In microeconomics, the elasticity of demand refers to the measure of how sensitive the demand for a good is to shifts in other economic variables. In practice, elasticity is particularly important in modeling the potential change in demand due to factors like changes in the good's price. Despite its importance, it is one of the most misunderstood concepts. To get a better grasp on the elasticity of demand in practice, let's take a look at a practice problem.

Before trying to tackle this question, you'll want to refer to the following introductory articles to ensure your understanding of the underlying concepts: a beginner's guide to elasticity and using calculus to calculate elasticities.

## Elasticity Practice Problem

This practice problem has three parts: a, b, and c. Let's read through the prompt and questions.

**Q:** The weekly demand function for butter in the province of Quebec is Qd = 20000 - 500Px + 25M + 250Py, where Qd is quantity in kilograms purchased per week, P is price per kg in dollars, M is the average annual income of a Quebec consumer in thousands of dollar, and Py is the price of a kg of margarine. Assume that M = 20, Py = $2, and the weekly supply function is such that the equilibrium price of one kilogram of butter is $14.

**a.** Calculate the cross-price elasticity of the demand for butter (i.e. in response to changes in the price of margarine) at the equilibrium. What does this number mean? Is the sign important?

**b.** Calculate the income elasticity of demand for butter at the equilibrium.

**c.** Calculate the price elasticity of demand for butter at the equilibrium. What can we say about the demand for butter at this price-point? What significance does this fact hold for suppliers of butter?

## Gathering the Information and Solving for Q

Whenever I work on a question such as the one above, I first like to tabulate all of the relevant information at my disposal. From the question we know that:

M = 20 (in thousands)

Py = 2

Px = 14

Q = 20000 - 500*Px + 25*M + 250*Py

With this information, we can substitute and calculate for Q:

Q = 20000 - 500*Px + 25*M + 250*Py

Q = 20000 - 500*14 + 25*20 + 250*2

Q = 20000 - 7000 + 500 + 500

Q = 14000

Having solved for Q, we can now add this information to our table:

M = 20 (in thousands)

Py = 2

Px = 14

Q = 14000

Q = 20000 - 500*Px + 25*M + 250*Py

Next, we'll answer a practice problem.

## Elasticity Practice Problem: Part A Explained

a. Calculate the cross-price elasticity of the demand for butter (i.e. in response to changes in the price of margarine) at the equilibrium. What does this number mean? Is the sign important?

So far, we know that:

M = 20 (in thousands)

Py = 2

Px = 14

Q = 14000

Q = 20000 - 500*Px + 25*M + 250*Py

After reading using calculus to calculate cross-price elasticity of demand, we see that we can calculate any elasticity by the formula:

## Elasticity of Z With Respect to Y = (dZ / dY)*(Y/Z)

In the case of cross-price elasticity of demand, we are interested in the elasticity of quantity demand with respect to the other firm's price P'. Thus we can use the following equation:

Cross-price elasticity of demand = (dQ / dPy)*(Py/Q)

In order to use this equation, we must have quantity alone on the left-hand side, and the right-hand side is some function of the other firm's price. That is the case in our demand equation of Q = 20000 - 500*Px + 25*M + 250*Py.

Thus we differentiate with respect to P' and get:

dQ/dPy = 250

So we substitute dQ/dPy = 250 and Q = 20000 - 500*Px + 25*M + 250*Py into our cross-price elasticity of demand equation:

Cross-price elasticity of demand = (dQ / dPy)*(Py/Q)

Cross-price elasticity of demand = (250*Py)/(20000 - 500*Px + 25*M + 250*Py)

We're interested in finding what the cross-price elasticity of demand is at M = 20, Py = 2, Px = 14, so we substitute these into our cross-price elasticity of demand equation:

Cross-price elasticity of demand = (250*Py)/(20000 - 500*Px + 25*M + 250*Py)

Cross-price elasticity of demand = (250*2)/(14000)

Cross-price elasticity of demand = 500/14000

Cross-price elasticity of demand = 0.0357

Thus our cross-price elasticity of demand is 0.0357. Since it is greater than 0, we say that goods are substitutes (if it were negative, then the goods would be complements). The number indicates that when the price of margarine goes up 1%, the demand for butter goes up around 0.0357%.

We'll answer part b of the practice problem on the next page.

## Elasticity Practice Problem: Part B Explained

b. Calculate the income elasticity of demand for butter at the equilibrium.

We know that:

M = 20 (in thousands)

Py = 2

Px = 14

Q = 14000

Q = 20000 - 500*Px + 25*M + 250*Py

After reading using calculus to calculate income elasticity of demand, we see that (using M for income rather than I as in the original article), we can calculate any elasticity by the formula:

## Elasticity of Z With Respect to Y = (dZ / dY)*(Y/Z)

In the case of income elasticity of demand, we are interested in the elasticity of quantity demand with respect to income. Thus we can use the following equation:

## Price Elasticity of Income: = (dQ / dM)*(M/Q)

In order to use this equation, we must have quantity alone on the left-hand side, and the right-hand side is some function of income. That is the case in our demand equation of Q = 20000 - 500*Px + 25*M + 250*Py. Thus we differentiate with respect to M and get:

## dQ/dM = 25

So we substitute dQ/dM = 25 and Q = 20000 - 500*Px + 25*M + 250*Py into our price elasticity of income equation:

Income elasticity of demand: = (dQ / dM)*(M/Q)

Income elasticity of demand: = (25)*(20/14000)

Income elasticity of demand: = 0.0357

Thus our income elasticity of demand is 0.0357. Since it is greater than 0, we say that goods are substitutes.

Next, we'll answer part c of the practice problem on the last page.

## Elasticity Practice Problem: Part C Explained

c. Calculate the price elasticity of demand for butter at the equilibrium. What can we say about the demand for butter at this price-point? What significance does this fact hold for suppliers of butter?

We know that:

M = 20 (in thousands)

Py = 2

Px = 14

Q = 14000

Q = 20000 - 500*Px + 25*M + 250*Py

Once again, from reading using calculus to calculate price elasticity of demand, we know that we can calculate any elasticity by the formula:

## Elasticity of Z With Respect to Y = (dZ / dY)*(Y/Z)

In the case of price elasticity of demand, we are interested in the elasticity of quantity demand with respect to price. Thus we can use the following equation:

Price elasticity of demand: = (dQ / dPx)*(Px/Q)

Once again, in order to use this equation, we must have quantity alone on the left-hand side, and the right-hand side is some function of price. That is still the case in our demand equation of 20000 - 500*Px + 25*M + 250*Py. Thus we differentiate with respect to P and get:

dQ/dPx = -500

So we substitute dQ/dP = -500, Px=14, and Q = 20000 - 500*Px + 25*M + 250*Py into our price elasticity of demand equation:

Price elasticity of demand: = (dQ / dPx)*(Px/Q)

Price elasticity of demand: = (-500)*(14/20000 - 500*Px + 25*M + 250*Py)

Price elasticity of demand: = (-500*14)/14000

Price elasticity of demand: = (-7000)/14000

Price elasticity of demand: = -0.5

Thus our price elasticity of demand is -0.5.

Since it is less than 1 in absolute terms, we say that demand is price inelastic, which means that consumers are not very sensitive to price changes, so a price hike will lead to increased revenue for the industry.