Consider public policy aimed at smoking.
Studies indicate that the price elasticity of demand for cigarettes in about 0.4. If a pack of cirarettes currently costs $2 and the government wants to reduce smoking by 20 percent, by how much should the govenment increase the price?
Answer in suppose to be $3.33. When I try to calculate using formulate I can not get the answer.
Formula is:
P2-P1
_________
(P2+P1)/2
I got some of the calulation:
Percentage change in P=
.20 Quantity/.4 Price Elasticity of Demand = 0.5
P2-2
_______ = 0.5
(P2+2)/2
I have tried the calculation several ways and I can not arrive at the answer of $3.33.
You are very close. Check your arithmatic.
To avoid confusion, I will use Pn to mean P2 (price in period 2)
You have, as stated (Pn - 2)/((Pn + 2)/2) = 0.5 So, solve for Pn.
First multiply numerator and denominator by 2/2 (=1). You get:
(2Pn - 4) / (Pn + 2) = 0.5
Now multiply through by the denominator. You get:
2Pn - 4 = .5Pn + 1
Move the Pn's to the left of the =, the constants to the right. so:
1.5Pn = 5
Divide both sides by 1.5:
Pn = 3.3333
QED
To calculate the increase in price needed to achieve a 20% reduction in smoking, you can use the formula for price elasticity of demand:
Price Elasticity of Demand = (% Change in Quantity Demanded) / (% Change in Price)
In this case, the price elasticity of demand is given as 0.4. We can assume that the percentage change in quantity demanded is -20% (since the government wants to reduce smoking by 20%). Let's denote the percentage change in price as P:
0.4 = (-20%) / P
First, convert the percentage change in quantity demanded to a decimal:
0.4 = (-0.2) / P
Now, cross-multiply and solve for P:
0.4P = -0.2
P = -0.2 / 0.4
P = -0.5
Since price cannot be negative, this result doesn't make sense. It seems there is an error or inconsistency in the provided information about the price elasticity of demand.
To arrive at the answer of $3.33, we can use an alternative approach. We can find the change in price needed to achieve the desired reduction in smoking by multiplying the current price by the desired percentage change:
% Change in Price = (-20%) * $2 = -$0.40
Since the government wants to increase the price, we take the absolute value of the change:
% Change in Price = $0.40
Now, to find the new price (P2), add the change in price to the current price:
P2 = $2 + $0.40
P2 = $2.40
Therefore, the government should increase the price by $0.40 to achieve a 20% reduction in smoking. It seems there is an error in the given answer of $3.33.
To find the increase in price needed to achieve a 20% reduction in smoking, you can use the price elasticity of demand formula. Let's break down the calculation step by step:
Step 1: Write down the known information:
- Price elasticity of demand (e) = 0.4
- Initial price (P1) = $2
- Desired reduction in smoking (%) = 20% = 0.2
Step 2: Substitute the known values into the formula:
(P2 - P1) / ((P2 + P1) / 2) = e
Step 3: Solve for P2 (the new price):
(P2 - 2) / ((P2 + 2) / 2) = 0.4
Step 4: Cross-multiply and simplify the equation:
2(P2 - 2) = 0.4(P2 + 2)
2P2 - 4 = 0.4P2 + 0.8
2P2 - 0.4P2 = 0.8 + 4
1.6P2 = 4.8
P2 = 4.8 / 1.6
P2 = $3
Step 5: Calculate the increase in price:
Increase in price = P2 - P1
Increase in price = $3 - $2
Increase in price = $1
Oops, it seems like there was an error in the calculation in my last response. I apologize for the confusion. Let me fix it.
To find the correct increase in price, let's re-calculate:
Step 1: Write down the known information:
- Price elasticity of demand (e) = 0.4
- Initial price (P1) = $2
- Desired reduction in smoking (%) = 20% = 0.2
Step 2: Substitute the known values into the formula:
(P2 - P1) / ((P2 + P1) / 2) = 0.4
Step 3: Solve for P2 (the new price):
(P2 - 2) / ((P2 + 2) / 2) = 0.4
Step 4: Cross-multiply and simplify the equation:
2(P2 - 2) = 0.4(P2 + 2)
2P2 - 4 = 0.4P2 + 0.8
2P2 - 0.4P2 = 0.8 + 4
1.6P2 = 4.8
P2 = 4.8 / 1.6
P2 = $3
Step 5: Calculate the increase in price:
Increase in price = P2 - P1
Increase in price = $3 - $2
Increase in price = $1
I apologize for my previous incorrect response. The correct answer is indeed an increase in price of $1.