The indirect utility function describes the maximum amount of utility a consumer can achieve, given a fixed amount of income and facing fixed prices. For two goods it is defined as
where $V$ denotes the indirect utility function, is the price of good 1,
is the price of good 2,
is the consumers income,
is the utility function,
is the quantity of good 1 consumed in equilibrium and
is the quantity of good 2 consumed in equilibrium.
Cobb Douglas Utility function
Consider the scenario where a consumer has the utility function described by Cobb Douglas Preferences, such that
and has the following budget constraint:
It is important to note that and
are just feasible options for good 1 and 2 and are not the equilibrium level of consumption
andÂ
.
To calculate the indirect utility function, we need to first find the demand equations andÂ
. Using our equilibrium condition
And substituting in the marginal utility for both good and good
gives us
which allows us to solve for in terms of
such that:
which we now substitute into the budget constraint, such that:
Which can be re-arranged to give us the demand equation, which we denote as
By symmetry, we should hopefully see that:
Substituting both these demand equations into our utility function gives the indirect utility function: