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.
If you catch a typo or error, or just have a suggestion, please submit a note here. Thanks!

# Demand for Perfect Substitutes

We can write a generic perfect substitutes utility function as $$u(x_1,x_2) = ax_1 + bx_2$$ This will have a constant MRS of $$MRS = {MU_1 \over MU_2} = {a \over b}$$ Since the MRS is constant and the price ratio is constant, one of the following three conditions must hold:

• If the MRS is greater than the price ratio $(a/b > p_1/p_2)$, the utility-maximizing choice will be to buy only good 1
• If the MRS is less than the price ratio $(a/b < p_1/p_2)$, the utility-maximizing choice will be to buy only good 2
• If the MRS is exactly equal to the price ratio $(a/b = p_1/p_2)$, all bundles along the budget constraint will give the same amount of utility

We therefore need to express the optimal bundle as a piecewise function, to delineate what happens in each of those three cases: \begin{aligned} x_1^\star(p_1,p_2,m) &= \begin{cases} {m \over p_1} & \text{ if }{a \over b} > {p_1 \over p_2 }\\ \\ \left[0, {m \over p_1}\right] & \text{ if }{a \over b} = {p_1 \over p_2 }\\ \\ 0 & \text{ if }{a \over b} < {p_1 \over p_2 } \end{cases}\\ \\ x_2^\star(p_1,p_2,m) &= \begin{cases} 0 & \text{ if }{a \over b} > {p_1 \over p_2 }\\ \\ \left[0, {m \over p_2}\right] & \text{ if }{a \over b} = {p_1 \over p_2 }\\ \\ {m \over p_2} & \text{ if }{a \over b} < {p_1 \over p_2 } \end{cases} \end{aligned} 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 substitutes

The behavior for goods that are perfect substitutes was different than these other kinds of goods, because it’s characterized by a discontinuity: below a certain price of good 1, you’ll spend all you money on good 1; but above that price, you’ll spend none. At the exact cutoff price, you’re indifferent between all the bundles on the budget line. This results in a demand curve that “jumps” at a single price from $x_1 = 0$ to $x_1 = m/p_1$:

To see how this works, check the box marked “Show $m/p_1$.” A dotted curve will appear showing this relationship. All along this curve, the consumer is spending all her money on good 1. Of course, along the vertical axis, she’s spending none of her money on good 1. The horizontal portion of her demand curve occurs at the price of good 1 such that $MRS = p_1/p_2$; that is, when $p_1 = MRS \times p_2$.

Try changing the price of good 2, and see how this affects the demand curve. Changing $p_2$ means this cutoff price shifts: the higher the price of good 2, the higher the price of good 1 at which you’re indifferent between buying these two goods.

Copyright (c) Christopher Makler / econgraphs.org