If 3000 two-bedroom houses are demanded per year, at a price of $500,000 per uni

Question

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.

Explanation / Answer

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.

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