Given 𝑄1 = 26 − 3/5𝑝1 + 2𝑝2 + 0.01𝐼 , where 𝑝1 is the price of good 1, 𝑝2 is the price of good 2, and 𝐼 is income

Question

a. Calculate the price elasticity of demand when
𝑝1 = 10, 𝑝2 = 5, 𝐼 = 1000. Is demand for good 1 elastic or inelastic? Explain.
b. What type of good is 2 in relation to good 1 (is it a
substitute, complement, neither)?

Answer 1

To calculate the price elasticity of demand for good 1, we need to use the formula:
Elasticity = (% change in quantity demanded) / (% change in price)

a. When p1 = 10, p2 = 5, and I = 1000, we need to calculate the % change in quantity demanded when the price of good 1 changes. Let's say the initial quantity demanded is Q1a, and when the price changes to p1 = 10, the new quantity demanded is Q1b.

% change in quantity demanded = ((Q1b - Q1a) / Q1a) * 100%

Now, we need to calculate Q1a and Q1b using the equation Q1 = 26 - (3/5)p1 + 2p2 + 0.01I.

When p1 = 10, p2 = 5, and I = 1000:
Q1a = 26 - (3/5)10 + 2(5) + 0.01(1000)
= 26 - 6 + 10 + 10
= 40

Now, let's choose a new price p1b for good 1. Let's say p1b = 12:
Q1b = 26 - (3/5)12 + 2(5) + 0.01(1000)
= 26 - 7.2 + 10 + 10
= 38.8

% change in quantity demanded = ((38.8 - 40) / 40) * 100%
= (-1.2 / 40) * 100%
= -3%

Now, we need to calculate the % change in price when the price of good 1 changes from p1a to p1b.

% change in price = ((p1b - p1a) / p1a) * 100%
= ((12 - 10) / 10) * 100%
= (2 / 10) * 100%
= 20%

Finally, we can calculate the price elasticity of demand:
Elasticity = (-3% / 20%)
= -0.15

Since the elasticity is negative, demand for good 1 is elastic. This means that a 1% increase in the price of good 1 leads to a 0.15% decrease in quantity demanded.

b. To determine if good 2 is a substitute, complement, or neither in relation to good 1, we need to look at the coefficient of p2 in the equation:

Coefficient of p2 = 2

Since the coefficient of p2 is positive, we can conclude that good 2 is a substitute for good 1. This means that when the price of good 1 increases, the demand for good 2 will also increase.