Reference no: EM133202711
There are two goods in the economy, meat and potatoes. The price of meat is 15 dollars per kilogram (kg) of meat. The price of potatoes is 10 dollars per kg of potatoes. The consumer has wealth 90 dollars to spend on meat and potatoes. Let x1 denote the quantity of meat in kgs she chooses to buy and let x2 denote the quantity of potatoes in 1 kgs she chooses to buy. Suppose throughout that she can only purchase non-negative quantities of either good.
Solve the following utility maximization problems when the preferences of a consumer are characterized by the following utility functions. For parts (a)to (c) you need to work out what is the quantity of each good the consumer wishes to purchase given her budget constraint and the value of λ∗(the marginal utility of wealth).
(a) U (x1, x2) = x31 × x22
(Hint: MU1 (x1, x2) = 3x21x22and MU2 (x1, x2) = 2x31x2).
(b) U (x1, x2) = 3 ln x1 + x2
(Hint: MU1 (x1, x2) = 3/x1 and MU2 (x1, x2) = 1).
(c) U (x1, x2) = 3x1 + 2x2
(Hint: MU1 (x1, x2) = 3, MU2 (x1, x2) = 2).
(d) Explain what would happen to the amount of meat and potatoes the consumer would choose if her utility were U (x1, x2) = 3x1 + 2x2, her wealth was (still) 90 dollars, the price of potatoes was (still) 10 dollars per kg but the price of meat was (now) greater than 15 dollars per kg.