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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.

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    $\begingroup$ 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. $\endgroup$ – Ubiquitous Jan 19 at 8:43
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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.

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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).

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