Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.

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1 Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while the vribles K nd L usully represent the quntity of cpitl nd lbor, respectively Although the specific vlue is not known the rnge of vlue is usully restricted, either by ssumption or economic intuition or logic For emple ll of the bove vribles will hve non negtive vlue, ie number greter thn or equl to zero A function is mthemticl eqution tht reltes two or more vribles together There re two types of functionl forms: generl functionl form nd specific functionl form A generl functionl form reltes the vribles together in non-specific formt For instnce, the generl functionl form for the production function is Y F( K, L) which reds Y is function of K nd L In this cse, it is known tht both cpitl nd lbor ffect the level of output but not the ect mnner in which they do so A specific functionl form reltes the vribles in precise formt There cn be mny different specific functions for generl function For emple, the Cobb-Dougls production function Y K L α α, where α (lph) is constnt between 0 nd, is one specific functionl form for the generl production function Y F( K, L) In the contet of n economic model, we mke distinction between two kinds of vribles: endogenous vribles nd eogenous vribles An endogenous vrible is vrible tht n economic model tries to eplin (determined within the model) An eogenous vrible, on the other hnd, is vrible tht n economic model tkes s given (determined outside the model) To illustrte, consider the following simple production function: Y KL This function summrizes ll the fesible production ptterns vilble; it will give the vlue of output Y (eogenous) s function of K nd L (endogenous) For instnce if K0 nd L5 then the firm will produce 50 units of output A constnt is mthemticl bbrevition for number tht does not chnge A constnt cn be represented by Greek letter like α, β (bet) or (more simply) by letter like, b or A, B For emple, let's introduce constnt representing (Hicks neutrl) technology prmeter in our production function: Plese do not distribute outside the clssroom Comments on this hndout re welcome

2 Y A F( K, L) It is cler tht the greter the vlue of A the greter the output of the firm for ny given level of cpitl nd lbor 2 Power Functions A power function is function where vrible is rised to constnt power A power function tkes the following generl form: ε f ( ) k, where k nd ε re constnts The constnt ε is clled the eponent of the function For emple, the vrible could be rised to the second, third or /2 power There re mny rules of eponents tht re very useful to simplify some epression Rule : 0 Rule 2: p Rule 3: p Rule 4: b b b Rule 6: Rule 5: ( ) b Rule 7: y ( y) Rule 8: y b + b y 3 Nturl Logrithms The nturl logrithm is logrithmic function tht tkes the Euler constnt e , s its bse In other words, the nturl logrithm y ln( ) is the inverse of the eponentil function y e For mny resons, economists often employ logrithmic trnsformtion to n eqution First, logrithmic trnsformtion converts product of two or more vribles into sum of those sme vribles A sum is often esier to del with thn product Second, logrithmic trnsformtion is strictly monotonic in the sense tht ll the peks (mimums) nd vlleys (minimums) of the originl function re retined Therefore, the trnsformed eqution preserves importnt properties of the originl eqution Third, the growth rte of vrible cn lso be pproimted by tking the log difference of the vrible (see section 4 below) The rules for nturl logrithms re s follows For ny constnt r nd vribles nd y: Rule : e ln ln e r r Rule 2: 2

3 Rule 3: ln ( y) ln + ln y Rule 4: ln( y) ln ln y Antoine Gervis University of Notre Dme Rule 5: ln r r ln For emple, the logrithmic trnsformtion of the function y Ae is simply ln y ln A + While the first is not liner function the second clerly is Let A0 nd rnge from 0 to 25, then grphiclly: Y lny Y lny 4 Levels vs Growth Rtes A level vrible records n mount For instnce if we let Y denote the level of rel GDP then is the mount of goods nd services produced in yer A growth rte vrible records the percentge chnge cross specific time period To clculte growth rte, you need to hve the level t the beginning of the time period nd t the end of the period For instnce, the nnul growth rte of Y for 2006 is given by: Y2006 Y2005 g( Y) % Y, Y where g ( Y) represents the nnul growth rte, Y 2006 is the level of rel GDP in 2006 nd Y 2005 is the level of rel GDP in The growth rte cn lso be pproimted by tking the log difference of the vrible Therefore, the growth rte of rel GDP in 2002 cn lso be clculted s: g( Y) ln Y2006 ln Y2005 3

4 ln Y ln Y Y Y Y Proof 2 : ln Y ln Y0 g( Y) Y Y Y To clculte n verge nnul growth rte, you need to divide the log difference by the number of yers For instnce, the verge nnul growth rte from 975 to 2000 is: ln Y2000 ln Y ( Y) 25 g 975 There eists couple of Mth Tricks tht use the rules for nturl logrithms nd derivtives to obtin the two simple rules of converting levels into growth rtes: Rule : The growth rte of sum is the sum of the growth rtes, g ( XY) g( X) + g( Y) Rule 2: The growth rte of rtio is the difference of the growth rtes, g( X Y) g( X) g( Y), where X nd Y re two vribles mesured in levels, g( XY) is the growth rtes of the product XY, while g( X) nd g ( Y) re the growth rtes of X nd Y, respectively Proof of : ln X ln Y ln( XY) ln( XY) 0 ( ln X ln X0 ) + ( ln Y ln Y0 ) + g( X) + g( Y) Proof of 2: similr 5 Derivtives If we define some function y f ( ), then for two distinct points nd c we cn form the difference quotient, f ( ) f( c) c The derivtive of the function y f ( ) with respect to is the limit of tht quotient when becomes very close to c Formlly, f ( ) lim c Hence the derivtive provides the rte of chnge in f() s the chnge in goes to 0 2 Do not pnic! The proofs re included for completeness only You will not be sked to reproduce them on the em 4

5 There re mny nottions used to represent derivtive Let y f ( ), then the derivtive of y with respect to cn be represented by ll the following: d y f ( ), f( ), or d A number of simple rules eist to compute derivtives Here we list subset of these rules: Rule : The Constnt rule: [ kf ( ) ] [ f ( ) ] k ε [ ] Rule 2: The eponent rule: ε ε Rule 3: The Product rule: Rule 4: The Rtio rule: Rule 5: The Log rule: Rule 6: The Chin rule: [ f ( ) g( ) ] f ( ) g( ) + f ( ) g ( ) [ f ( ) g( ) ] f ( ) g( ) f ( ) g ( ) [ g( ) ] 2 [ ln( ) ] { f[ g( ) ]} f [ g( ) ] g ( ) The prtil derivtive of some function y f (,z) with respect to revels how much f (, z) chnges when chnges by very smll mount holding z constnt Similrly, the prtil derivtive of y f (,z) with respect to z revels how much f (, z) chnges when z chnges by very smll mount holding constnt Fortuntely, the rules for prtil derivtives re the sme s those of the "regulr" derivtive Agin, there re mny nottions used to represent prtil derivtive Let y f (,z), then the derivtive of y with respect to cn be represented by ll the following: y (, z), f (, z), or f For emple, tking the prtil derivtive of the production function Y A F( K, L) with respect to L gives the chnge in output Y when the quntity of lbor L chnges by very smll mount, holding cpitl K constnt This derivtive is defined s the mrginl product of lbor (MPL) while the prtil derivtive of Y with respect to K is defined s the mrginl product of cpitl (MPK) 5

6 6 Optimiztion Antoine Gervis University of Notre Dme Mny problems in economics tke the form of n optimiztion problem: consumers mimize utility or minimize ependiture nd firms mimize profits or minimize costs Generlly speking, mimiztion problem is one in which the gent selects those vlues of the choice vrible(s) tht provides the highest vlue of the objective function (like Profit or Utility) potentilly fcing some constrints (like income, technology, or input vilbility) In generl the objective function of mimiztion problem will be hump-shped such tht the mimum will be ttined t the pek of the function (see grph below) Therefore, the optiml vlues cn be obtin by tking the first derivtive of the function with respect to the choice vrible(s) nd setting them equl to zero, these re clled the First-Order Conditions (FOC) For emple, the competitive firm (price tker) profit (π) mimiztion problem is sitution in which the firm selects vlues for the quntity of lbor (L) nd stock of cpitl (K) tht provides the highest level of (economic) profits Π PY RK WL, subject to the technology constrint tht Y AF( K, L) π π* F L ( K, *) 0 L L* π ( K, L) L M Π PY RK WL ( K,L ) st Y AF( K, L) Π R FOC(K) : PAFK ( K, L) R 0 AFK ( K, L) K P Π W FOC(L) : PAFL ( K, L) W 0 AFL ( K, L) L P The first-order conditions tell us tht the firm hires lbor until the mrginl product of lbor is equl to the rel wge ( w W P ) nd rents cpitl until the mrginl product of cpitl is equl to the rel rentl rte on cpitl ( r R P ) 6