Homogeneous and Homothetic Functions

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Homogeneous and Homothetic Functions
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Homogeneous functions

• A function is said to be 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

• In general, j can take any value.
• In economic applications the constant j is
  usually taken to be positive

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EXAMPLE

• The value of the function is not be affected at
  all by equal proportionate changes in all the
  independent variables
• This makes the function f a homogeneous function
  of degree zero.

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EXAMPLE
The function g is homogeneous of degree one (or,
of the first degree) multiplication of each
variable by j will alter the value of the
function exactly j-fold as well.
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EXAMPLE
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Linear Homogeneity
Linear homogeneity," means homogeneity of degree
one Let a production function be of the form,
Q f(K, L) Linear homogeneity means constant
returns to scale
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Properties of Linearly Homogeneous Functions
Property I. Given a linearly homogeneous function
Q f(K, L), the average physical product of
labor (APPL) and of capital (APPK) are functions
of capital-labor ratio (K/L k)
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Property II. Given a linearly homogeneous
function Q f(K, L), the marginal physical
product of labor (MPPL) and of capital (MPPK) are
functions of k alone.
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Property III. Eulers Theorem If Q f(K, L), is
linearly homogeneous, then
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Cobb-Douglas Production Function
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Signs of derivatives
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Special Case a ß 1
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Eulers Theorem
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Homothetic Functions

• Given a set of input prices, homogeneity (of any
  degree) of the production function produces a
  linear expansion path.
• But linear expansion paths can also result from
  homothetic functions.
• Homotheticity can arise from a composite
  function in the form
• H hQ(a,b) h'(Q) ?0
• where Q(a, b) is homogeneous of degree r.
• Although derived from a homogeneous function, the
  function
• H H(a, b) is in general not homogeneous in the
  variables a and b.
• Nonetheless, the expansion paths of H(a, b), are
  linear.
• The key to this result is that, at any given
  point in the ab plane, the H isoquant shares the
  same slope as the Q isoquant

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Homothetic Functions
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Elasticity of Substitution
We are interested in the effect of a change in
the Pa/Pb ratio upon the least-cost input
combination b/a for producing the same given
output Qo (that is, while we stay on the same
isoquant).
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Elasticity of Substitution
For a generalized Cobb-Douglas production
function