Selected homework solutions
• Show that if u represents %, then g(u) represents % for any strictly increasing
g : R → R. Show that this not true for non-decreasing g. If u is strictly concave,
is g(u) for a strictly increasing g?
Answer: To show that g(u) represents u we need to show that x % y ⇔
g(u(x)) ≥ g(u(y)). Assume x % y. Then u(x) ≥ u(y) since u represents % and
g(u(x)) ≥ g(u(y)) since g is strictly increasing. Conversely, assume g(u(x)) ≥
g(u(y)). Then u(x) ≥ u(y) (if u(x) < u(y), then g(u(x)) < g(u(y)) for a strictly
increasing g). It follows that x % y since we assume u represents %.
To show that this is not true for non-decreasing g, it is enough to come up with
a counter example. Let g(·) = 0 (a constant function) and u(x) = log(x).
√
To show that g(u) is not necessarily strictly concave, let u(x) = x and g(u) =
u2 . u is strictly concave, and g is strictly increasing, but g(u(x)) = x is not
strictly concave.
• Suppose utility is given by U (x, y) = au(x) + bu(y). Let w be a monotonic g
transformation of u and W (x, y) = aw(x) + bw(y). Do U and W represent the
same preferences?
Answer: No. Let a = 1, b = 21 , and U (x, y) = x + 12 y. Let g(u) = u2 . Then,
according to U, (2, 4) ∼ (0, 8). But according to W , (0, 8) (2, 4).
• % is said to be weakly monotone if x ≥ y implies x % y. Show that if % is
transitive, locally nonsatiated, and weakly monotone, then it is monotone.
Answer: Assume y >> x We need to show that y x. By local nonsatiation, we
can find z close enough to x so that z x and y >> z. By weak monotonocity,
y % z. But z x, so by transitivity y x, which is what needed to show.
• Verify that lexicographic preferences are complete, transitive, strongly monotone, and strictly convex.
Answer: Assume for simplicity that the consumption set has two dimensions.
All the proofs are done by cases. For example, to show that it’s transitive, let
x % y and y % z.
1. Case 1: x1 > y1 and y1 > z1 . Then x1 > z1 and hence x % z.
2. Case 2: x1 = y1 and y1 > z1 . Then x1 > z1 and hence x % z.
1
3. Case 3: x1 > y1 and y1 = z1 . Then x1 > z1 and hence x % z.
4. Case 4: x1 = y1 and y1 = z1 . Then x2 ≥ y2 and y2 ≥ z2 . Hence x2 ≥ z2
and hence x % z.
Etc...
• Exhibit an example of a preference relation that is not continuous but representable by a utility function.
Answer: Consider the function

1 if x ≤ 1
u(x) =
0 if x > 1
Then 2 % 1 + 1/n for all n. But 1 2. So the preferences are not continuous.
• Show that WARP implies GWARP.
Answer: Assume WARP, so xt Rxs and xt 6= xs implies not xs Rxt . We need to
show that xt Rxs implies NOT xs P xt . We do this by cases.
1. Case 1: xt 6= xs . Then by assumption we get not xs Rxt , i.e. not ps xs ≥
ps xt , i.e. ps xs < ps xt , which is what we needed to show.
2. Case 2: xt = xs Then ps xs = ps xt and hence xt is not strictly within the
choice set when xs was chosen.
• (From Exercise 3.D.5 in MWG) Show that the indirect utility function is quasiconvex for CES preferences.
Answer: The indirect utility function for the parameters of the problem is given
by
w
1 ,
(pδ1 + pδ2 ) δ
where δ =
ρ
.
ρ−1
We need to show that that the set {(p, w) : v(p, w) ≤ u} is convex for any u.
Note that it is enough to show that that the set {p : v(p, w) ≤ u} is convex for
any u and w > 0. We do this by cases.
Case 1: δ ∈ (0, 1). Then pδ1 + pδ2 is a concave function. Hence the set {p :
1
pδ1 + pδ2 ≥ u} is convex for any u. Hence the set {p : (pδ1 + pδ2 ) δ ≥ u} is convex
1
for any u. Hence the set {p : (pδ1 + pδ2 )− δ ≤ u} is convex for any u. Hence the
1
set {p : w(pδ1 + pδ2 )− δ ≤ u} is convex for any u and w > 0.
2
Case 2: δ < 0. Then pδ1 +pδ2 is a convex function. Hence the set {p : pδ1 +pδ2 ≤ u}
1
is convex for any u. Hence the set {p : (pδ1 + pδ2 )− δ ≤ u} is convex for any u.
1
Hence the set {p : w(pδ1 + pδ2 )− δ ≤ u} is convex for any u and w > 0.
3