# Maximization of CD production function

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 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

The first order condition for the problem is

$$w = \alpha p_t \left(\frac{K_t}{L_t^*}\right)^{1-\alpha},$$ where the star indicates the optimally chosen control variable. Solve for $$L_t^*$$

$$L_t^* = \left(\frac{\alpha}{w}\right)^{\frac{1}{1-\alpha}}p_t^{\frac{1}{1-\alpha}}K_t$$ and note that by definition $$\max_x f(x) = f(x^*)$$ for any objective function $$f$$ (as long as the maximum exists). Plugging in the above expression for $$L_t^*$$ into the objective function will then get you

\begin{align} p_t &\left(\frac{\alpha}{w}\right)^{\frac{\alpha}{1 - \alpha}}p_t^{\frac{\alpha}{1-\alpha}}K_t^\alpha K_t^{1 - \alpha} - w\left(\frac{\alpha}{w}\right)^{\frac{1}{1-\alpha}}p_t^{\frac{1}{1-\alpha}}K_t\\ &=\left(\frac{\alpha}{w}\right)^{\frac{\alpha}{1-\alpha}}p_t^{\frac{1}{1-\alpha}}K_t - \alpha \left(\frac{\alpha}{w}\right)^{\frac{\alpha}{1-\alpha}}p_t^{\frac{1}{1-\alpha}}K_t\\ &=(1 - \alpha) \left(\frac{\alpha}{w}\right)^{\frac{\alpha}{1-\alpha}}p_t^{\frac{1}{1-\alpha}}K_t \\ &= hp_t^{\frac{1}{1-\alpha}}K_t. \end{align}

I re-write the maximization problem (I neglect the subscript $$t$$):

$$\max_{L}\left\{ pL^{\alpha}K^{1-\alpha}-wL\right\}\quad (1)$$

Taking the derivative with respect to $$L$$ of the expression in $$(1)$$ and equating it to zero
we have:

$$\alpha pL^{\alpha-1}K^{1-\alpha} -w=0\qquad (2)$$

from which we have

$$L^{\alpha-1}= (\frac {w}{\alpha p}) K^{\alpha-1}$$

hence

$$L= \left(\frac {w}{\alpha p}\right)^{\frac{1}{\alpha -1}}K\qquad (3)$$

Substituting $$(3)$$ into the function to be maximized in $$(1)$$ yields

$$\max_{L}\left\{ pL^{\alpha}K^{1-\alpha}-wL\right\}= p(\frac{\alpha p}{w})^{\frac{\alpha}{1-\alpha}}K^{\alpha} K^{1-\alpha} -w (\frac{\alpha p}{w})^{\frac{1}{1-\alpha}} K=$$ $$p(\frac{\alpha p}{w})^{\frac{\alpha}{1-\alpha}}K - w (\frac{\alpha p}{w})^{\frac{1}{1-\alpha}} K=$$ $$=p^{\frac{1}{1-\alpha}} (\frac{\alpha }{w})^{\frac{\alpha}{1-\alpha}}K -p^{\frac{1}{1-\alpha}}w (\frac{\alpha }{w})^{\frac{1}{1-\alpha}} K=$$ $$=p^{\frac{1}{1-\alpha}}K [(\frac{\alpha}{w})^{\frac{\alpha}{1-\alpha}}-\alpha \frac{\alpha ^{\frac{1}{1-\alpha}} \frac{1}{\alpha}}{w ^{\frac{\alpha }{1-\alpha}}}]=$$ $$= p^{\frac{1}{1-\alpha}}K [(\frac{\alpha}{w})^{\frac{\alpha}{1-\alpha}}- \alpha (\frac{\alpha}{w}) ^{\frac{\alpha}{1-\alpha}}]=$$ $$= p^{\frac{1}{1-\alpha}} K(\frac{\alpha}{w})^\frac{\alpha}{1-\alpha}(1-\alpha) =$$

$$= hp^{\frac{1}{1-\alpha}}K$$