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# The Firm: Basics - PowerPoint PPT Presentation

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

The Firm: Basics

The setting

The environment for the basic model of the firm.

Input require-ment sets

Isoquants

Returns to scale

Marginal products

• 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...

• 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

• 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...

• 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.

The Firm: Basics

The setting

The structure of the production function.

Input require-ment sets

Isoquants

Returns to scale

Marginal products

• 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

The Firm: Basics

The setting

Contours of the production function.

Input require-ment sets

Isoquants

Returns to scale

Marginal products

• 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

z2

• A constant elasticity of substitution isoquant

• Increase the elasticity of substitution...

structure of the contour map...

z1

• The isoquants

• Draw any ray through the origin…

z2

• Get same MRTS as it cuts each isoquant.

z1

O

• 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

The Firm: Basics

The setting

Changing all inputs together.

Input require-ment sets

Isoquants

Returns to scale

Marginal products

• 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…

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

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

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

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

The Firm: Basics

The setting

Changing one input at time.

Input require-ment sets

Isoquants

Returns to scale

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.

q

• Now take a vertical “slice”

• The resulting path for z2 = constant

z2

0

Let’s look at its shape

z1

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…

• Technical efficiency

• Returns to scale

• Convexity

• MRTS

• Marginal product

Review

Review

Review

Review

Review

• Introduce the market

• Optimisation problem of the firm

• Method of solution

• Solution concepts.