Suppose you decide to produce a product and you discover it has a demand function given by;
Q = 10P^-0.5Y^0.75
(I) calculate the price elasticity of demand for the product.
(ii) calculate the income elasticity of demand.
Suppose you decide to produce a product and you discover it has a demand function given by;
Q = 10P^-0.5Y^0.75
(I) calculate the price elasticity of demand for the product.
(ii) calculate the income elasticity of demand.
Respond to this Question
Ed=10
Ed = 5
To calculate the price elasticity of demand (PED), we need to determine the percentage change in quantity demanded with respect to the percentage change in price. Here's a step-by-step guide to finding the PED:
Step 1: Differentiate the demand function to derive the demand curve equation. In this case, differentiate the function Q = 10P^(-0.5)Y^(0.75) with respect to P, assuming Y is constant.
dQ/dP = 10*(-0.5)P^(-0.5-1)Y^(0.75)
Simplifying this expression gives:
dQ/dP = -5P^(-1.5)Y^(0.75)
Step 2: Calculate the current values of Q and P for which you want to find the PED. Let's assume Q = Q0 and P = P0.
Step 3: Calculate the percentage change in quantity demanded. Use the formula:
%ΔQ = (Q1 - Q0) / Q0 * 100, where Q1 is the new quantity demanded.
Step 4: Calculate the percentage change in price. Use the formula:
%ΔP = (P1 - P0) / P0 * 100, where P1 is the new price.
Step 5: Substitute the values obtained from steps 2, 3, and 4 into the expression you derived in step 1. This gives you the derivative at the specific point.
dQ/dP = -5P^(-1.5)Y^(0.75)
Step 6: Calculate the PED using the expression:
PED = (dQ/dP) * (P0 / Q0)
Now, let's move on to calculating the income elasticity of demand (YED) for the product:
Step 1: Differentiate the demand function to derive the demand curve equation. In this case, differentiate the function Q = 10P^(-0.5)Y^(0.75) with respect to Y, assuming P is constant.
dQ/dY = 10P^(-0.5)*0.75Y^(-0.25)
Simplifying this expression gives:
dQ/dY = 7.5P^(-0.5)Y^(-0.25)
Step 2: Calculate the current values of Q and Y for which you want to find the YED. Let's assume Q = Q0 and Y = Y0.
Step 3: Calculate the percentage change in quantity demanded. Use the formula:
%ΔQ = (Q1 - Q0) / Q0 * 100, where Q1 is the new quantity demanded.
Step 4: Calculate the percentage change in income. Use the formula:
%ΔY = (Y1 - Y0) / Y0 * 100, where Y1 is the new income.
Step 5: Substitute the values obtained from steps 2, 3, and 4 into the expression you derived in step 1. This gives you the derivative at the specific point.
dQ/dY = 7.5P^(-0.5)Y^(-0.25)
Step 6: Calculate the YED using the expression:
YED = (dQ/dY) * (Y0 / Q0)
By following these steps, you will be able to calculate both the price elasticity of demand (PED) and the income elasticity of demand (YED) for the given demand function.