<p>Suppose an agent wants to maximize:</p>

<p>$Max \sum_{t=0}^\infty \beta^t \ln(C_t) $</p>

<p>s.t.</p>

<p>$C_t + I_t = Y_t$</p>

<p>$Y_t = K_t$</p>

<p>$K_{t+1} = K_t^\alpha I_t^\gamma$</p>

<p>where $\alpha, \gamma &gt;0$ and $\alpha + \gamma &lt; 1$.</p>

<p>One can combine the three constraints in: $C_t = K_t - K_{t+1}^{\frac{1}{\gamma}}K_{t}^{\frac{-\alpha}{\gamma}}$</p>

<p>Write the lagrangean as:</p>

<p>${\cal L} = \sum_{t=0}^\infty \beta^t \ln(C_t) - \lambda_t(C_t - K_t +K_{t+1}^{\frac{1}{\gamma}}K_{t}^{\frac{-\alpha}{\gamma}})$</p>

<p>I think the FOC yields:</p>

<p>$\frac{1}{\gamma C_t} K_{t+1}^{\frac{1}{\gamma}-1} K_t^{-\frac{\alpha}{\gamma}} = \frac{1}{\beta C_{t+1}}\left(1 + \frac{\alpha}{\gamma} K_{t+2}^{\frac{1}{\gamma}}K_{t+1}^{-\frac{\alpha+\gamma}{\gamma}}\right)$</p>

<p>If the FOC is correct, how to compute the steady state from here?</p>

CC BY-SA 3.0
2015-12-08T10:52:17.253Z
Steady state implies that variables do not change, so $K_t = K_{t+1}$, etc.
Giskard
CC BY-SA 3.0
2015-12-08T11:03:53.870Z
Not really. I think this model might imply endogenous growth. So in fact: $\frac{c_{t+1}}{c_t} = constant$, which is different from 0.
phdstudent
CC BY-SA 3.0
2015-12-08T11:41:46.177Z
I would call that the steady growth path, not the steady state, but I am unfamiliar with the naming conventions in this area. Whether growth is optimal is determined by the variables $\alpha, \gamma$. Are there any specific limits on those?
Giskard
CC BY-SA 3.0
2015-12-08T11:50:27.267Z
If $\alpha + \gamma < 1$ I am pretty sure you will have a steady state without growth.
Giskard
CC BY-SA 3.0
2015-12-08T12:36:15.880Z
Just edited. All $C_t$ should be capitalized. Why if $\gamma + \alpha <1 $ there is no growth?
phdstudent
CC BY-SA 3.0
2015-12-08T13:05:06.020Z
Yes, $\alpha > 0 $ and $\gamma > 0$ and $\alpha + \gamma < 1$
phdstudent
CC BY-SA 3.0
2015-12-08T15:08:19.007Z
Let us continue this discussion in chat
phdstudent
<macroeconomics><general-equilibrium>
Simple Macroeconomics problem
phdstudent
# Simple Macroeconomics problem

(Asked by: username_0 on 2015-12-08)

Suppose an agent wants to maximize:

$Max \sum\_{t=0}^\infty \beta^t \ln(C\_t) $

s.t.

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

$Y\_t = K\_t$

$K\_{t+1} = K\_t^\alpha I\_t^\gamma$

where $\alpha, \gamma >0$ and $\alpha + \gamma < 1$.

One can combine the three constraints in: $C\_t = K\_t - K\_{t+1}^{\frac{1}{\gamma}}K\_{t}^{\frac{-\alpha}{\gamma}}$

Write the lagrangean as:

${\cal L} = \sum\_{t=0}^\infty \beta^t \ln(C\_t) - \lambda\_t(C\_t - K\_t +K\_{t+1}^{\frac{1}{\gamma}}K\_{t}^{\frac{-\alpha}{\gamma}})$

I think the FOC yields:

$\frac{1}{\gamma C\_t} K\_{t+1}^{\frac{1}{\gamma}-1} K\_t^{-\frac{\alpha}{\gamma}} = \frac{1}{\beta C\_{t+1}}\left(1 + \frac{\alpha}{\gamma} K\_{t+2}^{\frac{1}{\gamma}}K\_{t+1}^{-\frac{\alpha+\gamma}{\gamma}}\right)$

If the FOC is correct, how to compute the steady state from here?

**Tags:** macroeconomics, general-equilibrium

> **Comments:**
> **username_1**: Steady state implies that variables do not change, so $K_t = K_{t+1}$, etc.  
> **username_0**: Not really. I think this model might imply endogenous growth. So in fact: $\frac{c_{t+1}}{c_t} = constant$, which is different from 0.  
> **username_1**: I would call that the steady growth path, not the steady state, but I am unfamiliar with the naming conventions in this area. Whether growth is optimal is determined by the variables $\alpha, \gamma$. Are there any specific limits on those?  
> **username_1**: If $\alpha + \gamma < 1$ I am pretty sure you will have a steady state without growth.  
> **username_0**: Just edited. All $C_t$ should be capitalized. Why if $\gamma + \alpha <1 $ there is no growth?  
> **username_0**: Yes, $\alpha > 0 $ and $\gamma > 0$ and $\alpha + \gamma < 1$  
> **username_0**: Let us continue this discussion in chat  

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## 0 answers

