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 to use the following formula:
E = (%ΔQ / %ΔP)
Where E is the price elasticity of demand, %ΔQ is the percentage change in quantity demanded, and %ΔP is the percentage change in price.
In this case, we are 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. We are asked to calculate the price elasticity of demand when 𝑝1 = 10, 𝑝2 = 5, 𝐼 = 1000 for good 1.
To calculate the percentage change in quantity demanded, we need to find the difference between the original quantity demanded and the new quantity demanded, and then divide it by the original quantity demanded:
%ΔQ = (Qnew - Qoriginal) / Qoriginal
To calculate the percentage change in price, we need to find the difference between the new price and the original price, and then divide it by the original price:
%ΔP = (Pnew - Poriginal) / Poriginal
Now let's calculate %ΔQ and %ΔP:
%ΔQ = (Qnew - Qoriginal) / Qoriginal
%ΔQ = ((26 - 35 * 10 + 2 * 5 + 0.01 * 1000) - (26 - 35 * 10 + 2 * 5 + 0.01 * 1000)) / (26 - 35 * 10 + 2 * 5 + 0.01 * 1000)
%ΔQ = 0 / (26 - 35 * 10 + 2 * 5 + 0.01 * 1000)
%ΔQ = 0
%ΔP = (Pnew - Poriginal) / Poriginal
%ΔP = ((10 - 10) / 10)
%ΔP = 0
Now let's plug in the values into the price elasticity of demand formula:
E = (%ΔQ / %ΔP)
E = (0 / 0)
As we can see, the numerator is 0 and the denominator is also 0. This indicates that the price elasticity of demand is undefined. Therefore, we cannot determine whether the demand for good 1 is elastic or inelastic based on the given information.
To calculate the price elasticity of demand (PED), we need to use the following formula:
PED = (% change in quantity demanded) / (% change in price)
In this case, we are given the demand function Q1 = 26 - 35p1 + 2p2 + 0.01I, where p1 is the price of good 1, p2 is the price of good 2, and I is the income. We need to calculate the PED at p1 = 10, p2 = 5, and I = 1000.
First, let's find the initial quantity demanded at these given prices:
Q1 = 26 - 35(10) + 2(5) + 0.01(1000)
= 26 - 350 + 10 + 10
= -304
Now, let's find the new quantity demanded when p1 changes by a percentage:
Suppose p1 decreases by 1%, then the new price would be 10 - (0.01 * 10) = 10 - 0.1 = 9.9.
Q1' = 26 - 35(9.9) + 2(5) + 0.01(1000)
= 26 - 346.5 + 10 + 10
= -300.5
Next, let's calculate the percentage change in quantity demanded:
% change in quantity demanded = (new quantity demanded - initial quantity demanded) / initial quantity demanded * 100
% change in quantity demanded = (-300.5 - (-304)) / (-304) * 100
= 3.5 / 304 * 100
≈ 1.15%
Now, let's calculate the percentage change in price:
% change in price = (new price - initial price) / initial price * 100
% change in price = (9.9 - 10) / 10 * 100
= -0.1 / 10 * 100
= -1.00%
Now, we can substitute these values into the formula to calculate the PED:
PED = (% change in quantity demanded) / (% change in price)
= 1.15 / -1.00
≈ -1.15
The PED is negative, which indicates an inverse relationship between price and quantity demanded. In this case, the magnitude of the PED is greater than 1, which means that demand for good 1 is elastic. This means that a 1% decrease in the price of good 1 would lead to a more than 1% increase in quantity demanded.
To calculate the price elasticity of demand (PED), we need to determine the percentage change in the quantity demanded of good 1 in response to a percentage change in its price.
The formula for price elasticity of demand is:
PED = [(∆Q/Q) / (∆P/P)]
Where ∆Q represents the change in quantity demanded, Q represents the original quantity demanded, ∆P represents the change in price, and P represents the original price.
In this case, we are given the demand function 𝑄1 = 26 − 35𝑝1 + 2𝑝2 + 0.01𝐼, so we need to find the partial derivative ∂Q1/∂P1.
Taking the partial derivative with respect to p1, we get:
∂Q1/∂P1 = -35 + 0 + 0 = -35
Now we can substitute the given values into the formula for PED:
PED = (∂Q1/∂P1) × (P1/Q1)
Plugging in the values ∂Q1/∂P1 = -35, P1 = 10, and Q1 = 𝑄1 = 26 − 35𝑝1 + 2𝑝2 + 0.01𝐼, we have:
PED = (-35) × (10 / (26 − 35(10) + 2(5) + 0.01(1000)))
Simplifying the denominator, we get:
PED = (-35) × (10 / (26 − 350 + 10 + 10))
PED = (-35) × (10 / (36 − 340))
PED = (-35) × (10 / (-304))
PED ≈ 1.15
Since the computed value of the price elasticity of demand (PED) is positive and less than 1, it indicates that the demand for good 1 is inelastic. This means that a 1% increase in the price of good 1 will result in less than a 1% decrease in the quantity demanded. In other words, the demand for good 1 is relatively unresponsive to changes in its price.