Production Theory lecture - Focusing on production function.ppt

Short-Run v. Long-Run
 Fixed input/factor of production: quantity
of input is fixed regardless of required
output level, e.g. capital or specialized
labour
 Variable input/factor of production:
quantity of input used depends on the
level of output
 Short run: at least one input/factor is fixed
 Long run: all inputs/factors are variable

Production Function
 A technology is a process by which
inputs (e.g. labour and capital) are
converted into output.
 The output level is denoted by y.
 The technology’s production function
states the maximum amount of output
possible from an input bundle.
y f x xn
 ( , , )
1 

Production Function
y = f(x) is the
production
function
x’ x
Input Level
Output Level
y’
y’ = f(x’) is the maximum
output level obtainable
from x’ input units.
One input

Technology Set
 The collection of all feasible
production plans is the technology set.

Technology Set
y = f(x) is the
production
function.
x’ x
Input Level
Output Level
y’
y”
One input
y” = f(x’) is an output level
that is feasible from x’
input units.

Technology Set
x’ x
Input Level
Output Level
y’
One input
y”
The technology
set

Technology Set
x’ x
Input Level
Output Level
y’
One input
y”
The technology
set
Technically
inefficient
plans
Technically
efficient plans

Technology: Multiple Inputs
 What does a technology look like
when there is more than one input?
 The two input case: Input levels are
x1 and x2. Output level is y.
 Example of production function is
3
/
1
2
3
/
1
1
2
1 2
)
,
( x
x
x
x
f
y 


PREVIEW: ISOQUANT
 An isoquant is the set of all combinations
of inputs 1 and 2 that are just sufficient to
produce a given amount of output.
 The slope of the isoquant = the marginal
rate of technical substitution (MRTS) = the
technical rate of substitution (TRS)
 MRTS (TRS): The number of units of K that
we can dispose of if one more unit of L
becomes available while remaining on the
original isoquant.

Technologies with Multiple
Inputs
 The complete collection of isoquants
is the isoquant map.
 The isoquant map is equivalent to
the production function.
 Example
3
/
1
2
3
/
1
1
2
1 2
)
,
( x
x
x
x
f
y 


Isoquants with Two Inputs
Y=20
Y=40
L
K

Isoquants with Two Inputs
 Properties
Y/K>0, Y/L>0
2
Y/K2
<0,2
Y/L2
<0
Diminishing marginal product
(Diminishing marginal utility)

x2
x1
All isoquants are hyperbolic,
asymptoting to, but never
touching any axis.
Cobb-Douglas Technology
y x x
a a
 1 2
1 2

Marginal (Physical) Product
 The marginal product of input i is the
rate-of-change of the output level as
the level of input i changes, holding
all other input levels fixed.
y f x xn
 ( , , )
1 
i
i
x
y
MP




y f x x x x
 
( , ) /
1 2 1
1/3
2
2 3
then the marginal product of input 1 is

y f x x x x
 
( , ) /
1 2 1
1/3
2
2 3
MP
y
x
x x
1
1
1
2 3
2
2 3
1
3
  


/ /

y f x x x x
 
( , ) /
1 2 1
1/3
2
2 3
MP
y
x
x x
1
1
1
2 3
2
2 3
1
3
  


/ /
and the marginal product of input 2 is

y f x x x x
 
( , ) /
1 2 1
1/3
2
2 3
MP
y
x
x x
1
1
1
2 3
2
2 3
1
3
  


/ /
and the marginal product of input 2 is
MP
y
x
x x
2
2
1
1/3
2
1/3
2
3
  


.

 The marginal product of input i is
diminishing if it becomes smaller as
the level of input i increases. That is,
if
0
2
2











i
i
i
i
i
x
y
x
y
x
x
MP









Technical Rate-of-Substitution
x2
x1
y
The slope is the rate at which
input 2 must be given up as
input 1’s level is increased so as
not to change the output level.
The slope of an isoquant is its
technical rate-of-substitution.
x2
'
x1
'

 How is a technical rate-of-substitution
computed?

 How is a technical rate-of-substitution
computed?
 The production function is
 A small change (dx1, dx2) in the input
bundle causes a change to the output
level of
y f x x
 ( , ).
1 2
dy
y
x
dx
y
x
dx
 




1
1
2
2.

dy
y
x
dx
y
x
dx
 




1
1
2
2.
Along an individual isoquant, dy = 0,
therefore the changes dx1 and dx2 must
satisfy the following,
0
1
1
2
2
 




y
x
dx
y
x
dx .

0
1
1
2
2
 




y
x
dx
y
x
dx
which rearranges to




y
x
dx
y
x
dx
2
2
1
1

or dx
dx
y x
y x
2
1
1
2

 
 
/
/
.

dx
dx
y x
y x
2
1
1
2

 
 
/
/
is the rate at which input 2 must be given
up as input 1 increases so as to keep
the output level constant. It is the slope
of the isoquant = MRTS = TRS.

Production Theory lecture - Focusing on production function.ppt

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Production Theory lecture - Focusing on production function.ppt