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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

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2 Answers 2

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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}

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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

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

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

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