Dirk Bergemann
Solutions prepared by Vincent Pohl
Department of Economics
Yale University
Economics 121b: Intermediate Microeconomics
Problem Set 6: Concavity
2/17/10
Common mistakes:
• Graphs were not labelled, for example f (λx + (1 − λ)y) and min {f (x),
f (y)}were not indicated in problem 1(a).
• In problem 1(d), the negative second deritive of a concave function was
used in the argument without showing that concavity implies a negative
second dericative.
• In problem 2(f), the price change led to a consumption bundle on a different indifference curve, which is not the case for compensated demand.
1. Concavity and quasi-concavity
(a) See Figures 1 and 2.
(b) The function in Figure 2 is quasi-concave, but not concave. It has
parts on the left and right that are convex. As we can see in this
graph,
λf (z) + (1 − λ) f (y) ≥ f (λz + (1 − λ) y) ,
which violates concavity. In terms of the second derivative the difference between concave and quasi-concave functions is that concave
functions have f !! (x) < 0, which is not necessarily the case for quasiconcave functions.
(c) Let’s start with the definition of concavity:
f (λx + (1 − λ) y)
≥ λf (x) + (1 − λ) f (y)
= f (y) + λ [f (x) − f (y)]
≥ f (y)
if f (x) ≥ f (y). We can also write this definition as follows:
f (λx + (1 − λ) y)
≥ λf (x) + (1 − λ) f (y)
= f (x) + (λ − 1)f (x) + (1 − λ)f (y)
= f (x) + (1 − λ) [f (y) − f (x)]
≥ f (x)
1
y
f(y)
f(!z+(1-!)y)
!f(z)+(1-!)f(y)
f(z)
!z+(1-!)y
z
Figure 1: A concave function
z
f(z)
!f(z)+(1-!)f(y)
!z+(1-!)y
f(!z+(1-!)y)
y
f(y)=min{f(y),f(z)}
Figure 2: A quasi-concave function
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if f (y) ≥ f (x). Hence, we have either
f (λx + (1 − λ) y) ≥ f (y)
or
f (λx + (1 − λ) y) ≥ f (x),
which implies
f (λx + (1 − λ) y) ≥ min {f (x), f (y)} ,
the definition of quasi-concavity.
(d) Note that the question should have read “Argue that all extrema
(critical points) of a concave function are global maxima.”
Start with the definition of concavity:
f (λx + (1 − λ) y) ≥ λf (x) + (1 − λ) f (y) ,
which can be rewritten as
f (y + λ(x − y)) ≥ f (y) + λ [f (x) − f (y)]
and further as
f (y + λ(x − y)) − f (y)
≥ f (x) − f (y).
λ
Taking the limit as λ goes to zero we get (note that the RHS does
not depend on λ):
f (y + λ(x − y)) − f (y)
= f ! (y) ≥ f (x) − f (y).
λ→0
λ
lim
(1)
Suppose y is a critical point so that f ! (y) = 0. Hence equation (1)
simplifies to
f (x) − f (y) ≤ 0 ⇔
f (x) ≤ f (y).
Since this inequality holds for all x, we get that the function f is
lower at any point other than the critical point y, implying that y is
a global maximum of f .
2. Quasi-concavity and the upper contour set
(a) See Figure 3. Note that for points y and z the linear combination
is identical to the (linear part of) the function. Hence the strict
inequality in the definition of strict quasi-concavity is violated.
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y !y+(1-!)z
f(!y+(1-!)z)
=min{f(y),f(z)}
z
!x+(1-!)y
x
Figure 3: A quasi-concave function that is not strictly quasi-concave
(b) From Figure 2, which depicts a strictly quasi-concave function, and
the quasi-concave function in Figure 3 it is clear that the former
can only have at most one global maximum, whereas the latter may
have many. More formally, the definition of strict quasi-concavity
contains a strict inequality, and this inequality has to hold for all
x %= y. Suppose the strictly quasi-concave function has a maximum
and calling it
z = λx + (1 − λ)y
we see that f (z) has to be strictly larger than both f (x) and f (y).
Since we can pick x and y as close to z as we want, the function
is lower at any point other than z. This means that z is a global
maximum. In Figure 3, however, we can pick x and y such that
f (z) = f (x) = f (y), which means that all three points are global
maxima.
(c) See Figure 4. The shaded area is the set of all consumption bundles
that yield higher utility than ū, which is the utility level associated
with the depicted indifference curve.
(d) Note that this question should have read “Argue that strict quasiconcavity (1) implies the convexity of the upper contour sets (2).”
Consider Figure 5, which shows the same strictly quasi-concave function as Figure 2. The upper contour set U C(z) is convex. Why are
these two definitions equivalent? A strictly quasi-concave function
has only at most one global maximum. Hence, the function cannot
raise once it started to drop. In other words, when the function
drops below the cutoff z, it stays below z. The upper contour set can
therefore not consist of several segments, which would make it non4
good 2
x
#x+(1-#)y"x
{x!R!|u(x)>"}
y
"
good 1
Figure 4: The upper contour set of a utility function
z
UC(z)
Figure 5: Equivalence of the two definitions of strict quasi-concavity
5
z
UC(z)
UC(z)
Figure 6: A non-convex upper contour set
convex. Consider Figure 6 where the upper contour set is not convex
as a counterexample. Obviously this function is not quasi-concave.
(e) In part (d) we showed that strict quasi-concave functions have convex
upper contour sets and in part (c) we illustrated a convex upper contour set for a utility function. In Figure 4, the convex combination
between two points on the same indifference curve is an element of
the upper contour set (note that the upper contour set does not include the indifference curve itself). Hence, this consumer has strictly
convex preferences.
(f) See Figure 7, which depicts an increase in the price of good 1. The
∂xc (p,U )
substitution effect or 1∂p1
is the change in demand for good 1
given the change in its price holding utility constant. This is labelled
as “SE” in the graph. We move along the indifference curve as the
slope of the budget line becomes steeper. With strictly convex preferences, the demand for good 1 has to decrease as its price increases. A
bundle to the right of the initial bundle also has to be below the initial bundle to be on the same indifference curve. This is inconsistent
with an increase in p1 . Hence we have that
!
"
∂xci p, U
< 0.
∂pi
(p,M )
The same does not hold for uncompensated demand because ∂xi∂p
i
consists of substitution and income effect. The income effect can
prevail as we will see in question 3, making the own-price effect of
uncompensated demand positive.
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good 2
bundle after price change
bundle with same utility as initial
initial bundle
IE
SE
good 1
Figure 7: Income and substitution effects
3. Giffen good. Starting with the definition of a Giffen good we have
∂xi (p, M )
> 0.
∂pi
Using the Slutsky equation for the own-price effect of good i this implies
!
"
∂xci p, U
∂xi (p, M )
− xi
>0
∂pi
∂M
or
!
"
∂xci p, U
∂xi (p, M )
> xi
.
(2)
∂pi
∂M
From question 2(f) we know that
!
"
∂xci p, U
< 0.
∂pi
Hence, for inequality (2) to hold we need at least
∂xi (p, M )
< 0.
∂M
This is the definition of an inferior good, which means that every Giffen
good is also an inferior good. On the other hand, not every inferior good
is a Giffen good because the RHS of equation (2) could be negative but
larger than the LHS. In that case the own-price effect would be negative.
In other words, for a good to be a Giffen good, we need the income effect
(p,M )
times demand xi ∂xi∂M
to be smaller (or larger in absolute value) than
∂xci (p,U )
the substitution effect
.
∂pi
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