Chapter 6 - Inputs and Production Functions

Height Exercise

We are going to take a sample of students from the class andcalculate a few statistics on their height.

I Person (P)

I Height in inches (Hi )

I Total Height

I Average Height

✓PHi

I Marginal Height

✓�P

Outline

Introduction

Theory of the Firm

Technology of Production

Firm Productivity

Isoquants

Input Substitutability

Elasticity of Substitution

Special Production Functions

Returns to Scale

Technological Progress

Introduction

I We have been talking about how consumers maximize utilityI =) That helped us derive the demand curve

I Now we are going to look at the theory behind how we derivethe supply curve

Theory of the Firm

We will address:

I How firms make cost-minimizing production decisions

I How (optimal) cost varies with output

I Characteristics of market supply

Production Process: What do firms do?I Combine inputs to achieve an output

I Example of inputs -

labor, capital, equipment, raw materials

I A certain “technology” is used which tells us how muchoutput is attainable from a given set of inputs

I Example of inputs - labor, capital, equipment, raw materials

Technology of ProductionSome Definitions

Factors of Production: resources (inputs), such as labor andcapital equipment, that firms use to manufacture goods andservices

Production Function: Tells us the maximum possible output thatcan be attainable by the firm for any given quantity of inputs

Q = f (L,K ,M) labor, capital, materialsQcorn = f (F , S ,A, L) fertilizer, seeds, acreage, labor

(L,K ,M)

(L,K ,M)firm has two technology choices

I Production function depends on a given technology process(for now assume technology can’t be changed)

Technology of ProductionSome Definitions

Factors of Production: resources (inputs), such as labor andcapital equipment, that firms use to manufacture goods andservices

Production Function: Tells us the maximum possible output thatcan be attainable by the firm for any given quantity of inputs

Q = f (L,K ,M) labor, capital, materialsQcorn = f (F , S ,A, L) fertilizer, seeds, acreage, labor

(L,K ,M)

(L,K ,M)firm has two technology choices

I Production function depends on a given technology process(for now assume technology can’t be changed)

Production FunctionsSuppose you only had one input: Labor

Q = f (L)

B - not being technically e�cient

Ccan get to here with same L

D - technically infeasible

Q = f (L)

- not being technically e�cient

Q = f (L)

- technically infeasible

Q = f (L)

More Definitions

Production Set: the set of all technically feasible combinations

I Points on or below the production function (A,B ,C )

I D not in production set because at that input, that output isunattainable

Technically E�cient

I Points A and C

I A technically e�cient firm attains the maximum possibleoutput from its inputs

Technically Ine�cient

I Point C

I Firms that get less output from its inputs than is technicallypossible

Technically Infeasible Point D

A Note on Inputs

It is important to know that the variables in the productionfunction are flows (amount of units used per unit of time) and notstocks

I Stock of capital: total factory installation

I Flow of capital: machine hours used per unit of time inproduction

What do we mean by captial?

I IS - physical goods - like robots, conveyor belts, ovens - thatcan produce other goods

I IS NOT - financial capital

Firm Productivity

1. Average Product (APL) =Total Product

Quantity of Labor=

LI Average amount of output per unit of laborI slope of ray from origin to point on curve

2. Marginal Product

(MPL) =Change in total product

Change in quantity of labor=

@LI additional output from a little more input (labor)I slope of the curve at a specific amount of labor

Let’s look at these on the Total Production Function

Firm Productivity

Quantity of Labor=

2. Marginal Product

Firm Productivity

Quantity of Labor=

2. Marginal Product

Total Production Function

QQ = f (L)

Total Production Function

I This function only depends on L. Could be that the firm alsouses K , but in this case capital is held fixed (only seeing halfof the story)

I Average Product vs. Marginal Product

I Increasing Return to Labor (MPL)

I Diminishing Return to Labor (MPL)

I Diminishing Total Return

I MPL and APL cross

Law of Diminishing Marginal Returns

Principle that as the usage of one input increases, holding thequantities of all other inputs fixed, a point will be reached beyondwhich the MP of that variable will decrease

I Example - Suppose I gave 3 people in this room a stack ofpapers and asked them to cut each sheet in half. How manypapers could they cut in half in 1 minute if I gave them:

I no scissors?I 1 pair of scissors?I 2 pair of scissors?I 3 pair of scissors?I 4 pair of scissors?I 10 pair of scissors?

I no scissors?

I 1 pair of scissors?I 2 pair of scissors?I 3 pair of scissors?I 4 pair of scissors?I 10 pair of scissors?

I no scissors?I 1 pair of scissors?

I 2 pair of scissors?I 3 pair of scissors?I 4 pair of scissors?I 10 pair of scissors?

I no scissors?I 1 pair of scissors?I 2 pair of scissors?

I 3 pair of scissors?I 4 pair of scissors?I 10 pair of scissors?

I no scissors?I 1 pair of scissors?I 2 pair of scissors?I 3 pair of scissors?

I 4 pair of scissors?I 10 pair of scissors?

I no scissors?I 1 pair of scissors?I 2 pair of scissors?I 3 pair of scissors?I 4 pair of scissors?

I 10 pair of scissors?

Relationship between MP and AP

I When average product is increasing in labor, the marginalproduct is greater than the average product, MPL > APL

I When average product is decreasing in labor, the marginalproduct is less than the average product, MPL < APL

I When average product neither decreases nor increases inlabor, the marginal product is equal to the average product,MPL = APL

Two Input Production Function

Marginal Product

MPL =Change in quantity of output

Change in quantity of labor

��K held constant

��¯K

MPK =�Q

��¯L

I will describe the steepness of the hill going in the “K”direction but holding level of L constant

Isoquant

These are similar to indi↵erence curves, but rather than holdingutility fixed, we fix the quantity

Isoquant (“same quantity”): a curve that shows all of thecombinations of labor and capital (or any 2 inputs) that canproduce a given level of output

Isoquants

Properties of Isoquants

I Downward sloping if MPL and MPK are both positive (or ifthey are both negative)

I Upward sloping if MPL is negative or MPK is negative (justone of them)

I Curvature tells how easily substitutable the two inputs are

How do I graph an isoquant?Let’s say the production function is Q =

pKL. Let’s graph the

isoquant for Q = 20.

I In general, solve for K in terms of Q and L.I General isoquant

Q2 = KL

I Isoquant for Q = 20pKL = 20

KL = 400

K =400

I Then, graph that line

isoquant for Q = 20.I In general, solve for K in terms of Q and L.

I General isoquant

Q2 = KL

KL = 400

K =400

isoquant for Q = 20.I In general, solve for K in terms of Q and L.I General isoquant

Q2 = KL

KL = 400

K =400

isoquant for Q = 20.I In general, solve for K in terms of Q and L.I General isoquant

Q2 = KL

KL = 400

K =400

Economic and Uneconomic Regions of Production

IUneconomic Region of Production: The region ofupward-sloping of backward-bending isoquants. In theuneconomic region, at least one input has a negative marginalproduct.

IEconomic Region of Production: The region whereisoquants are downward sloping.

Economic and Uneconomic Regions of Production

Marginal Rate of Technical Substitution of Labor forCapital

MRTSL,K : Rate at which the quantity of capital can be decreased(increased) for every one unit increase (decrease) in the quantity oflabor, holding the quantity of output constant

I Similar concept to marginal rate of substitution

MRTSL,K = ��K

�L= �dK

I Like MRS it is the opposite of the slope of our isoquants

MRTSL,K

I Point A on graph MRTSL,K = 2.5 =) starting from point Awe can substitute one man hour per day for 2.5 machine hoursper day and keep output constant

I Point B on graph MRTSL,K = 0.4 =) starting from point Bwe can substitute one man hour per day for 0.4 machine hoursper day and keep output constant

I As we move along this isoquant to right the MRTSL,Kdecreases (the slope becomes less negative, or flatter). As weget more labor, we are willing to give up more labor foranother unit of machine hours

MRTSL,K

Diminishing Marginal Rate of Technical Substitution:MRTSL,K diminishes as the quantity of labor increases along anisoquant.

I Isoquants that exhibit this behavior are convex, or bowed intoward the origin

MRTSL,K

But, there is more, we can relate this to the marginal products

Q = Q(L,K )

total di↵erentiate: dQ =@Q

dQ = MPLdL+MPKdK

along isoquant dQ = 0: 0 = MPLdL+MPKdK

MPLdL = �MPKdK

MPK= �dK

dL= MRTSL,K

Problem

Suppose a firm’s production function is Q = KL.

a) What is the MRTSL,K?

b) Is the MRTSL,K diminishing?

Firm’s Input Substitution Opportunities

Now, let’s look at 2 di↵erent production functions

1. Di�cult to substitute labor for capital

2. East to substitute labor for capital

Di�cult to substitute labor for capital

I Even if they increase labor from 100 to 400 man-hours, theyare only able to reduce machine-hours by 5

I Almost no substitutability between capital and labor

I Input mix

◆won’t change much

I Above point A, the MRTSL,K is almost 1 and below point A,it goes to 0 quickly

I L-shaped isoquants (similar to perfect complements, but notquite “perfect”)

Easy to substitute labor for capital

I When they increase man-hours from 100 to 400 (quadruple),they are able to reduce machine-hours by more than half (30machine-hours)

I MRTSL,K changes gradually as you move along the isoquant

I Nearly straight lined isoquants (like perfect substititues, butnot quite “perfect”)

=) curvature of the isoquant tell us about the substitutability ofK for L

Measuring Susbstitutability

I measures how quickly the marginal rate of technicalsubstitution of labor for capital (MRTSL,K ) changes as wemove along the isoquant

How can we look at this? Let’s think about the capital-labor ratio

I As we move along isoquant to right K -L ratio decreases

I Capital-labor ratio is measured by the ray from origin to point

on isoquant (e.g., Point AK0

Measuring Susbstitutability

I measures how quickly the marginal rate of technicalsubstitution of labor for capital (MRTSL,K ) changes as wemove along the isoquant

How can we look at this? Let’s think about the capital-labor ratio

I As we move along isoquant to right K -L ratio decreases

I Capital-labor ratio is measured by the ray from origin to point

on isoquant (e.g., Point AK0

K -L Ratio Graphically

K -L ratio =K0

K -L ratio =K1

Ldecreases moving along isoquant

K -L ratio =K0

K -L ratio =K1

K -L ratio =K0

K -L ratio =K1

K -L ratio =K0

K -L ratio =K1

K -L ratio =K0

K -L ratio =K1

K -L ratio =K0

K -L ratio =K1

K -L ratio =K0

K -L ratio =K1

K -L ratio =K0

K -L ratio =K1

Notes on K -L ratio and MRTSL,K

I Capital-labor ratio decreases moving right (increasing L) alongthe isoquant

I As you move along the isoquant MRTSL,K may also decreaseI If K -L ratio (input mix) changes a lot but the MRTSL,K does

notI Then the inputs are easily substitutable in the production

process

I On the other hand, if the K -L ratio (input mix) changes alittle bit and the MRTSL,K changes a lot

I Then substitution must be di�cult

Elasticity of SubstitutionI The elasticity of substitution is a scale-free (does not depend

on units) measure of this responsiveness of MRTSL,K tochanges in the K -L ratio

� =%� in K -L ratio

%�MRTSL,K=

d(K/L)

dMRTSL,K·MRTSL,K

I Note: Along an isoquant K/L and MRTSL,K always move inthe same direction. Therefore, � is always positive.

� � 0

I For a large �I Big numerator / small denominatorI Lots of substitutability

I For a small �I Small numerator / big denominatorI Less substitutability

%�MRTSL,K=

d(K/L)

dMRTSL,K·MRTSL,K

� � 0

%�MRTSL,K=

d(K/L)

dMRTSL,K·MRTSL,K

� � 0

%�MRTSL,K=

d(K/L)

dMRTSL,K·MRTSL,K

� � 0

Special Production Functions

1. Linear

2. Fixed-proportions

3. Cobb-Douglas

4. Constant Elasticity of Substitution

Linear Production Function

Q = aL+ bK where a and b are positive constants

Set-Up: You have a stand that sells french fries. It is calledFreddie’s French Fry stand. You can either use Idaho potatoes (I )or Maine potatoes (M) to make your french fries (Q). Yourproduction function is given as:

Q = 2I +M

Draw the isoquant for Q = 2 putting M on the horizontal

Q = 2I +M

Freddie’s French Fry Stand

I = 1� M

I = 0 =) M = 2

M = 0 =) I = 1

I = 1� M

I = 0 =) M = 2

M = 0 =) I = 1

I = 1� M

I = 0 =) M = 2

M = 0 =) I = 1

I = 1� M

I = 0 =) M = 2

M = 0 =) I = 1

I = 1� M

I = 0 =) M = 2

M = 0 =) I = 1

I also could have said: “Freddie can make 2 servings of french frieswith either 1 Idaho potato or 2 Maine potatoes.” And you wouldhave to come up with that production function.

What is the slope of the isoquant?

MRTSM,I =MPM

@Q/@I=

I Slope is -0.5 (a constant slope and constant MRTSM,I )

I You are willing to give up half an Idaho potato for one Mainepotato (the Maine potatoes must be smaller...)

What is the elasticity of substitution?

� =%�(K/L)

%�MRTSM,I=

I Note, the MRTSM,I is constant to the change is always 0.

I So, goods are infinitely (perfectly) substitutable for each other

I Perfect substitutes do not require that they have to besubstituted 1 for 1, only in some constant ratio (2 Maine for 1Idaho in this case)

Fixed-Proportions Production Function

Q = min(aK , bL) where a and b are positive constants

Set-Up: You have a factory that makes cerial. It is called Cindi’sCinnamon Crunch. Your factory puts 12oz. of cereal (C ) into abox (B). Your production function is given as:

Q = min(B ,C

Draw the isoquants for Q = 1 and Q = 2 putting C on the

horizontal axis.

Q = min(B ,C

horizontal axis.

Q = min(B ,C

horizontal axis.

Cindi’s Cinnamon Crunch

AQ = 11

Q = 22

AQ = 11

Q = 22

AQ = 11

Q = 22

AQ = 11

Q = 22

AQ = 11

Q = 22

AQ = 11

Q = 22

MRTSC ,B - Above A = 1Below A = 0At A it is undefined

So, moving from just above A to just below A the MRTSC ,B

changes an infinite amount, for a small change in C/B . Thus,

� = 0

There is no ability to substitute one input for the other. (perfectcomplements)

� = 0

Cobb-Douglas Production Function

Q = AL↵K� where ↵ � 0, � � 0, and A � 0

With Cobb-Douglas there is some substitutability between L and K

Cobb-Douglas Isoquants

Q = 1Q = 4

Cobb-Douglas

What is the MRTSL,K? Q = AL↵K�

↵AL↵�1K�

�AL↵K��1

MRTSL,K =↵K

↵MRTSL,K

What is the elasticity of substitution of Cobb-Douglas?

� =d(K/L)

dMRTSL,K

MRTSL,KK/L

✓�

◆MRTSL,K�↵MRTSL,K

✓�

◆↵

�= 1

Cobb-Douglas

↵AL↵�1K�

�AL↵K��1

MRTSL,K =↵K

↵MRTSL,K

� =d(K/L)

dMRTSL,K

MRTSL,KK/L

✓�

◆↵

�= 1

Cobb-Douglas

↵AL↵�1K�

�AL↵K��1

MRTSL,K =↵K

↵MRTSL,K

� =d(K/L)

dMRTSL,K

MRTSL,KK/L

✓�

◆↵

�= 1

Constant Elasticity of Substitution (CES)

We have already seen three types of production functions with aconstant elasticity of substitution:

1. Linear Production Function - � = 12. Fixed Proportions Production Function - � = 0

3. Cobb-Douglas - � = 1

There is a single function that contains these three special casefunctions (and every other � in between)

Q =haL

��1

� + bK��1

i ��1

where a, b, and � are positive constants and � is the elasticity ofsubstitution

Constant Elasticity of Substitution (CES)

Characteristics of Production Functions

Linear � = 1 Inputs are perfect substitutesIsoquants are straight lines

Fixed-proportions � = 0 Inputs are perfect complementsIsoquants are L-shaped

Cobb-Douglas � = 1 Isoquants are curvesCES 0 � 1 Includes above three as special

casesShape of isoquants varies

Note: If you can take the derivative of the CES function to findthe marginal product of K and L and you find that fun, then youshould study hard and get your PhD in Economics...

Practice - Production Functions

Suppose production technology takes the form Q = K 2L2 � L3.Let K = 10.

a) Over what range of L does the production function exhibitincreasing marginal returns, diminishing marginal returns, anddiminishing total returns?

Returns to Scale

Returns to Scale: The concept that tells us the % by whichoutput will increase (decrease) when all inputs are increased(decreased) by a given %

=%�(quantity of output)

%�(quantity of all inputs)

Note: When we were looking at the marginal product, it wasalways with respect to one input, holding the other constant. Here,we are thinking about what happens when BOTH inputs arechanged by the same percent (or proportional) amount

Returns to Scale, Example

For example, think of a two input production function K and LI Both K and L are increased by some proportion � > 1

I E.g., � = 2 (doubled)I K ! �K ! 2KI L ! �L ! 2L

I Define � as the corresponding proportionate increase in QI Q ! �Q

We want to know how � relates to �.� > �, increasing returns to scale

doubling inputs more than doubles output� = �, constant returns to scale

doubling inputs exactly doubles output� < �, decreasing returns to scale

doubling inputs less than doubles output

We want to know how � relates to �.

� > �, increasing returns to scaledoubling inputs more than doubles output

� = �, constant returns to scaledoubling inputs exactly doubles output

� < �, decreasing returns to scaledoubling inputs less than doubles output

doubling inputs more than doubles output

� = �, constant returns to scaledoubling inputs exactly doubles output

doubling inputs exactly doubles output

doubling inputs exactly doubles output� < �, decreasing returns to scale

doubling inputs less than doubles output

What type of returns to scale?

Returns to Scale - Cobb-DouglasDoes a Cobb-Douglas production function exhibit increasing,decreasing, or constant returns to scale?Q = AL↵K�

Let K1

and L1

denote the initial inputs of labor and capital whichyields quantity Q

. Both inputs are then increased by theproportional amount � > 1 to get a new quantity Q

. Then, wejust need to compare how Q

and Q2

are related.

= A(�L1

)↵(�K1

= �↵+�AL↵1

= �↵+�Q1

I If ↵+ � > 1 then �↵+� > �. So, increasing returns to scaleI if ↵+ � = 1 then �↵+� = �. So, constant returns to scaleI if ↵+ � < 1 then �↵+� < �. So, decreasing returns to scale

Let K1

and L1

and Q2

are related.

= A(�L1

)↵(�K1

= �↵+�AL↵1

= �↵+�Q1

Let K1

and L1

and Q2

are related.

= A(�L1

)↵(�K1

= �↵+�AL↵1

= �↵+�Q1

I If ↵+ � > 1 then �↵+� > �. So, increasing returns to scale

I if ↵+ � = 1 then �↵+� = �. So, constant returns to scaleI if ↵+ � < 1 then �↵+� < �. So, decreasing returns to scale

Let K1

and L1

and Q2

are related.

= A(�L1

)↵(�K1

= �↵+�AL↵1

= �↵+�Q1

I If ↵+ � > 1 then �↵+� > �. So, increasing returns to scaleI if ↵+ � = 1 then �↵+� = �. So, constant returns to scale

I if ↵+ � < 1 then �↵+� < �. So, decreasing returns to scale

Let K1

and L1

and Q2

are related.

= A(�L1

)↵(�K1

= �↵+�AL↵1

= �↵+�Q1

Practice - Returns to Scale

Suppose the production function is:

Q = 2K + 3L

a) What types of inputs are K and L?

b) Does this production function exhibit increasing, constant ordecreasing returns to scale?

Practice - Constant Elasticity of Substitution

Consider a CES production function given by Q =⇥K 0.5 + L0.5

a) What is the elasticity of substitution for this productionfunction?

b) Find the MPL, MPK , and MRTSL,K .

c) Does this production function exhibit increasing, decreasing, orconstant returns to scale?

d) Suppose that the production function took the form

Q =⇥100 + K 0.5 + L0.5

. Does this production functionexhibit increasing, decreasing, or constant returns to scale?

Returns to Scale vs. Diminishing Marginal Returns

So far, we have treated the firm’s technology as fixed. However,we know that overtime technology changes (improves) and thuschanges the firm’s production function.

ITechnological Progress: A change in a production processthat enables a firm to achieve more output from a givencombination of inputs, or get the same output for less inputs

Let’s assume that we have production technology that only uses Kand L. There are three categories of technological progress.

1. Neutral technological progressI Isoquant shifts inward (produce more output for less input)I MRTSL,K remains unchanged (curve of isoquant stays the

Neutral Technological Progress

2. Labor-saving Technological ProgressI Isoquant shifts inwardsI MPK increases relative to MPLI MRTSL,K flattens along a ray from originI Technical advances in capital equipment like robots and

computers

Labor-Saving Technological Progress

3. Capital-saving Technological ProgressI Isoquant shifts inwardsI MPL increases relative to MPKI MRTSL,K steepens along a ray from originI Education or skill level of workforce improves more quickly

than technical advances in capital

Capital-Saving Technological Progress

Skill-Biased Technical Change

Assume that there are two types of labor: high-skill and low-skill.Both inputs exhibit diminishing marginal products. Two thingshave been observed over the last half-century.

1. The ratio of high-skill workers to low-skill workers has beenincreasing (more people are graduating college)

2. The wages of high-skill workers relative to low-skill workershas also been increasing

How can this be if we have diminishing marginal product?

I Technological progress over last half-century (e.g., informationtechnology, computers) tended to be complementary to skilledlabor

Chapter 6 - Inputs and Production Functions

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