Given 𝑄1 = 26 − 35𝑝1 + 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 the formula:
Elasticity = (% Change in Quantity Demanded) / (% Change in Price)
First, let's find the initial quantity demanded when p1 = 10, p2 = 5, and I = 1000.
Q1 = 26 - 35p1 + 2p2 + 0.01I
Q1 = 26 - 35(10) + 2(5) + 0.01(1000)
Q1 = 26 - 350 + 10 + 10
Q1 = -304
Now, let's find the final quantity demanded when p1 increases by 1%.
Q1' = 26 - 35(p1 + 0.01) + 2p2 + 0.01I
Q1' = 26 - 35(10.01) + 2(5) + 0.01(1000)
Q1' = 26 - 350.35 + 10 + 10
Q1' = -304.35
Now, let's find the percentage change in quantity demanded.
% Change in Quantity Demanded = (Q1' - Q1) / Q1 * 100
% Change in Quantity Demanded = (-304.35 - (-304)) / (-304) * 100
% Change in Quantity Demanded = 0.35 / (-304) * 100
% Change in Quantity Demanded ≈ -0.115%
Now, let's find the percentage change in price.
% Change in Price = (p1' - p1) / p1 * 100
% Change in Price = (10.01 - 10) / 10 * 100
% Change in Price = 0.01 / 10 * 100
% Change in Price = 0.1%
Now, let's calculate the price elasticity of demand.
Elasticity = (% Change in Quantity Demanded) / (% Change in Price)
Elasticity = (-0.115%) / (0.1%)
Elasticity = -0.115
Since the price elasticity of demand is negative, demand for good 1 is elastic. This means that a small increase in price will lead to a relatively larger decrease in quantity demanded.
To calculate the price elasticity of demand, we need to find the derivative of 𝑄1 with respect to 𝑝1, and then multiply it by 𝑝1 divided by 𝑄1.
Step 1: Find the derivative of 𝑄1 with respect to 𝑝1.
𝑑𝑄1/𝑑𝑝1 = -35 + 0
Since the derivative of 2𝑝2 + 0.01𝐼 with respect to 𝑝1 is zero, we can ignore it.
Step 2: Calculate the price elasticity of demand.
Price elasticity of demand (𝑒) = (𝑑𝑄1/𝑑𝑝1) * (𝑝1 / 𝑄1)
Substituting the given values 𝑝1 = 10, 𝑝2= 5, and 𝐼 = 1000 into 𝑄1, we get:
𝑄1 = 26 − 35(10) + 2(5) + 0.01(1000) = 26 - 350 + 10 + 10 = -304
Now we can calculate the price elasticity of demand:
𝑒 = (𝑑𝑄1/𝑑𝑝1) * (𝑝1 / 𝑄1) = (-35 * 10 / -304) = 1.15
Step 3: Determine whether demand for good 1 is elastic or inelastic.
Since the absolute value of 𝑒 (1.15) is greater than 1, demand for good 1 is elastic. This means that a change in the price of good 1 will result in a relatively larger change in the quantity demanded.
To calculate the price elasticity of demand, we need to take 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 Q1 (the quantity of good 1).
Given that Q1 = 26 -35p1 + 2p2 + 0.01I, we can rearrange the equation to isolate p1:
35p1 = 26 + 2p2 + 0.01I - Q1
Now, let's differentiate both sides of the equation with respect to p1:
35(dp1/dp1) = 2(dp2/dp1) + 0.01(dI/dp1) - (dQ1/dp1)
Since dp1/dp1 = 1, we have:
35 = 2(dp2/dp1) + 0.01(dI/dp1) - (dQ1/dp1)
Now, substitute the given values p1 = 10, p2 = 5, and I = 1000 into the equation.
Next, calculate (dQ1/dp1) by substituting the given values into the equation:
Q1 = 26 - 35p1 + 2p2 + 0.01I
(dQ1/dp1) = -35
Substitute these values into the equation:
35 = 2(dp2/dp1) + 0.01(dI/dp1) - (-35)
35 = 2(dp2/dp1) + 0.01(dI/dp1) + 35
Simplify the equation:
0 = 2(dp2/dp1) + 0.01(dI/dp1)
Without the values of (dp2/dp1) and (dI/dp1), we cannot directly calculate the price elasticity of demand.
However, to determine whether the demand for good 1 is elastic or inelastic, we need to consider the magnitude of the price elasticity of demand. If the magnitude of the price elasticity of demand is greater than 1, the demand is elastic. If it is less than 1, the demand is inelastic.