The following demand and supply functions for a product are given:
- Demand function:
- Supply function:
Questions
- (a) Determine the price elasticity of demand and price elasticity of supply at the equilibrium point. Interpret the meaning of the demand elasticity.
- (b) Find the price at which the price elasticity of demand is equal to -1 (unitary elasticity).
- (c) Explain the interpretation of unitary elasticity () and its impact on total consumer expenditure when the price of the product increases.
Solutions
(a) Finding the Equilibrium Price and Elasticities
Step 1: Find the Equilibrium Price and Quantity
At equilibrium, demand equals supply:
Rearrange the equation:
Rearrange into quadratic form:
Solve for using the quadratic formula:
Solving for :
Thus, the equilibrium price is:
Substituting into the demand function to find the equilibrium quantity:
Thus, the equilibrium quantity is:
Step 2: Compute the Price Elasticity of Demand
The price elasticity of demand is given by:
Differentiate the demand function:
At :
Now, calculate elasticity:
Interpretation:
Since , demand is elastic, meaning that a 1% increase in price leads to a more than 1% decrease in quantity demanded.
Step 3: Compute the Price Elasticity of Supply
The price elasticity of supply is given by:
Differentiate the supply function:
At :
Now, calculate elasticity:
Interpretation:
Since , supply is inelastic, meaning that a 1% increase in price leads to a less than 1% increase in quantity supplied.
(b) Find When
Setting the price elasticity of demand equal to -1:
From the demand function:
Substituting:
Rearrange:
Solve using the quadratic formula:
Approximating:
Thus, when .
(c) Interpretation of Unit Elasticity and Consumer Expenditure
The total consumer expenditure is:
- When , total expenditure remains constant as price changes.
- If the price increases, the quantity demanded decreases proportionally, keeping total revenue stable.
- If the price decreases, the quantity demanded increases proportionally, again maintaining total revenue.
Thus, when demand is unit elastic, total expenditure does not change when price changes.
Final Answers
- Equilibrium Price & Quantity:
- , .
- (Elastic demand).
- (Inelastic supply).
- Unitary Elasticity at :
- .
- Interpretation of Unit Elasticity:
- When , total consumer expenditure remains constant despite price changes.