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Ch. 12 Optimization with Equality Constraints

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

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Ch. 12 Optimization with Equality Constraints

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  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. 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/ fyequal at the tangent • fx/ gx = fy /gy = equi-marginal principle

  3. 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

  4. 12.2-1 Lagrange-multiplier method

  5. 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/ fyequal at the tangent • fx/ gx = fy /gy = equi-marginal principle

  6. 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

  7. 11.4 n-variable soc principal minors test for unconstrained max or min

  8. 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.

  9. 12.3-1 Second-order total differential

  10. 12.3-1 Second-order total differential

  11. 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

  12. 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

  13. Quantity Q2 P0 If the price of Q1 increases, then the change in demand equals the substitution effect (AB) and the income effect (BC). P0 B A C U0 U1 Quantity Q1 P1 P1 P0 Graph: Substitution and Income Effects

  14. 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). B Y1'Y0 Y1 A C U0 U1 -Px1/Py0 -Px0/Py0 Quantity X Price X P1 P0 Ordinary demand Compensated demand X1 X1' X0 Quantity X Graph: Substitution and Income Effects

  15. 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

  16. 12.7-6 CES production function

  17. 12.7-6 CES production function

  18. 12.7-6 CES production function

  19. 12.7-6 CES production function

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