Respuesta :
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. [tex]p_1=700[/tex] and [tex]x_1=90[/tex]
If the unit price increases to $900, the quantity demanded decreases by 50 pounds.
i.e. [tex]p_2=900[/tex] and [tex]x_2=90-50=40[/tex]
The relation between the price and demand is given by,
[tex]\frac{x-x_1}{p-p_1}=\frac{x_2-x_1}{p_2-p_1}[/tex]
Substitute the values,
[tex]\frac{x-90}{p-700}=\frac{40-90}{900-700}[/tex]
[tex]\frac{x-90}{p-700}=\frac{-50}{200}[/tex]
[tex]\frac{x-90}{p-700}=\frac{-1}{4}[/tex]
Cross multiply,
[tex]4(x-90)=-1(p-700)[/tex]
[tex]4x-360=-p+700[/tex]
[tex]p=700+360-4x[/tex]
[tex]p=1060-4x[/tex]
1) The demand equation is [tex]p=1060-4x[/tex]
2) No consumer will buy commodity i.e. x=0
Substitute in the demand function,
[tex]p=1060-4(0)[/tex]
[tex]p=1060[/tex]
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,
[tex]0=1060-4x[/tex]
[tex]4x=1060[/tex]
[tex]x=\frac{1060}{4}[/tex]
[tex]x=265[/tex]
So, 265 pounds of this commodity would consumers take if it was free.
