We have a utility function Uxy with monotonic preferences Th

We have a utility function U(x,y) with monotonic preferences. The marginal utility U\'x is a function of X only, and U\'y is a function of Y only (for example, U(x,y)=f(x)+g(y). They are both decreasing marginal utilities.

What can I know about this function? are both products normal goods? is the MRS decreasing? are they substitutes, compliments or unrelated goods?

Solution

(a) An utility function of the form U(x, y) = f(x) + g(y), the goods are perfect substitutess and their indifference curve is a straight line. Optimal utility lies at the corner points of the straight line touching the axes.

(b) MRS for perect substitutes is a constant. In such cases, MRS violates the principle of diminishing MRS.

(c) The goods are normal if with increase in income, their demand increases. For perfect substitutes, if the MRS is higher (lower) that price ratio, the consumer will buy only the goods represented along horizontal (vertical) axis. If the change in price ratio (which changes real income) is large, then the consumer will switch his purchase pattern and buy the other good.

Therefore, whether the goods are normal or inferior can be ascertained if we have information about MRS and price ratio.