Agricultural Economics

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AgEc 301 Agricultural Economics I
Slide Set 7 Chapter 8
Production Analysis
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Factor Product - Factor Factor

• Thus far we have talked about the mathematical
  details in factor-product and factor-factor
  relationships.
• What we havent done is talked about the
  optimization rules in these situations.

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Optimization

• In order to determine the best course of action
  in these situations, we must have information on
  relative prices.

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Optimal Input Use

• NOTE
• I will use a slightly different terminology in
  this section than your text does. This is simply
  to avoid confusion later on.

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Optimal Input Use

• When determining the amount of input to be used,
  the decision rule is
• MVP MIC
• Where MVP is marginal value product, and MIC is
  marginal input cost. Your text calls MVP MRP.

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Optimal Input Use

• MVP Poutput MP
• Poutput
• MIC Pinput the price of the input.

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Optimal Input Use

• Note that for the single input case
• MIC
• In this case TC Pinput X so that the
  derivative is simply Pinput.

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MIC or MC?

• I reserve the use of the term MC or Marginal
  Cost, to refer to the optimal output situation
  or
• MC
• Or the derivative with respect to output, not
  input level.

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Optimal Input Use

• A profit maximizing firm will always set marginal
  value product equal to marginal input cost for
  every input.
• Optimal employment and economic efficiency are
  achieved if all firms employ resources according
  to this decision rule.

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What do you do if???

• So, what decision rule do you use if you have
  scarce resources and are unable to allocate them
  in such a manner that MVP MIC for all inputs?

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Illustration of Optimal Employment

• Example for Tax Advisers, Inc.
• Three CPAs can process 2.4 returns per hour
• Four CPAs would increase output to 2.8 per hour.
• The fourth CPA reduces MP from 1.4 to 0.4
• CPAs earn 35 per hour
• Tax Advisers charges 100 per hour

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Illustration (continued)

• In this example, MIC 35 per hour (the cost of
  an additional CPA per hour).
• MVP 100 MP
• 100 0.4
• 40

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Illustration (continued)

• In this case, the cost of an additional CPA is
  35 per hour, but that CPA will bring in 40 per
  hour in revenue. Or
• MVP gt MIC
• The decision should be to employ the fourth CPA.

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Input Demand Function

• The MVP of labor and wage rates present firms
  with a clear incentive regarding employment
  levels.
• If MVPL gt wage, then hire
• If MVPL lt wage, then reduce

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Input Demand

• When MVPL wage then employment is at the
  optimal level.
• It is unrealistic to assume that an unlimited
  pool of labor exists at a given wage rate. As
  labor demand increases, wages must also.

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Input Demand Illustration

• A manufacturing firm faces the following demand
  for one of its products
• Q 300,000 2,500P
• Or
• P 120 0.0004Q

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Input Demand Illustration

• Total costs, not including assembly labor are
• TC 1,810,000 24Q
• To assemble the product, the firm will need to
  hire and train staff. The labor supply curve is
• LS 10,000PL

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Input Demand Illustration

• Based on this information, we can derive the
  demand curve for labor from the firms profit
  function
• ? TR - TC
• ? (120 0.0004Q)Q 1,810,000 24Q 2PLQ
• Where 2PLQ is the cost of assembly. (it takes 2
  hours to assemble 1 unit).

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Input Demand Illustration

• To find the labor demand curve, we need to know
  the optimal level of output.
• Profit ? is maximized at the point where marginal
  profit M?0 (Why?)

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Input Demand Illustration

• M? - 0.0008Q 96 2PL 0
• Solving for PL we find
• 2PL 96 0.0008Q
• PL 48 0.0002Q
• This also equals the firms MVP at optimal
  production (Why?)

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Input Demand Illustration

• To find the optimal level of employment, simply
  determine the amount of labor needed to produce
  at the profit maximizing level. Recall L 2Q
  (from the assumptions at the beginning) or, Q
  .5L

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Input Demand Illustration

• With Q 0.5L, the firms demand curve for labor
  is
• PL 48 0.0004(0.5L)
• PL 48 0.0002L
• and thus,
• LD 240,000 5000 PL

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Input Demand Illustration

• At any given wage rate, this demand function
  indicates the optimal level of employment, and
  conversely, at any given employment level, the
  optimal wage rate.
• The equilibrium wage and employment level can be
  determined by setting demand equal to supply

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Equilibrium P and Q

• Labor Demand Labor Supply
• 240000 5000PL 10000PL
• 15000PL 240000
• PL 16
• The equilibrium wage rate for this company is
  16.00 per hour

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Equilibrium P and Q

• To calculate the equilibrium quantity, simply
  plug the wage rate in to supply and demand.
• 240000 5000(16) 10000(16)
• 160,000 160,000
• Equilibrium quantity is 160,000 hours.

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Optimal Combinations of Inputs

• A graph of the combinations of two inputs capable
  of producing the same level of output is called
  an isoquant.
• In order to determine the optimal combination, we
  need information about the relative input prices.

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Isocost lines

• The isocost line is a locus of points along which
  the cost of the input combinations is the same.

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Isocost lines

• The simplest way to draw an isocost line is to
  determine, for a given budget, how many units of
  each input can be purchased. Then draw a line
  between them.

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Budget Lines

• The budget can be expressed mathematically as
• B PX X PY Y
• With a little manipulation this is
• Y B/PY (PX/PY)X

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Decision Rule

• The slope of this budget line is
• - Px/PY
• And, our decision rule is
• Px/PY MPx/MPY

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Decision Rule

• Or, restating our decision rule
• MPX/PX MPY/PY
• Which basically says, optimal input proportions
  are employed when an additional dollar spent on
  any input yields the same increase in output

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Expansion Path

• If you connect the points of tangency between a
  set of budget lines and isoquants, you can plot
  the points of optimal input combinations as you
  move toward higher isoquants. A line connecting
  these points is called the expansion path.

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Expansion Path
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Optimal Levels of Multiple Inputs

• Cost minimization requires only that the ratios
  of marginal products to prices be equal for all
  inputs. That is, inputs have to be combined in a
  cost minimizing fashion for a given level of
  output.

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Profit Maximization

• Profit maximization requires that a firm employ
  optimal input proportions AND produce an optimal
  quantity of output.
• So, cost minimization and optimal input
  proportions are necessary but not sufficient for
  profit maximization.

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Profit Maximization

• Profit maximization requires that the firm employ
  all inputs up to the point where
• MCQ MRQ
• Profit maximization requires that for each and
  every input
• PX/MPX MRQ

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Profit Maximization

• So for our two-input one-output function Q
  f(X,Y), it would also require that
• PY/MPY MRQ
• Rearranging, we get
• PX MPX MRQ MVPX
• PY MPY MRQ MVPY

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Profit Maximization

• Profits are maximized when inputs are employed so
  that price equals marginal value product for each
  input.