The Firm: Basics

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The Firm Basics

• Microeconomia III (Lecture 1)
• Tratto da Cowell F. (2004),
• Principles of Microeoconomics

2
Overview...
The Firm Basics
The setting
The environment for the basic model of the firm.
Input require-ment sets
Isoquants
Returns to scale
Marginal products
3
The basics of production...

• We set out some of the elements needed for an
  analysis of the firm.
• Technical efficiency
• Returns to scale
• 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 output

For next presentation

• Prices

wi

• price of input i

w (w1, w2 , ..., wm )

• Input-price vector

p

• price of output

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Feasible production

• The basic relationship between output and
  inputs
• q f (z1, z2, ...., zm )

• single-output, multiple-input production relation

The production function

• This can be written more compactly as
• q f (z)

• Note that we use and not in the relation.
• Consider the meaning of f

Vector of inputs

• f gives the maximum amount of output that can
  be produced from a given list of inputs

distinguish two important cases...
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Technical efficiency

• Case 1
• q f (z)

• The case where production is technically efficient

• The case where production is (technically)
  inefficient

• Case 2
• q lt f (z)

Intuition if the combination (z,q) is
inefficient you can throw away some inputs and
still produce the same output
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The function ?
q

• The production function

• Interior points are feasible but inefficient

• Boundary points are feasible and efficient

q f (z)

• Infeasible points

0

• We need to examine its structure in detail.

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Overview...
The Firm Basics
The setting
The structure of the production function.
Input require-ment sets
Isoquants
Returns to scale
Marginal products
9
The input requirement set

• Pick a particular output level q
• Find a feasible input vector z

• remember, we must have q f (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 case.
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The input requirement set

• Feasible but inefficient

z2

• Feasible and technically efficient

• Infeasible points.

Z(q)
q lt f (z)
q gt f (z)
z1
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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)

• z

• Note important role of convexity.

• A combination of two techniques may produce more
  output.

• z²

• What if we changed some of the assumptions?

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Case 2 Z Convex (but not strictly)

• Pick two boundary points

• Draw the line between them

• Intermediate points lie in Z (perhaps on the
  boundary).

Z(q)

• z

• z²

• A combination of feasible techniques is also
  feasible

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Case 3 Z smooth but not convex

• Join two points across the dent

• Take an intermediate point

• Highlight zone where this can occur.

Z(q)

• in this region there is an indivisibility

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Case 4 Z convex but not smooth
q f (z)

• Slope of the boundary is undefined at this
  point.

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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
Only one efficient point and not smooth. But not
perverse.
Problems the "dent" represents an indivisibility
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Overview...
The Firm Basics
The setting
Contours of the production function.
Input require-ment sets
Isoquants
Returns to scale
Marginal products
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Isoquants

• 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)...

• 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

fj (z) fi (z)
f(z) fi(z) zi .

• Gives the rate at which you can trade off one
  input against another along the isoquant to
  maintain a constant q.

Lets look at its shape
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Isoquant, input ratio, MRTS

• The set Z(q).

• A contour of the function f.

• An efficient point.

• The input ratio

• Marginal Rate of Technical Substitution

• Increase the MRTS

z2 / z1 constant
MRTS21f1(z)/f2(z)

• The isoquant is the boundary of Z.

• z'

• Input ratio describes one particular production
  technique.

• z

z f (z)q

• 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.
• Responsiveness of input ratio to the MRTS is a
  key property of f.
• Given by the elasticity of substitution

?log(z1/z2) ? ???? ?log(f1/f2)

• Can be seen as the isoquants curvature

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Elasticity of substitution
z2

• A constant elasticity of substitution isoquant

• Increase the elasticity of substitution...

structure of the contour map...
z1
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Homothetic contours

• The isoquants

• Draw any ray through the origin

z2

• Get same MRTS as it cuts each isoquant.

z1
O
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Contours of a homogeneous function

• The isoquants

z2

• Coordinates of input z

• Coordinates of scaled up input tz

• tz

tz2
f (tz) trf (z)

• z

z2
trq
q
z1
O
O


tz1
z1
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Overview...
The Firm Basics
The setting
Changing all inputs together.
Input require-ment sets
Isoquants
Returns to scale
Marginal products
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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
• 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 (z1 and z2 vary in the same
  proportion z2/z1 constant)

z2
0

• tgt1 implies
• f(tz) gt tf(z)

• Double inputs and you more than double output

z1
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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

• tgt1 implies
• f(tz) lt tf(z)

• Double inputs and output increases by less than
  double

z1
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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

z2
0

• f(tz) tf(z)

• Double inputs and output exactly doubles

z1
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Relationship to isoquants
q

• A constant returns to scale function

• Pick a point on the surface

• The expansion path is a ray

• Take a horizontal slice

• Project down to get the isoquant

• Repeat to get isoquant map

z2
0

• f(tz) tf(z)

• Double inputs and output exactly doubles

z1
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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

z2
0

• The isoquant map is the projection of the set of
  feasible points

z1
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Overview...
The Firm Basics
The setting
Changing one input at time.
Input require-ment sets
Isoquants
Returns to scale
Marginal products
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Marginal products

• Pick a technically efficient input vector

• Remember, this means a z such that q f(z)

• Keep all but one input constant

• Measure the marginal change in output w.r.t.
  this input

• The marginal product

f(z) MPi fi(z) zi .
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CRTS production function again
q

• Now take a vertical slice

• The resulting path for z2 constant

z2
0
Lets look at its shape
z1
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MP for the CRTS function
f1(z)

• The feasible set

q

• Technically efficient points

f(z)

• Slope of tangent is the marginal product of
  input 1

• Increase z1

• A section of the production function

• Input 1 is essentialIf z10 then q0

• f1(z) falls with z1 (or stays constant) if f
  is concave

z1
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Relationship between q and z1

• Weve 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

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Key concepts

• Technical efficiency
• Returns to scale
• Convexity
• MRTS
• Marginal product

Review
Review
Review
Review
Review
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What next?

• Introduce the market
• Optimisation problem of the firm
• Method of solution
• Solution concepts.