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Note: These explanations are in the process of being adapted from my textbook.
I'm trying to make them each a "standalone" treatment of a concept, but there may still
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The CES Utility Function

A more general way of modeling substitutability is via a constant elasticity of substitution (CES) utility function, which may be written \(u(x_1,x_2) = \left(\alpha x_1^r + (1 - \alpha)x_2^r\right)^{1 \over r}\) A little math shows that the MRS of this utility function is \(MRS = {\alpha \over 1 - \alpha} \left( {x_2 \over x_1}\right)^{1 - r}\) There are two parameters in this utility function:

You can check to see that the marginal utilities for this are positive, so the preferences are monotonic. For any $r < 1$ the MRS is decreasing as you move down and to the right (i.e., as $x_1$ increases and $x_2$ decreases), so the preferences are also convex. For any $r > 1$ the MRS is increasing as you move down and to the right, so the preferences are concave.

Try playing with $\alpha$ and $r$ in the diagram below to see how the indifference map changes:

Note that when $r$ is an extremely large negative number, the indifference curves approach the L-shaped curves of the perfect complements utility function; when $r = 0$, the indifference curves resemble those of a Cobb-Douglas utility function; and when $r = 1$, the indifference curves are linear like a perfect substitutes utility function. In fact, if you compare the MRS of those utility functions, you can confirm that this is the case. It also illustrates that there is a wide range of preference that are complements but not perfect complements (with $-\infty < r < 0$) and substitutes but not perfect substitutes (with $0 < r < 1$).

Next: Quasilinear Utility Functions
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