Find the consumers’ surplus if the demand function for a particular beverage is given by D(q)=(8000)/(4q+1)^3 and if the supply and demand are in equilibrium at q=4
To find the consumers' surplus, we first need to find the equilibrium price at q=4. This can be done by finding the price corresponding to q=4 in the demand function.
Given demand function D(q) = (8000)/(4q+1)^3
At q=4,
D(4) = (8000)/(4(4)+1)^3
= (8000)/(16+1)^3
= (8000)/(17)^3
= 31.54
So, the equilibrium price at q=4 is $31.54.
Next, we calculate the area of the triangle formed by the demand curve and the equilibrium price line at q=4. This area represents the consumers' surplus.
Consumers' surplus = (1/2) * base * height
Consumers' surplus = (1/2) * 4 * (31.54 - 0)
Consumers' surplus = 2 * 31.54
Consumers' surplus = $63.08
Therefore, the consumers' surplus for this beverage is $63.08.