Econ 205 - Slides from Lecture 3Joel SobelAugust 25, 2010Econ 205 SobelDifferentiationCalculus ideas:1. Linear functions are easy to deal with.2. Not all functions are linear.3. Lots of functions are “locally” linear.4. So: sacrifice “global” analysis of linear functions with “local”analysis of a larger class of functions.For this to work you need to know how to approximate functionswith linear functions.This is what differentiation is about.Econ 205 SobelConstant ApproximationsDefinitionThe zero-th order approximation of the function f at a point a inthe domain of f is the function A0(x) ≡ f (a).Good news: It does approximate at a point.Bad news: It does nothing else.Econ 205 SobelLinear ApproximationReplace A0with the best linear approximation to f .A line that intersects the graph of f at the points (x, f (x)) and thepoint (x + δ, f (x + δ)). This line has slope given by:f (x + δ) − f (x)(x + δ) − x=f (x + δ) − f (x)δWhen f is linear, the line with this slope is f .Econ 205 SobelLimiting ArgumentWhen δ is small:limδ→0f (x + δ) − f (x)(x + δ) − x= limδ→0f (x + δ) − f (x)δNote:IThe denominator of the expression does not make sense whenδ = 0.IFor the limit to make sense x must be an interior point of thedomain of f .IIf the limit exists, we say that f is differentiable at x and callthe limit the derivative of f at x. There are two “one-sided”versions of this definition.Econ 205 SobelOne-sided versionsf : (a, b) −→ Rand x ∈ (a, b).Define if they existf0+≡ limy→x+f (y) − f (x)y − x( limδ→0+f (x + δ) − f (x)δ)andf0−≡ limy→x−f (y) − f (x)y − x( limδ→0−f (x + δ) − f (x)δ)Econ 205 SobelDefinitionIf both left and right hand derivative exist and are equal at a pointx, then the function is said to be differentiable at x and thecommon value is called the derivative.There are many ways to denote the derivative of the function f atthe point x: f0(x), Df (x), anddfdx(x) are all common.Econ 205 SobelExamplef (x) = x2f0(x) = limy→xf (y) − f (x)y − x= limy→xy2− x2y − x= limy→x(y − x)(y + x)y − x= limy→x(y + x)= 2xEcon 205 Sobelf (x) =x, if x ≥ 0,−x, if x < 0.So we get from this thatf0+(0) = 1 right hand derivativef0−(0) = −1 left hand derivativeNo derivative exists sincef0+(0) 6= f0−(0)Econ 205 SobelFirst-Order ApproximationA1(y) ≡ f (x) + f0(x)(y − x).This is the equation of the line with slope f0(x) that passes through(x, f (x)). It follows from the definition of the derivative thatlimy→xf (y) − A1(y)y − x= 0.A1is a better approximation to f than A0.Econ 205 SobelMore Approximation1. Linear approximation A1close to f when y is close to x.2. It is also the case that the linear approximation is close to f ifyou divide the difference by something really close to zero(y − x).Econ 205 SobelDifferentiable implies Continuous“It is harder to be differentiable than to be continuous.”TheoremIff : X −→ Rwhere X is an open interval is differentiable at a point x, then f iscontinuous at x.Econ 205 SobelProof.limy→xf (y) − f (x) = limy→xf (y) − f (x)y − x(y − x)= f0(x) · 0= 0becauselimy→xg(y )h(y ) =limy→xg(y)limy→xh(y )Econ 205 SobelSimple FactsTheoremSuppose f and g are defined on an open interval containing point xand both functions are differentiable at x. Then, (f + g), f · g , andf /g are differentiable at x (the last of these provided g0(x) 6= 0).1.(f + g)0(x) = f0(x) + g0(x)2.(f · g)0(x) = f0(x)g(x) + f (x)g0(x)3.(fg)0(x) =g(x)f0(x) − f (x)g0(x)[g(x)]2Econ 205 SobelProof.1. Easy.2. Let h = fg . Thenh(y ) − h(x) = f (y)[g(y) − g(x)] + g(x)[f (y) − f (x)]If we divide this by y − x and note that f (y) −→ f (x), andg(y) −→ g (x) as y −→ x, then the result follows.Then the result follows.3. Now let h = f /g. Thenh(y ) − h(x)y − x=1g(y )g(x)g(x)f (y) − f (x)y − x− f (x)g(y ) − g(x)y − xNow let y −→ x.Econ 205 SobelChain RuleTheorem (Chain Rule)Suppose thatg : X −→ Yandf : Y −→ Rthat g is differentiable at x and that f is differentiable aty = g(x) ∈ Y . Then(f ◦ g)0(x) = f0(y)g0(x)= f0(g(x))g0(x)Econ 205 SobelDefine the functionh(z) =(f (g(z))−f (g (x))g(z)−g (x)if g(z) − g(x) 6= 0f0(g(x)) if g(z) − g(x) = 0limz→xf (g(z)) − f (g(x))z − x= limz→xh(z) ·g(z) − g(x)z − x=hlimz→xh(z)ilimz→xg(z) − g(x)z − xEcon 205 SobelTo complete the proof it suffices to show thatlimz→xh(z) = f0(g(x)).Let ε > 0.There is δ1> 0 such that if0 <| y − g(x) |< δ1implies|f (y) − f (g(x))y − g(x)− f0(g(x)) |< ε.There is δ > 0 such that if0 <| z − x |< 0,then| g(z) − g(x) |< δ1.Hence if 0 <| z − x |< δ and g(z) 6= g (x), then|f (g(z))−f (g(x ))g(z)−g (x)− f0(g(x)) |< ε. This means that if 0 <| z − x |< δand g(z) 6= g(x), then | h(z) − f (g(x)) |< ε. To complete theproof just note that if g(z) = g(y) then h(z) = f0(g(x)).Econ 205 SobelDefinitionx∗is a global maximizer of the function f on the interval [a, b] ifx∗∈ [a, b] and f (x∗) ≥ f (x) for all x ∈ [a, b]. We say globallysince it maximizes the function over the whole domain. However,we say x∗is a local maximizer of the function f if ∃ a segment(a, b) such thatf (x∗) ≥ f (x), ∀ x ∈ (a, b)What we call global max here is normally just called max.Econ 205 SobelDefinitionWe say x∗is a local minimizer of the function f if ∃ a segment(a, b) such thatf (x∗) ≤ f (x), ∀ x ∈ (a, b)Econ 205 SobelIA maximum that occurs at the boundary of the domain is nota local maximum.IA maximum that occurs in the interior of the domain is alocal maximum.IAssume f is defined on [a, b]. If f has a local max atc ∈ (a, b) and if f0(c) exists, thenf0(c) = 0.Econ 205 SobelFirst Order ConditionProof.For | x − c |< δ, we havef (x) − f (c) ≤ 0Therefore, if x ∈ (c, c + δ), thenf (x) − f (c)x − c≤ 0 (1)while for x ∈ (c − δ, c)f (x) − f (c)x − c≥ 0. (2)That is, right derivative is not positive and left derivative is notnegative.Since f is differentiable at c, both left and right derivatives existand they are equal.Econ 205 SobelComments1. Same argument to conclude that if c is a local minimum andf0(c) exists, then f0(c) = 0.2. The equation f0(c) = 0 is called a first-order condition.3. Problems:3.1 Doesn’t distinguish local max from local min.3.2 Local (not global)3.3 f0(c) = 0 can be neither local min nor local max.4. If f0(x) 6= 0 for all x ∈