What is an example of a utility function where one good is inferior?

Question

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 $\frac{\partial A}{\partial y}<0$, at least under some circumstances?

This seems to me a very simple question but briefly Googling I'm unable to find anything.

Answer 1

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, $0<\alpha_1<\alpha_2$, and in the domain $x>\gamma_x$ and $0\leq y<\gamma_y$.

Update: $$U(x,v) = x + \ln(v)$$ If the budget is $w$, $v^* = \min(P_x / P_V, w)$ so for $w>P_x / P_V$ $v$ 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: $$ U = \begin{Bmatrix} \alpha X - \beta X^2 / 2 + \lambda Y + \delta Y^2 / 2 & for & 0\leq X\leq \alpha/\beta \\ \alpha^2 /2 \beta + \lambda Y + \delta Y^2 /2 & for& X > \alpha/\beta\end{Bmatrix} $$ 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.

Answer 2

Let's see what inferiority of one good in the two-good case implies. Look up Silberberg's "The Structure of Economics" (still one of the best undergraduate microeconomics textbooks ever written), ch. 10 for more details.

Utility maximization is described by (stars denote optimal levels)

$$U_A(A^*,B^*) - \lambda^*p_A \equiv 0$$ $$U_B(A^*,B^*) - \lambda^*p_B \equiv 0$$ $$y- p_AA^* - p_BB^* \equiv 0$$

and note the use of the identity symbol instead of simple equality -these relations always hold at the optimum. Then we can differentiate both sides and maintain the identity. Do that and solve the $3 \times 3$ system of equations to determine the various derivatives, and you will find that if good $A$ is inferior, $\frac {\partial A^*}{\partial y} <0$, then we must have that

$$p_AU^*_{BB}> p_BU^*_{AB}$$

If we are willing to accept $U_{BB} >0$, then the cross-partial $U_{AB}$ can be zero, and we can have a utility function as the one mentioned in @BKay 's answer.

But if we want to maintain $U_{BB} <0$, then it must be the case that $U_{AB}$, the cross-partial derivative of the utility function must also be strictly negative (and so not-zero). This in turn implies preferences that are not separable, additively or multiplicatively.

Perhaps you can consider something like

$$U(A,B) =\ln\left[aA^k + bB^h\right]$$

and all four parameters positive. For example, for values, $a=5, k=0.4, b=0.2, h=0.8$ the indifference map is

My conjecture is that for $0<h<1$ you may be able to have all the standard setup together with inferiority of $A$ (and for suitable values of prices and the other parameters of course). Find the first order conditions, substitute for $B$ in terms of $A$ in the budget constraint, and use the implicit function theorem to determine the conditions on the parameters required for $\frac {\partial A^*}{\partial y} <0$. And don't forget to check whether these conditions are compatible with the second-order conditions for utility maximization.

COMMENT October 7, 2015
Some comments in this answer appear to me to confound the issue of preference representation and preservation of preference ranking under monotonic transformations, with the "inferiority" property of a good. Preferences and their representation have nothing to do with the existence of a budget constraint. On the other hand, "inferiority" has everything to do with the existence of a budget constraint, and how it affects choices (not preferences) as it changes.

And monotonic transfomration do not leave everything "unchanged". Consider the utility function $V = A^k+ B^h$, and its monotonic transformation $U =\ln(A^k+ B^h)$. One can easily see that while $\frac {\partial^2 V}{\partial AB} = 0$, we have that $\frac {\partial^2 U}{\partial AB} \neq 0$. In other words, monotonic transformations may preserve the ranking of bundles, but this does not mean that they give the same relations between goods. And as I have written above, the property of "inferiority" depends on the signs and relative magnitudes of the second partial derivatives of the utility function used, signs and relative magnitudes that depend on the actual functional form used.

Answer 3

It is quite tricky to get tractable models with reasonable/realistic properties. A general $n$-goods case is given by Sørensen in Heijman et al. (2012), p. 100-3. Another example, for two goods and with limited domain, is given by Haagsma (2012).
Checking the references therein is the easiest way to get a substantial collection of utility functions for inferior goods - though it seems there is more literature on Giffen goods than the less demanding inferior ones.

Regarding the previous discussion on the convexity of preferences, utility functions which yield different demand functions upon a positive monotonic transformation are not quasiconcave and, hence, the preferences are not convex, given that quasiconcavity is preserved with any nondecreasing composition. That the function Alecos Papadopoulos suggested is not Cobb-Douglas should be easy to see.
Nevertheless, if it is quasiconcave, then $u(x_1,x_2)$ will yield the same demand functions (and the same price and income effects) as $v(x_1,x_2)=f(u(x_1,x_2)$ where $f$ is a positive monotonic transformation, regardless of $u$ being weakly separable or not. A caveat: caution for the effects on the domain.

Answer 4

The paper "Hunger and the Demands for Cheap Dietary Staples of the Extreme Poor" provides an example of a utility function in which one of the goods is Giffen and it can be calibrated to characterize demands in the real world (https://ssrn.com/abstract=3728187):

$$u\left( b , m \right) = \left[ b + m - \bar{c} \right]^{\beta}\left[ m \right]^{1-\beta}$$

where $$b$$ is the Giffen good (the dietary staple).

When this utility function is maximized subject to the budget constraint $$p_{b}b + p_{m}m \leqslant i $$

the following demand for the dietary staple is found:

$$b\left(p_{m},p_{b},i\right)=\frac{i\left[ \beta p_{m} - p_{b} \right]+p_{m}p_{b}\left( 1-\beta \right)\bar{c}}{p_{b}\left( p_{m}-p_{b} \right)}$$

When this demand function is calibrated to account for the elasticities that Jensen and Miller (2008) estimate in Hunan, China, the values of the parameters become $$ \beta \cong 0.06, \bar{c} \cong 4,900, p_{m} \cong 3.107, i \cong 9,046 $$.

Then, it is a simple exercise to map the staple as a function of its price using Excel. It is recommended to create a grid for the price of the staple that goes between 0.4 and 1.7 with a spacing of 0.01 in between values. That is, the column that characterizes the values of the price of the staple should be 0.40, 0.41, 0.42, ... and so on. The result is the following graph:

Notice that, in this graph, the price of the staple is in the X-axis, and the staple is Giffen when its price is above 0.75.

The indifference curves of the calibrated version of this utility function look life this: