The elasticity of demand with respect to price for a commodity is constant and is equal to 2. Find the total revenue function, given that when the price is 1,the denand is 4
To find the total revenue function, we need to know the relationship between price and quantity demanded.
We are given that the elasticity of demand with respect to price is constant and equal to 2. This means that when the price changes by 1%, the quantity demanded changes by 2%.
Let's assume that the demand function for the commodity is given by Q = f(P), where Q is the quantity demanded and P is the price.
Since we are given that when the price is 1, the demand is 4, we can substitute these values into the demand function to get:
Q = f(1) = 4
Now, let's find the derivative of the demand function with respect to price. Since the elasticity of demand with respect to price is constant and equal to 2, we have:
Elasticity of demand = % change in quantity demanded / % change in price
2 = (dQ/Q) / (dP/P)
Simplifying, we have:
2 = dQ / Q / dP / P
Rearranging, we get:
2 = dQ / dP * P / Q
Since dQ/dP is the derivative of the demand function, and it represents the rate at which quantity demanded changes with respect to price, we can replace it with df/dP:
2 = df/dP * P / Q
Now, let's solve this equation for df/dP:
df/dP = 2Q / P
Since we know that Q = 4 when P = 1, we can substitute these values into the equation to find the value of df/dP:
df/dP = 2 * 4 / 1 = 8
Now, we have the derivative of the demand function with respect to price, df/dP = 8. To find the total revenue function, we integrate this derivative with respect to price:
R(P) = ∫(df/dP) dP
R(P) = ∫8 dP
R(P) = 8P + C
where C is the constant of integration.
In the context of total revenue, the constant of integration represents the initial revenue when the price is 0. However, since the demand function is not given for prices less than 1, we cannot determine the exact value of C.
Therefore, the total revenue function is given by R(P) = 8P + C, where C is a constant.