ams/econ 11b
supplementary notes
ucsc
Constrained optimization.
c
2010, Yonatan Katznelson
1.
Constraints
In many of the optimization problems that arise in economics, there are restrictions on
the values that the independent variables may take.
Example 1.
A firm’s cost function is given by
C
= 0
.
05
Q
2
A
+ 0
.
01
Q
A
Q
B
+ 0
.
03
Q
2
B
+ 10
Q
A
+ 15
Q
B
+ 12000
,
(1)
where
Q
A
and
Q
B
are the quantities of the firm’s product that are produced in the firm’s
two facilities A and B, respectively.
If the firm has a contract to produce 2000 units of
output, then how many units should they produce in each facility to
minimize
their cost?
The condition that total output should equal 2000, imposes the following restriction on
the variables
Q
A
and
Q
B
Q
A
+
Q
B
= 2000
,
(2)
together with the conditions
Q
A
, Q
B
≥
0, since neither output can be negative.
The condition in equation (2) is called a
constraint
because it
constrains
(restricts) the
possible values of the free variables
Q
A
and
Q
B
. The function to be optimized (the cost
function, in the example above) is called the
objective function
in these problems.
Example 2.
A firm’s output is given by the CobbDouglas model
Q
=
AK
α
L
β
,
(3)
where
Q
is the firm’s output,
K
is quantity of the firm’s capital input and
L
is the quantity
of the firm’s labor input. The constants,
α
,
β
and
A
are all positive and we also assume
that
α
+
β
= 1. If the prices per unit of capital and labor are
p
K
and
p
L
, respectively, and
the firm’s production budget is
B
, then
how should the firm allocate its budget to maximize
its output?
The constraint in this case is given by the equation
p
K
·
K
+
p
L
·
L
=
B,
(4)
reflecting the facts that (a) it costs
p
K
·
K
+
p
L
·
L
to use
K
units of capital and
L
units of
labor, and (b) the total cost must equal
B
. The objective function in this example is the
output,
Q
.
In what follows, we’ll see two approaches to solving this type of optimization problem.
One approach, substitution, is more elementary, and uses the constraint to reduce the
number of variables.
The second approach,
the method of Lagrange multipliers
, is more
sophisticated and actually introduces a new variable to the problem.
†
On the other hand,
†
Or more than one new variable, if there is more than one constraint.
1
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this additional variable yields important information about the optimization problem that
is more difficult to derive using the substitution method.
Also, it is often the case, that
the algebra involved in finding the relevant critical points is actually easier in the Lagrange
multiplier approach.
2.
Substitution
One approach to solving a constrained optimization problem is to use the constraint (or
constraints, if there is more than one) to reduce the number of variables, and transform the
problem to an
unconstrained optimization problem in fewer variables
.
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 Spring '08
 BINICI
 Economics, Critical Point, Optimization, Yd, ﬁrst order conditions

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