Elasticity and Revenue

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 17

Elasticity

Why Elasticity?

Notation

\epsilon_{Y,X} = \frac{\% \Delta Y}{\% \Delta X}
\text{Elasticity }(\epsilon) = \frac{\% \text{ change in endogenous variable}}{\% \text{ change in exogenous variable}}

"X elasticity of Y"
or "Elasticity of Y with respect to X"

Examples:
"Price elasticity of demand"
"Income elasticity of demand"
"Cross-price elasticity of demand"
"Price elasticity of supply"

Perfectly Inelastic

Inelastic

Unit Elastic

Elastic

Perfectly Elastic

|\epsilon| = 0
|\epsilon| < 1
|\epsilon| = 1
|\epsilon| > 1
|\epsilon| = \infty

Doesn't change

Changes by less than the change in X

Changes proportionally to the change in X

Changes by more than the change in X

Changes "infinitely" (usually: to/from zero)

How does the endogenous variable Y respond to a
change in the exogenous variable X?

\text{Elasticity }(\epsilon) = \frac{\% \text{ change in endogenous variable}}{\% \text{ change in exogenous variable}}

(note: all of these refer to the ratio of the perentage change, not absolute change)

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Using Elasticities

  • Suppose the price elasticity of demand is -2.
  • This means that each % increase in the price
    leads to approximately a 2% decrease in the quantity demanded
  • Example 1: a 3% increase in price would lead to a ~6% decrease in quantity
  • Example 2: a 0.5% decrease in price would lead to a ~1% increase in quantity
  • These are approximations in the same way as if \(dy/dx = -2\) along a function, increasing \(x\) by 3 would cause \(y\) to decrease by approximately 6.
\epsilon_{Y,X}
= \frac{\% \Delta Y}{\% \Delta X}
= \frac{\Delta Y / Y}{\Delta X / X}
= \frac{\Delta Y}{\Delta X}\times \frac{X}{Y}
\text{Suppose }Y \text{ is a function of }X.

General formula:

Linear relationship:

Using calculus:

Multiplicative relationship:

Y = mX + b
Y = f(X)
Y = aX^b
\text{General formula: }\epsilon_{Y,X} = \frac{\Delta Y}{\Delta X}\times \frac{X}{Y}
\text{If }Y = a + bX \text{, then }\frac{\Delta Y}{\Delta X} =
b
\epsilon_{Y,X} = b\times \frac{X}{a + bX}

Note: the slope of the relationship is \(b\).

Elasticity is related to, but not the same thing as, slope.

= \frac{bX}{a + bX}
\epsilon_{y,x}
= \frac{\% \Delta y}{\% \Delta x}
= \frac{\Delta y / y}{\Delta x / x}
= \frac{\Delta y}{\Delta x}\times \frac{x}{y}
\text{Suppose }y \text{ is a function of }x.
\text{In the limit, as }\Delta x \rightarrow 0:
\epsilon_{y,x} = \frac{dy}{dx}\times \frac{x}{y}
\text{General formula: }\epsilon_{y,x} = \frac{dy}{dx}\times \frac{x}{y}
\text{If }y = ax^b \text{, then }\frac{dy}{dx} =
abx^{b-1}
\epsilon_{y,x} = abx^{b-1} \times \frac{x}{ax^b}
= b

This is related to logs, in a way that you can explore in the homework.

This is a super useful trick and one that comes up on exams all the time!

Demand Elasticities

x_1^*(p_1,p_2,m)

How much of a good a consumer wants to buy, as a function of:

  • the price of that good
  • the price of other goods
  • their income

We can ask: how much does the amount of this good change, when one of those determinants changes?

Own-Price Elasticity

What is the effect of a 1% change
in the price of good 1 \((p_1)\) on the quantity demanded of good 1 \((x_1^*)\)?

no change

perfectly inelastic

less than 1%

inelastic

exactly 1%

unit elastic

more than 1%

elastic

  1. How does \(x_1^*\) change with \(p_1\)?
    • ​​Own-price elasticity
    • Elastic vs. inelastic
x_1^*(p_1,p_2,m)

Three Relationships

Cross-Price Elasticity

What is the effect of an increase
in the price of good 2 \((p_2)\) on the quantity demanded of good 1 \((x_1^*)\)?

no change

independent

decrease

complements

increase

substitutes

  1. How does \(x_1^*\) change with \(p_1\)?
    • ​​Own-price elasticity
    • Elastic vs. inelastic
  2. How does \(x_1^*\) change with \(p_2\)?
    • ​​Cross-price elasticity
    • Complements vs. substitutes
x_1^*(p_1,p_2,m)

Three Relationships

Income Elasticity

What is the effect of an increase
in income \((m)\) on the quantity demanded of good 1 \((x_1^*)\)?

decrease

good 1 is inferior

increase

good 1 is normal

  1. How does \(x_1^*\) change with \(p_1\)?
    • ​​Own-price elasticity
    • Elastic vs. inelastic
  2. How does \(x_1^*\) change with \(p_2\)?
    • ​​Cross-price elasticity
    • Complements vs. substitutes
  3. How does \(x_1^*\) change with \(m\)?
    • ​​Income elasticity
    • Normal vs. inferior goods
x_1^*(p_1,p_2,m)

Three Relationships

Revenue for a Firm

Profit

The profit from \(q\) units of output

\pi(q) = r(q) - c(q)

PROFIT

REVENUE

COST

is the revenue from selling them

minus the cost of producing them.

Revenue

We will assume that the firm sells all units of the good for the same price, \(p\). (No "price discrimination")

r(q) = p(q) \times q

The revenue from \(q\) units of output

REVENUE

PRICE

QUANTITY

is the price at which each unit it sold

times the quantity (# of units sold).

The price the firm can charge may depend on the number of units it wants to sell: inverse demand \(p(q)\)

  • Usually downward-sloping: to sell more output, they need to drop their price
  • Special case: a price taker faces a horizontal inverse demand curve;
    can sell as much output as they like at some constant price \(p(q) = p\)

Demand and Inverse Demand

Demand curve:

quantity as a function of price

Inverse demand curve:
price as a function of quantity

QUANTITY

PRICE

\text{revenue} = r(q) = p(q) \times q
\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p(q)
\displaystyle{\text{average revenue} = \frac{r(q)}{q} = p(q)}

If the firm wants to sell \(q\) units, it sells all units at the same price \(p(q)\)

Since all units are sold for \(p\), the average revenue per unit is just \(p\).

By the product rule...
let's delve into this...

Total, Average, and Marginal Revenue

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p
dr = dp \times q + dq \times p
p
p(q)
q

The total revenue is the price times quantity (area of the rectangle)

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p
dr = dp \times q + dq \times p
p
p(q)
q
dp
dq

Note: \(MR < 0\) if

dq \times p
{dq \over q} < {dp \over p}
\% \Delta q < \% \Delta p
|\epsilon| < 1
dp \times q
<

The total revenue is the price times quantity (area of the rectangle)

If the firm wants to sell \(dq\) more units, it needs to drop its price by \(dp\)

Revenue loss from lower price on existing sales of \(q\): \(dp \times q\)

Revenue gain from additional sales at \(p\): \(dq \times p\)

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p

Marginal Revenue and Elasticity

= \left[{dp \over dq} \times {q \over p}\right] \times p + p
= {p \over {dq \over dp} \times {p \over q}} + p
= \left(1- {1 \over |\epsilon_{q,p}|}\right)p

(multiply first term by \(p/p\))

(simplify)

(since \(\epsilon < 0\))

Notes

Elastic demand: \(MR > 0\)

Inelastic demand: \(MR < 0\)

In general: the more elastic demand is, the less one needs to lower ones price to sell more goods, so the closer \(MR\) is to \(p\).

Next Time

  • Bring cost and revenue together to find the profit-maximizing quantity for a firm
  • Compare results when the firm has market power (faces a downward-sloping demand curve) and when it is a price taker (faces a horizontal demand curve)