Utility function with two goods with declining spending share on one good

Question

My question is somewhat related to utility functions with inferior goods (some answers here).

Suppose there are two goods, $a, b$. I am looking for a utility function such that it satisfies the usual condition of positive but diminishing marginal utilities. However, I would want that the share of income spent on good $a$ is decreasing with income. For example, spending share on food may decline with income even though it may not be an inferior good.

Clearly, if good $a$ is an inferior good, the above requirement is satisfied (see proof below). But as the linked answers mention, such utility functions become rapidly intractable. I am hoping that since my requirement is less stringent, it may be possible to come up with somewhat easier-to-deal with utility functions.

Basically, let $a^*(p_a,p_b,m), b^*(p_a,p_b,m)$ be the demand functions derived from $U(a,b)$. What I want is:

$$\begin{align} \frac{\partial}{\partial m}\left(\frac{p_a a^*}{m}\right) &< 0 \\ \implies \frac{p_a}{m}\frac{\partial a^*}{\partial m}-p_a a^*\frac{1}{m^2} &<0 \\ \implies \frac{\partial a^*}{\partial m}&<\frac{m}{a^*} \end{align}$$

Clearly, if the LHS is non-positive (inferior or zero income effect good) the inequality is satisfied. My requirement is therefore less stringent.

I also found a function for zero income effect here. This is so far my best alternative but I do not like the fact that second good is constant marginal utility (I would prefer diminishing). Given that what I have in mind is quite realistic for some goods, there should certainly be literature on this.

Please do share the references if your answer is motivated by some paper/book.

Answer 1

For a formal treatment (if you aim at estimating some demand data), I recommend Deaton and Muellbauer (1980)'s almost ideal demand system. Also see the Wikipedia page. To make the expenditure share on some good $i$ ($w_i$) decrease in wealth $m$, simply choose $\beta_i < 0$.

However, according to your references, you might be looking for some simple examples. I constructed a simple example as the following: $$u(x,y) = \sqrt{x} + \sqrt{y+a}, \; a > 0$$ subject to $$px + y = m$$ The interior demand is $$x(p,m) = \frac{m+a}{p+p^2}, y(p,m) = \frac{p^2m - ap}{p+p^2}$$ which exists when $mp \geq a$. Although both goods are normal, the share of expenditure on $x$ decreases in $m$, i.e., $$w_x = \frac{px(p,m)}{m} = \frac{p+a\frac{p}{m}}{p+p^2}$$

On the other hand, if $mp < a$, we have a corner solution with $$x(p,m) = \frac{m}{p}, y(p,m) = 0$$ In this case, $w_x \equiv 1$ as $m$ increases. Fix $p$, increasing $m$ from $0$ to $\infty$, we see a continuous decrease of $w_x$.

Finally, it is easy to verify that the utility function exhibits diminishing marginal utility in both goods.