Utility Maximization and Expenditure Minimization Problems in A Simple Case Quiz Questions y solving the utility maximization and expenditure minimization

Utility Maximization and Expenditure Minimization Problems in A Simple Case Quiz Questions y solving the utility maximization and expenditure
minimization problems in a simple case. PROBLEM SET 9 – due Tue at 11pm
(do your best to solve it by Sunday)
First, let’s get some practice for the final quiz by solving the utility maximization and expenditure
minimization problems in a simple case. Then we can use these results to calculate the effect of
a price change on consumer welfare.
1. Suppose Tom likes both ketchup (x) and mayo (y) on his hotdogs: ( , ) = 2
a. Solve the utility maximization problem to find the ordinary (uncompensated) demand
functions ∗ ( , , ) and ∗ ( , , ), and the indirect utility function ( , , ).
b. Solve the expenditure minimization problem to find the compensated demand
functions ( , , ) and ( , , ), and the expenditure function ( , , )
c. Tom spends $18 each year on these condiments. As usual in this class, assume items
are infinitely divisible (i.e. Tom can buy any fraction of a ketchup or mayo bottle he
chooses). Initially, bottles of ketchup and mayo are equally priced at $3 per bottle.
Find Tom’s optimal basket of condiments ( , ), and the utility he gets from
consuming them.
d. Suppose that ketchup becomes more popular and its price increases to $6/bottle.
(Tom’s preferences are not affected.) What is the new bundle (call it “C”) Tom will
buy, and what utility will it bring him?
e. If Tom’s condiments budget could be increased so that his utility did not decrease as
a result of the price change, what bundle (“B”) would he buy? How much would this
transfer – this compensating variation in his budget – have to be? (You can use that
3
√32 ≅ 3.17)
f. If prices were kept at their initial level ( = = 3), but some money was taken from
Tom so that his utility became the same as at the bundle C you found above, how
much would this equivalent variation in his budget have to be, and what basket (“D”)
would he purchase?
g. [Optional: how might you calculate the change in consumer surplus Tom experiences
as a result of the price change? See how far you can get. (Hint: you will need to
calculate an integral – gasp!)]
The next two problems are about budget constraints. It will help to watch this video first:

2. Nadia consumes frozen yogurt cups (x) and pretzels (y) whenever she goes out. Suppose she
allocates $60 each month to just these 2 goods.
a. If yogurt cups cost $5 each and pretzels cost $2.50, draw Nadia’s budget constraint.
Next, use the graphs on the next page to show how the budget constraint changes
when the shops implement each of the different pricing strategies (consider them
separately, at least at first)
Next to each line segment of each graph, write the corresponding budget constraint
equation (you don’t have to specify the range, as long as the coordinates of the end
points are clearly visible in the graph)
1
b. The pretzel shop introduces a 60% discount for any pretzels that a consumer buys
beyond the first 20.
c. The owner of the yogurt shop introduces reward stamp cards: after every time
Nadia buys 4 yogurt cups, she gets a 5th one for free.
How does this case compare to a simple 20% discount? Which would consumers
prefer?
d. The yogurt shop introduces yet another pricing strategy: they will offer a
membership, which costs $15/month, and which gives the consumer a 50% discount
on every cup of yogurt they buy. (neither of the earlier promotions are in place
anymore)
2
3. Electricity pricing:
Unlike many goods for which there are quantity discounts (the more you buy, the less
you pay per unit), electricity pricing can include quantity surcharges (the more you
consume, the more you pay per unit). Electricity usage is measured in kilowatt hours
(kWh). To give you an idea, 1 kWh is how much a 10-watt LED bulb uses in 100 hours, or
how much an old-school 100-watt light bulb uses in 10 hours.
The current pricing structure for a house in San Diego can be approximated as follows:
– Tier 1 consumption: 20 cents per kWh for the first 400 kWh
– Tier 2: 40c/kWh on the interval 400-1200kWh
– Tier 3: 50c/kWh above 1200kWh
Consider a household that has a monthly budget of $1000 after paying rent. Let
electricity be on the horizontal axis (good x) and group all other purchases into an
“other” category, placing it on the vertical axis (y). Assuming for convenience that =1,
draw the household’s monthly budget constraint in the graph below.
Just as at problem 2, write the corresponding budget constraint equation next to each
line segment.
3

Purchase answer to see full
attachment