Class 9 Stat701 Fall 1997

Learning, productivity and cost models.


From last time: transformation in regression.

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For next time read Berndt 3.5,3.6.


Todays class.

The game plan

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1. Model learning curves
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2. Model production functions
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3. Model costs associated with production function
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4. Specialize cost function to include the learning curve model as a special case

1. Learning curves

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The motivation
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Unit costs decrease as cumulative output increases.
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Strategic implications for pricing and marketing strategy
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Formulation
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displaymath115

where

  • tex2html_wrap_inline117 is unit cost in time period t (adjusted for inflation)
  • tex2html_wrap_inline119 is unit cost in initial time period
  • tex2html_wrap_inline121 cumulative production up to but not including time t
  • tex2html_wrap_inline123 is unit cost elasticity with respect to cumulative volume
  • tex2html_wrap_inline125 stochastic disturbance term (our tex2html_wrap_inline127 )

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Note. Response is unit cost. A multiplicative model.
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Make linear by taking logs.
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displaymath129

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Estimate tex2html_wrap_inline131 from a simple regression.


2. Cobb Douglas production function.

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Model:

displaymath133

where

  • y is the output
  • A denotes the state of technical knowledge
  • tex2html_wrap_inline139 denotes the quantity of input i
  • tex2html_wrap_inline143 is the parameter to be estimated (like an elasticity of output with respect to input i)
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Note: the response is output. Another multiplicative model.
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Define returns to scale as tex2html_wrap_inline147


3. The cost function

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The cost function is tex2html_wrap_inline149 .
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Just the quantity of inputs times their prices.
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Relates the minimum cost of producing a level of output y to the prices of the inputs and the state of technical knowledge.

Objective; Find the input levels that minimze the production cost for a given level of output. (Cost minimizer assumption.)

This is an optimization problem, in particular choose input levels to minimize costs. But subject to a constraint: the inputs must produce a given level of output, y.

Mathematical technique for solution of constrained optimization: Lagrange multipliers.

It turns out that, assuming the Cobb Douglas production function, then the optimal level of inputs produce a COST FUNCTION of the form

displaymath155

where

displaymath157

It looks a mess, but notice that it is multiplicative, so taking logs will achieve a linear expression ready for regression.

Further, using the fact that tex2html_wrap_inline159 the logged version can be rewritten as

displaymath161

where

From this lot we can get at what's of interest, tex2html_wrap_inline177 .


4. Putting together the Learning Curve and the Cost Function

Objective: make assumptions that incorporate the learning curve into the cost function as a special case.

Then the question becomes can we put restrictions and assumptions on the cost function so that the learning curve is a special case?

Here's how it goes.

This leads to a simpler equation:

displaymath189

Here tex2html_wrap_inline191 is a real total cost because it as been adjusted by the GNP deflator.

Finally move to unit real costs rather than total real costs and you obtain

displaymath193

which for r = 1 is the learning curve model.

How much sense does the previous equation make?

It says that the log of your average real cost at time t depends on two things.


Summary

We have seen a variety of econometric models in action.



Richard Waterman
Wed Oct 1 20:47:26 EDT 1997