<p>You have obtained</p>

<p>$$ \frac{c_{t+1}}{c_t}=[\beta(\alpha+1-\delta)]^{\frac{1}{\gamma}} \equiv 1+g$$</p>

<p>and</p>

<p>$$\frac{k_{t+1}}{k_t}=1+(1-\delta)-\frac{c_t}{k_t}$$</p>

<p>By equating you can show that there is a unique rule that maintains a balanced growth path</p>

<p>$$\frac{k_{t+1}}{k_t}=\frac{c_{t+1}}{c_t} \implies c_t =( 1-\delta-g)k_t$$ </p>

<p>(too much consumption, by the way). This shows that the model has a unique balanced growth path. </p>

<p>If you want further to argue that the economy will indeed choose this path, you have to invoke the Transversality condition (which constraints the consequences that a chosen path should have  on capital accumulation), and maybe the Inada condition that your chosen utility function satisfies. </p>

CC BY-SA 3.0
2016-04-13T02:09:50.970Z
Is $g$ just a constant or does it have a specific meaning?
DornerA
CC BY-SA 3.0
2016-04-13T02:25:02.240Z
@Dorner just a compact way to write the constant growth rate of consumption obtained through the Euler equation.
Alecos Papadopoulos
CC BY-SA 3.0
2016-04-13T01:04:06.610Z
2016-04-13T01:04:06.610Z
Alecos Papadopoulos
<p>I've got a model of endogenous growth due to spillovers.</p>

<p>$\textbf{Model:}$
$$K_t=\frac{1}{n}\sum_{t=1}^nk_t$$
In this model, $k_t$ is chosen by agents, and $K_t=\bar{k}_t$ (the average of all $k_t$).</p>

<p>Now, agents want to dynamically maximize utility (under certain constraints) and they have CRRA (constant relative risk aversion) utility, so the maximization looks like:
$$\sum_{t=0}^\infty\beta^t\bigg(\frac{c_t^{1-\gamma}}{1-\gamma}\bigg)$$
$$s.t.\;Y_t=k_t^\alpha(E_tL)^{1-\alpha}$$
$$c_t+i_t=Y_t$$
$$k_{t+1}=(1-\delta)k_t+i_t$$
$$c_t,i_t\geq0$$</p>

<p>$E_tL$ is effective labor and the rest of the variables are typical  (I can give their definitions if requested).</p>

<p>One last addition to the model is that there are two equilibrium constraints:
$$E_t=\frac{K_t}{L}$$
$$k_t=K_t$$
$\textbf{Solution:}$</p>

<p>Using an Euler equation approach, two terms in the objective:
$$...\frac{\beta^t[k_t^\alpha K_t^{1-\alpha}+(1-\delta)k_t-k_{t+1}]^{1-\gamma}}{1-\gamma}+\frac{\beta^{t+1}[k_{t+1}^\alpha K_{t+1}^{1-\alpha}+(1-\delta)k_{t+1}-k_{t+2}]^{1-\gamma}}{1-\gamma}...$$
FOC:</p>

<p>w.r.t  $k_{t+1}$ and substituting in consumption:</p>

<p>$$\beta^tc_t^{-\gamma}=\beta^{t+1}c_{t+1}^{-\gamma}[\alpha k_{t+1}^{\alpha-1}K_{t+1}^{1-\alpha}+1-\delta]$$ 
Substituting in the equilibrium constraints:
$$c_t^{-\gamma}=\beta c_{t+1}^{-\gamma}[\alpha+1-\delta]$$
$$\implies \frac{c_{t+1}}{c_t}=[\beta(\alpha+1-\delta)]^{\frac{1}{\gamma}}$$
This implies that consumption grows at a constant rate which depends on preference parameters.</p>

<p>The next thing I want to prove that we have a balanced growth path. By this I mean that all variables grow at the same constant rate. $$\frac{k_{t+1}}{k_t}=\frac{c_{t+1}}{c_t}?$$
I have started to answer the question, but I have gotten stuck. Here is what I have so far:
$$k_{t+1}=k_t^\alpha K_t^{1-\alpha}+(1-\delta)k_t-c_t$$
In equilibrium $K_t=k_t$:
$$k_{t+1}=k_t+(1-\delta)k_t-c_t$$
$$\implies \frac{k_{t+1}}{k_t}=1+(1-\delta)-\frac{c_t}{k_t}$$
If we have a constant rate of capital growth, suppose:
$$\frac{k_{t+1}}{k_t}&lt;\frac{c_{t+1}}{c_t}$$
If this is true:
$$\underset{t\rightarrow \infty}{lim}\;\frac{k_{t+1}}{k_t}=\underset{t\rightarrow \infty}{lim}\;1+(1-\delta)-\frac{c_t}{k_t}=-\infty$$
This means that the growth rate will continue decreasing. Now there are two ways this can happen. The first way is if $\underset{t\rightarrow \infty}{lim}\;k_t=-\infty$, which would clearly show that it couldn't be the case because capital cannot be negative. The second way which this could happen is if $\underset{t\rightarrow \infty}{lim}\;k_t=D$ where $D$ is some positive horizontal asymptote. If it did approach some asymptote, there is no reason why it could not be the case that consumption grows at a faster rate than capital. This is where I am stuck. How can I show that it cannot converge to some asymptote, and that $\underset{t\rightarrow \infty}{lim}\;k_t=-\infty$? Also, if there is an easier way to show that this model exhibits balanced growth paths, what is it? Any help would be greatly appreciated!</p>

<macroeconomics><economic-growth><endogenous-growth>
Endogenous Growth: Balanced Growth Path with CRRA Utility
DornerA
# Endogenous Growth: Balanced Growth Path with CRRA Utility

(Asked by: username_0 on 2016-04-12)

I've got a model of endogenous growth due to spillovers.

$\textbf{Model:}$
$$K\_t=\frac{1}{n}\sum\_{t=1}^nk\_t$$
In this model, $k\_t$ is chosen by agents, and $K\_t=\bar{k}\_t$ (the average of all $k\_t$).

Now, agents want to dynamically maximize utility (under certain constraints) and they have CRRA (constant relative risk aversion) utility, so the maximization looks like:
$$\sum\_{t=0}^\infty\beta^t\bigg(\frac{c\_t^{1-\gamma}}{1-\gamma}\bigg)$$
$$s.t.\;Y\_t=k\_t^\alpha(E\_tL)^{1-\alpha}$$
$$c\_t+i\_t=Y\_t$$
$$k\_{t+1}=(1-\delta)k\_t+i\_t$$
$$c\_t,i\_t\geq0$$

$E\_tL$ is effective labor and the rest of the variables are typical (I can give their definitions if requested).

One last addition to the model is that there are two equilibrium constraints:
$$E\_t=\frac{K\_t}{L}$$
$$k\_t=K\_t$$
$\textbf{Solution:}$

Using an Euler equation approach, two terms in the objective:
$$...\frac{\beta^t[k\_t^\alpha K\_t^{1-\alpha}+(1-\delta)k\_t-k\_{t+1}]^{1-\gamma}}{1-\gamma}+\frac{\beta^{t+1}[k\_{t+1}^\alpha K\_{t+1}^{1-\alpha}+(1-\delta)k\_{t+1}-k\_{t+2}]^{1-\gamma}}{1-\gamma}...$$
FOC:

w.r.t $k\_{t+1}$ and substituting in consumption:

$$\beta^tc\_t^{-\gamma}=\beta^{t+1}c\_{t+1}^{-\gamma}[\alpha k\_{t+1}^{\alpha-1}K\_{t+1}^{1-\alpha}+1-\delta]$$
Substituting in the equilibrium constraints:
$$c\_t^{-\gamma}=\beta c\_{t+1}^{-\gamma}[\alpha+1-\delta]$$
$$\implies \frac{c\_{t+1}}{c\_t}=[\beta(\alpha+1-\delta)]^{\frac{1}{\gamma}}$$
This implies that consumption grows at a constant rate which depends on preference parameters.

The next thing I want to prove that we have a balanced growth path. By this I mean that all variables grow at the same constant rate. $$\frac{k\_{t+1}}{k\_t}=\frac{c\_{t+1}}{c\_t}?$$
I have started to answer the question, but I have gotten stuck. Here is what I have so far:
$$k\_{t+1}=k\_t^\alpha K\_t^{1-\alpha}+(1-\delta)k\_t-c\_t$$
In equilibrium $K\_t=k\_t$:
$$k\_{t+1}=k\_t+(1-\delta)k\_t-c\_t$$
$$\implies \frac{k\_{t+1}}{k\_t}=1+(1-\delta)-\frac{c\_t}{k\_t}$$
If we have a constant rate of capital growth, suppose:
$$\frac{k\_{t+1}}{k\_t}<\frac{c\_{t+1}}{c\_t}$$
If this is true:
$$\underset{t\rightarrow \infty}{lim}\;\frac{k\_{t+1}}{k\_t}=\underset{t\rightarrow \infty}{lim}\;1+(1-\delta)-\frac{c\_t}{k\_t}=-\infty$$
This means that the growth rate will continue decreasing. Now there are two ways this can happen. The first way is if $\underset{t\rightarrow \infty}{lim}\;k\_t=-\infty$, which would clearly show that it couldn't be the case because capital cannot be negative. The second way which this could happen is if $\underset{t\rightarrow \infty}{lim}\;k\_t=D$ where $D$ is some positive horizontal asymptote. If it did approach some asymptote, there is no reason why it could not be the case that consumption grows at a faster rate than capital. This is where I am stuck. How can I show that it cannot converge to some asymptote, and that $\underset{t\rightarrow \infty}{lim}\;k\_t=-\infty$? Also, if there is an easier way to show that this model exhibits balanced growth paths, what is it? Any help would be greatly appreciated!

**Tags:** macroeconomics, economic-growth, endogenous-growth

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## 1 answers

### Answer by username_1 on 2016-04-13. Score: 3 | Accepted answer

You have obtained

$$ \frac{c\_{t+1}}{c\_t}=[\beta(\alpha+1-\delta)]^{\frac{1}{\gamma}} \equiv 1+g$$

and

$$\frac{k\_{t+1}}{k\_t}=1+(1-\delta)-\frac{c\_t}{k\_t}$$

By equating you can show that there is a unique rule that maintains a balanced growth path

$$\frac{k\_{t+1}}{k\_t}=\frac{c\_{t+1}}{c\_t} \implies c\_t =( 1-\delta-g)k\_t$$

(too much consumption, by the way). This shows that the model has a unique balanced growth path.

If you want further to argue that the economy will indeed choose this path, you have to invoke the Transversality condition (which constraints the consequences that a chosen path should have on capital accumulation), and maybe the Inada condition that your chosen utility function satisfies.

> **Comments:**
> **username_0**: Is $g$ just a constant or does it have a specific meaning?  
> **username_1**: @Dorner just a compact way to write the constant growth rate of consumption obtained through the Euler equation.  

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