When can I say that a utility function has constant marginal utility?

Question

Does this utility function have increasing/decreasing or constant marginal utility?

$ U(x,y) = x^2 y^2 $

Now, $ f_x = 2xy^2 $, $ f_{xx} = 2y^2 $, $ f_y = 2yx^2 $, $f_{yy} = 2x^2 $

$ f_{xx} $ has no $ x $ term in it -- so is the marginal utillity of $ x$ constant or increasing? It increases as $ y $ increases, of course, but it stays constant if we increase $ x $. My textbook says that this is a case of increasing marginal utility, but I don't understand why. Similar problem for $ f_{yy} $ , which has no $ y $ term.

Answer 1

Marginal utility (of $x$) in your case is $U_x(x,y)=2xy^2$. You use the sign of the derivative of MU, namely $U_{xx}$, to tell whether MU is increasing, constant, or decreasing.

Specifically, you have

• increasing MU if $U_{xx}>0$,
• constant MU if $U_{xx}=0$, and
• decreasing MU if $U_{xx}<0$.

In your case, assuming that $y>0$, you'd have $U_{xx}>0$, hence increasing MU.

Answer 2

Marginal utility tells you how the utility changes as you alter x. That is the first derivative, which here is a function of x. This means it is increasing. The rate of that increase is constant as long as y is fixed (second derivative).