Demand and Supply Elasticity Exercise

by Electra Radioti
Market Equilibrium

 

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

  • Demand function:

    QD=560−30P−P2Q_D = 560 – 30P – P^2

  • Supply function:

    QS=140+P+0.1P2Q_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 PP at which the price elasticity of demand is equal to -1 (unitary elasticity).
  3. (c) Explain the interpretation of unitary elasticity (Ed=−1E_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:

QD=QSQ_D = Q_S 560−30P−P2=140+P+0.1P2560 – 30P – P^2 = 140 + P + 0.1P^2

Rearrange the equation:

560−140=30P+P2+P+0.1P2560 – 140 = 30P + P^2 + P + 0.1P^2 420=31P+1.1P2420 = 31P + 1.1P^2

Rearrange into quadratic form:

1.1P2+31P−420=01.1P^2 + 31P – 420 = 0

Solve for PP using the quadratic formula:

P=−31±312−4(1.1)(−420)2(1.1)P = \frac{-31 \pm \sqrt{31^2 – 4(1.1)(-420)}}{2(1.1)} P=−31±961+18482.2P = \frac{-31 \pm \sqrt{961 + 1848}}{2.2} P=−31±28092.2P = \frac{-31 \pm \sqrt{2809}}{2.2} P=−31±532.2P = \frac{-31 \pm 53}{2.2}

Solving for PP:

P1=−31+532.2=222.2=10P_1 = \frac{-31 + 53}{2.2} = \frac{22}{2.2} = 10 P2=−31−532.2=−842.2≈−38.18(not feasible)P_2 = \frac{-31 – 53}{2.2} = \frac{-84}{2.2} \approx -38.18 \quad (\text{not feasible})

Thus, the equilibrium price is:

P∗=10P^* = 10

Substituting into the demand function to find the equilibrium quantity:

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

Thus, the equilibrium quantity is:

Q∗=160Q^* = 160


Step 2: Compute the Price Elasticity of Demand EdE_d

The price elasticity of demand is given by:

Ed=dQDdP×PQDE_d = \frac{dQ_D}{dP} \times \frac{P}{Q_D}

Differentiate the demand function:

dQDdP=−30−2P\frac{dQ_D}{dP} = -30 – 2P

At P∗=10P^* = 10:

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

Now, calculate elasticity:

Ed=(−50)×10160E_d = (-50) \times \frac{10}{160} Ed=−50×0.0625=−3.125E_d = -50 \times 0.0625 = -3.125

Interpretation:
Since Ed<−1E_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 EsE_s

The price elasticity of supply is given by:

Es=dQSdP×PQSE_s = \frac{dQ_S}{dP} \times \frac{P}{Q_S}

Differentiate the supply function:

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

At P∗=10P^* = 10:

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

Now, calculate elasticity:

Es=3×10160E_s = 3 \times \frac{10}{160} Es=3×0.0625=0.1875E_s = 3 \times 0.0625 = 0.1875

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


(b) Find PP When Ed=−1E_d = -1

Setting the price elasticity of demand equal to -1:

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

From the demand function:

QD=560−30P−P2Q_D = 560 – 30P – P^2

Substituting:

(30+2P)×P560−30P−P2=1(30 + 2P) \times \frac{P}{560 – 30P – P^2} = 1

Rearrange:

(30+2P)P=560−30P−P2(30 + 2P)P = 560 – 30P – P^2 30P+2P2=560−30P−P230P + 2P^2 = 560 – 30P – P^2 30P+2P2+P2+30P=56030P + 2P^2 + P^2 + 30P = 560 3P2+60P−560=03P^2 + 60P – 560 = 0

Solve using the quadratic formula:

P=−60±602−4(3)(−560)2(3)P = \frac{-60 \pm \sqrt{60^2 – 4(3)(-560)}}{2(3)} P=−60±3600+67206P = \frac{-60 \pm \sqrt{3600 + 6720}}{6} P=−60±103206P = \frac{-60 \pm \sqrt{10320}}{6}

Approximating:

P=−60±101.66P = \frac{-60 \pm 101.6}{6} P1=−60+101.66=41.66≈6.93P_1 = \frac{-60 + 101.6}{6} = \frac{41.6}{6} \approx 6.93 P2=−60−101.66≈−27(not feasible)P_2 = \frac{-60 – 101.6}{6} \approx -27 \quad (\text{not feasible})

Thus, P≈6.93P \approx 6.93 when Ed=−1E_d = -1.


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

The total consumer expenditure is:

TE=P×QTE = P \times Q

  • When Ed=−1E_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∗=10P^* = 10, Q∗=160Q^* = 160.
    • Ed=−3.125E_d = -3.125 (Elastic demand).
    • Es=0.1875E_s = 0.1875 (Inelastic supply).
  2. Unitary Elasticity at PP:
    • P≈6.93P \approx 6.93.
  3. Interpretation of Unit Elasticity:
    • When Ed=−1E_d = -1, total consumer expenditure remains constant despite price changes.

 

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