Dr' Elaine S' Tan

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• Dr. Elaine S. Tan
• EC 1102 Math Economics
• Topic 7- Constrained Optimisation
• Learning outcomesBy the end of this module, you
  should be able to
• 1.Understand the difference between constrained
  and unconstrained optimisation.
• 2.Use the Lagrange Multiplier method to optimise
  functions subject to a constraint.
• 3.Interpret the meaning of the Lagrange
  Multiplier in an economic context.

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Constrained Optimization

• How can you find the optimum if you have a
  constraint on the possible answers?
• Some possible economic applications
• Maximizing consumer utility, given a budget
  constraint
• Maximizing profits, given firms output
  constraint
• We are only going to be looking at linear
  constraints though the techniques covered here
  can be extended to deal with other types of
  constraints (non linear, inequality constraints,)

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• To find the optimum values of a function zf(x,y)
  subject to a constraint, axbyM, we define the
  Lagrangian Multiplier, ?, where
• LL(x,y,?)f(x,y) ?(M-ax-by)
• ? is called a Lagrange multiplier and is treated
  as a variable (interpretation how z changes when
  constraint increases by 1 to M 1.)
• L is the sum of the original function to be
  optimized and ?(constraint0).
• The optimum value of ? may be determined in
  addition to the optimum values of the variables x
  and y.
• The method of finding the optimum value(s) of L
  is the same as in the previous section, but
  extended slightly, since the problem now consists
  of three independent variables.

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Example (Application 1)

• The total revenue function for two goods is given
  by
• TR36x-3x256y-4y2
• Find the number of units o each good which must
  be sold if profit is to be maximized when the
  firm is subject to a budget constraint 5x10y80
• Define the Lagrangian
• Step 1 Find the first order partial derivatives

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• Step2 Equate the first order partial
  derivatives to zero and solve for x, y and ?.
• Step 3 Show that solutions for x and y are the
  appropriate nature (i.e. minimum or maximum, not
  saddle or inflection)

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Economic Application 2

• A firm produces goods X and Y.
• (a) Find the optimal level of X and Y it should
  produce so profits are maximized, given the
  following total profit function capacity
  constraint
• ? 80x 2x2 xy 3y2 100y
• x y 12
• (b) What is the effect on ? if capacity increases
  by 1-unit?

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Economic Application 3

• Avner has 120 to be spent on 2 goods, x and y.
  His utility function is
• U xy when Px 1, Py 4
• How many units of x and y will Avner purchase?
• What is the effect on his utility with a 1
  increase in his budget?