I have a question
Suppose there are $2$ goods, $X$ and $Y$ where, $X$ is Good good, $Y$ is Bad good
Now Can the Utility be
1.) $U = x-y$
2.) $U = \frac{x}{y}$
3.) $U = \frac{\ln x}{y}$
They all are satisfying the simple definition of good and bad commodity
But For
1.) $U_x=1$ and $U_y=-1$
$U_{xx}=0$ and $U_{yy}=0$
So as $x$ goes up, $U_x$ is the same, Does that violate the Law of Marginal Diminishing Utility?
And What about $y$, Does that also have to hold the LMDU(Law of Marginal Diminishing Utility)?
And does $y$ violate it here?
2.) $U_x=\frac{1}{y}$ and $U_y=\frac{-x}{y^2}$
$U_{xx}=0$ and $U_{yy}=\frac{-2x}{y^3}$
Here, I have same question for $x$
And Here $U_{yy}<0$, That make LMDU hold, again does that matter for bad good?
3.) $U_x=\frac{1}{x.y}$ and $U_y=\frac{-\ln x}{y^2}$
$U_{xx}=\frac{-1}{x^2.y}$ and $U_{yy}=\frac{-2.\ln x}{y^3}$
This seems to be the ideal case, and LMDU holds for both $X$ and $Y$
The thing that cause confusion here is $U=xy$ (normal cobb-douglas)
For cobb-douglas, $U_x=y$, $U_y=x$, $U_{xx}=0=U_{yy}$
Does this even holds Law of Marginal Diminishing Utility, Of course it does I have been using it for a whole year now but then why is $U_{xx}$ and $U_{yy}$ not negative?
And I have another simple question about bad good, In any normal case where both goods are good we say MRS = $-|\frac{U_x}{U_y}|$ but what if $y$ is Bad good that does change anything
Does that change this to MRS = $|\frac{U_x}{U_y}|$ ?
I am getting really confused in these basics, Any help will be really really appreciated
In the context of ordinal utility, there is no reason for utility to obey diminishing marginal utility. Suppose $x,y\geq 0$ represent the amounts of goods 1 and 2 respectively. Then the utility function
$$U(x,y):=\sqrt{xy}$$
represents exactly the same preferences as the utility function
$$V(x,y):=x^2y^2,$$
but $U$ exhibits diminishing marginal utility in each good, while $V$ exhibits increasing marginal utility in each good.
Diminishing MRS, on the other hand, is a meaningful feature of preferences. This is because MRS reflects the maximum amount of one good an agent would be willing to trade for an additional unit of another good. Note both $U,V$ above exhibit diminishing MRS.