# If one always spends the same amount, what's one's price elasticity of demand?

Source: p 109, Question 5.9, Principles of Microeconomics, 7 Ed, 2014, by N Gregory Mankiw
= Question 5.7, Principles of Microeconomics, 4 Ed, 2008, by N Gregory Mankiw

9. $\color{green } { \text { Before looking at the price,} }$ Jessie spends $d$ dollars on something (abbreviate this S). What is Jessie’s price elasticity of demand?

Given Answer: Jessie's price elasticity of demand is one, because he spends the same amount on S, no matter what the price, which means $\color{darkred} { \text { his percentage change in quantity } }$ is equal to the percentage change in price.

From p 91: Price elasticity of demand $= \dfrac{ \text{ Percentage change in quantity demanded }} { \text{ Percentage change in price } }$

The green implies Jessie's ignorance of prices, so % change in price = 0. Yet please explain the answer? Especially the red, because the question implies nothing about the % change in quantity demanded?

Footnote: I slightly generalised the question. Mankiw writes 'gas' instead of 'something'.

• The textbook question implies everything about what happens to the quantity, and nothing about what happens to price. Just imagine Jessie spending $d$ today when the price is $p$, and also, per assumptions, $d$ tomorrow, when the price has risen to , say, $bp$. Then you can arrive at a conclusion about quantities demanded, and calculate the price elasticity of demand (and post the lot as an answer to your question). – Alecos Papadopoulos Mar 3 '15 at 3:12
• possible duplicate of Intuition: Difference in price elasticities of demand due to different bases – FooBar Mar 3 '15 at 15:32

The intuition is that if Jessie spends the same amount on $S$ no matter the price $p$, then a % change in price of price will reduce the demand by the same % . The thing that might confuse you is that the formula you are given is an approximation for the true price elasticity of demand given by \begin{align} \eta=\frac{dS\Big/ S}{dp\Big/ p} \end{align} where $d$ is the derivative operator. The approximation \begin{align} \frac{\%\Delta S}{\% \Delta p} \end{align} only works for very small changes in $p$ and $S$. To see why the elasticity of demand is $\mid 1 \mid$, let $b$ be the fixed amount spend on $S$. Then the quantity demanded $S$ is given by \begin{align} S=\frac{b}{p} \end{align}

Assume that there is a small price change from $p$ to $p+\epsilon, \, \epsilon>0$. Then we can define $S_1=\frac{b}{p}$ as the demand before the price change and $S_2=\frac{b}{p+\epsilon}$ as the demand after the price change. Thereby we see that \begin{align} \frac{S_2\Big/ S_1 -1}{\frac{p+\epsilon}{p}-1} = \frac{\frac{b}{p+\epsilon}\Big/\frac{b}{p}-1}{\frac{p+\epsilon}{p}-1}=\frac{\frac{-\epsilon}{p+\epsilon}}{\frac{e}{p}}=-\frac{p}{p+\epsilon}\approx -1 \end{align} when $\epsilon$ is small enough.

• I wouldn't call the point elasticity of demand the "true" one. It is just the point elasticity of demand. One can obtain the exact result of unitary elasticity for non infinitesimal changes by using the arc elasticity. – Alecos Papadopoulos Mar 7 '15 at 3:19
• Okay. Maybe "true" should be replaced. My point was just to make clear that the percentage ratio is an approximation. – Rud Faden Mar 7 '15 at 5:59
• Hum, Rud, the quantity we're interested in is a limit, not an approximation. The limit is $1$, not about one or approximately one. – VicAche May 25 '15 at 16:17
• The point is that the percentage ratio is an approximation for the elasticity and it only works for small changes. – Rud Faden May 25 '15 at 16:37

The Arc Elasticity is the one that will work using the percentage formula (see also this post)

$$\eta_{arc} = \frac {q_b-q_a}{q_b+q_a}\cdot \frac{p_b+p_a}{p_b-p_a} \tag{1}$$

Quantity demanded is $$S = E/p$$

where $E$ is total expenditure.

Assume that price increases from $p_a$ to $p_b = (1+c)p_a$, where $c$ is not "small". The second quotient in the Arc Elasticity formula is

$$\frac{p_b+p_a}{p_b-p_a} = \frac{(1+c)p_a+p_a}{(1+c)p_a-p_a} = \frac {2+c}{c} \tag{2}$$

Total expenditure remains the same, by assumption. So for the quotient involving quantities we have

$$q_b-q_a = \frac {E}{(1+c)p_a} - \frac {E}{p_a} = \frac {E - (1+c)E}{(1+c)p_a} = \frac {-c}{1+c}\cdot \frac {E}{p} \tag{3}$$

and

$$q_b+q_a = \frac {E}{(1+c)p_a} + \frac {E}{p_a} = \frac {E + (1+c)E}{(1+c)p_a} = \frac {2+c}{1+c}\cdot \frac {E}{p} \tag{4}$$

So

$$\frac {q_b-q_a}{q_b+q_a} = \frac{\frac {-c}{1+c}\cdot \frac {E}{p}}{\frac {2+c}{1+c}\cdot \frac {E}{p}} = \frac {-c}{2+c} \tag {5}$$

Combining $(2)$ and $(5)$ we get

$$\eta_{arc} = \frac {-c}{2+c} \cdot \frac {2+c}{c} = -1$$