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

Question

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.

Answer 1

To calculate the price elasticity of demand, we can use the following formula:

E = (%ΔQ1 / %Δp1)

Where E is the price elasticity of demand, %ΔQ1 is the percentage change in quantity demanded of good 1, and %Δp1 is the percentage change in the price of good 1.

To calculate the percentage change in quantity demanded (%ΔQ1), we need to know the initial quantity demanded and the new quantity demanded. However, these values are not given in the question. Without this information, we cannot calculate the price elasticity of demand.

Therefore, we cannot determine whether the demand for good 1 is elastic or inelastic based on the given information.

Answer 2

Well, let's put on our economic detective hats and solve this mystery! We're dealing with the price elasticity of demand here, which sounds as mysterious as a magician's disappearing rabbit.

First, we've got this equation that's longer than a CVS receipt: 𝑄1 = 26 − 35𝑝1 + 2𝑝2 + 0.01𝐼. Now, you might think it's a secret code, but it's just economics trying to be all cryptic and fancy.

Now, when 𝑝1 = 10, 𝑝2 = 5, and 𝐼 = 1000, let's plug these values into our magic equation and see what pops out. Drumroll, please!

𝑄1 = 26 − 35(10) + 2(5) + 0.01(1000)

After some quick math (and a couple of cups of coffee), we find that 𝑄1 = -115. Yep, you read that right, negative! Demand for good 1 has gone into hiding faster than a cat when it hears a vacuum cleaner.

Now, let's talk elasticity. If demand is as elastic as a bungee cord, it means a small change in price sends people running for the hills, screaming, "I'm not paying that much!" But if it's inelastic, people are more loyal than a dog guarding its favorite chew toy.

In our case, with a negative quantity, we've got some crazy elastic demand! People are fleeing from good 1 like it's the haunted house at the carnival. So, demand for good 1 is elastic, and that's why it's doing the disappearing act – it can't handle those high prices! 🎩💫

Answer 3

Apologies for the confusion caused earlier in my response. It seems there was a misunderstanding regarding the interpretation of the equation and calculation of quantity.

Given the equation 𝑄1 = 26 − 35𝑝1 + 2𝑝2 + 0.01𝐼, we can interpret it as a demand function for good 1, where 𝑝1 represents the price of good 1, 𝑝2 is the price of good 2, and 𝐼 is income.

To calculate the price elasticity of demand, we need to take the derivative of 𝑄1 with respect to 𝑝1 and then multiply it by 𝑝1/Q1 (the ratio of the price of good 1 to the quantity demanded of good 1). This will give us the price elasticity of demand for good 1.

Taking the derivative of 𝑄1 with respect to 𝑝1, we get:

d𝑄1/d𝑝1 = -35

Now, to calculate the price elasticity of demand, 𝐸, we need to multiply d𝑄1/d𝑝1 by 𝑝1/Q1, where 𝑝1 = 10 and 𝑝2 = 5. However, we do not have the values for 𝑞1 (quantity demanded of good 1) and 𝑄1 (initial quantity demanded of good 1) given in the question, so we cannot calculate 𝐸.

Unfortunately, without the necessary information, we cannot determine the price elasticity of demand or whether the demand for good 1 is elastic or inelastic.

Answer 4

To calculate the price elasticity of demand for good 1, we need to use the following formula:

Eᵢ = (∂𝑄ᵢ/∂𝑝ᵢ) * (𝑝ᵢ / 𝑄ᵢ)

Where:
- Eᵢ represents the price elasticity of demand for good i
- 𝑄ᵢ represents the quantity demanded of good i
- 𝑝ᵢ represents the price of good i

In this case, we are looking at good 1, so we are interested in calculating E₁.

Given:
𝑄₁ = 26 − 35𝑝₁ + 2𝑝₂ + 0.01𝐼
𝑝₁ = 10
𝑝₂ = 5
𝐼 = 1000

Let's calculate E₁ step-by-step.

Step 1: Calculate ∂𝑄₁/∂𝑝₁
To do this, we differentiate 𝑄₁ with respect to 𝑝₁ and hold all the other variables constant.

∂𝑄₁/∂𝑝₁ = -35

Step 2: Calculate 𝑄₁
To do this, we substitute the given values of 𝑝₁, 𝑝₂, and 𝐼 into the equation 𝑄₁ = 26 − 35𝑝₁ + 2𝑝₂ + 0.01𝐼.

𝑄₁ = 26 − 35(10) + 2(5) + 0.01(1000)
= 26 − 350 + 10 + 10
= -304

Step 3: Calculate E₁
Now we can substitute the values obtained in steps 1 and 2 into the price elasticity formula.

E₁ = (∂𝑄₁/∂𝑝₁) * (𝑝₁ / 𝑄₁)
= (-35) * (10 / -304)
= 1.1513

Since the price elasticity of demand (E₁) is greater than 1, demand for good 1 is elastic. This means that a 1% increase in price would lead to a more than 1% decrease in quantity demanded, indicating a relatively big sensitivity of demand to price changes.

Answer 5

To calculate the price elasticity of demand (PED), 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 the quantity demanded:

PED = (∂Q1/∂p1) * (p1/Q1)

First, let's find the derivative of the demand function with respect to p1:

∂Q1/∂p1 = -35

Next, substitute the values into the formula:

PED = (-35) * (10 / Q1)

Now, we need to find the value of Q1 by substituting the given values (p1 = 10, p2 = 5, I = 1000) into the demand function:

Q1 = 26 - 35p1 + 2p2 + 0.01I
= 26 - 35(10) + 2(5) + 0.01(1000)
= 26 - 350 + 10 + 10
= -304

Finally, substitute the value of Q1 into the PED formula:

PED = (-35) * (10 / -304)
≈ 1.15

The price elasticity of demand for good 1 is approximately 1.15. Since the magnitude of the PED is greater than 1, demand for good 1 is considered elastic.

Explanation:
Price elasticity of demand measures the responsiveness of quantity demanded to a change in price. If the magnitude of the PED is greater than 1, it indicates that demand is elastic, meaning a small change in price will result in a proportionally larger change in quantity demanded. In this case, a 1% increase in the price of good 1 would lead to approximately a 1.15% decrease in quantity demanded, indicating that demand is relatively sensitive to price changes.