The demand and supply functions of a good are given by{ P = 7 QD + 18P = −13 QS + 40where P denotes the price, QD the quantity demanded and QS the quantitysupplied.(a) Determine the equilibrium price and quantity.(b) Sketch the demand and supply functions on the same graph. Label theaxes and the intercepts clearly.
Question
The demand and supply functions of a good are given by{ P = 7 QD + 18P = −13 QS + 40where P denotes the price, QD the quantity demanded and QS the quantitysupplied.(a) Determine the equilibrium price and quantity.(b) Sketch the demand and supply functions on the same graph. Label theaxes and the intercepts clearly.
Solution
(a) To find the equilibrium price and quantity, we need to set the quantity demanded equal to the quantity supplied. This means we set QD = QS and solve for P.
Given the equations:
P = 7QD + 18 P = -13QS + 40
Setting QD = QS gives us:
7QD + 18 = -13QD + 40
Solving for QD (or QS since they are equal at equilibrium) gives:
20QD = 22 QD = QS = 22/20 = 1.1
Substituting QD = 1.1 into either the demand or supply equation will give us the equilibrium price. Using the demand equation:
P = 7(1.1) + 18 = 25.7
So, the equilibrium price is 25.7 and the equilibrium quantity is 1.1.
(b) To sketch the demand and supply functions, you would typically use graph paper or a graphing software. Since I can't draw here, I'll describe it:
- The demand curve starts at P = 18 (when QD = 0) and slopes upwards, since the coefficient of QD is positive.
- The supply curve starts at P = 40 (when QS = 0) and slopes downwards, since the coefficient of QS is negative.
- The two curves intersect at the equilibrium point, which is (1.1, 25.7). This is where the quantity demanded equals the quantity supplied.
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