<p>Consider the following production function:
<span class="math-container">$$
Y=A\left[\alpha K^{\rho}+\left(1-\alpha\right)L^{\rho}\right]^{\frac{1}{\rho}}
$$</span>
In the above, <span class="math-container">$\rho$</span> is the substitution parameter, and <span class="math-container">$\sigma=\frac{1}{1-\rho}$</span>
is the elasticity of substitution. One definition of the elasticity
of substitution is that this parameter is defined as the substitution
of <span class="math-container">$K/L$</span> with respect to <span class="math-container">$MRS_{K,L}$</span> <strong>holding output constant</strong>. In other words:
<span class="math-container">$$
\sigma=\frac{d\text{log}\left(K^{*}/L^{*}\right)}{d\text{log}\left(F_{L^{*}}/F_{K^{*}}\right)}
$$</span>
 Notice, that the optimal choice on part of the firm implies that
<span class="math-container">$F_{L}/F_{K}=w/r$</span> . As such:
<span class="math-container">$$
\sigma=\frac{d\text{log}\left(K^{*}/L^{*}\right)}{d\text{log}\left(w/r\right)}
$$</span></p>

<p>My question is as follows: how important is the fact that we hold
output constant? I understand that if we consider the problem of the
firm in two stages, i.e. in the first stage as a cost minimization
problem (choosing optimal <span class="math-container">$K^{*}$</span> and <span class="math-container">$L^{*}$</span> for a given level
of output), and in the second stage, choosing an output level to maximize
profits, then if relative prices change, it is not necessary that
the profit maximizing output stays the same. In other words, if relative
prices change, we shift to a different isoquant, and this definition
of <span class="math-container">$\sigma$</span> is not the correct one. </p>

CC BY-SA 4.0
2020-02-04T23:05:58.623Z
Well... the simple answer is that the MRS is meaningless if you don't hold output (and therefore, the location in K-L space) constant, as it is constructed from the ratio of a couple of partial derivatives.  This really isn't even economics - it's just math.

Are you worried about whether/how this would be a problem if prices change?  You don't have prices in this model.
heh
CC BY-SA 4.0
2020-02-05T14:26:36.830Z
Well the appropriate prices are interest rate and wages. The issue here is that if relative prices change, then an optimal bundle changes. However, the optimal bundle can change in a way such that output is not constant anymore. I guess the question is, how  can we impose that output remain constant when relative prices change?
ChinG
CC BY-SA 4.0
2020-02-05T15:15:49.217Z
"However, the optimal bundle can change in a way such that output is not constant anymore."  This doesn't make sense. "I guess the question is, how can we impose that output remain constant when relative prices change?"  You can introduce an additional constraint to the system $Y == constant$, but why would you want to do that?  As you note, shifting prices means that the optimal output level may change in response.  But the definition of $\sigma$ doesn't change - what changes are $w$, $r$, $K*$ and $L*$.
heh
CC BY-SA 4.0
2020-02-05T15:22:48.667Z
It's a little unclear why you're stuck, so I'll just emphasize that the elasticity of substitution is only defined at equilibrium, wherein an optimal output level is specified in the basis of given prices.  If those prices change, then the optimal output level is expected to change - but the definition of $\sigma$ will be just as valid at the new equilibrium as it was at the old.  Perhaps what you're really asking is how to decompose the transition between equilibria into a substitution effect and an income effect?
heh
CC BY-SA 4.0
2020-02-05T15:26:07.660Z
Sorry for the confusion. Youre getting to the heart of the problem. If relative price change, then capital and labor change, and output potentially changes. So, how can you define the EOS along an isoquant? In other words, if the output changes, how does the concept of optimal substitution along an isoquant make sense?
ChinG
CC BY-SA 4.0
2020-02-05T15:40:08.833Z
It sometimes helps to take a step back from the economics and consider the math - what *is* the EOS?  It's just the slope of the isoquant (weighted by price) at a given point.  That's all.  There is no reason to expect the slope of a generic isoquant (e.g., a curve in abstract mathematical space) to be the same at all points.  There is certainly no reason to expect the slopes of two different isoquants to be the same.  But it remains true that once one finds oneself occupying a point along an isoquant, one can quantify the slope there.
heh
<microeconomics><elasticity-of-substitution>
Constant Elasticity of Substitution- profit maximization vs cost minimization
ChinG
# Constant Elasticity of Substitution- profit maximization vs cost minimization

(Asked by: username_0 on 2020-02-02)

Consider the following production function:
$$
Y=A\left[\alpha K^{\rho}+\left(1-\alpha\right)L^{\rho}\right]^{\frac{1}{\rho}}
$$
In the above, $\rho$ is the substitution parameter, and $\sigma=\frac{1}{1-\rho}$
is the elasticity of substitution. One definition of the elasticity
of substitution is that this parameter is defined as the substitution
of $K/L$ with respect to $MRS\_{K,L}$ **holding output constant**. In other words:
$$
\sigma=\frac{d\text{log}\left(K^{\*}/L^{\*}\right)}{d\text{log}\left(F\_{L^{\*}}/F\_{K^{\*}}\right)}
$$
Notice, that the optimal choice on part of the firm implies that
$F\_{L}/F\_{K}=w/r$ . As such:
$$
\sigma=\frac{d\text{log}\left(K^{\*}/L^{\*}\right)}{d\text{log}\left(w/r\right)}
$$

My question is as follows: how important is the fact that we hold
output constant? I understand that if we consider the problem of the
firm in two stages, i.e. in the first stage as a cost minimization
problem (choosing optimal $K^{\*}$ and $L^{\*}$ for a given level
of output), and in the second stage, choosing an output level to maximize
profits, then if relative prices change, it is not necessary that
the profit maximizing output stays the same. In other words, if relative
prices change, we shift to a different isoquant, and this definition
of $\sigma$ is not the correct one.

**Tags:** microeconomics, elasticity-of-substitution
> **Comments:**
> **username_1**: Well... the simple answer is that the MRS is meaningless if you don't hold output (and therefore, the location in K-L space) constant, as it is constructed from the ratio of a couple of partial derivatives.  This really isn't even economics - it's just math.  Are you worried about whether/how this would be a problem if prices change?  You don't have prices in this model.  
> **username_0**: Well the appropriate prices are interest rate and wages. The issue here is that if relative prices change, then an optimal bundle changes. However, the optimal bundle can change in a way such that output is not constant anymore. I guess the question is, how  can we impose that output remain constant when relative prices change?  
> **username_1**: "However, the optimal bundle can change in a way such that output is not constant anymore."  This doesn't make sense. "I guess the question is, how can we impose that output remain constant when relative prices change?"  You can introduce an additional constraint to the system $Y == constant$, but why would you want to do that?  As you note, shifting prices means that the optimal output level may change in response.  But the definition of $\sigma$ doesn't change - what changes are $w$, $r$, $K*$ and $L*$.  
> **username_1**: It's a little unclear why you're stuck, so I'll just emphasize that the elasticity of substitution is only defined at equilibrium, wherein an optimal output level is specified in the basis of given prices.  If those prices change, then the optimal output level is expected to change - but the definition of $\sigma$ will be just as valid at the new equilibrium as it was at the old.  Perhaps what you're really asking is how to decompose the transition between equilibria into a substitution effect and an income effect?  
> **username_0**: Sorry for the confusion. Youre getting to the heart of the problem. If relative price change, then capital and labor change, and output potentially changes. So, how can you define the EOS along an isoquant? In other words, if the output changes, how does the concept of optimal substitution along an isoquant make sense?  
> **username_1**: It sometimes helps to take a step back from the economics and consider the math - what *is* the EOS?  It's just the slope of the isoquant (weighted by price) at a given point.  That's all.  There is no reason to expect the slope of a generic isoquant (e.g., a curve in abstract mathematical space) to be the same at all points.  There is certainly no reason to expect the slopes of two different isoquants to be the same.  But it remains true that once one finds oneself occupying a point along an isoquant, one can quantify the slope there.  

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## 0 answers

