# Constant Elasticity of Substitution- profit maximization vs cost minimization

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.

• 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 Feb 4 '20 at 23:05
• 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 Feb 5 '20 at 14:26
• "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 Feb 5 '20 at 15:15
• 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 Feb 5 '20 at 15:22
• 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 Feb 5 '20 at 15:40