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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.