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

There is one agent with utility function given by:

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

With budget constraint:

$$$$c_{t} + k_{t+1} = (1+r_t)k_{t} + l_tw_t \tag{2}$$$$

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

$$$$Y = K^{\alpha}L^{1-\alpha}\tag{3}$$$$

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

$$$$r = \frac{1-\beta}{\beta}\tag{8}$$$$

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

$$$$C = rK + wL \tag{9}$$$$

$$$$C^{\sigma} L^{\gamma} =w\tag{10}$$$$

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

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

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?

• Is population constant? What is the assumption on capital depreciation? – Alecos Papadopoulos Jan 25 '19 at 20:06
• Also, in eq. 2, it appears we should have $w_tl_t$, not just $w_t$. – Alecos Papadopoulos Jan 25 '19 at 20:11
• @Alecos, I corrected equation 2, thanks. About depreciation, it is zero. – Tecon Jan 25 '19 at 21:32
• 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 Jan 26 '19 at 17:44
• There isn’t population growth and it is constantly equal to 1. I corrected the problem with the consumption. Thanks. – Tecon Jan 26 '19 at 18:05

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.

• Thanks @Alecos. I don’t understand what you do with equation (10). At the end, $C^{\sigma}$ is equal to what? – Tecon Jan 26 '19 at 20:41
• @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 Jan 26 '19 at 22:46
• 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 Jan 28 '19 at 11:02
• This last expression is identical to the one obtained by "solving for the ratio", as I show here. How do you interpret this? – Tecon Jan 28 '19 at 11:03