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?
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1$\begingroup$ Could you explain the capital stock externality? I assume its something to do with the $b$ in the exponent of $K_{t}$ $\endgroup$– BrennanAug 14, 2019 at 5:11
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$\begingroup$ 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. $\endgroup$– zelloAug 14, 2019 at 12:52
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$\begingroup$ 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 $b$. $\endgroup$– BrennanAug 14, 2019 at 16:14
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$\begingroup$ 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 $\endgroup$– BrennanAug 14, 2019 at 16:16
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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.
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$\begingroup$ 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? $\endgroup$– BrennanAug 16, 2019 at 4:16
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1$\begingroup$ @Brennan simple math! a+b+(1−a) = 1+b > 1. $(zK_t)^{a+b} (L_tz)^{1-a} = K_t^{a+b}z^{a+b} L_t^{1-a} z^{1-a}=z^{1+b}(K_t)^{a+b} (L_t)^{1-a}$ $\endgroup$ Aug 16, 2019 at 11:55
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$\begingroup$ @Brennan Just currious! What do you think is the steady state of a system with DRS and linear depreciation? $\endgroup$ Aug 16, 2019 at 12:13
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$\begingroup$ 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 $\endgroup$– BrennanAug 16, 2019 at 15:39
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$\begingroup$ @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? $\endgroup$ Aug 16, 2019 at 15:43