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.
To calculate the price elasticity of demand, we need to use the following formula:
Price Elasticity of Demand = (% Change in Quantity Demanded) / (% Change in Price)
In this case, we are given the demand equation Q1 = 26 - (3/5)p1 + 2p2 + 0.01I, so to calculate the price elasticity of demand, we need to find the percentage change in quantity demanded and the percentage change in price.
Using the given values:
p1 = 10
p2 = 5
I = 1000
We can substitute these values into the demand equation to find the quantity demanded:
Q1 = 26 - (3/5)p1 + 2p2 + 0.01I
Q1 = 26 - (3/5)(10) + 2(5) + 0.01(1000)
Q1 = 26 - 6 + 10 + 10
Q1 = 40
Now, we can calculate the percentage change in quantity demanded when p1 changes by 1%:
% Change in Quantity Demanded = (Q1_new - Q1_old) / Q1_old
% Change in Quantity Demanded = (40 - 39) / 39
% Change in Quantity Demanded = 1 / 39
% Change in Quantity Demanded ≈ 0.0256
Next, we can calculate the percentage change in price when p1 changes by 1%:
% Change in Price = (p1_new - p1_old) / p1_old
% Change in Price = (10.1 - 10) / 10
% Change in Price = 0.1 / 10
% Change in Price = 0.01
Now we can calculate the price elasticity of demand:
Price Elasticity of Demand = (% Change in Quantity Demanded) / (% Change in Price)
Price Elasticity of Demand = 0.0256 / 0.01
Price Elasticity of Demand ≈ 2.56
Since the price elasticity of demand is greater than 1, demand for good 1 is considered to be elastic. This means that a 1% change in the price of good 1 will result in a greater than 1% change in the quantity demanded.
To calculate the price elasticity of demand, we need to find the percentage change in demand with respect to the percentage change in price.
First, let's differentiate the demand function 𝑄1 = 26 − (3/5)𝑝1 + 2𝑝2 + 0.01𝐼 with respect to 𝑝1:
d𝑄1/d𝑝1 = -(3/5)
Now, let's find the value of d𝑄1/d𝑝1 at 𝑝1 = 10, 𝑝2 = 5, 𝐼 = 1000:
d𝑄1/d𝑝1 = -(3/5) = -0.6
Next, let's find the value of 𝑄1 at 𝑝1 = 10, 𝑝2 = 5, 𝐼 = 1000:
𝑄1 = 26 − (3/5)𝑝1 + 2𝑝2 + 0.01𝐼
= 26 - (3/5)(10) + 2(5) + 0.01(1000)
= 26 - 6 + 10 + 10
= 40
Now, let's calculate the percentage change in demand (𝑄1) with respect to the percentage change in price (𝑝1):
% change in demand = (d𝑄1/d𝑝1) * (𝑝1 / 𝑄1)
% change in demand = (-0.6) * (10 / 40) = -0.15
The price elasticity of demand is given by the absolute value of the percentage change in demand divided by the percentage change in price:
Price elasticity of demand = |% change in demand / % change in price|
Price elasticity of demand = |-0.15 / 1| = 0.15
Since the price elasticity of demand is less than 1, demand for good 1 is inelastic. This means that a change in the price of good 1 will have a relatively small effect on the quantity demanded of good 1.
To calculate the price elasticity of demand, we need to determine the percentage change in the quantity of good 1 demanded in response to a percentage change in the price of good 1.
The formula for price elasticity of demand is as follows:
E = (% change in quantity demanded) / (% change in price)
To calculate the price elasticity of demand when p1 = 10, p2 = 5, and I = 1000, we need to determine the percentage change in quantity demanded due to a percentage change in the price of good 1.
Let's assume the price of good 1 increases by 1%, from 10 to 10.1.
Therefore, the change in price is: (10.1 - 10)/10 = 0.01
Now, we need to determine the corresponding change in quantity demanded. To do this, we plug the initial values (p1 = 10, p2 = 5, I = 1000) into the demand function and calculate the quantity demanded at those prices and income level.
So, Q1 = 26 - (3/5 * 10) + (2 * 5) + (0.01 * 1000) = 23
Next, let's determine the new quantity demanded when the price of good 1 increases by 1% to 10.1.
Q1_new = 26 - (3/5 * 10.1) + (2 * 5) + (0.01 * 1000) = 22.98 ≈ 23
Now, we can calculate the percentage change in quantity demanded:
% change in quantity demanded = (Q1_new - Q1) / Q1 * 100
= (23 - 23) / 23 * 100
= 0
Lastly, we can calculate the price elasticity of demand using the formula mentioned earlier:
E = (% change in quantity demanded) / (% change in price)
= 0 / 0.01
= 0
Therefore, when p1 = 10, p2 = 5, and I = 1000, the price elasticity of demand for good 1 is 0.
Based on the value of elasticity, we can determine whether demand for good 1 is elastic or inelastic.
If the price elasticity of demand is greater than 1, demand is considered elastic.
If the price elasticity of demand is less than 1, demand is considered inelastic.
If the price elasticity of demand is equal to 1, demand is considered unit elastic.
Since the price elasticity of demand for good 1 is 0, which is less than 1, demand for good 1 is considered inelastic in this case. This means that a 1% increase in the price of good 1 will lead to a less than 1% decrease in the quantity demanded.