Part I Analysis in Economics

Transcription

1 Part I Analysis in Economics D 1 1 (Function) A function f from a set A into a set B, denoted by f : A B, is a correspondence that assigns to each element A eactly one element y B We call y the image of under f and denote it by ( ) f The domain of f is the set ( ) { ( ) } f A = f A B R 1 1 In this part, we only work with functions whose domains and ranges are sets of real numbers We call such functions real functions of one real variable E 1 1 Some important economic functions: 1 (Demand functions) p = 5 e i) ( ) phl ii) ( p) 36 = p HpL p Here are: p : : the price per unit of product the demand 1

2 (Supply functions) i) p ( ) = phl p = 5 + 8p ii) ( ) 5 Here are: p : : the price per unit of product the supply HpL p 3 (Revenue functions) i) R ( ) : = p( ) = (1 15 ) = 1 15 RHL

3 ii) R ( p) = p ( p) = p (8 8 p) = 8p 8p RHpL p iii) (A discrete revenue function) The following table shows the revenue [thousand ] of a firm in the first si month of a year: Month Revenue (Production functions) Let r : : Factor input Output i) (Cobb-Douglas production function): ( r) = 7r 5 HrL r 3

4 5 ii) (CES production function): ( r) = ( r + 5) HrL r iii) (Limitational production function): 5 (Total cost functions) Es sei : C : C : f Production Total costs Fied costs C v : Variable costs C( ) = C + C ( ) f v 75r für r ( r) = 15 für r > i) (Neoclassical production function): C( ) = e 1 ii) (Piecewise defined cost function): für < für 4 < 8 C( ) = für 8 < 1 + < für (Average, average variable and average cost functions) C( ) c( ) : =, >, ( ) = = c 4

5 chl Cv( ) cv( ) : =, >, v ( ) = = c cvhl c f C f :, = >, ( ) 8 c f =, > cfhl (Profit functions) P( ) : R( ) C( ) =, P( ) = 55 ( ) 3 = PHL

6 8 (Average profit function) P( ) p( ) : =, >, 3 + ( ) p = = + phl E 1 Find the domain of the 1 production function ( r) = r demand function 1 p( ) = function E( Y ) = ln( Y + 1) 75, ( E : monthly ependiture on energy; Y : monthly family income) Solution: 1 r 1 > 3 Y > D 1 (Limit of a Function) Let f : A B be a real function and, l R Then ( ) lim f ( ) = b ε > δ > : < a < f ( ) b < ε a D 1 3 (One-sided Limits) 1 Let f : A B, +, δ > be a real function defined in ] δ [ A number l r is called the limit of f ( ) as approaches from the right (or simply called the right-hand limit of f at, symbolically 6

7 + Let f : A B ( ) lim f ( ) = l, if ε > δ > : < < + δ f ( ) - l < ε r be a real function defined in ] δ [,, δ > A number l l is called the limit of f ( ) as approaches from the left (or simply called the left-hand limit of f at, symbolically r ( ) lim f ( ) = l, if ε > δ > : δ < < f ( ) - l < ε l T 1 1 A function f : A B has a limit as approaches if and only if the left-hand and right-hand limits at eist and are equal In symbols, we write l T 1 (Uniqueness of Limits) lim f ( ) = lim f ( ) = lim f ( ) + lim f ( ) = a lim f ( ) = b a = b D 1 4 (Infinite Limits) 1 ( δ ) lim f ( ) =, if n N δ > : < < f ( ) > n 3 4 lim f ( ), if n Z + = \ δ ( δ ) ( N {}) > : < < f ( ) < n ( δ ) lim f ( ) =, if n N δ > : < < + f ( ) > n lim f ( ), if n Z + = \ δ ( δ ) ( N {}) > : < < + f ( ) < n 5 6 lim f ( ) =, if n N δ > : < - < δ f ( ) > n lim f ( ), if n Z = \ δ ( δ ) ( N {}) > : < < f ( ) < n 7

8 E 1 3 The following function epresses the dependency of the butter consumption B [ /month] of a family on its monthly income Y [ /month}: 15 Y B( Y ) = 6 e, Y > Investigate the change in butter consumption when the family income approaches zero? Solution: 15 1 Y lim B( Y ) = lim 6 e = 6 lim = 6 = Y Y Y Y e D 1 5 (Limit at Infinity) 1 lim f ( ) = l, if ε > n N : > n f ( ) - l < ε lim f ( ) = l, if ε > n Z \ ( Z {}) δ > : < n f ( ) l < ε E 1 4 Given the function described in E 13, find its point of satiation for the income approaching infinity Solution: 15 lim B( Y ) = lim 6 e Y = 6 e = 6 1 = 6 Y Y 8

9 T 1 3 Let a, b, R 1 lim b = b lim = 3 lim ( ) lim a + b = a + b = lim sin = sin lim cos = cos T 1 4 (The Algebra of Limits) Let f, g be two real functions If hold: lim f ( ) = a and lim g( ) = b, then the following rules 1 lim ( ( ) ( )) f + g = a + b lim ( ( ) ( )) f g = a b 3 lim k f ( ) = k a ( k is any constant) 4 lim f ( ) g( ) = a b f ( ) a 5 lim = if b ` g( ) b T 1 5 (The Sandwich Theorem) Let A R and f, g, h : A R be three real functions g( ) f ( ) h( ), A lim g( ) = lim h( ) = a lim f ( ) = a T 1 6 (Elementary Limits of Circular Functions) sinθ 1 lim = 1 θ θ cosθ 1 lim = θ θ tanθ 3 lim = 1 θ θ T 1 7 (Limit Theorem for Composites) Let f and g be two real functions or, equivalently, 9 ( ) lim f ( ) = b lim g( u) = g( b) lim g f ( ) = g( b) a a b a

10 ( ) lim g f ( ) = g lim f ( ) a a D 1 6 (Continuity) Let f : A R be a real function f is said to be continuous at = if lim f ( ) = f ( ) or, equivalently ε >, δ > : - < δ f ( ) f ( ) < ε R 1 Here and throughout, the symbol f C( ) means f is continuous at Thus, we obtain a rule to test for continuity at a given Point: R 1 3 (Continuity Test) f C( ) ( i) A, ( ii) lim f ( ) eists, and ( iii) lim f ( ) = f ( ) E 1 5 A firm that produces some amount of output has the following cost function: ( ) 4 if > C( ) = if = 1 What is the right-hand limit of this function as output approaches zero? What is the function actually equal to when output is zero? 3 Is the function continuous? If not, eplain why not Solution: 1 lim C( ) = 19 + C () = 3 No, since lim C() C() 1

11 T 1 8 If f is continuous at =, then the following combinations are also continuous at = : ( i) f + g, ( ii) f g, ( iii) k f ( k R), ( iv) f g, f ( v), provided g( ) g T 1 9 If f is continuous at, and g is continuous at f ( ), then the composite g f is continuous at D 1 7 (One-sided Continuity) 1 A function f is called continuous from the left at, if lim f ( ) = f ( ) A function f is called continuous from the right at,if lim f ( ) = f ( ) + 3 If f is defined on [ a, b ], continuous on ], [ then f is said to be continuous on [ a, b ] T 1 1 a b and lim f ( ) = f ( a) and lim f ( ) = f ( b), + a b f C( ) lim f ( ) = lim f ( ) = f ( ) + 11

12 T 1 11 (Fundamental Theorem on Continuous Real Functions) f : a, b R f ( a, b ) = c, d for some suitable c, d If [ ] is a continuous function, then [ ] [ ] In addition, if a real function is continuous on [ a, b ], then f attains an absolute maimum value M and an absolute minimum value m somewhere on this interval T 1 1 (Min-Ma Theorem for Continuous Functions) f : a, b R If [ ] is a continuous function, then there eist, [ a, b] 1 1 [ ] f ( ) f ( ) f ( ), a, b T 1 13 (Intermediate Value Theorem for Continuous Functions) f : a, b R is a continuous function and if f ( a) f ( b) If [ ] (or [ f ( b), f ( a )]), there eists a number c [ a, b] such that f ( c) T 1 14 (Intermediate Zero Theorem) If f : [ a, b] R is a continuous function and f ( a) f ( b) such that f ( c ) = such that R, then for any k [ f ( a), f ( b) ] = k <, then there eists c [ a, b] T 1 15 (Boundedness Theorem) f : a, b R is a continuous function, then there eists a positive number M such that If [ ] [ ] f ( ) M, a, b D 1 8 (Uniform Continuity) f : a, b R be a real function f is said to be uniformly continuous on f if If [ ] R 1 4 ε >, δ > :, y A, y < δ f ( ) f ( y) < ε Uniform continuity continuity, but not the converse D 1 9 (Derivative) Let f : A R and ] a, b[ A The derivative of f is a function f ' whose value at is the number f '( ) : = lim f ( ) f ( ) or, equivalently f '( ) : = lim f ( + ) f ( ) with = + 1

13 Generally speaking, the derivative of f at any ] [ a b A is given by, f f ( + h) f ( ) '( ) : = lim h h If the derivative f '( ) eists, we say that f has a derivative (or, is differentiable) at If f has a derivative at every point of its domain, then f is said to be differentiable E 1 6 A firm has the following revenue function E( ) 15 5 = Find its derivative at = Solution: E( + h) E() E '() = lim h h = ( ) ( ) h h lim h h 148h h = lim h h h ( h) = lim 148 = 148 D 1 1 (One-sided Derivative) Let f : A R be a real function Then f is said to be differentiable on [, ] all ] a, b[ and the limits a b if f ' eists for (i) ' f ( a + h) f ( a) f+ ( a) : = lim (right-hand derivative at a) + h h or, equivalently, and (ii) ' f ( ) f ( a) f+ ( a) : = lim+ a a ' f ( b + h) f ( b) f ( a) : = lim (left-hand derivative at b) h h or, equivalently, 13

14 ' f ( ) f ( b) f ( a) : = lim b b eists at the endpoints a and b T 1 16 A real function f : A R is differentiable at = if and only if ' ' f+ f f ( ) = ( ) = '( ) T 1 17 (Differentiability-Continuity Theorem) If a real function f is differentiable at, then f is continuous at T 1 18 (Algebra of Derivatives) If f and g are real functions that are differentiable at, then f ± g, f g and f / g are differentiable at (in the case of f / g provided that g( ) ) Moreover, (i) ( f ± g)'( ) = f '( ) ± g '( ) (Sum-Difference Rule) (ii) ( f g)'( ) = f ( ) g '( ) + g( ) f '( ) (Product Rule) (iii) ( ) f ( ) g '( ) g( ) f '( ) f / g '( ) = (Quotient Rule) ( g( ) ) T 1 19 (Chain Rule) Let the function F be defined as f composed with g, that is F = f g = f ( g( )) Then F ' is given by df F ' = = f '( g( )) g '( ) d R 1 5 Alternatively, in Leibniz notation, the chain rule can be epressed as dy d dy dz = dz d E 1 7 A firm s cost function is given by 1+ 4 C( ) = e, Find the marginal cost function Solution: dz z = 1 + 4, 1 d = 14

15 dc C( ) = e z, e z dz = dc( ) dz dc( ) z 1+ 4 C '( ) = = = 1 e = e d d dz D 1 11 (Elasticity Function) Let f be a differentiable function on A The function ε f, ( ) : = f '( ) f ( ) will be called the elasticity function of f on A R The elasticity ε f, can be interpreted as follows: A percentage change of leads to an approimate change of f by ε f, ε f, > means that both and f change in the same direction; ε f, < means that and f change in opposite directions 3 Following cases can be distinguished: ε f, < 1: f is inelastic, ε f, > 1: f is elastic ε f, = 1: f is proportional elastic, ε f, : f is completely elastic, ε f, f is completely inelastic 4 Let f be denoted by y = f ( ) und its inverse by = g( y) Then ε ε = y,, y 1 E 1 8 A firm has the revenue function R( ) 3 5 = Find the elasticity function ( ) ε at the point = 1 and interpret your result E, Solution: ε R, ( ) = (3 5 ) 3 5, ε R, (1) 91 15

16 An increase of demand from 1 units by 1% will lead to an increase of the revenue of the firm by approimately 91% The revenue function is inelastic at 1 = D 1 1 (Differential) Let f be a differentiable function at and The difference between f ( + ) and f ( ), denoted by f, f : = f ( + ) f ( ) is called the increment of f from to + The product f '( ) is called differential of f at with increment, and is denoted by df, or, equivalently, df : = f '( ) dy : = f '( ) R 1 7 In the above definition, can be any nonzero value However, in most applications of differentials, we choose d = Thus, we also write We have or, equivalently E 1 9 The total cost function of a firm is given by dy : = f '( ) d f df y f '( ) = f '( ) d 3 C( ) = The firm would like to increase its production from 1 to 1 units Find the approimate increase of costs using the differential of the cost function Solution: ( ) dc( ) = C '( ) d = d = 76 = 1, d= = 1, d= D 1 13 (Maimum and Minimum of a Function) Let f : A R be a function 1 A number M is called the maimum value of f over A if 16

17 f ( ) M, A f ( c) = M for some c A A number m is called the minimum value of f over A if f ( ) m, A f ( c) = m for some c A T 1 Let f : A R be a function and c A If f '( c) eists and f '( c), then f ( c) is neither a maimum nor a minimum value of f in any neighbourhood of c Proof: If f '( c ) and f '( c ) >, then f f ( ) f ( c) '( c) = lim > c c The there eists an Interval ] c δ, c δ [ + such that f ( ) f ( c) >, c, c c, c + c This implies that f ( ) f ( c) and c ] δ [ ] δ [ have the same sign in ] c δ, c[ ] c, c δ [ +, that is, and f ( ) < f ( c) if < c f ( ) > f ( c) if > c Hence, f ( c) is neither a maimum nor a minimum value of f in any neighbourhood of c A similar argument holds if f '( c ) < T 1 1 (Contrapositive of T 1 ) Let f : A R be a function and c A If f ( c) is either a maimum or a minimum value of f in some neighbourhood of c, then either f '( c) does not eist or f '( c ) = D 1 14 (Critical Number) Let f : A R be a function A number c A is called a critical number of f if either f '( c ) does not eist or f '( c ) = T 1 (Rolle ) a, b Aand f : A R be a function If Let [ ] (i) f is continuous on [ a, b ], (ii) f is differentiable on ] a, b [, and (iii) f ( a) = f ( b), then there eists an element c ] a, b[ such that f '( c ) = 17

18 T 1 3 (Mean Value Theorem) a, b Aand f : A R Let [ ] be a function If f is continuous on [, ] on ] a, b [, then there eists a number c ] a, b[ such that a b and differentiable f ( b) f ( a) = f '( c) b a D 1 15 (Monotonic Functions) Let f : A R be a function (i) f is said to be monotonic increasing if f ( 1) f ( ), 1, A : 1 < (ii) f is said to be monotonic decreasing if f ( 1) f ( ), 1, A : 1 < (iii) f is said to be strictly increasing if f ( 1) < f ( ), 1, A: 1 < (iv) f is said to be strictly decreasing if f ( 1) > f ( ), 1, A : 1 < (v) f is said to be monotonic if f is either increasing in A or decreasing in A (vi) f is said to be strictly monotonic if f is either strictly increasing in A or strictly decreasing in A T 1 4 (Monotonicity Theorem) Let f : A R then be a function If f is continuous on [ a, b] Aand differentiable on ], [ (i) f '( ) >, ] a, b[ f is strictly increasing on [ a, b] (ii) f < ] a b[ f [ a b] '( ),, is strictly decreasing on, E 1 1 Find the intervals in which the profit function a b, P = + ( ) is strictly monotonic and sketch the graph of f Solution: P '( ) = + 9, P '( ) > < 9 f is strictly increasing in ], 9[ and strictly decreasing for > 9 PHL

19 D 1 16 (Etrema of a Function) Let f : A R be a function and A We say that (i) f ( ) is a relative maimum of f on A if f ( ) f ( ), A U ( ) (ii) f ( ) is a relative minimum of f on A if f ( ) f ( ), A U ( ) (iii) f ( ) is an absolute maimum of f on A if f ( ) f ( ), A (iii) f ( ) is an absolute minimum of f on A if f ( ) f ( ), A T 1 5 (A Necessary Condition for Relative Etrema) If a function f has a relative etremum at a number, then f '( ) = or f '( ) does not eist T 1 6 If f is a differentiable function on a set A and has a relative etremum at, then f '( ) = T 1 7 (First Derivative Test for Etrema) Let be a critical number of f and f be continuous at If there eists an ε > ε ε such that (i) (ii) (iii) ] ε [ ] ε[ f '( ) <,, f '( ) >,, + then f ( ) is a relative minimum ] ε [ ] ε[ f '( ) >,, f '( ) <,, + then f ( ) is a relative maimum ] ε [ ] + ε[ f '( ) keeps the same sign on,, then f ( ) is not an maimum T 1 8 (Second Derivative Test for Etrema) If is a critical number of a function f which is twice differentiable on an interval ε, + ε for someε >, then ] [ (i) (ii) f ''( ) < f ( ) is a relative maimum of f f ''( ) > f ( ) is a relative minimum of f 19

20 R 1 8 An algorithm for obtaining the etrema of a given continuous function f on the interval [ a, b] is given as follows: 1 Determine all critical numbers of f on ] a, b [ Calculate the value of f at all its critical numbers and also f ( a) and f ( b ) 3 Compare all the value of f in and the largest one is the maimum value of f on[ a, b ] while the smallest value is the minimum value of f on[ a, b ] E 1 11 The profit function of a firm depending on the price charged for its product is given by 3 P( p) = p + 48p-119, p 45 Find the price for which its profit will be maimised Solution: 1 P '( p) = 3p + 48 : =, < p < 45 p = 4 P ''( p) = 6 p < Hence, the only critical numbers of P on ], 45[ is p = 4 with P (4) = 9 P () = 119, P (45) = Because of 119 < 5875 < 9 the firm s profit will be maimised for p = 4 : PHpL p D 1 17 (Conve and Concave Function) Let f be a function which is continuous on [ a, b ] and is differentiable on ], [ a b We say that (i) f is conve if the graph of f lies above the tangent lines to f throughout ] a, b [ (ii) f is concave if the graph of f lies below the tangent lines to f throughout ] a, b [

21 T 1 9 (Test for Conveity and Concavity) Let f : A R be a function whose second derivative eists on ], [ f ''( ) >, a, b f is conve (i) ] [ (ii) ] [ f ''( ) <, a, b f is concave a b Then we have: D 1 18 (Inflection Points), f ( ) is called a point of inflection of a function f if there eists an interval The point ( ) ] ε, + ε[ such that f ''( ) >, ], [ and f ''( ), ], ε[ < + T 1 3 (Test for Inflection Points) If (, f ( ) ) is an inflection point of f, then either f ''( ) = or f ''( ) does not eist E 1 1 A firm has the total cost function ( ) C( ) = , 1 Discuss the most important properties of its average cost function Solution: 4 3 c( ) = ( ) 1 Domain C( ) c( ) : = _ D( c( )) = 1, ; _ : maimum production capacity _ Let us assume that the firm has a maimum capacity of = 8 1 Continuity c( ) ist continuous on D 3 Points of Intersection with the Aes a) with the ais 4 3 c( ) : = ( ) = It can be (numerically) shown that the graph of c( ) does not cut the ais b) With the y ais Because of 1 there is also no point of intersection with the y ais 4 Monotonicity 1

22 c 4 '( ) = 1 ( ) c '( ) : = 1 = 8, = 4 ( 4) ( 8) c( ) is non-decreasing The solution of the above inequality leads to the following results: c( ) is non-decreasing for ] 1, 8[ ]4, 8[ c( ) is non-increasing for ]8, 4[ 5 Etrema 4 c ''( ) = 1 ( ) ; c ''(8) = 18 < ; c ''(4) = 18 > Thus, c( ) assumes a relative maimum at = 8 with c (8) = 1919 and a relative minimum at = 4 with c (4) = 1896 Because of c (1) = 157 and c (8) = 468 the absolute minimum and the absolute maimum of ( ) c are ( 1, 157 ) and ( ) 6 Conveity and Concavity 8, 468, respectively 4 c( ) is conve, ]34, 8[ c ''( ) = 1 ( ) c( ) is cocave, ]1, 34[, ie 34 ( 34, 19896) is a point of inflection of c( ) The average costs increase progressively in ] 34, 8[ and regressively in ] 1, 34[ 7 Graph of the Function: chl

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