MICROECONOMICS Principles and Analysis Frank Cowell July 2015 1 Overview July 2015 2 The setting Input requirement sets Returns to scale Marginal products The Firm Basics The environment for the basic model of the firm ID: 316656
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Slide1
The Firm: Basics
MICROECONOMICSPrinciples and Analysis Frank Cowell
April 2018
1Slide2
Overview
April 2018
2
The setting
Input require-ment sets
Returns to scale
Marginal products
The Firm: Basics
The environment for the basic model of the firm
IsoquantsSlide3
The basics of production
Some elements needed for an analysis of the firmTechnical efficiencyReturns to scale
ConvexitySubstitutability
Marginal products
This is in the context of
a single-output firm
assuming a competitive environment
First we need the building blocks of a model
April 2018
3Slide4
Notation
April 2018
4
Quantities
z
i
amount of input
i
z
= (
z
1
, z
2 ,
, zm
)
input vector
amount of output
q
Prices
price of input
i
w
= (w1
, w2
,
, w
m
)
Input-price vector
price of output
p
For next presentation
w
i
Slide5
Feasible production
April 2018
5
The basic relationship between output and inputs:
q
£
f
(
z
1
, z2
, , zm )
This can be written more compactly as:
q £
f (z)
single-output, multiple-input production relation
f
gives the maximum amount of output that can be produced from a given list of inputs
Note that we use “
£” and not “=” in the relation. Why?Consider the meaning of f
Vector of inputs
The production function
distinguish two important cases...Slide6
Technical efficiency
April 2018
6
The case where production is
technically efficient
The case where production is (technically) inefficient
Case 1:
q
=
f
(
z
)
Case 2:
q
<
f
(
z
)
Intuition: if the combination
(
z
,q
) is inefficient, you can throw away some inputs and still produce the same output Slide7
z
2
z
1
q
>
f
(
z
)
Boundary: feasible
and
efficient points
The function
April 2018
7
q
0
Cone:
production function
“Inside”:
feasible but
inefficient points
“Outside”: Infeasible
points
q <
f
(
z
)
q
=
f
(
z
)
We need to examine its structure in detailSlide8
Overview
April 2018
8
The setting
Input require-ment sets
Returns to scale
Marginal products
The Firm: Basics
The structure of the production function
IsoquantsSlide9
The input requirement set
April 2018
9
remember, we must have
q
£
f
(
z
)
Pick a particular output level
qFind a feasible input vector
z
Repeat to find all such vectors
Yields the input-requirement set
Z
(
q
) := {z: f
(z) ³
q}
The set of input vectors that meet the technical feasibility condition for output
q
The shape of
Z
depends on the assumptions made about production
We will look at four cases
First, the “standard” caseSlide10
z
1
z
2
The input requirement set
April 2018
10
“Inside”: feasible
but inefficient
Boundary: feasible
and technically efficient
“Outside”: Infeasible
Z
(
q
)
q <
f
(
z
)
q
=
f
(
z
)
q >
f
(
z
) Slide11
z
1
z
2
Case 1:
Z
smooth, strictly convex
April 2018
11
Pick two boundary points
Draw the line between them
Intermediate points lie in the interior of
Z
A combination of two techniques may produce more output
What
if we changed some of the assumptions?
z
¢
z
²
Z
(
q
)
q<
f
(
z
)
q
=
f
(
z
"
)
q
=
f
(
z
'
)
Note important role of convexitySlide12
Case 2: Z
Convex (but not strictly)April 2018
12
z
1
z
2
A combination of feasible techniques is also feasible
Z
(
q
)
z
¢
z
²
Pick two boundary points
Draw the line between them
Intermediate points lie in
Z
(perhaps on the boundary)Slide13
Case 3: Z
smooth but not convexApril 2018
13
z
1
z
2
in this region there is an indivisibility
Join two points across the “dent”
Z
(
q
)
Take an intermediate point
Point lies in infeasible zone
This point is infeasible
Slide14
z
1
z
2
Case 4:
Z
convex but not smooth
April 2018
14
Only one technically efficient point
Slope of the boundary is undefined at this point
q =
f
(
z
) Slide15
z
1
z
2
z
1
z
2
z
1
z
2
z
1
z
2
Summary: 4 possibilities for
Z
April 2018
15
Only one efficient point and not smooth
Problems: the dent represents an indivisibility
Standard case, but strong assumptions about divisibility and smoothness
Almost conventional: mixtures may be just as good as single techniquesSlide16
Overview
April 2018
16
The setting
Input require-ment sets
Returns to scale
Marginal products
The Firm: Basics
Contours of the production function
IsoquantsSlide17
Isoquants
17
Pick a particular output level
q
Find the input requirement set
Z
(
q
)
The
isoquant
is the boundary of Z
: {z : f
(z) = q
}
Think of the isoquant as an integral part of the set Z
(q)
¶
f(z)
f
i(z) :=
——
¶z
i .
f
j (
z
)
——
f
i
(
z
)
If the function
f
is differentiable at
z
then the
marginal rate of technical substitution
is the slope at
z
:
Where appropriate, use subscript to denote partial derivatives. So
Gives rate at which you trade off one input against another along the isoquant, maintaining constant
q
Let’s look at its shape
April 2018Slide18
z
1
z
2
Draw input-requirement
set
Z
(
q
)
Isoquant, input ratio, MRTS
April 2018
18
Boundary:
contour
of the function
f
{
z
:
f
(
z
)=
q
}
An efficient point
Input ratio describes one production technique
z
2
°
z
1
°
z
°
Slope of ray:
input ratio
z
2
/
z
1
= constant
Slope of
boundary: Marginal
Rate of Technical Substitution
The isoquant is the boundary of
Z
Higher slope: increased MRTS
z
′
MRTS
21
=
f
1
(
z
)/
f
2
(
z
)
MRTS
21
: implicit “price” of input 1 in terms of 2
Higher “price”: smaller relative use of input 1Slide19
MRTS and elasticity of substitution
Responsiveness of inputs to MRTS is elasticity of substitution
April 2018
19
∂log(
z
1
/
z
2
)
=
∂log(
f1/
f2
)
prop
change input ratio
=
prop
change in MRTS
input-ratio MRTS
input-ratio MRTS
z
1
z
2
s
= ½
z
1
z
2
s
= 2Slide20
Elasticity of substitution
April 2018
20
z
1
z
2
structure of the contour map...
A constant elasticity of substitution isoquant
Flatter isoquant: higher elasticity
of substitution...
*
detail on slide can
only be seen if you run the slideshowSlide21
Homothetic contours
April 2018
21
Curves: the isoquant map
O
z
1
z
2
Ray
through the
origin: a given input ratio
Same
MRTS
where ray
cuts each isoquantSlide22
Contours of a homogeneous function
April 2018
22
Curves: the isoquant map
O
z
1
z
2
Point
z
°
: inputs that will produce
q
Point
t
z
°
:
inputs
that will produce
t
rq
tz
1
°
tz
2
°
z
2
°
z
1
°
t
z
°
z
°
q
t
r
q
f
(
t
z
)
=
t
r
f
(
z
)Slide23
Overview
April 2018
23
The setting
Input require-ment sets
Returns to scale
Marginal products
The Firm: Basics
Changing all inputs together
IsoquantsSlide24
Let's rebuild from the isoquants
The isoquants form a contour map If we looked at the “parent” diagram, what would we see?Consider returns to scale of the production functionExamine effect of varying all inputs together:
Focus on the expansion pathq plotted against proportionate increases in zTake three standard cases:
Increasing Returns to Scale
Decreasing Returns to Scale
Constant Returns to Scale
Do this for 2 inputs, one output
April 2018
24Slide25
Case 1: IRTS
April 2018
25
z
2
q
z
1
0
An increasing returns to scale function
t >
1
implies
f
(
t
z
)
>
t
f
(
z
)
Point
on the
surface: feasible, efficient
Dotted line plots the
expansion
path
Double inputs and you more than double outputSlide26
Case 2: DRTS
April 2018
26
q
z
1
A decreasing returns to scale function
Point on the surface: feasible, efficient
Dotted line plots
the
expansion path…
z
2
0
t
>
1
implies
f
(
t
z
)
<
t
f
(
z
)
Double inputs and output increases by less than double Slide27
Case 3: CRTS
April 2018
27
z
2
q
0
A constant returns to scale function
Point on the surface: feasible, efficient
The expansion path is a ray
z
1
f
(
t
z
)
=
t
f
(
z
)
Double inputs and output exactly doubles Slide28
Relationship to isoquants
April 2018
28
z
2
q
0
Take any
production function
Horizontal
“slice
”: given q level
z
1
Project down to get the isoquant
Repeat to get isoquant map
Isoquant
map is the projection of
the
set of
technically efficient pointsSlide29
Overview
April 2018
29
The setting
Input require-ment sets
Returns to scale
Marginal products
The Firm: Basics
Changing one input at time
IsoquantsSlide30
Marginal products
April 2018
30
Measure the marginal change in output w.r.t. this input
¶
f
(
z
)
MP
i
=
fi
(z) =
——¶zi
.
Pick a technically efficient input vector
Keep all but one input constant
Remember, this means a
z such that
q = f(
z)
The marginal productSlide31
CRTS production function again
April 2018
31
z
2
q
0
Vertical slice: keep one input constant
z
1
Broken line: path
for
z
2
=
const
Let’s look at its shapeSlide32
Marginal Product
for CRTS production functionApril 2018
32
z
1
q
f
(
z
)
Shaded area: feasible
set
A section of the production function
Boundary: technically
efficient points
Input 1 is essential:
If
z
1
= 0
then
q
= 0
Slope of
tangent: MP
of input 1
f
1
(
z
)
Slope depends on value of
z
1
…
f
1
(
z
)
falls with
z
1
(or stays constant) if
f
is concaveSlide33
Relationship between q
and z1April 2018
33
z
1
q
z
1
q
z
1
q
z
1
q
We’ve just taken the
conventional case
But in general this curve depends on the shape of
Some other possibilities for the relation between output and one input…Slide34
Key concepts
Technical efficiencyReturns to scaleConvexityMRTSMarginal product
April 2018
34Slide35
What next?
Introduce the marketOptimisation problem of the firmMethod of solutionSolution concepts
April 2018
35