principles and analysis the firm: basics
TRANSCRIPT
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THE FIRM: BASICS
MICROECONOMICSPrinciples and Analysis
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OVERVIEW...
The setting
Input require-
ment sets
The Firm: Basics
The environment
for the basic ment sets
Returns to scale
Marginal
products
for the basic
model of the firm.Isoquants
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THE BASICS OF PRODUCTION...
� We set out some of the elements needed for an analysis of the firm.� Technical efficiency� Returns to scale� Convexity� Convexity� Substitutability� Marginal products
� This is in the context of a single-output firm...
� ...and assuming a competitive environment.
� First we need the building blocks of a model...
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NOTATION
� Quantities
zi •amount of input i
z = (z1, z2 , ..., zm ) •input vector
q •amount of outputq
n Prices
•price of input i
w = (w1, w2 , ..., wm ) •Input-price vector
•price of outputp
wi
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MOTIVATION OF THE FIRM
� Almost without exception we shall assume that
the objective of the firm is to maximise profits:
this assumes either that the firm is run by owner-
managers or that the firm correctly interprets
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shareholders’interests.
� More formally, we define the expression for
profits as
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i
n
i
i zwpq ∑=
−=Π1
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FEASIBLE PRODUCTION
n The basic relationship between
output and inputs:
q ≤ φ (z1, z2, ...., zm )
•single-output, multiple-input
production relationThe production
function
n This can be written more compactly
as:
q ≤ φ (z)
n φ 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 φ
Vector of inputs
distinguish two
important cases...
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TECHNICAL EFFICIENCY
•The case where
production is technically
efficient
n Case 1:
q = φ (z)
•The case where
production is (technically)
inefficient
n Case 2:
q < φ (z)
Intuition: if the combination (z,q) is inefficient you can throw
away some inputs and still produce the same output 77
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q >φ (z)
� Boundary points are feasible
and efficient
THE FUNCTION φ
q� The production function� Interior points are feasible
but inefficient
� Infeasible points
q <φ (z) q =φ (z)
z20
�We need to
examine its
structure in
detail.
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OVERVIEW...
The setting
Input require-
ment sets
The Firm: Basics
The structure of
the production ment sets
Returns to scale
Marginal
products
the production
function. Isoquants
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PROPERTIES OF THE PRODUCTION FUNCTION
� Let us examine more closely the production
function given in .
� We will call a particular vector of inputs a
technique.
q ≤ φ (z)
technique.
� It is useful to introduce two concepts relating to
the techniques available for a particular output
level q:
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THE INPUT REQUIREMENT SET
n remember, we must
have q ≤ φ (z)
Pick a particular output level q
n Find a feasible input vector z
Repeat to find all such vectors
n Yields the input-requirement set
Z(q) := {z: φ (z) ≥ q}n The set of input vectors
that meet the technical
feasibility condition for
output q... n The shape of Z depends on the
assumptions made about production...
nWe will look at four cases. First, the
“standard” case.
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z2
THE INPUT REQUIREMENT SET
� Infeasible points.
�Feasible but inefficient�
�Feasible and technically
efficient Z(q)
z1
q < φ (z)
q = φ (z)
q > φ (z)
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z2
CASE 1: Z SMOOTH, STRICTLY CONVEX
� Pick two boundary points
� Draw the line between them
� Intermediate points lie in the
interior of Z.Z(q)
z1
�A combination of two
techniques may produce
more output.
�What if we changed some of
the assumptions?
� z′
� z″
q< φ (z)
q = φ (z")
q = φ (z') �Note important role of
convexity.
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CASE 2: Z CONVEX (BUT NOT STRICTLY)
z2
Z(q)
� Pick two boundary points
� Draw the line between them
� Intermediate points lie in Z
(perhaps on the boundary).
z1
� A combination of feasible
techniques is also feasible
� z′
� z″
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CASE 3: Z SMOOTH BUT NOT CONVEX
z2� Join two points across the
“dent”
Z(q)
� Take an intermediate point
� Highlight zone where this can
occur.
z1
� in this region there is an
indivisibility
This point is
infeasible
�
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z2
CASE 4: Z CONVEX BUT NOT SMOOTH
z1
� � Slope of the boundary is
undefined at this point.
q = φ (z)
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z2z2
SUMMARY: 4 POSSIBILITIES FOR Z
Standard case,
but strong
assumptions
about
divisibility and
smoothness
Almost
conventional:
mixtures may
be just as
good as single
techniques
z1
z2
z1
z2
z1z1
Only one
efficient point
and not
smooth. But
not perverse.
Problems:
the "dent"
represents an
indivisibility
smoothness techniques
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OVERVIEW...
The setting
Input require-
ment sets
The Firm: Basics
Contours of the
production ment sets
Returns to scale
Marginal
products
production
function. Isoquants
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ISOQUANTS
n Pick a particular output level q
n Find the input requirement set Z(q)
n The isoquant is the boundary of Z:
{ z : φ (z) = q }
n Think of the isoquant as
an integral part of the set
Z(q)...
{ z : φ (z) = q }
∂φ(z)φi(z) := ——
∂zi .
φj (z)——φi (z)
n If the function φ is differentiable at z
then the marginal rate of technical
substitution is the slope at z:
n Where appropriate, use
subscript to denote partial
derivatives. So
n Gives the rate at which you can trade
off one output against another along
the isoquant – to maintain a constant q.
Let’s look at its
shape
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z2
� The set Z(q).
ISOQUANT, INPUT RATIO, MRTS
� A contour of the function φφφφ.
� An efficient point.
� The input ratio
� Marginal Rate of Technical
Substitution
� Increase the MRTS
z1
{z: φ (z)=q}
� Input ratio describes
one particular
production technique.
z2°
z1°
• z°
z2 / z1= constant � The isoquant is the
boundary of Z.
� Increase the MRTS
• z′
MRTS21=φ1(z)/φ2(z)
� Higher input ratio
associated with higher
MRTS..
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INPUT RATIO AND MRTS
� MRTS21 is the implicit “price” of input 1 in terms
of input 2.
� The higher is this “price”, the smaller is the
relative usage of input 1.relative usage of input 1.
� Responsiveness of input ratio to the MRTS is a
key property of φ.
� Given by the elasticity of substitution
�� Can think of it as measuring the Can think of it as measuring the isoquant’sisoquant’s “curvature” or “curvature” or
“bendiness”“bendiness”
∂∂log(log(zz11//zz22))σσijij == --
∂∂log(log(φφ11//φφ22))
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ELASTICITY OF SUBSTITUTION
z2
� A constant elasticity of
substitution isoquant
� Increase the elasticity of
substitution...
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ELASTICITY OF SUBSTITUTION
� Higher values of σ mean that the production
function is more “flexible” in that there is a
proportionately larger change in the production
technique in response to a given proportionate technique in response to a given proportionate
change in the implicit relative valuation of the
factors:
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HOMOTHETIC CONTOURS
� With homothetic contours, each isoquant appears
like a photocopied enlargement; along any ray
through the origin all the tangents have the same
slope so that the MRTS depends only on the slope so that the MRTS depends only on the
relative proportions of the inputs used in the
production process.
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HOMOTHETIC CONTOURS
� The isoquants
z2� Draw any ray through the
origin…
� Get same MRTS as it cuts
each isoquant.
Oz1
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CONTOURS OF A HOMOGENEOUS FUNCTION
� An important subcase of the family of homothetic
functions is the homogeneous production
functions, for which the map looks the same but
where the labelling of the contours has to satisfy where the labelling of the contours has to satisfy
the following rule: for any scalar t > 0 and any
input vector z≥0:
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)()( zttzrφφ =
where r is a positive scalar.where r is a positive scalar.
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CONTOURS OF A HOMOGENEOUS FUNCTION
� If φ(.) satisfies the property in the above
equation then it issaid to be homogeneous of
degree r. Clearly the parameter r carries important
information about the way output responds to a information about the way output responds to a
proportionate change in all inputs together:
� If r > 1, for example then doubling more inputs
will more than double output.
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CONTOURS OF A HOMOGENEOUS FUNCTION
� The isoquants
z2 � Coordinates of input z°
� Coordinates of “scaled up”
input tz°
Oz1
O tz1°
tz2°
z2°
z1°
• tz°
• z°
q
trq
φ (tz) = trφ (z)
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OVERVIEW...
The setting
Input require-
ment sets
The Firm: Basics
Changing all
inputs together. ment sets
Returns to scale
Marginal
products
inputs together.
Isoquants
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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 function.
� Examine effect of varying all inputs together:� Examine effect of varying all inputs together:� Focus on the expansion path.
� q plotted against proportionate increases in z.
� Take three standard cases:� Increasing Returns to Scale
� Decreasing Returns to Scale
� Constant Returns to Scale
� Let's do this for 2 inputs, one output…
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CASE 1: IRTS
q� An increasing returns to
scale function� Pick an arbitrary point on
the surface
� The expansion path…
z2
�
0� t>1 implies
φ(tz) > tφ(z)
�Double inputs
and you more
than double
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CASE 2: DRTS
q� A decreasing returns to scale
function� Pick an arbitrary point on
the surface
� The expansion path…
z2
�
0� t>1 implies
φ(tz) < tφ(z)
�Double inputs
and output
increases by less
than double 3232
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CASE 3: CRTS
q� A constant returns to scale
function� Pick a point on the surface
� The expansion path is a ray
z20
�
� φ(tz) = tφ(z)
�Double inputs
and output exactly
doubles
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RELATIONSHIP TO ISOQUANTS
q� Take any one of the three
cases (here it is CRTS)
� Take a horizontal “slice”
� Project down to get the
isoquant� Repeat to get isoquant map
z20 �The isoquant map
is the projection of
the set of feasible
points
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OVERVIEW...
The setting
Input require-
ment sets
The Firm: Basics
Changing one
input at time. ment sets
Returns to scale
Marginal
products
input at time.
Isoquants
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MARGINAL PRODUCTS
n Pick a technically efficient
input vector
Keep all but one input constant
n Remember, this means a z
such that q= φ(z)
n Measure the marginal change in
output w.r.t. this input
∂φ(z)MPi = φi(z) = ——
∂zi .
n Keep all but one input constant
n The marginal product
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CRTS PRODUCTION FUNCTION AGAIN
q
� Now take a vertical “slice”
� The resulting path for z2 =
constant
z20
Let’s look at its
shape
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MP FOR THE CRTS FUNCTION
q
φ(z)
� The feasible set
� Technically efficient points
� Slope of tangent is the
marginal product of input 1
φ1(z)
� Increase z1…
z1
� A section of the
production function
�Input 1 is essential:
If z1= 0 then q = 0
�φ1(z) falls with z1 (or
stays constant) if φ is
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RELATIONSHIP BETWEEN Q AND Z1
q q
� We’ve just taken the
conventional case
� But in general this
curve depends on the
z1
z1
q
z1
q
z1
curve depends on the
shape of φ.
� Some other
possibilities for the
relation between output
and one input…
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KEY CONCEPTS
� Technical efficiency
� Returns to scale
� Convexity
MRTS� MRTS
� Marginal product
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WHAT NEXT?
� Introduce the market
� Optimisation problem of the firm
� Method of solution
Solution concepts.� Solution concepts.
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