Suppose that an individual consumes three goods -- food, clothing, automobiles. Denote the quantities of these goods consumed by X, Y, and Z respectively. Suppose the individual utility function is given by U = 5 ln x + 4 ln y = ln (1+z) and the prices of the goods are given by Price X = $1 . Price Y = $2. and Price Z = $2,000.
Let the individual's total income be $56,000. Automobiles, however, have the property that they must be bought in discrete untis (that is, Z must equal 0, 1, 2, and so on). It is impossible in this simple model to buy one half a car. Given these contraints, how will an individual choose to allocate income so as to maximize utility?