Revenue from Two Products A company manufactures and sells two competing products, I and II , that cost $ p 1 and $ p 2 per unit, respectively, to produce. Let R ( x , y ) be the revenue from marketing x units of product I and y units of product II . Show that if the company’s profit is maximized when x = a , y = b , then ∂ R ∂ x ( a , b ) = p I and ∂ R ∂ y ( a , b ) = p II .
Revenue from Two Products A company manufactures and sells two competing products, I and II , that cost $ p 1 and $ p 2 per unit, respectively, to produce. Let R ( x , y ) be the revenue from marketing x units of product I and y units of product II . Show that if the company’s profit is maximized when x = a , y = b , then ∂ R ∂ x ( a , b ) = p I and ∂ R ∂ y ( a , b ) = p II .
Solution Summary: The author explains the formula used to prove that the company's profit is maximized when x, y, and p = p_1, respectively.
Revenue from Two Products A company manufactures and sells two competing products,
I
and
II
, that cost
$
p
1
and
$
p
2
per unit, respectively, to produce. Let
R
(
x
,
y
)
be the revenue from marketing
x
units of product
I
and
y
units of product
II
. Show that if the company’s profit is maximized when
x
=
a
,
y
=
b
, then
∂
R
∂
x
(
a
,
b
)
=
p
I
and
∂
R
∂
y
(
a
,
b
)
=
p
II
.
Maximize
f(x, y, z) = xyz
Constraint:
x + y + z − 24 = 0
Maximum of f(x, y, z) =
at (x, y, z) =
Calculus III - Given the function f(x,y)=x^3-12x+y^3-3y find the global maximum and minimum on the region bounded by x=5, y=10x, and y=0. There is a critical point of (2,1) within the region.
Prove convexity of the following:
the risk function of portfolio x,
Risk(x) := E[R(x)] + δE[|R(x) - E[R(x)]|]
where R(x) is the portfolio return and δ is any positive scalar.
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