DEPARTMENT OF ECONOMICS /11. dy =, for each of the following, use the chain rule to find dt

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1 SCHOO OF ORIENTA AND AFRICAN STUDIES UNIVERSITY OF ONDON DEPARTMENT OF ECONOMICS / MSc Economics PREIMINARY MATHEMATICS EXERCISE 4 (Skech answer) Course websie: hp://mercur.soas.ac.uk/users/sm97/eaching_msc_premah.hm d e d ( ) 1. Given e (a) d, for each of he following, use he chain rule o find d 5 e, (b) 4e, (c) 6e. Soluion (a) Use he chain rule or he naural eponenial funcion rule: g ( ) Given ( ) e g is a differeniable funcion of, he derivaive f, where ( ) ( ) is f ( ) e g g ( ). In his case, d 5 e 5 5e 5 d (b) d 4 e 1e d (c) d 6e 1e d. Wrie an eponenial epression of he value of Soluion (a) 1 coninuousl compounded a an ineres rae of 5% for ears; (b) 75 coninuousl compounded a an ineres rae of 4% for ears. (a) The formula of coninuous compounding is Pn P In his case, P 1, i. 5, and n..5( ) Hence, he eponenial epression is P n 1e If we are o obain a numerical value for he presen value,.5 ( 1e ) 1 ( ) P n abou 11.6 (b) In his case, P 75, i. 4, and n..4( ) Hence, he eponenial epression is P n 75e If we are o obain a numerical value for he presen value,.4 ( 75e ) 75 ( ) P n abou Wha is he insananeous rae of growh of in each of he following (a).7 e, (b). 1e, (c) e in. e. 1

2 Soluion (a) The insananeous rae of growh is defined as d.7. 7 dv r d V Here, e.7.7e d d.7.7 Divide b, d e. 7.7 e Hence r. 7 Answer: The insananeous rae of growh is 7 per cen. (b) (c) Alernaivel, Take log on boh sides, ln.7 ln e. 7 Take derivaive wih respec o, d ln.7 d Hence r. 7 Answer: The insananeous rae of growh is 7 per cen. d.. 1 e..6e d..6e Hence r.. 1e Answer: The insananeous rae of growh is per cen. d d.e 1.e.e Hence r 1.e Answer: The insananeous rae of growh is 1 per cen. 4. Differeniae each of he following wih respec o (a) e, (b) Soluion (a) B he produc rule d e + e e ( + 1) d d b+ c b+ c b+ c a b e + a e ae d (c) ln ln( 1+ ) d ( ) ( 1) d ln. 1+ b+ c ae, (c) (b) ( )( ) ( ) ( b +1) 1 ( + 1) H H and 5. Suppose populaion grows according o he funcion ( ) b consumpion grows according o C C a e. Use naural logarihms o find he raes of growh of (a) populaion

3 (b) consumpion, and (c) consumpion per head. Soluion (a) Take he naural log ln H ln H + b ln ( ) Differeniae he log funcion d ln H b ln( ) d Hence he insananeous rae of growh is ln( ) Check he resul: e, H H ( ) b which enables us o see ha ln( ) ln Since b. can be wrien as b ln ( ) ln( )b H H e H e, b is he insananeous rae of growh of H. (b) Take he naural log ln C ln C + a ln( e) ln C + a Differeniae he log funcion d lnc a d Hence he insananeous rae of growh is a. (c) Consumpion per head is defined as a C C e H H ( ) b Take he naural log a C C e a b ln ln ln( C e ) ln H ( ) lnc + a ln H b ln b H H ( ) Differeniae he log funcion C d ln H a bln( ) d The insananeous rae of growh is bln( ) ( ) ( ) a. 6. If is relaed o b relaed? k, how will he raes of growh of an be Soluion Take he naural log ln k ln Given ha an are boh funcion of Differeniae he log funcion wih respec o d ln d ln k d d or r k r where r and r are he insananeous rae of growh of an. Hence he rae of growh of is k imes he growh rae of.

4 7. Find he saionar values of he following funcions, and esablish wheher he are relaive maima or relaive minima or poins of infleion (a) , (b) 5 +, 1 (c) Soluion (a) Firs order condiion: d The funcion has a saionar poin a 1 Second order condiion: d 4 < The saionar value is a local/ relaive maima (b) Firs order condiion: d The funcion has a saionar poin a 1 Second order condiion: d 1 > The saionar value is a local/ relaive minima. 4

5 (c) Firs order condiion: d + 1 ( 1) The funcion has a saionar poin a 1 Second order condiion: d d A 1, Since we canno eclude he possibili ha he funcion has a local maima or minima, he nh order derivaive es is carried ou. Third order derivaive: d Since he non-zero higher order derivaive is obained a he rd order (odd number) derivaive, he saionar value is a poin of infleion. 5

6 e T φ( ) be a oal funcion (for eample, oal cos or oal produc). (a) Wrie epressions for he marginal funcion, M, and he average funcion, A. (b) Show ha when A reaches a relaive eremum, M A. (c) Wha general principal does his sugges for drawing marginal and average funcions in he same diagram? (d) Evaluae he elasici of he oal funcion, T, a he poin where A reaches an ereme value. Soluion d T (a) M φ ( ) T φ( ) A (b) A reaches relaive eremum a da φ ( ) φ( ) (firs order necessar condiion) d This can be rewrien as φ ( ) ( ) φ which implies M A Q.E.D. (c) In a graph of average and marginal funcions, he marginal curve cus he average curve a is minimum or maimum poins. See eamples below for cases of producion and cos funcions. (d) Elasici of T can be wrien as 6

7 dt M ε d T T A A A s relaive eremum M A. Hence ε 1. T Noe: Eample of producion funcion: e us verif he above resuls using a producion funcion Given he oal produc (producion) funcion TP 9 The average produc is given b TP AP 9 and he marginal produc is dtp MP 18 d TP MP AP From he graph we can see ha (1) MP increases when TP is conve and increasing a an increasing rae, is a a maima where TP is a is inflecion poin, and decreases when TP is concave and increasing a a decreasing rae. () When TP is a is maima MP is zero (necessar condiion of local maima). () When AP is a is maima MP AP. This confirms he resul in (b). (4) When AP is a is maima and MP AP, he slope of he line from he origin o he TP curve is angen o he TP curve. Wh is his? The slope of he line 7

8 9 from he origin o he TP curve is defined as, which is he AP. The slope of he line angenial o he TP curve is nohing bu he MP. Hence his is simpl confirming ha MP AP. dtp dtp MP (5) Since he elasici of TP is d, a MP AP, elasici d TP TP AP is one. This confirms (c). Eample of cos funcion: Alernivel we can confirm he resul using oal/average/marginal cos funcions. Given he oal cos funcion TC Q + 7Q + 1 The average cos is given b TC 1 AC Q Q Q and he marginal cos is dtc MC 6 Q + 7 dq TC Q + 7Q AC 1 Q Q MC 6 Q From he graph we can see ha when AC is a is maima and MC AC, he slope of he line from he origin o he TC curve is angen o he TC curve. 8

9 9. A perfecl compeiive firm has a single variable inpu (labour) wih wage rae W per period. Is fied inpus cos he firm a oal of F per period. The price of he produc is P. Soluion (a) (a) Wrie he firm s (i) producion funcion, (ii) revenue funcion, (iii) cos funcion, and (iv) profi funcion. (b) Wha is he firs-order condiion for profi maimizaion? (c) Wha economic circumsances ensure ha profi is maimized? (i) Since he funcional form is no specified, he producion funcion can be wrien in general form as Q q K, ( ) where K is fied. (ii) The revenue funcion can be wrien as R PQ Pq( K, ) (iii) The cos funcion can be wrien as C F + W (iv) The profi is given b π R C Pq( K, ) ( F + W) (b) Firs order condiion dπ Pq ( ) W d or q ( ) P W W q ( ) P which implies ha he marginal produc of labour is equal o real wage a he saionar poin. (c) The second order sufficien condiion for profi maimizaion requires Q d π d Pq ( ) < d Q This implies ha for P >, q ( ) <, ha is, here is a diminishing d marginal produc of labour, which implies a sricl concave producion funcion. 9

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 6 SECTION 6.1: LIFE CYCLE CONSUMPTION AND WEALTH T 1. . Let ct. ) is a strictly concave function of c

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 6 SECTION 6.1: LIFE CYCLE CONSUMPTION AND WEALTH T 1. . Let ct. ) is a strictly concave function of c John Riley December 00 S O EVEN NUMBERED EXERCISES IN CHAPER 6 SECION 6: LIFE CYCLE CONSUMPION AND WEALH Eercise 6-: Opimal saving wih more han one commodiy A consumer has a period uiliy funcion δ u (