# Solow Model with a Capital Stock Externality

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?

• Could you explain the capital stock externality? I assume its something to do with the $b$ in the exponent of $K_{t}$ – Brennan Aug 14 '19 at 5:11
• 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 Aug 14 '19 at 12:52
• 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 Aug 14 '19 at 16:14
• 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 Aug 14 '19 at 16:16

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.
• 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 Aug 16 '19 at 4:16
• @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 Aug 16 '19 at 11:55