# Introduction At the same time, if good A

Introduction

The Law
of Demand is a fundamental principle of Economics, which states that as the
price of a product falls, the quantity demanded of the product will usually
increase, all other things remaining unchanged1. The change in the
price of a product results in a change in its quantity demanded due to two main
reasons, known as “price determinants of demand”: the income effect and the
substitution effect1. Having learned about the income effect and the
substitution effect in my Economics class during our Microeconomics unit, I
became interested in exploring their roles in determining the change in the
quantity demanded of a good from a change in relative price from a mathematical
perspective.

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Definitions and
Concepts

As
mentioned above, when the price of a good changes, two effects occur: the
income effect and the substitution effect. If the price of good A were to
decrease, for example, you would have to give up less of good B to purchase
good A. Therefore, the rate at which you can replace good B for good A changes,
which is known as the change of relative price. The change in quantity demanded
of good B for good A caused by this is the substitution
effect2. At the same time, if good A becomes cheaper, your
income will buy you more of good A. Therefore, the purchasing power of your
money increases, while your total income remains the same. You will want to
purchase more of good A. This is known as the income effect2.

To observe these two
effects in isolation, we will first change the price of good A relative to good
B and then adjust income to hold purchasing power constant and observe what
happens in the graph below. Then, we will change the purchasing power while
holding the relative price of good A to good B constant.

When
the price of good A decreases, the original
budget line, which represents all of the combinations of goods A and B that
a consumer can afford to buy with their income3, pivots around the
original choice. The original choice
is the combination of goods A and B where the consumer would choose to maximize
their satisfaction, or utility, for
their given income2. The pivoted budget line will then shift out to
keep the cost of all the new combinations the same as before the price change. The
final choice, is where the consumer now
maximizes their utility2.

The
pivot represents the change in the relative price of good A to good B while
purchasing power remains constant: the substitution effect2. The
shift represents the change in purchasing power while the relative price of
good A to good B remains constant: the income effect2.

Since
the original choice lies on the pivoted budget line, it is just affordable2.
The purchasing power has remained constant, since the original choice is just
affordable at the new pivoted line2. But a new trade-off of relative
prices exists, which drives the substitution effect.

We
need to calculate how much we have to adjust income in order to keep the old choice
just affordable.

m is the
initial amount of income that makes the original choice just affordable.

m’ is the
amount of income that will make the original choice just affordable.

pa is the
original price of good A.

p’a is the
new price of good A.

pb is the price of good B.

So,
the change in money income necessary to make the old choice affordable at the
new prices is just the original amount of consumption of good A times the
change in price2. Therefore, when a price decreases, a consumer’s
purchasing power goes up, the consumer’s income will need to be decreased in
order to keep purchasing power fixed2. The opposite will be true
when a price increases.

Although
the original choice is still affordable, it is no longer the optimal purchase
at the pivoted budget line. The optimal purchase on the pivoted budget line is
represented by the pivoted choice2.
In order to determine
the substitution effect, we must use the consumer’s demand to calculate the
optimal choices at (p’a, m’) and (pa, m).

All
we have to remember is to treat y (the other variable) in the first case like a
constant, and x in the second case as a constant, and apply the rules for
ordinary differentiation4. Knowing how to calculate partial derivatives
will be useful when we calculate the slope of the final indifference curve to
determine its equation, given a final budget line in the utility maximization
exercise that follows.

When
the price of good A changes, it will lead to a change in the quantity demanded
of good B. Specifically, an increase in the price of good A will cause an
increase in the quantity demanded of good B, and a reduction in the quantity
demanded of good A. This is very different than the Cobb-Douglas-type indifference
curve examined above, where a change in the price of good A did not impact the
quantity demanded of good B, and vice-versa. So, the shape of indifference
curves between two perfect substitutes will be a linear function, as opposed to
a curved linear function. It is important here to not confuse the linear
function representing the budget line with the linear function representing the
indifference curves.

Since
good A and good B are perfect
substitutes, this means the consumer has no preference of one good over the
other. For example, pens with dark blue ink and pens with black ink1.
Intuitively, if one is cheaper than the other, it is reasonable to expect that
the consumer will always choose the cheaper of the two. In mathematical terms,
this means that we always have a corner solution, unless the price of good A
and good B are identical, in which case, any combination of good A and good B
can logically be chosen.

Now, we look to
understand the nature of the change in quantity demanded in good A and B as the
relative price changes to determine whether the income or substitution effect
dominates, using the Slutsky identity.

Since both the budget
line and the indifference curve functions are linear, we will be looking for an
intersection point. However, this intersection point must cross the
indifference curve that provides the highest utility. Intuitively, the answer
should be all of good A or all of good B. The actual result will depend on the
slope of the budget line relative to the slope of the indifference curve. The
slope of the budget line is the relative price and the slope of the indifference
curve is the marginal rate of substitution (MRS).