Mathematical Preliminaries

Transcription

1 Mathematical Preliminaries Economics Intermediate Macroeconomics Aaron Hedlund Baylor University Fall 2013 Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

2 Outline I: Sequences and Series II: Continuity and Differentiation III: Optimization and Comparative Statics IV: Basic Probability and Statistics Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

3 Sequences and Series A sequence is a function whose domain is the positive integers. Examples: f (t) = 2t or 2, 4, 6, 8, 10,... f (t) = ( 2) t or 2, 4, 8, 16, 32,... A sequence is convergent with limit L if, for any ɛ > 0, there is some T such that a t L < ɛ whenever t > T. We write lim t a t = L. If {a t } has no limit, it is divergent. If {a t } is a sequence, then s T = T t=t 0 a t, T = 1, 2, 3, is a series. Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

4 Sequences and Series Theorem If s T = T t=t 0 a t is the series associated with sequence a t and a lim T +1 T a T = L it follows that: 1 if L < 1, then s T converges 2 if L > 1, then s T diverges 3 if L = 1, then s T may converge or diverge A geometric series is a series s T of the form s T = T t=t 0 ar t = ar T0 + ar T0+1 + ar T ar T Applying the theorem above, a geometric series converges if ar T +1 ar = r < 1. T Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

5 Sequences and Series The partial sum of a geometric series can be written explicitly as T ar t = ar T0 (1 r T T0+1 ) 1 r t=t 0 Taking the limit when r < 1, we get lim T T ar t = ar T0 1 r t=t 0 The present value PV of a stream of payments {π t } T t=1, discounted at rate r, is given by PV = T t=1 π t (1 + r) t 1 Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

6 Continuity and Differentiation A function f : (a, b) R is continuous at x 0 (a, b) if, for any ɛ > 0, there exists δ > 0 such that f (x) f (x 0 ) < ɛ whenever x x 0 < δ. We say that f (x) is differentiable at x (a, b) if the limit f (x) = f (x + x) f (x) lim x 0 x exists and is finite. The derivative of y = f (x) can also be written as dy df dx or dx. A function is continuously differentiable on a set S if it is differentiable and its derivative is continuous on S. Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

7 Differentiation Rules 1 f (x) = c f (x) = 0. 2 f (x) = x n f (x) = nx n 1. 3 g(x) = cf (x) g (x) = cf (x). 4 h(x) = f (x) ± g(x) h (x) = f (x) ± g (x). 5 h(x) = f (x)g(x) h (x) = f (x)g(x) + f (x)g (x). 6 h(x) = f (x) g(x) h (x) = g(x)f (x) f (x)g (x) [g(x)] 2. 7 h(x) = f (g(x)) h (x) = f (g(x))g (x), or dh dx = df dx g(x) dg dx. 8 h(x) = f 1 (x) h 1 dh (x) =, or f (f 1 (x)) dx = 1 df dx f 1 (x) Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25.

8 Partial Differentiation The partial derivative of a function y = f (x 1, x 2,..., x n ) with respect to x i is written as y x i, f x i, or f i, and is defined as f f (x 1,..., x i + x i,..., x n ) f (x 1,..., x i,..., x n ) = lim x i x i 0 x i Most of the partial differentiation rules are simple extensions of the single variable rules, except for the chain rule. Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

9 Multivariable Chain Rule and Total Differentiation Chain rule: Let y = f (u 1, u 2,..., u m ) and u i = g i (x 1, x 2,..., x n ) for all i = 1, 2,..., m. Denote x = (x 1,..., x n ) and define h(x) = f (g 1 (x),..., g m (x)). Then h x j = f u 1 g 1 x j + f u 2 g 2 x j + + f u m g m x j = m f u i i=1 y = y u 1 + y u y u m = x j u 1 x j u 2 x j u m x j The total differential of a function f (x) is i=1 ui =g i (x) m y u i g i x j, i.e. ui =g i (x) u i x j df = n i=1 f x i dx i Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

10 Second-Order Partial Derivatives The second-order partial derivative of f (x) with respect to x i and then x j is Theorem (Young s Theorem) f ij = f i(x) x j If f (x) has continuous first-order and second-order partial derivatives, the order of differentiation in computing the cross-partial is irrelevant, i.e. f ij = f ji. Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

11 Concavity and Convexity A set S is convex if (1 θ)x + θx S for all x, x S and θ (0, 1). A function f (x) is concave if, for all x, x, and θ (0, 1), f ((1 θ)x + θx ) (1 θ)f (x) + θf (x ) A function f (x) is quasi-concave if, for all x, x, and θ (0, 1), f ((1 θ)x + θx ) min{f (x), f (x )} Equivalently, f is quasi-concave if S(a) = {x: f (x) a} are convex for all a. All concave functions are quasi-concave but not vice-versa. Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

12 Concavity and Differentiability Univariate functions: Let f (x) be twice continuously differentiable. If f is concave, then f 0. Bivariate functions: Let f (x 1, x 2 ) be twice continuously differentiable. If f is concave, then f11 0 and f 11 f 22 (f 12 ) 2 0. If f (x 1, x 2 ) is quasi-concave, then f 11 f 2 2 2f 1f 2 f 12 + f 22 f 2 1 < 0. Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

13 Constrained Optimization This section establishes necessary and sufficient conditions for solutions to problems of the following form: g 1 (x) 0 g 2 (x) 0... max f (x) such that g l (x) 0 h 1 (x) = 0... h k (x) = 0 Define the Lagrangian to the above problem as l k L(x, λ, γ) = f (x) + λ i g i (x) + γ j h j (x) i=1 j=1 Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

14 Constrained Optimization Theorem (Kuhn-Tucker) Suppose that f, {g i } l i=1, and {h j} k j=1 are continuously differentiable and x is a local constrained optimizer of f. Also, assume that the constraint qualification is satisfied. Then there exist multipliers λ and γ such that L First-Order Conditions: (x, λ, γ ) = 0,..., L (x, λ, γ ) = 0 x 1 x { n g1 (x Constraints: ) 0,..., g l (x ) 0 h 1 (x ) = 0,..., h k (x ) = 0 Complementary Slackness: λ 1g 1 (x ) = 0,..., λ l g l(x ) = 0 Nonnegative g-multipliers: λ 1 0,..., λ l 0 If f is concave and if {g i } l i=1 and {h j} k j=1 are quasi-concave, then the above conditions are also sufficient for x to be an optimal solution. Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

15 Constrained Optimization: A Few Remarks An inequality constraint g i (x) is binding at a solution x if loosening the constraint and re-optimizing causes f to increase. Mathematically, f (x (m i )) m i = λ i (m i) > 0 where x (m i ) is the solution with multiplier λ i (m i) when the constraint is loosened to g i (x) m i. A binding constraint has g i (x ) = 0 and λ i > 0. A non-binding constraint has g i (x ) > 0 and λ i = 0. In rare instances λ i = 0 and g i (x ) = 0, in which case the constraint is not binding because λ i = 0. Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

16 Constrained Optimization: A Simple Example A Binding Constraint: max x 2 + 4x 4 such that x 3 0 L = x 2 + 4x 4 + λ(x 3) Solution Conditions: L x = 0 = 2x λ x 3 0 λ (x 3) = 0 λ 0 x = 3, λ = 2 A Non-Binding Constraint: max x 2 + 4x 4 such that 3 x 0 L = x 2 + 4x 4 + λ(3 x) Solution Conditions: L x = 0 = 2x + 4 λ 3 x 0 λ (3 x ) = 0 λ 0 x = 2, λ = 0 Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

17 Comparative Statics Comparative statics analyzes how optimal solutions respond to changes in underlying parameters. Example: how does labor supply change in response to a wage increase? First, an important theorem: Theorem (Implicit Function Theorem) Let F (x, y) be a continuously differentiable function around (x, y ) with F (x, y ) = 0 and F y (x, y ) 0. Then there is a continuously differentiable function y = f (x) defined in a neighborhood B of x such that 1 F (x, y(x)) = 0 for all x B 2 y = f (x ) y 3 x i (x ) = f i (x ) = F xi (x, y )/F y (x, y ) Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

18 Comparative Statics - One Endogenous Variable Assume that we have an optimization problem that gives the following solution condition: F (x, α) = 0 Assume that F and the implicit solution x (α) are differentiable. As a function of α, we have Differentiating by α gives F (x (α), α) = 0 F x dx dα + F α = 0 dx dα = F α(x, α) F x (x, α) Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

19 Comparative Statics - Two Endogenous Variables Suppose we have the following solution conditions: F 1 (x 1, x 2, α) = 0 F 2 (x 1, x 2, α) = 0 Assume that the solution gives differentiable implicit functions x1 (α) and x2 (α). Differentiating with respect to α gives F1 1 x1 F 2 1 α + F 2 1 x1 α + F 2 2 x 2 Solving the system of equations gives α + F α 1 = 0 x2 α + F α 2 = 0 x 1 α = F 2 αf 1 2 F 1 αf 2 2 F 1 1 F 2 2 F 2 1 F 1 2 and x 2 α = F 1 αf 2 1 F 2 αf 1 1 F 1 1 F 2 2 F 2 1 F 1 2 Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

20 Probability and Statistics Let Ω = {ω 1, ω 2,..., ω n } denote the sample space of an experiment, where each ω i is an outcome. Example: tossing a single die can yield any of the following outcomes: {1, 2, 3, 4, 5, 6}. An event is a subset of outcomes in the sample space E Ω. Example: getting an even-numbered toss is the event that consists of the following outcomes: {2, 4, 6}. We can assign a probability P(E) to events, where P satisfies 1 0 P(E) 1 for all E. 2 P(Ω) = 1. 3 P(E 1 E m ) = m i=1 P(E i) if E 1,..., E m are mutually exclusive events. Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

21 Conditional Probability and Independence The conditional probability of E 2 given E 1 is the probability that E 2 will occur given that E 1 has occurred. It is represented by P(E 2 E 1 ) = P(E 1 E 2 ) P(E 1 ) Events E 1 and E 2 are independent if P(E 2 E 1 ) = P(E 2 ), or equivalently, if P(E 1 E 2 ) = P(E 1 )P(E 2 ). In applications it is useful to know Bayes rule: P(E F ) = P(E F ) P(F ) = P(F E)P(E) P(F E)P(E) + P(F E c )P(E c ) Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

22 Random Variables and Expectations A random variable X is a function defined on a sample space, X : Ω R. Example: Ω = {Heads, Tails}, X (Heads) = 0, X (Tails) = 1. The expected value or mean of a random variable X is E(X ) = n x i P(ω : X (ω) = x i ) i=1 where X takes on values {x 1,..., x n }. Let P(x i ) = P(ω : X (ω) = x i ). Above example: E(X ) = = 0.5. We often write µ X instead of E(X ). Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

23 Example: Expected Utility Set of events each period is S = {s 1,..., s S }. Event histories s t = (s 0, s 1,..., s t ) with probabilities π(s t ). Consumers value random consumption streams {c t (s t )} using expected utility: U({c t(s t )}) = E β t u(c t) = t=0 β t π(s t )u(c t(s t )) t=0 s t S t Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

24 Variance, Covariance, and Correlation The variance of a random variable is Var(X ) = E[(X µ X ) 2 ] = n (x i µ X ) 2 P(x i ) i=1 The covariance of two random variables X and Y is Cov(X, Y ) = E[(X µ X )(Y µ Y )] = n (x i µ X )(y i µ Y )P(x i, y i ) i=1 The correlation between X and Y is Corr(X, Y ) = Cov(X,Y ) SD(X )SD(Y ), where SD(X ) = Var(X ) is the standard deviation of X. Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25

25 Sample Statistics When confronting actual data, the underlying probabilities are not readily observable, forcing us to compute sample statistics. Suppose we have data {(x 1, y 1 ), (x 2, y 2 ),..., (x n, y n )}. The sample mean and variance of X are X = 1 n n i=1 x i and Var(X ) = 1 n 1 n i=1 (x i X ) 2. The sample covariance between X and Y is Cov(X, Y ) = 1 n 1 n i=1 (x i X )(y i Y ). Econ 3307 (Baylor University) Mathematical Preliminaries Fall / 25