Production optimization : an example from last lecture
Suppose that x units of labor and y units of capital can produce
1
3
f (x, y ) = 60x 4 y 4 units of a certain product. Also suppose that each unit
of labor costs $100, whereas each unit of capital costs $200. Assume that
$30,000 is available to spend on production. How many units of labor and
how many units of capital should be utilized to maximize production?
A. 37.5 units of labor and 225 units of capital
B. 50 units of capital and 60 units of labor
C. 225 units of labor and 37.5 units of capital
D. 60 units of capital and 50 units of labor
Math 105 (Section 204)
Multivariable Calculus – Extremization
2011W T2
1/5
Production possibilities
Suppose that a firm makes two products A and B that use the same raw
materials. Given a fixed amount of raw materials and a fixed amount of
manpower, the firm must decide how much of its resources should be
allocated to the production of A and how much to B. If x units of A and y
units of B are produced, often x and y satisfy an equation
g (x, y ) = 0,
for example,
x2 y2
+ 2 = 1.
a2
b
The graph of this equation for x ≥ 0, y ≥ 0 is called a production
possibilities curve. A point (x, y ) on this curve represents a production
schedule for the firm, committing it to produce x units of A and y units of
B. The equation g = 0 (i.e., the relation between x and y ) is determined
by limitations on personnel and raw material available to the firm.
Math 105 (Section 204)
Multivariable Calculus – Extremization
2011W T2
2/5
An example
A firm produces two products A and B, where each unit of A yields a $3
profit, and each unit of B yields a $4 profit. If the firm operates with
9x 2 + 4y 2 = 18, 000
as the production possibilities curve, find the production schedule that
maximizes the profit function.
√
A. (0, 450)
√
B. (20 5, 0)
C. (20, 60)
D. (60, 20)
Here the first and second components denote units of A and B respectively.
Math 105 (Section 204)
Multivariable Calculus – Extremization
2011W T2
3/5
Sigma notation
A shorthand notation for expressing long sums: if m > n, we write
an + an+1 + · · · + am :=
m
X
ak
k=n
For instance,
1.
4+5+6+7+8+9=
9
X
k.
k=4
Math 105 (Section 204)
Multivariable Calculus – Extremization
2011W T2
4/5
Sigma notation
A shorthand notation for expressing long sums: if m > n, we write
an + an+1 + · · · + am :=
m
X
ak
k=n
For instance,
1.
4+5+6+7+8+9=
9
X
k.
k=4
2.
2
2
2
2
1 + 4 + 9 + 16 = 1 + 2 + 3 + 4 =
2
X
j 2.
j=1
Math 105 (Section 204)
Multivariable Calculus – Extremization
2011W T2
4/5
Sigma notation
A shorthand notation for expressing long sums: if m > n, we write
an + an+1 + · · · + am :=
m
X
ak
k=n
For instance,
1.
4+5+6+7+8+9=
9
X
k.
k=4
2.
2
2
2
2
1 + 4 + 9 + 16 = 1 + 2 + 3 + 4 =
2
X
j 2.
j=1
3.
49
X
1
1
1
1
+
+ ··· +
=
.
1·2 2·3
49 · 50
k(k + 1)
k=1
Math 105 (Section 204)
Multivariable Calculus – Extremization
2011W T2
4/5
An example
Write down the following sum in sigma notation:
4 + 7 + 10 + · · · + 22.
A.
22
X
k
k=4
B.
7
X
(4 + 3k)
k=0
C.
7
X
(1 + 3k)
k=1
D.
18
X
Math 105 (Section 204)
(22 − k)
Multivariablek=0
Calculus – Extremization
2011W T2
5/5