Workbook. Michael Sampson. An Introduction to. Mathematical Economics Part 1 Q 2 Q 1. Loglinear Publishing

Transcription

1 Workbook An Introduction to Mathematical Economics Part U Q Loglinear Publishing Q Michael Sampson

2 Copyright 00 Michael Sampson. Loglinear Publications: Terms of Use This document is distributed "AS IS" and with no warranties of any kind, whether express or implied. Until November, 00 you are hereby given permission to print one () and only one hardcopy version free of charge from the electronic version of this document (i.e., the pdf file) provided that:. The printed version is for your personal use only.. You make no further copies from the hardcopy version. In particular no photocopies, electronic copies or any other form of reproduction.. You agree not to ever sell the hardcopy version to anyone else. 4. You agree that if you ever give the hardcopy version to anyone else that this page, in particular the Copyright Notice and the Terms of Use are included and the person to whom the copy is given accepts these Terms of Use. Until November, 00 you are hereby given permission to make (and if you wish sell) an unlimited number of copies on paper only from the electronic version (i.e., the pdf file) of this document or from a printed copy of the electronic version of this document provided that:. You agree to pay a royalty of either $.00 Canadian or $.00 US per copy to the author within 60 days of making the copies or to destroy any copies after 60 days for which you have not paid the royalty of $.00 Canadian or $.00 US per copy. Payment can be made either by cheque or money order and should be sent to the author at: Professor Michael Sampson Department of Economics Concordia University 4 de Maisonneuve Blvd W. Montreal, Quebec Canada, HG M8. If you intend to make five or more copies, or if you can reasonably expect that five or more copies of the text will be made then you agree to notify the author before making any copies by at: sampson@loglinear.com or by fax at You agree to include on each paper copy of this document and at the same page number as this page on the electronic version of the document: ) the above Copyright Notice, ) the URL: and the address sampson@loglinear.com. You may then if you wish remove this Terms of Use from the paper copies you make.

3 Contents The Mathematical Method. ProofbyContradiction..... ProofbyInduction... Univariate Calculus 9. Maximization,Minimization, Concavity, Convexity ThePerfectlyCompetitiveFirm CostMinimization....4 Utility Maximization MarkMaximization Monopoly PriceDiscrimination InverseFunctions Product,QuotientandChainRules....0PolynomialsandTaylorSeries... 4.ExponentialFunctionsandLogarithms... 9.LinearRegression... 8.MaximumLikelihood Matrix Algebra 9. Multiplication, Transpose, Determinants and Inverse ApplicationstoEconometrics SystemsofEquations,Cramer srule....4 QuadraticForms,EigenvaluesandEigenvectors Multivariate Calculus 4. UnconstrainedOptimization Pro tmaximization LinearRegression MaximumLikelihood MultivariateChainRule LagrangiansandConstrainedOptimization Utility Maximization CostMinimization...78 i

4 CONTENTS ii 4.6. Eigenvectors, Eigenvalues and Constrained Optimization. 8

5 Chapter The Mathematical Method. Proof by Contradiction Problem Prove that p p is irrational where p is any prime number (a prime number has only one and itself as factors). Hint: it is a fact that any integer n can be factored uniquely into r prime numbers; for example 60 where ; ; and are prime numbers. Assume that p p is rational so that: p p a b Then: where a and b are integers. p a p b ) a b p: Now if integer a has r prime factors (for example 0 has r factors; i:e:; 0 )thena has r prime factors (for example 0 has r 6factors; i:e:; 0 )andsoa hasanevennumber of prime factors: r. The same argument applies to b ; that is b must have an even number of prime factors; say s. From this it follows that b p has an odd number of prime factors s +since p is odd. But a b p and so a has an oddnumberofprimefactors:s +.Thuswehaveshownthata has both an even and an odd number of prime factors, which is a contradiction. Therefore p p is irrational. Problem Prove that there are an in nite number of prime numbers. Hint: suppose that p max is the largest prime number so that the prime numbers are ; ; ; 7;:::p max and consider the number: n 7 p max +:

6 CHAPTER. THE MATHEMATICAL METHOD Under the assumption that there are a nite number of primes: ; ; ; 7;:::p max construct the number: n 7 ::: p max +: Clearly n is an integer. Either n is a prime number or it is not a prime number. If n prime then clearly n>p max which contradicts the assumption that p max is the largest prime number. If n is not prime then it has at least one prime factor say ^p >: Now ^p being a divisor of n implies that ^p 6 ; ; ; 7;:::p max since otherwise: n ^p {z} integer 7 ::: p max ^p {z } integer + ^p {z} non-integer which is not possible. Since ; ; ; 7;:::p max is the list of all primes up to p max it must be that ^p >p max ; which contradicts p max being the largest prime. Thus we derive a contradiction if either n is prime or not prime. It follows that there is no largest prime number. To make the proof a little more concrete, suppose that one thought that was the largest prime number so that the primes are ; ; 4 and : Then: n +: But itself is a prime number (no integer besides and divides ) and is greater than which contradicts our belief that was the largest prime number. Problem Consider the function y p x which is plotted below: x y p x Note that the function gets atter and atter as x increases. A natural conjecture then is that p x never gets bigger than any number so that p x c: Prove that this is not the case; that p x is in fact unbounded. :

7 CHAPTER. THE MATHEMATICAL METHOD Assume to the contrary that p x is bounded so that p x c for all x for some c < : Now we are free to choose any x we like so let x (c +) : The then have: p x q(c +) c + > c ) > 0: Since > 0 contradicts the fact that 0 < the assumption that p x c is false and so p x is unbounded... Proof by Induction Problem 4 Use mathematical induction to prove that: S n n n + n + n 6 : Proof by Induction. Let S n n : Clearly S : Now make the induction hypothesis so we assume that the result is true for i<n;in particular for we can assume that: S n (n ) + (n ) + n 6 : Thus: S n (n ) + n {z } S n (n ) + (n ) n n +n + n 6 + n + n n + + n + n 6 n + n + n 6 + n + n +( n + n)+ n + n + n 6 : µ + 6 Problem Show that the sum of the rst n odd numbers is n : For example thesumofthe rst4 odd numbers is 4 6since: +++76:

8 CHAPTER. THE MATHEMATICAL METHOD 4 Problem 6 Show that Problem 7 Isitthecasethat n (+++:::+ n) : n :::+ n? Problem 8 Suppose that the binomial coe cients are de ned by: µ µ µ n n n + k k k where µ 0 : 0 This is illustrated below in Pascal s triangle: where the rst row gives 0 0, the second gives 0 and ; the third 0, and and so on. To illustrate the generation n k consider n and k : Goingtothe fthrowwe ndthat: µ µ µ : Use induction then to prove that: µ n n! k k!(n k)! : where for example! 4 0and! well that 0! :!! 0: Recall as For n 0and k 0we have: µ 0 0! 0 0!0! :

9 CHAPTER. THE MATHEMATICAL METHOD Nowassumetheresultistrueforn : We then have: µ µ n n (n )! + k k k!(n k)! + (n )! (k )! (n k)! (n )! k (k )! (n k)! + (n )! (k )! (n k)(n k)! µ (n )! (k )! (n k)! k + n k µ (n )! n k + k (k )! (n k)! k (n k) µ (n )! n (k )! (n k)! k (n k) n (n )! (k (k )!) ((n k) (n k)!) n! k!(n k)! : Problem 9 Use induction and the previous result to prove the binomial theorem; that: µ µ µ µ (x + y) n n n n n x n y 0 + x n y + x n y + + x 0 y n : 0 n It is clearly true for n since: (x + y) x + y µ x y Now assume it is true for n : We then have: µ x 0 y : (x + y) n (x + y) (x + y) n µµ µ µ µ n n n n (x + y) x n y 0 + x n y + x n y + + x 0 y n 0 n µ µ µ µ n n n n x n y 0 + x n y + x n y + + x y n 0 n µ µ µ µ n n n n + x n y + x n y + x n y + + x 0 y n 0 n µ µµ µ µµ µ n n n n n x n y x n y + + x n y 0 0 µµ µ µ n n n x y n + x 0 y n n n n

10 CHAPTER. THE MATHEMATICAL METHOD 6 By multiplying the two terms in brackets you will nd that the coe cient on x n j y j is µ µ µ n n n + : j j j The two terms on x n y 0 x n y 0 have to be treated a little di erently but these work as well since: µ µ µ µ n n n n ; : 0 0 n n Thus: (x + y) n µ n x n y µ n x n y + µ n x n y + + Problem 0 The Fibonacci numbers are de ned as with f f : Thus: f n f n + f n f f + f + f 4 f + f + f f 4 + f +. µ n x 0 y n : n There is a close link between these numbers and Á +p :68: For example the ratio of subsequent numbers approaches Á; for example f f 4 :6 ¼ Á: Use proof by induction to show that: + p n p n f n p and from this show that: lim n! f n + f n p : For n we have: + p p p p p f

11 CHAPTER. THE MATHEMATICAL METHOD 7 and so the result is true for n : Now we assume the result is true for ; ;:::n andattempttoshowitistrueforn: Thus: f n f n + f n n + p µ + p p n p + n p n p n + p + + p n + p p p + p n p µ p p p n n p n + n p + p n p p : But since: Ã + p! Ã +p p! ; p we have: f n + p n + p n p + p p p n p n p and the result is proven. Now we have: Since f n f n + p + p p 0:68 is a fraction, n n p n p n : p n! 0 as n!and p n!

12 CHAPTER. THE MATHEMATICAL METHOD 8 0 as n!so that: lim n! f n f n lim n! lim n! +p : + p + p + p n + p n n n p p n n

13 Chapter Univariate Calculus. Maximization,Minimization, Concavity, Convexity Problem Consider the following function: f (x) x x for x>0: Find f 0 (x) and show that f (x) is globally concave. Find the global maximum x explaining how you know it is a global maximum. Is f (x) globally increasing or globally decreasing? Since: f 0 (x) x x ) f 0 (x )0) (x ) (x ) 0 ) (x ) (x ) ) x (multipling both sides by (x ) ) Similarly: f 00 (x) 4 x 4 x < 0 (since x > 0 and x > 0) so that f (x) is globally concave which in turn implies that x is a global maximum. Note that f (x) is neither globally 9

14 CHAPTER. UNIVARIATE CALCULUS 0 increasing nor decreasing since f 0 (x) > 0 for x< and f 0 (x) < 0 for x>. See the plot of f (x) x x below: x f (x) x x : Problem Consider the following function: f (x) x 4 x for x>0: Find f 0 (x) and show that f (x) is globally concave. Find the global maximum x explaining how you know it is a global maximum. If f (x) 0 then what is x? We have: f 0 (x) 4x and f (x) is globally concave since for all x: To solve for x note that: f 00 (x) 4 x 6 < 0: f 0 (x ) 0 4(x ) ) (x ) µ ) x : Since f (x) is globally concave x is a global maximum. (Note it is not enough to say that f 00 () < 0 since this only proves that x is a local maximum. )

15 CHAPTER. UNIVARIATE CALCULUS f (x) is neither globally increasing nor globally decreasing since f 0 (x) 4x µ 4 x 4 x 4 x (x ) so that f 0 (x) > 0 for x< since: x > (x ) while f 0 (x) < 0 for x> since x < (x ) : Let r be the root which satis es f (r) 0for r>0: Now: f (r) 0 ) r 4 r 0 ) r 4 r: Since r>0 we can divide both sides by ~x to obtain: r ) r The function f (x) is plotted below: µ 97:66: x f (x) x 4 x Problem Consider the following function: f (x) x x; for x>0: Find f 0 (x) and show that f (x) is globally concave. Find the global maximum x explaining how you know it is a global maximum. If g (x) + x and h (x) g x x ; then nd the x which solves the rstorder conditions for h (x) :

16 CHAPTER. UNIVARIATE CALCULUS Since: f 0 (x) x ) f 0 (x )0) (x ) 0 ) (x ) Similarly: ) x µ 8 : f 00 (x) x 4 < 0 (since x 4 > 0)so that f (x) is globally concave which in turn implies that x 8 is a global maximum. Note that f (x) is neither globally increasing nor decreasing since f 0 (x) > 0 for x< 8 and f 0 (x) < 0 for x> 8.Seethe plot of f (x) below: x f (x) x x : If g (x) + x then: Therefore: g 0 (x) > 0: + x x h 0 (x ) g 0 (f (x )) f 0 (x )0 {z } + ) g 0 (f (x )) (x ) 0 {z } f 0 (x ) ) x 8

17 CHAPTER. UNIVARIATE CALCULUS from work with f (x) above. Alternatively working directly with h (x) we nd that: h (x) g x x µ + x x x x x x + : Using the quotient rule it follows that: x x + x h 0 (x) x x x x x + x x x + x x x x + x x x + so that: h 0 (x ) 0 ) (x ) 0 ) x 8 : Problem 4 Suppose f (x) is given by: f (x) x x for x>0: Find f 0 (x) and show that f (x) is globally concave. Determine the rst-order conditions and use these to nd that x which maximizes f (x) :Prove that x is a global maximum. Is f (x) globally increasing or globally decreasing? If h (x) g ( p x x) where g (x) is monotonically increasing, nd the value of x which maximizes h (x) : We have: f 0 (x) x and f (x) is globally concave since for all x: f 00 (x) 4 x < 0:

18 CHAPTER. UNIVARIATE CALCULUS 4 To solve for x note that: f 0 (x ) 0 (x ) ) (x ) ) x () 4 : Since f (x) is globally concave x 4 is a global maximum. (Note it is not enough to say that f 00 4 < 0 since this only proves that x 4 is a local maximum. f (x) is neither globally increasing nor globally decreasing since f 0 (x) > 0 for x< 4 while f 0 (x) < 0 for x> 4 : Now: h 0 (x) g 0 p x x µ x using the chain rule. But since g 0 ( p x x) > 0 it follows that if h 0 (x )0 then (x ) 0or x 4 as above. Problem Consider the following function: f (x) 8x 4 x; for x > 0: Is f (x) a monotonically increasing or decreasing function? Is f (x) concave or convex? Using the rst order conditions nd any local maxima or minima of f (x) : What is the global maximum of f (x)? Explain. Since: f 0 (x) 6x 4 ) f 0 (x )06(x ) 4 ) (x ) 4 6 µ 4 ) x 6:

19 CHAPTER. UNIVARIATE CALCULUS Since f 0 (x) > 0 for x<6 and f 0 (x) < 0 for x>6 as illustrated in the graph below: x f 0 (x) the function f (x) is neither monotonically increasing nor decreasing. Since z} { + f 00 (x) z} { x 4 < 0 the function f (x) is globally concave with a unique global maximum at x 6: This is illustrated in the graph below: Problem 6 Consider the function: x f (x) 8x 4 x f (x) x a x b for x>0 where 0 <a< and b>: Show that f (x) has a unique global maximum x, ndx ; and show that f (x) is globally concave.

20 CHAPTER. UNIVARIATE CALCULUS 6 We have: f 0 (x) ax a bx b f 0 (x ) 0 ) a (x ) a b (x ) b 0 ) a (x ) a b (x ) b ) (x ) a (x ) b b a ) (x ) a b b a µ b ) x a b : a To show that f (x) has a unique global maximum at x note that: + f 00 z} { z } { z} { z } { (x) a (a )x a b (b )x b < 0 where a < 0 since a< and b > 0 since b>. Thereforef (x) is globally concave so that x b a a b Problem 7 Consider the function: + is a global maximum. f (x) Ax a Bx b for x>0 where 0 <a<; A>0; B>0 and b>: Show that f (x) has a unique global maximum x, ndx ; and show that f (x) is globally concave. We have: f 0 (x) Aax a Bbx b f 0 (x ) 0 ) A a (x ) a B b (x ) b 0 ) a (x ) a b (x ) b ) (x ) a (x ) b B b A a ) (x ) a b B b ) x µ B b A a A a a b : +

21 CHAPTER. UNIVARIATE CALCULUS 7 To show that f (x) has a unique global maximum at x note that: + + z} { f 00 z} { z } { z} { z} { z } { (x) A a (a )x a B b (b )x b < 0 where a < 0 since a< and b > 0 since b>. Thereforef (x) is globally concave so that x B b A a a b + + is a global maximum. Problem 8 Consider the following function de ned on x>0: f(x) x 4x x +; Use rst order conditions to nd the possible local maxima or minima, Use second order conditions to determine which are local maxima and which are local minima. Is the function convex or concave and what are the global minima and maxima? s We have: since x 4 Now: so that: f 0 (x) x 8x ) f 0 (x )0(x ) 8(x ) ) x 4 + p 46 0:788 p 46 0:849 is ruled out by the requirement that x>0: f 00 (x) 0x 8 f 00 (0:788) 0 0:788 8:6 > 0 and hence x 0:788 is a local minimum. The function is neither globally concave nor convex since f 00 (x) < 0 for x< 8 0 and f 00 (x) > 0 for x> 8 0 with x 8 0 and in ection point where f (x) changes from being concave to convex. +

22 CHAPTER. UNIVARIATE CALCULUS 8 This is illustrated in the plot below: x f (x) x 4x x + : From the graph it is clear that x 4 + p 46 is a global minimum and that f (x) has no global maximum. To prove that x 4 + p 46 is a global minimum note that: f 0 (x) x 8x µ x µ 4 µ µ p 4 46 x + p 46 and that x 4 p 46 > 0 for x>0: It follows that f 0 (x) < 0 for x< 4 + p 46 and f 0 (x) > 0 for x> 4 + p 46 so that x 4 + p 46 is a global minimum.. The Perfectly Competitive Firm The next two problems are based on the following information: Suppose in the short-run a perfectly competitive rm faces a price P; anominal wage W; a rental cost of capital R; and a xed amount of capital ¹K: The rm s short-runproductionfunctionrelatingq output to L labour is given by: Q f (L) L : Problem 9 What is the marginal product of labour? Show that this production function satis es the assumption of a diminishing marginal product of labour. Find pro ts as a function of L or ¼ (L) and show that ¼ (L) is globally concave. Derive the rm s demand curve for labour L and determine the elasticity of demand.

23 CHAPTER. UNIVARIATE CALCULUS 9 We have a positive marginal product of labour since: + MP L (L) f 0 (L) z} { L > 0 and a diminishing marginal product of labour since: + MPL 0 (L) f 00 (L) z} { L 4 < 0: 9 Pro ts as a function of L is found by replacing Q with L as: so that: ¼ 0 (L) PL W ¼ (L) PL WL R ¹K ¼ 00 (L) 9 PL 4 < 0) ¼ (L) is globally concave: From the rst-order conditions for pro t maximization we have for real wage w W P : ¼ 0 (L ) 0 P (L ) W ) (L ) W P w µ µ ) L w w : Since L has the form: y Ax b the elasticity of demand is the exponent on w or : Problem 0 If instead of the production function Q f (L) one writes L g (Q) ; then what is g (Q) given the rm s production function above? Use this to obtain the rm s cost function and show that the cost function is convex. Derive ¼ (Q) (pro tsasafunctionofq) and Q ; the rm s supply curve. By solving for L from Q L we have: Q L ) L Q :

24 CHAPTER. UNIVARIATE CALCULUS 0 The cost function is then found by replacing L with Q function of Q : to obtain costs as a Di erentiation of C (Q) then yields: C (Q) WL+ R ¹K WQ + R ¹K: MC (Q) C 0 (Q) WQ + ) MC 0 (Q) C 00 (Q) z} { 4 W Q > 0; ) C (Q) is globally convex. Pro ts are found by replacing WL+ R ¹K with C (Q) as: ¼ (Q) PQ WL+ R ¹K PQ C (Q) PQ WQ R ¹K so that letting p P W be the real price of Q : ¼ 0 (Q ) 0 ) P W (Q ) 0 ) (Q ) P W p µ µ ) Q p p : Since Q has the form y Ax b with p the 0 x 0 variable and Q the 0 y 0 variable, the elasticity of supply is the exponent on p or : Problem Suppose in the short run a perfectly competitive rm faces a price P and nominal wage W: The rm s short run production function relating Q output to L labour is given by: Q f (L) 9L : What is the marginal product of labour? Show that this production function satis es the assumption of a diminishing marginal product of labour. If one rewrote f (L) in the form: f (L) e a+b ln(l), then what are a and b? Derive ¼ (L) (pro tsasafunctionofl) and show that it is concave. Derive the rm s demand curve for labour and determine the elasticity of the demand curve. The marginal product of labour is MP L (L) f 0 (L) L :

25 CHAPTER. UNIVARIATE CALCULUS Since MP 0 L (L) f 00 (L) L < 0 there is a diminishing marginal product of labour. This is illustrated in the graphs below L f (L) L MP L (L) If a+b ln(l) f (L) e ) e a+b ln(l) e a e b ln(l) e a e ln(l) b e a L b 9L : Therefore e a 9or a ln(9)and b :

26 CHAPTER. UNIVARIATE CALCULUS Pro ts are given by replacing Q with 9L as: ¼ (L) P 9L WL so that: Now from ¼ 0 (L) PL W ¼ 00 (L) PL < 0 ) ¼ (L) is globally concave. ¼ 0 (L ) 0 P (L ) W ) (L ) W P w where w W P is the real wage. It follows that: (L ) w ) L (w) µ w : Since this is of the form y Ax b the elasticity of demand is : Problem If the rm s production function is: Q f (L) 9L nd L as a function of Q or L (Q) : Use this to obtain the rm s cost function when labour is the only factor of production and W is the nominal wage. Show that for this rm that marginal cost: MC (Q) is positive, that MC 0 (Q) is also positive and that the cost function is convex. From Q 9L it follows that: L Q ) L (Q) 9 µ Q : 9 Therefore cost as a function of Q is found by replacing WL with 9 Q as: C (Q) WL µ W Q : 9

27 CHAPTER. UNIVARIATE CALCULUS Thus: MC(Q) C 0 (Q) W so that marginal cost is positive. Since: C 00 (Q) MC 0 (Q) 6W µ Q > 0: 9 µ Q>0 9 for all Q it follows that C (Q) is globally convex and marginal cost is increasing. The is illustrated in the plot below for Q C (Q) W 9 Q : Problem Suppose a rm faces a cost function C (Q) Q 4 +0: Find the marginal cost function MC (Q) C 0 (Q) and the average cost function AC (Q) C (Q) Q : Show that the cost function is globally convex. Show that the average cost function AC (Q) is also globally convex and nd that Q at which average cost is at a minimum. If P 0then what will the pro t maximizing Q be? Marginal MC (Q) and average cost AC (Q) are: MC(Q) C 0 (Q) 0Q AC (Q) C (Q) Q Q4 +0 Q + 0 Q Q

28 CHAPTER. UNIVARIATE CALCULUS 4 and C 00 (Q) MC 0 (Q) 60Q > 0 for all Q so that C (Q) is globally convex. Now: AC 0 (Q) Q 0 Q AC 00 (Q) 0Q + 40 Q > 0 for all Q so that AC (Q) is globally convex as illustrated in the plot below: Q AC (Q) Q + 0 Q : To nd the minimum of AC (Q) note that: Now: AC 0 (Q ) 0 (Q ) 0 (Q ) ) (Q ) ) Q (8) 4 :688: ¼ (Q) 0Q C (Q) 0Q Q 4 +0 ) ¼ 0 (Q )00 0 (Q ) ) (Q ) 0 0 µ ) Q 0:797:

29 CHAPTER. UNIVARIATE CALCULUS Alternatively set P MC (Q) : Note that: ¼ 00 (Q) 60Q < 0 so that ¼ (Q) is globally concave and Q is a global maximum. ¼ (Q) is plotted below: Q ¼ (Q) 0Q Q 4 +0 Problem 4 Suppose that the rm has a production function Q f (L) L where 0 < <: Derive the rm s demand curve for labour L L (w) and show that the elasticity of demand is while the elasticity of supply is : We have: ¼ (L) PL WL ) ¼ (L ) P (L ) W 0 ) (L ) W P w w µ ) L w : The elasticity of labour demand is then the exponent on the real wage < 0 since <: Now Q f (L )(L ) µ w µ W P µ W P :

30 CHAPTER. UNIVARIATE CALCULUS 6 The elasticity of supply is the exponent on P : > 0 since 0 < <. Problem Suppose a rm has a production function given by: Q f (L) µ + L : Show that this production function is globally increasing and concave. Show that the marginal product of labour: MP L (L) is positive and diminishing. Find the pro t maximizing L when the price the rm receives is P and the nominal wage is W: Find the rm s labour demand curve and prove that the labour demand curve is downward sloping. Is the elasticity constant? We have: µ f 0 (L) + L µ L Since f 0 (L) (+L) +L L +L L (+L) it follows that: ( + L) > 0: f 00 (L) (+L) ( + L) < 0: Therefore f (L) is globally concave. Since MP L (L) f 0 (L) > 0 and MP 0 L (L) f 00 (L) < 0 by above the marginal product of labour in positive and diminishing. Since from pro t maximization we know that MP L (L )w or ( + L ) w ) ( + L ) w ) ( + L )w ) L w : Therefore: dl dw w < 0

31 CHAPTER. UNIVARIATE CALCULUS 7 so the labour demand curve is downward sloping. The elasticity is not constant since: (!) dl dw w L w w w or as illustrated in the plot below: w w w : (w) w w Thenextproblemsarebasedonthefollowinginformation: Suppose the rm s production function is given by: Q (L + K ) where L is labour and K is capital. In the short-run the amount of capital is given by K 6: Problem 6 Find the short-run production function. Show that the marginal product of labour is positive and that the diminishing marginal product of labour assumption is satis ed. The short-run production function is: Therefore: Q f (L) (L +6 ) L +8L +6: z} { MP L (L) f 0 (L) +4L > 0 +

32 CHAPTER. UNIVARIATE CALCULUS 8 and z} { MPL 0 (L) f 00 (L) L < 0: Problem 7 Assume that the rm is perfectly competitive in the goods market and the labour market. Derive the rm s demand curve for labour from the pro t maximizing condition and show that it is downward sloping. From the pro t maximizing condition + we have: +4(L ) w MP L (L )w W P ) (L ) (w ) 4 µ ) L 6 (w ) 4 (w ) : Note that this requires that w>: We then have: dl dw (w ) < 0: {z } + Problem 8 Find the rm s cost function C(Q), marginal cost and show that C 00 (Q) > 0: Solving Q (L +6 ) for L we nd that: Q (L +4) ) L Q 4 ) L Q 8Q +6: Replacing L in C WL+ R K ¹ with Q 8Q +6we obtain: C (Q) W Q 8Q +6 + R K ¹ ) MC(Q) C 0 (Q) W 4Q ) MC 0 (Q) C 00 (Q) WQ > 0:

33 CHAPTER. UNIVARIATE CALCULUS 9 Problem 9 Suppose that a rm s short-run production is given by: Q f (L) L ln(l 4 e 0 )80 ln L 6 ln( 4) where Q is output and L is the number of workers. Calculate the marginal product of labour and show that there is a diminishing marginal product of labour. Find the rm s demand curve for labour and the elasticity of labour demand. and Simplifying (in excruciating detail) we have: f (L) L ln(l 4 e 0 )80 ln L 6 ln( 4) L 4 e 0 ln(l)80+ 6 ln(l)+ln(p 4) L 4 e 4 ln(l)+ln() L 4 e 4 ln(l) e ln() ln(l) L 4 4 e L 4 L 4 L : Thus f (L) is a disguised Cobb-Douglas production function. Therefore: MP L (L) f 0 (L) L MP 0 L (L) f 00 (L) L < 0 so there is a diminishing marginal product of labour. From the pro t maximizing condition: we have: MP L (L )w (L ) w so that the elasticity of demand is : ) L w w Problem 0 Suppose in the short-run a perfectly competitive rm faces a price P; anominalwagew; a rental cost of capital R; anda xedamountofcapital ¹K: The rm s short-run production function relating Q output to L labour is given by: Q f (L) +L :

34 CHAPTER. UNIVARIATE CALCULUS 0 What is the marginal product of labour? Show that this production function satis es the assumption of a diminishing marginal product of labour. Derive the rm s demand curve for labour L and show that the demand curve for labour is downward sloping if W>P. Find the rm s cost function C (Q), calculate MC (Q) and show that C (Q) is globally convex. Would the rm ever produce where Q<? The derivations are a little easier if f (L) is rst simpli ed as: Q f (L) +L +L + L: The marginal product of labour is then given by: MP L (L) f 0 (L) +L and the assumption of a diminishing marginal product of labour is satis ed since: + MPL 0 (L) f 00 (L) z} { L < 0: To nd pro ts as a function of L replace Q with +L + L as: ¼ (L) P +L + L WL R ¹K ) ¼ 0 (L )0P +(L ) W ) (L ) W P w ) L (w ) (w ) : Therefore assuming w>: dl dw (w ) < 0: To obtain C (Q) note that: Q +L ) L Q so that replacing L with Q yields cost as a function of Q as: C (Q) WL++R K ¹ W Q + R K ¹ W Q Q + + R ¹K:

35 CHAPTER. UNIVARIATE CALCULUS Thus: MC(Q) W Q and C 00 (Q) WQ > 0) C (Q) is globally convex. Note that MC(Q) < 0 for Q< and since MC(Q) P>0 the rm would always produce at Q>:. Cost Minimization Problem Suppose that a rm has a Cobb-Douglas production function: Q L K and is faced with the problem of choosing L and K (weareinthelong-runnow!) to minimize cost of producing a xed Q or: WL+ RK: Since Q is xed, once L is chosen so too is K since: Q L K ) K QL ) K Q L Therefore we can write cost solely as a function of L as: Thus: C (L) WL+ RK WL+ RQ L : C 0 (L ) W RQ (L ) 0 ) (L ) Q W R ) L QW R : This gives the cost minimizing amount of labour L : To nd the optimal K calculate the required K need to produce Q as: K Q (L ) Q QW R QW R :

36 CHAPTER. UNIVARIATE CALCULUS The rm s long-run cost function is then: Note that C WL + RK WQW R + RQW R QW R + QW R QW R : which illustrates Shephard s lemma. dc dw QW R L dc dr QW R K Problem Suppose that a rm has a Cobb-Douglas production function: Q L K and is faced with the problem of choosing L and K (we are in the long-run) to minimize cost of producing a xed Q or: WL+ RK: Since Q is xed, once L is chosen so too is K since: Q L K ) K QL ) K Q L Therefore we can write cost solely as a function of L as: Thus: C (L) WL+ RK WL+ RQ L : C 0 (L ) W RQ (L ) 0 ) (L ) W Q R ) (L ) ( + ) W Q R µ µw ) L + R µ ) L + W + Q + + R + Q + :

37 CHAPTER. UNIVARIATE CALCULUS This gives the cost minimizing amount of labour L : To nd the optimal K calculate the required K needed to produce Q as: K Q (L ) Ã µ Q µ µ µ µ + W + R + Q + + W + R + Q + + W + R + Q ( + ) + W + R + Q ( + ) + W + R + Q + :.4 Utility Maximization! Problem Suppose that a household consuming apples Q and oranges Q gets utility: U 0:4ln(Q )+0:6ln(Q ) : The household has income Y 0; the price of Q is P and the price of Q is P so that the budget constraint is: 0 P Q +Q : Solve for Q as a function of Q and write utility then as a function of Q ; say f (Q ) : Find the utility maximizing Q from the rst-order conditions and show that Q is a global maximum. What is the elasticity of demand? From the budget constraint: so that Q 0 P Q P Q µ U (Q )0:4ln(Q )+0:6ln P Q

38 CHAPTER. UNIVARIATE CALCULUS 4 so that using the chain rule: U 0 (Q ) 0:4 P Q 0:6 P Q ) U 0 (Q )00:4 Q P 0:6 P Q ) 0:4 0:6 Q P 0 P Q ) 0:4 (0 P Q )0:6 P Q ) 4P Q ) Q 4 P 4P : The elasticity of demand is the exponent on P which is : Taking the second derivative we nd that: U 00 (Q ) 0:4 Q µ P 0:6 P < 0 Q so that U (Q ) is globally concave and hence Q is a global maximum. Problem 4 Suppose that a household consuming apples Q and oranges Q gets utility: U 0:ln(Q )+0:7ln(Q ) : The household has income Y; the price of Q is P and the price of Q is P so thattheincomeconstraintis: Y P Q + P Q : Find the utility maximizing Q. What is the elasticity of demand for Q and the income elasticity of demand for Q? First solve for Q as a function of Q as: Q Y P Q P Y P P P Q and use this to write U as a function of Q only as: µ Y U (Q )0:ln(Q )+0:7ln P Q : P P

39 CHAPTER. UNIVARIATE CALCULUS Di erentiating U (Q ) with respect to Q and using the chain rule for the second term yields: U 0 (Q ) 0: P 0:7 Q P Y P P P Q so that the rst-order conditions for utility maximization require: U 0 (Q ) 0 0: Q P 0:7 P Y P P P Q ) 0: Q P 0:7 P Y P P P Q ) 0: 0:7 Q P Y P Q ) 0:(Y P Q )0:7P Q ) 0:Y P Q ) Q 0:Y 0:Y P P : The income elasticity is the exponent on Y or while the own price elasticity is the exponent on P or : Problem Suppose that a household consuming apples Q and oranges Q gets utility: U ln (Q )+( )ln(q ) : where: 0 < <: The household has income Y; the price of Q is P and the price of Q is P so that the income constraint is: Y P Q + P Q : Find the utility maximizing Q. What is the elasticity of demand for Q and theincomeelasticityofdemandforq? What does the taste parameter determine? First solve for Q as a function of Q as: Q Y P Q P Y P P P Q and use this to write U as a function of Q only as: µ Y U (Q ) ln (Q )+( )ln P Q : P P

40 CHAPTER. UNIVARIATE CALCULUS 6 Di erentiating U (Q ) with respect to Q and using the chain rule for the second term yields: U 0 (Q ) P ( ) Q P Y P P P Q so that the rst-order conditions for utility maximization require: U 0 (Q ) 0 Q P ( ) P Y P P P Q ) Q P ( ) P Y P P P Q ) ( ) Q P Y P Q ) (Y P Q )( ) P Q ) Y P Q ) Q Y Y P P : The income elasticity is the exponent on Y or while the own price elasticity is the exponent on P or : Since: P Q Y the taste parameter determines the budget share of good : Thus if 0:4 then the household always spends 40% of its income on good : Problem 6 Suppose that a household consuming apples Q and oranges Q gets utility: U 0: p Q +0:7 p Q : The household has income Y; the price of Q is P and the price of Q is P so thattheincomeconstraintis: Y P Q + P Q : Find the utility maximizing Q. What is the elasticity of demand for Q and the income elasticity of demand for Q? First solve for Q as a function of Q as: Q Y P Q P Y P P P Q

41 CHAPTER. UNIVARIATE CALCULUS 7 and use this to write U as a function of Q only as: U (Q )0: p r Y Q +0:7 P Q : P P Di erentiating U (Q ) with respect to Q and using the chain rule for the second term yields: U 0 (Q ) 0: p P 0:7 q Q P Y P P P Q so that the rst-order conditions for utility maximization require: U 0 (Q ) 0 0: p P 0:7 q Q P Y ) p 0: P 0:7 q Q P Y ) (0:) Q µ P P P P P Q (0:7) ) (0:) µ Y P P P Q Y P P P P P Q P Q (0:7) µ P P ) Q (0:) Y P (0:) P P +(0:7) P P ) Q (0:) P P Y (0:) P +(0:7) P The income elasticity is the exponent on Y or : Q Problem 7 Suppose that a household consuming apples Q and oranges Q gets utility: U U (Q )+U (Q ) : where U 00 (Q ) < 0 and U 00 (Q ) < 0: The household has income Y; the price of Q is P and the price of Q is P so that the income constraint is: Y P Q + P Q : Find the rule for utility maximization. Show that Q and Q are normal goods. First solve for Q as a function of Q as: Q Y P Q P Y P P P Q

42 CHAPTER. UNIVARIATE CALCULUS 8 and use this to write U as a function of Q only as: µ Y U (Q )U (Q )+U P Q : P P Di erentiating U (Q ) with respect to Q and using the chain rule for the second term yields: U 0 (Q )U 0 (Q ) P µ Y U 0 P Q P P P so that the rst-order conditions for utility maximization require: U 0 (Q ) 0 U 0 (Q ) P µ Y U 0 P Q P P P µ ) U 0 (Q )P Y U 0 P Q P P P since Q Y P ) U 0 (Q )P P U 0 (Q ) P P Q : We can write this in turn as: U 0 (Q ) U 0 (Q ) P P or the household allocates goods so that the marginal utility of each good divided by its price is the same for both goods. To show that both goods are normal write Q (Y ); that is the demand for good as a function of income Y: From the rst-order condition then we have: U 0 (Q (Y )) P µ Y U 0 P Q (Y ) : P P P Di erentiating both sides with respect to Y and using the chain rule yields: U 00 (Q (Y )) dq (Y ) P µ Y U 00 P µ Q dy P P P (Y ) P dq (Y ) P P dy ) dq (Y ) P (P ) U 00 Y P P P Q (Y ) dy U 00 (Q (Y )) + P P U 00 P U 00 Y P P P Q (Y ) P U 00 (Q (Y )) + P U 00 Y P P P Q (Y ) Y P P P Q (Y ) > 0 since U 00 < 0 and U 00 < 0 implies that both the numerator and denominator are

43 CHAPTER. UNIVARIATE CALCULUS 9 negative. For Q (Y ) we have: dq (Y ) dy d dy µ Y P P P Q (Y ) P dq (Y ) P P dy P U 00 Y P P P U 00 (Q (Y )) + P U 00. Mark Maximization (Y ) ) > 0: Y P P P Q (Y P Q Problem 8 Suppose you are writing an exam with two questions. The mark you receive on question is M (t ) where: M (t )60( e t ) and t is thetimespentonquestion: Similarly the mark on question is: M (t ) 40 ( e t ) where t is the time spent on question : Show for question that the marginal mark is positive and diminishing. If the exam is hour long so that t + t ; nd the optimal amount of time t and t that you would spend on each question. What grade would you receive? For question : + z} { M 0 (t ) 60e > 0) marginal mark is positive + z} { M 00 ) 60e < 0) diminishing marginal mark. Since t t we can write the mark as a function of t alone as: so that: M (t ) M (t )+M ( t ) 60 e t +40 ( t) e M 0 (t ) 60e t 40e ( t) ) M 0 (t )060e t 40e ( t ) ) 60e t 40e ( t ) ) e ( t ) e ( t ) e t µ ) t ln ) t + µ ln 0:70:

44 CHAPTER. UNIVARIATE CALCULUS 40 Thus t t 0:70 0:97: Thus the student spends about 70% of the time on question (about 4 minutes) and about 0% ofthetimeonquestion (about 8 minutes). The student receives a grade of: M (t ) 60 e 0: ( 0:70) e 40:7 or a grade of about 4%: The grade as a function of t is plotted below: M (t )60( e t )+40 e ( t) Problem 9 Suppose you are writing an exam with two questions. Let t be theamountoftimeyouspendonquestion and t theamountoftimeyour spend on question : Let M (t )0:6ln(t ) bethemarkyougetonquestion and let M (t )0:4ln(t ) bethemarkyougetonquestion so that your total grade is: Let M 0:6ln(t )+0:4ln(t ) : T t + t be the total time you have to write the exam. Find t and t ; the optimal amount of time to spend on each question, and prove that what you nd is a global maximum. By replacing t with T t we have: M (t )0:6ln(t )+0:4ln(T t )

45 CHAPTER. UNIVARIATE CALCULUS 4 so that: M 0 (t ) 0:6 0:4 t T t ) M 0 (t )0 0:6 t 0:4 T t ) 0:6 t 0:4 T t ) 0:6(T t )0:4t ) t 0:6T: Since t T t we have: t 0:4T so that it optimal to spend 60% ofthetimeonquestion and 40% on question : t 0:6T is a global maximum since: M 00 (t ) 0:6 t so that M (t ) is globally concave. 0:4 (T t ) < 0 Problem 40 Suppose you are writing an exam with two questions. The mark you receive on question is M (t ) and the mark on question is: M (t ) where t isthetimespentonquestion: Find rule the optimal amount of time t and t that you would spend on each question. Show that the solution to the rst-order conditions is a global maximum if M (t ) and M (t ) are concave. Show that t and t are normal goods, that is if T increases (the instructor gives you more time) then both t and t increase. We have: t T t so that we can write the mark as a function of t alone as: M (t )M (t )+M (T t ) so that using the chain rule for M (T t ): M 0 (t ) M 0 (t ) M 0 (T t ) ) M 0 (t )M 0 (T t ) ) M 0 (t )M 0 (t )

46 CHAPTER. UNIVARIATE CALCULUS 4 where t T t is the optimal amount of time spent on question : Thus the student should allocate his or her time so as to equate the marginal mark in each question. Given that M 00 (t ) < 0 and M 00 (t ) < 0 we have: M 0 (t ) M 0 (t ) M 0 (T t ) ) M 00 (t )M 00 )+M 00 ) < 0 so that M (t ) is globally concave and the solution to the rst-order conditions is a global maximum. We can write t (T ) ; that is the optimal amount of time spent on question will depend on the amount of time available. This satis es: M 0 (t (T )) M 0 (T t (T )) : Di erentiating both sides with respect to T and using the chain rule we nd that: M 00 (t (T )) dt (T ) µ M 00 (T t (T )) dt (T ) dt dt ) dt (T ) dt M 00 (t M 00 (T t (T )) (T )) + M 00 (T t (T )) > 0 since given M 00 (t ) < 0 and M 00 (t ) < 0 both the numerator and denominator are negative. Furthermore since t (T )T dt (T ) dt we have: dt (T ) dt.6 Monopoly dt (T ) dt M 00 (t M 00 (t M 00 M 00 (T t (T )) (T )) + M 00 (T t (T t (T )) (T )) + M 00 (T t (T )) (T )) > 0: Thenexttwoproblemsarebasedonthefollowinginformation: Consider a monopolist who faces an inverse demand function of the form: P P (Q) ;P 0 (Q) < 0 so that unlike the perfectly competitive rm a monopolist must reckon with the fact that selling more will cause the price he receives to drop. Revenue for the monopolist is given by: R (Q) P (Q) Q;

47 CHAPTER. UNIVARIATE CALCULUS 4 so that marginal revenue is: MR(Q) R 0 (Q) : Suppose that the cost function of the monopolist is of the form: which is increasing and convex so that: C (Q) MC(Q) C 0 (Q) > 0 MC 0 (Q) C 00 (Q) > 0: Problem 4 What rule should the rm use to decide how much of Q to produce. Pro ts are given by: ¼ (Q) R (Q) C (Q) P (Q) Q C (Q) so that the rst-order condition for pro t maximization is: ¼ 0 (Q ) R 0 (Q ) C 0 (Q ) MR(Q ) MC (Q ) ) MR(Q )MC (Q ) : Problem 4 Use proof by contradiction to show that if P 0 (Q) < 0 then the monopolist who equates marginal revenue with marginal cost will produce less output than a perfectly competitive industry where price equals marginal cost. Let P M and Q M be the monopolist s price and output and let P C and Q C be the competitive level of output. Assume for the sake of argument that Q M Q C.SinceP 0 (Q) < 0 we have: Since MC 0 (Q) > 0 we have: Therefore: P (Q M ) P (Q C ) ) P (Q C ) P (Q M ) 0: Q M Q C ) MC(Q M ) MC(Q C ) : MR(Q M ) MC(Q M ) MC (Q C )P (Q C ) ) MR(Q M )P (Q M )+P 0 (Q M ) Q M P (Q C ) ) P 0 (Q M ) Q M P (Q C ) P (Q M ) 0 ) P 0 (Q M ) 0:

48 CHAPTER. UNIVARIATE CALCULUS 44 But by we know that P 0 (Q M ) < 0 so we have a contradiction. Therefore Q M <Q C and since P 0 (Q) < 0 we have: P M P (Q M ) > P (Q C )P C : Problem 4 If the inverse demand is P (Q) AQ for 0 < < and the monopolists technology is Q f (L) BL for 0 < <: Find the pro t maximizing level of output. so that: We have: ¼ (L) P (Q) Q WL A (BL ) WL AB L ( ) WL ) ¼ 0 (L )0 ( ) AB (L ) ( ) W ) (L ) ( ) W ( ) AB ) µ L ( ) W ( ) AB Q B (L ) Ã µ B W ( ) AB! ( ) : Problem 44 If the inverse demand is P (Q) Q ;Wand the monopolists technology is Q f (L) L nd the pro t maximizing level of output. We have: ¼ (L) P (Q) Q WL L 6 L ) ¼ 0 (L )0 6 (L ) 6 ) (L ) 6 ) L () 6 so that: Q (L ) () 6 () :

49 CHAPTER. UNIVARIATE CALCULUS 4 Problem 4 Given the demand curve: Q 0 P what is the elasticity of demand? Is demand elastic or inelastic? If Q (P ) 0 P + then what is the elasticity of demand when P?Prove that if revenue is given by: R (P ) P Q (P ) then for any demand curve R 0 (P) Q (P)(+ (P)) where (P ) is the elasticity of demand. If a rm maximizes sales (instead of pro ts) at P ; then what will the elasticity of demand be? The elasticity of demand is : Demand is elastic since < : If Q (P ) 0 P +Solution: then Q 0 (P ) 0 P so that: 6 and hence: (P ) Q0 (P ) P Q (P ) 0P 0P + () :: In general (P ) varies with P as shown below: 0 P Plot of (P) 0P 0P + Problem 46 Suppose a monopolist faces an inverse demand curve: P (Q) Q 4 and cost function C (Q) Q +0: What are the marginal revenue, marginal cost and average cost functions? Determine Q ; the pro t maximizing level of output for the monopolist and P ; the resulting price. Revenue and marginal revenue are given by: R (Q) P (Q) Q Q 4 ) MR(Q) R 0 (Q) Q 4 :

50 CHAPTER. UNIVARIATE CALCULUS 46 The inverse demand curve P (Q) and marginal revenue curve MR(Q) are plotted below: Q Marginal MC(Q) and average cost AC (Q) are: and are plotted below: MC(Q) C 0 (Q) Q AC (Q) C (Q) Q Q +0 Q + 0 Q Q Q MC(Q) and AC (Q) From MR(Q )MC (Q ) we have: (Q ) 4 (Q ) ) (Q ) + 4 µ 4 ) Q 9 0:6: 0

51 CHAPTER. UNIVARIATE CALCULUS 47 Therefore: P (Q ) 4.7 Price Discrimination à µ 4! :8: 0 Consider a rm (a monopolist) who is able to price discriminate between two markets; for example an airline which can charge one price for an economy fare to a tourist and another price for business class. Let P and Q bethepriceandquantitysoldtothe rstmarketandlet P and Q be the price and quantity sold to the second market. The inverse demand curves are given respectively by: P P (Q ) ;P 0 (Q ) < 0 P P (Q ) ;P 0 (Q ) < 0 so that revenue for the two markets is given by: R (Q ) P (Q ) Q ; R (Q ) P (Q ) Q and marginal revenue for market is given by: MR (Q ) R 0 (Q ) P (Q )+P 0 (Q ) Q P (Q ) P (Q )(+ (Q )) µ P (Q ) + ~ µ +P 0 Q (Q ) P (Q ) where (Q ) < 0 is the elasticity of the inverse demand curve which is ~ the inverse of the elasticity of the demand curve ~ : Similarly for market : MR (Q ) P (Q )(+ (Q )) µ P (Q ) + ~ where (Q ) < 0 is the elasticity of the inverse demand curve which is ~ the inverse of the elasticity of the demand curve ~ : Suppose that the rm has already chosen the pro t maximizing total level of output Q Q + Q : Problem 47 What rule should the rm use to decide how much of Q to allocate to the two markets; for example with an airline, how many seats should be economy and how many should be business class.

52 CHAPTER. UNIVARIATE CALCULUS 48 Since Q is xed so too are costs. The rm therefore needs to allocate Q amongst Q and Q so as to maximize the revenue it makes given by: R R (Q )+R (Q ) : We can write R as a function of Q alone using: so that: Q Q Q R (Q )R (Q )+R (Q Q ) : Di erentiating with respect to Q and using the chain rule to di erentiate R (Q Q ) with respect to Q we nd that : R 0 (Q ) R 0 (Q ) R 0 (Q Q ) MR (Q ) MR (Q Q ) ) R 0 (Q )0MR (Q ) MR (Q Q ) ) MR (Q )MR (Q ) where: Q Q Q : Thus the rst should allocate Q and Q between the two markets so as to equate marginal revenue in each market. This in turn implies that: P (Q )(+ (Q )) MR (Q )MR (Q )P (Q )(+ (Q )) or: P (Q )(+ (Q )) P (Q )(+ (Q )) or: P (Q ) P (Q ) ( + (Q )) ( + (Q )) + ~ + ~ : From this it follows that if ~ < ~ so that demand is more elastic in market than market ; or equivalently if (Q ) > (Q ) ; then P <P : Thus the rm charges the higher price to the market where demand is more inelastic or more price insensitive. Problem 48 Suppose that the inverse demand for economy seats and business class seats is given respectively by: P A Q 4, P A Q : What will P P ; the ratio of the price of economy seats to business class seats be? If Q 00then what will Q and Q be?

53 CHAPTER. UNIVARIATE CALCULUS 49 Here since the demand curves take the form y Ax b the elasticity of demand is constant for both markets with ~ 4 and ~ : We therefore have: P P + ~ + ~ so that P P : Thus economy seats will be two-thirds the price of business class seats. To calculate Q and Q we use: A Q ( ) A (Q Q ) ( ) µ ) (Q ) A ( ) (Q Q ) so that: (Q ) + ) (Q ) + µ A ( ) A ( ).8 Inverse Functions A ( ) µ A ( ) A ( ) Q µ A ( ) A ( ) µ Q A ( ) Q 0 A ( ) Q 0: Problem 49 If f 0 (x) > 0 and f 00 (x) < 0 so that f (x) is concave prove that the inverse function g (x) will be convex or g 00 (x) > 0: Di erentiating both sides of f (g (x)) x with respect to x and using the chain rule leads to: f 0 (g (x)) g 0 (x) : Since f 0 (g (x)) > 0 it follows that g 0 (x) f 0 (g (x)) > 0: Di erentiating both sides of f 0 (g (x)) g 0 (x) with respect to x and using the chain and product rules leads to: f 00 (g (x)) g 0 (x) + f 0 (g (x)) g 00 (x) 0 z } { z } { ) g 00 f 00 (g (x)) g 0 (x) (x) f 0 > 0: (g (x)) {z } + +

54 CHAPTER. UNIVARIATE CALCULUS 0 Problem 0 If f (x) µ px then what is the inverse function and what is the elasticity of the inverse function? If f (x) µ px x since: g (f (x)) f (g (x)) ) g (x) x x x x x x x x x: The elasticity of g (x) is the exponent on x or : Problem If the demand curve takes the form Q Q (P )AP what is the inverse demand curve P P (Q)?. What is the elasticity of Q (P ) and what is the elasticity of P (Q) and how are the two elasticities related? Here: P (Q) A Q : Since both Q (P ) and P (Q) have the form Ax b the elasticity of Q (P ) is while the elasticity of P (Q) is : Problem If a demand curve takes the form Q Q (P ) where f 0 (P ) < 0 show that it has an inverse demand curve P P (Q) : If (P ) is the elasticity of the demand curve and ~ (Q) is the elasticity of the inverse demand curve, show that: (P ) ~ (Q) ~ (Q (P )) ~ (P (Q)) :

55 CHAPTER. UNIVARIATE CALCULUS Problem If g (x) and f (x) are two functions, then what does the chain rule give as the derivative of h (x) f (g (x)) is? If g (x) is the inverse function of f (x) then what is f (g (x)) equal to? Under what conditions does an inverse function exist? What is the inverse function of f (x) x for <x<? What is the inverse function of f (x) x for 0 <x<? Show that f (x) x +x + for x>0 has an inverse function. Show that if f (x) is a globally decreasing and convex function then its inverse function g (x) is convex. We have using the chain rule: h 0 (x) f 0 (g (x)) g 0 (x) and if f (x) and g (x) are inverses then f (g (x)) x: To have an inverse we require f 0 (x) > 0 or f 0 (x) < 0 for all x in the domain of f (x) : If f (x) x for <x< then f (x) does not have an inverse since f 0 (x) x>0 for x>0 and f 0 (x) x<0 for x<0: For f (x) x with 0 <x< the inverse function is f (x) x since f 0 (x) > 0 for x>0: Now di erentiating both sides of f (g (x)) x with respect to x we obtain: f 0 (g (x)) g 0 (x) : Since f 0 (g (x)) < 0 it follows that g 0 (x) f 0 (g(x)) with respect to x we obtain: f 00 (g (x)) g 0 (x) + f 0 (g (x)) g 00 (x) 0 so that g (x) is convex. < 0: Again di erentiating z } { z } { ) g 00 f 00 (g (x)) g 0 (x) (x) f 0 > 0 (g (x)) {z } Problem 4 Suppose that for x>0 that f (x) is given by: f (x) x +x +: Show that f (x) has an inverse function g (x) and nd g (x). Since: f 0 (x) x +> 0 since x>0 + +

56 CHAPTER. UNIVARIATE CALCULUS it follows that f (x) has an inverse function. If y g (x) then f (g (x)) x or: f (y) y +y +x ) (y +) x ) y +x ) g (x) y x : A more careful analysis would explain why we take the positive and not the negative square root of x in the third line. Problem Suppose that for x>0 that f (x) is given by: f (x) x x +: Show that f (x) does not have an inverse function. Show how by restricting the domain of f (x) further you can allow for the existence of an inverse function. Since: f 0 (x) x so that f 0 (x) < 0 for x< and f 0 (x) > 0 for x> it follows that f (x) is not monotonic and hence has no inverse function. If we restrict the domain of f (x) so that x> (or 0 <x< will do too) then f 0 (x) < 0 and so f (x) is monotonic. If we de ne y g (x) then: f (y) y y +x ) (y ) x ) y x ) g (x) y x +:.9 Product, Quotient and Chain Rules Problem 6 Suppose that f (x) and g(x) are two monotonically increasing convex functions so that for all x; f 0 (x) > 0; g 0 (x) > 0; f 00 (x) > 0 and g 00 (x) > 0. Construct a new function h(x) g(f(x)): Show that h (x) is monotonically increasing and convex. We have from the chain rule that: h 0 (x) g 0 (f (x)) f 0 (x) > 0

57 CHAPTER. UNIVARIATE CALCULUS since g 0 (f (x)) > 0 and f 0 (x) by assumption. Di erentiating yet again we nd that: h 00 (x) g 00 (f (x)) (f 0 (x)) + g 0 (f (x)) f 00 (x) > 0 since: g 00 (f (x)) > 0; (f 0 (x)) > 0;g 0 (f (x)) > 0 and f 00 (x) > 0: Problem 7 If C (Q) is the cost function and AC(Q) C(Q)Q is the average cost function, show that average cost is increasing whenever marginal cost exceeds average cost. Di erentiating AC(Q) and using the quotient rule we nd that: dac(q) dq QC0 (Q) C (Q) Q Q C 0 (Q) C(Q) Q Q Q MC(Q) AC (Q) : Q Since the denominator Q is positive we have: dac(q) dq > 0, MC (Q) >AC(Q) dac(q) dq < 0, MC (Q) <AC(Q) : Problem 8 If h (x) f (g (x)) then what does the chain rule say h 0 (x) is? If h (x) F (g (x) ;g (x)) then what is h 0 (x)? Suppose Q f (L) where f 0 (L) > 0 and f 00 (L) < 0: If the demand for labour is: L g (w) de ned by: f 0 (g (w)) w; then show that g 0 (w) < 0: Problem 9 Suppose that W (t) is the nominal wage at time t and P (t) is the price level at time t so that the real wage at time t is: w (t) W (t) P (t) : The growth rate of w (t) is! (t) w0 (t) w(t) ; thegrowthrateofw (t) is (t) W 0 (t) W (t) and the growth rate of P (t) (i.e.,therateofin ation)is (t) P 0 (t) P (t) : Prove that:! (t) (t) (t) : using ) the quotient rule ) the product rule, and ) the chain rule. For example if nominal wages increase by 0% and in ation is 4% then real wages increase by 0% 4% 6%:

58 CHAPTER. UNIVARIATE CALCULUS 4 Using the quotient rule we have: w 0 (t) P (t) W 0 (t) W (t) P 0 (t) P (t) )! (t) w0 (t) w (t) P (t)w 0 (t) W (t)p 0 (t) P (t) W (t) P (t) W 0 (t) W (t) P 0 (t) P (t) (t) (t) : To use the product rule write W (t) w (t) P (t) and apply the product rule. To use the chain rule apply ln () to both sides of w (t) W (t) P (t) to obtain: ln (w (t)) ln (W (t)) ln (P (t)) and di erentiate both sides with respect to t using the chain rule. Problem 60 Suppose f 0 (x) > 0, f 00 (x) < 0; g 0 (x) > 0 and g 00 (x) < 0: If h (x) g (f (x)) then show that h 0 (x) > 0 and determine if h (x) is concave or convex. We have using the chain rule that: + z } { z } { h 0 (x) g 0 (f (x)) f 0 (x) > 0: Di erentiating again with respect to x we obtain: + z } { z } { z } { z } { h 00 (x) g 00 (f (x)) f 0 (x) + g 0 (f (x)) f 00 (x) < 0:.0 Polynomials and Taylor Series Problem 6 Consider the following functions de ned on x>0: + + f(x) x 4x x +: Use the rst order conditions to nd the possible local maxima or minima. Use the second order conditions to determine which are local maxima and which are local minima. Is the function globally convex or globally concave. Is x aglobal maximum or minimum?