Suppose Antonio gets utility from consuming two goods, burgers and fries. His utility function is given by U = √ BF = B 0.5F 0.5 , where B is the amount of burgers he eats and F the servings of fries. Antonio’s marginal utility of a burger is MUB = 0.5B −0.5F 0.5 and his marginal utility of an order of fries is MUF = 0.5B 0.5F −0.5 . Antonio’s income is $20, and the prices of burgers and fries are $5 and $2, respectively. What are Antonio’s utility maximizing quantities of burgers and fries?

The optimal solution to the maximization problem of a consumer is equivalent to the ratio of marginal utilities of goods and it is also equal to the price ratio of the goods. Mathematically:

[tex]MRS_{BF} = \frac{MU_{B}}{MU_{F}} = \frac{P_{B} }{P_{F}}[/tex]

The burgers and fries marginal utilities are [tex]MU_{B}[/tex] and [tex]MU_{F}[/tex] respectively while their prices are [tex]P_{B}[/tex] and [tex]P_{F}[/tex] respectively. Thus,

[tex]\frac{0.5B^{-0.5}F^{0.5} }{0.5B^{0.5}F^{-0.5}} = \frac{5}{2}[/tex]

Further simplification:

F/B = 5/2 ; F = 2.5B

Using Antonio's budget income,

B = income/ [tex]P_{B}[/tex] - ([tex]P_{F}[/tex] / [tex]P_{B}[/tex])*F

If we use the values in the problem, we have:

B = 20/5 - (2/5)*F = 4 - 0.4F

if we substitute F = 2.5B

B = 4 - 0.4*2.5B

B = 4 - B

B = 4/2 = 2

F =2.5B = 2.5*B = 2.5*2 = 5

Thus, given the budget constraint of Antonio, he can maximize his utility by eating 5 servings of fries and 2 burgers.