Joh Riley 3 August 6 Basic mathematics of ecoomic models Fuctios ad derivatives Limit of a fuctio Cotiuity 3 Level ad superlevel sets 3 4 Cost fuctio ad margial cost 4 5 Derivative of a fuctio 5 6 Higher order derivatives 3 7 Partial derivatives 3 8 Cocave fuctios 5 Techical Appedi pages
Fuctios ad limits Cosider a firm that uses quatities of m differet iputs to produce a sigle output q A particular iput pla (or iput choice) is the a vector (,, m ) The set of all possible m -dimesioal vectors is writte as m Sice iputs caot be egative the feasible set of iput vectors is the set of positive vectors i m This is writte as m For ay iput vector there is some maimum techologically feasible output q Such a mappig q f () from a vector i m (the domai) to a sigle umber i (the rage) is called a fuctio Ecoomists call this maimum output fuctio a productio fuctio More geerally, suppose that the domai of the fuctio S is a subset of Uless both the domai of the fuctio is clear we write f : S where S vector Typically, the fuctios used i ecoomics are cotiuous Iformally, a small chage i the results i a small chage i the mappig f() While we will ot dwell much o mathematical formalities i this course, cotiuity plays such a importat role that we ow provide a formal defiitio We first eed to defie the limit of a fuctio f( ) for some value Here is the idea Cosider first a mappig of oe variable, f : Pick ay, as small as you like The there eists a such that if lies i the ope iterval (, ), the f( ) lies i the ope iterval ( f ( ), f ( ) ) For fuctios of variables the defiitio is essetially the same We simply replace the iterval (, ) by the ope -ball of radius, that is N (, ) { } Defiitio: Limit of a fuctio f : The fuctio f( ) has a limit L at if, for ay there eists a such that f( ) lies i the iterval ( L, L ) for all i the ope -ball N(, )
Cotiuity A fuctio f is cotiuous at f ( ) Epressed i mathematical shorthad if the limit of the fuctio at is equal to the value of the fuctio Defiitio: Cotiuity of f : at f( ) is cotiuous at if f ( ) lim f ( ) Eample: a, f( ) g( ), where g ( ) Note that g ( ) is ot defied at However for all, ( )( ) g( ) Thus g ( ) has a limit of at It follows that the fuctio f( ) is cotiuous at if ad oly if a Defiitio: Cotiuity of f : o S f is cotiuous o S if it is cotiuous for all S
3 Level set, superlevel set ad sublevel set The set of all vectors i the domai of the fuctio that yield the same value of the fuctio is called a level set m { f ( ) q} The set of all vectors for which f() is at least q is called a superlevel set m { f ( ) q} The set of all vectors for which f() is at most q is called a sublevel set m { f ( ) q} Eample : Level ad superlevel sets of the productio fuctio f ( ) The set of iput vectors for which f() (the maimum output) is satisfies the equatio Level ad Superlevel sets of a productio fuctio These are also called cotour, upper cotour ad lower cotour sets 3
Dividig by yields the mappig This level set is depicted above For all the vectors to the right of this curve the maimum output is higher Thus the shaded regio is a superlevel set Note that i this eample the two aes form the level set { f ( ) } Eample : Level ad superlevel sets of the utility fuctio f () / / The set of cosumptio vectors for which utility is u are the vectors (, ) satisfyig the equatio f () u / / Rearragig this equatio yields the mappig ( u ) The level set with / u is depicted below For all the vectors to the right of this curve the maimum output is higher Thus the shaded regio is a superlevel set Note that i cotrast with eample, f (, ) ad f(,) are both strictly icreasig Thus each level set itersects with both aes Level ad Superlevel sets of a productio fuctio 4
4 Cost fuctio of a firm Suppose that the firm is a price taker i all markets Rather tha cosider the full profit-maimiatio problem directly, it ofte proves helpful to break the problem ito two stages Stage : Solve for the least cost iput vector * ( q, r) for every output q The resultig miimied cost * C( q, r) r ( q, r) is called the cost fuctio of the firm Stage : Solve for the output q that maimies TR( q) C( q) Profit-maimiig output Cosider the secod stage of the maimiatio problem Sice the firm is also a price taker i the output market, total reveue is TR( q) pq As ecoomists, we are traied to look o the margi ad compare margial beefit ad margial cost of switchig from oe pla to aother Sice we are ot cosiderig iput price chages, we write the cost fuctio as Cq ( ) The the cost of producig oe more uit is MC( q) C( q ) C( q) We will assume that the margial cost is higher whe more uits are produced Eample: C( q) q Sice ( a b) a ab b it follows that ( ) q q q Hece MC( q) C( q ) C( q) 4q We use the Greek letter to idicate profit The for a price takig firm the profit is TR TC pq C( q) For the price takig firm the margial reveue from sellig oe more uit is the price Therefore the margial profit is M MR MC( q) p MC( q) 5
Both MR ad MC are depicted below for the eample whe the output price is For all outputs of 4 or less, the margial profit is strictly positive ad for all higher outputs, the margial profit is strictly egative Thus the profit-maimiig output is * q 5 Margial cost ad margial reveue Cosider ay p satisfyig 8 p That is, p is i the closed iterval [8,] Note that MC(4) 8 ad MC(5) Thus for ay price p [8,], * q 5 maimies the profit of the firm If p 8 the MR MC(4) so the firm makes the same profit if it produces 4 or 5 uits Similarly, if p the MR MC(5) so the firm makes the same profit if it produces 5 or 6 uits The complete mappig from prices to profit-maimiig outputs is depicted below 6
Mappig from price to profit-maimiig output Sice the mappig is multi-valued for the prices {,6,,4,}, it caot be reduced to a simple formula So i ecoomic modellig ecoomists make a further simplificatio They assume that there is o atural miimum uit Istead the output of a firm is assumed to be ay positive real umber To compute margial cost we cosider some icremetal output which cost chages is q q q The rate at C C q C q q q q ( ) ( ) We the defie margial cost to be the rate at which cost rises as the icremetal output approaches ero For our eample C C( q ) C( q ) ( q ) ( q ) ( q q )( q q ) q q q q q q q q q I the figure below this is the slope of the chord AB for some Thus the margial cost at output q is q I the limit q approaches q MC( q ) q 7
I the figure ote that the firm s margial cost is the slope of the lie just touchig the cost curve This is called the taget lie It is ow easy to solve for the profit maimiig output of the firm The firm caot effect the price so MR p Margial profit is therefore M MR( q) MC( q) p q Thus margial profit is positive it p q ad egative if p q Thus the firm maimies profit by choosig the output q p This is the firm s supply fuctio * Eercise: Idustry supply Suppose that the cost fuctio for firm i,, is C ( q ) a q i i i i (a) Solve for firm i 's supply fuctio (b) Hece show that with two firms the idustry supply fuctio is q * ( ) q a a (c) If there are firms show that the idustry supply fuctio is * q ( ) q a i i 8
5 Derivative of a fuctio Of course by takig a limit of small icremets i outputs we are simply followig the lead of Newto ad Leibit ad have reiveted calculus For ay fuctio lie touchig the graph of the fuctio as fuctio Cq ( ) dc ( q ) dq Cq ( ) we write the slope of the Mathematicias call this the derivative of the We use the idea of a limit to write dow a formal defiitio of the derivative of a fuctio For ay q q, defie the slope fuctio C C q C q sq ( ) q q q ( ) ( ) I the figure above, this is the slope of the chord AB Defiitio: Derivative of Cq ( ) at The derivative of Cq ( ) at q q is the limit of the slope fuctio sq ( ) at q 9
The derivative of a fuctio is typically writte i oe of the followig ways d dq df f ( q) ( q) f ( q) dq We have already established that if As a secod eample, suppose that C( q) C( q) q q 3 dc the ( q ) q dq The C q ( q ) sq ( ) q q q 3 3 As ca be readily checked, 3 3 q ( q ) ( q q )( q qq ( q ) ) Therefore s( q) q qq ( q ) Takig the limit at q, dc ( q ) 3( q ) dq The followig rule for the derivatives of power fuctios is the most used rule i ecoomic modelig Derivative of a power fuctio If d ( Aq ) Abq dq b b q ad q b b are both defied, the logarithm Ecoomists ofte use two other fuctios, the epoetial fuctio ad its iverse, the atural Review: The epoetial ad logarithmic fuctios
Suppose that a bak offers a aual iterest rate of per dollar of savigs If this is computed aually, the at the ed of the year the each dollar grows to If the iterest is computed semiaually the dollar grows to If the compoudig takes place case i which the iterest rate is As rises the ed of year value, after si moths ad ( )( ) ( ) by the ed of the year times the ed of year amout is ( ) ( ) Cosider first the special rises more ad more slowly ad approaches a limitig value of approimately 78 The actual limit is slightly larger ad is writte as e The ed of your value for other iterest rates ca also be computed For ay defie m / where is the umber of times that iterest is computed The m ad so m m ( ) ( ) [( ) ] m m r The epressio i brackets has a limitig value of e so the limitig value of ( ) is e The graph of the epoetial fuctio y e is depicted below The iverse of this fuctio, that is, the mappig from y to is the logarithmic fuctio It is ot too difficult to show that the logarithmic fuctio has the followig very ice properties:
a l l l (ii) l al (i) Epoetial fuctio Logarithmic fuctio Derivative of the epoetial fuctio ad atural logarithm a If f ( ) e the df d a ad if g ( ) l ae the dg d You are epected to review the basics of sigle variable calculus prior to takig this course Here are some importat thigs to review Derivative of a sum If h( q) f ( q) g( q) the dh ( q) df ( q) dg ( q) dq dq dq You ca always check your work by goig to http://wwwwolframalphacom/calculators/derivative-calculator/
Derivative of a product dh df dg If h( q) f ( q) g( q) the ( q) ( q) g( q) f ( q) ( q) dq dq dq The et rule follows directly Derivative of a ratio If df dg ( q) g( q) f ( q) ( q) f( q) dh dq dq hq ( ) the ( q) gq ( ) dq g( q) Chai Rule dh dg d If h( q) g( f ( q)) the ( q) where dq d dq f ( q) Eample : q q df dg h( q) qe Let f ( q) q ad g( q) e The ( q) ad ( q ) e q dq dq Therefore dh df dg ( q) ( q) g( q) f ( q) ( q) e qe ( q) e dq d dq q q q Eample : q df dg hq ( ) Let f ( q) q ad g( q) q The ( q) ( q) q dq dq Therefore df dg ( q) g( q) f ( q) ( q) dh dq dq ( q) q ( q) dq g( q) ( q) ( q) Eample 3: h q Defie h( ) l ad ( ) l( q ) ( q) q 3
The dh ( ) dh d q q q dq d dq q 6 Higher order derivatives If a fuctio is differetiable for all S df, the the derivative ( ) d ca therefore use the defiitio of a derivative to defie the secod derivative is a fuctio defied o S We d df ( ( )) d d Ecoomists typically use oe of the followig short-had for the secod derivative: d df d f ( ( )) f ( ) ( ) d d d Higher order derivatives are defied i the same maer 7 Partial derivatives of f : If the fuctio f() compoet of the vector is a mappig of vectors oto the real lie we ca cosider chagig just the ad defiig the partial derivative with respect to this variable j-th We defie the slope fuctio as follows s ( ) j j f (,,,,,, ) f (,,,,,, ) j j j m j j j m j j Defiitio: The j -th partial derivative of f() at j -th partial derivative of f() at s at is the limit of the slope fuctio j( j) 4
The vector of all the partial derivatives of a fuctio is called the gradiet vector Gradiet vector f f f ( ) ( ( ),, ( )) m Eample : Cosumer s utility fuctio U ( ) l l Sice d U j l q if follows that, j,, dq q j j Eample : Productio fuctio of a firm f () A Sice d ( q ) q dq if follows that f the partial derivative with respect to A f () Usig a idetical argumet for f it follows that the gradiet vector is (, ) f( ) 5
Cocave fuctios For ay pair of -vectors ad, a cove combiatio of these two vectors is a weighted sum where the weights are both strictly positive ad sum to oe Thus we ca write the cove combiatios of two vectors as follows: Defiitio: Cove combiatio of ad ( ), where Defiitio : Cocave fuctio For ay set S the fuctio f : S is cocave o S if for all ad S, ad every cove combiatio f f f ( ) ( ) ( ) ( ) I words, the value of fuctio f( ) eceeds ( ) f ( ) f ( ), that is the correspodig weighted sum of the values fuctio at ad Oe dimesioal case f : We begi by cosiderig the oe dimesioal case i which is a poit o the real lie the iterval The graph of a cocave fuctio is as depicted below As you ca check, [, ] ito three subitervals of equal legth 3 /3 ad /3 partitio Cosider the taget lie at taget lie The slope of the taget lie is df y f d ( ) ( )( ) As is ituitively clear, the graph of the fuctio must lie below this df d ( ), thus the formula for the taget lie is 3 Cofirm that /3 /3 3 6
Thus if a fuctio f : is cocave, the for all cove combiatios df f ( ) f ( ) ( )( ) d It is ot difficult to show that the coverse is also true If a fuctio is differetiable we have a secod equivalet defiitio of a cocave fuctio Defiitio : Cocave fuctio f : S For ay set S the differetiable fuctio f : S is cocave o S if for all every cove combiatio, S ad df f ( ) f ( ) ( ) ( ) d 7
From the diagram, ote that icreases 4 The coverse is also true If at some depicted below) the fuctio caot be cocave df f ( ) f ( ) ( )( ) if the slope of the fuctio decreases as d ab the slope is icreasig (as i the iterval [, ] Thus we have a third equivalet defiitio for differetiable fuctios Defiitio 3: Cocave fuctio f : S For ay set S the differetiable fuctio f : S is cocave o S if for all S, the derivative of f is decreasig, that is f( ) 4 Techical footote: From the defiitio of a itegral, derivative is decreasig, f ( ) f ( ) f ( ) d f ( )( ) ( ) ( ) ( ) f f f d Therefore if the 8
The geeral case: f : S where S While we shall ot prove it here 5, the followig iequality holds for higher dimesios as well f f f ( ) ( ) ( ) ( ) It is ot difficult to show that the coverse is also true Thus for the geeral case we have a secod equivalet defiitio of a cocave fuctio Defiitio: Cocave fuctio For ay set S the differetiable fuctio f : S is cocave o S if for all ad S ad every cove combiatio f f f ( ) ( ) ( ) ( ) Sufficiet coditios for a fuctio to be cocave Give the great simplificatios for characteriig a maimum for fuctios that are cocave, we ow summarie sufficiet coditios for a fuctio to be cocave S A fuctio f : S is cocave if f( ) S The sum of cocave fuctios is cocave S3 A cocave fuctio of a icreasig cocave fuctio is cocave S4: If f( ) is homogeeous of degree (that is f ( ) f ( ) ad the superlevel sets of f are cove the f( ) is cocave We have already discussed the first sufficiet coditio The secod follows directly from the first defiitio of a cocave fuctio 6 The third required the applicatio of Defiitio twice Eample : f ( ) l j j j 5 A proof is sketched i the Techical Appedi The proof is NOT required readig 6 Simply write dow the defiitio for the cocave fuctios ad the add the iequalities 9
Each term i the summatio has a egative secod derivative By S each term is cocave The by S the sum is cocave The derivatio of the fourth set of sufficiet coditios is sigificatly more subtle Eample : f ( ) ( ) where j j ad Each term i the summatio has a egative secod derivative By S each term is cocave The by S the sum is cocave Sice the sum s is a icreasig fuctio, the by S3, s is cocave For may fuctios commoly used i ecoomics, the followig result ca be sued to show that superlevel sets are cove Sufficiet coditio for cove superlevel sets f The superlevel sets of are cove if, for some icreasig fuctio g :, the fuctio h( ) g( f ( ) is cocave Eample: f ( ) ( ) a / Defie g( y) / y The g( f ( )) Each term o the right had side has a egative secod derivative sice Thus by S each term is cocave The by S the sum is cocave Thus the superlevel sets of f are cove Fially we ote that f ( ) (( ) ( ) ) ( ) ( ) f ( ) a / a / a /
TECHNICAL APPENDIX Propositio: The differetiable fuctio f is cocave if ad oly if for ay, Proof (oly if): f f f ( ) ( ) ( ) ( ) Defie ( ) ( ) ( ), ad y( ) f ( ( )) Note that d j j j d so d d Therefore dy f d f d d ( ) ( ( )) ( ( )) ( ) (*) Also
y( ) y() f ( ( )) f ( ()) Sice f( ) is cocave, for all i the ope iterval (,) Therefore f f f f f f ( ( )) ( ) ( ) ( ) ( ) ( ( ) ( )) y( ) y() ( f ( ) f ( )) We have argued that y( ) y() ( f ( ) f ( )) Divide both sides by (,) y( ) y() f Take the limit as dy dt ( ) f ( ) () f ( ) f ( ) Appealig to (*), settig ad otig that dy d Therefore f f () ( ) ( ) ( ) ( ) f ( ) f ( ) () it follows that
Proof: (if) For ay y, f f ( y) f ( ) ( ) ( y ) (i) Set ad (ii) Set ad y y The The f f f f f ( ) f ( ) ( ) ( ) ( ) ( ) ( ) ( ) Eercise: Complete the proof Hit: Multiply the first iequality by ( ) iequalities ad the secod by The add the resultig 3