Functions. Paul Schrimpf. October 9, UBC Economics 526. Functions. Paul Schrimpf. Definition and examples. Special types of functions

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1 of Functions UBC Economics 526 October 9, 2013

2 of of 3.. 4

3 of Functions UBC Economics 526 October 9, 2013

4 of Section 1

5 Functions of A function from a set A to a set B is a rule that assigns to each a A one and only one b B f : A B A is the domain B is the target space Image of A under f {y B : f(x) = y for some x A}

6 of..1 Production : f : R 2 R Linear f(x 1, x 2 ) = a 1 x 1 + a 2 x 2 Cobb-Douglas: f(x 1, x 2 ) = Kx α 1 2 Constant elasticity of substitution: f(x 1, x 2 ) = K(c 1 x a 1 + c 2 x a 2 ) b/a.2 Utility : u : R T R Constant relative risk aversion: u(c 1,..., c T ) = T t=1 c1 γ t βt 1 γ Constant absolute risk aversion: u(c 1,..., c T ) = T t=1 βt ( e αc t ) 1 xα 2.3 Demand function with constant elasticity, D : R 3 R 2 ( Mp α 11 D(p 1, p 2, y) = 1 pα 12 2 yβ 1 Mp α 21 1 pα 22 2 yβ 2 )

7 Visualizing function of The level sets of a function f : X Y are sets of the form {x X : f(x) = y} for some fixed y Y. Indifference curves Isoquants

8 x x 1 y Functions Figure : CES, a = 2, b = 4 5 of x 2 x 1

9 x x 1 y Functions of Figure : CRRA, γ = 2, β = 0.95 x 2 x 1

10 x x 1 y Functions of Figure : CARA, α = 1, β = 0.95 x 2 x 1

11 of Section 2 of

12 Types of of A function f : V W where V and W are vector spaces is linear if f preserves addition and scalar multiplication, ie f(x + y) = f(x) + f(y) f(αx) = αf(x)

13 of R R: a 0 + a 1 x + a 2 x 2 q : R n R is a quadratic if q(x 1,..., x n ) = a 0 + n i=1,j i Quadratic a ij x i x j Written using matrix: q(x 1,..., x n ) = a 0 + x T Ax 1 a 11 2 a a 1n 1 2 where A = a 1 12 a 22 2 a 2n (not unique). 1 2 a 1n a nn

14 of Polynomials A monomial f : R n R is any function of the form f(x 1,..., x n ) = cx a 1 1 xa 2 2 xan n where a i are nonnegative integers. n i=1 a i is the degree of the monomial. A polynomial f : R n R is the sum of finitely many monomials, i.e. f(x 1,..., x n ) = k k=1 c k x a 1k 1 x a nk n The maximum degree of the monomials making up a polynomial is the degree of the polynomial.

15 of A function f : V W which V and W are real vector spaces is homogenous of degree k if f(tx) = t k f(x) for all x V, t R.

16 . Functions of Example Linear are homogenous of degree 1. Example A production function that is homogenous of degree 1 has constant returns to scale because doubling each of the inputs doubles the output. A production function that is homogenous of degree less than 1 has decreasing returns to scale. A production of that is homogenous of degree greater than 1 has increasing returns to scale. Example An affine transformation, f(x) = Ax + b, is not homogenous if b 0.

17 of Let f : R R. f is strictly increasing if for all x 1 > x 2, f(x 1 ) > f(x 2 ). f is strictly decreasing if for all x 1 > x 2, f(x 1 ) < f(x 2 ). f is strictly monontonic if it is either strictly increasing or decreasing. If the strict inequalities (< and >) are replaced with weak inequalities ( and ), then we would say f is weakly increasing / decreasing / monotonic.

18 of Let f : V R where V is a vector space. f is homothetic if a homogenous g : V R and a monotonic h : R R asuch that h g : V R defined by (h g)(x) = h(g(x)) is equal to f.

19 of Section 3

20 of A function f : X Y where X and Y are metric spaces is continuous if whenever {x n } n=1 converges to x in X, then f(x n ) f(x) in Y. No jumps or holes.

21 . ϵ δ definition of continuity of Lemma f : X Y is continuous at x if and only if for every ϵ > 0 δ > 0 such that d(x, x ) < δ implies d(f(x), f(x )) < ϵ. Proof. On problem set.

22 of. Topological definition of continuity Let f : X Y. The preimage of V Y is the set in X, f 1 (V) defined by f 1 (V) = {x X : f(x) V} Lemma f : X Y is continuous if and only if f 1 (V) is open for all open V Y. Corollary f : X Y is continuous if and only if f 1 (V) is closed for all closed V Y.

23 and arithmetic of Theorem Let f : X Y and g : X Y be continuous and X and Y be vector spaces. Then (f + g)(x) = f(x) + g(x) is continuous. Proof. If f and g are continuous, then by definition f(x n ) f(x) and g(x n ) g(x) whenever x n x. From the previous lecture the limit of a (finite) sum is the sum of limits, so f(x n ) + g(x n ) f(x) + g(x), and f + g is continuous. Same for subtraction, multiplication, etc

24 and composition of (f g)(x) = f(g(x)) is the composition of f and g. Theorem Let f : X Y and g : Y Z be continuous where X, Y, and Z are metric spaces. Then f g is continuous. Proof. Let x n x. g is continuous, so g(x n ) g(x). f is also continuous, so f(g(x n )) f(g(x)).

25 One-to-one of f : X Y is one-to-one or injective if for all x 1, x 2 X, if and only if x 1 = x 2. f(x 1 ) = f(x 2 ) f is injective if for each y Y, the set {x : f(x) = y} is either a singleton or empty If f is one-to-one, then f(x) = b has at most one solution

26 of Onto f : X Y is onto or surjective if y Y, x X such that f(x) = y. If f is onto, then f(x) = b has at least one solution When f is one-to-one and onto, f is bijective. If f : X Y is bijective, then the inverse of f, written f 1 satisfies f(f 1 (y)) = y and f 1 (f(x)) = x.

27 of Section 4

28 Correspondence of A correspondence from a set X to a set Y, is a rule that assigns to each a x X a subset of Y. We denote a correspondence by ϕ : X Y.

29 of Example (Budget correspondence) n goods with prices p R n. Income of m, a consumer can afford χ(p, m) = {x X R n : p x m} Consumer s problem Indirect utility function max u(x) x χ(p,m) v(p, m) = max u(x). x χ(p,m) The demand correspondence (usually function) is x (p, m) = arg max u(x). x χ(p,m)

30 of INSERT PICTURE OF upper and lower hemicontinuous

31 of A correspondence, ϕ : X Y is upper hemicontinuous at x if for all sequences x n x and y n ϕ(x n ) with y n y, then y ϕ(x). A correspondence, ϕ : X Y is lower hemicontinuous at x if for all sequences x n x and y ϕ(x), there exists a subsequence, x nk and y k ϕ(x nk ) with y k y. A correspondence is continuous at x if it is both upper and lower hemicontinuous at x

The Consumer, the Firm, and an Economy

Andrew McLennan October 28, 2014 Economics 7250 Advanced Mathematical Techniques for Economics Second Semester 2014 Lecture 15 The Consumer, the Firm, and an Economy I. Introduction A. The material discussed