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 the two equations equal to each other:
7QD + 18 = -13QS + 40
Since QD = QS at equilibrium, we can simplify this to:
7Q + 18 = -13Q + 40
Combine like terms:
20Q = 22
Solve for Q:
Q = 22/20 = 1.1
Substitute Q = 1.1 into either the demand or supply equation to find P. If we substitute into the demand equation we get:
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 on the price axis (when QD = 0) and slopes upwards, since the coefficient of QD is positive.
- The supply curve starts at P = 40 on the price axis (when QS = 0) and slopes downwards, since the coefficient of QS is negative.
- The two curves intersect at the equilibrium point (P = 25.7, Q = 1.1).
The price is on the y-axis and the quantity is on the x-axis. The intercepts are clearly labeled as described above.
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