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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)

- z¢

- Note important role of convexity.

- A combination of two techniques may produce more output.

- z²

- 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)

- z¢

- z²

- 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

z2

z2

z2

z1

z1

z1

z1

Summary: 4 possibilities for ZStandard 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°

{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)

- 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

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.

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