Suppose The Market Demand Function For Good X Is Given By Qx = 30 – 2P, And The Supply Function For Good X Is Given By Qx = 6 + 4P. Then, Calculate The:(2 Point)
A. Market Clearing Price And Quantity Respectively.
B. Determine Market Equilibrium
C. Price And Supply Elasticity Of Demand
Suppose the market demand function for good X is given by Qx = 30 – 2P,
and the supply function for good X is given by Qx
Suppose the market demand function for good X is given by Qx = 30 – 2P, and the supply function for good X is given by Qx = 6 + 4P. Then, calculate the:(2 point) A. Market clearing price and quantity respectively. B. Determine market equilibrium C. Price and supply elasticity of demand.
The Correct Answer and Explanation is:
Given:
- Demand Function: Q<sub>D</sub> = 30 – 2P
- Supply Function: Q<sub>S</sub> = 6 + 4P
A. Market Clearing Price and Quantity
At market equilibrium (clearing point), quantity demanded = quantity supplied: 30−2P=6+4P30 – 2P = 6 + 4P30−2P=6+4P
Solving for P: 30−6=2P+4P⇒24=6P⇒P=430 – 6 = 2P + 4P \Rightarrow 24 = 6P \Rightarrow P = 430−6=2P+4P⇒24=6P⇒P=4
Now, substitute P = 4 into either the demand or supply equation to find Q:
- Q = 30 – 2(4) = 30 – 8 = 22
✅ Answer A:
Market clearing price = 4,
Market clearing quantity = 22
B. Determine Market Equilibrium
Market equilibrium occurs where Q<sub>D</sub> = Q<sub>S</sub>, which we just found:
- At P = 4, both demand and supply equal Q = 22
✅ Answer B:
Market equilibrium is at P = 4 and Q = 22
C. Price Elasticity of Demand and Supply
Price Elasticity of Demand (E<sub>D</sub>):
Formula: ED=dQDdP×PQE_D = \frac{dQ_D}{dP} \times \frac{P}{Q}ED=dPdQD×QP
From demand: Q<sub>D</sub> = 30 – 2P → dQ<sub>D</sub>/dP = -2
At equilibrium P = 4, Q = 22: ED=(−2)×422=−822≈−0.36E_D = (-2) \times \frac{4}{22} = -\frac{8}{22} \approx -0.36ED=(−2)×224=−228≈−0.36
(Demand is inelastic in absolute value since |E<sub>D</sub>| < 1)
Price Elasticity of Supply (E<sub>S</sub>):
From supply: Q<sub>S</sub> = 6 + 4P → dQ<sub>S</sub>/dP = 4 ES=4×422=1622≈0.73E_S = 4 \times \frac{4}{22} = \frac{16}{22} \approx 0.73ES=4×224=2216≈0.73
(Supply is inelastic as well)
✅ Answer C:
- Price Elasticity of Demand ≈ -0.36
- Price Elasticity of Supply ≈ 0.73
Explanation (300 Words)
To find the market equilibrium for good X, we equate the market demand and supply functions. The demand function, QD=30−2PQ_D = 30 – 2PQD=30−2P, shows that as price increases, quantity demanded decreases. The supply function, QS=6+4PQ_S = 6 + 4PQS=6+4P, indicates that quantity supplied increases with price. Setting QD=QSQ_D = Q_SQD=QS allows us to determine the market-clearing price, which balances the interest of consumers and producers.
Solving the equation 30−2P=6+4P30 – 2P = 6 + 4P30−2P=6+4P, we find the equilibrium price P=4P = 4P=4. Substituting back into either function gives an equilibrium quantity Q=22Q = 22Q=22. This equilibrium reflects a stable point in the market where there is neither excess demand nor excess supply.
We further analyze how sensitive demand and supply are to price changes using elasticity. The price elasticity of demand measures how much quantity demanded responds to a change in price. Using the formula ED=(dQD/dP)×(P/Q)E_D = (dQ_D/dP) \times (P/Q)ED=(dQD/dP)×(P/Q), we calculate ED=−0.36E_D = -0.36ED=−0.36, indicating that demand is inelastic—consumers do not significantly reduce quantity demanded when price rises.
Similarly, we calculate the price elasticity of supply as ES=(dQS/dP)×(P/Q)=0.73E_S = (dQ_S/dP) \times (P/Q) = 0.73ES=(dQS/dP)×(P/Q)=0.73, also inelastic. This suggests that producers do not rapidly increase output when price rises. Inelastic elasticities are common in the short run due to fixed consumer habits and production limitations.
Understanding these values is crucial for policymakers and businesses when forecasting market behavior or implementing pricing strategies.
