The demand for rice in Japan for a particular year was estimated by the general function q = f (p) =Ap^(-0.13), where p represents the price of a unit of rice, A represents a constant that can be calculated uniquely for a particular year, and q represents the annual per capita rice demand. Calculate and interpret the elasticity of demand.
To calculate the elasticity of demand, we need to differentiate the demand function, q = A * p^(-0.13), with respect to the price (p):
dq/dp = d/dp (A * p^(-0.13))
To find dq/dp, apply the power rule of differentiation. The power rule states that if we have a function of the form f(x) = A * x^n, then the derivative of that function with respect to x is given by f'(x) = A * n * x^(n-1).
Applying the power rule to our demand function, we get:
dq/dp = (-0.13) * A * p^(-0.13 - 1) = -0.13A * p^(-1.13)
Now, we can calculate the elasticity of demand (E) using the formula:
E = (dq/dp) * (p/q)
Substituting the values we have:
E = (-0.13A * p^(-1.13)) * (p / (Ap^(-0.13)))
Simplifying:
E = -0.13A * p^(-1.13) * p / (Ap^(-0.13))
E = -0.13 * p^( -1.13 - 1) / A^( -0.13 + 1)
E = -0.13 * p^(-2.13) / A^0.87
Interpreting the elasticity of demand:
The value of the elasticity of demand (E) tells us how sensitive the quantity demanded (q) is to changes in the price (p). Specifically, the magnitude of E indicates the responsiveness of demand to price changes.
If E > 1, demand is elastic, meaning a change in price will lead to a proportionately larger change in quantity demanded. This suggests that rice is relatively price-sensitive in Japan, and small price changes can have a significant impact on rice demand.
If E = 1, demand is unitary elastic, indicating that the percentage change in price equals the percentage change in quantity demanded.
If E < 1, demand is inelastic, meaning a change in price will result in a proportionately smaller change in quantity demanded. This implies that rice is relatively price-insensitive in Japan, and price changes will have a limited effect on demand.
Additionally, if E = 0, demand is perfectly inelastic, indicating that the quantity demanded remains unchanged regardless of price changes. Conversely, if E = infinity, demand is perfectly elastic, meaning a slight change in price will lead to an infinite change in quantity demanded.
Therefore, to fully interpret the elasticity of demand, we need to know the specific value of the constant A and the price of rice (p) in Japan.