If 3000 twobedroom houses are demanded per year at a price o

If 3000 two-bedroom houses are demanded per year, at a price of $500,000 per unit. However, for every $40,000 increase in the price per unit, the quantity of units demanded falls by 200. Contractors are unwilling to build and sell any two-bedroom houses for any price less than $100,000 but are willing to sell as much as 7700 unit per year at a price of $1 million. If the demand and supply curves are assumed to be linear.

(i) Find the demand curve.

(ii) Find the supply curve.

(iii) Find the approximate equilibrium quantity and price.

Solution

We are given that both the demand and the supply functions are linear.

Let the demand function be y = ax + b where x is the price and y, the demand.

Since the demand is 3000 units when the price is $ 50000, we have 500000a + b = 3000 (1)

When the price increases by $ 4000, the demand falls by 200 to 2800. Therefore, 540000a + b = 2800...(2)

On subtracting the 1st equation from the 2nd equation, we get 40000a = - 200 or, a = -200/40000 = -1/200. Then from the 1st equation, we have b = 3000 - 500000a = 3000 + 500000/200 = 3000 + 2500 = 5500. Thus, the demand function is y =( -1/200) x + 5500.... (3)

Let the supply function be y = cx + d, where x is the price and y is the supply of 2 bedroom units.

Since there is a supply of 7700 units at $ 1000000, we have 1000000c + d = 7700 ....(4)

Further, since there is no supply at $ 100000, we have 100000c + d = 0 ....(5)

On subtracting the 5th equation from the 4th equation, we have 900000c = 7700. Therefore, c = 7700/900000 = 77/9000. Now, from the 5th equation, we have d = - 100000c = - 100000* 77/9000 = - 7700/9 . Thus the equation of the supply function is y = cx + d or, y = (77/9000)x - 7700/9 ...(6)

The equilibrium occurs when supply is equal to the demand. Then, we have   ( -1/200) x + 5500 = (77/9000)x - 7700/9 or, ( 77/9000) x + (1/200)x = 55500 + 7700/9 or, 244x/18000 = 56500/9 or, x = ( 56500/9) *18000/244) = $ 463114.75 Then the quantity of equilibrium ( from the 3rd equation) is ( -1/200) x + 5500 =   ( -1/200)* 463114.75 + 5500 = - 2315.57+ 5500 = 3184.43 = 3184 ( on rounding off). Thus the equilibrium price and quantity are $ 463114.75 and 3184 units respectively.