Chapter 6: Agricultural Production Economics Production with One Input and One Output

Chapter 6: Agricultural Production Economics

Production with One Input and One Output

A Production Function:

Transformation of

input into output

A technical relationship

(not behavioral)

Output:

CornTobaccoWheatBeefMilk

Input:

SeedFertilizerFeedMachinery

FERTILIZER

11-48-0

P205 N K20

JOHN DEERE

Fixed versus Variable InputsFixed--

Farmer does not expectto vary

Over the planning horizon

Variable--

Farmer expects to vary

Over the planning horizon

???

??

?

Length of Planning Horizon:in the mind of the farmer6 months?The Growing Season?2 years?10 years (for Christmas trees)?Only the farmer knows for sure

6 months ?

2 years ? 50 years ?

Old idea--

Inputs could be categorizedLand--fixedLabor--variableMachinery--fixed (sort of!)

Not a correct idea

JOHN DEERE

Correct idea:Planning horizon determines whether inputs

Short Run--All inputs fixedIntermediate Run--Some fixed,

some variableLong Run--All inputs variable

are fixed or variable

Inputs:Traditional list

LandLabor

CapitalManagement

With capital you can purchaseland and laborIs management an input??

A Production Function:

Y = f(X)Y = output such as bu. of corn

X = input such as fertilizer

f(x) = rule for transforming X into Y

such as:

Y = 3X

Y = X

Y = .3X + .05X - .002X

Each of theseare production functions

0.5

2 3

Y = f(X | X X X )

The Variable inputThe output

Inputs treated as fixed

Y

X | X X X3

Y or TPP

TPP = TotalPhysical

Product

1 2 3 4

1 2 3 4

Y

X | X X X

Y or TPP

Y'

Y''

Y'''

X' X'' X'''

Specific amount of output froma specific amount of input

1 2 3 4 1 1 1

Marginal ProductThe incremental change in output

associated with a1 unit change

in the use of the input

Marginal Product of input x:

x = change in x

y = change in y

y = change in y

x = change in x= Marginal Product

Also called Marginal Physical Product

or MPP for short

Diminishing,

Constant

and Increasing

Marginal Product

ConstantMarginal Product

Case 1:

Output

Input (x)0 1 2 3 4

2

4

6

8

(y)

Constant slope

y

Constant Marginal Product

Output

Input (x)1 2 3 4

2

4

6

8

(y)

0

Constant slope

y

Triangles all thesame size and slope

= 2x

2

1

1 unit across2 units up2

2

2

1

1

1


Output

Input (x)1 2 3 4

2

4

6

8

(y)

0

Constant slope

y = 2x

2

1

2

2

2

1

1

1Each additionalunit of Xproduces twoadditional unitsof Y


Input (x)1 2 3 4Constant Marginal Product of b

Output (y)

0

Constant slope of by

1

1

1

1

b

b

b

b

b

b

b

b

=bx

Each additionalunit of x

additional Unitsof y

produces b

The MarginalProduct of anadditional unitof x is b


x x y y y/ xMPP


x x y y

0 0

1 2

2 4

4 8

5 10

3 6

y / xMPP


x x y y

0 0

1 2

2 4

4 8

5 10

3 6

1

1

1

1

1

y /MPP

x


x x y y

0 0

1 2

2 4

4 8

5 10

3 6

1

1

1

1

1

Y /MPP

2

2

2

2

2

x


x x y y

0 0

1 2

2 4

4 8

5 10

3 6

1

1

1

1

1

Y /MPP

2

2

2

2

2

2/1

2/1

2/1

2/1

2/1

MPP = 2 everywhere

x

Constant MPP

x

y = b

x

y

b

y = bx

b = MarginalProduct of anAdditionalUnit of x

Marginal Product

Case 2:Increasing

Output (y)

Input (x)0 1 2 3 4 5

0.72

3.5

0.7 1.3

4.5

Increasingmarginalreturnsto the

variableinput

Increasing Marginal Product

3

6.5

11

1.5

x x y y

0 0

Y / xMPP

1 0.7

2 2.0

3 3.5

4 6.5


5 11.0

x x y y

0 01

1

1

1

1

Y / xMPP

1 0.7

2 2.0

3 3.5

4 6.5


5 11.0

x x y y

0 01

1

1

1

1

Y / xMPP

.7

1.3

1.5

3.

4.5

1 0.7

2 2.0

3 3.5

4 6.5


5

MPP increases as x increases

11.0

x x y y

0 01

1

1

1

1

Y / xMPP

.7

1.3

1.5

3.

4.5

1 0.7

2 2.0

3 3.5

4 6.5

.7/1

1.3/1

1.5/1

3.0/1

4.5/1


5

MPP increases as x increases

11.0

Case 3:

Decreasing(Diminishing)MarginalProduct

Output (y)

Input (x)0 1 2 3 4 5

y = f(x)

5

2

1

.5

.3

1

1

1

1

1

5

7

88.58.8

Slope increasesbut at adecreasing rateAdditional unitsof x produceless and lessadditional y

Decreasing (Diminishing) Marginal Product

x x y y y / x MPP

Decreasing Marginal Product

x x y y y / x MPP

0 0

1 5

2 7

3 8

4 8.5

5 8.8


x x y y y / x MPP

0 0

1 5

2 7

3 8

4 8.5

5 8.8

1

1

1

1

1


x x y y y / x MPP

0 0

1 5

2 7

3 8

4 8.5

5 8.8

1

1

1

1

1

5

2

1

0.5


0.3

x x y y y / x MPP

0 0

1 5

2 7

3 8

4 8.5

5 8.8

1

1

1

1

1

5

2

1

0.5

5/1

2/1

1/1

.5/1

.3/1


As the use of x increases, MPP decreases

0.3

A Neoclassical ProductionFunction

X | X X X X 1 2 3 4 5


Y

X | X X X X 1 2 3 4 5


Y

Increasing MPP(and TPP)

X | X X X X 1 2 3 4 5


Y


InflectionPoint

X | X X X X 1 2 3 4 5


Y


InflectionPoint

Decreasing MPPIncreasing TPP

X | X X X X 1 2 3 4 5


Y


InflectionPoint


Maximum TPP0 MPP

X | X X X X 1 2 3 4 5


Y


InflectionPoint


Maximum TPP0 MPP

Negative MPPDeclining TPP

X | X X X X 1 2 3 4 5

Law of Diminishing

(Marginal) ReturnsAs units of the variable input (X )are added to units

of the fixed inputs ( X , X , X , X )we eventually reach a pointwhere each ADDITIONAL unitof the variable input (X )produces Less and Less ADDITIONAL output!

1

2 3 4 5

1

Y


InflectionPoint

Decreasing MPP Increasing TPP

Maximum TPP0 MPP


Law of DiminishingReturns holdsStarting Here

X | X X X X 1 2 3 4 5

Y

Increasing MPP

(and TPP)

InflectionPoint

Decreasing MPP

Increasing TPP

Maximum TPP0 MPP


X | X X X X 1 2 3 4 5

Y

Increasing MPP

(and TPP)

InflectionPoint

Decreasing MPP

Increasing TPP

Maximum TPP0 MPP


MPP

0

X | X X X X 1 2 3 4 5

Y

Increasing MPP

(and TPP)

InflectionPoint

Decreasing MPP

Increasing TPP

Maximum TPP0 MPP


MPP

0

X | X X X X 1 2 3 4 5

MPP

X | X X X X 1 2 3 4 5

Y

Increasing MPP

(and TPP)

InflectionPoint

Decreasing MPP

Increasing TPP

Maximum TPP0 MPP


MPP

0

X | X X X X 1 2 3 4 5

MPP

X | X X X X 1 2 3 4 5

Y

Increasing MPP

(and TPP)

InflectionPoint

Decreasing MPP

Increasing TPP

Maximum TPP0 MPP


MPP

0

X | X X X X 1 2 3 4 5

MPP

X | X X X X 1 2 3 4 5

AveragePhysical

ProductThe ratio of output to variable input

Y/XY/X | X X X X

Average productof ALL units of X used(not the incremental unit)

1 2 3 4 5

X Y Y/X

2 16 83 21 74 24 65 25 56 18 3

0 0 undefined1 7 7

Input Output (TPP) APP

TPP and APP

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7

XY (Scatter) 1 XY (Scatter) 2

Y

TPP

APP

PointInflection

X

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7


Y

Line out of Origin

TPP

APP

PointInflection

X

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7


Y

Line out of Origin

Point of Tangency

TPP

APP

PointInflection

X

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7


Y

Maximum APP

Line out of Origin

Point of Tangency

TPP

APP

PointInflection

X

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7


Y

Maximum APP

Line out of Origin

Point of Tangency

TPP

APP

PointInflection

X

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7


Y

Line out of Origin

Ratio Y/X

= Slope of Line

From OriginTPP

APP

APP = Y/X

Y

X

X

YAPP MAXIMUM

InflectionPoint

X

X

APPAPP,MPP

0

APP:Never Negative

YAPP MAXIMUM

InflectionPoint

MPP = 0

MPP MAXIMUM X

X

MPP=APP

MPP = APP

APP

MPP

APP,MPP

0

Do They have a Relationship???

MPP

APP

Marginal Physical ProductAverage Physical ProductMPP

X X

MPP APP

APP

0X | X X X X 1 2 3 4 5

MPP,

APP

APP

0X | X X X X 1 2 3 4 5

and Increasng APP

Positive

APP

MPP,

APP

APP

0X | X X X X 1 2 3 4 5

and Increasng APP

Positive

MPP,

APP

Maximum

APP

APP

0X | X X X X 1 2 3 4 5

and Increasng APP

Positive but Decreasing APP

Positive

MaximumAPP

MPP,

APP

APP

0X | X X X X 1 2 3 4 5

and Increasng APP


Positive

MaximumAPP

MPP,

APP

APP

0X | X X X X 1 2 3 4 5

and Increasng APP

InflectionPoint of

TPPMaximum

MPP


Positive

MaximumAPP

MPP,

APP

APP

0X | X X X X 1 2 3 4 5

IncreasingMPP

DecreasingMPP

0 MPPMaximum TPP

Positive

and Increasng APP

InflectionPoint of

TPPMaximum

MPP

Positive

but


Positive

MaximumAPP

MPP=APP

MPP,

APP

APP

MPP

0X | X X X X 1 2 3 4 5

IncreasingMPP

DecreasingMPP

0 MPPMaximum TPP

Positive

Negative andDecreasing MPP

and Increasng APP

InflectionPoint of

TPPMaximum

MPP

Positive

but


Positive

MaximumAPP

MPP=APP

MPP,

APP

measures:responsiveness of outputto changes in the useof Inputs

Elasticity of Production

A pure number(has no units)

Elasticity of Production% Change in output (Y)

divided by% Change in input (X)

% in output Y% in input X

=

Elasticity of Production


Y/YX/X

=

YX

XY

.

MPP 1/APP

= = MPP/APP


= MPP/APP

The Elasticity of Production (Ep)is the Ratio

of MPP to APP

AVP

Ep = 0

$

MVP

Ep = 1

0X | X X X X 1 2 3 4 5

Ep > 1(MPP>APP)

0<Ep<1 Ep < 0

IncreasingMPP

DecreasingMPP

0 MPPMaximum TPP

Positive


and Increasng APP

When the elasticity of production is greaterthan one, MPP lies above APP, APP is increasing,but MPP may be either increasing or decreasing.

When the elasticity of production is betweenzero and 1, both MPP and APP are decreasing.However, MPP is positive here.

Wnen the elasticity of production is negative,MPP is negative, and TPP is falling. However,

APP still remains positive.

Profit Maximixation:

and 1 output (Y)

1 input (X)

Assumptions:

1. Constant Input Price

The producer can purchaseas much or as littleof the needed input

at the going market price.

No producer canaffect input prices

by the amount of the purchase.

2. Constant Output PriceNo producer can affectthe price of the output (Y)because of theindividual production decision.

The price of the input is V.The price of the output is P.

3. Production Function Known

with CertaintyThis is an unrealistic assumption for agriculture!

Profit =Total Revenue - Total Cost

= TR - TC

= PY –V X. but Y = f(X)

so= Pf(X) – V X.

Total Value of Product Total Factor Cost

.

.

P f(X) - V X

Total Value of Product Total Factor Cost

Maximizing Profit:Maximize the difference

between

TVP and TFC

TVP TFC

. .

What is the appearance of a

TVP CURVE?

The TVP curve is a production function

with the vertical axis measured in dollar value

of output, not physical units

TVP = P TPP.

such as bushels or pounds.

TPPY

Production Function

TPP P .$

TPP P.

=TVP

TVP Curve

XX

TPP

What is the appearance of a

Total Factor Cost (TFC)Curve?

Total Factor Cost (TFC) Curve

TFC = V X

TFC

.

$

V

1

x

TFC = V X

TFC TVP

TPP and TVP max

.

$

V

1

x

Now Superimpose TVP Curve

TFC = V X

TFC TVP

Tangent

Tangent

TPP and TVP max

.

$

V

1

x

TFC = V X

TFC TVP

Tangent

Tangent

TPP and TVP max

.

$

V

1

x

Right of APP maxLeft of TPP Max

APP Max

TFC = V X

TFC TVP

Tangent

Tangent

TPP and TVP max

.

$

Maximum Vertical Distance= Maximum Profit

Maximum Vertical Distance= Maximum Loss

V

1

x

TFC = V X

TFC

1

V

TVP

Tangent

Tangent

TPP max

.

$

Profit is maximumwhere slope of TVP= Slope of TFC

X

Slope of TVP = Slope of TPP P .

= MPP P.

= MVP

= Marginal Value of the Product

So profits are maximum where:Slope of TVP = Slope of TFCMVP = MFCMVP = VMVP = the input price,assuming constant input and output prices

$MVP

0

MVP= MPP P

MFC = V

Profit MaxMVP=MFC=V

Profit Min

AVP=APP P

TFC = V X

TFC

1

V

TVP

Tangent

Tangent

TPP max

.

$

MVP=MFC=V

AVP Max

X

X

Stagesof

Production

Stage I

0 units of Xto level of X whichMaximizes AVP

Stage II

Level of X that Maximizes AVP

toLevel of X that Maximizes TPP

(0 MVP and 0 MPP)

Stage III

Level of X that Maximizes TPP (0 MPP)

and Beyond ......

Y

Stage III

X

The Rational Producer...1. Never produces beyond

the point of maximum TPP(input prices are never negative)

2. Produces at the point of maximum TPPonly if the input is free!

3. Does not normally producein stage I of Production

Stage II is theRational Stage of Production

Where the profit maximizing pointis found

$AVP

AVP=APP P.

Why not stage I?

Pick any point on the AVP curve.Draw an AVP curve.

Average Value of the Product= Average Physical Producttimes the product price

0X

$AVP

AVP=APP P.

X'

Area enclosed by rectangleis total revenue

from the use of X' units of X

X0

$

X

AVP=APP P.

MVP= MPP P.

Now add MVP curve

Marginal Value Product

= Marginal Physical Product

times the product price

0

$

MVP

Maximum ProfitTotal Factor Cost

of Input X

at profit max

Now add MFC curve (MFC = V)

Marginal Factor Cost= the price (V) of the input (X)

AVP=APP P

0X

MFC=V

$

AVP=APP P.

MVP

Maximum Profit

Total Revenuefrom sale of the product

using profit maximizinglevel of X

MFC=V

X0

$

X

AVP=APP P.

MVP

Maximum ProfitTotal Factor Costof Input X

at Profit MaxCost of X

Revenue-Cost=Profit

MFC=V

MFC=V

0

$

X

MVP

AVP

But if MFC > Maximum AVPCosts > RevenueLose money where MVP=MFC, andshut down instead!

Revenue

MFC= V

0

$

X

MVP

AVPRevenueCost of X

MFC= V

0

$

X

MVP

AVPRevenue

MFC= V

Revenue fails to cover costsresulting in a loss as indicated

Revenue

Loss

0

Stages of Productionand Elasticities of Production

Stage I Ep > 1Stage II 0 <Ep < 1

Stage III Ep < 0Rational Stage where0 <Ep < 1

AVP

Ep = 0

Stage I Stage II Stage III

$

MVP

Ep = 1

0X | X X X X 1 2 3 4 5

Ep > 1(MPP>APP)

0<Ep<1 Ep < 0

IncreasingMPP

DecreasingMPP

0 MPPMaximum TPP

Positive


and Increasng APP

AVP

Ep = 0

Stage I Stage II Stage III

Demand Curve for input X$

MVP

Ep = 1

0X | X X X X 1 2 3 4 5

Ep > 1(MPP>APP)

0<Ep<1 Ep < 0

The Demand Curve for a Singe Input

All Points of Intersection BetweenMFC and MVP that liein Stage II of Production

The Quantity of Input the ProducerWould Use to Maximize Profitsat Each Possible Input Price

Documents

Chapter 6: Agricultural Production Economics Production with One Input and One Output