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We have that: $$\dot K = s K^\alpha L_0^b e^{nbt}$$ Rewriting the differential equation gives: $$K^{-\alpha} \frac{dK}{dt} = s L_0^b e^{nbt}$$ Integrate both sides with respect to $t$ from $0$ to $T$ gives: $$\frac{1}{b} [K^b]^T_0 = s L_0^b \frac{1}{nb}[e^{nbt}]^T_0$$ So: $$K^b_T = K^b_0 - \frac{s}{n} L_0^b + \frac{s}{n} L_0^b e^{nbT}$$ Equivalently:...
The direction of $\frac{\partial c^{*}}{\partial n}$ is not ambiguous. An easy way to show this is taking derivative of $c^*=(1-s)f(k^*)$ so that $\frac{\partial c^{*}}{\partial n}=(1-s)f'\frac{\partial k^{*}}{\partial n}$ and because $f'>0$ and we can prove $\frac{\partial k^{*}}{\partial n}<0$ we thus have $\frac{\partial c^{*}}{\partial n}<0$. ...