Simon Fraser University Prof. Karaivanov
Department of Economics Econ 301
MARGINAL UTILITY AND MRS (detailed notes)
Knowing about utility, a natural question is by how much a consumerās utility would increase
if she consumes one more unit of some good. This increment in utility is called marginal utility.
Definition: Marginal Utility (MU) - the change in utility associated with a small change
in the amount of one of the goods consumed holding the quantity of the other good fixed.
There are two important things above:
1. First, notice that marginal utility measures the rate of change in utility when we vary
the quantity of a good consumed. Thus it basically measures the āslopeā of the utility function
with respect to changes in this good. However the utility function has two arguments so there
will be two āslopesā i.e. when we talk about marginal utility we should always specify with
respect to which good.
2. Notice that the quantity of one of the goods is always held constant when computing the
marginal utility with respect to the other.
Given the above definition we can write the marginal utility with respect to good 1 (MU1)
as the ratio:
MU1=4U
4x1
=u(x1+4x1,x
2)āu(x1,x
2)
4x1
(1)
that measures the rate of change in utility (4U) associated with a small change in the amount
of good 1 (4x1).Thus to calculate the change in utility resulting from a change in consumption
in good 1 we should just multiply the marginal utility (MU1) by the change in consumption:
4U=MU14x1(2)
the marginal utility thus measures the rate at which consumption units are converted into
utility units, i.e. the āpriceā of one more unit of consumption in terms of utility units.
Marginal utility is even easier to understand using simple calculus - notice that the above
expression (1) is exactly the partial derivative of uwith respect to x1when we let the change
4x1go to zero i.e. become infinitesimally small. Thus more formally we can write:
MU1=āu(x1,x
2)
āx1
and MU2=āu(x1,x
2)
āx2
Remember what partial derivatives are: you diļ¬erentiate the function with respect to one
of the variables holding the other fixed (i.e. treating it as a constant).
Some examples of marginal utilities:
1
