Microeconomia: Preferenze Lessicografiche (Dimostrazioni Delle Proprietà)

Problem Set 2 - Answer Key
Professor: Paul Glewwe TA: Julieth Santamaria
September 21,2017
1. Show whether lexicographic preferences are complete, transitive, (strongly) monotone, and (strictly)
convex.
Sol.
1. Completeness means that for all x and y, either x y or y x. Suppose that without loss of
generality only two goods, x1 ≥ y1 , then x y. If x1 = y1 and x2 > y2 , then x y; if x1 = y1 and
x2 = y2 , then x ∼ y. Then lexicographic preferences are complete.
2. To show transitivity, suppose that x y z. If x1 > y1 > z1 , transitivity holds, since x1 > z1 imply
x z. If x1 = y1 > z1 , again transitivity holds, since x1 > z1 imply x z. Suppose x1 = y1 = z1 .
Then if x2 > y2 ≥ z2 or x2 = y2 > z2, we have x z. The only remaining case is x ∼ y ∼ z, for which
the transitivity relationship is trivial
3. Strong monotonicity: suppose x ≥ y and x 6= y. Then either x1 > y1 , or x1 = y1 and x2 > y2 , and
in either case x y.
4.To show convexity, suppose that y x and z x. This implies that y1 and z1 are at least as good
as x1 , so that if either y1 or z1 is strictly greater than x1 , we have αz + (1α)y x. Suppose then that
y1 = z1 = x1 . Then since y x and z x, we must have y2 , z2 > x2 , and αz + (1 − α)y x.
2. State and prove whether the following statements are true or false.
• Any homothetic, continuous, and monotonic preference relation on the commodity bundle space
can be represented by a continuous utility function that is homogeneous of degree one.
Sol.
True. Continuous and monotonic preferences imply that if x ∼ y, then U (x) = U (y) ⇒ αU (x) =
αU (y). On the other hand it also implies that if αx ∼ αy, then U (αx) = U (αy).
Preferences are homothetic if x ∼ y implies αx ∼ αy. Then it must be true that U (αx) = U (αy) =
αU (x) = αU (y)
• If preferences are transitive, convex, and continuous, they must be locally non-satiated
Sol.
False. One example is the utility function in 4.(c)
1
• If the utility function is homothetic, the marginal utility of income is independent of prices and
depends only on income
Sol.
False. First, let’s show that x is HD1 in w by contradiction. We know that x(p, αw) and αx(p, w)
are feasible on αw. But x(p, αw) is the optimal choice. Then, it must be true that:
U (x(p, αw)) > U (αx(p, w))
U (x(p, αw)) U (αx(p, w))
>
α α
x(p, αw) αx(p, w)
U >U
α α
x(p, αw)
U > U (x(p, w))
α
The third line is because when preferences are homothetic, then U is HD1. The latter line implies
a contradiction because the term on the right should be the optimum, yet, there is another bundle
that is affordable and that reports more utility. So, it must be true that x(p, αw) = αx(p, w) if
preferences are homothetic.
If that is true, then x(p, w) = wx(p). Then, the marginal utility of income can be written as follows
∂v(x(p, w)) ∂v(wx(p))

= = v(x(p))
∂w ∂w
• Lexicographic preferences are homothetic.
Sol.
True. The proof is trivial. If x1 > y1 , then x y.∀α > 0, then it is also true that αx1 > αy1 ,
which implies αx αy. Similarly, if x1 = y, then x ∼ y, and αx1 = αy, then αx ∼ αy
3. A consumer has preferences for goods 1 and 2 defined by: (x1 , x2 ) (z1 , z2 ) ⇐⇒ x1 ≥ z1 and
x1 + x2 ≥ z1 + z2 . Show whether these preferences are complete, reflexive, transitive, locally non-
satiable, and convex. Justify your answer and draw the indifference, upper contour, and lower contour
sets.
x2
z1 + z2 U CS
z2 •
LCS
x1
z1 z1 + z2
2
Sol.
These preference are not complete (because there are areas without color) but they are reflexive,
transitive, locally non satiated, convex but not continuous. The upper contour set is the green area,
the lower contour set is the yellow area, the indifference ”curve” is the red point.
4. Sketch indifference curves for the following utility functions. If it helps, you can do a transformation of
the utility function before sketching the indifference curves. In each case, say whether the underlying
preference ordering is convex or strictly convex
2
(a) U (x1 , x2 ) = emin{x1 ,x2 }
1
(b) U (x1 , x2 ) = x21 + ln x2
2
(c) U (x1 , x2 ) = 4 − [(x1 − 2)2 + (x2 − 3)2 ]
√
(d) U (x1 , x2 ) = min{ x1 x2 , x2 }

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Problem Set 2 - Answer Key

Professor: Paul Glewwe TA: Julieth Santamaria

∂v(x(p, w)) ∂v(wx(p))

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