Question

Introduction to Numerical Solutions for Equilibrium in Demand and Supply Modes Market for Housing $120,000 0, $100,000 10000,$10000p $100,000 $80,000 $60,000 0, $50000 Demand $40,000 pply $20,000 $0 0 10000, $0 10000 6000 8000 4000 2000 12000 Quantity of HOMES Using the mathematical concept of slope-intercept form of a line find the equilibrium price and Quantity of homes in this market. RISE m = slope= RUN AY y mx+b Slope-intercept form: b-yintercept AX For simplicity, round answer for both Equilibrium Price and Quantity of Homes to whole numbers, Given we are talking about thousands of homes rounding to whole numbers instead of decimals has negligible impact Let P Price, Hdem = Quantity demanded of homes. and Hsup = Quantity supplied of homes

Answer 1

1. The equation of the demand curve would be as $\dpi{80} y = m x + b$ or $\dpi{80} P = m H_{dem} + b$ . The slope would be as $\dpi{80} m = \frac{100000 - 0}{0 - 10000}$ or $\dpi{80} m = - 10$ . The equation would now be $\dpi{80} P = -10 H_{dem} + b$ . For demand be zero, the price is $100,000, and we have $\dpi{80} 100000 = -10 *0 + b$ or $\dpi{80} b = 100000$ . The demand equation would hence be $\dpi{80} P = -10 H_{dem} + 100000$ or $\dpi{80} 10 H_{dem} = 100000 - P$ or $\dpi{80} H_{dem} = 10000 - 0.1 P$ , which is the required demand function.

The equation of the supply curve would be as $\dpi{80} y = m x + b$ or $\dpi{80} P = m H_{sup} + b$ . The slope would be as $\dpi{80} m = \frac{100000 - 50000}{10000 - 0}$ or $\dpi{80} m = 5$ . The equation would now be $\dpi{80} P = 5 H_{sup} + b$ . For supply be zero, the price is $50,000, and we have $\dpi{80} 50000 = 5 *0 + b$ or $\dpi{80} b = 50000$ . The supply equation would hence be $\dpi{80} P = 5 H_{dem} + 50000$ or $\dpi{80} 5 H_{sup} = P - 50000$ or $\dpi{80} H_{sup} = 0.2 P - 10000$ , which is the required demand function.

The equilibrium price would be where $\dpi{80} H_{dem} = H_{sup}$ or $\dpi{80} 10000 - 0.1 P = 0.2 P - 10000$ or $\dpi{80} 20000 = 0.3 P$ or $\dpi{80} P = 66667$ dollars, and for $\dpi{80} H_{dem} = 10000 - 0.1 P$ , we have $\dpi{80} H^* = 10000 - 0.1*(200000/3)$ or $\dpi{80} H^* = 3333$ houses. These are the required market equilibrium price and quantity.

2. The equilibrium price would be $\dpi{80} Q_{DEM} = Q_{SUP}$ or $\dpi{80} - 4 P + 400 = 4 P - 40$ or $\dpi{80} 8 P = 440$ or $\dpi{80} P = 55$ dollars and since $\dpi{80} Q_{DEM} = - 4 P + 400$ , we have $\dpi{80} Q^* = - 4 P^* + 400$ or $\dpi{80} Q^* = - 4 *55 + 400$ or $\dpi{80} Q^* = 180$ units.

The equilibrium price is 55 (dollars) and equilibrium quantity is 180 (units).