Unformatted text preview: ){ ) As another example, if we tripled all inputs and: ( ){ ( ) ( ) ( ) If a production function
Optional: A function ( ) is said to be homogeneous of degree
if ( )
exhibits constant returns to scale then scaling up all inputs by a factor means that:
( ) ( ) As such, a production function that is homogenous of degree 1 exhibits constant returns to scale.
The following notation is convenient for the discussion below:
(
( ) ) Output with inputs and Output with twice the initial inputs and , i.e. the new level of inputs and How can we tell a firm’s returns to scale? Here are two examples. First, the complements production function where the
) is:
output from a bundle of inputs (
(
The output with twice the inputs
( ( ) ) is:
) ( ) ( ) ( ) Doubling inputs always doubles output so that the complements production function has constant returns to scale.
The second example is with the CobbDouglas production function. The output with inputs
( is: )
16 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. The output with twice the inputs is: (
For what parameter values and
returns to scale? Comparing ( ) ( ( )
( (
( ) (
) )
( ) ) ( ) ) exhibits increasing returns to scale (doubling all inputs quadruples ( ) ( )
( ) The following graph for (
)
(with
output) along the inputs expansion path
:
) ( ) ( ( ) does this CobbDouglas production have increasing, constant, and decreasing
)with
(
) we see that: { When )( (
( ) )( ) ) (
( ) ( ) ) Expansion path In Wolfram Alpha type plot L, (1/L), (4/L), (16/L) from L=0,5 K=0,5 The following graph for (
)
(with
doubles output) along the inputs expansion path
( ) (
( ) exhibits constant returns to scale (doubling all inputs
:
) ) (
( )
) ( )
( ( ) ) 17
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 L, (1/L^0.25)^1/(0.75), (2/L^0.25)^1/(0.75), (4/L^0.25)^1/(0.75) from L=0,5 K=0,5 The following graph for (
)
(with
than doubles output) along the inputs expansion path
( ) (
( ) exhibits decreasing returns to scale (doubling all inputs less
: )
) (
( )
) ( )
( ( ) ) In Wolfram Alpha type plot L, (1/L^0.1)^1/(0.5), (1.515717/L^0.1)^1/(0.5), (2.297397/L^0.1)^1/(0.5) from L=0,5 K=0,5 There are several reasons why economists are interested in returns to scale:
● For some firms, the optimal (cost minimizing) inputs expansion path is the 45 degree line and as such we would like to
know what happens to output when inputs are scaled up along the 45 degree inputs expansion path 18
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. ● It may help a firm decide whether to produce more output by building a new identical facility versus doubling the
existing facility. If the cost of doubling the existing facility is the same as the cost of building another identical facility
then10:
If there are increasing returns to scale, the firm should double the existing facility...
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 Fall '14
 Economics, Microeconomics, Economics of production, producer, S. Ajaz Hussain, Sayed Ajaz Hussain

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