Answer and explanation:
Given : At a unit price of $700, the quantity demanded of a certain commodity is 90 pounds. If the unit price increases to $900, the quantity demanded decreases by 50 pounds.
To find : 1) The demand equation ?
2) At what price are no consumers willing to buy this commodity?
3) According to the above model, how many pounds of this commodity would consumers take if it was free?
Solution :
Let 'p' is the unit price and 'x' is the quantity demanded for this commodity in pounds.
At a unit price of $700, the quantity demanded of a certain commodity is 90 pounds.
i.e.
and ![x_1=90](https://tex.z-dn.net/?f=x_1%3D90)
If the unit price increases to $900, the quantity demanded decreases by 50 pounds.
i.e.
and ![x_2=90-50=40](https://tex.z-dn.net/?f=x_2%3D90-50%3D40)
The relation between the price and demand is given by,
![\frac{x-x_1}{p-p_1}=\frac{x_2-x_1}{p_2-p_1}](https://tex.z-dn.net/?f=%5Cfrac%7Bx-x_1%7D%7Bp-p_1%7D%3D%5Cfrac%7Bx_2-x_1%7D%7Bp_2-p_1%7D)
Substitute the values,
![\frac{x-90}{p-700}=\frac{40-90}{900-700}](https://tex.z-dn.net/?f=%5Cfrac%7Bx-90%7D%7Bp-700%7D%3D%5Cfrac%7B40-90%7D%7B900-700%7D)
![\frac{x-90}{p-700}=\frac{-50}{200}](https://tex.z-dn.net/?f=%5Cfrac%7Bx-90%7D%7Bp-700%7D%3D%5Cfrac%7B-50%7D%7B200%7D)
![\frac{x-90}{p-700}=\frac{-1}{4}](https://tex.z-dn.net/?f=%5Cfrac%7Bx-90%7D%7Bp-700%7D%3D%5Cfrac%7B-1%7D%7B4%7D)
Cross multiply,
![4(x-90)=-1(p-700)](https://tex.z-dn.net/?f=4%28x-90%29%3D-1%28p-700%29)
![4x-360=-p+700](https://tex.z-dn.net/?f=4x-360%3D-p%2B700)
![p=700+360-4x](https://tex.z-dn.net/?f=p%3D700%2B360-4x)
![p=1060-4x](https://tex.z-dn.net/?f=p%3D1060-4x)
1) The demand equation is ![p=1060-4x](https://tex.z-dn.net/?f=p%3D1060-4x)
2) No consumer will buy commodity i.e. x=0
Substitute in the demand function,
![p=1060-4(0)](https://tex.z-dn.net/?f=p%3D1060-4%280%29)
![p=1060](https://tex.z-dn.net/?f=p%3D1060)
So, $1060 is the price where no consumers willing to buy this commodity.
3) If it is free means price became zero.
Substitute p=0 in the demand function,
![0=1060-4x](https://tex.z-dn.net/?f=0%3D1060-4x)
![4x=1060](https://tex.z-dn.net/?f=4x%3D1060)
![x=\frac{1060}{4}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B1060%7D%7B4%7D)
![x=265](https://tex.z-dn.net/?f=x%3D265)
So, 265 pounds of this commodity would consumers take if it was free.