Agricultural Economics

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AgEc 301 Agricultural Economics I
Slide Set 8 Chapter 8
Returns to Scale
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Returns to Scale

• We have examined changes in output in response to
  changes in individual inputs. A related and
  important question is how output changes when all
  inputs are changed proportionately.

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Returns to Scale

• There are three categories for the change in
  production in response to proportionate changes
  in all inputs
• Constant Returns to Scale
• Increasing Returns to Scale
• Decreasing Returns to Scale

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Constant RTS

• Constant Returns to Scale exists when a given
  percentage increase in all inputs, leads to the
  same percentage increase in output.
• - Linear homogeneity

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Increasing RTS

• Increasing Returns to Scale exists when a given
  percentage increase in all inputs, leads to a
  larger percentage increase in output.

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Decreasing RTS

• Decreasing Returns to Scale exists when a given
  percentage increase in all inputs, leads to a
  smaller percentage increase in output.

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Returns to Scale

• It is possible for a production function to
  exhibit increasing, constant and decreasing RTS
  over ranges of production.

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Output Elasticity

• Returns to scale can be determined for any
  function through analysis of its output
  elasticity eQ.
• eQ

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Output Elasticity

• In terms of calculus, this is
• eQ
• Where X is all inputs.
• Does this look familiar?

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Output Elasticity

• eQ is the multiple input analog of the function
  coefficient (you recall that the function
  coefficient is equal to MP over AP)

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Returns to Scale
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Estimating Returns to Scale

• Returns to scale can be estimated by simply
  increasing all inputs by a constant and observing
  how output changes.
• Consider the function
• Q f(X, Y, Z)

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Estimating Returns to Scale

• If all inputs are multiplied by a constant, k,
  and we write the equations as
• hQ f(kX, kY, kZ)
• The relationship between k and h is a measure of
  RTS

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Estimating Returns to Scale

• In this example

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Estimating Returns to Scale

• Some of you may recognize this as very similar to
  the definition of homogeneity.