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The Firm: Basics. MICROECONOMICS Principles and Analysis Frank Cowell . October 2011 . 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. The basics of production.

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the firm basics

The Firm: Basics

MICROECONOMICS

Principles and Analysis

Frank Cowell

October 2011

overview
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

the basics of production
The basics of production...
  • 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...
notation
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
feasible production
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. Why?
  • 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...

technical efficiency
Technical efficiency
  • Case 1:

q =f (z)

  • The case where production is technically efficient
  • The case where production is (technically) inefficient
  • 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

the function

q >f(z)

q <f(z)

z2

z1

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.
overview1
Overview...

The Firm: Basics

The setting

The structure of the production function.

Input require-ment sets

Isoquants

Returns to scale

Marginal products

the input requirement set
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.

the input requirement set1

q =f(z)

The input requirement set
  • Feasible but inefficient

z2

  • Feasible and technically efficient
  • Infeasible points.

Z(q)

q <f(z)

q >f(z)

z1

case 1 z smooth strictly convex

z2

q=f(z\')

q< f(z)

q =f(z")

z1

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)

  • Note important role of convexity.
  • A combination of two techniques may produce more output.
  • What if we changed some of the assumptions?
case 2 z convex but not strictly

z2

z1

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)

  • A combination of feasible techniques is also feasible
case 3 z smooth but not convex

z2

This point is infeasible

z1

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
case 4 z convex but not smooth

z2

z1

Case 4: Z convex but not smooth

q =f(z)

  • Slope of the boundary is undefined at this point.
summary 4 possibilities for z

z2

z2

z2

z2

z1

z1

z1

z1

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

overview2
Overview...

The Firm: Basics

The setting

Contours of the production function.

Input require-ment sets

Isoquants

Returns to scale

Marginal products

isoquants
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 constant output q

Let’s look at its shape

isoquant input ratio mrts

z2

°

z2

z1

°

z1

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

MRTS21=f1(z)/f2(z)

  • The isoquant is the boundary of Z
  • z′
  • Input ratio describes one production technique

{z: f (z)=q}

  • MRTS21: implicit “price” of input 1 in terms of 2
  • Higher “price”: smaller relative use of input 1
mrts and elasticity of substitution

z2

z2

z1

z1

MRTS and elasticity of substitution

∂log(z1/z2)

=  

∂log(f1/f2)

  • Responsiveness of inputs to MRTS is elasticity of substitution

prop change input ratio

 

prop change in MRTS

D input-ratio D MRTS

=    

input-ratio MRTS

s = ½

s = 2

elasticity of substitution
Elasticity of substitution

z2

  • A constant elasticity of substitution isoquant
  • Increase the elasticity of substitution...

structure of the contour map...

z1

homothetic contours
Homothetic contours
  • The isoquants
  • Draw any ray through the origin…

z2

  • Get same MRTS as it cuts each isoquant.

z1

O

contours of a homogeneous function
Contours of a homogeneous function
  • The isoquants

z2

  • Coordinates of input z°
  • Coordinates of “scaled up” input tz°
  • tz°

°

tz2

f (tz) = trf (z)

°

z2

trq

q

z1

O

O

°

°

tz1

z1

overview3
Overview...

The Firm: Basics

The setting

Changing all inputs together.

Input require-ment sets

Isoquants

Returns to scale

Marginal products

let s rebuild from the isoquants
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…
case 1 irts
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
  • f(tz) > tf(z)
  • Double inputs and you more than double output

z1

case 2 drts
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
  • f(tz) < tf(z)
  • Double inputs and output increases by less than double

z1

case 3 crts
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

relationship to isoquants
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

overview4
Overview...

The Firm: Basics

The setting

Changing one input at time.

Input require-ment sets

Isoquants

Returns to scale

Marginal products

marginal products
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.

crts production function again
CRTS production function again

q

  • Now take a vertical “slice”
  • The resulting path for z2 = constant

z2

0

Let’s look at its shape

z1

mp for the crts function
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 essential:Ifz1= 0 thenq = 0
  • f1(z) falls withz1 (or stays constant) iffis concave

z1

relationship between q and z 1

q

q

q

z1

z1

z1

q

z1

Relationship between q and z1
  • 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…
key concepts
Key concepts
  • Technical efficiency
  • Returns to scale
  • Convexity
  • MRTS
  • Marginal product

Review

Review

Review

Review

Review

what next
What next?
  • Introduce the market
  • Optimisation problem of the firm
  • Method of solution
  • Solution concepts.
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