Solve for the steady state with CRS Cobb-Douglas, problem with the system of equations

Question

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?

Answer 1

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.