Ch. 12 Optimization with Equality Constraints
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Transcript of Ch. 12 Optimization with Equality Constraints
1
Ch. 12 Optimization with Equality Constraints
• 12.1 Effects of a Constraint• 12.2 Finding the Stationary Values• 12.3 Second-Order Conditions• 12.4 Quasi-concavity and Quasi-
convexity• 12.5 Utility Maximization and
Consumer Demand• 12.6 Homogeneous Functions• 12.7 Least-Cost Combination of Inputs• 12.8 Some concluding remarks
2
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6
12.2-2 Total-differential approach
• dL = fxdx + fydy = 0 differential of L=f(x,y)
• dg = gxdx + gydy = 0 differential of g=g(x,y)
• dx & dy dependent on each other
• dy/dx = -fx/ fy slope of isoquant curve
• dy/dx = -gx/gy slope of the constraint line
• -gx /gy = -fx/ fy equal at the tangent
• fx/ gx = fy /gy = equi-marginal principle
7
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12
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0
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0
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Where
min:)definite positive0,...,0,0,0H
max(:definite negative0)1,...(0,0,0H
361) (p. soc, case variablend)
min:)definite positive0,0H
max(:definite negative0,0H
soc of test variable3c)
min:)definite positive0H
max(:definite negative0H
soc of test variable2b)
constraint than variablemore one
bemust therebecause test variable-one No a)
min andmax dconstrainefor test minors principal SOC variable-n 12.3
333
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14
12.2 Finding the Stationary Values
• 12.2-1 Lagrange-multiplier method
• 12.2-2 Total-differential approach• 12.2-3 An interpretation of the
Lagrange multiplier• 12.2-4 n-variable and multi-
constraint case
15
12.2-1 Lagrange-multiplier method
0
0
LL
)5(
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),(λ),()1(
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16
12.2-2 Total-differential approach
• dL = fxdx + fydy = 0 differential of L=f(x,y)
• dg = gxdx + gydy = 0 differential of g=g(x,y)
• dx & dy dependent on each other
• dy/dx = -fx/ fy slope of isoquant curve
• dy/dx = -gx/gy slope of the constraint line
• -gx /gy = -fx/ fy equal at the tangent
• fx/ gx = fy /gy = equi-marginal principle
17
0)5(
0)4(
0)3(
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multiplier Lagrange theoftion interpretaAn 3-12.2
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18
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yx
19
12.3 Second-Order Conditions
• 12.3-1 Second-order total differential
• 12.3-2 Second-order conditions• 12.3-3 The bordered Hessian• 12.3-4 n-variable case• 12.3-5 Multi-constraint case
20
11.4 n-variable soc principal minors test for unconstrained max or min
:)0,...,0,0,0H:min
(:0)1,...(0,0,0H:max
317) (p. soc, case variablen
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soc of test variable2
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(:,0H:max
tminor tes principal soc, of test variable-1
321
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21
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1
1
n
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21
12.3-1 Second-order total differential
has no effect on the value of Z* because the constraint equals zero but …
• A new set of second-order conditions are needed
• The constraint changes the criterion for a relative max. or min.
22
12.3-1 Second-order total differential
0h β 2 iff 0)5(
h β 2)4(
2(3)
yfor constraint thesolve α
(2)
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22
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22
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23
12.3-1 Second-order total differential
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bα-h β α 2aβ -a
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24
361 p. W,&C;
0
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0
H
Where
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25
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26
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format Matrix
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22
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27
12.4 Quasi-concavity and Quasi-convexity
• 12.4-1 Geometric characterization• 12.4-2 Algebraic definition• 12.4-3 Differentiable functions• 12.4-4 A further look at the
bordered Hessian• 12.4-5 Absolute vs. relative
extrema
28
12.5 Utility Maximization and
Consumer Demand • 12.5-1 First-order condition• 12.5-2 Second-order condition• 12.5-3 Comparative-static analysis• 12.5-4 Proportionate changes in
prices and income
29
02
0
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conditionsorder -First 15.12
22
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30
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,
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31
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32
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21
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33
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34
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analysis static-eComparativ 3-12.5
35
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36
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22
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39
Graph: Substitution and Income Effects
B
A
C
U1
U0
Quantity Q1
Quantity Q2
P0
P0P1
P1
P0
If the price of Q1 increases, then the change in demand equals the substitution effect (AB)and the income effect (BC).
40
B
A
C
U1
U0
Quantity X
Quantity Y
-Px1/Py0
If the price of Q1 increases, then the change in ordinary demand equals the sum of the substitution effect (AB) and the income effect (BC).
X1 X1' X0
Price X
P1
P0Ordinary demand
Compensated demand
Quantity X
-Px1/Py0 -Px0/Py0
Y1'Y0
Y1
Graph: Substitution and Income Effects
52
12.7 Least-Cost Combination of Inputs
• 12.7-1 First-order condition• 12.7-2 Second-order condition• 12.7-3 The expansion path• 12.7-4 Homothetic functions• 12.7-5 Elasticity of substitution• 12.7-6 CES production function• 12.7-7 Cobb-Douglas function as a
special case of the CES function
53
0
0
0
00,
,
0& ,, subject to Minimize
constant held are P & P where,,,;,,
conditionsorder -First 1-12.7
0
ba0
bbbab
abaaa
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b
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54
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,
linebudget and constraintBudget
onsubstituti technicalof rate marginal
Isoquant0
,
function Production
conditionsorder -First 17.12
0
0
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55
b
abbaba
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