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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


– 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

– The consumer can always compare/rank bundles.

– 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

(1)   \begin{equation*} \nonumber \boxed  { MRS =  - \frac{MU_x\downarrow}{MU_y\uparrow}\Downarrow} \end{equation*}

–As we get more q1 each additional unit is worth less to us (so U_1 goes down).

–As we get less q2 each additional unit is worth more to us (so U_2 goes up).

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