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.

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).