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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

Using the same logic, the total derivative of a function U(x,y) is given by \Delta U = \frac{\partial U}{\partial y} \Delta y + \frac{\partial U}{\partial x}\Delta x

Since, an indifference curve represents no change in utility, i.e. \Delta U=0 we can say that \Delta U = \frac{\partial U}{\partial y} \Delta y + \frac{\partial U}{\partial x}\Delta x =0

The slope of the indifference curve is given by \frac{\Delta y}{\Delta x} which can be manipulated from the above expression. By rearranging, we can write \frac{\partial U}{\partial y} \Delta y = - \frac{\partial U}{\partial x}\Delta x. This can be further re-written as:

(1)   \begin{equation*} MRS= \frac{\Delta y} {\Delta x} = - \dfrac{\dfrac{\partial U}{\partial x}}{\dfrac{\partial U}{\partial y}} \end{equation*}

(2)   \begin{equation*} MRS = - \frac{\frac{\partial U}{\partial x}}{\frac{\partial U}{\partial y}} = - \frac{MU_x}{MU_y} \end{equation*}

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

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

This makes the indifference curve convex as the MRS \Downarrow as x increases. That is, the slope of the indifference curve decreases as x increases.

As shown in figure above, strictly convex preferences show that the marginal rate of substitution (MRS) decreases as we move down the indifference curve. The more of a good you have, the more willing you are to sacrifice it to gain an additional unit of another good.

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Axioms of Preferences

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.

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