When the price of oranges increases from $1.00 per pound to $1.50 per pound and quantity demanded falls from 500 pounds to 400 pounds, what is the price elasticity of demand, what happens to total revenue, and what is the income elasticity if income rises from $3,000 to $3,300 and quantity demanded rises from 400 to 450?

What you are being asked to measure

You are comparing how responsive quantity demanded is to (1) a change in the price of oranges, and (2) a change in consumer income. The clean way to do this is the midpoint (arc elasticity) method, because it avoids getting different answers depending on which starting point you pick.

Price elasticity of demand (PED) for oranges

Given:

• $P_1 = 1.00$, $P_2 = 1.50$
• $Q_1 = 500$, $Q_2 = 400$

Midpoint percent changes:

$$ \%\Delta Q = \frac{Q_2 - Q_1}{\frac{Q_1 + Q_2}{2}} = \frac{400-500}{\frac{500+400}{2}} = \frac{-100}{450} \approx -0.2222 $$

$$ \%\Delta P = \frac{P_2 - P_1}{\frac{P_1 + P_2}{2}} = \frac{1.50-1.00}{\frac{1.50+1.00}{2}} = \frac{0.50}{1.25} = 0.4 $$

Elasticity:

$$ PED = \frac{\%\Delta Q}{\%\Delta P} = \frac{-0.2222}{0.4} \approx -0.5556 $$

Interpretation: $|PED| \approx 0.56 < 1$, so demand is inelastic. Quantity demanded falls, but proportionally less than price rises.

Total revenue check (ties directly to elasticity)

Total revenue is $TR = P \times Q$.

Before the price change: $$ TR_1 = 1.00 \times 500 = 500 $$ After the price change: $$ TR_2 = 1.50 \times 400 = 600 $$

So total revenue increases from $\$500$ to $\$600$. When demand is inelastic, raising price tends to raise total revenue because the drop in $Q$ is relatively small.

Income elasticity of demand (YED)

Now income changes and quantity changes (using the new quantity level range provided).

Given:

• $Y_1 = 3000$, $Y_2 = 3300$
• $Q_1 = 400$, $Q_2 = 450$

Midpoint percent changes:

$$ \%\Delta Q = \frac{450-400}{\frac{400+450}{2}} = \frac{50}{425} \approx 0.11765 $$

$$ \%\Delta Y = \frac{3300-3000}{\frac{3000+3300}{2}} = \frac{300}{3150} \approx 0.09524 $$

Income elasticity:

$$ YED = \frac{\%\Delta Q}{\%\Delta Y} = \frac{0.11765}{0.09524} \approx 1.24 $$

Interpretation: $YED > 0$, so oranges are a normal good. Since $YED \approx 1.24 > 1$, demand is income-elastic over this range (quantity responds more than proportionally to income).

Quick summary of what each number means

• $PED \approx -0.56$: price up 1%, quantity down about 0.56%.
• $TR$ rises $500 \to 600$: consistent with inelastic demand.
• $YED \approx 1.24$: oranges are normal, and income-elastic here.