4 - Economic applications

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Simultaneous Equations

Section 4 Practise questions

Question 1

Equilibrium in the market for a good occurs at the price $(P)$ at which the quantity supplied $(Q_s)$ and the quantity demanded $(Q_s)$ are equal, that is, when $Q_s=Q_d$. Find the equilibrium price and quantity for the following market: $$Q_{s} = -45 + 8P$$ $$Q_{d} = 125 - 2P$$

Market equilibrium requires $Q_s=Q_d$: $$-45 + 8P = 125 - 2P$$ $$10P = 170$$ $$\;\; P = 17$$ Substitute $P=17$ in the supply function (or, in the demand function): $$Q = -45 + 8(17)$$ $$Q = -45 + 136 \;\;$$ $$Q = 91 \qquad\qquad$$ The equilibrium price and quantity are $17$ and $91$, respectively.

Figure 6

Question 2

Suppose that the demand function is $$Q_{d} = 400 - 40P + 5I$$ Where $I$ is the average income per person.
Further, suppose that the supply function is $$Q_{s} = -400 + 50P$$ What is the market equilibrium quantity when $I=20$?

When $I = 20$ $$Q_{d} = 400 - 40P + 5(20)$$ $$Q_{d} = 500 - 40P \qquad\quad\;$$ Market equilibrium requires that the quantity demanded and the quantity supplied are equal: $$Q_{d} = Q_{s}$$ $$\;\;\; 500 - 40P = -400 + 50P$$ $$90P = 900$$ $$P = 10$$ Substitute $P=10$ in the supply equation: $$Q_{s} = -400 + 50P$$ $$\quad Q_{s} = -400 + 50(10)$$ $$Q_{s} = 100 \qquad\quad\;\;\;$$ Equilibrium quantity is: $$Q = 100$$

Question 3

Given $Y=C+I+G, C=135+0.8 Y, I=75$ and $G=30$. Find the equilibrium level of income.

$$Y = C + I + G$$ Substitute for $C$ $$Y = 135 + 0.8Y + I + G$$ Substitute for $I$ and $G$ $$Y = 135 + 0.8Y + 75 + 30$$ Solving for $Y$ $$Y - 0.8Y = 240$$ $$Y(1 - 0.8) = 240$$ $$0.2Y = 240$$ $$Y = \frac{240}{0.2} = 1200$$ Equilibrium income is $$Y = 1200$$