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## Microeconomics Question from Walter E. Williams:Edit

Assume that the cost of a unit of search is the same for all goods and all have the same price variability in the absence of search. (a) Show that if the cost of search is proportional to income, the rich will search more and thus pay less for goods with income elasticities greater than unity and search less and pay more for goods with income elasticities less than unity. (b) On the basis of these results, do you expect the poor to pay more for good[s]? Are there policy implications in your findings which may increase the welfare of the poor?

The fundamental principle here is that one should cease to search when the gains from additional search are equal to the costs of additional search.

Let $s$ represent the units of search, $I$ represent income, $P$ the price in the absence of search, $p$ the price paid post-search, and $X$ the amount of a certain good. Then, an individual $i$ will set this equality:

$s_i(kI_i)=(P-p_i)X_i$

where $k$ is some constant between 0 and 1.

Let's assume that all individuals are the same except for income, and consider purchases of a good X by a rich individual (indicated by a subscript $r$) and a poor individual (a subscript $p$). If the good's income elasticity is greater than unity, then we have the arc elasticity:

$\frac{\frac{x_r-x_p}{(x_r+x_p)/2}}{\frac{I_r-I_p}{(I_r+I_p)/2}}>1$

which reduces to:

$\frac{x_r}{I_r}>\frac{x_p}{I_p}$.

The poor and rich individuals will each set the cost of search equal to the expected gains from search:

$s_p(kI_p)=(P-p_p)X_p$

$s_r(kI_r)=(P-p_r)X_r$

Combining, we have:

$\frac{(P-p_p)X_p}{s_pI_p}=\frac{(P-p_r)X_r}{s_rI_r}$

And since all terms are positive, we know that

$\frac{P-p_p}{s_p}>\frac{P-p_r}{s_r}$.

In other words, the average amount saved per unit of search is higher for the poor individual than for the rich individual. Given strictly diminishing returns to search, this implies that the rich individual searches more.

The solution for the situation where income elasticity is less than unity is similar.

Goods with income elasticity above one are most often considered luxury goods, while goods with income elasticity below one are other normal goods (often called "necessities") or inferior goods. Relative to their rich brethren, the poor will spend more time in search of savings on, and will ultimately spend less on, necessities. This seems to be a reasonable prediction, or even a desirable outcome.

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