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), while if there are decreasing returns to scale then the cost function is strictly convex (i.e.
). For example: 14 Note to self: maybe this section should come earlier
42 ECO 204 Chapter 12: a Firm’s Cost Minimization Problem (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. “Cobb-Douglas” Cost Function ( )
Increasing RTS Constant RTS Decreasing RTS Strictly concave cost function Linear cost function Strictly convex cost function In fact, this is a general result (true of any production function). Here is a “simple” graphical proof – take a look at this
) with increasing returns to scale15:
rather odd graph of a production function
Left side Right side
) 45 degree
line 45 degree
line If there are increasing returns to scale it means that doubling inputs more than doubles output; put another way, to
produce double the output we need to less than double inputs. Now, the following graph shows the output ,
“optimal” cost ( ) and the cost of inputs to produce 15 Note to self: for next version do math proof.
43 ECO 204 Chapter 12: a Firm’s Cost Minimization Problem (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Left side Right side
() ( ) Production
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- Fall '14