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The graph above represents a typical Cobb-Douglas utility function given by $U(x,y)=\ x^\alpha y^\beta$ . Higher level of utility is presented by a higher utility function such that $U_5 > U_4$ , meaning the consumer is better off at $U_5 than U_4$ .

Recall that total derivative of a function $z(x,y)$ is given by $\Delta z = \frac{\partial z}{\partial y} \Delta y + \frac{\partial z}{\partial x}\Delta x$

Axioms of Preferences

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Monotonicity (non-satiation)
– We are talking about goods and not bads >> More is better!
– Consider two bundles X and Y. If Y has at least as much of both goods, and more of one, then

(y1,y2)≥(x1,x2)

Convexity
– Averages are better than extremes (or at least not worse)
– An average of two bundles on the same indifference curve will be (at least weakly) preferred

Completeness
– The consumer can always compare/rank bundles.

Continuous
– If X is preferred to Y, and there is a third bundle Z which lies within a small radius of Y, then X will be preferred to Z.

Marginal Rate of Substitution

So the slope of an indifference curve is equal to the ratio of marginal utilities (will become important later).
If indifference curves are convex, so that the MRS is diminishing:

–We can notice a few things about the $MRS$ function. As $x\uparrow$ $MU_x\downarrow$ , conversely as $x\downarrow$ $MU_x\uparrow$ . What happens to the MRS in this case? As a consumer moves along the $x axis$