<p>There is no steady state in the Solow framework. The system is explosive because <span class="math-container">$a,b &gt;0 \wedge a+b&lt;1 \Rightarrow a+b + (1-a) &gt; 1$</span>. Thus <span class="math-container">$(zK_t)^{a+b} (L_tz)^{1-a} = z^{1+b}(K_t)^{a+b} (L_t)^{1-a}$</span>. I.e. this production function has increasing returns to scale. Such capital stock externalities are discussed in the AK-Models.</p>

CC BY-SA 4.0
2019-08-16T04:16:01.247Z
If $a+b$ is less than 1 then individually they are less than one. And if they are strictly positive, this implies they are decimals bound between 0 and 1. I dont see how this is increasing returns to scale? There are cases with IRS in which there are no steady states but this is DRS im pretty sure. And generally, there _is_ usually a steady state in the Solow framework. I dont understand how you work out the exponent of $z$ to be $1+b$ either, could you clarify?
Brennan
CC BY-SA 4.0
2019-08-16T11:55:28.167Z
@Brennan simple math! a+b+(1−a) = 1+b > 1. $(zK_t)^{a+b} (L_tz)^{1-a} = K_t^{a+b}z^{a+b} L_t^{1-a} z^{1-a}=z^{1+b}(K_t)^{a+b} (L_t)^{1-a}$
Grada Gukovic
CC BY-SA 4.0
2019-08-16T12:13:26.940Z
@Brennan Just currious! What do you think is the steady state of a system with DRS and linear depreciation?
Grada Gukovic
CC BY-SA 4.0
2019-08-16T15:39:25.367Z
Ah ok i see now! And do you mean linear depreciation in the “investment” curve of the Solow system of equations? I am familiar with the system of differentials in the Solow model
Brennan
CC BY-SA 4.0
2019-08-16T15:43:14.347Z
@Brennan lets formulate the question differently. Can you provide an example where the production function is not CRS and you have a steady state with output larger than 0 in a Solow model?
Grada Gukovic
CC BY-SA 4.0
2019-08-15T21:52:49.563Z
2019-08-15T22:27:35.967Z
2019-08-15T22:27:35.967Z
Grada Gukovic
<p>Suppose we have the following form of the Solow Model: <span class="math-container">$$ Y_t=K_t^{a+b} L_t^{1-a} $$</span> where a,b >0, and a+b&lt;1.</p>

<p>Is it possible to determine the steady state growth rate of k (k=K/L), and the steady state growth rates of MPL and MPK?</p>

CC BY-SA 4.0
2019-08-14T05:11:52.307Z
Could you explain the capital stock externality? I assume its something to do with the $b$ in the exponent of $K_{t}$
Brennan
CC BY-SA 4.0
2019-08-14T12:52:32.687Z
Initially this model looked like $$ Y_t=\bar{K}_t^b K_t^{a} L_t^{1-a} $$ where $\bar{K}$ is the aggregate capital stock. Since all households are identical (or $\bar{K} = K$) I can write as $$ Y_t= K_t^{a+b} L_t^{1-a} $$
Now I assume that I need to rewrite this function somehow that I can use the results of the standard Solow model.
zello
CC BY-SA 4.0
2019-08-14T16:14:24.800Z
Hmm ok, what is this capital externality or an example of one if its general? Judging by your subscript notation I assume you're working with difference equations as opposed to differential equations (e.g. would be $Y(t)$) and so I would start by defining the growth rate of $k$ which I believe is $$\frac{k_{t+1}-k_{t}}{k_{t}}$$ and then using your Solow model equation you can solve for $\frac{K_{t}}{L_{t}}=k_{t}$ as some function of $Y$ and your parameters $a$ and $b$.
Brennan
CC BY-SA 4.0
2019-08-14T16:16:03.513Z
Then MPL and MPK are similar but you have to find them first (partials of $Y$ with respect to $L$ and $K$, respectively) then also find their growth rates in the same manner. I have not done this with difference equations only differentials so I cannot provide a complete answer for you, so I am just trying to give you so guide for moving forward until someone else can fully answer (or correct me) or you figure it out
Brennan
<macroeconomics><economic-growth>
Solow Model with a Capital Stock Externality
zello
# Solow Model with a Capital Stock Externality

(Asked by: username_0 on 2019-08-14)

Suppose we have the following form of the Solow Model: $$ Y\_t=K\_t^{a+b} L\_t^{1-a} $$ where a,b >0, and a+b<1.

Is it possible to determine the steady state growth rate of k (k=K/L), and the steady state growth rates of MPL and MPK?

**Tags:** macroeconomics, economic-growth
> **Comments:**
> **username_1**: Could you explain the capital stock externality? I assume its something to do with the $b$ in the exponent of $K_{t}$  
> **username_0**: Initially this model looked like $$ Y_t=\bar{K}_t^b K_t^{a} L_t^{1-a} $$ where $\bar{K}$ is the aggregate capital stock. Since all households are identical (or $\bar{K} = K$) I can write as $$ Y_t= K_t^{a+b} L_t^{1-a} $$ Now I assume that I need to rewrite this function somehow that I can use the results of the standard Solow model.  
> **username_1**: Hmm ok, what is this capital externality or an example of one if its general? Judging by your subscript notation I assume you're working with difference equations as opposed to differential equations (e.g. would be $Y(t)$) and so I would start by defining the growth rate of $k$ which I believe is $$\frac{k_{t+1}-k_{t}}{k_{t}}$$ and then using your Solow model equation you can solve for $\frac{K_{t}}{L_{t}}=k_{t}$ as some function of $Y$ and your parameters $a$ and $b$.  
> **username_1**: Then MPL and MPK are similar but you have to find them first (partials of $Y$ with respect to $L$ and $K$, respectively) then also find their growth rates in the same manner. I have not done this with difference equations only differentials so I cannot provide a complete answer for you, so I am just trying to give you so guide for moving forward until someone else can fully answer (or correct me) or you figure it out  

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## 1 answers

### Answer by username_2 on 2019-08-15. Score: 1

There is no steady state in the Solow framework. The system is explosive because $a,b >0 \wedge a+b<1 \Rightarrow a+b + (1-a) > 1$. Thus $(zK\_t)^{a+b} (L\_tz)^{1-a} = z^{1+b}(K\_t)^{a+b} (L\_t)^{1-a}$. I.e. this production function has increasing returns to scale. Such capital stock externalities are discussed in the AK-Models.

> **Comments:**
> **username_1**: If $a+b$ is less than 1 then individually they are less than one. And if they are strictly positive, this implies they are decimals bound between 0 and 1. I dont see how this is increasing returns to scale? There are cases with IRS in which there are no steady states but this is DRS im pretty sure. And generally, there _is_ usually a steady state in the Solow framework. I dont understand how you work out the exponent of $z$ to be $1+b$ either, could you clarify?  
> **username_2**: @Brennan simple math! a+b+(1−a) = 1+b > 1. $(zK_t)^{a+b} (L_tz)^{1-a} = K_t^{a+b}z^{a+b} L_t^{1-a} z^{1-a}=z^{1+b}(K_t)^{a+b} (L_t)^{1-a}$  
> **username_2**: @Brennan Just currious! What do you think is the steady state of a system with DRS and linear depreciation?  
> **username_1**: Ah ok i see now! And do you mean linear depreciation in the “investment” curve of the Solow system of equations? I am familiar with the system of differentials in the Solow model  
> **username_2**: @Brennan lets formulate the question differently. Can you provide an example where the production function is not CRS and you have a steady state with output larger than 0 in a Solow model?  

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