BETA
Note: These explanations are in the process of being adapted from my textbook.
I'm trying to make them each a "standalone" treatment of a concept, but there may still
be references to the narrative flow of the book that I have yet to remove.
This work is under development and has not yet been professionally edited.
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# Demand for Perfect Complements

We can write a generic perfect complements utility function as $$u(x_1,x_2) = \min\left\{ {x_1 \over a}, {x_2 \over b}\right\}$$ As we’ve argued before, the optimal bundle for this sort of utility function will occur where the minimands are equalized: that is, $${x_1 \over a} = {x_2 \over b}$$ or $$x_2 = {b \over a}x_1$$ Plugging this back into the budget constraint, we can solve for $x_1$: \begin{aligned} p_1x_1 + p_2x_2 &= m\\ p_1x_1 + p_2\left[{b \over a}x_1\right] &= m\\ ap_1x_1 + bp_2x_1 &= am\\ p_1x_1 &= {am \over ap_1 + bp_2}\\ x_1^\star(p_1,p_2) &= {am \over ap_1 + bp_2} \end{aligned} and therefore $$x_2^\star = {b \over a}x_1^\star = {bm \over ap_1 + bp_2}$$ Intuitively, one way of thinking about this is that you want to always buy $a$ units of good 1 for every $b$ units of good 2 you buy. A bundle of $a$ units of good 1 and $b$ units of good 2 costs $ap_1 + bp_2$; therefore, the maximum number of such bundles you can buy with $m$ dollars of income is $$N = \text{Max bundles of }(a, b) = {m \over ap_1 + bp_2}$$ Since each of those bundles has $a$ units of good 1, your demand for good 1 is $$x_1^\star = a \times N = a \times {m \over ap_1 + bp_2}$$ and since each bundle contains $b$ units of good 2, your demand for good 2 is $$x_2^\star = b \times N = b \times {m \over ap_1 + bp_2}$$

Try playing around with the graph below to see how $a$, $b$, $p_1$, $p_2$, and $m$ affect the optimal choice:

## Demand curves for perfect complements

We can plot the demand curve for perfect complements is constructed in the same way as other utility functions: for each budget line, find the corresponding quantity demanded, and plot quantity as a function of price:

The interesting part about this demand curve is that it has a horizontal intercept. That is, even at a price of zero, you won’t buy an infinite amount of the good. This is because you always want to consume these two goods in a specific ratio, so even if one of the goods is free, you’ll stop consuming it when you achieve that desired ratio.

This can help explain, for example, why restaurants put salt and pepper out for free on a table. Salt, in fact, used to be one of the most valuable commodities in the world, before refrigeration, because it acts as a preservative. Nowadays, you can get it for free! Why? Because even though it’s free, you’re not going to just pour it over your meal in a gleeful frenzy at all this free salt; rather, you’ll just use a sprinkle or two to adjust the taste of your meal. Because you’re only interested in salt as a complement to your other food, you don’t use an infinite amount even if it’s free.

Copyright (c) Christopher Makler / econgraphs.org