What is an example of a utility function where one good is inferior? Say the consumer has a standard convex, monotonic preference over Apples and Bananas. 

(Update: I'd like the preference to be as 'standard' as possible. So ideally we have diminishing MRS everywhere and we also have "more is better" everywhere.)

Say his preference can be represented by some utility function $u(A,B)$. He must satisfy some budget constraint $p_AA+p_BB=y$, where $y$ is his income. 

Then what is an example of a utility function in which $\frac{\partial A}{\partial y}&lt;0$, at least under some circumstances?

This seems to me a very simple question but briefly Googling I'm unable to find anything.
A good cannot be inferior over the entire income range.

The paper A Convenient Utility Function with Giffen Behaviour shows that for a person with utility of the form:

$$U(x,y) = \alpha_1 \ln(x-\gamma_x)- \alpha_2 \ln(\gamma_y - y) $$

X is inferior if $\gamma_x$ and $\gamma_y$ are positive, $0&lt;\alpha_1&lt;\alpha_2$, and in the domain $x&gt;\gamma_x$ and $0\leq y&lt;\gamma_y$.  

Update:
$$U(x,v) = x + \ln(v)$$
If the budget is $w$, $v^* = \min(P_x / P_V, w)$ so for $w&gt;P_x / P_V$ $v$ is inferior sticky good. Realized this is actually a zero income elasticity not a negative one so it is not inferior.

I found another funky functional form for a utility function where one good is inferior but it too has increasing marginal utility of the other good: An Inferior Good and a Novel Indifference Map 

$$U = A_1 \ln(x) + y^2 /2 $$ That function gives a crazy indifference map. 

The classic example to me of inferior goods are things like cheap food, where delicious food which is much more expensive crowds it out because there is an additional constraint (stomach capacity) which eventually binds. It should be readily possible to make an example where inferiority is a consequence of this second constraint rather than the utility function. 

Update with another example:

The paper The Case of a “Giffen Good” (Spiegel (2014)) shows that for a person with utility of the form:
$$ U = \begin{Bmatrix} \alpha X - \beta X^2 / 2 + \lambda Y + \delta Y^2 / 2 &amp; for &amp; 0\leq X\leq \alpha/\beta \\ \alpha^2 /2 \beta + \lambda Y + \delta Y^2 /2 &amp; for&amp; X &gt; \alpha/\beta\end{Bmatrix} $$
where $\alpha, \beta, \lambda,$, and $\delta$ are constant and positive values. 

But as in the above functions, this utility function has increasing MU in one good (Y). This is apparently common in Giffen settings:


  In the case of an additive utility function where the marginal
  utilities of all goods are diminishing with the consumption of the
  goods, that is, the marginal utility of income is dimin­ishing, all
  goods are normal and net-substitutes for each other. However, if for
  some good (in our case, good Y) the marginal utility is positive and
  increasing and for the other good(s) the marginal utility(ies) is
  (are) diminishing (in our case, good X), then the marginal utility of
  income is increasing. The good that exhibits increasing marginal
  utility is a luxury good, whereas the good that exhibits dimin­ishing
  marginal utility is an inferior good. These characteristics were
  proved by Liebhafsky (1969) and Silberberg (1972) and wen: used to
  develop the utility function above that illustrates the case of a
  Giffen good.