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 equationsÂ 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: