Lecture 5' KuhnTucker II some FAQs

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Lecture 5. Kuhn-Tucker II some FAQs

• Learning objectives. By the end of todays
  lecture you should
• Understand better how to do non-linear
  programming
• Introduction
• Last lecture we met the Kuhn-Tucker approach.
• Today we extend it.
• FAQ1. What happens if you have more than one
  constraint?
• Then you have Lagrange multiplier for each
  constraint and a complementary slackness
  condition for each constraint (see later)
• FAQ2. Why does Chiang give slightly different
  conditions?
• Answer recall that we were trying to maximize
  the objective, f(x,y) subject to the
  constraint(s), g(x,y) 0. We set up the
  Lagrangean L f(x,y) ?g(x,y) and presented
  the following necessary conditions

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2. Why different.

• Chiang adds in the constraint that x and y must
  not be negative and gives the following
  conditions

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2. Why different.

• One way to think about this is to add new
  constraints with associated multipliers µ (mu)
  and ? (rho). We then have the modified
  Lagrangean
• L f(x,y)?g(x,y) µx ?y
• which is maximized with respect to x,y, µ,? and
  ?. Necessary conditions

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2. Why different.

• We can rewrite these terms using the fact that L
  L µx ?y and the fact that and
• We can now see that when µ is positive x must be
  zero and
• Likewise, when ? is positive, y must be zero and
• Which is what Chiangs version says.

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2. More FAQs.

• FAQ 3. So which version should we use?
• Answer. If x 0 is a constraint, then you can
  either use Chiangs formulation or you can put it
  in as an explicit constraint in the Kuhn-Tucker
  problem. If there arent constraints on the sign
  of the variables, then dont use Chiangs
  formulation.
• FAQ 4. How can we tell which constraints are
  binding when theres more than one constraint?
• Answer.
• Formally you have to check all the possibilities.
  All the wrong ones will produce contradictions in
  your solutions. Only the right one will produce
  an answer that doesnt suffer from
  inconsistencies.
• FAQ 5. How many possibilities are there?
• Answer If there is one constraint, then there
  are two possibilities either the constraint is
  binding or it is not binding.
• If there are two constraints there are four
  possibilities because each of the constraints can
  be binding or not.
• If there are three constraints, there are 8
  possibilities.
• If there are n constraints then there are 2n
  possibilities.

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2. More FAQs.

• FAQ 6. Any shortcuts.
• Answer.
• Economic intuition. Generally if the problem is
  an economic one you can use economic reasoning to
  show that the constraints binds or doesnt.
• Example If the utility function is increasing in
  x and y then the budget constraint will be
  binding.
• Geometry. If you have two variables and several
  constraints, it can help to draw them along with
  indifference curves for the objective function.
• you still need to prove that your answer is
  consistent
• But
• Since only one pattern of constraint is
  consistent, if you have a possible solution that
  is consistent then you have the right solution.

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Example

• The objective 10x-x2 14y-y2
• The constraints x 0, y 0, x 10, xy 8
• The Lagrangean 10x-x2 14y-y2?x µy ?(10-x)
  ?(8-x - y)
• Finding the binding constraints
• Common sense the derivatives of the objective
  function are positive at zero. So we expect x gt
  0, y gt 0.
• Graphical
• (the unshaded area is known as the constraint set)

So our first guess is that 8xy is the only
binding constraint
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Example - solution

• The Lagrangean 10x-x2 14y-y2?x µy ?(10-x)
  ?(8-x - y)
• First order conditions

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Example - solution

• But use and so on.

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Example - solution

• We try the solution where only 8-x-y0 is binding
  in which case ??µ0
• From the first two equations we get x 2 y. From
  the third equation we get x 3, 5 y, but is it
  consistent with our assumption about the
  solution?
• X lt 10 so 10-x0 is not binding therefore ? 0
• X gt 0, so x0 is not binding therefore ? 0
• Y gt 0, so y0 is not binding therefore µ 0

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A cautionary note.

• Suppose the constraints were x 0, y 0, x 5,
  xy 8
• It would then be much harder to spot which
  constraints are likely to be binding.
• So you would want to check that x 5 wasnt a
  binding constraint.

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A cautionary note II.

• Suppose the constraints were x 0, y 0, x
  10, xy 8
• Then the constraint is empty
• There is no solution to the problem.

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You try

• The objective is 16x-2x2 12y-y2
• The constraints x 0, y 0, y 6, x2y 8
• Draw the constraint set and write down the
  Lagrangean.

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You try

• The Lagrangean is 16x-2x2 12y-y2 ?x µy
  ?(6-y) ?(8-x -2y)
• Differentiate and write down the first order
  conditions.

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Summing up Solving Kuhn-Tucker problems

• First write all the constraints in the
  appropriate format 0.
• (Optional draw the constraint set or use
  economic intuition to identify which constraints
  are likely to be binding)
• Write down the Lagrangean with one lagrangean
  multiplier for each constraint. (you dont need
  to use greek letters for the multipliers)
• Write down the first order conditions including
  the complementary slackness conditions.
• Solve by guessing which constraints are binding
  and finding the resulting solution. Or just try
  all combinations of constraints.
• Check your answer is consistent with your initial
  guess.

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4. Utility maximisation.

• 2. The lagrangean 0.4log (x) 0.6log(y)
  ?m-?px-?qy
• 3. First order conditions.