Consider a consumer with the utility function Ux y min3x 5y

Consider a consumer with the utility function U(x, y) = min(3x, 5y), that is, the two goods are perfect complements in the ratio 3:5. The prices of the two goods are Px = $5 and Py = $10, and the consumer’s income is $220. Determine the optimum consumption basket.

Solution

Here we have Leontief preferences and so the optimal is at the kink. And so we have 3X=5Y. The budget constraint is given by 5X+10Y=220. Thus substituting the optimal we have 25/3Y+10Y=220. Thus 55/3Y=220 and hence Y=12,X=20. The optimum consumption basket is thus (20,12).