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Unformatted text preview: ervices company is in the short run since some costs are “fixed”.
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ECO 204 Chapter 11: Producer Theory— the Basics (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Another example: suppose a firm is in the long run and uses variable capital and labor as nonessential inputs (in the
sense that output can be produced by using labor and/or capital) and where the firm can produce the same level of
output by substituting labor for capital at a constant rate of substitution. Again borrowing intuition from consumer
theory, this production process can be modeled by the long run “Linear” or “Perfect Substitutes” production function:
{ } Notice that the firm can produce positive output by using labor only (
⏟ ), or by using both labor and capital ( ⏟ ⏟ ), or by using capital only ( ⏟ ). We will demonstrate the “constant rate of substitution” below. Another example: suppose a firm is in the long run and uses variable capital and labor as inputs as essential,
complementary, inputs. Again using intuition from consumer theory, we see that this production process can be
modeled by the long run “complements” production function:
{ } Labor and capital are essential inputs because to produce positive output the firm needs both workers and capital (i.e.
) because otherwise:
{
}
{
}
On the other hand, if the firm is in the short run, then its output is:
(
⏟ ) For example, suppose a firm has a CobbDouglas production function with variable labor but the amount of capital fixed
at the level . This short run CobbDouglas production function is: Both inputs are essential for production and that given that capital is fixed we see that the only way to produce more
output is by hiring more labor.
As another example, suppose a firm has a CobbDouglas production function with variable labor and materials but the
amount of capital fixed at the level . This short run CobbDouglas production function is: Here, all inputs are essential for production and given that capital is fixed we see that the only way to produce more
output is by hiring more labor and/or using more materials.
From these examples it should be obvious that the production function in producer theory is the counterpart of the
utility function in consumer theory. As such, it shouldn’t come as a surprise that the “usual suspects” from consumer
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ECO 204 Chapter 11: Producer Theory— the Basics (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. theory – CobbDouglas, Linear, Complements, QuasiLinear, etc. utility functions – have counterparts in producer theory:
CobbDouglas, Linear, Complements, QuasiLinear, etc. production functions. The parallels between consumer and
producer theory permit a facile and intuitive development of producer theory.
Let’s begin by looking at “3D production surface plots”. For example, suppose a firm’s long run production function is:
defined over {( ) } The 3D production surface plot is:
This contour shows
all
combinations
that will produc...
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