. The equation for a demand curve has been estimated to be Q = 100 - 10P + 0.5Y, where
Q is quantity, P is price, and Y is income. Assume P = 7 and Y = 50.
a. Interpret the equation.
b. At a price of 7, what is price elasticity?
c. At an income level of 50, what is income elasticity?
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a. The equation Q = 100 - 10P + 0.5Y represents a demand curve, where Q is the quantity of a good or service demanded, P is the price of the good or service, and Y is the income of the consumers. This equation suggests that the quantity demanded is influenced by three factors - the price of the good or service, the income of the consumers, and a constant term of 100.
b. To determine the price elasticity of demand at a price of 7, we need to calculate the percentage change in quantity demanded with respect to a percentage change in price. The formula for price elasticity of demand is given as:
Price Elasticity = % Change in Quantity Demanded / % Change in Price
In this case, we want to find the price elasticity at P = 7. To calculate the percentage change in quantity demanded, we need to compare it to a different price level. Let's assume another price, P' = 8. Using these values, we can calculate the quantity demanded for P = 7 and P' = 8:
Q = 100 - 10P + 0.5Y
Q = 100 - 10(7) + 0.5(50)
Q = 100 - 70 + 25
Q = 55
Similarly, for P' = 8:
Q' = 100 - 10P' + 0.5Y
Q' = 100 - 10(8) + 0.5(50)
Q' = 100 - 80 + 25
Q' = 45
Now, we can calculate the percentage change in quantity demanded and the percentage change in price:
% Change in Quantity Demanded = (Q' - Q) / Q * 100
% Change in Quantity Demanded = (45 - 55) / 55 * 100
% Change in Quantity Demanded = -10 / 55 * 100
% Change in Quantity Demanded ≈ -18.18%
% Change in Price = (P' - P) / P * 100
% Change in Price = (8 - 7) / 7 * 100
% Change in Price = 1 / 7 * 100
% Change in Price ≈ 14.29%
Price Elasticity = % Change in Quantity Demanded / % Change in Price
Price Elasticity ≈ -18.18% / 14.29%
Price Elasticity ≈ -1.27
Therefore, at a price of 7, the price elasticity is approximately -1.27. The negative sign indicates that the demand is elastic, i.e., a change in price leads to a relatively larger change in quantity demanded.
c. To calculate the income elasticity at an income level of 50, we use a similar approach. The formula for income elasticity of demand is given as:
Income Elasticity = % Change in Quantity Demanded / % Change in Income
Let's assume a different income level, Y' = 60. Using these values, we can calculate the quantity demanded for Y = 50 and Y' = 60:
Q = 100 - 10P + 0.5Y
Q = 100 - 10(7) + 0.5(50)
Q = 100 - 70 + 25
Q = 55
Similarly, for Y' = 60:
Q' = 100 - 10P + 0.5Y'
Q' = 100 - 10(7) + 0.5(60)
Q' = 100 - 70 + 30
Q' = 60
Now, we can calculate the percentage change in quantity demanded and the percentage change in income:
% Change in Quantity Demanded = (Q' - Q) / Q * 100
% Change in Quantity Demanded = (60 - 55) / 55 * 100
% Change in Quantity Demanded = 5 / 55 * 100
% Change in Quantity Demanded ≈ 9.09%
% Change in Income = (Y' - Y) / Y * 100
% Change in Income = (60 - 50) / 50 * 100
% Change in Income = 10 / 50 * 100
% Change in Income = 20%
Income Elasticity = % Change in Quantity Demanded / % Change in Income
Income Elasticity ≈ 9.09% / 20%
Income Elasticity ≈ 0.45
Therefore, at an income level of 50, the income elasticity is approximately 0.45. The positive sign suggests that the good is a normal good, i.e., an increase in income leads to a proportionate increase in demand.