Economic Optimization

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Economic Optimization

• Optimal decision- the choice alternative that
  produces a result most consistent with managerial
  objectives, which we presume is profit
  maximization.

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Steps in the Decision Process

• Economic relations must be expressed in a form
  suitable for analysis the managerial decision
  problem must be expressed in analytical terms.
• Optimization techniques must be applied to
  determine the best, or optimal, solution in light
  of managerial objectives.

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Maximizing the Value of the Firm

• The value of the firm is impacted by
• Total Revenue.... which is a function of
  marketing strategies, pricing and distribution
  policies, nature of competition....
• Total Cost .... which is a function of the price
  and availability of inputs, alternative
  production methods...

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Basic Economic RelationsOutline

• Functional Relations Equations
• Total, Average, and Marginal Relations
• Graphing Total, Average, and Marginal Relations

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Functional Relations EquationsDefinitions

• Equations Analytical representation of
  functional relationships.
• Dependent Variable The variable on the left
  side of the equation...the y-variable...the
  variable whose value is dependent upon changes in
  the x-variables...the endogenous variable.
• Independent Variable The variables on the right
  side of the equation...the x-variables...the
  variables who determine the value of the
  y-variable...the exogenous variable.

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Functional Relations EquationsMore Definitions

• Endogenous variable A factor that is determined
  by the independent variables in the model.
• Exogenous variables A factor determined outside
  the model, yet impacts the dependent variable in
  the model.

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Functional Relations Equations

• TR f(Q)
• In words....
• Total Revenue is a function of output.
• What is endogenous?
• What is exogenous?
• Are there other variables that will impact total
  revenue?

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Total, Average, and Marginal RelationsDefinitions

• Marginal change in a dependent variable caused
  by a one unit change in an independent variable.
• A Marginal Change is represented as
• ?Y / ?X
• Alternative definitions
• Rate of change
• Slope

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Examples of Marginal Relationships

• Marginal Revenue change in total revenue
  associated with a one-unit change in output.
• Marginal Cost Change in total cost associated
  with a one-unit change in output.
• Marginal Profit Change in profit associated
  with a one-unit change in output.

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Examples of Average Relationships

• Average Revenue Total Revenue/Output
• What is another name for average revenue?
• Average Cost Total Cost/Output
• Average Fixed Cost TFC / Q
• Average Variable Cost TVC/Q
• Average Profit Total Profit/ Output

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Graphing Total, Marginal, and Average Relations

• Slope a measure of the steepness of a line.
• OR The rate of change
• The marginal relation
• Tangent a line that touches but does not
  intersect a given curve.
• This is used to find the slope of a nonlinear
  curve.
• Inflection Point a point of maximum slope.

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Review of Basic Microeconomics

• Law of Demand As the price of a good rises,
  quantity demand will fall, ceteris paribus.
• In equation form P a bQ
• Total Revenue Price Quantity
• In equation form TR PQ
• OR TR aQ bQ2

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Review of Basic Microeconomics

• Revenue Maximization Activity level that
  generates the highest revenue.
• If P 100 2Q, how much should I produce to
  maximize total revenue?
• Marginal Revenue 100 4Q
• Total Revenue is maximized when marginal revenue
  is zero. WHY???
• When Marginal Revenue is positive, Total Revenue
  is rising.
• When Marginal Revenue is negative, Total Revenue
  is falling.
• When Marginal Revenue is zero, Total Revenue is
  neither rising or falling, therefore it is
  maximized.
• Therefore, total revenue is maximized at 25 units.

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Review of Basic Microeconomics

• What is the objective of the firm?
• PROFIT MAXIMIZATION
• Profit Total Revenue Total Cost
• To understand profit, you need to understand both
  revenue and cost. Understanding total revenue
  begins with the Law of Demand. Understanding
  total cost begins with the Law of Diminishing
  Returns.

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Review of Basic Microeconomics

• The Law of Diminishing Returns As a variable
  input increases, holding all else constant, the
  rate of increase in output will eventually
  diminish.
• OR..... As a firm increases its employment of
  labor, holding capital and all other factors
  constant, the productivity of each worker hired
  will eventually diminish.
• WHY? Each additional worker has less and less
  capital to work with.
• The Law of Diminishing Returns is the foundation
  from which we build the relationship between
  output and total cost, marginal cost, and average
  cost.

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Review of Basic Microeconomics

• Average Cost Minimization
• Why not Total Cost Minimization?
• If Marginal Cost is less than Average Cost,
  Average Cost will be declining.
• If Marginal Cost is greater than Average Cost,
  Average Cost will be increasing.
• When MC AC, Average Cost is Minimized.
• THE MATH OF AC MINIMIZATION

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Review of Basic Microeconomics

• Profit Total Revenue Total Cost
• Marginal Profit MR MC
• Profit is maximized when marginal profit equals
  zero. WHY?
• When marginal profit is positive (MRgtMC), profit
  is rising.
• When marginal profit is negative (MRltMC), profit
  is falling.
• When marginal profit is zero (MRMC), profit is
  not rising or falling, it is maximized.

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The Law of Demand
Law of Diminishing Returns

• Total Cost and Output
• Marginal Cost and Average Cost
• AVERAGE COST
• MINIMIZING LEVEL OF
• OUTPUT

• Price and Output
• via Own-Price Elasticity
• Total Revenue and Output
• Marginal Revenue and Output
• REVENU E MAXIMIZING LEVEL
• OF OUTPUT

Profit Total Revenue Total Cost Marginal
Profit Marginal Revenue Marginal Cost PROFIT
MAXIMIZING LEVEL OF OUTPUT
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Constrained Optimization

• Often we wish to optimize, but are faced with a
  constraint.
• In such a case, we need to use a Lagrangian
  Multiplier.

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Lagrangian Multipliers

• L f(x,z) ?(Y g(x,z)
• To find the optimal values of x and z, we take
  the derivative of the Lagrangian with respect to
  x, z, and ? setting these derivatives equal to
  zero.
• We then solve for the optimal values of x, z, and
  ?.

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Lagrangian Example (1)

• Profit p -10,000 400Q -2Q2
• Why is the constant term negative?
• M p 400 4Q
• Profit is maximized when Q 100
• What if each unit of output required 4 hours of
  skilled labor, and the firm only had 300 hours of
  skilled labor available?

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Solving Lagrangian (1)

• L -10,000 400Q 2Q2 ?(300 4Q)
• dL/dQ 400 4Q - 4?
• dL/d? 300 4Q
• 300 4Q 0
• Q 75
• 400 4(75) - 4? 0
• 100 - 4? 0
• ? 25

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Interpretation of Lagrangian (1)

• Because of the labor constraint, profit
  maximizing output has fallen from 100 to 75.
• What does ? mean?
• The is the marginal impact on the objective
  function of increasing or decreasing the
  constraint by one unit.
• If a one unit increase in output was possible,
  profit would rise by 25.
• How much would the firm pay for an extra hour of
  skilled labor? 6.25, since it takes four hours
  to produce one unit.
• What if the firm had 400 hours of skilled labor?
  Then the constraint would not be binding and
  profit would be maximized at 100 units.

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Lagrangian Example (2)

• A firm faces the following cost function
• Cost C f(x,z) 8x2 12z2 4xz
• The firm will produce 80 units of x and z, with
  any mix of x and z being acceptable.
• The objective is to minimize cost while producing
  these units.
• L 8x2 12z2 4xz ?(80 x z)
• To solve the Lagrangian, take the derivative of
  the function with respect to x, z, and ?.

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Solving the Lagrangian (2)Part One

• dL/dx 16x 4z ? 0
• dL/dz 24z 4x ? 0
• dL/d? 80 x z 0
• If you solve the first equation of ? you get
• ? 16x 4z
• If you substitute this in the second equation you
  get
• 14z 4x 16x 4z 10z 20x 0
• z 2x

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Solving the Lagrangian (2)Part Two

• If you substitute the value of z into the third
  equation you get
• 80 x 2x 0
• x 26.67
• z 53.33
• Note x z 26.67 53.33 80
• ? 16(26.67) 4(53.33) 213.40