Econ 600: Mathematical Economics

1
Econ 600 Mathematical Economics

• July/August 2006
• Stephen Hutton

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Why optimization?

• Almost all economics is about solving constrained
  optimization problems. Most economic models
  start by writing down an objective function.
• Utility maximization, profit maximization, cost
  minimization, etc.
• Static optimization most common in
  microeconomics
• Dynamic optimization most common in
  macroeconomics

3
My approach to course

• Focus on intuitive explanation of most important
  concepts, rather than formal proofs.
• Motivate with relevant examples
• Practice problems and using tools in problem sets
• Assumes some basic math background (people with
  strong background might not find course useful)
• For more details, see course notes, textbooks,
  future courses
• Goal of course introduction to these concepts

4
Order of material

• Course will skip around notes a bit during the
  static course specifically, Ill cover the first
  half of lecture 1, then give some definitions
  from lecture 3, then go back to lecture 1 and do
  the rest in order.
• Sorry! ?

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Why not basic optimization?

• Simplest method of unconstrained optimization
  (set deriv 0) often fails
• Might not identify the optima, or optima might
  not exist
• Solution unbounded
• Function not always differentiable
• Function not always continuous
• Multiple local optima

6
Norms and Metrics

• It is useful to have some idea of distance or
  closeness in vector space
• The most common measure is Euclidean distance
  this is sufficient for our purposes (dealing with
  n-dimensional real numbers)
• General requirements of norm anything that
  satisfies conditions 1), 2), 3) (see notes)

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Continuity

• General intuitive sense of continuity (no gaps or
  jumps). Whenever x is close to x, f(x) is close
  to f(x)
• Formal definitionsA sequence of elements, xn
  is said to converge to a point, x in Rn if for
  every ? gt 0 there is a number, N such that for
  all n lt N, xn-x lt ?.
• A function fRn?Rn is continuous at a point, x if
  for ALL sequences xn converging to x, the
  derived sequence of points in the target space
  f((xn) converges to the point f(x).
• A function is continuous if it is continuous at
  all points in its domain.
• What does this mean in 2d? Sequence of points
  converging from below, sequence of points
  converging from above. Holds true in higher
  levels of dimensionality.

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Continuity 2

• Why continuity? Needed to guarantee existence of
  solution
• So typically assume continuity on functions to
  guarantee (with other assumptions) that a
  solution to the problem exists
• Sometimes continuity is too strong. To guarantee
  a maximum, upper semi-continuity is enough. To
  guarantee a minimum, lower semi-continuity
• Upper semi-continuity For all xn ? x, limn??
  f(xn) f(x)
• Lower semi-continuity For all xn ? x, limn??
  f(xn) ? f(x)
• Note that if these hold with equality, we have
  continuity.
• Note, figure 6 in notes is wrong

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Open sets(notes from lecture 3)

• For many set definitions and proofs we use the
  concept of an open ball of arbitrarily small
  size.
• An open ball is a set of points (or vectors)
  within a given distance from a particular point
  (or vector). FormallyLet e be a small real
  number. Be(x)y x-ylt e.
• A set of points S in Rn is open if for all points
  in S, there exists an open ball that is entirely
  contained within S. Eg (1,2) vs (1,2.
• Any union of open sets is open.
• Any finite intersection of open sets is open.

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Interior, closed set(notes in lecture 3)

• The interior of a set S is the largest open set
  contained in S. Formally, Int(S) UiSi where Si
  is an open subset of S.
• If S is open, Int(S)S
• A set is closed if all sequences within the set
  converge to points within the set. Formally, fix
  a set S and let xm be any sequence of elements
  in S. If limm??xmr where r is in S, for all
  convergent sequences in S, then S is closed.
• S is closed if and only if SC is open.

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Boundary, bounded, compact(notes in lecture 3)

• The boundary of a set S denoted B(S) is the set
  of points such that for all egt0, Be(x)nS is not
  empty and Be(x)nSC is not empty. Ie any open
  ball contains points both in S and not in S.
• If S is closed, SB(S)
• A set S is bounded if the distance between all
  objects in the set is finite.
• A set is compact if it is closed and bounded.
• These definitions correspond to their commonsense
  interpretations.

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Weierstrasss Theorem(notes in lecture 3)

• Gives us a sufficient condition to ensure that a
  solution to a constrained optimization problem
  exists. If the constraint set C is compact and
  the function f is continuous, then there always
  exists at least one solution tomax f(x) s.t. x
  is in C
• Formally Let fRn?R be continuous. If C is a
  compact subset of Rn, then there exists x in C,
  y in C s.t. f(x)?f(x)?f(y) for all x in C.

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Vector geometry

• Want to extend intuition about slope 0 idea of
  optimum to multiple dimensions. We need some
  vector tools to do this
• Inner product xy(x1y1x2y2xnyn)
• Euclidean norm and inner product related
  x2xx
• Two vectors are orthogonal (perpendicular) if xy
  0.
• Inner product of two vectors v, w is vw in
  matrix notation.
• vw gt 0 then v, w form acute angle
• vw lt 0 then v, w form obtuse angle.
• vw 0 then v, w orthogonal.

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Linear functions

• A function fV?W is linear if for any two real
  numbers a,b and any two elements v,v in V we
  have f(avbv) af(v)bf(v)
• Note that our usual interpretation of linear
  functions in R1 (f(x)mxb) are not generally
  linear, these are affine. (Only linear if b0).
• Every linear function defined on Rn can be
  represented by an n-dimensional vector
  (f1,f2,fn) with the feature that f(x) Sfixi
• Ie value of function at x is inner product of
  defining vector with x.
• Note, in every situation we can imagine dealing
  with, functionals are also functions.

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Hyperplanes

• A hyperplane is the set of points given by
  xf(x)c where f is a linear functional and c
  is some real number.
• Eg1 For R2 a typical hyperplane is a straight
  line.
• Eg2 For R3 a typical hyperplane is a plane.
• Think about a hyperplane as one of the level sets
  of the linear functional f. As we vary c, we
  change level sets.
• The defining vector of f(x) is orthogonal to the
  hyperplane.

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Separating Hyperplanes

• A half-space is the set of points on one side of
  a hyperplane. Formally HS(f) xf(x)?c or
  HS(f) xf(x)c.
• Consider any two disjoint sets when can we
  construct a hyperplane that separates the sets?
• Examples in notes.
• If C lies in a half-space defined by H and H
  contains a point on the boundary of C, then H is
  a supporting hyperplane of C.

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Convex sets

• A set is convex if the convex combination of all
  points in a set is also in the set.
• No such thing as a concave set. Related but
  different idea to convex/concave functions.
• Formally a set C in Rn is convex if for all x, y
  in C, for all ? between 0,1 we have ?x(1-?)y
  is in C.
• Any convex set can be represented as intersection
  of halfspaces defined by supporting hyperplanes.
• Any halfspace is a convex set.

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Separating Hyperplanes 2

• Separating hyperplane theorem Suppose X, Y are
  non-empty convex sets in Rn such that the
  interior of YnX is empty and the interior of Y is
  not empty.Then there exists a vector a in Rn
  which is the defining vector of a separating
  hyperplane between X and Y.Proof in texts.
• Applications general equilibrium theory, second
  fundamental theorem of welfare economics.
  Conditions where a pareto optimum allocation can
  be supported as a price equilibrium. Need convex
  preferences to be able to guarantee that there is
  a price ratio (a hyperplane) that can sustain an
  equilibrium.

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Graphs

• The graph is what you normally see when you plot
  a function.
• Formally the graph of a function from V to W is
  the ordered pair of elements,

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Derivatives

• We already know from basic calculus that a
  necessary condition for x to be an unconstrained
  maximum of a function f is that its derivative be
  zero (if the derivative exists) at x.
• A derivative tells us something about the slope
  of the graph of the function.
• We can also think about the derivative as telling
  us the slope of the supporting hyperplane to the
  graph of f at the point (x,f(x)).(see notes)

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Multidimensional derivativesand gradients

• We can extend what we know about derivatives from
  single-dimensional space to multi-dimensional
  space directly.
• The gradient of f at x is just the n-dimensional
  (column) vector which lists all the partial
  derivatives if they exist.
• This nx1 matrix is also known as the Jacobian.
• The derivative of f is the transpose of the
  gradient.
• The gradient can be interpreted as a supporting
  hyperplane of the graph of f.

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Second order derivatives

• We can think about the second derivative of
  multidimensional functions directly as in the
  single dimension case.
• The first derivative of the function f was an nx1
  vector the second derivative is an nxn matrix
  known as the Hessian.
• If f is twice continuously differentiable (ie all
  elements of Hessian exist) then the Hessian
  matrix is symmetric (second derivatives are
  irrespective of order).

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Homogeneous functions

• Certain functions in Rn are particularly
  well-behaved and have useful properties that we
  can exploit without having to prove them every
  time.
• A function fRn?R is homogeneous of degree k if
  f(tx1,tx2,.,tkf(x).In practice we will deal
  with homogeneous functions of degree 0 and degree
  1.Eg demand function is homog degree 0 in
  prices (in general equilibrium) or in prices and
  wealth double all prices and income has no
  impact on demand.
• Homogeneous functions allow us to determine the
  entire behavior of the function from only knowing
  about the behavior in a small ball around the
  originWhy? Because for any point x, we can
  define x as a scalar multiple of some point x it
  that ball, so xtx
• If k1 we say that f is linearly homogeneous.
• Eulers theorem if f is h.o.d. k then

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Homogenous functions 2

• A ray through x is the line (or hyperplane)
  running through x and the origin running forever
  in both directions.Formally a ray is the set
  x in Rnxtx, for t in R
• The gradient of a homogenous function is the
  essentially the same along any ray (linked by a
  scalar multiple). Ie the gradient at x is
  linearly dependent with the gradient at x.Thus
  level sets along any ray have the same
  slope.Application homogeneous utility functions
  rule out income effects in demand. (At constant
  prices, consumers demand goods in the same
  proportion as income changes.)

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Homothetic functions

• A function fRn?R is homothetic if f(x)h(v(x))
  where hR?R is strictly increasing and vR?R
  is h.o.d. k.
• Application we often assume that preferences are
  homothetic. This gives that indifference sets
  are related by proportional expansion along rays.
• This means that we can deduce the consumers
  entire preference relation from a single
  indifference set.

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More properties of gradients(secondary
importance)

• Consider a continuously differentiable function,
  fRn?R. The gradient of f (Df(x)) is a vector in
  Rn which points in the direction of greatest
  increase of f moving from the point x.
• Define a (very small) vector v s.t. Df(x)v0 (ie
  v is orthogonal to the gradient). Then the
  vector v is moving us away from x in a direction
  that adds zero to the value of f(x). Thus, any
  points on the vector v are at the same level of
  f(x). So we have a method of finding the level
  sets of f(x) by solving Df(x)v0. Also, v is
  tangent to the level set of f(x).
• The direction of greatest increase of a function
  at a point x is at right angles to the level set
  at x.

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Upper contour sets

• The level sets of a function are the set of
  points which yield the same value of the
  function. Formally, for fRn?R the level set is
  xf(x)cEg indifference curves are level sets
  of utility functions.
• The upper contour set is the set of points above
  the level set, ie the set xf(x)? c.

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Concave functions

• For any two points, we can trace out the line of
  points joining them through tx(1-t)y, varying t
  between 0 and 1. This is a convex combination of
  x and y.
• A function is concave if for all x, yie line
  joining any two points is (weakly) less than the
  graph of the function between those two points
• A function is strictly concave if the inequality
  is strict for all x,y.

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Convex functions

• A function is convex if for all x, yie line
  joining any two points is (weakly) greater than
  the graph of the function between the points.
• A function is strictly convex if the inequality
  is strict for all x,y.
• A function f is convex if f is concave.
• The upper contour set of a convex function is a
  convex set. The lower contour set of a concave
  function is a convex set.

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Concavity, convexity and second derivatives

• If fR?R and f is C2, then f is concave iff
  f(x)0 for all x. (And strictly concave for
  strict inequality).
• If fR?R and f is C2, then f is convex iff
  f(x)?0 for all x. (And strictly convex for
  strict inequality).

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Concave functions and gradients

• Any concave function lies below its gradient (or
  below its subgradient if f is not C1).
• Any convex function lies above its gradient (or
  above subgradient if f is not C1.
• Graphically function lies below/above line
  tangent to graph at any point.

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Negative and positive (semi-) definite

• Consider any square symmetric matrix A.
• A is negative semi-definite if xAx0 for all
  x.If in addition xAx0 implies that x0, then A
  is negative definite.
• A is positive semi-definite if xAx?0 for all
  x.If in addition xAx0 implies that x0, then A
  is positive definite.

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Principal minors and nsd/psd

• Let A be a square matrix. The kth order leading
  principal minor of A is the determinant of the
  kxk matrix obtained by deleting the last n-k rows
  and columns.
• An nxn square symmetric matrix is positive
  definite if its n leading principal minors are
  strictly positive.
• An nxn square symmetric matrix is negative
  definite if its n leading principal minors are
  alternate in sign with a11 lt 0.
• There are conditions for getting nsd/psd from
  principal minors.

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Reminder determinant of a 3x3 matrix

• You wont have to take the determinant of a
  matrix bigger than 3x3 without a computer, but
  for 3x3

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Concavity/convexity and nd/pd

• Any ease way to identify if a function is convex
  or concave is from the Hessian matrix.
• Suppose fRn?R is C2. Then
• f is strictly concave iff the Hessian matrix is
  negative definite for all x.
• f is concave iff the Hessian matrix is negative
  semi-definite for all x.
• f is strictly convex iff the Hessian matrix is
  positive definite for all x.
• f is convex iff the Hessian matrix is positive
  semi-definite for all x.

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Quasi-concavity

• A function is quasi-concave if f(tx
  (1-t)y)?minf(x),f(y) for x,y in Rn, 0t1
• Alternatively a function is quasi-concave if its
  upper contour sets are convex sets.
• A function is strictly quasi-concave if in
  addition f(tx (1-t)y)minf(x),f(y) for 0lttlt1
  implies that xy
• All concave functions are quasi-concave (but not
  vice versa).
• Why quasi-concavity? Strictly quasi-concave
  functions have a unique maximum.

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Quasi-convexity

• A function is quasi-convex if f(tx (1-t)y)
  maxf(x),f(y) for x,y in Rn, 0t1
• Alternatively a function is convex if its lower
  contour sets are convex sets.
• A function is strictly quasi-convex if in
  addition f(tx (1-t)y)maxf(x),f(y) for 0lttlt1
  implies that xy
• All convex functions are quasi-convex (but not
  vice versa).
• Why quasi-convexity? Strictly quasi-convex
  functions have a unique miniumum.

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Bordered Hessian

• The bordered hessian matrix H is just the hessian
  matrix next to the Jacobian and its transpose
• If the leading principal minors of H from k3
  onwards alternate in sign with the first lpmgt0,
  then f is quasi-concave. If they are all
  negative, then f is quasi-convex.

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Concavity and monotonic transformations

• (Not in the lecture notes, but useful for solving
  some of the problem set problems).
• The sum of two concave functions is concave
  (proof in PS2).
• Any monotonic transformation of a concave
  function is quasiconcave (though not necessarily
  concave). Formally, if h(x)g(f(x)), where f(x)
  is concave and g(x) is monotonic, then h(x) is
  quasi-concave.
• Useful trick the ln(x) function is a monotonic
  transformation.

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Unconstrained optimization

• If x is a solution to the problem maxxf(x), x is
  in Rn, what can we say about characteristics of
  x?
• A point x is a global maximum of f if for all x
  in Rn, f(x)?f(x).
• A point x is a local maximum of f if there exists
  an open ball of positive radius around x, Be(x)
  s.t. for all x in the ball, f(x) ? f(x).
• If x is a global maximum then it is a local
  maximum (but not necessarily vice versa).
• If f is C1, then if f is a local maximum of f,
  then the gradient of f at x 0. Necessary but
  not sufficient.This is the direct extension of
  the single dimension case.

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Unconstrained optimization 2

• If x is a local maximum of f, then there is an
  open ball around x, Be(x) s.t. f is concave on
  Be(x).
• If x is a local minimum of f, then there is an
  open ball around x, Be(x) s.t. f is convex on
  Be(x).
• Suppose f is C2. If x is a local maximum, then
  the Hessian of f at x is negative semi-definite.
• Suppose f is C2. If x is a local minimum, then
  the Hessian of f at x is positive semi-definite.
• To identify a global max, we either solve for all
  local maxima and then compare them, or look for
  additional features on f that guarantee that any
  local max are global.

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Unconstrained optimization 3

• If f Rn?R is concave and C1, then Df(x)0
  implies that x is a global maximum of f. (And x
  being a global maximum implies that the gradient
  is zero.) This is both a necessary and
  sufficient condition.
• In general, we only really look at maximization,
  since all minimization problems can be turned
  into maximization problems by looking at f.
• x solves max f(x) if and only if x solves min
  f(x).

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Non-differentiable functions(secondary
importance)

• In economics, we rarely have to deal with
  non-differentiable functions normally we assume
  these away.
• The superdifferential of a concave function f at
  a point x is the set of all supporting
  hyperplanes of the graph of f at the point
  (x,f(x)).
• A supergradient of a function f at a point x is
  an element of the superdiffential of f at x.
• If x is an unconstrained local maximum of a
  function fRn?R, then the vector of n zeros must
  be an element of the superdifferential of f at
  x.
• And equivalently subdifferential, subgradient,
  local minimum for convex functions.

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Constrained optimization

• General form of constrained optimization
• Normally we write the constraint by writing out
  restrictions (eg x ?1) rather than using set
  notation.
• Sometimes (for equality constraints) it is more
  convenient to solve problems by substituting the
  constraint(s) into the objective function, and so
  solving an unconstrained optimization problem.
• Most common restrictions equality or inequality
  constraints.
• Eg Manager trying to induce worker to provide
  optimal effort (moral hazard contract).

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Constrained optimization 2

• No reason why can only have one restriction. Can
  have any number of constraints, which may be of
  any form. Most typically we use equality and
  inequality constraints these are easier to solve
  analytically than constraints that x belong to
  some general set.
• These restrictions define the constraint set.
• Most general notation, while using only
  inequality constraints
• where G(x) is a mx1 vector of inequality
  constraints (m is number of constraints).
• Eg For the restrictions 3x1x210, x1?2, we
  have

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Constrained optimization 3

• We will need limitations on the constraint set to
  guarantee solution of existence (Weierstrass
  theorem).
• What can happen if constraint set not convex,
  closed? (examples)
• Denoting constraint setscharacterizes all
  values of x in Rn where f(x) ? c

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General typology of constrained maximization

• Unconstrained maximization. C is just the whole
  vector space that x lies in (usually Rn). We
  know how to solve these.
• Lagrange Maximization problems. Here the
  constraint set is defined solely by equality
  constraints.
• Linear programming problems. Not covered in this
  course.
• Kuhn-Tucker problems. These involve inequality
  constraints. Sometimes we also allow equality
  constraints, but we focus on inequality
  constraints. (Any problem with equality
  constraints could be transformed by substitution
  to deal only with inequality constraints.)

48
Lagrange problems

• Covered briefly here, mostly to compare and
  contrast with Kuhn-Tucker.
• Canonical Lagrange problem is of form
• Often we have a problem with inequality
  constraints, but we can use economic logic to
  show that at our solution the constraints will
  bind, and so we can solve the problem as if we
  had equality constraints.
• Eg Consumer utility maximization if utility
  function is increasing in all goods, then
  consumer will spend all income. So budget
  constraint pxw becomes pxw.

49
Lagrange problems 2

• Lagrange theorem in the canonical Lagrange
  problem (CL) above, suppose that f and G are C1
  and suppose that the nxm matrix DG(x) has rank
  m. Then if x solves CL, there exists a vector
  ? in Rn such that Df(x) DG(x) ?0. Ie
• This is just a general form of writing what we
  know from solving Lagrange problems we get n
  FOCs that all equal zero at the solution.
• Rank m requirement is called Constraint
  qualification, we will come back to this with
  Kuhn Tucker. But this is a necessary (not
  sufficient) condition for the existence of
  Lagrange Multipliers.

50
Basic example

• max f(x1,x2) s.t. g1(x1,x2) c1, g2(x1,x2)c2
• L f(x1,x2)?1(g1(x1,x2)-c1)?2(g2(x1,x2)-c2)
• FOCsx1 f1(x1,x2) ?1g11(x1,x2) ?2g21(x1,x2)
  0x2 f2(x1,x2) ?1g12(x1,x2) ?2g22(x1,x2)
  0
• Plus constraints?1 g1(x1,x2) c1 0?2
  g2(x1,x2) c2 0

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Lagrange problems 3

• We can also view the FOCs from the theorem as
• Ie we can express the gradient of the objective
  function as a linear combination of the gradients
  of the constraint functions, where the weights
  are determined by ?. (see diagram in notes)
• Note that no claims are made about the sign of ?
  (but sign will be more important in KT).

52
Kuhn Tucker 1

• The most common form of constrained optimization
  in economics takes the form
• (Note that we can include non-negativity
  constraints inside the G(x) vector, or not.)
• Examples utility maximization.
• Cost minimization

53
Kuhn Tucker 2

• Key problem with inequality constraints solution
  to problem might be on boundary of constraint, or
  might be internal. (see diagram in notes)
• Main advance of KT sets up necessary conditions
  for optimum in situations where constraints bind,
  and for situations where they dont. Then
  compare between these cases.
• Basic idea if constraints bind at a solution,
  then the value of the function must decrease as
  we move away from the constraint. So if at
  constraint xc, we cant be at a maximum unless
  f(x)?0 at that point. If constraint is x ?c, we
  cant be at a maximum unless f(x)0 at that
  point. Otherwise, we could increase the value of
  the function without violating any of the
  constraints.

54
Kuhn-Tucker 3

• We say a weak inequality constraint is binding if
  the constraint holds with equality.
• Unlike Lagrange problems, in KT problems,
  constraints might bind a solution, or they might
  not (if we have an internal solution). If a
  particular constraint does not bind, then its
  multiplier is zero if the constraint does bind,
  then the multiplier is non-zero (and is gt0 or lt0
  depending on our notational formulation of the
  problem).
• We can think of the multiplier on a constraint as
  being the shadow value of relaxing that
  constraint.
• Main new thing to deal with complementary
  slackness conditions. Complementary slackness
  conditions are a way of saying that either a) a
  particular constraint is binding (and so the
  respective multiplier for that constraint is
  non-zero), which implies a condition on the slope
  of the function at the constraint (it must be
  increasing towards the constraint) b) a
  constraint does not bind (so we must be in an
  internal solution, with a FOC that equals zero).

55
Example 1

• Max f(x) s.t. 10-x?0, x ?0L f(x)
  ?(10-x)FOCsx f(x)- ? 0? 10-x ?0CSCS1
  (f(x)-?)x0CS2 (10-x)?0

56
Example 1, contd

• Case 1, strict interior. xgt0, xlt10 From CS2, we
  have ?0.From CS1, we have f(x) 0. (ie
  unconstrained optimum)
• Case 2, left boundary, x0.From CS2, we have
  ?0.From FOC1 (x) we need f(x) 0.
• Case 3, right boundary, x0.From CS1, we have
  f(x) ?, and we know ??0 by construction, so we
  must have f(x) ?0.
• Thus, we can use the KT method to reject any
  candidate cases that dont have the right slope.

57
Solving KT problems

• Two methods, basically identical but slightly
  different in how they handle non-negativity
  constraints.
• Method 1 (treat non-negativity constraints as
  different from other conditions)
• Write the Lagrangean with a multiplier for each
  constraint other than non-negativity constraints
  on choice variables. If we write the constraints
  in the Lagrangean as g(x)?0, we should add (not
  substract) the multipliers in the Lagrangean,
  assume the multipliers ??0, and this will make
  the FOCs for x non-positive, and the FOCs for the
  multiplers ? non-negative.
• Take FOCs for each choice variable and each
  multiplier.
• Take CS conditions from the FOC for each choice
  variable that has a non-negativity constraint,
  and for each multiplier.
• Take cases for different possibilities of
  constraints binding reject infeasible cases,
  compare feasible cases.

58
Solving KT problems 2

• Second method treat non-negativity constraints
  as the same as any other constraint functionally
  the same but doesnt take shortcuts.
• Write the Lagrangean with a multiplier for each
  constraint. This will give us more multipliers
  than the previous method.
• Take FOCs for each choice variable and each
  multiplier.
• Take CS conditions for each multiplier. This
  gives us the same number of CS conditions as the
  previous method.
• Take cases for different possibilities of
  constraints binding reject infeasible cases,
  compare feasible cases.

59
Example 2, method 1

• Max x2 s.t. x?0, x2L x2 ?(2-x)FOCs x 2x
  - ? 0 ? (2-x) ? 0CS (2x ?)x 0 (2-x)?
  0
• Case 1, internal solution, xgt0, ?0
  contradiction from FOC1 rules this case out.
• Case 2, left boundary, x0, ?0. Consistent, but
  turns out to be a minimum.
• Case 3, right boundary, ? gt 0, xgt0. CS2 implies
  x2.

60
Example 2, method 2

• Max x2 s.t. x?0, x2L x2 ?1(2-x) ?2(x)
  FOCs x 2x ?1 ?2 0 ?1 (2-x) ? 0 ?2
  x ? 0 CS (2-x)?1 0 x?2 0
• Case 1, internal solution, ?10, ?20 From FOC1,
  consistent only if x0 (ie actually case 2)
• Case 2, left boundary, ?10, ?2gt0 From CS2, x0.
  Consistent, but turns out to be a minimum.
• Case 3, right boundary, ?1 gt 0, ?20. CS1
  implies x2.
• Case 4, ?1 gt 0, ?2gt0 from CS1 and CS2, clearly
  contradictory (0x2).

61
Sign issues

• There are multiple ways of setting up the KT
  Lagrangean using different signs.
• One way is as above in the Lagrangean, add ?g(x)
  terms, write the g(x) terms as ? 0, assume the
  multipliers ?i?0, which implies that the FOC
  terms are 0 for choice variables and ?0 for
  multipliers. The lecture notes (mostly) use this
  method.
• Another way is to subtract the ?g(x) terms in L,
  and write the g(x) terms as 0, assume implies
  ?i?0, which implies the FOC terms are 0 for
  choice variables and ?0 for multipliers.SB uses
  this method.
• Whatever method you choose, be consistent.

62
Example 1, SB signing

• Max f(x) s.t. 10-x?0, x ?0L f(x) -
  ?(x-10)FOCsx f(x)- ? 0?
  -(x-10)?0CSCS1 (f(x)-?)x0CS2 -(x-10)?0

63
Kuhn Tucker 4

• Formal treatment start with Lagrangian. When
  fRn?R and GRn?Rm, the Lagrangian of the KT
  problem is a new function LRnm?R.
• Important to note the domain limit on L the
  Lagrangian is non-negative (and so (We could
  rewrite the problem restricting the multipliers
  to be negative by changing the in the
  Lagrangian to - .)(We could also rewrite the
  problem without the implicit non-negativity
  constraints in general KT problems not in
  economic settings, we need not require x
  non-negative.)

64
Kuhn-Tucker 5

• As in the Lagrange method case, we can rewrite
  the Lagrangian asdecomposing G into its
  components.
• For any fixed point x, define indices of GK
  igi(x)0 and M ixigt0.
• Define by differentiating G with only the K
  components wrt components j in M . This is MxK
  matrix.

65
Kuhn Tucker Theorem

• Suppose that x solves the canonical KT as a
  local maximum and suppose that H(x) has maximal
  rank (Constraint Qualification). Then there
  exists ??0 s.t. (ie FOCs for choice
  vbles) for i1,..n (ie CS conditions for
  non-negativity constraints) (ie FOCs for
  multipliers) (ie CS conditions for
  multipliers)

66
KT theorem notes

• The constraint qualification (H(x) has maximal
  rank) is complex and is typically ignored. But
  technically we need this to guarantee the
  theorem, and that the solution method yields
  actual necessary conditions
• These are necessary conditions for a solution.
  Just because they are satisfied does not mean we
  have solved the problem we could have multiple
  candidate solutions, or multiple solutions, or no
  solution at all (if no x exists).

67
KT and existence/uniqueness

• Suppose G(x) is concave, and f(x) is strictly
  quasi-concave (of G(x) strictly concave, and f(x)
  quasi-concave), then if x solves KT, x is
  quasi-concave. Furthermore, if xG(x)?0,x?0 is
  compact and non-empty and f(x) is continuous,
  then there exists x which solves KT.
• Proof Existence from Weierstrass theorem. For
  uniqueness Suppose there are some x, x that
  both solve KT. Then f(x) f(x) and G(x)?0,
  G(x)?0. Since G is concave, for t in 0,1 we
  have G(tx (1-t)x) ? tG(x) (1-t)G(x) ? 0.
  So tx (1-t)x is feasible for KT. But f
  strictly quasi-concave implies f(tx(1-t)x) gt
  minf(x),f(x)f(x). So we have a feasible x
  (tx (1-t)x) which does better than x and x.
  Which contradicts x, x both being optimal
  solutions.

68
The constraint qualification

• Consider the problemmax x1 s.t. (1-x1)3-x2 ?0,
  x1 ?0, x2 ?0.(see picture in notes, (1,0) is
  soln)At solution, x2 ?0 is a binding
  constraint.Note that gradient of constraint at
  (1,0) isDg(1,0) (2(x1-1),-1) (0,-1) at
  soln.This gives H matrix of which has a rank
  of 1.
• The gradient of f(1,0) is (1,0), which cannot be
  expressed as a linear combination of (0,1) or
  (0,-1). So no multipliers exist that satisfy the
  KT necessary conditions.

69
Non-convex choice sets

• Sometimes we have non-convex choice sets
  typically these lead to multiple local optima.
• In these cases, we can go ahead and solve the
  problem separately in each case and then compare.
  OR we can solve the problem simultaneously.

70
Example labour supply with overtime

• Utility function U(c,l)calß
• Non-negativity constraint on consumption. Time
  constraints l ? 0 and 24 l ? 0 on leisure (note
  l is leisure, not labour).
• Overtime means that wage rate w per hour for
  first 8 hours, 1.5w per hour for extra hours.
  This meansc w(24-l) for l ? 16c 8w
  1.5w(16-l) for l 16.

71
Overtime 2

• The problem is that we have different functions
  for the boundary of the constraint set depending
  on the level of l. The actual problem we are
  solving has either the first constraint OR the
  second constraint if we tried solving the
  problem by maximising U(x) s.t. both constraints
  for all l then we would solve the wrong
  problem.(see figures in notes)
• To solve the problem, note that the complement of
  the constraint set is convex.c ? w(24-l) for l
  ? 16c ? 8w 1.5w(16-l) for l 16
• So consider the constraint set given by(c
  w(24-l))(c-8w-1.5w(16-l)) ? 0(see figure in
  notes)

72
Overtime 3

• Then, without harm we could rewrite the problem
  asmaxc,l calß s.t. c ?0, l ?0, 24-l ?0 -(c
  w(24-l))(c-8w-1.5w(16-l)) ? 0
• Note that this is not identical to the original
  problem (it omits the bottom left area), but we
  can clearly argue that the difference is
  harmless, since the omitted area is dominated by
  points in allowed area.
• Note that if x solves max f(x), it solves max
  g(f(x)) where g(.) is a monotonic transformation.
• So lets max log(calß) instead s.t. the same
  constraints.
• This gives the LagrangeanL alog(c)ßlog(l)µ(24
  -l)?(c-w(24-l))(c-(8w1.5w(16-l)))

73
Overtime 4

• We can use economic and mathematical logic to
  simplify the problem. First, note that since the
  derivative of the log function is infinity at c0
  or l0, this clearly cant be a solution, so µ0
  at any optimum and we can ignore CS conditions on
  c and l.
• So rewrite Lagrangean dropping µ termL
  alog(c)ßlog(l)?(c-w(24-l))(c-(8w1.5w(16-l)))
• Now lets look at the FOCs.

74
Overtime 5

• FOCscl?
• CS conditionnoting that the equalities occur
  in the FOCs because we argued that non-negativity
  constraints for c and l dont bind.

75
Overtime 6

• If l and c were such that the FOC for ? were
  strictly negative, we must have ?0 by CS, but
  this makes the first two FOCs impossible to
  satisfy.So (c-8w-1.5w(16-l))0 and/or
  (c-w(24-l))0In other words, we cant have an
  internal solution to the problem (which is good,
  since these are clearly dominated).
• Case 1 (c-w(24-l))0 (no overtime worked)From
  first two FOCs, we get awlßc, which with
  c24w-wl gives us c 24a/(aß)
• Case 2 (c-8w-1.5w(16-l))0 (overtime)From the
  first two FOCs, we get 3awl2ßc, which we can
  combine with c 8w1.5w(16-l)) to get an
  expression for c in terms of parameters.
• Actual solution depends on particular parameters
  of utility function (graphically could be
  either).

76
The cost minimization problem

• Cost minimization problem what is the cheapest
  way to produce at least y output from x inputs at
  input price vector w.
• C(y,w) -maxx wx s.t. f(x) ? y, y?0, x ?0.
• If f(x) is a concave function, then the set
  xf(x)?y is a convex set (since this is an
  upper contour set).
• To show that C(y,w) is convex in yConsider any
  two levels of output y, y and define
  ytty(1-t)y (ie convex combination).

77
Convexity of the cost function

• Let x be a solution to the cost minimization
  problem for y, xt for yt, x for y.
• Concavity of f(x) impliesf(tx(1-t)x)?tf(x)(1-
  t)f(x).
• Feasibility implies f(x) ? y, f(x) ?y.
• Together these implyf(tx(1-t)x)?tf(x)(1-t)f(x
  )?ty (1-t)ytt
• So the convex combination tx(1-t)x is feasible
  for yt.

78
Convexity of the cost fn 2

• By definitionC(y,w) wxC(y,w)
  wxC(yt,w) wxt
• But C(yt,w) wxt w(tx(1-t)x)twx(1-t)wx
  t C(y,w) (1-t)C(y,w)where the inequality
  comes since xt solves the problem for yt.
• So C(.) is convex in y.

79
Implicit functions(SB easier than lecture notes)

• So far we have been working only with functions
  in which the endogenous variables are explicit
  functions of the exogenous or independent
  variables.Ie y F(x1,x2,xn)
• This is not always the case frequently we have
  economic situations with exogenous variables
  mixed in with endogenous variables.G(x1,x2,xn,y)
  0
• If for each x vector this equation determines a
  corresponding value of y, then this equation
  defines an implicit function of the exogenous
  variables x.
• Sometimes we can solve the equation to write y as
  an explicit function of x, but sometimes this is
  not possible, or it is easier to work with the
  implicit function.

80
Implicit functions 2

• 4x 2y 5 expresses y as an implicit function
  of x. Here we can easily solve for the explicit
  function.
• y2-5xy4x20 expresses y implicitly in terms of
  x. Here we can also solve for the explicit
  relationship using the quadratic formula but it
  is a correspondence, not a function, y4x OR
  x.
• Y5-5xy4x20 cannot be solved into an explicit
  function, but still implicitly defines y in terms
  of x.Eg x0 implies y0. x1 implies y1.

81
Implicit functions 3

• Consider a profit-maximizing firm that uses a
  single input x at a cost of w per unit to make a
  single output y using technology yf(x), and
  sells the output for p per unit.Profit function
  p(x)pf(x)-wxFOC pf(x)-w0
• Think of p and w as exogenous variables. For
  each choice of p and w, the firm will choice a
  value of x that satisfies the FOC.To study
  profit-maximising behaviour in general, we need
  to work with this FOC defining x as an implicit
  function of p and w.
• In particular, we will want to know how the
  choice of x changes in response to changes in p
  and w.

82
Implicit functions 4

• An implicit function (or correspondence) of y in
  terms of x does not always exist, even if we can
  write an equation of the form G(x,y)cEg
  x2y21. When xgt1 there is no y that satisfies
  this equation. So there is no implicit function
  mapping xs greater than 1 into ys.
• We would like to have us general conditions
  telling us when an implicit function exists.

83
Implicit functions 5

• Consider the problemmaxx?0f(xq) s.t.
  G(xq)?0where q is some k dimensional vector of
  exogenous real numbers.
• Call a solution to this problem x(q), and the
  value the solution attains V(q) f(x(q)q).
• Note that x(q) may not be unique, but V(q) is
  still well-defined (ie there may be multiple xs
  that maximise the function, but they all give the
  same value (otherwise some wouldnt solve the
  maximisation problem))
• Interesting question how do V and x change with
  q?
• We have implicitly defined functions mapping qs
  to Vs.

84
Implicit functions 6

• The problem above really describes a family of
  optimization problems each different value of
  the q vector yields a different member of the
  family (ie a different optimization problem).
• The FOCs from KT suggest that it will be useful
  to be able to solve generally systems of
  equations where (why? Because the FOCs
  constitute such a system.)
• Eg Finding the equation for a level set, is to
  find z(q) such that T(z(q),q)-c0. Here, z(q) is
  an implicit function
• As noted previously, not all systems provide
  implicit functions. Some give correspondences,
  or give situations where there is mapping x(q).
• The implicit function theorem tells us when it is
  possible to find an implicit function from a
  system of equations.

85
Implicit function theorem(for system of
equations)

• Let TRkp?Rk be C1. Suppose that T(z,q)0.
  If the kxk matrix formed by stacking the k
  gradient vectors (wrt z) of T1,T2,Tk is
  invertible (or equivalently has full rank or is
  non-singular), then there exist k C1 functions
  each mapping Rp ?Rk such thatz1(q)z1,
  z2(q)z2, . zk(q)zk andT(z(q),q) 0 for
  all q in Be(q) for some egt0.

86
IFT example

• Consider the utility maximisation problemmaxx
  in Rn U(x) s.t. pxI, U strictly quasi-concave,
  DU(x)gt0, dU(0)/dxi?.
• We know a solution to this problem satisfies xi gt
  0 (because of dU(0)/dxi?) and I-px0 (because
  DU(x)gt0) and the FOCsdU/dx1-?p10dU/dxn-?pn0
  I-px0

87
IFT example contd

• This system of equations maps from the space
  R2n2 (because x and p are nx1, ? and I are
  scalars) to the space Rn1 (the number of
  equations).
• To apply the IFT, set z (x, ?), q(p,I)Create
  a function T R2n2? Rn1 given by

88
IFT example contd

• If this function T is C1 and if the n1xn1
  matrix of derivatives of T (wrt x and ?) is
  invertible, then by the IFT we know that there
  exist n1 C1 functionsx1(p,I), x2(p,I), .
  xn(p,I), ?(p,I)s.t. T(x1(p,I), x2(p,I), .
  xn(p,I), ?(p,I)) 0 for all p,I in a
  neighborhood of a given price income vector
  (p,I).
• Ie, the IFT gives us the existence of
  continuously differentiable consumer demand
  functions.

89
Theorem of the maximum

• Consider the family of lagrangian problemsV(q)
  maxxf(xq) s.t. G(xq)0This can be
  generalized to KT by restricting attention only
  to constraints that are binding at a given
  solution.
• Define the function TRnmp?Rnm by

90
Theorem of the Maximum 2

• The FOCs for this problem at an optimum are
  represented by T(x,?q)0. We want to know
  about defining the solutions to the problem, x
  and ?, as functions of q.
• The IFT already tells when we can do this if the
  (nm)x(nm) matrix constructed by taking the
  derivative of T wrt x and ? is invertible, then
  we can find C1 functions x(q) and ?(q) s.t.
  T(x(q), ?(q)q)0 for q in a neighborhood
  of q.
• Ie we need the matrix below to have full rank

91
Theorem of the Maximum 3

• Suppose the Lagrange problem above satisfies the
  conditions of the implicit function theorem at
  x(q),q. If f is C1 at x(q),q, then V(q) is
  C1 at q.
• Thus, small changes in q around q will have
  small changes in V(q) around V(q).

92
Envelope Theorem

• Applying the IFT to our FOCs means we know (under
  conditions) that x(q) that solves our FOCs exists
  and is C1, and that V(.) is C1.
• The envelope theorem tells us how V(q) changes in
  response to changes in q.
• The basic answer from the ET is that all we need
  to do is look at the direct partial derivative of
  the objective function (or of the Lagrangian for
  constrained problems) with respect to q.
• We do not need to reoptimise and pick out
  different x(q) and ?(q), because the fact that we
  were at an optimum means these partial derivs are
  already zero.

93
Envelope theorem 2

• Consider the problemmaxxf(xq) s.t.
  G(xq)0, GRn?Rm.where q is a p-dimensional
  vector of exogenous variables.Assume that, at a
  solution, the FOCs hold with equality and that we
  can ignore the CS conditions.(Or assume that we
  only include constraints that bind at the
  solution in G() )
• Suppose that the problem is well behaved, so we
  have that at a particular value q, the solution
  x(q), ?(q) are C1 and V(q)f(x(q)q) is
  C1.(Note that we could get these from the IFT
  and the Theorem of the Maximum)

94
Envelope theorem 3

• Suppose the problem above satisfies the
  conditions of the IFT at x(q). If f is C1 at
  x(q),q thenie the derivative of the
  value function V(q) is equal to the derivative of
  the Lagrangean

95
Envelope theorem 4

• So, to determine how the value function changes,
  we merely need to look at how the objective
  function and constraint functions change with q
  directly.
• We do not need to include the impact of changes
  in the optimization variables x and ?, because we
  have already optimized L(x,?,q) with respect to
  these.
• So, for an unconstrained optimization problem,
  the effect on V(.) is just the derivative of the
  objective function.
• For a constrained optimization problem, we also
  need to add in the effect on the constraint.
  Changing q could effect the constraint (relaxing
  or tightening it), which we know has shadow value
  ?.
• Proof is in lecture notes.

96
Envelope theorem example

• Consider a problem for the formmaxxf(x) s.t.
  q-g(x) ?0
• Thus, as q gets bigger, the constraint is easier
  to satisfy. What would we gain from a small
  increase in q, and thus a slight relaxation of
  the constraint?
• The Lagrangian is L(x,?q) f(x) ?(q-g(x))
• The partial deriv of the Lagrangian wrt q is ?.
  Thus, dV(q)/dq ?.
• A small increase in q increases the value by
  ?.Thus, the lagrange multiplier is the shadow
  price. It describes the price of relaxing the
  constraint.
• If the constraint does not bind, ?0 and dV(q)/dq
  0.

97
Envelope theorem example 2

• We can use the envelope theorem to show that in
  the consumer max problem, ? is the marginal
  utility of income.
• Consider the cost min problemC(y,w) maxx
  -wx s.t. f(x)-y?0.
• Lagrangian is L(x,?y,w) -wx
  ?(f(x)-y)Denote the optimal solution to be
  x(y,w).
• From the ET, we get

98
ET example 2, contd

• This is known as Shephards lemma the partial
  derivative of the cost function with respect to
  wi is just xi, the demand for factor i.
• Also note thatie the change in demand for
  factor i with respect to a small change in price
  of factor j is equal to the change in demand for
  factor j in response to a small change in the
  price of factor i.

99
Correspondences

• A correspondence is a transformation that maps a
  vector space into collections of subsets in
  another vector space.
• Eg a correspondence FRn??R takes any n
  dimensional vector and gives as its output a
  subset of R. If this subset has a only one
  element for every input vector, then the
  correspondence is also a function.
• Examples of correspondences solution to the cost
  minimization problem, or the utility maximization
  problem.

100
Correspondences 2

• A correspondence F is bounded if for all x and
  for all y in F(x), the size of y is bounded.
  That is, yM for some finite M. For bounded
  correspondences we have the following
  definitions.
• A correspondence F is convex-valued if for all x,
  F(x) is a convex set. (All functions are
  convex-valued correspondences).
• A correspondence F is upper hemi-continuous at a
  point x if for all sequences xn that converge
  to x, and all sequences yn such that yn in
  F(xn) converge to y, then y is in F(x).
• For bounded correspondences, if a correspondence
  is uhc for all x, then its graph is a closed set.

101
Correspondences 3

• A correspondence F is lower hemi-continuous at a
  point x, if for all sequences xn that converge
  to x and for all y in F(x), there exists a
  sequence yn s.t. yn is in F(xn) and the
  sequence converges to y.
• See figure in notes.

102
Fixed point theorems

• A fixed point of a function fRn?Rn is a point x,
  such that xf(x). A fixed point of a
  correspondence FRn??Rn is a point x such that x
  is an element of F(x).
• Solving a set of equations can be described as
  finding a fixed point. (Suppose you are finding
  x to solve f(x) 0. Then you are looking for a
  fixed point in the function g(x), where g(x) x
  f(x), since for a fixed point x in g, x
  g(x) x f(x), so f(x) 0.)
• Fixed points are crucial in proofs of existence
  of equilibriums in GE and in games.

103
Fixed point theorems

• If fR?R, then a fixed point of f is any point
  where the graph of f crosses the 45 degree line
  (ie the line f(x)x).
• A function can have many fixed points, a unique
  fixed point, or none at all.
• When can we be sure that a function possesses a
  fixed point? We use fixed point theorems.

104
Brouwer fixed point theorem

• Suppose fRn?Rn and for some convex, compact set
  C (that is a subset of Rn) f maps C into itself.
  (ie if x is in C, then f(x) is in C). If f is
  continuous, then f possesses a fixed point.
• Continuity
• Convexity of C
• Compactness of C
• C maps into itself.

105
Kakutani fixed point theorem

• Suppose FRn??Rn is a convex-valued
  correspondence, and for some convex compact set C
  in Rn, F maps C into itself. (ie if x is in C,
  then F(x) is a subset of C). If F is upper
  hemicontinuous, then F possesses a fixed point.
• These FPTs give existence. To get uniqueness we
  need something else.

106
Contraction mappings

• Suppose fRn?Rn such that f(x)f(y) ?
  x-y for some ? lt 1 and for all x,y. Then f
  ix a contraction mapping.
• Let Ca,b be the set of all continuous functions
  f0,1?R with the supnorm metric f) maxx
  in a,b f(x). Suppose TC?C (that is, T takes a
  continuous function, does something to it and
  returns a new, possibly different continuous
  function). If, for all f,g in C, Tf-Tg ?
  f-g for some ? lt 1, then T is a contraction
  mapping.

107
Contraction mapping theorem

• If f or T (as defined above) is a contraction
  mapping, it possesses a unique fixed point, x.

108
Dynamic optimisation

• Up to now, we have looked at static optimisation
  problems, where agents select variables to
  maximise a single objective function.
• Many economic models, particularly in
  macroeconomics (eg saving and investment
  behaviour), use dynamic models, where agents make
  choices each period that affect their potential
  choices in future periods, and often have a
  total objective function that maximises the
  (discounted) sum of payoffs in each period.
• Much of the material in the notes is focused on
  differential and difference equations (lectures
  1-4), but we will attempt to spend more time on
  lectures 5-6, which are the focus of most dynamic
  models.

109
Ordinary differential equations

• Differential equations are used to model
  situations which treat time as a continuous
  variable (as opposed to in discrete periods,
  where we use difference equations).
• An ordinary differential equation is an
  expression which describes a relationship between
  a function of one variable and its derivatives.
• Formallywhere is a vector of
  parametersF if a function Rm1p?R

110
Ordinary differential equations 2

• The solution is a function x(t) that, together
  with its derivatives, satisfies this equation.
• This is an ordinary differential equation because
  x is a function of one argument, t, only. If it
  was a function of more than one variable, we
  would have a partial differential equation, which
  we will not study here.
• A differential equation is linear if F is linear
  in x(t) and its derivatives.
• A differential equation is autonomous if t does
  not appear as an independent argument of F, but
  enters through x only.
• The order of a differential equation is the order
  of the highest derivative of x that appears in it
  (ie order m above).

111
First order differential equation

• Any differential equation can be reduced to a
  first-order differential equation system by
  introducing additional variables.
• Consider x3(t) ax2(t) bx1(t) x(t)Define
  y(t) x1(t), z(t) x2(t)
• Then y1(t) x2(t) z(t), z1(t)x3(t).
• So we have the system

112
Particular and general solutions

• A particular solution to a differential equation
  is a differentiable function x(t) that satisfies
  the equation for some subinterval I0 of the
  domain of definition of t, I.
• The set of all solutions is called the general
  solution, xg(t).
• To see that the solution to a differential
  equation is generally not unique, considerx1(t)
  2x(t).One solution is x(t) e2t. But for any
  constant c, x(t) ce2t is also a solution.
• The non-uniqueness problem can be overcome by
  augmenting the differential equation with a
  boundary condition x(t0) x0.

113
Boundary value problems

• A boundary value problem is defined by a
  differential equationx1(t) ft,x(t)and a
  boundary conditionx(t0) x0, (x0,t0) is an
  element of X x I
• Under some conditions, every boundary value
  problem has a unique solution.
• Fundamental Existence Uniqueness theoremLet F
  be C1 in some neighborhood of (x0,t0). Then in
  some subinterval I0 of I containing t0 there is a
  unique solution to the boundary value problem.

114
Boundary values problems 2

• If F is not C1 in some neighborhood of (x0,t0),
  the solution may not be unique. Consider x1(t)
  3x(t)2/3 x,t in R x(0)0Both x(t) t3 and
  x(t) 0 are solutions.Note f(x) 3x(t)2/3 is
  not differentiable at x0.
• The solution may not exist globally.Consider x1
  (t) x(t)2 x,t in R x(0) 1x(t) 1/(1-t)
  is a solution, but is only defined for t in -?,1)

115
Steady states and stability

• When using continuous time dynamic models, we are
  often interested in the long-run properties of
  the differential equation.
• In particular, we are interested in the
  properties of its steady state (our equilibrium
  concept for dynamic systems, where the system
  remains unchanged from period to period), and
  whether or not the solution eventually converges
  to the steady state (ie is the equilibrium
  stable, will we return there after shocks).
• We can analyze the steady state without having to
  find an explicit solution for the differential
  equation.

116
Steady states and stability 2

• Consider the autonomous differential
  equation x1(t) fx(t)
• A steady state is a point such that
• Phase diagrams to illustrate this.
• Steady states may not exist, may not be unique,
  may not be isolated.
• Stability consider an equation that is initially
  at rest at an equilibrium point , and suppose
  that some shock causes a deviation from .We
  want to know if the equation will return to the
  steady state (or at least remain close to it), or
  if it will get farther and farther away over time.

117
Steady states and stability 3

• Let be an isolated (ie locally unique) steady
  state of the autonomous differential
  equation x1(t)fx(t),
• We say that is stable if for any e gt 0, there
  exists d in (0,e such thatie any solution
  x(t) that at some point enters a ball of radius d
  around remains within a ball of (possibly
  larger) radius e forever after.

118
Steady states and stability 4

• A steady state is asymptotically stable if it is
  stable AND d can be chosen in such a way that any
  solution that satisfiesfor some t0 will also
  satisfy
• That is, any solution that gets sufficiently
  close to not only remains nearby but converges
  to as t ??.

119
Phase diagrams arrows of motion

• The sign of x1(t) tells us about the direction
  that x(t) is moving (see diagram).
• x1(t) gt 0 implies that x(t) is increasing (arrows
  of motion point right).
• x1(t) lt 0 implies that x(t) is decreasing (arrows
  of motion point left).
• Thus x1 and x3 in diagram are locally
  asymptotically stable x2 is unstable.
• x1 in the second diagram (see notes) is globally
  asymptotically stable.

120
Phase diagrams arrows of motion 2

• We can conclude that if for all x in some
  neighborhood of a steady state
• x(t) lt implies x1(t) gt 0 AND x(t) gt implies
  that x1(t)lt0, then is asymptotically stable.
• x(t)lt implies x1(t) lt 0 and x(t) gt implies
  x1(t)gt0 then is unstable.
• Therefore, we can determine the stability
  property of a steady state by checking the sign
  of the derivative of fx(t) at .
• is (locally) asymptotically stable if
• is unstable if
• If , then we dont know.

121
Grobman-Hartman theorem

• Let be a steady state of out standard
  autonomous differential equation
• We say that is a hyperbolic equilibrium if
• The previous analysis suggests we can study the
  stability properties of a nonlinear differential
  equation by linearizing it, as long as the
  equilibrium is hyperbolic.
• Theorem If is a hyperbolic equilibrium of the
  autonomous differential equation above, then
  there is a neighborhood U of such that the
  equation is topologically equivalent to the
  linear equationin U.(Note that this is a
  first-order Taylor series approximation of f
  around ). (See notes