer

1) Instead of $(9)$ use

$$C = K^{\alpha}L^{1-\alpha} \tag{9\*}$$

2) Equation $(10)$ is not correct. For constant population $N$ and identical agents,

$$c^{\sigma} l^{\gamma} = w \implies (C/N)^{\sigma} (L/N)^{\gamma} = w \implies C^{\sigma} L^{\gamma} = N^{\sigma +\gamma}w \tag{10\*}$$

Insert $(9^\*)$ in $(11)$ and solve for $C$ as a function of $K$, using $(8)$ for $r$. Substitute back in $(9^\*)$ and solve for $K$ as a function of $L$. Insert into $(12)$ and you will get $w$ as a function of parameters. Etc.

> **Comments:**
> **username_0**: Thanks @username_1. I don’t understand what you do with equation (10). At the end, $C^{\sigma}$ is equal to what?  
> **username_1**: @username_0 I start with the per capita expression and then write  $c=C/N, l = L/N$, which gives a denominator of $N^{\sigma + \gamma}$ that I then move to the other side of the equation.  
> **username_0**: Sorry for my previous stupid question, I was reading from the phone and I couldn't see the end of the equation. I followed your steps. Given that $N=1$, equation (10*) is equal to equation (10). I substitute (9*) in (11) and i get: $C=\frac{r}{\alpha}K$. I then substitute $C$ in (9*) and obtain: $K=(\frac{\alpha}{r})^{(\frac{1}{1-\alpha})}L$. I plug this into (12) and obtain $r = \alpha (1-\alpha)^{\frac{1-\alpha}{\alpha}}w^{\frac{\alpha-1}{\alpha}}$.  
> **username_0**: This last expression is identical to the one obtained by "solving for the ratio", as I show [here](https://economics.stackexchange.com/questions/26580/cobb-douglas-production-function-given-w-get-r-regardless-of-input-levels). How do you interpret this?  

---

