To determine the own-price, cross-price, X, Y, and income elasticities of demand for soft drinks, we need to follow the formula for elasticity. In general, the elasticity of demand (E) is calculated as:
E = (% change in quantity demanded) / (% change in price/income/cross-price)
Where:
E > 1 indicates elastic demand (quantity is sensitive to changes in price/income/cross-price).
E = 1 indicates unitary elasticity (percentage changes in price/income/cross-price lead to the same percentage changes in quantity demanded).
E < 1 indicates inelastic demand (quantity is relatively insensitive to changes in price/income/cross-price).
Now, let's calculate the specific elasticities:
Own-price elasticity (Ep):
Ep = (% change in quantity demanded) / (% change in price)
Given the demand function as: Q = 20P^0.25 * P^0.45 * M^2
We'll differentiate the demand function with respect to price (P) to get the percentage change in quantity demanded (∆Q/Q) divided by the percentage change in price (∆P/P):
Ep = (∆Q/Q) / (∆P/P)
Cross-price elasticity (Exy):
Exy = (% change in quantity demanded of Y) / (% change in price of X)
Given the demand function, we'll differentiate it with respect to the price of a related good (Y) to get the percentage change in quantity demanded of Y (∆Qy/Qy) divided by the percentage change in the price of the soft drink (X):
Exy = (∆Qy/Qy) / (∆Px/Px)
Income elasticity (Ei):
Ei = (% change in quantity demanded) / (% change in income)
Given the demand function, we'll differentiate it with respect to income (M) to get the percentage change in quantity demanded (∆Q/Q) divided by the percentage change in income (∆M/M):
Ei = (∆Q/Q) / (∆M/M)
Interpretation of results:
If Ep > 1, it indicates that soft drinks have elastic demand, meaning consumers are very responsive to price changes. A price increase will result in a proportionally larger decrease in quantity demanded, and vice versa.
If Exy > 0, it indicates that soft drinks and the related good (Y) are substitutes. If Exy < 0, they are complements. The magnitude of Exy indicates the strength of the relationship.
If Ei > 0, it indicates that soft drinks are a normal good. An increase in income will lead to a proportionally larger increase in the quantity demanded of soft drinks.
Please provide the specific values for P and M to calculate the elasticities accurately.