Errata and hanges for Leture Note (I would like to thank Tomasz Sulka for the following hanges): Page 5 of LN: f f ' lim should be g g' f f ' lim lim g g ' Page 8 of LN: the following words (in RED) have been added to make eplanations better The above equation shows that points on the urve of f() an be epanded by the point A and its derivatives Note that both definitions give you the same results with slightly different meanings: The first one shows how to represent the nearby points ff( + ) around ff(); the seond one shows how to represent the neighbour points ff() around ff( ) with differene denoted by You would use both definitions in your ore/option modules Please see the file QM Review of -Variable Calulus_7_98pdf for updated version from LEARN or module website Brief answer to eerise f ' ( + ) + + ; f ' ; 4 + ( + ) ( + f ' e + e ) ( ) f ' ( ) ln ( ) + (a) f ' ; f '' b (b) First take log on both sides of f a We have ln f b ln a Now it is easier to differentiate: d ln f d( b ln a) df df b( ln a) b( ln a) f b( ln a) a b d d f d d The seond derivative is then followed: d df df b b( ln a) b ( ln a) a d d d () df g ( + g' ) e + Note that hain rules involved d + g d f d[ ( + g' ) e ] + g + g g'' e + ( + g' ) e d d (d) df 4 + d( + ) d + + Thus
df 4 + d + nd derivative: an eerise for you (e) This is about quotient rule If you forget about it, you ould always use produt rule instead: f e ( + ) df e e ( + ) d ( + ) nd derivative: an eerise for you The usual trik for omputing growth rates are taking log and differentiating with respet to time Taking log of prodution funtion gives ln y ln a + ln k and differentiating both sides with respet to t gives the growth rate of prodution dy da dk funtion: + + 4 4 y dt a dt k dt 4 The first order ondition (FOC) gives f ' ( ) or The SOC gives f '' f '' < and f '' > Thus, we have a loal min at and a loal ma at (Why not a global one?) 5 First alulate the first order ondition: f ' The seond order ondition: f '' ( ) f '' < for all domain of Thus, the funtion f has a global maimum at For the minimum of f, we need to ompare the values at endpoints f() 5 7 Global minimum 55 7 Global minimum Global maimum
(a) This leads to a definition of the elastiity of demand, ε D, as: ( ) dy p Y Z p ε D ; Y dp Yp' ( ) Y Z if the elastiity is defined as positive number R py ZK r Z (b) The FOC for K: ZK r K K The SOC Z r onfirms that this is profit maimisation problem 7 (a) e d e d de e + d d( + ) (b) d ( + ) ( + ) + + + ln ln 4 4 5 5 () ( + ) d ( + ) d( + ) d( + ) ( + ) + 5 5 dy dy da (d) Let y a Take log and differentiate: ln ad yd a d y ln a ln a da a Thus, a d + ln a ln a ln (e) d d d + ln ln ln ln (f) Need to use the integration by parts: d ( ln ) ln d + d Integrating both sides gives ln ln d + d + ln d ln d ln 4 8
8 8 8 ( 8 ) 8 8 4d 4 d d ; T T T T t t t t T e ρ ρ ρ ρ ρ ρ ρ ρ ρ T ρ e dt e dt de e ( e e ) e d de e 9 d 4 d ; 4 4 4 4 y 4 e d e d de e +, sine We have y e + y e + + e This one is a bit hard at the first sight This is standard intertemporal revenue integral, where g denotes produtivity fator of prodution funtion and N denotes the number of employees The differential equations for g and N are: dg dg dn dn η ηdt ; λ λdt g dt g N dt N Suppose that the initial values for g and N at t are represented by g and N Integrating both sides of equations gives, similar to the eample on page 8 of LN : ηt λt g g e and N N e, Substituting into the integral gives θ η t θ λθt ρt θ ( ρ + λθ η ) t gn ( ge N e e dt N N, g gn e dt, ρ + λθ η with N N and g g Of ourse we need the assumption of ρ + λθ η > The first dividend is at the end of the year, thus *5, and grows the same rate after Or more general: d( + d( + d( + P + + + ( + ( + ( + where d is the dividend at the beginning of the year, g is dividend growth rate, and r is the disount rate Multiplying both sides by (+/(+ gives ( + d( + d( + d( + P d + + + + d + P ( + ( + ( + ( + Thus, we have ( + + r r g d 5 P d P d P 4, + g + g r g 5 whih is the share priing by onstant Gordon growth model in undergraduate finane 4
The equilibrium prie is q and p p 4 q The onsumer surplus is omputed by the area below the demand urve (p4-q/) and above the p, integrating from q to q: q q q CS 4 dq d q q q Sine it is a linear system, we an just also omputer the area by (4-)*/ For a funtion f f ( ) f + f ' + f '' ( ) q, the seond-order Taylor s epansions are denoted by + Let f,, and 57 We then have f ( + ) f ( 57) 57 + ( ) 8 + 57 ( 57) 8 + 85 4 84 It an be seen that for small, the first order approimation is already quite aurate Note that you an also use f f ( ) + f ' + f '' to! obtain the same results: f f ( 57) 57 + ( ) ( ) + 57 ( 57) 8 + 85 4 84 8 5
4 For f f ( ) + f ' + f '', we need to substitute f at! bak into the taylor s epansions Anyway, do the first and seond derivatives we have f e e + 4 e! Substituting gives : f 5 The onstant growth for labour input gives (tt) nnnn(tt) Take log and differentiate the relationship of kk(tt) KK(tt) with respet to time, LL(tt) (tt) (tt) (tt) kk(tt) KK LL Substituting kk(tt) KK(tt), (tt) nnnn(tt), and (tt) ss (tt) ββ LL(tt) ββ δδδδ(tt) bak LL(tt) to the above equation gives (tt) kk(tt) kk(tt) KK(tt) ss KK(tt) ββ LL(tt) ββ δδδδ(tt) nn (tt) ss KK(tt) ββ LL(tt) ββ nn δδ (tt) ss kk(tt) ββ (nn + δδ)kk(tt) By using the linear algebrai definition, we an know diretly by just looking at the diagram that the funtion is both quasionave and quasionve Or we an let a and b be two different values of assume that f (a) a f (b) b, whih implies a b By the definition on page : f is ( ) ( )( ) f ' f ' b for f ( ) f ( ) We have f ' f ' a Thus, the funtion is both quasionave and quasionve quasi - onave iff quasi - onve a b b( a b) a b a( a b) 7 This question is easy Here it is used for the preparation of Leture note 5 The optimal point is However, the funtion is faing a onstraint This implies that is the maimum/etreme point Note that at the maimum point of this simple onstrained problem implies and f '( ) ( + ) <, whih shows the ondition of the Kukm-Tuker Theorem The easiest way to see this is to draw the diagram for the funtion 8 First and seond order derivatives give u'( ) σ Therefore, we have u'' σ σ and
σ σ u'' ( ) u' ( ) σ σ As σ approahes, both σ and σ are approahing We need l'hôpital's rule σ f f ' ln lim lim lim lim ln, g g' σ σ See page 5 of leture note for details of a similar ase 7