The demand function for a good is q = 56.6 − 0.25p − 0.03m + 0.45ps + 0.6n where q is the quantity demanded per week, p is the price per unit, m is the average weekly income, ps is the price of a competing good and n is the population in millions. Given values are p = 65, m = 350, ps = 60 and n = 24. a. Calculate the price elasticity of demand. b. Find out what would happen to (a) if n rose to 26. c. Explain why this is an inferior good
(a) q = 56.6 - 0.25*65 - 0.03*350 + 0.45*60 + 0.6*24 = 71.25
(b) q = 72.45
∆n = 2/24 = 8.3%
∆q = 1.2/71.25 = 1.7%
So, what does that mean for elasticity?
To calculate the price elasticity of demand, we follow the formula:
E = (% change in quantity demanded) / (% change in price)
a. Calculate the price elasticity of demand:
Given:
q = 56.6 − 0.25p − 0.03m + 0.45ps + 0.6n
p = 65
m = 350
ps = 60
n = 24
Substituting the given values into the demand function, we have:
q = 56.6 − 0.25(65) − 0.03(350) + 0.45(60) + 0.6(24)
q = 56.6 − 16.25 − 10.5 + 27 + 14.4
q = 71.25
Now, let's calculate the quantity demanded when the price changes slightly:
q1 = 56.6 − 0.25(66) − 0.03(350) + 0.45(60) + 0.6(24)
q1 = 56.6 − 16.5 − 10.5 + 27 + 14.4
q1 = 71
Percentage change in quantity demanded: ((q1 - q) / q) x 100
= ((71 - 71.25) / 71.25) x 100
= -0.35%
Percentage change in price: ((p1 - p) / p) x 100
= ((66 - 65) / 65) x 100
= 1.54%
Applying these values in the price elasticity of demand formula:
E = (-0.35% / 1.54%) = -0.227
The price elasticity of demand is approximately -0.227.
b. To find out what would happen to the price elasticity of demand if n rose to 26, we need to recalculate the quantity demanded and the percentage change in quantity demanded using the new value of n.
Given:
n = 26
q2 = 56.6 − 0.25(65) − 0.03(350) + 0.45(60) + 0.6(26)
q2 = 56.6 − 16.25 − 10.5 + 27 + 15.6
q2 = 73.45
Percentage change in quantity demanded: ((q2 - q) / q) x 100
= ((73.45 - 71.25) / 71.25) x 100
= 3.09%
Applying this new percentage change in quantity demanded to the original percentage change in price:
E2 = (3.09% / 1.54%)
E2 = 2
If n rose to 26, the price elasticity of demand would increase to approximately 2.
c. This is an inferior good because the coefficient of n (population) in the demand function is positive. Usually, when the population increases, people tend to have higher incomes, resulting in an increase in demand for most goods. However, in this case, as n (population) increases, the quantity demanded decreases. This suggests that the good is considered less desirable or inferior as people's incomes rise and they can afford more expensive or better-quality alternatives.
To calculate the price elasticity of demand, we need to use the following formula:
E = (% change in quantity demanded) / (% change in price)
a. Calculate the price elasticity of demand:
First, let's calculate the initial quantity demanded using the given values:
q = 56.6 - 0.25p - 0.03m + 0.45ps + 0.6n
q = 56.6 - 0.25(65) - 0.03(350) + 0.45(60) + 0.6(24)
q = 56.6 - 16.25 - 10.5 + 27 + 14.4
q = 71.25
Now, let's calculate the quantity demanded if we change the price by 1 unit (δp = 1):
q' = 56.6 - 0.25(p + δp) - 0.03m + 0.45ps + 0.6n
q' = 56.6 - 0.25(65 + 1) - 0.03(350) + 0.45(60) + 0.6(24)
q' = 56.6 - 16.5 - 10.5 + 27 + 14.4
q' = 71
Now, let's calculate the percentage change in quantity demanded:
% change in quantity demanded = (q' - q) / q * 100
% change in quantity demanded = (71 - 71.25) / 71.25 * 100
% change in quantity demanded ≈ -0.35%
Next, let's calculate the percentage change in price:
% change in price = (δp / p) * 100
% change in price = (1 / 65) * 100
% change in price ≈ 1.54%
Now, we can calculate the price elasticity of demand:
E = (% change in quantity demanded) / (% change in price)
E = (-0.35% / 1.54%)
E ≈ -0.227
b. Find out what would happen to (a) if n rose to 26:
To find out how a change in n would affect the price elasticity of demand (E), we need to recalculate the price elasticity using the new value of n:
q' = 56.6 - 0.25(65) - 0.03(350) + 0.45(60) + 0.6(26) (using n = 26)
q' = 56.6 - 16.25 - 10.5 + 27 + 15.6
q' = 72.45
% change in quantity demanded = (q' - q) / q * 100
% change in quantity demanded = (72.45 - 71.25) / 71.25 * 100
% change in quantity demanded ≈ 1.68%
E = (% change in quantity demanded) / (% change in price)
E = (1.68% / 1.54%)
E ≈ 1.091
c. Explain why this is an inferior good:
An inferior good is a product for which demand decreases as income increases. In the given demand function, we can observe that the coefficient of income (m) is negative (-0.03). This means that as average weekly income (m) increases, the quantity demanded per week (q) will decrease. When income rises, people tend to shift their consumption towards higher-quality products or substitutes, reducing the demand for the inferior good.