Ch' 12 Optimization with Equality Constraints

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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

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12.2-2 Total-differential approach

• dL fxdx fydy 0 differential of Lf(x,y)
• dg gxdx gydy 0 differential of gg(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

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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

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12.2-1 Lagrange-multiplier method
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12.2-2 Total-differential approach

• dL fxdx fydy 0 differential of Lf(x,y)
• dg gxdx gydy 0 differential of gg(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

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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

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

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12.3-1 Second-order total differential
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12.3-1 Second-order total differential
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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

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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

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Graph Substitution and Income Effects
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Graph Substitution and Income Effects
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12.5-4 Proportionate changes in prices and income
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12.6 Homogeneous Functions

• 12.6-1 Linear homogeneity
• 12.6-2 Cobb-Douglas production function
• 12.6-3 Extension of the results

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12.6-1 Linear homogeneity

• A function f(x1, ..., xn) is homogeneous of
  degree r if multiplication of each of its
  independent variables by a constant j will alter
  the value of the function by the proportion jr,
  that is
• if f (jx1, ..., jxn) jrf(x1, ... xn)
• for all f (jx1, ... jxn) in the domain of f
• If r 0, j0 1,, the function is homogeneous of
  degree zero (e.g., utility function subject to a
  budget constraint)
• If r 1, j1 j, the function is homogeneous of
  degree one, (e.g., production function with
  constant returns to scale)

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12.6-1 Linear homogeneity

• Given the linearly homogeneous production
  function Q f(K, L),
• The average physical product of labor (APPL) and
  of capital (APPK) can be expressed as the
  capital-labor ratio, k ? K/L, alone. Let j 1/L

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12.6-1 Linear homogeneity

• Given the linearly homogeneous production
  function Q f(K, L),
• The marginal physical products MPPL and MPPK can
  be expressed as functions of k alone

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12.6-1 Linear homogeneity
Given the linearly homogeneous production
function Q f(K, L), if each input is paid the
amount of its marginal product the total product
will be exactly exhausted by the distributive
shares for all the inputs, i.e., no residual.
Eulers theorem (No fixed factors?)
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12.6-2 Cobb-Douglas production function

• Q AK?L? is homogeneous of degree j??

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12.6-2 Cobb-Douglas production function

• Q AK?L?
• Function is homogeneous of degree j??
• ? ? gt 1 increasing returns (paid lt share)
• ? ? lt 1 decreasing returns (paid gt share)
• 2) If ? ?1, function is linearly homogeneous
• 3) Isoquants are negatively sloped and strictly
  convex (K, L gt 0)
• 4) Function is quasi-concave (K, L gt 0)

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12.6-2 Cobb-Douglas production function
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12.6-2 Cobb-Douglas production function
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12.6-2 Cobb-Douglas production function
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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

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12.7-2 Second-order condition

• ?(QaaQb2-2QabQaQbQbbQa2)lt0

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12.7-1 First-order condition
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12.7-3 The expansion path

• Points of tangency that describes the least cost
  combinations required to produce varying levels
  of Qo
• All points on the expansion path must show the
  same fixed input ratio, i.e., the expansion path
  must be a straight line emanating from the
  origin.

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12.7-4 Homothetic functions

• Suppose H is meat, Q is feed grain, and a, b
  are fertilizer and land respectively
• Although a homothetic function is derived from a
  homogeneous function, the function H (a,b) itself
  is not necessarily homogeneous in the variables a
  b. None the less, the expansion paths of H
  (a,b) are linear like those of Q (a,b)

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12.7-4 Homothetic functions

• Let H Q2

• Q isoquants share the same slopes so therefore
  the expansion path of H (a,b) like those of Q
  (a,b) are linear

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12.7-5 Elasticity of substitution
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12.7-6 CES production function
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12.7-6 CES production function
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12.7-6 CES production function
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12.7-6 CES production function
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12.7-6 CES production function
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12.7-6 CES production function
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12.7-6 CES production function
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12.7-6 CES production function