Question: Suppose $C_t=(1-s)Y_t-\lambda G_t$ where $s>\sigma$ as in the basic Solow model. Out of the government expenditure , proportion $\phi$ is invested in public capital formation. Hence we assume $K_{t+1}=I_t+\phi G_t+(1-\delta)K_t$
$Y_t=K^{\alpha}_tL^{1-\alpha}_t$
$Y_t=C_t+I_t+G_t$
$K_{t+1}=I_t+(1-\delta)K_t$
$L_{t+1}=(1+n)L_t$
$G_t=\sigma Y_t$
In what case will the steady state level of capital per capita increase in $\sigma$
Attempt:
step (1) $\frac{Y_t}{L_t}=\frac{K^{\alpha}_t}{L_t}\frac{L^{1-\alpha}_t}{L_t}$ which equals $y_t=k^{\alpha}_t$
Using the same process for the national identity we get
The evolution of capital per capita is given by the following equation:
$(1+n)k_{t+1}=i_t+\phi g_t+(1-\delta)k_t$ which then goes to:
$k_{t+1}\approx i_t+\phi g_t+(1-\delta-n)k_t$
$k_{t+1}=[k^{\alpha}_t-c_t-g_t]+\phi g_t+(1-\delta-n)k_t$
Now, we can subtract $k_t$ from $(1-\delta-n)k_t$ and we get:
$k_{t+1}-k_t=-(n+\delta)k_t+\phi g_t+k^{\alpha}_t-c_t-g_t$
=$(n+\delta)k_t+\phi g_t+k^{\alpha}_t - [(1-s)k^{\alpha}_t-\lambda g_t]-[\sigma k^{\alpha}_t]$
= $-(\delta +n)k_t+\phi [k^{\alpha}_t\sigma]+k^{\alpha}_t-[(1-s)k^{\alpha}_t-\lambda [k^{\alpha}_t \sigma]]$
We can simplify this algebraically to:
$k_{t+1}-k_t= \phi [k^{\alpha}_t \sigma] + sk^{\alpha}_t+ \lambda[k^{\alpha}_t\sigma]+(\delta+n)k_t$
Finally divide both sides by $k_t$ and set the LHS equal to 0 and get the steady state equilibrium as:
$0 = \frac{\phi[k^{\alpha}_t\sigma]}{k_t}+\frac{sk^{\alpha}_t}{k_t}+\frac{\lambda[k^{\alpha}_t\sigma]}{k_t}-(\delta+n)k_t$
Comments: I am pretty sure my calculus is correct (perhaps save for the last equation. Can anyone confirm this for me?