Suppose a demand function is given by Q=10P^-0.5Y^0.75. Calculate the price elasticity of demand and income elasticity of demand.
To calculate the price elasticity of demand and income elasticity of demand, we need to use the formulas for each elasticity.
1. Price Elasticity of Demand (PED):
The formula for PED is:
PED = (% change in quantity demanded) / (% change in price)
To calculate the price elasticity of demand, we need to differentiate the demand function with respect to price (P) and then multiply it by (P/Q) to get the actual elasticity value.
Given the demand function: Q = 10P^(-0.5)Y^(0.75)
Differentiating with respect to P, we get:
dQ/dP = -0.5 * 10P^(-1.5)Y^(0.75)
Now, we calculate the percentage change in quantity demanded and the percentage change in price:
% change in quantity demanded = (dQ/dP) * (P/Q)
% change in price = (∆P / P)
Plugging in the values into the PED formula:
PED = (% change in quantity demanded) / (% change in price)
PED = (dQ/dP) * (P/Q) / (∆P / P)
2. Income Elasticity of Demand (IED):
The formula for IED is:
IED = (% change in quantity demanded) / (% change in income)
To calculate the income elasticity of demand, we need to differentiate the demand function with respect to income (Y) and then multiply it by (Y/Q) to get the actual elasticity value.
Differentiating the demand function with respect to income (Y), we get:
dQ/dY = 0.75 * 10P^(-0.5)Y^(0.75 - 1)
Now, we calculate the percentage change in quantity demanded and the percentage change in income:
% change in quantity demanded = (dQ/dY) * (Y/Q)
% change in income = (∆Y / Y)
Plugging in the values into the IED formula:
IED = (% change in quantity demanded) / (% change in income)
IED = (dQ/dY) * (Y/Q) / (∆Y / Y)
By calculating these two formulas, we can obtain both the price elasticity of demand (PED) and the income elasticity of demand (IED).