I was reading the paper "Optimal Investment Under Uncertainty" (Abel, 1982). At one point the author addresses the following problem:
$$\max_{L_{t}}=\left\{ p_{t}L_{t}^{\alpha}K_{t}^{1-\alpha}-wL_{t}\right\}=hp_{t}^{\frac{1}{1-\alpha}}K_{t}$$
Where:
$$h=(1-\alpha)\left(\frac{\alpha}{w}\right)^{\frac{\alpha}{1-\alpha}}$$
I have tried solving the first derivative with respect to L equal to zero and resubstituting the result in the production function (indirect production function), but I cannot get the result ($hp_{t}^{\frac{1}{1-\alpha}}K_{t}$). Can someone explain to me how to proceed?
https://repository.upenn.edu/cgi/viewcontent.cgi?article=1206&context=fnce_papers
*Pag. 4
We maximize $Y_t=p_tL_t^{\alpha}K_t^{1-\alpha}-wL^t$ w.r.t $L_t$. The derivative is
$$\frac{\partial Y_t}{\partial L_t} =\alpha\cdot p_tL_t^{\alpha-1}K_t^{1-\alpha}-w=0$$
Solving the equation for $L_t$
$$L_t^{\alpha-1}=\frac{w}{\alpha p_t}\cdot K_t^{\alpha-1}\Rightarrow L_t^*=\left(\frac{w}{\alpha p_t}\cdot K_t^{\alpha-1} \right)^{\frac{1}{\alpha-1}}$$
Now we insert the optimal value into $Y_t$.
$$Y_t(L_t^*)=p_t\left(\left(\frac{w}{\alpha p_t}\cdot K_t^{\alpha-1} \right)^{\frac{1}{\alpha-1}}\right)^{\alpha}K_t^{1-\alpha}-w\left(\frac{w}{\alpha p_t}\cdot K_t^{\alpha-1} \right)^{\frac{1}{\alpha-1}}$$
$$=p_t\left(\frac{w}{\alpha p_t}\cdot K_t^{\alpha-1} \right)^{\frac{\alpha}{\alpha-1}}K_t^{1-\alpha}-w\left(\frac{w}{\alpha p_t}\cdot K_t^{\alpha-1} \right)^{\frac{1}{\alpha-1}}$$
Focussing on the exponent of $K_t$ at the first summand: $(\alpha-1)\cdot \frac{\alpha}{\alpha-1}+(1-\alpha)=1$. Second summand: $(\alpha-1)\cdot \frac{1}{\alpha-1}=1$
Exponents $p_t$ at the first summand: $1-{\frac{\alpha}{\alpha-1}}={\frac{\alpha-1}{\alpha-1}}-{\frac{\alpha}{\alpha-1}}=-\frac{1}{\alpha-1}=\frac{1}{1-\alpha}$. Second summand: $-\frac{1}{\alpha-1}=\frac{1}{1-\alpha}$
Exponents of $w$ at the second summand: $1+\frac{1}{\alpha-1}=\frac{\alpha-1}{\alpha-1}+\frac{1}{\alpha-1}=\frac{\alpha}{\alpha-1}$
I omit the variables $p_t$ and $K_t$ to get a better overview. The corresponding exponents should be clear now.
$$\left(\frac{w}{\alpha }\right)^{\frac{\alpha}{\alpha-1}}-w^{\frac{\alpha}{\alpha-1}}\left(\frac{1}{\alpha }\right)^{\frac{1}{\alpha-1}}$$
Exponents for $w$: At the solution w is in the denominator. Thus we add a negative sign to bring it in the denominator: $-\frac{\alpha}{\alpha-1}=\frac{\alpha}{1-\alpha}$
$$\left(\frac{1}{\alpha }\right)^{\frac{\alpha}{\alpha-1}}-\left(\frac{1}{\alpha }\right)^{\frac{1}{\alpha-1}}=\alpha ^{\frac{\alpha}{1-\alpha}}-\alpha ^{\frac{1}{1-\alpha}}$$
Factoring out $\alpha ^{\frac{\alpha}{1-\alpha}}$. That gives for the exponents of the second term $\frac{1}{1-\alpha}-\frac{\alpha}{1-\alpha}=1$
$$\left(1- \alpha^1\right)\cdot \alpha ^{\frac{\alpha}{1-\alpha}}=\left(1- \alpha\right)\cdot \alpha ^{\frac{\alpha}{1-\alpha}}$$
Therefore $$h=\left(1- \alpha\right)\cdot \left(\frac{\alpha}{w}\right) ^{\frac{\alpha}{1-\alpha}}$$