The Firm: Basics

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The Firm: Basics. MICROECONOMICS Principles and Analysis Frank Cowell . 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

MICROECONOMICS

Principles and Analysis

Frank Cowell

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 
• Some 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
• 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
• 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
• 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

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
Overview

The Firm: Basics

The setting

Input require-ment sets

The structure of the production function

Isoquants

Returns to scale

Marginal products

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

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

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?

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

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

z2

z1

Case 4: Z convex but not smooth

q =f (z)

• Slope of the boundary is undefined at this point

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

Problems: the dent represents an indivisibility

Overview

The Firm: Basics

The setting

Contours of the production function

Input require-ment sets

Isoquants

Returns to scale

Marginal products

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 rate at which you trade off one input against another along the isoquant, maintaining constant q

Let’s look at its shape

z2

°

z2

z1

°

z1

Isoquant, input ratio, MRTS
• The set Z(q)
• A contour of the function 
• 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

z2

z2

z1

z1

MRTS and elasticity of substitution

∂log(z1/z2)

=  

∂log(f1/f2)

input-ratio  MRTS

  

input-ratio MRTS

prop change input ratio -  =

prop change in MRTS

s= ½

s= 2

Responsiveness of inputs to MRTS is 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
• The isoquants
• Draw any ray through the origin…

z2

• Get same MRTS as it cuts each isoquant

z1

O

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

Overview...

The Firm: Basics

The setting

Input require-ment sets

Changing all inputs together

Isoquants

Returns to scale

Marginal products

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
• An increasing returns to scale function
• Pick an arbitrary point on the surface

q

• The expansion path…
• t>1 implies f(tz)> tf(z)

z2

0

• Double inputs and you more than double output

z1

Case 2: DRTS
• A decreasing returns to scale function

q

• Pick an arbitrary point on the surface
• The expansion path…
• t>1 implies f(tz)< tf(z)

z2

0

• Double inputs and output increases by less than double

z1

Case 3: CRTS
• A constant returns to scale function

q

• Pick a point on the surface
• The expansion path is a ray
• f (tz) = tf (z)

z2

0

• Double inputs and output exactly doubles

z1

Relationship to isoquants
• Take any one of the three cases (here it is CRTS)

q

• 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

Overview 

The Firm: Basics

The setting

Input require-ment sets

Changing one input at time

Isoquants

Returns to scale

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

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

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) iff is concave

z1

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

Review

Review

Review

Review

Review

Technical efficiency

Returns to scale

Convexity

MRTS

Marginal product

What next?

Introduce the market

Optimisation problem of the firm

Method of solution

Solution concepts