Question

Suppose you have $16 and your utility function is U(x,y) = xy. What do prices have to be for your optimal consumption bundle to be x = 2 and y = 1? Show your work.

Answer 1

Given:

Assuming price of x is px and price of y is py and the income is $16, the budget constraint equation will be given as:

Also, the utility function is given by:

Now, we know that the condition for the optimal bundle in case of two goods is that marginal utility per dollar spent on each good should be the same:

Rearranging the terms, we obtain:

where is the Marginal utility of good x

and    is the Marginal utility of good y.

Now, we can calculate marginal utilities as:

Similarly,

It is given that the optimal consumption bundle is: x = 2 and y = 1 so, substituting these values in the optimal condition:

So, .

Now, if we substitute this condition in the budget constraint along with the optimal bundle values:

So,

We get:

So, in order to obtain the given optimal consumption bundle, px should be $ 4 and py should be $ 8.