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.
To calculate the price elasticity of demand (PED) and income elasticity of demand (YED), we need to use the following formulas:
PED = (% change in quantity demanded / % change in price)
YED = (% change in quantity demanded / % change in income)
Let's break down the steps to calculate each elasticity:
(i) Price Elasticity of Demand (PED):
1. Start by taking the natural logarithm of the demand function to linearize it:
ln(Q) = ln(10) - 0.5ln(P) + 0.75ln(Y)
2. Differentiate the linearized demand function with respect to the natural logarithm of price (ln(P)):
d(ln(Q))/d(ln(P)) = d(ln(10))/d(ln(P)) - 0.5d(ln(P))/d(ln(P)) + 0.75d(ln(Y))/d(ln(P))
3. Simplify the expression:
PED = -0.5
Therefore, the price elasticity of demand for the product is -0.5.
(ii) Income Elasticity of Demand (YED):
1. Again, start by taking the natural logarithm of the demand function to linearize it:
ln(Q) = ln(10) - 0.5ln(P) + 0.75ln(Y)
2. Differentiate the linearized demand function with respect to the natural logarithm of income (ln(Y)):
d(ln(Q))/d(ln(Y)) = d(ln(10))/d(ln(Y)) - 0.5d(ln(P))/d(ln(Y)) + 0.75d(ln(Y))/d(ln(Y))
3. Simplify the expression:
YED = 0.75
So, the income elasticity of demand for the product is 0.75.
Therefore, the price elasticity of demand is -0.5, and the income elasticity of demand is 0.75 based on the given demand function.