506
APPENDIX A. MATHEMATICS BACKGROUND
Figure A.9: A strictly concave-contoured (strictly quasiconcave) function
There are functions for which the contours look like those of a concave
function but which are not themselves concave. An example here would
be ' (f (x)) where f is a concave function and is an arbitrary monotonic
transformation.
These remarks lead us to the de…nition:
De…nition A.23 A function f is (strictly) concave-contoured if all the sets
B(y0 ) in (A.31) are (strictly) convex.
A synonym for (strictly) concave-contoured is (strictly) quasiconcave. Try
not to let this (unfortunately necessary) jargon confuse you. Take, for example,
a “conventional” looking utility function such as
U (x) = x1 x2 :
(A.32)
According to de…nition A.23 this function is strictly quasiconcave: if you draw
the set of points B( ) := f(x1 ; x2 ) : x1 x2
g you will get a strictly convex
set. Furthermore, although U in (A.32) is not a concave function, it is a simple
transformation of the strictly concave function
^ (x) = log x1 + log x2 ;
U
(A.33)
^ . But when we draw those contours
and has the same shape of contour map as U
on a diagram with the usual axes we would colloquially describe their shape
A.7. MAXIMISATION
507
as being “convex to the origin”! There is nothing seriously wrong here: the
de…nition, the terminology and our intuitive view are all correct; it is just a
matter of the way in which we visualise the function. Finally, the following
complementary property is sometimes useful:
De…nition A.24 A function f is (strictly) quasiconvex if
siconcave.
A.6.6
f is (strictly) qua-
The Hessian property
Rn to R. Let fij (x) denote
Consider a twice-di¤erentiable function f from D
@ 2 f (x)
@xi @xj . The symmetric matrix
2
f11 (x) f12 (x)
6 f21 (x) f22 (x)
6
4 :::
:::
fn1 (x) fn2 (x)
is known as the Hessian matrix of f .
3
::: f1n (x)
::: f2n (x) 7
7
5
:::
:::
::: fnn (x)
De…nition A.25 The Hessian matrix of f at x is negative semide…nite if, for
any vector w 2 Rn , it is true that
n X
n
X
wi wj fij (x)
0:
i=1 j=1
A twice-di¤erentiable function f from D to R is concave if and only if f is
negative semi-de…nite for all x 2 D.
De…nition A.26 The Hessian matrix of f at x is negative de…nite if, for any
vector w 2 Rn ,w 6= 0, it is true that
n X
n
X
wi wj fij (x) < 0:
i=1 j=1
A twice-di¤erentiable function f from D to R is strictly concave if f is
negative de…nite for all x 2 D; but the reverse is not true – a strictly concave
function f may have a negative semi-de…nite Hessian.
If the Hessian of f is negative de…nite for all x 2 D we will say that f has
the Hessian property.
A.7
Maximisation
Because a lot of economics is concerned with optimisation we will brie‡y overview
the main techniques and results. However this only touches the edge of a very
big subject: you should consult the references in section A.9 for more details.
508
APPENDIX A. MATHEMATICS BACKGROUND
A.7.1
The basic technique
The problem of maximising a function of n variables
max f (x)
x2X
(A.34)
X Rn is straightforward if the function f is di¤erentiable and the domain X
is unbounded. We adopt the usual …rst-order condition (FOC)
@f (x)
= 0; i = 1; 2; :::; n
@xi
(A.35)
and then solve for the values of (x1 ; x2 ; :::; xn ) that satisfy (A.35). However the
FOC is, at best, a necessary condition for a maximum of f . The problem is
that the FOC is essentially a simple hill-climbing rule: “if I’m really at the top
of the hill then the ground must be ‡at just where I’m standing.” There are a
number of di¢ culties with this:
The rule only picks out “stationary points” of the function f . As Figure
A.10 illustrates, this condition is satis…ed by a minimum (point C) as well
as a maximum (point A), or by a point of in‡ection (E). To eliminate
points such as C and E we may look at the second-order conditions which
essentially require that at the top of the hill (a point such as A) the slope
must be (locally) decreasing in every direction.
Even if we eliminate minima and points of in‡ection the FOC may pick
out multiple “local” maxima. In Figure A.10 points A and D are each
local maxima, but obviously A is the point that we really want. we may
be able to eliminate. This problem may be sidestepped by introducing
a priori restrictions on the nature of the function f that eliminate the
possibility of multiple stationary points –for example by requiring that f
be strictly concave.
If we have been careless in specifying the problem then the hill-climbing
rule may be completely misleading. We have assumed that each x-component
can range freely from 1 to +1. But suppose – as if often in the case
in economics – that the de…nition of the variable is such that only nonnegative values make sense. Then it is clear from Figure A.10 that A is
an irrelevant point and the maximum is at B. In climbing the hill we have
reached a logical “wall” and we can climb no higher.
Likewise if we have overlooked the requirement that the function f be
everywhere di¤erentiable the hill-climbing rule represented by the FOC
may be misleading. If we draw the function
8
x 1
< x
f (x) =
:
2 x
x>1
A.7. MAXIMISATION
509
Figure A.10: Di¤erent types of stationary point
it is clear that it is continuous and has a maximum at x = 1. But the
FOC as stated in (A.35) is useless because the di¤erential of f is unde…ned
exactly at x = 1.
If we can sweep these di¢ culties aside then we can use the solution to the
system of equations provided by the FOC in a powerful way. To see what is
usually done, slightly rewrite the maximisation problem (A.34) as
max f (x; p)
x2Rn
(A.36)
where p represents a vector of parameters, a set of numbers that are …xed for the
particular maximisation problem in hand but which can be used to characterise
the di¤erent members of a whole class of maximisation problems and their
solutions. For example p might represent prices (outside the control of a small
…rm and therefore taken as given) and might x represent the list of quantities
of inputs and outputs that the …rm chooses in its production process; pro…ts
depend on both the parameters and the choice variables.
We can then treat the FOC (A.35) as a system of n equations in n unknowns
(the components of x).Without further regularity conditions such a system is
not guaranteed to have a solution nor, if it has a solution, will it necessarily
be unique. However, if it does then we can write it as a function of the given
510
APPENDIX A. MATHEMATICS BACKGROUND
parameters p:
x1 = x1 (p)
x2 = x2 (p)
:::
xn = xn (p)
9
>
>
=
(A.37)
>
>
;
We may refer to the functions x1 ( ) in (A.37) as the response functions in that
they indicate how the optimal values of the choice variables (x ) would change
in response to changes in values of the given parameters p.
A.7.2
Constrained maximisation
By itself the basic technique in section A.7.1 is of limited value in economics:
optimisation is usually subject to some side constraints which have not yet been
introduced. We now move on to a simple case of constrained optimisation that,
although restricted in its immediate applicability to economic problems, forms
the basis of other useful techniques. We consider the problem of maximising a
di¤erentiable function of n variables
max f (x; p)
(A.38)
x2Rn
subject to the m equality constraints
G1 (x; p) = 0
G2 (x; p) = 0
:::
Gm (x; p) = 0
9
>
>
=
(A.39)
>
>
;
There is a standard technique for solving this kind of problem: this is to incorporate the constraint in a new maximand. To do this introduce the Lagrange
multipliers 1 ; :::; m , a set of non-negative variables, one for each constraint.
The constrained maximisation problem in the n variables x1 ; :::; xn , is equivalent
to the following (unconstrained) maximisation problem in the n + m variables
x1 ; :::; xn ; 1 ; :::; m ,
L(x; ; p) := f (x; p)
m
X
jG
j
(x; p)
(A.40)
j=1
, where L is the Lagrangean function. By introducing the Lagrange multipliers
we have transformed the constrained optimisation problem into one that is of
the same format as in section A.7.1, namely
max L(x; ; p)
x;
(A.41)
A.7. MAXIMISATION
511
The FOC for solving (A.41) are found by di¤erentiating (A.40) with respect
to each of the n + m variables and setting each to zero.
@L(x ; ; p)
@xi
@L(x ; ; p)
@ j
=
0; i = 1; :::; n
(A.42)
=
0; j = 1; :::; m
(A.43)
where the “ ” means that the di¤erential is being evaluated at a solution point
(x ; ). So the FOC consist of the n equations
m
@f (x ; p) X
=
@xi
j=1
@Gj (x ; p)
; i = 1; :::; n
j
@xi
(A.44)
plus the m constraint equations (A.39) evaluated at x . We therefore have a
system of n + m equations (A.44,A.39) in n + m variables.
As in section A.7.1, if the system of equations does have a unique solution
(x ; ), then this can be written as a function of the parameters p:
9
x1 = x1 (p) >
>
=
x2 = x2 (p)
(A.45)
:::
>
>
;
xn = xn (p)
1
2
m
= 1 (p)
= 2 (p)
:::
= m (p)
9
>
>
=
>
>
;
(A.46)
Once again the functions x1 ( ) in (A.45) are the response functions and have the
same interpretation. The Lagrange multipliers in (A.46) also have an interesting
interpretation which is handled in A.7.4 below.
If the equations (A.44,A.39) yield more than one solution, but f in (A.38)
is quasiconcave and the set of x satisfying (A.39) is convex then we can appeal
to the commonsense result in Theorem A.12.
A.7.3
More on constrained maximisation
Now modify the problem in section A.7.2 in two ways that are especially relevant
to economic problems
Instead of allowing each component xi to range freely from 1 to +1.we
restrict to some interval of the real line. So we will now write the domain
restriction x 2 X where we will take X to be the non-negative orthant of
Rn . The results below can be adapted to other speci…cations of X.
512
APPENDIX A. MATHEMATICS BACKGROUND
We replace the equality constraints in
equality constraints
G1 (x; p)
G2 (x; p)
:::
Gm (x; p)
(A.39) by the corresponding in9
0 >
>
=
0
(A.47)
>
>
;
0
This is reasonable in economic applications of optimisation. For example
the appropriate way of stating a budget constraint is “expenditure must
not exceed income” rather than “...must equal...”.
So the problem is now
max f (x; p)
x2X
subject to (A.47). The solution to this modi…ed problem is similar to that
for the standard Lagrangean – see Intriligator (1971), pages 49-60. Again we
transform the problem by forming a Lagrangean (as in A.40):
max
x2X; >0
L(x; ; p)
(A.48)
However, instead of (A.42, A.43)we now have the following FOCs:
@L(x ; ; p)
@xi
@L(x ; ; p)
xi
@xi
=
0; i = 1; :::; n
(A.49)
0; i = 1; :::; n
(A.50)
0; j = 1; :::; m
(A.51)
0; j = 1; :::; m
(A.52)
and
@L(x ; ; p)
@ j
@L(x ; ; p)
j
@ j
=
This set of equations and inequalities is conventionally known as the KuhnTucker conditions. They have important implications relating the values of the
variables and the Lagrange multipliers at the optimum.
Applying this result we …nd
@f (x ; p)
@xi
m
X
j=1
@Gj (x ; p)
; i = 1; :::; n
j
@xi
(A.53)
with (A.44) if xi > 0. Note that if, for some i, xi = 0 we could have strict
inequality in (A.53). Figure A.11 illustrates this possibility for a case where the
objective function is strictly concave: note that the conventional condition of
“slope=0” (A.42) (which would appear to be satis…ed at point A) is irrelevant
here since a point such as A would violate the constraint xi 0; at the optimum
(point B) the Lagrangean has a strictly decreasing slope. Similar interpretations
will apply to the Lagrange multipliers:
A.7. MAXIMISATION
513
Figure A.11: A case where xi = 0 at the optimum
1. If the Lagrange multiplier associated with constraint j is strictly positive
at the optimum ( j > 0), then it must be binding (Gj (x ; p) = 0).
2. Conversely one could have an optimum where one or more Lagrange multiplier ( j = 0) is zero in which case the constraint may be slack – i.e.
not binding –(Gj (x ; p) < 0).
So, for each j at the optimum, there is at most one inequality condition: if
there is a strict inequality on the Lagrange multiplier then the corresponding
constraint must be satis…ed with equality (case 1); if there is a strict inequality
on the constraint then the corresponding Lagrange multiplier must be equal
to zero (case 2). These facts are conventionally known as the complementary
slackness condition. However, note that one can have cases where both the
Lagrange multiplier ( j = 0) and the constraint is binding (Gj (x ; p) = 0).
Again if the system (A.53,A.47) yields a unique solution it can be written as
a function of the parameters p which in turn determines the response functions;
but if it yields more than one solution, but f in (A.38) is quasiconcave and the
set of x satisfying (A.47) is convex then we can use the following.
Theorem A.12 If f : Rn 7! R is quasiconcave and A
set of values x that solve the problem
max f (x) subject to x 2 A
is convex.
Rn is convex then the