Optimization

1
Optimization

• Lecture 7
• The Maximum Value Function

2
Maximum value function

• How does the value of an optimization problem
  change to changes in the severity of the
  constraints?
• Idem for parameters in the objective function.
• Idem for the addition of several new constraints
  important application envelope theorem

3
1 Sensitivity in standard constrained
optimization

• Max f(x) subject to g(x)?b, where x, g and b are
  vectors and ?j is the Lagrange multiplier
  associated with the jth restriction
• How does the value of the problem depend on the
  parameters?
• Suppose we have a more general set A where a is
  an element of and is free to range for all values
  of b g(x)?b
• We have V(a) as the value of the problem it is
  the maximum value function for V(b)

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Maximum value functions

• V(a) is non-decreasing in each aj
• If f is concave and each gj is convex, V(a) is
  concave (and can so easily be optimized)
• Handy result. Let ? be the Lagrange multipliers.
  The j-th ? is ?jVj(b) dV/daj (at ab), so we
  can use the maximum value function and substitute
  for b!

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Example exercise 6.1.5

• Max x2ln(x1)
• Subject to p1x1p2x2 m x2?a and mltp2
• Lagrangean L x2ln(x1)-?1(p1x1p2x2-m)?2(x2a)
  
• For x1 1/x1-?1p1 0
• For x2 1- ?1p2 ?2 0
• ?2?0, x2?-a and ?2(x2a) 0

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Example exercise 6.1.5

• Suppose x2gt-a in that case ?2 0, ?1 1/p2 and
  x1 p2/p1. Using the budget constraint x2
  (m-p2)/p2. This holds if (m-p2)/p2gt-a. or
  agt(p2-m)/p2. We get V(m,a)(m-p2)/p2ln(p2/p1)
• If x2 -a then x1 (map2)/p1 , so we get
  V(m,a) -aln(map2)/p1
• Now we can use the theorem to see that
  ?1 V1(m,0) 1/m and ?2 V2(m,0) 0

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2 Maximum value and comparative statics

• Now we have max f(x,c) subject to g(x,c)?0
• Lagrangean L(?,x) f(x,c)- ?g(x,c)
• Let xx(c) be the solution. The maximum value
  function is V(c) f x(c),c
• V(c) is concave if f is concave and g is convex
  in general
• For g(x) 0 we get ?V/?cj ?f/?cj

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Exercise 6.2.6

• Max x1ax2b st p1x1p2x2 m
• Lagrangean L x1ax2b- ?(p1x1-p2x2-m)
• For x1 ax1a-1x2b- ?p1 0
• For x2 bx1ax2b-1- ?p2 0
• With the budget constraint we get x1 (a/(ab))
  m/p1 and x2 (b/(ab))m/p2
• So V (a/p1)a (b/p2)b m/(ab)ab
• Next we can analyze the properties of V

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3 Addition of new constraints

• Suppose we have a standard problem max
  f(x,c) st g(x,c)?0 and we want to add more
  constraints in g. How does this affect the
  optimum?
• Let V(c) be the value function. x x(c0) is the
  optimum. If constraints are added that satisfy
  this optimum condition, the V(c) will remain the
  same in the optimum

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Example
V
V1
V2
c
c0
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Envelope property

• The more restricted V(c)s are more concave (a
  greater curvature at c0) but the same tangent at
  c0.
• This problem is said to have the envelope
  property. This property has various handy
  consequences
• Interesting is the case where we max f(x,c) st
  g(x,c)?0 and keep some of the xi fixed

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What do we learn from this?

• Suppose we are in the optimum. We can maximize
  either V(c) or V(c) as long as c c0! This is
  the envelope theorem
• This can be handy in static and dynamic
  optimization
• So adding constraints that satisfy the optimum
  conditions do not change the optimum solution but
  does change the maximum value function

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History of the envelope theorem

• Early 1930s Jacob Viner analyzes the behaviour
  of firms in the short and long run. In the short
  run capital is fixed, while labour variable
• Viner posited a series of short-run cost curves,
  whose minimum points first fall and then rise. If
  both production factors were variable, long-run
  average cost would always be less than or equal
  to the corresponding short-run costs the envelope

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Short- and long-run costs
C(y)
SRAC1
LRAC
SRAC2
y
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The Puzzle

• It is impossible to draw an envelope that passes
  the minimum value of the short-run cost curves
• The tangency to the short-run curves and the
  envelope is identical for a fixed capital stock
  so costs are falling at the same rate no matter
  whether we look at the short or long run

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Envelope property example

• Suppose we have a firm that minimizes costs rKwL
  subject to a technology QF(K,L). Suppose capital
  is fixed in the short run KK0. K0 is optimal if
  QQ0. We can then simply minimize over L (see the
  plot hereafter).

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Envelope Theorem
V
VrKwL
V(KK1)
V(KK0)
Q
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The envelope theorem

• Let f(x,c) and gj(x,c) be real-valued
  continuously differential functions on Rn1
• Choose x for a given c so as to
• Max f(x,c) subject to gj(x,c)?0, x ?0,
  j 2,,m

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Envelope theorem (2)

• fx(x,c)?gx(x,c)?0 with
  fx(x,c) ?gx(x,c).x 0
• ?g(x,c) 0, g(x,c)?0
• x ? 0 and ? ? 0
• How does c affect x and ? ?
• Suppose we have an interior solution
  fx(x,c)?gx(x,c) 0, and g(x,c) 0

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Maximum value function

• Define F(c) f x(c),c MVF
• B(c) f x(c),c?(c).gx(c),c
• C(x, ?, c) f(x,c) ?.g(x,c) is the Lagrangean
• If F and B are continuously differentiable, all
  partial derivatives of F(c), B(c),
  C(x, ?, c) with respect to c are identical across
  the various functions

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Example simple static model

• The firm maximizes profits P(p,w)
  max pf(x)-wx with respect to x
• Let x be the optimal choice
• The envelope theorem says that the derivative of
  P with respect to p is simply f(x) evaluated at x
  x, so f(x(p,w))
• For w it is ?P/?w -x(p,w)

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Example consumer theory

• Max u(x) st p.x?m. Let x(p,m) be the optimum and
  the maximum value function V(p,m) ux(p,m) (
  indirect utility function)
• We can rewrite the problem into min p.x st u(x)?u
  with the minimum value function
  Ep,u p.x(p,u) expenditure function
• Duality implies Vp,E(p,u) u and Ep,V(p,m)
  m holds for all m and u

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Example consumer theory (2)

• xk(p,u)?E/?pk Hicksian demand for good k is
  the slope in the pk-direction of the expenditure
  function Shephards lemma!
• ?V/?m ? and ?E/?u ? and ? 1/?
• ?V/?pk - ? xk(p,m)
• xk(p,m) - (?V/?pk)/(?V/?m) Roys identity

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Applications of the envelope theorem

• Suppose we have a consumer who optimizes utility
  Ut ?st??s-t u(Cs) subject to Bs1
  (1r)BsYs-Cs, s?t, and a so-called non-Ponzi
  game condition
• Some intertemporal adding up gives Wt1
  (1r)(Wt-St)
• We formulate a value function
• J(Wt) max u(Ct)?J(Wt1)

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Application (2)

• FOC u(Ct) (1r) ?J(Wt1)
• Now we use the envelope theorem to say something
  more on the marginal utility of wealth. An
  increment to wealth on any date has the same
  effect on lifetime utility regardless of the use
  wealth is put
• So we can write J(W) u(C )which leads to the
  famous Euler equation u(Ct) (1r) ?u(Ct1)