2. FUNCTIONS

Function

Given two sets A and B, a function defined on A and with values in B, denoted by f : A→B, is a rule that associates to each element of A one and only one element of B

Cobb-Douglas function

The Cobb-Douglas function f : ℝⁿ₊→ℝ is defined by f (x₁, x₂,…,x_n) = ∏ⁿ_i=1 x_i^α_i , with α_i > 0 for every i= 1,2…n, such that Σⁿ_i=1α_i= 1

Graph of a function

The graph of a function f : A → B, denoted by Gr f, is a subset of the Cartesian product A × B, whose elements are the ordered pairs (x, f(x)).

Gr f= {(x, f(x)) : x ∈ A} ⊆ A x B

Utility function

Interpreting the vectors in ℝⁿ₊ as bundles of goods, denoting preference with the symbol ≿

The utility function u : ℝⁿ₊→ℝ is used to represent the preference over the available bundles of goods, such that x ≿ y ←→ u(x) ≥ u(y)

Production function

The production function f : A ⊆ ℝⁿ₊→ℝ describes how much output f(x) is obtained by starting from a vector x ∈ ℝⁿ ₊ of input

Preimage of a point

Let f : A → B. The preimage of a point y ∈ B is the set of all points x ∈ A such that f(x) = y

f^-1(y) = {x ∈ A : f(x) = y}

Level set of a function

Let f : A ⊆ ℝⁿ→ ℝ. The level set of level k ∈ ℝ of f is the preimage f^-1(k) = {x ∈ A : f(x) = k}

Indifference curves

The indifference sets are the level sets of a utility function. If A ⊆ ℝ², the indifference sets are called indifference curves.

Given u : A ⊆ ℝⁿ₊→ℝ, indifference curve u^-1(k)={x ∈ A : u(x) = k} is formed by all the bundles x ∈ A that have utility k

Sum function in ℝ^A

Let f, g ∈ ℝ^A. The sum function f + g is constructed by adding the images of each x ∈ A under f and g

f + g is the element of ℝ^A such that (f+g)(x) = f(x) + g(x) for every x ∈ A

Product function in ℝ^A

Let f, g ∈ ℝ^A. The product function, denoted by (fg), is constructed by multiplying the images of each x ∈ A under f and g

fg is the element of ℝ^A such that (fg)(x) = f(x) g(x) for every x ∈ A

Composite function

Given four sets A, B, C and D, let f : A → B and g : C→ D be two functions.

If Im f ⊆ C, the composite function g◦f : A → D is defined by (g◦f)(x) = g(f(x)) for every x ∈ A.

Injective function

Given two sets A and B, a function f : A → B is injective if each element of A is mapped to a different element of B: x ≠ y → f(x) ≠ f(y) for every x, y ∈ A

Surjective function

Given two sets A and B, a function f : A → B is surjective if, for each element of B, there is at least one element of A such that f(x) = y: Im f = B

Bijective function

Given two sets A and B, a function f : A → B is bijective if it is both surjective and injective

Inverse function

Let f : A → B be an injective function.

The inverse function f^-1 : Im f → A of f is defined by f^-1 (y) = x, if and only if f(x) = y

Bounded functions in ℝ^A

Let f : A → ℝ. The function is:

Bounded above if Im f is a set bounded above: there exists a scalar M ∈ ℝ such that M ≥ f(x) for every x ∈ A

Bounded below if Im f is a set bounded below: there exists a scalar M ∈ ℝ such that M ≤ f(x) for every x ∈ A

Bounded if it is bounded above and below

Characterization of bounded functions in ℝ^A

A function f : A → ℝ is bounded if and only if there exists a scalar k>0 such that |f(x)| ≤ k, for every x ∈ A

Increasing & decreasing real-valued functions of one real variable

Let f :A ⊆ ℝ→ℝ. The function is:

Increasing if x > y → f(x) ≥ f(y) for every x, y ∈ A
Strictly increasing if x > y → f(x) > f(y) for every x, y A

Decreasing if x > y → f(x) ≤ f(y) for every x, y ∈ A
Strictly decreasing x > y → f(x) < f(y) for every x, y ∈ A

Constant if there exists k ∈ ℝ such that f(x) = k for every x ∈ A

Monotonic if it is either increasing or decreasing

A monotone function is invertible iff it is strictly monotone

An increasing function f : A ⊆ ℝ→ℝ is strictly increasing if and only if f(x) = f (y) → x = y for every x, y ∈ A; i.e., if and only if it is injective (and therefore invertible)

ONLY IF: Let f be strictly increasing, and let f(x) = f(y). By contradiction, suppose x ≠ y - wlog x > y. Since f is strictly increasing, we have f(x) ≠ f(y), which contradicts f(x) = f(y). So it follows that x = y. Therefore, f is injective

IF: Let f(x) = f(y) → x = y for every x, y ∈ A (i), and let f be increasing.
Let x > y. Since f is increasing, we have f(x) ≥ f(y). But f(x) ≠ f(y) (i), so it must be that f(x) > f(y). Therefore, f is strictly increasing.

Characterization of strict monotonicity as an order isomorphism

A function A ⊆ ℝ→ℝ is strictly increasing if and only if x ≥ y ←→ f(x) ≥ f(y) for every x, y ∈ A

A function A ⊆ ℝ→ℝ is strictly decreasing if and only if x ≥ y ←→ f(x) ≤ f(y) for every x, y ∈ A

Concave & convex functions of one real variable

Let I ⊆ ℝ be an interval.

A function f : I → ℝ is concave if the graph of the function always lies on or above the line segment connecting any two points in the graph: f(αx + (1-α)y) ≥ αf(x) + (1-α)f(y) for every x, y ∈ I and every α ∈ [0,1]

A function f : I → ℝ is convex if the graph of the function always lies on or below the line segment connecting any two points in the graph: f(αx + (1-α)y) ≤ αf(x) + (1-α)f(y) for every x, y ∈ I and every α ∈ [0,1]

Global maximizer of one real variable

Let f : A ⊆ ℝ→ ℝ. An element x̂ ∈ A is called a global maximizer of f on A if f(x̂) ≥ f(x) for every x ∈ A

Characterization of order-equivalent functions of n real variables

Let f,g : A ⊆ ℝⁿ→ ℝ. We have that f(x) ≥ f(y) ←→ g(x) ≥ g(y) for every x, y ∈ A if and only if there exists a function h : Im f → ℝ that is strictly increasing such that g = h ◦ f

Concave & convex functions of n real variables

Let C ⊆ ℝⁿ be a convex set.

A function f : C → ℝ is concave if the graph of the function always lies on or above the line segment connecting any two points in the graph: f(αx + (1-α)y) ≥ αf(x) + (1-α)f(y) for every x, y ∈ C and every α ∈ [0,1]

A function f : C → ℝ is convex if the graph of the function always lies on or below the line segment connecting any two points in the graph: f(αx + (1-α)y) ≤ αf(x) + (1-α)f(y) for every x, y ∈ C and every α ∈ [0,1]

Affine functions

Let C ⊆ ℝⁿ be a convex set.

A function f : C → ℝ is affine if it is both concave and convex: f(αx + (1-α)y) = αf(x) + (1-α)f(y) for every x, y ∈ C and every α ∈ [0,1]

Upper & lower contours of a function

Let f : C → ℝ. For every k ∈ ℝ:

The upper contours of f, denoted by (f ≥ k), are the sets {x ∈ C : f(x) ≥ k}

The lower contours of f, denoted by (f ≤ k), are the sets {x ∈ C : f(x) ≤ k}}

Concave/convex functions have convex upper/lower contour sets

If f : C → ℝ is concave, all its upper contour sets (f ≥ k) are convex

Given k ∈ ℝ, let (f ≥ k) be a non-empty set. Let x, y ∈ (f ≥ k) and α ∈ [0,1].

Since f is concave, f(αx + (1-α)y) ≥ αf(x) + (1-α)f(y) ≥ αk + (1-α)k = k. Therefore, αf(x) + (1-α)f(y) ∈ (f ≥ k), and (f ≥ k) is convex.

Preservation of concavity & convexity with respect to addition

The sum of two concave functions is concave

The sum of two convex functions is convex

Preservation of concavity & convexity with respect to scalar multiplication

Let f : C → ℝ be concave. The function αf is concave for every α ≥ 0

Let f : C → ℝ be convex. The function αf is convex for every α ≥ 0

Quasi-concave & quasi-convex functions

Let C ⊆ ℝⁿ be a convex set.

A function f : C → ℝ is quasi-concave if, between any two points in the graph of the function, the graph always lies on or above the height of the lower of the two points: f(αx + (1-α)y) ≥ min{f(x), f(y)}, for every x, y ∈ C and every α ∈ [0,1]

A function f : C → ℝ is quasi-convex if, between any two points in the graph of the function, the graph always lies on or below the height of the higher of the two points: f(αx + (1-α)y) ≤ max{f(x), f(y)}, for every x, y ∈ C and every α ∈ [0,1]

A function f : C → ℝ is quasi-affine if it is both quasi-concave and quasi-convex

Concave/convex functions are quasi-concave/quasi-convex

A concave function is quasi-concave: f(αx + (1-α)y) ≥ αf(x) + (1-α)f(y) → f(αx + (1-α)y) ≥ min{f(x), f(y)}, for every x, y ∈ C and every α ∈ [0,1]

Let C ⊆ ℝⁿ be a convex set, let f : C → ℝ be a concave function. Since f(x) ≥ min{f(x), f(y)}, and f(y) ≥ min{f(x), f(y)}, their weighted average αf(x) + (1-α)f(y) ≥ min{f(x), f(y)}.

Since f(αx + (1-α)y) ≥ αf(x) + (1-α)f(y), by (i) f(αx + (1-α)y) ≥ min{f(x), f(y)}. So f is quasi-concave.

Affinity of scalar functions

Monotone scalar functions on an interval are quasi-affine
[Let I ⊆ ℝ be an interval. If f : I → ℝ is monotone, f is both quasi-concave and quasi-convex]

Characterization of quasi-concavity & quasi-convexity

Let C ⊆ ℝⁿ be a convex set.

A function f : C → ℝ is quasi-concave if and only if all its upper contour sets are convex.

A function f : C → ℝ is quasi-convex if and only if all its lower contour sets are convex.

Preservation of quasi-concavity & quasi-convexity with respect to addition

The sum of two quasi-concave functions is not necessarily quasi-concave

The sum of two quasi-convex functions is not necessarily quasi-convex

Preservation of concavity/convexity & quasi-concavity/quasi-convexity with respect to increasing composition

Let C ⊆ ℝⁿ and A ⊆ ℝ be convex sets, let f : C → ℝ and g : A → ℝ, with Im f ⊆ A

If f is quasi-concave and g is increasing, g◦f : C → ℝ is quasi-concave
If f is concave and g is both concave and increasing, g◦f : C → ℝ is concave

If f is quasi-convex and g is increasing, g◦f : C → ℝ is quasi-convex
If f is convex and g is both convex and increasing, g◦f : C → ℝ is convex

Global maximizer of n real variables

Given a function f : A ⊆ ℝⁿ → ℝ, let C ⊆ A.

A point x̂ ∈ C is a global maximizer of f on C if f(x̂) ≥ f(x) for every x ∈ C

Argmax

Given a function f : A ⊆ ℝⁿ → ℝ, let C ⊆ A.

The solution set for a maximization problem max_x∈Cf(x), denoted by arg max_x∈Cf(x), is the set of global maximizers of f on C: arg max_x∈Cf(x)= {x̂ ∈ C : f(x̂) = max_x∈Cf(x)}

Strong maximizer of n real variables

Given a function f : A ⊆ ℝⁿ → ℝ, let C ⊆ A.

A point x̂ ∈ C is the strong global maximizer of f on C if f(x̂) > f(x) for every x ∈ C different from x̂

Characterization of the strong maximizer

A maximizer is strong if and only if it is unique

Preservation of maximizers & minimizers with respect to strictly increasing composition

Given a function f : A ⊆ ℝⁿ → ℝ, let C ⊆ A. Let g : B ⊆ ℝ→ ℝ be strictly increasing, with Im f ⊆ B. The optimization problems max_x∈Cf(x) and max_x∈C(g◦f)(x) have the same solution set: arg max_x∈Cf(x) = arg max_x∈C(g◦f)(x)

Since g is strictly increasing, a ≥b ←→ g(a) ≥ g (b) for every a, b ∈ Im f. Substituting a = f(x) and b = f(y), we obtain f(x) ≥ f(y) ←→ g(f(x)) ≥ g(f(y)) for every x, y ∈ C.

Therefore f(x̂) ≥ f(x) for every x ∈ C ←→ (g◦f)(x̂) ≥ (g◦f)(x) for every x ∈ C. So x̂ ∈ C solves max_xf(x) if and only if it solves max_x(g◦f)(x)

Convexity of the solution set of a concave or convex maximization problem

The set of maximizers of a concave function on a convex set is convex.
[Let C ⊆ ℝⁿ be a convex set. If f : C → ℝ is concave, then argmax_x∈Cf(x) is convex.]

The set of minimizers of a convex function on a convex set is convex.
[Let C ⊆ ℝⁿ be a convex set. If f : C → ℝ is convex, then argmin_x∈Cf(x) is convex.]

Uniqueness of the global optimizers under strict concavity & strict convexity

A strictly concave function on a convex set can have at most one global maximizer
[Let C ⊆ ℝⁿ be a convex set. If f : C → ℝ is strictly concave, then arg max_x∈Cf(x) is at most a singleton]

A strictly convex function on a convex set can have at most one global minimizer
[Let C ⊆ ℝⁿ be a convex set. If f : C → ℝ is strictly concave, then arg min_x∈Cf(x) is at most a singleton]

Local maximizer of n real variables

Given a function f : A ⊆ ℝⁿ → ℝ, let C ⊆ A.

A point x̂ ∈ C is a local maximizer of f on C if there exists a neighborhood B_ε(x̂) of such that f(x̂) ≥ f(x) for every x ∈ C∩B_ε(x̂)

Fenchel’s theorem

Let C ⊆ ℝⁿ be a convex set, let f : C → ℝ be concave. If x̂ ∈ C is a local maximizer, then it is a global maximizer.
Let C ⊆ ℝⁿ be a convex set, let f : C → ℝ be convex. If x̂ ∈ C is a local minimizer, then it is a global minimizer.

Let x̂ ∈ C be a local maximizer. Take any x ∈ C, and consider the convex linear combination x_α= αx + (1-α)x̂, for every α ∈ [0,1]. We see that x_α→ x̂ as α → 0⁺:

lim_α→0 ||x_α-x̂|| = lim_α→0 ||(αx + (1-α)x̂) -x̂|| = ||x-x̂|| lim_α→0 α = 0

So, given ε > 0, there exists ᾱ > 0 such that ||x_ᾱ-x̂|| < ε. So x_ᾱ ∈ B_ε(x̂)∩C.
Since f is concave, f(x̂) ≥ f(x_ᾱ ) ≥ ᾱf(x) + (1-ᾱ) f(x̂). So ᾱf(x̂) ≥ ᾱf(x) and f(x̂) ≥ f(x). Therefore, x̂ is a global maximizer.