2. The equation for a demand curve has been estimated to be Q = 100 - 10 P + 0.5 Y, where Q is quantity, P is price, and K 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?
a. The equation represents a demand curve, which relates the quantity demanded (Q) to the price (P) and the income (Y). According to the equation, the quantity demanded decreases as the price increases (negative coefficient for P), and increases as the income increases (positive coefficient for Y). The equation suggests that there is an inverse relationship between prices and quantity demanded, meaning that as prices go up, the quantity demanded decreases, and as income goes up, the quantity demanded also increases.
b. To calculate the price elasticity, we need to determine the percentage change in quantity demanded divided by the percentage change in price. However, since we are given a specific price (P = 7) and not the change in price, we cannot directly calculate the price elasticity. We would need additional information about the change in price to determine the price elasticity at that specific price.
c. To calculate the income elasticity, we need to determine the percentage change in quantity demanded divided by the percentage change in income. Again, since we are given a specific income level (Y = 50) and not the change in income, we cannot directly calculate the income elasticity. We would need additional information about the change in income to determine the income elasticity at that specific income level.
a. The equation for the demand curve is Q = 100 - 10P + 0.5Y. This equation represents the relationship between quantity demanded (Q) and two factors: price (P) and income (Y). It suggests that the quantity demanded of a particular good or service is influenced by both price and income.
b. To find the price elasticity at a price of 7, we need to calculate the percentage change in quantity demanded relative to a percentage change in price. The formula for price elasticity of demand is:
Price Elasticity of Demand = (% change in quantity demanded) / (% change in price)
Since the price is given as 7, we need to determine the quantity demanded at this price. Plugging in P = 7 into the demand equation Q = 100 - 10P + 0.5Y, we get:
Q = 100 - 10(7) + 0.5Y
Q = 100 - 70 + 0.5(50)
Q = 100 - 70 + 25
Q = 55
So, at a price of 7, the quantity demanded is 55.
To calculate the % change in quantity, we need to compare it to the quantity demanded at a different price. Let's assume the quantity demanded was 60 at a price of 6. Therefore, the percentage change in quantity demanded would be:
% change in quantity demanded = ((55 - 60) / 60) * 100
% change in quantity demanded = (-5 / 60) * 100
% change in quantity demanded = -8.33%
The % change in price is given as:
% change in price = ((7 - 6) / 6) * 100
% change in price = (1 / 6) * 100
% change in price = 16.67%
Now, we can calculate the price elasticity of demand:
Price Elasticity of Demand = (-8.33% / 16.67%)
Price Elasticity of Demand = -0.5
Therefore, at a price of 7, the price elasticity of demand is -0.5.
c. To find the income elasticity at an income level of 50, we need to calculate the percentage change in quantity demanded relative to a percentage change in income. The formula for income elasticity of demand is:
Income Elasticity of Demand = (% change in quantity demanded) / (% change in income)
Since the income is given as 50, we need to determine the quantity demanded at this income level. Plugging in Y = 50 into the demand equation Q = 100 - 10P + 0.5Y, we get:
Q = 100 - 10P + 0.5(50)
Q = 100 - 10P + 25
Q = 125 - 10P
Now, we need to compare it to the quantity demanded at a different income level. Let's assume the quantity demanded was 120 at an income level of 40. Therefore, the percentage change in quantity demanded would be:
% change in quantity demanded = ((125 - 120) / 120) * 100
% change in quantity demanded = (5 / 120) * 100
% change in quantity demanded = 4.17%
The % change in income is given as:
% change in income = ((50 - 40) / 40) * 100
% change in income = (10 / 40) * 100
% change in income = 25%
Now, we can calculate the income elasticity of demand:
Income Elasticity of Demand = (4.17% / 25%)
Income Elasticity of Demand = 0.17
Therefore, at an income level of 50, the income elasticity of demand is 0.17.