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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?

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  • $\begingroup$ Is population constant? What is the assumption on capital depreciation? $\endgroup$ – Alecos Papadopoulos Jan 25 at 20:06
  • $\begingroup$ Also, in eq. 2, it appears we should have $w_tl_t$, not just $w_t$. $\endgroup$ – Alecos Papadopoulos Jan 25 at 20:11
  • $\begingroup$ @Alecos, I corrected equation 2, thanks. About depreciation, it is zero. $\endgroup$ – Tecon Jan 25 at 21:32
  • $\begingroup$ 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. $\endgroup$ – Alecos Papadopoulos Jan 26 at 17:44
  • $\begingroup$ There isn’t population growth and it is constantly equal to 1. I corrected the problem with the consumption. Thanks. $\endgroup$ – Tecon Jan 26 at 18:05
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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.

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  • $\begingroup$ Thanks @Alecos. I don’t understand what you do with equation (10). At the end, $C^{\sigma}$ is equal to what? $\endgroup$ – Tecon Jan 26 at 20:41
  • $\begingroup$ @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. $\endgroup$ – Alecos Papadopoulos Jan 26 at 22:46
  • $\begingroup$ 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 $r = \alpha (1-\alpha)^{\frac{1-\alpha}{\alpha}}w^{\frac{\alpha-1}{\alpha}}$. $\endgroup$ – Tecon Jan 28 at 11:02
  • $\begingroup$ This last expression is identical to the one obtained by "solving for the ratio", as I show here. How do you interpret this? $\endgroup$ – Tecon Jan 28 at 11:03

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