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Question: Suppose $C_t=(1-s)Y_t$ where $s>\sigma$ as in the basic Solow model. Solve for capital per capita in the steady state.

$Y_t=K^{\alpha}_tL^{1-\alpha}_t$

$Y_t=C_t+I_t+G_t$

$K_{t+1}=I_t+(1-\delta)K_t$

$L_{t+1}=(1+n)L_t$

$G_t=\sigma Y_t$

Attempt:

$sy=sky^{\alpha}$: savings and investment per capita

$(n+d)k$: investment needed to keep capital per capita constant

n: population growth rate

d: depreciation

$\dot{k}= sky{\alpha}-(n+d)k=0$

Comment(s): I am unsure if this is even correct. My problem is that I didn’t even utilize the information given about consumption above. Any suggestions that could lead me down the right path?

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  • $\begingroup$ what exactly is the production function in this case? Is here everything assumed to be as in standard solow swan model? also how does technology enter? Can you please provide some clarity on the assumptions behind the question? $\endgroup$ – 1muflon1 Nov 29 '20 at 13:30
  • $\begingroup$ Hello there. I am sorry for not providing clarity. We assume the Solow growth model with government expenditure $G_t=\sigma Y_t$ $\endgroup$ – Tony456 Nov 29 '20 at 13:35
  • $\begingroup$ I hope this helps $\endgroup$ – Tony456 Nov 29 '20 at 13:35
  • $\begingroup$ could you please just write out all assumptions/set up? $\endgroup$ – 1muflon1 Nov 29 '20 at 13:37
  • $\begingroup$ Sure, I will edit them into my question $\endgroup$ – Tony456 Nov 29 '20 at 13:52
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There are some mistakes in the steps you took although they went in right direction. The correct steps are below.

First we can start by expressing output in per capita terms by dividing first two equations by $L_t$:

$$ \frac{Y_t}{L_t}=\frac{K^{\alpha}_t L^{1−\alpha}_t}{L_t} \implies y_t = k_t^{\alpha}$$

and

$$\frac{Y_t}{L_t}=\frac{C_t}{L_t}+\frac{I_t}{L_t}+\frac{G_t}{L_t} = y_t = c_t + i_t + g_t$$

where lower case letter refer to per capita variable (e.g. $\frac{Y_t}{L_t} = y_t$). Using this and the $L_{t+1}=(1+n)L_t $ the evolution of capital in per capita terms will be given by:

$$ (1 + n)k_{t+1} = (1 − \delta )k_t + i_t \implies k_{t+1} \approx (1 − n -\delta )k_t + i_t $$

Here we can use the fact that:

$$i_t = y_t - c_t - g_t = k_t^{\alpha} - c_t - g_t$$.

Hence:

$$k_{t+1} = (1 − n -\delta )k_t + k_t^{\alpha} - c_t - g_t $$

and as a consequence

$$k_{t+1} - k_t= -( n + \delta )k_t + k_t^{\alpha} - c_t - g_t $$

Now since $c_t=(1−s)(y_t-g_t)$, where ($-g_t$) is there because consumption depends only on disposable income after taxes and $g_t = \sigma y_t$ and remember $y_t= k^{\delta}_t$ we get:

$$k_{t+1} - k_t = -( n + \delta )k_t + k_t^{\alpha} - ( (1−s)(k_t^{\alpha}-\sigma k_t^{\alpha})- \sigma k_t^{\alpha}) \\ = -( n + \delta )k_t + k_t^{\alpha} - ( k_t^{\alpha} - s (1-\sigma) k_t^{\alpha} ) \\ = s (1-\sigma) k_t^{\alpha} -( n + \delta )k_t $$

where the last equation is really just this models version of $sy-(n+\delta)k_t$. Now finally dividing by $k_t$ to get to growth rates we get:

$$ \frac{k_{t+1} - k_t }{k_t}= s (1-\sigma) \frac{k_t^{\alpha}}{k_t} -( n + \delta ) $$

Now solving the above for steady state where $\frac{k_{t+1} - k_t }{k_t}=0$ (by definition a steady state), gives us:

$$ 0 = s (1-\sigma) \frac{k_t^{\alpha}}{k_t} -( n + \delta ) $$

which finally can be solved for optimal steady state per capita capital:

$$ k^* = \left( \frac{s(1-\sigma)}{n+\delta} \right)^{\frac{1}{1-\alpha}}$$

In addition very similar models to this one are covered by Barro & Sala-i-Martin Economic Growth 2nd ed. So you can check it out for further details.

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  • $\begingroup$ Thank you very much mate $\endgroup$ – Tony456 Nov 29 '20 at 15:43
  • $\begingroup$ @Tony456 you are welcome if you think this answer solved your question consider accepting it $\endgroup$ – 1muflon1 Nov 29 '20 at 15:44
  • $\begingroup$ How do I accept an answer exactly? I’m still new to this process. BTW, this method would also work if I changed say $C_t$ or similar yes? $\endgroup$ – Tony456 Nov 29 '20 at 15:45
  • $\begingroup$ @Tony456 you can click on check mark - I think you already figured it out (btw consider accepting also some of the older answers you got on previous q if they solved your issue - I see you have some good answers elsewhere as well). Yes the above would work also if there are changes in $C$ I expressed everything in per capita terms and using the parameters provided but exactly the same step with just different definitions would be followed if $c$ would be some other function. You can have look at some basic Solow model derivation in that book I recommended - steps are always pretty much same $\endgroup$ – 1muflon1 Nov 29 '20 at 15:49
  • $\begingroup$ I will definitely accept the answers from previous questions. This is ll really difficult (at least mathematical economics and macro) fortunately I have you guys to assist me with gaps. $\endgroup$ – Tony456 Nov 29 '20 at 15:54

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