EC400 Part II, Math for Micro: Lecture 3 Leonardo Felli NAB.SZT 13 - - PowerPoint PPT Presentation

EC400 Part II, Math for Micro: Lecture 3

Leonardo Felli

NAB.SZT

13 September 2010

Sufficient Conditions for Global Optimum

Second order sufficient conditions for global maximum (minimum) in Rn : Suppose that x∗ is a critical point of a function f (x) with continuous first and second order partial derivatives on Rn. Then x∗ is: A global maximizer (minimizer) for f (x) if D2f (x) is negative (positive) semi-definite on Rn. A strict global maximizer (minimizer) for f (x) if D2f (x) is negative (positive) definite on Rn.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 2 / 23

Why are concave functions so useful in economics?

The property that critical points of concave functions are global maximizers is an important one in economic theory. Recall:

Definition

A real valued function f defined on a convex subset U of Rn is concave, if for all x, y in U and for all t ∈ [0, 1] : f (tx + (1 − t)y) ≥ t f (x) + (1 − t) f (y)

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 3 / 23

Theorem

Let f1, ..., fk be concave functions, each defined on the same convex subset U of Rn. Let a1, a2, ..., ak be positive numbers. Then a1f1 + a2f2 + ... + akfk is a concave function on U.

Proof of Theorem

Consider x ∈ U, y ∈ U, and t ∈ [0, 1]. Let k = 2, then: a1f1(t x + (1 − t) y) + a2f2(t x + (1 − t) y) ≥ a1[t f1(x) + (1 − t) f1(y)] + a2[t f2(x) + (1 − t) f2(y)] ≥ t[a1 f1(x) + a2 f2(x)] + (1 − t) [a1 f1(y) + a2 f2(y)]

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 4 / 23

Application

Consider a firm’s profit maximization. The firm’s production function is y = g(x), where y denotes output and x denote the input bundle. Let p denote the price of output and wi the per-unit cost of input i, then the firm’s profit function is Π(x) = p g(x) − (w1x1 + w2x2 + ... + wnxn) The result above implies that the profit function is concave if the production function is concave.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 5 / 23

It follows from the concavity of −(w1x1 + w2x2 + ... + wnxn) and the concavity of g plus the result above. The first order conditions are p ∂g ∂xi = wi for i = 1, 2, ..., n. Concavity of the firm’s profit function implies that they are both necessary and sufficient for an interior profit maximizer.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 6 / 23

Preliminaries

Definition

The level k set of function f defined on U in Rn is: X f

k = {x ∈ U|f (x) = k}

This could be a point, a curve, a plane or an hyperplane.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 7 / 23

Quasi-Concavity

Definition

A function f defined on a convex subset U of Rn is quasi-concave if for every real number k, C +

k = {x ∈ U|f (x) ≥ k}

is a convex set. In other words, for any level k the set of points in the function domain that map to values of the function greater of equal than k is a convex set. Alternatively, the level sets of the function bound convex subsets from below.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 8 / 23

Example: Quasi-Concave Function f (x)

✻ ✲

f (x) x k a b

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 9 / 23

Example: Quasi-Concave Function F(x, y) = xy

✻ ✲

y x F(x, y) = k F(x, y) = k′, k′ > k

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 10 / 23

Quasi-Convexity

Definition

A function f defined on a convex subset U of Rn is quasi-convex if for every real number h, C −

h = {x ∈ U|f (x) ≤ h}

is a convex set. In other words, for any level h the set of points in the function domain that map to values of the function smaller or equal than h is a convex set. Alternatively, the level sets of the function bound convex subsets from above.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 11 / 23

Example: Quasi-Convex Function g(x)

✻ ✲

g(x) x k a b

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 12 / 23

Properties

Theorem

Every concave function is quasi-concave and every convex function is quasi-convex.

Proof of Theorem

First statement: Let x and y be two points in C +

k so that f (x) ≥ k and

f (y) ≥ k. Then f (tx + (1 − t)y) ≥ t f (x)+(1 − t) f (y) ≥ t k+(1 − t) k = k By definition of C +

k therefore tx + (1 − t)y ∈ C + k and hence this set is

convex.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 13 / 23

This is the second advantage of concave functions in economics: concave functions are quasi-concave. Quasi-concavity is simply a desirable property when we talk about economic objective functions such as preferences. The property that the set above any level set of a concave function is a convex set is a natural requirement for utility and production functions. Consider an indifference curve C of the concave utility function U. Implied quasi-concavity means that for every two bundles on C the set of bundles which are preferred to them, is a convex set. Then, given any two bundles, a consumer with a concave utility function will always prefer a mixture of the bundles to any of them.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 14 / 23

Indifference Curve

✻ ✲

x z

r

C α x + (1 − α) z y x

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 15 / 23

A more important advantage is the shape of the indifference curve is that it displays a diminishing marginal rate of substitution. As one moves left to right along the indifference curve C increasing consumption of good x, the consumer is willing to give up more and more units of good x to gain an additional unit of good y.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 16 / 23

Diminishing Marginal Rate of Sunstitution

✻ ✲

C y x

✲ ❄ ✲ ❄ ✲ ❄ ✲ ❄ ✲ ✲ ❄ ❄

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 17 / 23

Properties: Increasing Function in R1

Let y = f (x) be an increasing function on R1. The function is both quasi-concave and quasi-convex. The same applies to a decreasing function.

✻ ✲

ψ(x) x k a

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 18 / 23

Not true for Increasing Function in Rn, n > 1

✻ ✲

y x F(x, y) = k F(x, y) = k′, k′ > k

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 19 / 23

Other Properties

Any monotonic transformation of a concave function is quasi-concave. A single peaked function is quasi-concave (bell-shaped function—i.e. normal distribution).

✻ ✲

φ(x) x k a b

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 20 / 23

One More Example

Consider the following utility function Q(x, y) = min{x, y}. The typical indifference curve is L shaped—Leontief utility function. The region above and to the right of any of this function’s level sets is a convex set and hence Q is quasi-concave.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 21 / 23

Example: Leontief Q(x, y) = min{x, y}

✻ ✲

y x Q(x, y) = k Q(x, y) = k′, k′ > k

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 22 / 23

Theorem

Let f be a function defined on a convex set U in Rn. Then, the following statements are equivalent: (i) f is a quasiconcave function on U (ii) For all x, y ∈ U and t ∈ [0, 1], f (x) ≥ f (y) implies f (tx + (1 − t)y) ≥ f (y) (iii) For all x, y ∈ U and t ∈ [0, 1], f (tx + (1 − t)y) ≥ min{f (x), f (y)} You will prove this in class.

Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 23 / 23