The Firm: Basics

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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

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...
• 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
• 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

The structure of the production function.

Input require-ment sets

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. But not perverse.

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 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

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

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

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

Changing all inputs together.

Input require-ment sets

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

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

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

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

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

Overview...

The Firm: Basics

The setting

Changing one input at time.

Input require-ment sets

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

Review

Review

Review

Review

Review

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