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Unformatted text preview: one) expansion path where we’re interested in the impact on output from
expanding variable labor by a factor
along this inputs expansion path (see below): 20
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
increased A firm in the short run with fixed capital and variable labor As another example, suppose a firm has the production function:
. With fixed capital, to produce more
output the firm would have to use more variable labor and materials. We are interested in the impact on output from
expanding variable labor and materials by a factor
For example, the following graphs show the labor and
materials isoquants of
holding capital fixed at
and
In each case, we want to know
what happens to output when labor and materials are increased along the “45 degrees” inputs expansion path by a
factor
(in the graphs below variable inputs are being increased by a factor of
): Here, variable inputs labor and materials are being increased by a factor
along the 45 degree inputs expansion path (each graph is for a fixed level of ) Here are the definitions of returns to variable inputs: 21
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 variable inputs and: ( (
(
( ){ )
)
) As another example, if we tripled all variable inputs and: ( (
(
( ){ )
)
) The following notation is convenient for the discussion below (notice how we double the variable inputs only):
(
( ) ) Output with variable input and fixed input Output with twice the initial variable inputs and same amount of the fixed input How can we tell a firm’s returns to variable inputs? Here are two examples. First, the CobbDouglas production function:
(
The output with twice the variable inputs and the same amount of the fixed input (
Comparing ( )with ( ) ( ) ( is: ) ) we see that:
( )
( { When ) (
) ( )
( ) (
) ( ) )
( ( )
) The following graph for ( )
(with
and
) exhibits increasing returns to labor (doubling labor
expands output by a factor greater than 2) along the labor expansion path
(by the way, this production function
has increasing returns to scale – check this):
( ) (
( ) ) (
( )
) () (
( ) ( ) )
22 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. In Wolfram Alpha type plot (10/L^1.5)^(1/1), (28.28427/L^1.5)^(1/1) , (80/L^1.5)^(1/1) The following graph for ( )
(with
doubles output) along the labor expansion path
scale – check this):
( ) (
( and ) exhibits constant returns to labor (doubling labor
(by the way, this production function has increasing returns to ) ( ) ( )( )
) (
( ) ( The following graph for ( )
(with
and
less than doubles output) along the labor expansion path
returns to scale – check this):
) ( ) ( ) ) ) In Wolfram Alpha type plot (10/L^1)^(1/1), (20/L^1)^(1/1) , (40/L^1)^(1/1) ( from L=0,5 K=0,5 from L=0,5 K=0,5 ) exhibits decreasing returns to la...
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This document was uploaded on 01/19/2014.
 Fall '14

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