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Given $F$ is a constant elasticity of substitution (CES) production function: $$F(K,AL) = \left [ \alpha K^{\rho} + (1-\alpha) (AL)^{\rho} \right ]^{\frac{1}{\rho}},$$ where $\alpha \in \left ( 0,1 \right )$.

I want to find the values of $\rho$ for which the function , $F$, satisfies the Inada conditions. The Inada conditions are as follows: $$\lim_{X\rightarrow \infty } F'(X) \rightarrow 0$$ and $$\lim_{X\rightarrow 0 } F'(X) \rightarrow \infty,$$where $X \in \left \{ K,L\right \}$.

So far I have $$lim_{X\rightarrow \infty } F'(X) = lim_{X\rightarrow \infty } = \alpha K^{\rho -1} \left [ \alpha K^{\rho} + (1-\alpha) (AL)^{\rho} \right ]^{\frac{1-\rho}{\rho}}.$$

I know that for CES production function $\rho \leq 1$, so $\rho$ can't be $\infty$, but have a gut feeling that $\rho$ can be $0$. Is there a simple procedure to finding the values of $\rho$?

OGC
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1 Answers1

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$$F(K,AL) = \left [ \alpha K^{\rho} + (1-\alpha) (AL)^{\rho} \right ]^{\frac{1}{\rho}},$$

So $K$ and $AL$ are the variables here.

$lim_{K\rightarrow 0}\dfrac{\partial F}{\partial K}=lim_{K\rightarrow 0}\alpha K^{\rho -1} \left [ \alpha K^{\rho} + (1-\alpha) (AL)^{\rho} \right ]^{\frac{1-\rho}{\rho}}=0\cdot[0+(1-\alpha)(AL)^\rho]^{\frac{1-\rho}{\rho}}$

This expression has to approach infinity, so by intuition, the expression on the right $[(1-\alpha)(AL)^\rho]^{\frac{1-\rho}{\rho}}$ should be infinity for the limit to approach infinity!

The most reasonable value for $\rho$ as you said is 0 since $\dfrac{1-0}{0}\rightarrow \infty$.

So let us find the limit as $\rho\rightarrow 0$ $$Y=lim_{K\rightarrow 0}lim_{\rho\rightarrow 0}\dfrac{\partial F}{\partial K}=lim_{K\rightarrow 0}lim_{\rho\rightarrow 0}\quad\!\!\alpha K^{\rho -1} \left [ \alpha K^{\rho} + (1-\alpha) (AL)^{\rho} \right ]^{\frac{1-\rho}{\rho}}$$ \begin{align} ln(Y)&=lim_{K\rightarrow 0}lim_{\rho\rightarrow 0}\quad\!\! ln\left(\alpha K^{\rho -1} \left [ \alpha K^{\rho} + (1-\alpha) (AL)^{\rho} \right ]^{\frac{1-\rho}{\rho}}\right)\\ &=lim_{K\rightarrow 0}lim_{\rho\rightarrow 0}\quad\!\! \alpha\left[(\rho-1)ln(K)+\dfrac{1-\rho}{\rho}ln(\alpha K^\rho+(1-\alpha)(AL)^\rho) \right]\\ &=\alpha\left[ \quad\!\!\!lim_{K\rightarrow 0}lim_{\rho\rightarrow 0}\quad\!\!(\rho-1)ln(K)+ lim_{K\rightarrow 0}lim_{\rho\rightarrow 0} \quad\!\!\dfrac{1-\rho}{\rho}ln(\alpha K^\rho+(1-\alpha)(AL)^\rho) \right]\\ &=\alpha\left[-1\cdot (-\infty)\quad +\quad\infty\cdot 0 \right] \end{align}

While one can use L'Hopital's to calculate the limit on the right hand side $(lim_{K\rightarrow 0}lim_{\rho\rightarrow 0} \quad\!\!\dfrac{1-\rho}{\rho}ln(\alpha K^\rho+(1-\alpha)(AL)^\rho) )$ the actual value of this limit is irrelevant as calculating the limit with an indeterminate form of $0\cdot \infty$ would give you any value from $[0,\infty)$.

Thus, $ln(Y)\rightarrow \infty$ and $Y\rightarrow \infty$. The rest of the problem (partial $AL$, the other limit condition) would be similar calculations with different limits.

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Matt
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  • I didn't clearly understand why $$Y=lim_{K\rightarrow 0}lim_{\rho\rightarrow 0}\quad!!\alpha K^{\rho -1} \left [ \alpha K^{\rho} + (1-\alpha) (AL)^{\rho} \right ]^{\frac{1-\rho}{\rho}}?$$ This doesn't make much sense to me, as $Y=F(K,AL) = \left [ \alpha K^{\rho} + (1-\alpha) (AL)^{\rho} \right ]^{\frac{1}{\rho}}$. – OGC Jan 17 '16 at 19:12
  • @sisphus68 Also I can't make sense of your second last comment. Does $\rho$ take values from $\left [0,\infty \right )$? – OGC Jan 17 '16 at 19:36
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    Hi, because the question was looking to satisfy the Inada conditions (which deal with the derivative of the CES production function) that you provided, I let $Y=$the limit of the partial derivative of the CES production function with respect to $K$ since that is what we are looking to solve. I did not mean the $\rho$ takes those values (we found that $\rho$ should approach $0$ for the Inada conditions to be satisfied) but that the limit takes values from any non-negative real number. I made some adjustments to my originally post above. – Matt Jan 17 '16 at 21:42