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Unformatted text preview: ’s do an example to demonstrate why positive monotonic transformations work in
consumer theory but not in producer theory. First, suppose your utility function is: Suppose you prefer bundle
since ( )
( ): ( ) to bundle ( ). This preference ranking is represented by () () ()
To say ( )
because ( ) ()
() () does not mean that you are getting 62 units of utility from bundle
( ) you prefer bundle to bundle . Now consider this utility function gotten from applying a PMT to : { } to bundle According to this new utility function you prefer bundle rather, all it says is that since ( ) () () () () () ( ): () Contrast this with a producer theory example: suppose your production function is (for simplicity let Below are the outputs from two bundles of inputs: ( ) ( () ( ) ( ): () () ) and ): () Now look what happens if we apply a PMT on the above production function:
{ () } ()
13 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. () () Notice that the production functions yield two different levels of output from the same bundle of inputs – as such these
production functions represent two different production processes and demonstrates how a PMT on a production
function does not represent the same production process.
3. “Returns to Scale” in the Long Run
In this section we discuss an important concept in producer theory: the “returns to scale” in the long run. For the sake of
illustration consider the isoquant curves of a CobbDouglas production function with two inputs labor and capital: In theory, this firm in the long run can expand production by increasing inputs along any of the following (amongst an
infinite number of) input expansion paths (remember that in the long run all inputs are variable)9: 9 Note to self: in the CMP chapter show the case of ( ) which generates a nonlinear expansion.
14 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. Labor and
capital
increased
nonlinear
path Labor and
capital
increased in
1:1 ratio
Labor and
capital
increased
nonlinear
path Labor and
capital
increased in
1:2 ratio Labor and
capital
increased in
1:0 ratio In the next chapter we’ll show that while a firm in the long run has an infinite number of input expansion paths, only of
one paths of these is the cost minimizing path (which can be can be linear or nonlinear).
Here, for reasons to be explained below, instead of looking at the optimal cost minimizing input expansion path, let’s
focus on the 45 degree input expansion path and look at the returns to scale which measures the impact on output from
scaling up all inputs by a (common) factor larger than 1 (i.e. scaling up all inputs by a factor
For example, in the
following graph, we labor and capital are being doubled (i.e. scaled up by a factor
): ( ) We are interested in knowing the impact on output when all inputs are scaled up by a factor
expansion path. Now: along the 45 degree 15
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. ( ( ){ )
( ) ( ) For example, if we doubled all inputs and: ( (
(
( )
)
) (...
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This document was uploaded on 01/19/2014.
 Fall '14

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