Exercise Set 1
1) Show that Lexiogarphic preferences are transitive and complete, but not continous.
2) On a recent doctors visit, you have been told that you must watch your calorie
intake and must make sure you get enough vitamin E in your diet.
A. You have decided that, to make life simple, you will from now on eat only steak
and carrots. A nice steak has 250 calories and 10 units of vitamins, and a serving of
carrots has 100 calories and 30 units of vitamins. Your doctors instructions are that
you must eat no more than 2,000 calories and consume at least 150 units of vitamins
per day.
a. In a graph with servings of carrots on the horizontal axis and servings of steak
on the vertical axis, illustrate all combinations of carrots and steaks that make up a
2,000-calorie-a-day diet.
b. On the same graph, illustrate all the combinations of carrots and steaks that
provide exactly 150 units of vitamins.
c. On this graph, shade in the bundles of carrots and steaks that satisfy both of
your doctors requirements.
d. Now suppose you can buy a serving of carrots for $2 and a steak for $6.
You have $26 per day in your food budget. In your graph, illustrate your budget
constraint. If you love steak and dont mind eating or not eating carrots, what bundle
will you choose (assuming you take your doctors instructions seriously)?
B. Continue with the scenario as described in part A, letting carrots be denoted by
x1 and steak by x2 .
a. Define the line you drew in A(a) mathematically.
b. Define the line you drew in A(b) mathematically.
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c. In formal set notation, write down the expression that is equivalent to the
shaded area in A(c).
d. Derive the exact bundle you indicated on your graph in A(d).
3) Airlines offer frequent flyers different kinds of perks that we will model here
as reductions in average prices per mile flown. A. Suppose that an airline charges 20
cents per mile flown. However, once a customer reaches 25,000 miles in a given year,
the price drops to 10 cents per mile flown for each additional mile. The alternate way
to travel is to drive by car, which costs 16 cents per mile.
a. Consider a consumer who has a travel budget of $10,000 per year, a budget
that can be spent on the cost of getting to places as well as other consumption while
traveling. On a graph with miles flown on the horizontal axis and other consumption
on the vertical, illustrate the budget constraint for someone who only considers flying
(and not driving) to travel destinations.
b. On a similar graph with miles driven on the horizontal axis, illustrate the budget
constraint for someone that considers only driving (and not flying) as a means of
travel.
c. By overlaying these two budget constraints (changing the good on the horizontal axis simply to miles traveled), can you explain how frequent flyer perks might
persuade some to fly a lot more than he or she otherwise would?
B. Determine where the air-travel budget from A(a) intersects the car budget from
A(b).
3) A. Suppose now that your grandparents set up a trust fund that pays you $300
per week. In addition, you have up to 60 hours of leisure that you could devote to
work at a wage of $20 per hour.
a. On a graph with leisure hours per week on the horizontal axis and weekly
consumption in dollars on the vertical, illustrate your weekly budget constraint.
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b. How does your graph change when your wage falls to $10?
c. How does the graph change if instead the trust fund gets raided by your parents,
leaving you with only a $100 payment per week?
B. How would you write your budget constraint described in part A?
4) Suppose you are a farmer whose land produces 50 units of food this year and
is expected to produce another 50 units of food next year. (Assume that there is no
one else in the world to trade with.) A. On a graph with food consumption this year
on the horizontal axis and food consumption next year on the vertical, indicate your
choice set assuming there is no way for you to store food that you harvest this year
for future consumption.
a. Now suppose that you have a barn in which you can store food. However, over
the course of a year, half the food that you store spoils. How does this change your
choice set?
b. Now suppose that, in addition to the food units you harvest off your land, you
also own a cow. You could slaughter the cow this year and eat it for 50 units of food.
Or you could let it graze for another year and let it grow fatter, then slaughter it next
year for 75 units of food. But you dont have any means of refrigeration and so you
cannot store meat over time. How does this alter your budget constraint (assuming
you still have the barn from part (a))?
B. How would you write the choice set you derived in A(b) mathematically, with
indicating this years food consumption and indicating next years food consumption?
5) Different Interest Rates for Borrowing and Lending: Suppose we return to the
example from the text in which you earn $5,000 this summer and expect to earn
$5,500 next summer.
A. In the real world, banks usually charge higher interest rates for borrowing than
they will give on savings. So, instead of assuming that you can borrow and lend at
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the same interest rate, suppose the bank pays you an interest rate of 5 percent on
anything you save but will lend you money only at an interest rate of 10 percent.
a. Illustrate your budget constraint with consumption this summer on the horizontal and consumption next summer on the vertical axis.
b. How would your answer change if the interest rates for borrowing and lending
were reversed?
c. A set is defined as convex if the line connecting any two points in the set also
lies in the set. Is the choice set in part (a) a convex set? What about the choice set in
part (b)?
d. Which of the two scenarios would you prefer? Give both an intuitive answer
that does not refer to your graphs and demonstrate how the graphs give the same
answer.
B. Suppose more generally that you earn e1 this year and e2 next year and that the
interest rate for borrowing is rB and the interest rate for saving is rS . Let c1 and c2
denote consumption this year and next year.
a. Derive the general expression for your intertemporal choice set under these
conditions.
6) I hate grits so much that the very idea of owning grits makes me sick. I do, on
the other hand, enjoy a good breakfast of Coco Puffs Cereal.
A: In each of the following, put boxes of grits on the horizontal axis and boxes of
cereal on the vertical. Then graph three indifference curves and number them. (a)
Assume that my preferences satisfy the convexity and continuity assumptions and
otherwise satisfy the description above.
(b) How would your answer change if my preferences were non-convexi.e. if
averages were worse
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(c) How would your answer to (a) change if I hated both Coco Puffs and grits but
we again assumed my preferences satisfy the convexity assumption.
(d)What if I hated both goods and my preferences were non-convex?
B: Now suppose you like both grits and Coco Puffs, that your preferences satisfy
our five basic assumptions and that they can be represented by the utility function
u(x1 , x2 ) = x1 x2 .
(a) Consider two bundles, A=(1,20) and B=(10,2). Which one do you prefer?
(b) Use bundles A and B to illustrate that these preferences are in fact convex.
7) Consider my preferences for consumption and leisure.
A: Begin by assuming that my preferences over consumption and leisure satisfy
MON, CONV, CONT assumptions.
(a) On a graph with leisure hours per week on the horizontal axis and consumption dollars per week on the vertical, give an example of three indifference curves
(with associated utility numbers) from an indifference map that satisfies our assumptions.
(b) Now redefine the good on the horizontal axis as labor hours rather than leisure
hours. How would the same tastes look in this graph?
(c) How would both of your graphs change if tastes over leisure and consumption
were nonconvex i.e. if averages were worse than extremes.
B: Suppose your tastes over consumption and leisure could be described by the
utility function u(l, c) = l1 /2c1 /2.
(a) Do these tastes satisfy our 5 basic assumptions?
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