Demand and Supply Elasticity Exercise

The following demand and supply functions for a product are given:

• Demand function:

  $$Q_D = 560 – 30P – P^2$$

• Supply function:

  $$Q_S = 140 + P + 0.1P^2$$

Questions

1. (a) Determine the price elasticity of demand and price elasticity of supply at the equilibrium point. Interpret the meaning of the demand elasticity.
2. (b) Find the price $P$ at which the price elasticity of demand is equal to -1 (unitary elasticity).
3. (c) Explain the interpretation of unitary elasticity ($E_d = -1$) 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:

$$Q_D = Q_S$$ $$560 – 30P – P^2 = 140 + P + 0.1P^2$$

Rearrange the equation:

$$560 – 140 = 30P + P^2 + P + 0.1P^2$$ $$420 = 31P + 1.1P^2$$

Rearrange into quadratic form:

$$1.1P^2 + 31P – 420 = 0$$

Solve for $P$ using the quadratic formula:

$$P = \frac{-31 \pm \sqrt{31^2 – 4(1.1)(-420)}}{2(1.1)}$$ $$P = \frac{-31 \pm \sqrt{961 + 1848}}{2.2}$$ $$P = \frac{-31 \pm \sqrt{2809}}{2.2}$$ $$P = \frac{-31 \pm 53}{2.2}$$

Solving for $P$:

$$P_1 = \frac{-31 + 53}{2.2} = \frac{22}{2.2} = 10$$ $$P_2 = \frac{-31 – 53}{2.2} = \frac{-84}{2.2} \approx -38.18 \quad (\text{not feasible})$$

Thus, the equilibrium price is:

$$P^* = 10$$

Substituting into the demand function to find the equilibrium quantity:

$$Q_D = 560 – 30(10) – (10)^2$$ $$Q_D = 560 – 300 – 100 = 160$$

Thus, the equilibrium quantity is:

$$Q^* = 160$$

Step 2: Compute the Price Elasticity of Demand $E_d$

The price elasticity of demand is given by:

$$E_d = \frac{dQ_D}{dP} \times \frac{P}{Q_D}$$

Differentiate the demand function:

$$\frac{dQ_D}{dP} = -30 – 2P$$

At $P^* = 10$:

$$\frac{dQ_D}{dP} = -30 – 2(10) = -50$$

Now, calculate elasticity:

$$E_d = (-50) \times \frac{10}{160}$$ $$E_d = -50 \times 0.0625 = -3.125$$

Interpretation:
Since $E_d < -1$, 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 $E_s$

The price elasticity of supply is given by:

$$E_s = \frac{dQ_S}{dP} \times \frac{P}{Q_S}$$

Differentiate the supply function:

$$\frac{dQ_S}{dP} = 1 + 0.2P$$

At $P^* = 10$:

$$\frac{dQ_S}{dP} = 1 + 0.2(10) = 1 + 2 = 3$$

Now, calculate elasticity:

$$E_s = 3 \times \frac{10}{160}$$ $$E_s = 3 \times 0.0625 = 0.1875$$

Interpretation:
Since $E_s < 1$, supply is inelastic, meaning that a 1% increase in price leads to a less than 1% increase in quantity supplied.

(b) Find $P$ When $E_d = -1$

Setting the price elasticity of demand equal to -1:

$$– (30 + 2P) \times \frac{P}{Q_D} = -1$$ $$(30 + 2P) \times \frac{P}{Q_D} = 1$$

From the demand function:

$$Q_D = 560 – 30P – P^2$$

Substituting:

$$(30 + 2P) \times \frac{P}{560 – 30P – P^2} = 1$$

Rearrange:

$$(30 + 2P)P = 560 – 30P – P^2$$ $$30P + 2P^2 = 560 – 30P – P^2$$ $$30P + 2P^2 + P^2 + 30P = 560$$ $$3P^2 + 60P – 560 = 0$$

Solve using the quadratic formula:

$$P = \frac{-60 \pm \sqrt{60^2 – 4(3)(-560)}}{2(3)}$$ $$P = \frac{-60 \pm \sqrt{3600 + 6720}}{6}$$ $$P = \frac{-60 \pm \sqrt{10320}}{6}$$

Approximating:

$$P = \frac{-60 \pm 101.6}{6}$$ $$P_1 = \frac{-60 + 101.6}{6} = \frac{41.6}{6} \approx 6.93$$ $$P_2 = \frac{-60 – 101.6}{6} \approx -27 \quad (\text{not feasible})$$

Thus, $P \approx 6.93$ when $E_d = -1$.

(c) Interpretation of Unit Elasticity $(E_d = -1)$ and Consumer Expenditure

The total consumer expenditure is:

$$TE = P \times Q$$

• When $E_d = -1$, 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

1. Equilibrium Price & Quantity:
   • $P^* = 10$, $Q^* = 160$.
   • $E_d = -3.125$ (Elastic demand).
   • $E_s = 0.1875$ (Inelastic supply).
2. Unitary Elasticity at $P$:
   • $P \approx 6.93$.
3. Interpretation of Unit Elasticity:
   • When $E_d = -1$, total consumer expenditure remains constant despite price changes.