THE THEORY AND ESTIMATION OF PRODUCTION

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THE THEORY AND ESTIMATION OF PRODUCTION
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Production

• The transformation of resources into products.
• The process whereby inputs are turned into
  outputs.
• Economic efficiency of the production process is
  the issue under analysis.
• Economic efficiency calls for minimizing the cost
  of producing any output level during a period of
  time.

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The Production Function

• For a profit maximizing firm, the revenues and
  the costs are the two important components.
• The costs will be related to the production of
  the good or service by using the different
  categories of inputs.
• The production function gives a mathematical
  representation of the relationship between
• 1. the output produced and,
• 2. the inputs used for production.

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The Production Function

• Q f (X1, X2, , Xk)
• where Q output
• X1, , Xk inputs used in the
  production process
• Q is a measure of output at a specific point in
  time.
• The production relationship holds for a given
  level of technology.
• Q is the maximum amount that can be produced with
  a given level of inputs.

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The Production Function

• The production function defines
• the relationship between inputs and the maximum
  amount that can be produced
• within a given period of time and
• with a given level of technology.
• Traditionally, the production function is written
  for two categories of inputs, capital (K) and
  labor (L)
• Q f (K, L)

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The Production Function

• The exact mathematical specification of the
  production function depends upon the productivity
  of the inputs at various levels of employment.
• The productivity of the inputs depends on the
  state of the technology.
• State of the technology is the inherent ability
  of inputs to produce output, given the
  simultaneous efforts of all other inputs in the
  production process.

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State of the Technology

• E.g. Labor can be more productive if it works
  with modern mechanical and computer-assisted
  equipment.
• E.g. Plant or equipment can be more productive if
  it is being operated by highly-skilled and
  well-trained workers.

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Short-run versus Long-run

• A SR production function shows the maximum
  quantity of a good or service that can be
  produced by a set of inputs, assuming that the
  amount of at least one of the inputs used remains
  constant.
• A LR production function shows the maximum
  quantity of a good or service that can be
  produced by a set of inputs, assuming that the
  firm is free to vary the amount of all the inputs
  being used.

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Short-run versus Long-run

• Long-run does not refer to a long period of time.
• The distinction has no direct connection with
  time at all.
• When changing the scale of production, the firm
  must operate under short-run conditions until its
  most-fixed input becomes variable.

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Short-run versus Long-run

• E.g. Assembly of an automobile production.
• Fixed inputs land and building, assembly lines,
  computerized plant and equipment.
• Variable inputs worker-hours, component parts,
  energy.

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Short-Run AnalysisTotal, Average, and Marginal
Product

• Terminology
• Inputs Factors, Factors of production,
  Resources
• Output Quantity (Q), Total Product (TP),
  Product

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• Recall Q Total product f (X, Y)
• Marginal product of X (MPX) ?Q / ?X,
  holding Y constant
• Average product of X (MPX) Q / X,
  holding Y constant
• Marginal product is the change in total product
  resulting from a unit change in a variable input.
• Average product is the total product per unit of
  input used.

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When Y is held constant at the level of 2 units
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Law of Diminishing Returns

• As additional units of variable input are
  combined with a fixed input, at some point, the
  additional output (i.e., marginal product) starts
  to diminish.
• In the example, the law of diminishing marginal
  returns occurs at 3.5 units of input.
• If input X is not divisible, then we would have
  to add the 4th unit to observe the diminishing
  marginal returns.

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TP
TP
diminishing returns begins to take effect
AP
marginal product becomes negative
MP
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Diminishing Marginal Returns

• Examples
• 1. Production sorting refillable glass bottles
• Fixed input machinery and working area
• Variable input people working as sorters
• 2. Production development of applications
  software
• Fixed input programming language and
  hardware
• Variable input software programmers

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Three Stages of Production in the Short-Run
Stage III
Stage I
Stage II
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Stages of Production

• Stage I
• 1. Fixed input grossly underutilized
• 2. Specialization and teamwork cause AP to
  increase when additional X is used.
• Stage II
• 1. Specialization and teamwork continue to
  result in greater output when additional X is
  used
• 2. Fixed input is being properly utilized
• Stage III
• 1. Fixed input capacity is reached
• 2. Additional X causes output to fall

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Stages of Production

• At which state should the firm produce?
• What level of variable input should the firm use?
• The firm needs information in order to decide how
  many units of variable input to use
• how many units of output it could sell
• the price of the product
• monetary cost of employing various amounts of the
  X input

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P Product price 2 W Cost per unit of labor
10,000
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• The firm should employ additional units of X up
  to the point where
• additional marginal labor cost (MLC) of adding X
  is more than made up for by the additional
  marginal revenue product (MRP) brought in by the
  sale of the increased output.
• Where MRP MLC

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The Multiple Input Case

• In the more general case, the firm is faced with
  the decision regarding the choice of the optimal
  combination of inputs.
• For simplicity, we will consider the two-input
  case.
• The assumption is that all inputs of the firm can
  be divided into two basic categories.

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Isoquant Curves

• An isoquant curve represents the various
  combinations of two inputs that produce the same
  amount of output.
• In the example, the following combinations of
  inputs X and Y produce 52 units of output
• 2,6
• 3,4
• 4,3
• 6,2
• 8,2

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Example Isoquant Curve
Isoquant TP 52 units
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Isoquant Curves

• The slope and the shape of the isoquant curves
  depend on the degree of substitutability between
  the two inputs.
• The degree of substitutability is a measure of
  the ease with which one input can be used in
  place of the other in producing a given amount of
  output.

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Isoquant Curves
Perfect Substitution
Perfect Complementarity
Imperfect Substitution
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Optimal Choice and Substitutability

• Easiest is the perfect complementarity X and Y
  need to be used in a fixed proportion.
• Trivial is the perfect substitutability X and Y
  can be used in any proportion.
• Difficult is the imperfect substitutability X
  and Y need to be used such that the combination
  is optimal by taking into account two factors
• 1. Degree of substitutability (how far can X be
  substituted for Y?)
• 2. Relative prices of X and Y

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Marginal Rate of Technical Substitution

• MRTS is a measure of the degree of
  substitutability between two inputs
• MRTS (X for Y) ?Y / ?X
• Numerator shows the amount of Y removed from the
  production process and denominator shows the
  amount of X needed to be added to the production
  process in order to maintain the same amount of
  output.

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MRTS Along the Isoquant Curve

• From the example
• Movement
• from to MRTS
• 2,6 3,4 -2 / 1 -2.0
• 3,4 4,3 -1 / 1 -1.0
• 4,3 6,2 -1 / 2 -0.5
• 6,2 8,2 0 / 2 0.0

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MRTS Along the Isoquant Curve

• As we move along the isoquant curve, the absolute
  value of the MRTS declines.
• This phenomenon is called The Law of Diminishing
  MRTS.
• This law implies that increasingly more of input
  X is needed to compensate for the loss of a given
  amount of input Y to maintain the same output.

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MRTS Along the Isoquant Curve

• Why would the Law of Diminishing MRTS hold?
• The answer comes from the marginal products of
  the two inputs
• Recall
• MPX ?Q / ?X
• MPX is the change in output relative to the
  change in some given input.

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• From point 1 to point 2 along the example
  isoquant, we move from (2,6) to (3,4). This
  movement implies two MPs
• We first change Y and keep X constant
• Q for (2,6) 52 and Q for (2,4) 39.
• Therefore, MPY -13 / -2 6.5
• Then, we change X and keep Y constant at its new
  level
• Q for (2,4) 39 and Q for (3,4) 52.
• Therefore, MPX 13 / 1 13

?Q
?Y
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• From point 2 to point 3 along the example
  isoquant, we move from (3,4) to (4,3). This
  movement implies two MPs
• We first change Y and keep X constant
• Q for (3,4) 52 and Q for (3,3) 41.
• Therefore, MPY -11 / -1 11
• Then, we change X and keep Y constant at its new
  level
• Q for (3,3) 41 and Q for (4,3) 52.
• Therefore, MPX 11 / 1 11

?Q
?Y
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• Lets look at the MRTS for points 1 and 2
• From point 1 to point 2
• -MPY . ?Y MPX . ?X
• needs to be true in order to maintain the same
  level of output.
• From this it follows that

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• Therefore, the MRTS along an isoquant curve is
  equal to the ratio of the marginal products for
  the two inputs.
• MRTS diminishes along the isoquant curve because
  the relative marginal products change.
• Recall from earlier
• From point 1 to point 2
• MPX / MPY 13 / 6.5 2
• From point 2 to point 3
• MPX / MPY 11 / 11 1

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• As seen, the MRTS is declining as we move down on
  the isoquant curve.
• The reason the Law of Diminishing MRTS holds is
  that as we move to the extreme points along the
  isoquant curve, the marginal product of the input
  being added (X) declines relative to the marginal
  product of the input being reduced (Y).

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Optimal Combination of Multiple Inputs

• The budget limit imposed upon the firm will be
  the second factor to consider in choosing the
  optimal combination.
• The isocost curve shows the different
  combinations of the two inputs that can be
  purchased with a given level of monetary budget
• E PX.X PY.Y

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• Rearrange the isocost curve equation
• For example, suppose a firm has a budget of
  1,000 to spend on inputs X and Y. Also,
  suppose PX 100 and PY 200. Then,
• 1,000 100 X 200 Y

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• Combination X Y
• A 0 5
• B 2 4
• C 4 3
• D 6 2
• E 8 1
• F 10 0
• These are the possible combinations of X and Y
  that lie on the isocost curve of 1,000.

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ISOCOST1 E 1,000
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Optimal Combination of Multiple Inputs

• Recall that there are two factors in determining
  the optimal input
• 1. Degree of substitutability (how far can X be
  substituted for Y?)
• MRTS -MPX / MPY
• 2. Relative prices of X and Y
• PX / PY

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• For decision making, we need to combine the
  information from the isoquant curve with the
  information from the isocost curve

optimal combination
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• The optimal combination occurs at the point where
  the isoquant curve is tangent to the isocost
  curve.
• At the point of tangency, the slopes of the two
  curves need to be equal to each other

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• Rearranging terms,
• At the best (optimal) combination each input has
  the same amount of marginal product per
  dollar/lira spent on that input.

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• If the following is true
• then, we can improve our utilization of the
  budget by employing more units of X and fewer
  units of Y.
• As X ?, MPX ? and as Y ?, MPY ?.
• As a result, the equality is reached where

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• If the following is true
• then, we can improve our utilization of the
  budget by employing more units of Y and fewer
  units of X.
• As Y ?, MPY ? and as X ?, MPX ?.
• As a result, the equality is reached where

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The Long-Run Production Function

• In the long-run, the firm has enough time to
  change the amount of all of its inputs.
• When the firm changes the amount of all inputs,
  the resulting change in the total product is
  called the returns to scale.

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Returns to Scale

• Increasing returns to scale are observed when
• ? in inputs gt ? in Q
• Decreasing returns to scale are observed when
• ? in inputs lt ? in Q
• Constant returns to scale are observed when
• ? in inputs ? in Q

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Returns to Scale and Output Elasticity

• Output elasticity is the measure of returns to
  scale
• If EQ gt 1, increasing returns to scale (IRTS)
• If EQ lt 1, decreasing returns to scale (DRTS)
• If EQ 1, constant returns to scale (CRTS)

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Q
Q
Q
X,Y
X,Y
X,Y
IRTS
CRTS
DRTS