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 R n : Suppose that x ∗ is a critical point of a function f ( x ) with continuous first and second order partial derivatives on R n . Then x ∗ is: A global maximizer (minimizer) for f ( x ) if D 2 f ( x ) is negative (positive) semi-definite on R n . A strict global maximizer (minimizer) for f ( x ) if D 2 f ( x ) is negative (positive) definite on R n . 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 R n is concave, if for all x , y in U and for all t ∈ [0 , 1] : f ( t x + (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 f 1 , ..., f k be concave functions, each defined on the same convex subset U of R n . Let a 1 , a 2 , ..., a k be positive numbers. Then a 1 f 1 + a 2 f 2 + ... + a k f k is a concave function on U . Proof of Theorem Consider x ∈ U, y ∈ U, and t ∈ [0 , 1] . Let k = 2 , then: a 1 f 1 ( t x + (1 − t ) y ) + a 2 f 2 ( t x + (1 − t ) y ) ≥ a 1 [ t f 1 ( x ) + (1 − t ) f 1 ( y )] + a 2 [ t f 2 ( x ) + (1 − t ) f 2 ( y )] ≥ t [ a 1 f 1 ( x ) + a 2 f 2 ( x )] + (1 − t ) [ a 1 f 1 ( y ) + a 2 f 2 ( 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 w i the per-unit cost of input i , then the firm’s profit function is Π( x ) = p g ( x ) − ( w 1 x 1 + w 2 x 2 + ... + w n x n ) 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 − ( w 1 x 1 + w 2 x 2 + ... + w n x n ) and the concavity of g plus the result above. The first order conditions are p ∂ g = w i for i = 1 , 2 , ..., n . ∂ x i 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 R n 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 R n 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 ) k ✲ x b a 0 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 ✻ F ( x , y ) = k ′ , k ′ > k F ( x , y ) = k ✲ x 0 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 R n 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 ) k ✲ x b a 0 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 ( t x + (1 − t ) y ) ≥ t f ( x )+(1 − t ) f ( y ) ≥ t k +(1 − t ) k = k By definition of C + k therefore t x + (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 y ✻ C x α x + (1 − α ) z z r ✲ 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 y ✻ C ✲ ❄ ✲ ✲ ❄ ✲ ❄ ❄ ❄ ❄ ✲ ✲ ✲ x Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 17 / 23

Properties: Increasing Function in R 1 Let y = f ( x ) be an increasing function on R 1 . The function is both quasi-concave and quasi-convex. The same applies to a decreasing function. ✻ ψ ( x ) k ✲ x a 0 Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 3 13 September 2010 18 / 23

Not true for Increasing Function in R n , n > 1 y ✻ F ( x , y ) = k ′ , k ′ > k F ( x , y ) = k ✲ x 0 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 ) ✻ k ✲ x a b 0 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 ✻ Q ( x , y ) = k ′ , k ′ > k Q ( x , y ) = k ✲ x 0 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 R n . 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 ( t x + (1 − t ) y ) ≥ f ( y ) (iii) For all x , y ∈ U and t ∈ [0 , 1] , f ( t x + (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