Lecture 5
Multivariable Optimization
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1. Necessary Conditions
Optimization problems:
maxx1,x2 y = f(x1,x2)
minx1,x2 y = f(x1,x2)
E.g., profit maximization, cost
minimization
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When x2 is kept constant,
f(x1, c) is a function of x1 only.
When x1 is kept constant,
f(c, x2) is a function of x2 only
First-order conditions for a stationary point
y
x n

x1  x1* ,x 2  x 2*

f x , x
*
1
x n
*
2
  0,
ECON 1150, Spring 2013
n = 1,2.
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Example 5.1: Find stationary values of
the following functions:
(a) y = 2x1² + x2²;
(b) y = 4x1² - x1x2 + x2² - x1³.
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Possible cases of stationary points
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2. Sufficient Conditions
Second-order conditions for a local
maximum at a point (SOC max)
f11 < 0; f11·f22 – (f12)2 > 0
Second-order conditions for a local
minimum at a point (SOC min)
f11 > 0; f11·f22 – (f12)2 > 0.
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Classifying a stationary point
If f1(x1*,x2*) = f2(x1*,x2*) = 0, then
a. SOC max  local maximum
b. SOC min  local minimum
c. f11f22 – (f12)2 = 0  no conclusion
d. f11f22 – (f12)2 < 0  saddle point
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Example 5.2: Identify the nature of the
stationary points of the following function,
y = f(x1,x2) = x1³ + 5x1x2 – x2².
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Example 5.3: Identify the nature of the
stationary points of the following
functions:
a. f(x1,x2) = x12 + x24;
b. f(x1,x2) = x12 – x24;
c. f(x1,x2) = x12 + x23;
d. f(x1,x2) = x14 + x24;
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If for all (x1,x2),
f11  0 and f11f22 – (f12)2  0.
then f(x1,x2) is a concave function.
Example 5.4: Concave functions
a. y = x1 + x2;
b. y = x10.4x20.6 for positive x1 and x2;
c. y = lnx1 + lnx2.
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If for all (x1,x2),
f11  0 and f11f22 – (f12)2  0.
then f(x1,x2) is a convex function.
Example 5.5: Convex functions
a. y = x1 + x2;
b. y = 3x12 + 3x1x2 + x22.
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For a stationary point (x10, x20),
concave function  global maximum
convex function  global minimum
Example 5.6: Show that the function
f(x1,x2)
= -2x12 – 2x1x2 – 2x22 + 36x1 + 42x2 – 158
is a concave function. Find the global
maximum point.
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Remarks:
a. 2nd order conditions hold at a point
 Local extremum
b. 2nd order conditions hold for all points
 Global extremum
c. SOC are sufficient, but not necessary
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3. Economic Applications
A firm produces 2 different kinds A and B of a
commodity. The daily cost of producing x units
of A and y units of B is
C(x,y) = 0.04x2 + 0.01xy + 0.01y2 + 4x + 2y + 500.
Suppose that that firm sells all its output at a
price per unit of 15 for A and 9 for B. Find the
daily production levels x* and y* that maximize
profit per day.
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Long-run profit maximization of a competitive
firm
Output price: 200
Inputs: K with a price of 42
L with a price of 5
Production function: 3.1K0.3L0.25.
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General profit maximization problem
maxK,L (K,L) = pf(K,L) – wKK – wLL
FOC:
K(K,L) = pMPK – wK = 0
L(K,L) = pMPL – wL = 0
1. pMPn = wn
(n = K,L)
2. wK / MPK = wL / MPL = p = MC
3. wK/wL = MPK/MPL = MRTS
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Profit maximization of a 2-product firm
A two-product firm faces the demand and
cost functions below:
Q₁ = 40 - 2P₁ - P₂
Q₂ = 35 - P₁ - P₂
TC = Q₁² + 2Q₂² + 10
Find the profit-maximizing output levels and
the maximum profit.
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