L11: Functions R R

1
L11 Functions R R

• Required reading
• Lecture and lecture notes,
• Simon and Blume (Ch 13, 14,20)
• Contents
• Partial and Directional Derivatives
• Homogenous functions
• Cardinal vs Ordinal Property
• and Homothetic Functions

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Functions f X R where X

• Set X open
• Important examples
• Utility function
• Production function
• Examples Cobb-Douglass

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Functions f X R

• Geometric representation f
• Level set
• Utility function upper contour set

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Differentiable function

• Function
• Q Generalization to fX R? Partial
  derivative
• Change one variable at the time!
• A vector of N partial derivatives Gradient
• Direction of fastest growth of f

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Directional derivative

• f is C if is continuous
• Slope of f in a particular direction v at x?
• Let x,v
• D The directional derivative
• Theorem Let f be C. Then

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Homogenous function on XR

• Returns to Scale
• Nicely behave along the rays
• Let r,-1,0,1, be an integer
• D f is homogenous of degree r if
• Important cases r0, r1
• Returns to Scale
• Demand functions
• Homothetic preferences
• Geometric interpretation 3D, 2D

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Homogenous function

• Geometry of level sets
• Let X and X
• D X is a radial expansion of X if

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Homogenous function

• Theorem Let f be homogenous and X is a radial
  expansion of X. X is a level set iff X is a
  level set
• Can we say something about their slopes?

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Partial derivatives of homogenous functions

• Lemma Partial derivative of a r-homogenous
  function is homogenous of degree r-1

10
Partial derivatives of homogenous functions

• Suppose level sets are smooth curves
• Example Cobb-Douglass
• Slopes of level sets are constant along the rays

11
Euler theorem

• Theorem Suppose f is homogenous of degree r
• Then
• Constant returns to scale
• Proof

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Ordinal and cardinal property

• Utility function defines rational preferences on
  R
• Let gR R be strictly increasing
• v(x)g(u(x)) is a monotone transformation
• Q Preferences defined by v(x)?
• Intui
• D A property of a function is ordinal if
• f has the property any g(f(x)) has
  the property
• Property is cardinal if it is not ordinal

13
Ordinal version of homogeneity Homothetic
function

• Q Is homogeneity an ordinal property?
• Cobb Douglass example
• D vR R is homothetic if it is a monotone
  transformation of some homogenous function
• This is ordinal property