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The graph above represents a typical Cobb-Douglas utility function given by 
. Higher level of utility is presented by a higher utility function such that 
, meaning the consumer is better off at 
.
Recall that total derivative of a function 
is given by 
Using the same logic, the total derivative of a function 
is given by 
Since, an indifference curve represents no change in utility, i.e. 
we can say that 
The slope of the indifference curve is given by 
which can be manipulated from the above expression. By rearranging, we can write 
. This can be further re-written as:
(1) 
(2) 
We can notice a few things about the 
function. As 
, conversely as 
. What happens to the MRS in this case? As a consumer moves along the 
(3) 
This makes the indifference curve convex as the MRS 
as 
increases. That is, the slope of the indifference curve decreases as 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.
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
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.
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