Asked by Austin Bogle on Apr 26, 2024
Verified
Ms.Quasimodo has the utility function U(x, m) 100x x2/2 m, where x is her consumption of earplugs and m is money left over to spend on other stuff.If she has $10,000 to spend on earplugs and other stuff and if the price of earplugs rises from $50 to $70, then her net consumer's surplus
A) increases by 400.
B) falls by 2,800.
C) falls by 600.
D) falls by 800.
E) increases by 1,600.
Utility Function
A mathematical representation that translates the satisfaction or happiness a consumer gains from consuming quantities of goods and services into a numerical value.
Consumer's Surplus
The difference between what consumers are willing to pay for a good or service and what they actually pay.
Earplugs
Devices inserted in the ear canal to protect the ears from loud noises, water, foreign bodies, or excessive wind.
- Ascertain the relationship between alterations in consumer surplus and price modifications.
Verified Answer
FD
fatemeh dashtbozorgApr 28, 2024
Final Answer :
D
Explanation :
To find the net consumer's surplus, we need to first find the utility-maximizing consumption bundle before and after the price increase. The utility-maximizing consumption bundle can be found by setting the marginal utility per dollar of each good equal to each other. Mathematically, it can be represented as:
MUx/Px = MUm/Py
where MUx is the marginal utility of earplugs, Px is the price of earplugs, MUm is the marginal utility of other goods, and Py is the price of other goods.
So, the utility-maximizing consumption bundle before the price increase can be found by setting MUx/Px = MUm/Py = 100/50 = 2. And, we know that Ms. Quasimodo has $10,000 to spend, so she can buy 200 earplugs and have $0 left over for other stuff.
After the price increase, the new price of earplugs is $70. So, the new marginal utility per dollar of earplugs becomes:
MUx/Px = 100/70 = 10/7
Setting MUx/Px = MUm/Py, we can find the new optimal consumption bundle as:
10/7x = Umx = 1000 - m
m = 1000 - (10/7)x
Total spending is $10,000, so we have:
70x + m = 10,000
Substituting m from above, we get:
70x + 1000 - (10/7)x = 10,000
Solving for x, we get:
x = 105
So, Ms. Quasimodo can now buy 105 earplugs and have $1,450 left over for other stuff.
To find the net consumer's surplus, we need to find the difference between the total utility before and after the price increase and subtract the increase in spending:
Net consumer's surplus = U(200, 0) - U(105, 1450) - (70-50)*105
Net consumer's surplus = 20,000 - 19,200 - 2100 = 700
Therefore, Ms. Quasimodo's net consumer's surplus falls by $800 (option D).
MUx/Px = MUm/Py
where MUx is the marginal utility of earplugs, Px is the price of earplugs, MUm is the marginal utility of other goods, and Py is the price of other goods.
So, the utility-maximizing consumption bundle before the price increase can be found by setting MUx/Px = MUm/Py = 100/50 = 2. And, we know that Ms. Quasimodo has $10,000 to spend, so she can buy 200 earplugs and have $0 left over for other stuff.
After the price increase, the new price of earplugs is $70. So, the new marginal utility per dollar of earplugs becomes:
MUx/Px = 100/70 = 10/7
Setting MUx/Px = MUm/Py, we can find the new optimal consumption bundle as:
10/7x = Umx = 1000 - m
m = 1000 - (10/7)x
Total spending is $10,000, so we have:
70x + m = 10,000
Substituting m from above, we get:
70x + 1000 - (10/7)x = 10,000
Solving for x, we get:
x = 105
So, Ms. Quasimodo can now buy 105 earplugs and have $1,450 left over for other stuff.
To find the net consumer's surplus, we need to find the difference between the total utility before and after the price increase and subtract the increase in spending:
Net consumer's surplus = U(200, 0) - U(105, 1450) - (70-50)*105
Net consumer's surplus = 20,000 - 19,200 - 2100 = 700
Therefore, Ms. Quasimodo's net consumer's surplus falls by $800 (option D).
Learning Objectives
- Ascertain the relationship between alterations in consumer surplus and price modifications.