Subject: Economics / General Economics
Question
Intermediate Microeconomics
Submission options:
• in person during lectures or TA sessions;
• in Xiaoye Liao’s mail box at 19 West 4th, 6th Floor;
• (less recommended) e-mail Xiaoye Liao at xl775@nyu.edu
– in that case, be sure to use the subject line: “Submission Problem Set 2”.
Exercise 1
Consider the following utility functions:
(1) u(x1 , x2 ) = x1 + 2×2 .
0.75
(2) u(x1 , x2 ) = 2×0.25
1 x2 . (3) u(x1 , x2 ) = ? 12 x21 ? x2 .
(4) u(x1 , x2 ) = min{x1 , 2×2 }.
(5) u(x1 , x2 ) = max{x1 , 2×2 }.
Give the equation of the indifference curves corresponding to utility level k for (1)-(3)
expressing x2 as a function of x1 . Use separate graphs to sketch indifference curves for
(1)-(5).
Exercise 2
Jim’s utility function is u(x1 , x2 ) = x1 x2 . Jerry’s utility function is u(x1 , x2 ) = 1000×1 x2
+2000. Tammy’s utility function is u(x1 , x2 ) = x1 x2 (1?x1 x2 ). Oral’s utility function is
u(x1 , x2 ) = ?1/(10+2×1 x2 ). Marjoe’s utility function is u(x1 , x2 ) = x1 (x2 +1000). Pat’s
utility function is u(x1 , x2 ) = 0.5×1 x2 ? 10000. Billy’s utility function is u(x1 , x2 ) =
x1 /x2 . Francis’s utility function is u(x1 , x2 ) = ?x1 x2 .
(1) Who has the same preferences as Jim?
(2) Who has indifference curves whose general shape is similar to Jim’s indifference
curves?
(3) Explain why the above answers differ.
Exercise 3 1 Consider a two-period economy with a single commodity (say leisure): x1 is the consumption of leisure in period 1, and x2 is the consumption of leisure in period 2. When
Peter evaluates consumption streams, he cares only about the best time in his life. On
the other hand, Christine only cares about the worst time in her life. Provide utility
representation for their preferences.
Exercise 4
1 1 Martina’s preferences are represented by u(x1 , x2 ) = x12 + x23 , where x1 > 0, x2 > 0.
(1) What is the marginal utility of good 1 at (x1 , x2 )? (2, 3)?
(2) What is the marginal utility of good 2 at (x1 , x2 )? (3, 2)?
(3) What is the marginal rate of substitution of good 2 for good 1 at (x1 , x2 )? (2, 2)?
1 1 (4) Redo (1), (2), (3) for another representation v(x1 , x2 ) = ln(x12 + x23 ).
Exercise 5
Burt’s utility function is u(x1 , x2 ) = (x1 + 2)(x2 + 6), where x1 is the number of cookies
and x2 is the number of glasses of milk that he consumes.
(1) What is the slope of Burt’s indifference curve at the point where he is consuming
(4, 6)? Draw a line with this slope through (4, 6).
(2) The indifference curve through (4, 6) passes through the points (·, 0), (7, ·) and
(2, ·). Give the missing coordinates. What is the equation for Burt’s indifference
curve through (4, 6)? Sketch it.
(3) Burt currently has the bundle (4, 6). Ernie offers to give Burt 9 glasses of milk if
Burt will give Ernie 3 cookies. If Burt makes this trade, which bundle would he have?
Mark that bundle on your graph. Burt refuses to trade. Is this a wise decision?
(4) Ernie says to Burt, “Burt, your marginal rate of substitution (MRS) is -2. That
means that an extra cookie is worth only twice as much to you as an extra glass of
milk. I offered to give you 3 glasses of milk for every cookie you give me. If I offer to
give you more than your MRS, then you should want to trade with me.” Complete
Burt’s reply: “Ernie, you are right that my MRS is -2. But. . . ” Would Burt be
willing to give up 1 cookie for 3 glasses of milk? Would Burt object to giving up 2
cookies for 6 glasses of milk?
Exercise 6
Assume that your preference relation % can be represented by a utility function.
Show that your preference relation must then be rational, i.e. complete, transitive, and
reflexive. 2 Exercise 7 (Bonus Question) This question will not be graded. This exercise is given
just in case there would be people, who for some reason, however strange it might be,
decide to learn a bit more about the theory of rational preferences.
Let X = R2+ be the set of all bundles that the consumer could consider. Let be the
consumer’s weak preference relation on X. Suppose that is rational, i.e. complete,
transitive, and reflexive. In the class, we briefly discussed what it means that the
preference relation is continuous. Namely:
Definition 1 (Continuous Preferences) The weak preference relation on X =
R2+ is called continuous if for any two bundles x ? X and y ? X and any sequence of
bundles {xn } converging to x, i.e. xn ? x as n ? ?, the following two implications
hold:
• if for all n we have that xn y, then x y;
• if for all n we have that xn y, then x y.
Continuity of preferences is closely related to the continuity of utility function representing these preferences. In particular:
(1) Show that if the preference relation on X is represented by a continuous utility
function u, then itself is continuous.
What about the reverse? Is it true that continuous rational preferences on X can
always be represented by a continuous utility function? Remarkably, the answer is
“Yes” as was shown by Gerard Debreu:
Theorem 1 (Debreu) If the weak preference relation on X = R2+ is complete,
transitive, reflexive, and continuous, then it could be represented by a continuous
utility function.
Typically, we will be considering continuous rational preferences. Therefore, such
preferences would always have a continuous utility representation. In the absence of
continuity, however, the preferences may have no utility representation! The following
example illustrates this point.
Example. The lexicographic preference relation L is defined as x L y if either
“x1 > y1 ” or “x1 = y1 and x2 ? y2 .” The name derives from the way a dictionary is
organized; that is commodity 1 has the highest priority in determining the preference
ordering, just as the first letter of a word does in the ordering of a dictionary. When
the level of commodity is the same in two bundles, the amount of commodity 2 comes
into play.
(2) Verify that L is complete, transitive, reflexive, and monotone.
(3) For each x ? R2+ , find the indifference set of x,
I(x) = {y ? R2+ : y ? x}.
(4) Show that L is not continuos.
(5) (Harder) Show that L can not be represented by a utility function (even by a
noncontinuous one). (Hint: argue by contradiction.) 3
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