Econ 301, Chapter 4, Utility
Utility is a way, an old-fashioned way, of describing preferences...but it has its uses.
We think of a U(.) function that maps (x1,x2,...xn) to R. That is, it assigns a number to every
bundle. The bigger the number, the higher the “utility”.
Or u(x1,x2) > u(y1,y2) if and only if (x1,x2) ™ (y1,y2). The nice thing about a utility function is
that it lets you do math with > rather than more abstract thought with ™.
We use “ordinal utility” meaning we only care if U(X) is greater than, less than, or equal to some
other U(Y), we don’t attach any meaning to being twice as great, etc.
So if U(.) Is a utility function, so is f(U(.)), when f is any monotonic transformation mapping U
to R, i.e., df/du > 0. Example: If U( . ) is a utility function, you get the same information from 6
+ 3U( . )
Examples of utility functions
u(x1, x2) = x1x2
u(x1,x2) = x12x22
Here is what this looks like:
Here’s what that one looks
like:
u(x1,x2) = 2x1 + 3x2
Perfect substitutes
Note in this case that 3x1 is equivalent to 2x2.
u(x1,x2) = min{x1,x2}
u(x1,x2) = %x1 + x2
Perfect complements
Quasi linear (partly
linear)
This one is often used to keep a problem
simple, since each indifference curve is shaped exactly
like the one above and below, i.e., the slope of a
tangent line to an indifference curve is the same on
each curve above a given x1 value.
u(x1,x2) = x1ax2b
Cobb-Douglas
This may
seem ugly, but
it has some
properties that
make it easy
to work with.
The a and b
terms bias
your
preferences in
favor (or disfavor) of the goods. The picture on the left has a = .1 and b = .9, while the one on
the right has a = .9 and b = .1. Draw some budget constraints and see how people behave.
u(x1,x2) = a ln x1 + b ln x2 This is the natural log version of Cobb-Douglas
Marginal Utility and MRS
MU1 = Mu/Mx1
The MRS gives the slope of the indifference curve, and it tells you the rate at which you are just
barely willing to give up good 2 to get one more good 1. Read more in the text.