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I have been reading into generalizations of the concept of elasticity of substitution for more goods/inputs and three main possibilities emerged:

  • Hicksian EOS
  • Allen-Uzawa EOS
  • Morishima EOS

HICKS

As I understand it, the Hicksian EOS is just an application of EOS on pairs of goods while holding others constant:

$$\sigma_{i,j}^{H} = \frac{\frac{\partial x_j/x_i}{x_j/x_i}}{\frac{\partial MRS_{i,j}}{MRS_{i,j}}}$$

where $MRS_{i,j} = MU_i/MU_j$. This can be computed for any $i$ and $j$.

The Hicksian EOS should therefore measure the EOS between two goods/inputs while others are considered constant.

However, further than this, things start to get weird. For example, I have stumbled upon this:

$$\sigma_{i,j}^{??} = \frac{\frac{\partial U}{\partial x_i} \frac{\partial U}{\partial x_j}}{U \cdot \frac{\partial^2 U}{\partial x_i \partial x_j}}$$

which has different implication than the previous elasticity, however it could still belong to Hicks?

Allen-Uzawa

What concerns Allen-Uzawa EOS, it is strange how many different computations are offered there:

There is this:

$$ \sigma_{i,j}^{AU} = \frac{x_1 \frac{\partial U}{\partial x_1} + \dots x_n \frac{\partial U}{\partial x_n}}{\frac{\partial U}{\partial x_i} \frac{\partial U}{\partial x_j}} \frac{F_{i,j}}{F} $$

where $F$ should be determinant and $F_{i,j}$ should be co-factor of $ \frac{\partial^2 U}{\partial x_i \partial x_j}$ in this determinant.

However, there is also this:

$$\sigma_{i,j} = \frac{XPD_{i,j}}{S_j}$$

where $XPD_{i,j}$ is cross-price elasticity of demand and $S_j$ is a share of $j$ input in total cost.

Morishima

The last one is the weirdest:

$$\sigma_{i,j}^{M} = XPD_{j,i} - PD_{j,j} $$

Where $XPD$ is a crossprice elasticity of $j$ input according to $i$ price and $PD$ is own-price elasticity of demand.

The question:

Right now, this seems to me to be a little too chaotic. Almost all of these almost seem to be completely unrelated. Then, there is of course the fact that in later definition the prices were brought in, while the original definition defines the elasticity of substitution as a propriety of utility function (concretely, the curvature of the indifference curve). So I am a little bit confused right now and I would like to ask the following:

  • What do these have in common?
  • How do they differ?
  • How do those latter ones really generalize the former one?
  • Do these serve as a measure of how much goods are substitutes or complements?
  • Why the later ones operate with prices if the original is solely the propriety of a function?
  • What would be the connection between crossprice and EOS?
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2 Answers 2

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This note only answers the last of your question. All these elasticities tend to disappear from the empirical literature since the publication of the influencial paper by

Blackorby, C. and R. R. Russel, 1989, Will the real elasticity of substitution please stand up? (A comparison ofthe Allen/Uzawa and Morishima elasticities). The American Economic Review, 79, 882-888.

The (own- or) cross-price elasticity of $x^*_j$ wrt $p_k$ is defined by

$$ \frac{\partial x^*_j}{\partial p_k}(p,w) \frac{ p_k }{ x^*_j (p,w)} $$ which has an unambiguous interpretation, and supplanted these weird alternative concepts.

The Morishima elasticity is directly obtained from these elementarity elasticities and also has a clear interpretation as it measures the sensibility of the optimal quantity ratio, $x^*_j/x^*_k$ wrt a percentage change in the price ratio $p_j/p_k$.

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  • $\begingroup$ My biggest problem with cross-price elasticity was that it defines substitutes and complements with respect to price, which did not seem that sound to me for a primary concept. After all, substitutes and complements must ensue from tastes (how you like something, whether you need products together or you can exchange one for another). Therefore, I have expected that the primary definitional concept should be derived from preferences only and that is why I searched for elasticity of substitution (for which, the Robinson's satisfied this realism). $\endgroup$
    – Athaeneus
    Apr 2 at 5:05
  • $\begingroup$ @Athaeneus: complementarity or subsitutability relationships are not universal properties but conditional to price and budget. Two goods may be complements for low budget, and become substitutes when the budget increases (or when relative prices change). In other words, if you want to consider only the utility function, you may have to evaluate the cross derivatives of U over an infinity of virtual consumption bundles, without being sure which ones will effectively be chosen when a budget constraint will restrict the choice field. Is this convenient? It depends on your objective. $\endgroup$
    – Bertrand
    Apr 2 at 19:29
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To everyone interested in the same problem, I have found a perfect article that goes into great length and detail explaining what are the interconnections and attributes/characteristics of each of these elasticities. It is a very good starting point and I can recommend it by all means.

It is by João Eustáquio de Lima (2000) ALTERNATIVE DEFINITIONS OF ELASTICITY OF SUBSTITUTION: REVIEW AND APPLICATION

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