AAEC 3315 Agricultural Price Theory

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AAEC 3315Agricultural Price Theory

• CHAPTER 5
• Theory of Production
• The Case of One Variable Input
• in the Short-Run

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Objectives

• To gain understanding of
• Theory of Production
• Production Curves
• Total Physical Product
• Average Physical Product
• Marginal Physical Product
• Law of Diminishing Physical Product
• Stages of Production
• Production Functions

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Production Relationships

• Definition The technical relationship between
  inputs output indicating the maximum amount of
  output that can be produced using alternative
  amounts of variable inputs in combination with
  one or more fixed inputs under a given state of
  technology.
• Or, simply speaking, it is the technical
  relationship between inputs output

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Product Curves The Case of One Variable Input

• Total Physical Product (TPP) - illustrates the
  technological or physical relationship that
  exists between output and one variable input,
  ceteris paribus
• Starts increasing at an increasing rate.
• Continues to increase but at a decreasing rate
• Reaches the maximum, then decreases
• The functional form of a production function is
• Y f (X), where Y is the quantity of output and
  X is the quantity of input

Y
TPP
X
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Product Curves

• The point where TPP changes from increasing at an
  increasing rate to increasing at a decreasing
  rate is called the Inflection Points.
• Points A, B, and C Indicate total amount of
  output produced at each level of input use

Maximum Point
Y3
C
B
Y2
Inflection Point
Y1
A
X1
X2
X3
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Product Curves (Cont.)

• Average Physical Product (APP) - shows how much
  production, on average, can be obtained per unit
  of the variable input with a fixed amount of
  other inputs
• Indicates average productivity of the inputs
  being used - how productive is each input level
  on average
• APP Y / X
• Drawing a line from the origin which is tangent
  to the TPP curve gives APP max

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Product Curves (Cont.)

• Marginal Physical Product (MPP) - represents the
  amount of additional (i.e., marginal) output
  obtained from using an additional unit of
  variable input (X).
• MPP ?Y / ?X ?Y/?X
• or the slope of the TPP curve. Thus, MPP
  represents the rate of change in output resulting
  from adding one more unit of input
• Since MPP is the slope of TPP, it reaches a
  maximum at inflection point
• It reaches zero at the maximum point of TPP

MPP
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Product Curves
MPP is negative
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Relationships between Product Curves

• MPP reaches a maximum at inflection point
• MPP 0 occurs when TPP is maximum
• MPP is negative beyond TPP max
• Drawing a line from the origin which is tangent
  to the TPP curve gives APP max
• At point where APP is max, MPP crosses APP
  (MPPAPP)
• When MPP gt APP, APP is increasing
• When MPP APP, APP is at a max
• When MPP lt APP, APP is decreasing
• The relationship between TPP, APP, MPP is very
  specific. If we have COMPLETE information about
  one curve, the other two curves can be derived.

MPP is negative
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Law of Diminishing Marginal Physical Product

• Law of Diminishing Marginal Physical Product As
  additional units of one input are combined with a
  fixed amount of other inputs, a point is always
  reached where the additional product received
  from the last unit of added input (MPP) will
  decline
• This occurs at the inflection point

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Stages of ProductionRational Irrational

• The stage I of the production function is between
  0 and X1 units of X.
• In stage I
• TPP is increasing
• APP is increasing
• MPP increases, reaches a maximum decreases to
  APP
• Stage I is an irrational stage because APP is
  still increasing

I
0
X1
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Stages of ProductionRational Irrational

• The stage II of the production function is
  between X1 and X2 units of X.
• In Stage II
• TPP is increasing
• APP is decreasing
• MPP is decreasing and less than APP, but still
  positive
• RATIONAL STAGE BECAUSE TPP IS STILL INCREASING

I
II
X2
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Stages of ProductionRational Irrational

• Stage III of the production function is beyond X2
  level X
• In Stage III
• TPP is decreasing
• APP is decreasing
• MPP is decreasing and negative
• IRRATIONAL STAGE BECAUSE TPP IS DECREASING

I
II
III
X2
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A Hypothetical Production Function Schedule
Stage I
Stage II
Stage III
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Effects of Technological Change

• We know that the PF gives the max amount of
  output that can be produced by a firm using a
  given technology.
• The PF can shift over time as a result of a
  technological change
• Technological change refers to the introduction
  of new technology that increases output with the
  same amount of resources.

Y
TPP1
X
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Elasticity of Production

• The elasticity of production measures the degree
  of responsiveness between output and input.
• Using Calculus
• Like any other elasticity, elasticity of
  production is independent of units.
• It measures the percentage change in production
  in response to a percentage change in variable
  input.

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A Hypothetical Production FunctionA Mathematical
Example

• Consider a Production Function
• TP X2 1/30X3,
• where TP (Y) is quantity of output and X is the
  quantity of input.
• AP TP/X X (1/30)X2
• MP ?TP/?X
• 2X (3/30)X2
• 2X (1/10) X2

I
II
III
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A Hypothetical Production FunctionA Mathematical
Example

• Given
• TP X2 (1/30)X3,
• AP TP/X X (1/30)X2
• MP ?TP/?X 2X (1/10)X2
• At what levels of X does the MP
• reach its maximum?
• MP reaches its maximum
• where ?MP/?X 0
• That is, where
• 2 (2/10)X 0
• Or, 0.2 X 2
• Or, X 10

I
II
III
10
0
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A Hypothetical Production FunctionA Mathematical
Example

• Given
• TP X2 (1/30)X3,
• AP TP/X X (1/30)X2
• MP ?TP/?X 2X (1/10)X2
• At what levels of X does the AP
• reach its maximum?
• AP reaches its maximum
• where ?AP/?X 0
• That is, where
• 1 (2/30)X 0
• Or, (1/15) X 1
• Or, X 15

I
II
III
15
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A Hypothetical Production FunctionA Mathematical
Example

• Given,
• TP X2 (1/30)X3,
• AP TP/X X (1/30)X2
• MP ?TP/?X 2X (1/10)X2
• At what levels of X does the TP
• reach its maximum?
• TP reaches its maximum
• where ?TP/?X MP 0
• That is, where
• MP 2x (1/10)X2 0
• Using the quadratic
• formula of
• X 20

I
II
III
15
20
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A Hypothetical Production FunctionA Mathematical
Example

• Given
• TP X2 (1/30)X3,
• AP TP/X X (1/30)X2
• MP ?TP/?X 2X (1/10)X2
• What is the range of X values for
• Stage II?
• Stage II is the stage that
• begins where AP is at its
• maximum and ends where TP
• is at its maximum.
• Thus, the range of X values or
• Stage II is 15 and 20.

I
II
III
15
20
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A Hypothetical Production FunctionA Mathematical
Example

• Given
• TP X2 (1/30)X3,
• AP TP/X 2X (1/30)X2
• MP ?TP/?X 2X (1/10)X2
• At what level of X does the Law
• of Diminishing Returns set in?
• It sets in where MP reaches
• its maximum.
• Thus at X 10 the law of
• Diminishing returns sets in.

I
II
III
15
20
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A Hypothetical Production FunctionA Mathematical
Example

• Given TP X2 (1/30)X3,
• At Y 112.5 and X 15, what is the elasticity
  of production?
• Applying
• Ep (2X (1/10) X2) (X/Q)
• Ep ((215) (225/10)) (15/112.5)
• Ep (30 22.5) (0.133)
• Ep 7.5 0.133 0.997