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

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.

• A note of caution: although $U_{xx}=0$ implies constant marginal utility, this observation is totally meaningless. We know that utility functions areinvariant to monotone transformations. So take the function $f(x)=x^2$ on domain $\mathbb{R}_+$. Then $f(U(x))$ preserves the order (because $f'(x)>0$) and is also a valid utility function for the same preferences. But $\partial^2[f(U(x))]/\partial x^2=(U_x)^2f''(U(x))+f'(U(x))U_{xx}=(U_x)^2f''(U(x))>0$. So the same preferences can be represented by both a constant MU function and an increasing MU function; it's entirely arbitrary. Jan 19 '19 at 8:43

## 2 Answers

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.

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).