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
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)?
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.
To calculate the price elasticity of demand for good 1, we need to use the following formula:
E = (% change in quantity demanded) / (% change in price)
a. To calculate the price elasticity of demand when 𝑝1 = 10, 𝑝2 = 5, 𝐼 = 1000, we need to determine the percent change in quantity demanded and the percent change in price.
Let's assume we have data for two different scenarios, Scenario 1 and Scenario 2.
Scenario 1:
𝑝1 = 10
𝑝2 = 5
𝐼 = 1000
Using these values, we can plug them into the demand equation 𝑄1 = 26 − 3/5𝑝1 + 2𝑝2 + 0.01𝐼 to calculate the quantity demanded for good 1.
𝑄1 = 26 − 3/5(10) + 2(5) + 0.01(1000)
𝑄1 = 26 - 6 + 10 + 10
𝑄1 = 40
Now let's calculate the total revenue for Scenario 1:
Revenue1 = 𝑝1 * 𝑄1
Revenue1 = 10 * 40
Revenue1 = 400
Scenario 2:
Now let's change the price of good 1 to 𝑝1 = 9 and calculate the corresponding quantity demanded 𝑄1.
Using the same demand equation, we get:
𝑄1 = 26 − 3/5(9) + 2(5) + 0.01(1000)
𝑄1 = 26 - 5.4 + 10 + 10
𝑄1 = 40.6
Now let's calculate the total revenue for Scenario 2:
Revenue2 = 𝑝1 * 𝑄1
Revenue2 = 9 * 40.6
Revenue2 = 365.4
Now, let's calculate the percent change in quantity demanded and the percent change in price:
% Change in quantity demanded = ((𝑄2 - 𝑄1) / (𝑄1)) * 100
% Change in quantity demanded = ((40.6 - 40) / 40) * 100
% Change in quantity demanded = (0.6 / 40) * 100
% Change in quantity demanded = 1.5%
% Change in price = ((𝑝2 - 𝑝1) / (𝑝1)) * 100
% Change in price = ((9 - 10) / 10) * 100
% Change in price = (-1 / 10) * 100
% Change in price = -10%
Now let's plug in the values into the price elasticity of demand formula:
E = (% change in quantity demanded) / (% change in price)
E = (1.5% / -10%)
E = -0.15
Since the price elasticity of demand (E) is negative, we need to take its absolute value to determine the elasticity.
|E| = 0.15
The demand for good 1 is inelastic because the absolute value of the price elasticity is less than 1. An inelastic demand means that a change in price will result in a proportionately smaller change in the quantity demanded.
b. To determine if good 2 is a substitute or complement to good 1, we can compare the coefficients in the demand equation. In this case, the coefficient for 𝑝2 is positive (2) and not zero, which indicates that good 2 is a substitute for good 1. When the price of good 1 increases, the demand for good 2 will increase, and vice versa, suggesting that these two goods are substitutes for each other.
To calculate the price elasticity of demand for good 1, we need to find the derivative of the demand function with respect to the price of good 1 (p1) and then multiply it by the ratio of p1 to the demand function.
a. The demand function is given by 𝑄1 = 26 − 3/5𝑝1 + 2𝑝2 + 0.01𝐼. Let's find the derivative of 𝑄1 with respect to p1:
d𝑄1/dp1 = -3/5
Now, let's substitute the given values: 𝑝1 = 10, 𝑝2 = 5, and 𝐼 = 1000 into the equation to find 𝑄1:
𝑄1 = 26 - 3/5(10) + 2(5) + 0.01(1000)
= 26 - 6 + 10 + 10
= 40
Now, let's calculate the price elasticity of demand:
Price Elasticity of Demand = (d𝑄1/dp1) * (p1 / 𝑄1)
Elasticity = (-3/5) * (10 / 40)
Elasticity = -3/20
The price elasticity of demand is -3/20. Since elasticity is a measure of the responsiveness of demand to price changes, the demand for good 1 can be classified as inelastic. Inelastic demand means that the percentage change in the quantity demanded is less than the percentage change in price.
b. To determine if good 2 is a substitute, complement, or neither in relation to good 1, we need to consider the impact of changing the price of good 2 (p2) on the demand for good 1 (p1).
If an increase in the price of good 2 results in an increase in the demand for good 1, they are substitutes. If an increase in the price of good 2 results in a decrease in the demand for good 1, they are complements. If changing the price of good 2 has no effect on the demand for good 1, they are neither substitutes nor complements.
In this case, we do not have enough information to determine the relationship between good 1 and good 2. The demand function only provides information on how the price of good 2 affects the demand for good 1 but does not give us enough information to determine whether they are substitutes or complements.