# Price elasticity of demand when total expenditure is constant

I was recently given the following question: assuming that a consumer's total expenditure for a good does not change when the price of that good declines, what is the price elasticity of that consumer's demand?

Here's my initial thought process: Let their total expenditure $$E = PQ$$, where $$P$$ is the price and $$Q$$ is the quantity of that good. Suppose that $$P$$ decreases by $$20\%$$. Denote $$P^{\prime}$$ and $$Q^{\prime}$$ as the new price and quantity after the price decline. $$P^{\prime} = 0.8P$$, so $$Q^{\prime} = \frac{1}{0.8} Q$$ to make sure the final expenditure is the same. $$\frac{1}{0.8} = 1.25$$, which means quantity demanded increases by $$25\%$$. Thus, the price elasticity for his demand is relatively elastic.

However, the correct answer is that his demand is unit elastic. Could someone explain why this is true? Thank you!

The question is apparently trying to test the concept of total expenditure test. You can read about it here or here. Using the total expenditure test the demand is inelastic when total expenditure increases, unit elastic when it stays same, and elastic when it decreases.

I would particularly refer you to the graph on AmosWeb: • If $P = 25$ and $\Delta P = -0.2$, then $P^{\prime} = 25-0.2= 24.8$. Similarly, $Q^{\prime} = 20.25$. Multiplying them, $P^{\prime}Q^{\prime} = 502.2$, which is not the same as $PQ = 25 * 20$. So, even though $|e_d| = 1$, then isn't total expenditure is no longer the same?
– asdf
Oct 22 at 22:08
• Also, I calculated the percent change in $Q$ needed to offset the change in $P$, not $\Delta Q$.
– asdf
Oct 22 at 22:13
• @asdf 1. you just calculated two changes for which total expenditure is same. That alone does not tell you what elasticity is as my answer explains. 2. By definition any change whether change to offset something or not is $\Delta Q$ so your change to offset change in P is $\Delta Q$ by definition
– 1muflon1
Oct 22 at 22:16
• the change I calculated is a percent change, while $\Delta Q$ is just the change in the value of $Q$. Also, isn't elasticity just defined as the percent change in demand divided by the percent change in price? In that case, finding the percent changes for $P$ and $Q$ are sufficient to determine the elasticity of demand.
– asdf
Oct 22 at 22:21
• @asdf oh sorry, I did not noticed you were talking about percentages my bad. In any case the first part of my answer still applies to you.
– 1muflon1
Oct 22 at 22:29