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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361) (p. soc, case variablend)

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soc of test variable3c)

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soc of test variable2b)

constraint than variablemore one

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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

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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

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0)4(

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18

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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

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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


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2(3)

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23


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24

361 p. W,&C;

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0

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Where

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361) (p. soc, case variablend)

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soc of test variable3c)

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26

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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

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30

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31

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35

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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

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ch. 12 optimization with equality constraints

Documents

totaldifferential approachdl

differential of g

totaldifferential approach12

order total differential12

order conditions12

order condition12

constraint linegx gy

homogeneous functions12