ESCUELA LATINOAMERICANA DE COOPERACIÓN Y DESARROLLO Especialización en Cooperación Internacional para el Desarrollo

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1 ESCUELA LATINOAMERICANA DE COOPERACIÓN Y DESARROLLO Especialización en Cooperación Internacional para el Desarrollo A SURVIVAL KIT IN CASE O.MATHEMATICS b Marco Missaglia * Universit of Pavia September * Tese notes are tougt for tose wo need to know te ver basis of matematics for instance because te ave to better understand economics but do not ave time to go troug a serious book of matematics. Tat said, remember tat a serious book of matematics is alwas te best solution!

2 unctions A function is a relation between two or more variables. We ma sa, for eample, tat te income level enjoed b a person in a mont depends for a given level of te prevailing ourl wage rate - on te number of ours e/se as worked during tat mont: te more te ours of work, te iger te income. Or, again, we ma sa tat te demand for potatoes in a given period depends on is a function of te price of potatoes, te income of potential consumers, te price of goods tat satisf te same need and could be consumed instead of potatoes, etc. As te ave been written, te first is an eample of a function of one variable, te second of a function of more tan one variable. A bit more formall, we could write: Income f ours of work and Demand for potatoes g price of potatoes, consumers income, price of substitutes, Tese epressions ave to be read, respectivel: income is a function f is te tpe of function, see below of te ours of work and te demand for potatoes is a function g is anoter tpe of function, see below of te price of potatoes, consumers income, etc.. Te variable on te left of te equalit sign Left Hand Side: LHS, i.e. te variable we want to eplain is called te dependent variable, wereas te variables on te rigt and side RHS are called te independent variables. Let us start our stud from te simplest possible case, te case of functions of onl one variable. unctions of one variable On top of te eample given above, tere are several possible cases of function of one variable. or a given tecnolog, te consumption of fuel of a plane depends on te distance from te departure to te arrival place; m annual epense in car insurance basicall depends on te tpe of car I own; etc.. Plane consumption f Distance Insurance epense f Tpe of car In general, a function of one variable is written as f, were is te quantit measuring te level of dependent variable litres of fuel, dollars per monts, etc. and is te quantit measuring te level of te independent variable kilometres, ours of work, etc.. Matematicall tere are so man possible functions relating to ; for eample:

3 - log and so on and so fort. One can build millions of eamples of tis kind. Wen ou stud a function, i.e. te relation between some variables, te first question tat naturall arises is: wat appens to te dependent variable wen te independent variable increases decreases? Will it increase decrease as well or, viceversa, will it decrease increase? Answering tis question means calculating te derivative of wit respect to. Tere are several smbols used to write derivative of wit respect to. Here are te most widespread: ; f ; ' f ; f In our books of matematics ou will meet all tese smbols, but do not worr, te all ave te same meaning: te derivative of wit respect to. Now let s turn to a more substantive and complicated issue, ow to calculate a derivative. Te derivative of a function Wen an increase decrease of prompts an increase decrease of, ten te grap of te function will be someting like: igure a b c In all tese cases, te iger lower te iger lower. So, a positive relation between and is associated to an increasing curve in te grap. On te contrar, wen an increase decrease of prompts a decrease increase of, ten te grap of te function will be someting like:

4 ig. a b c In all tese cases te curve relating and in te grap is a decreasing curve. Te foregoing is still too general to be useful in te calculation of te derivative. To tis purpose, let us remind te kind of question we want to answer: wat appens to te dependent variable,, wen te independent variable increase? igure A natural wa of measuring te reaction of to a variation in is to calculate te ratio between te variation in prompted b a variation in and te variation in. ormall, for te function f we ave:

5 f f f o f were is noting but te distance between and. Te ratio is called incremental ratio. In te case illustrated in igure, it can be seen tat > and > and terefore te incremental ratio is positive, being te ratio between two positive quantities. We can terefore conclude tat in te case of an increasing function igure, te incremental ratio is positive. A rise decline in prompts a rise decline in. Wat appens wit a decreasing function? igure It is eas to see tat in tis case f f f o f since < te numerator is negative wereas > te denominator is positive. We can terefore conclude tat in case of a decreasing function, te incremental ratio is negative. A rise decline in prompts a decline rise in. rom te foregoing it follows tat te incremental ratio is a roug measure of te slope of a function: a positive incremental ratio signals a positive slope and viceversa. However, it is a not a precise measure of te slope, since in general te slope of a function is not constant. Consider for instance igure. Te slope is negative bot in te point, and in te point,, but it is clearl lower more negative in te latter as can be seen b te tangent lines drawn in igure. So, if we want to 5

6 measure more precisel te slope we ave to do it in a single point of te function rater tan over an interval of te function itself. Calculating te incremental ratio, as it sould be alread clear, means measuring te slope over an interval: an interval of lengt. As a consequence, if we want sa to measure te slope in te single point, we ave to reduce so tat te interval collapses into a single point. More formall: we ave to take te limit of te incremental ratio for progressivel tending to zero. Tis wa, we will ave calculated te derivative of f in one of its specific points. We are no read to understand tis etremel important definition: Te derivative of te function f in a specific point, is defined as lim f f In words: te derivative of wit respect to calculated in is equal to te limit for tending to zero of te incremental ratio calculated in. Eamples Calculate te derivative of te function Let s appl te previous definition lim f f lim lim wic simplifies to lim It is eas to see tat te derivative we ave just calculated is positive for eac possible value of. or instance, for - we ave a derivative equal to ; for a derivative equal to, etc. We can conclude tat te slope of te function is alwas positive, for eac possible point. Or, to put it in oter words, tis function is alwas increasing. Calculate te derivative of - lim lim lim In tis case te value of te derivative does not depend on. Tis means tat in eac point te slope of te function is te same, -. It is a straigt line wit a negative slope. 6

7 Calculating te derivatives as we ave just done, i.e. b appling te formal definition is etrel boring and in some cases quite difficult as well. Do not worr: instead of appling te definition, we can use te following rules of derivation. Some useful rule of derivation iven a function f n te corresponding derivative is n n Eample: take te function. Te application of te derivation rule gives Tis result is not surprising at all: tis is eactl wat we got in Eample in te previous section b appling te general definition of derivative. iven a function f α te corresponding derivative is α Eample: take te function -. Te application of te derivation rule gives Again, te result is not surprising at all: tis is eactl wat we got in Eample in te previous section b appling te general definition of derivative. iven a function f α n te corresponding derivative is n α n, wic is noting but a combination of rules and. Eample: take te function -. Te application of te derivation rule gives 7

8 8 iven a function f k were k is a constant, te corresponding derivative is. Tis result is not surprising at all. Indeed, remember tat te derivative calculated in a point measure te slope of te relevant function in tat point. Now, b looking at te grap of a function like k wic means tat irrespective of te value of, alwas takes te same value k k it is immediatel clear tat te slope of te k line is alwas nil. 5 iven te function f log te corresponding derivative is Te five derivation rules we ave just illustrated are enoug to understand most of te economic applications we will deal wit. However, for a better understanding of tese applications, te ave to be complemented wit some sub-rules wic are noting but a furter elaboration on te five main rules discussed above. A Te derivative of a sum difference Consider te function f g. Te corresponding derivative is 8

9 9 g f In words: te derivative of a sum is equal to te sum of te derivatives Eample: - We ave g and -. Te application of te derivation rule gives B Te derivative of a product Consider te function f g.. Te corresponding derivative is g g f Eample: [log ] were g log and. Te application of te derivation rule gives: ] log [ log. C Te derivative of a quotient Consider te function f g/. Te corresponding derivative is ] [ g g Eample stupid but useful: / were g and Te application of te derivation rule gives Anoter eample less stupid, as useful as before. Take te function log /. Te application of te derivation rule gives:

10 log log log log D Te derivative of a composite function Consider te function -. Tis function can be tougt of as composed of two sub-functions. To see w, let us define w w -, wic is clearl a function of, and g gw w, wic is again a function of w, wic in turn is a function of. Our original function, -, ma be terefore rewritten as gw, since gw g- -. In general, given a composite function gw, te corresponding derivative is g w w Eamples: iven -, te corresponding derivative is w. w, were we used te definition w - iven log, let us define w and g log w. Te application of te derivation rule gives. w E Te teorem of te implicit function Imagine ou ave a function f,. or instance; or log, etc. Tis kind of function is said to be implicit since it implicitl fies as a function of. or instance, log can be rewritten as log or, again /log, were is clearl a function of. Is it possible to calculate te derivative of wit respect to witout making an eplicit function of, but using te implicit form? Yes, it is. It can be sown tat given te implicit function f,

11 f f Eample: for log we ave Derivatives and percentage canges More or less anone is able to calculate a percentage cange. Imagine tat toda te temperature is degrees and tomorrow. We ma sa tat tere will be a percentage increase of Variation/Original level / / 5/ 5% Te same stor could be said differentl: tere is a variable, in tis case te temperature, tat canges as time goes on. In oter words, tis variable is a function of time, ft. We alread know tat a derivative measures te variation of te dependent variable; ence te percentage cange of suc a variable can be epressed as Variation Orig. Level t were a dot over a variable indicates its time derivative. Tis formula is etremel useful and widel applied in economics. Optimisation It often appens in economics to meet some function tat as to be maimised or minimised in general: optimised. or instance, a firm wises to maimise its profits or minimise its costs, bot of tem being a function of te quantit produced. Consider te case of a firm wose structure of cost is given b a fied cost component sa, equal to C. B definition C is a constant and a variable cost component, wose amount depends on te quantit produced: for instance, VC a.q, were Q is te output produced and a is a constant parameter Q is squared to capture te well establised idea in economics of decreasing returns of variable factors, see below. Total costs are te sum of fied and variable costs, i.e. TCQ C VC Q C aq Clearl, total costs are a function of Q, and tat s w we wrote TCQ instead of putting simpl TC. Wat is te average total cost ATC for tis firm, i.e. te average cost of producing one unit of output? Obviousl, we ave

12 ATC Q TCQ/Q C/Q aq Well, wat is te amount of Q suc tat te average total cost is minimised? Of course, tis is a ver relevant information for a firm tring to make as muc mone as possible. To answer our question we ave to calculate te derivative of ATC wit respect to Q and ten set te derivative equal to zero. W? Consider te generic functions drawn in te diagrams below b Tis function is decreasing until b and ten it is increasing. In oter words, it reaces a minimum wen b. To put it differentl: till be, te slope of te function is negative, from b onward te slope is positive and for b te slope is zero. But we alread know tat te slope a function in a point is noting but its derivative in tat point! Hence, a necessar condition for getting te minimum of a function is to put its derivative equal to zero. Mutatis mutandis, te same rule applies for getting a maimum. In general: A necessar condition to get an optimal point maimum, minimum of a function is to calculate its derivative and ten put it equal to zero. Let s go back to our problem of ATC minimization. It is eas to see, simpl b appling te rules we learnt before, tat te derivative of ATC Q TCQ/Q C/Q aq C.Q - aq is ATC Q C Q C a a Q Now let s put tis derivative equal to zero and solve for Q. We ave C aq C Q Q a C a So, wit C 9 and a 9, we get Q ; wit C 9 and again a9, we get Q, etc.. Not surprisingl, a firm wit iger fied costs must produce more in order to minimise te average cost of producing one unit of output.

13 unctions of two or more variables Remember te eample we gave at te ver beginning: Demand for potatoes g price of potatoes, consumers income, price of substitutes, In general, most of te relevant economic events ave to be described as functions of two or more variables. To give anoter, classical eample: te output produced b a firm is a function of te quantities of inputs emploed in te production process. Let us call Q te output produced, L te quantit of labour emploed b te firm and K te amount of capital maciner and te alike emploed in te production process. Terefore we ma write a generic production function: Q fl, K Q In tis new and ricer framework, wat is te meaning of derivative? or instance, wat is? In L te ligt of wat we learnt before it is ver tempting to sa tat tis is tis quantit is te variation of output Q induced b a small variation in L, te quantit of labour. True, but incomplete: it is te variation of output induced b a small variation in te quantit of labour for a given quantit of capital. Q Tis is w it is called te partial derivative of Q wit respect to L. Equivalentl, is te partial K derivative of Q wit respect to K and measures te variation of output induced b a small variation in te quantit of capital for a given quantit of labour. And ow can te partial derivatives be calculated? It is not difficult at all. Consider for instance te partial derivative wit respect to L. We know tat in tis case te quantit of capital is fied, i.e. K is to be considered constant wic greatl simplifies te calculation. To be concrete, let s give te production function a specific form Q L α K β Since te quantit of capital is constant we can put β K A, wit A constant. Hence α Q AL It is now etremel eas to calculate te relevant derivative. Just appl te eas rules we learnt in te stud of te functions of one variable: Q L AαL K al α β α Q α L Wat about te partial derivative of Q wit respect to K? Now it s L tat as to be considered constant, α so we can put L B, wit B constant. Hence: Q K BβK α L βk β β Q β K

14 Te notion of total differential Te eercises we ave done so far are in a sense artificial. Usuall te variation of te quantit produced is due to bot canges in L and K. Terefore we sould write someting like: Total variation of Q Variation of Q induced b variation in L Variation of Q induced b variation in L ormall: Q Q dq dl dk L K Te epression above is called total differential of Q. B plugging into tis epression te partial derivatives we calculated before, we get Q Q dq α dl β dk L K or αq dl L βq dk K dl dk Q α β L K dq Q dl dk α β L K In words, and not surprisingl: te percentage cange of output left and side is a weigted average of te percentage canges of labour and capital, were te weigts are given b te powers of te production functions. In economics, one of te most useful application of te concept of total differential is related to tose economic models epressed in forms of sstem of equations. Consider a sstem of two equations in two unknowns, and. In bot equations a parameter z appears:,, z,, z Imagine tat we want to know wat appens to due to a variation in z, te eogenous parameter. To calculate te relevant derivative, totall differentiate te sstem -. You get: d d dz d z d d dz d z Now, from we ave:

15 5 d dz Z d 5 Now just put 5 into dz z d dz Z d 6 or dz z z d 7 from wic we get wat we are looking for z z z z dz d 8 Te algebra if tis formula is a bit intimidating, but a simple eample will sow tat tings are simpler tan te seem to be. Consider te sstem z z B straigtforward application of te derivation rules we alread know, we ave 6, z -, ; -, z and. Now just appl formula 8:

16 6 dz d Note tat in order to calculate tis derivative we did not solve te sstem of equations: tis is te great advantage of te differential, it elp calculate a derivative witout eplicitl solving a sstem of equations wic sometimes is ver difficult, not to sa impossible, to solve.