Demand Function: Elasticity E=3, Find P=D(x)
Understanding elasticity of demand is crucial in economics and business for pricing strategies and predicting consumer behavior. In this article, we'll delve into how to determine the demand function given the elasticity of demand and a specific point on the demand curve. Specifically, we'll explore how to find the demand function p = D(x) when the elasticity E = 3 and D(64) = 36. This involves utilizing the formula for elasticity of demand, setting up a differential equation, solving it, and then using the given point to find the constant of integration. This comprehensive approach will provide a clear and concise method to solve this type of problem.
Understanding Elasticity of Demand
Elasticity of demand measures how much the quantity demanded of a good responds to a change in the price of that good. It's a crucial concept for businesses because it helps them understand how price changes will affect their revenue. The formula for price elasticity of demand (E) is given by:
E = - (dQ/dP) * (P/Q)
Where:
- E is the price elasticity of demand.
- Q is the quantity demanded.
- P is the price.
- dQ/dP is the derivative of the quantity demanded with respect to price.
The negative sign is included to ensure that the elasticity is usually a positive number, as demand typically decreases when price increases.
In our specific problem, we are given that E = 3. This means that a 1% change in price will lead to a 3% change in quantity demanded, in the opposite direction. We also have the information that D(64) = 36, which tells us that when the quantity demanded is 64, the price is 36. Our goal is to find the function p = D(x) that describes the relationship between price and quantity demanded.
Setting Up the Differential Equation
Given the elasticity of demand E = 3, we can set up a differential equation to relate the price p and quantity x. We know that:
3 = - (dx/dp) * (p/x)
To solve for the demand function p = D(x), we need to rearrange this equation and integrate. First, let's rewrite the equation as:
3 = - (dx/dp) * (p/x)
Multiply both sides by -1:
-3 = (dx/dp) * (p/x)
Now, we can separate the variables. Multiply both sides by dp and by x:
-3x dx = p dp
This separates the variables, with x on one side and p on the other, which allows us to integrate both sides.
Solving the Differential Equation
Now that we have separated the variables, we can integrate both sides of the equation:
∫ -3x dx = ∫ p dp
Integrating both sides, we get:
-3 * (x^2 / 2) = (p^2 / 2) + C
Where C is the constant of integration. We can multiply both sides by 2 to simplify the equation:
-3x^2 = p^2 + 2C
Let's replace 2C with a new constant K:
-3x^2 = p^2 + K
Now, we want to solve for p^2:
p^2 = -3x^2 - K
To find p, we take the square root of both sides:
p = √(-3x^2 - K)
However, since price p must be a real number and usually positive in the context of demand, we need to consider the constant K carefully. Also, note that the initial equation seems to have an error because it leads to a negative value inside the square root for positive x. The correct setup should reflect that elasticity is usually expressed as a positive value, and we need to adjust for the inverse relationship between price and quantity. Let's correct the setup.
Given E = 3, the correct equation should be:
E = - (dx/dp) * (p/x)
3 = - (dx/dp) * (p/x)
-3 (dx/x) = dp/p
Integrating both sides:
∫ -3 (dx/x) = ∫ dp/p
-3 ln|x| = ln|p| + C
Exponentiate both sides:
e^(-3 ln|x|) = e^(ln|p| + C)
e^(-3 ln|x|) = e^(ln|p|) * e^C
(x^(-3)) = p * e^C
Let A = e^(-C)
p = A / x^3
Finding the Constant of Integration
We are given that D(64) = 36. This means when x = 64, p = 36. We can use this information to find the value of the constant A.
36 = A / (64^3)
Now, solve for A:
A = 36 * (64^3)
A = 36 * 262144
A = 9437184
So, the demand function is:
p = 9437184 / x^3
The Demand Function
Therefore, the demand function p = D(x) is:
p = 9437184 / x^3
This function describes the relationship between the price p and the quantity demanded x, given the elasticity of demand E = 3 and the point D(64) = 36.
In conclusion, determining the demand function involves understanding the concept of elasticity of demand, setting up and solving a differential equation, and using given conditions to find the constant of integration. The demand function we found, p = 9437184 / x^3, provides a clear mathematical relationship between price and quantity demanded for this specific scenario. Understanding these principles is invaluable for anyone involved in economics, business, or market analysis. For further reading on the elasticity of demand, you can visit Investopedia's article on Elasticity of Demand.
