Would someone be able to help me solve capital per capita in the steady state (check my work)

Question

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?

Answer 1

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.