Intermediate Microeconomics Cheat Sheets
- 1. a > b . perfect Compte - . MRTX . ,×z=maxamt Strictly preferred merits : fixed ratios of good Zthatcan a Ib U=mih( ax , ,bXz ) purchase WII unit weakly preferred . perfect subs : reduction in good 't , a is atleast as good total amount negslopeof budget as b U=aX , tbxz line , Px . /p× a a~b . MRS , , ,×z= the . Optimal bundle : indifferent Max amount of MRSAPEMRTAB . Or . Complete : either Good 2 to be given at kink , or corner xdyoryhorbothupfor1moreof.HU/p,7V2/pz-s . Symmetric : similar good 1. the neg . of Spend more on X , to indifference the derivative of - diminishing MRS : - transitive :ifXdy indifference curve strictly convex , more E. yd 2. then Xd 2 Xz=fCX , ) , '/MRS×a,×, Of good compared to . locally nonsatiated : = U , 102 other , less " Valuable " no bliss point . Utility Function : . Lagrangian : . Monotonicity : more X1yiff UNRULY ) 1. L= UH . ,xzHXLgCx ) ) is preferred DUIDHLO → decreasing 2942×942×2,24×3=0 . Strict convexity : dUldH70→ increasing 3. Solve for X X 12,412 , then monotonic 4. X. = f C Xz ) XX t LI - a) y > 2 . Monotonic Trans - 5. Plug into Mx . rational : complete formations : same 6. Solve for Xz and transitive preferences L derma - Complements . indifference curve : five must be positive ) equally desirable . Cobb - douglas : ( w/ rational , mono - U= X. a Xz ' - a tonic , E convex pref - MRSx.x.FM/u.asxzerencesmustbethin.Asa- 1. x. → important Substitutes non - increasing , non - . Budget Line : set of crossing , preferred all bundles that causes farther from origin , an individual to convex E. everywhere ) spend all income
- 2. - Individual demand curve : the graphical 3. To find EV , . Steps to determine optimal decision under relationship btw p , EX , when optimizing i. Calculate Utility of new bundle Wnew ) uncertainty When U ( X )=XFXzLl - a ) Over budget set x. = a. ¥. ×z=u - as . ¥ ii. Solve for X. equiv and Xzequiv I. Determine the possible States of the world L= ULXITXCI - pix , - pzxz ) ( Mlk , 1MHz ) = Pxilpxz 2. Under each state , determine the - Law of demand : If p.lv , then X. T Uncw = ULX , , Xz ) t Use Old prices wealth the individual would have If ¥170 , Law of demand holds iii. Solve for compensated income 3. Setup expected utility . Engle Curve : function that shows an I equiv =p , orig X. equivtpzxzequivEU-p.ULX.lt/2zULXzlti..tpnULXn ) individual 's demand fora good at diff . iv. EV= I equiv - long 4. Take first order conditions ) of EU Levels of income 3. TO find CV , function wrtthevariablels ) the individual - Elasticity A. B : EA , , , elasticity of A Wrt i. Calc . Orig utility ( Uorig ) can choose B , B elasticity of A. the f. change in ii. Solve for Xsubsdaysubs 5. Solve for variables ) to determine the A that results from a I 't change in B t.MU//MUy)=px1py choice that would maximize EU Uorig = ULX , y ) - Risk Adverse : avoid an even bet , dislike EA ,B=fF3 . Ben = - BF iii. Solve for compensated income risk 's would pay to avoid it , ULX ) is strictly . magnitude Of Ea , B → whether effect Of lump = phew Xsubtpyysub concave , U' ' LN LO B becomes " magnified " ( IEA , 13171 ) or iv. C ✓ =/ orig - lamp . Risk Neutral : indifferent to even bet . " dampened " ( IEA , BILL ) as it Changes A CVEEV can be used to evaluate policies . indifferent to risk da wouldn't pay to . sign of EA.rs → A moves in same l t ) Or Ex : subsidy cost SIX E. est . sum of society 's avoid it . UCX ) is linear , U' ' C XI=0 opposite ( - ) direction as B CVIEV is Sly → inefficient if $4 > SIX . Risk Loving : would take an even bet , like - Demand Elasticity : Ex , , . Marginal Rate of Time Preference : risk 's would pay for a gamble , ULX ) is - Own Price Demand Elasticity :E x. p 1. The rate at which a consumer is willing to concave , U' ' LX ) > O If law of demand holds , Ex ,pL0 substitute current consumption for future . Risk Premium : the amount Less than EV If IEx.pl > I → elastic , < I → inelastic , =l→ consumption a consumer would accept in exchange unit elastic ( where rev . Maximized ) 2. MRS x. ixz When X , is con . today and for the lottery Cobb - Douglas : - I Xz is con . tomorrow 2. the ' ' fee " paid to avoid risk inherent . Cross Price Elasticity of Demand : Ex ; ,pj Usually MRSX . ,xz > I ( impatient ) to the lottery Exi , pj LO → complements ↳ U=X9 ' XYZ Where a , > Az 3. The value , r , such that ULEV - r )=EU Exi , pj > O → substitutes * Need Units of money insametimeper . where EU is the expected utility of the - Income Demand Elasticity :E x. I lntom . 's $ , budget constraint : lottery E x. I > O → normal ( Iti )IitIz= L Iti )XitX2 . To solve for risk premium : Ex , ISO → inferior ' Marginal Rate of Intertemporal Trans : I. solve for Ev MRT× , ,×z= Iti ( " luz = Iti ) 2. Solve for EU. Substitution Effect : the Change in an . L=UL× , ,×z)t×uHilI , + Iz - Lltilx , - Xz ) 3. plug into equation , ULEV - r )=EU . E individual 'S consumption Of a good due . Human capital production Function : a solve for r to the fact the relative prices have Changed math function describing how consumption U , Uz ) U , s - s - - = - good I pm, ) ¥22 Purchase more today affects future income . 1. Find optimal bundle before . z Draw new budget line p , pz becomes of good 1 price change cheaper - Intertemporal Utility Maximization with " - " - . Income Effect : change in consumption - - human capital development 's access to s - s - Of a good due to change in budget set , n n controlling for substitution effect financial markets : go , go , -1€, - Compensated budget line : shows all 1. Maximize Lifetime Income subject to , : y , . , : y ↳ bundles an individual can afford at new human cap . constraint . , . . 2. Max lifetime Utility subject to making . i . . i . . . orig . . . i . . i . . loris. . I 2 3 4 S I 2 3 4 S prices assuming they were compensated SO rig good , rig Good ' Max amount Of lifetime income s - 3. Find optimal bundle after s - 4. To calc . W . shift new they can afford Orig . level Of happiness Ex : constraint : zoo -2×12×22 , i 10%0=+1213×2113 - price change - budget line until it is Any changes in optimal bundle must be 4 - 4 - tangent to original L = ( 1. 1) X. t Xztx ( 300-2×12 - X } ) - - indifference curve due to change in prices C Sub . effect ) s - s - X. = 7.52×2=13.67 N N . Giffen Good : a good that is inferior 's o - o - f- X , "3Xz"3tX( ( 1.174.523+13.67 - I. IX , - Xz ) 802! 802-• ewtiescincmmag .EE#Es9Fwdo:tYumsan3nx.=issoxz=isi . . Lottery : Bundle of goods ( X , ,Xz , . . . , Xn ) - I - • - Bhim . I I I I l - l l toricI I , , , , I, , I , orig , , - Compensating Variation : amount Of w/ a lottery ( p , , pz , . . . , pm ) rig ' Z gig? " s n .ge#ebcti3eIIgIEIew3 4 s income someone would be willing to give s - s - Good ' I . p , t pzt . . . t pm =/ - 5. Label intersection . 4. To calc EV , shift Blorig Up ( need ) after a price reduction ( increase , z . pizo 4 - Of BLUEY - axis , 4 . until itistangenttolnew . " Xznew " G Blcompda to maintain Utility before change y-axis as " xzcomp " - 3. pi is the prob . of receiving bundle Xi , - 6. CV=p{ Xznew - Xzcomp ) 3 - new price , Old utility Level . Degenerate lottery : all prob . on I outcome o - 512-1.25=3.75 - - . equivalent variation : amount of income . expected value :p ,×,+pz×z+ . . . + pn× , I÷÷u¥•§↳;§o, .§ ,a consumer would need ( be willing to give . Mixing : Given 2 lotteries Lp , , Pz , . . . . pm ) E = i - a - • I Blcoymp - IBLequiv I new Up ) before a price reduction ( increase ) to ( t , ,tz , . . .tn ) and OLXLI , a MIX Of . . . . . . Kris. . , , . Kori. , iorig. ?rig I Zxcompxnew } 4 S I 2 = 3 4 S give the same level of utility as after the these 2 lotteries is the new lottery : good , rig yoga, S - 5. Label intersection of > rice change ( AP .tl/-X)ti,XPztll-X)t2 , . . . , April ' - d) th ) n - Blom .gg y-axis " xzorig " 's Sub . Effect :X sub - Xorig/ Old prices , new utility . Independence : If lottery PI lottery q E I " - Bleauivday - axis as Income Effect :X new - Xsubs or - " Xz equiv ' ' CV=pz( Xznew - Xzcomp ) both are Mixed with lottery tither lottery xzeausi-6.eu-pzlxzeauiu-xzon.gl s I 2178 p is still preferred to lottery q - Scs -23 . - s EV = Pz ( Xzequiv - Xzorig ) xzorig - Find CVEEV mathematically . Expected Utility Theorem : VNM Can be 2 , ; ¥ , Yg , 1. Use Lagrangian to find optimal bundle interpreted as the cardinal Utility received . Iaea " s > i i , Phon, , Iorio, , ( giffen before price change → X orig from good I such that ( p , , Pz , . . . , Pm ) I rig , z Inns4 s I ICO 2. Use Lagrangian to find optimal bundle ( q , ,qz , . . . , qn ) Iff p . U , tpzuzt . . . tpnun Good ' III > 151 after price change → Knew Iq , U , tqzuzt . . . tqnvn
- 3. Production Fundamentals ' Cost minimizing input blend : f ' Az = W' lwz ' Production Function : y = f CX . , Xz , . . . , Xn ) is the ( if using both goods ) amount of output , y , that can be efficiently .tk/Wk=f4WL L marginal productivity per produced using X . . Xz , . . . , Xn dollar spent ) . Efficient Production : given inputs , firm produces Cost Minimization largest amount of output possible . L = W , X , t Wz Xz t X ( q - f- LX , , Xz ) . NO free lunch : impossible to produce output - Factor Demand Function : specifies w/o using inputs relationship btw prices of input goods , ' Possibility of inaction : Xi 20 quantity of output produced . E amount ' Free disposal : inputs can be disposed of at no Of an input good a firm will select cost ; dfldxi O for every input . Total Cost = w , X , t Wzxz . Decreasing Returns to scale : a production set . Cost Function : Clq , w , , wz ) = displays decreasing returns to scale if fctxlctfcx ) W , X , ( q . W , , Wz ) t Wz Xz ( q , w , , Wz ) for all t 71 Where Xn ( q . W . , Wz ) are the firm 'S . Increasing Returns to Scale : f Ctx ) > tf Cx ) factor demand functions . Constant Returns to Scale : f Lt X ) = tf Cx ) . Market Demand Function : sum of ' Cobb - Douglas Production Function : FCK , L ) = aka L 's individual demand functions ( be ↳ If a t BC I → decreasing careful of corner solutions ! ) ↳ If a t B > I → increasing Profit Maximization - Fixed Proportions Production Function : . IT L p ) =p Dcp ) - C L Dcp ) ) f Lk , L ) = min Lak , BL ) . IT ( q ) =p L q ) q - C Cq ) . Linear Production Function : perfect subs , ' IT =p ( q ) q - C Cq ) Marginal f- C K , L ) = a K t BL = Req ) - C C q ) Revenue = . No monotonic transformations for production R ' ( q ) - C ' ( 91=0 Marginal functions ! R ' ( q ) = C ' ( q ) Cost . 15090 ants : graphical set of bundles that allow . DIT 1dg =p C q ) t p ' ( q ) q - C ' ( q ) = O a firm to produce the same level of output . Marginal Rate of Technical substitution ' lE÷pI= - ¥ - To = =L ( MRTS × , , xz ) : Max amount of input 2 firm . p = MC ( Markup = ¥ ) would be willing to give up to get one more of input I while keeping total output the same ; . Profit w/ Fixed Prices : ( negative of ) derivative of isoquant Xz=fCx , ) ; IT L q ) = pq - C Cq ) MRTS a , B = fa HB Lfa: marginal productivity Wrt Al p = MR = MC . ISO cost Line : graphical set of input good ' Market Supply Function Scp ) : sum of bundles that cost the same amount individual supply functions . Factor Price Ratio btw Input I da Input 2 : . Market Price in PC amount of input 2 the firm must give up 1 . Solve for each consumer 's demand to get one more of input I I maintain function for the specified good the same cost level ; L negative of ) the slope 2 . Find market demand of the boost line w/ input I on x - axis ; if 3. Solve for q each firm will produce at prices are fixed da nothing is being given a given price away for free = w ' lwz 4 . Find market supply 5 . Find p where Qs = QD
- 4. . Der feet Price Discrimination : firm 1. Determine firm 's profit function sells each unit at maximum amount as function of quantity Cor price ) . Cournot game ex each customer is willing to pay IT = q , p , t qzpz - C ( q , t ga ) - producers get all surplus , more units 2. Take first order condition Wrt MC for each firm : $2 sold than w/o price disc rim . , more both quantity L price ) variables p ( Q ) = 10 - Q total surplus generated , MR re pre - 3 . Solve for profit maximizing prices sent ed by demand curve L quantities ) IT , = ( 10 - q , - q z ) q , - 29 , . Non - linear price discrimination : price 4 . Plug into market demand to IT z = ( 10 - q , - q z ) q z - Zqz varies w/ quantity purchased but all determine profit maximizing prices consumers purchasing the same low anti ties ) DIT , 1dg , = 8 - of z - 2g , = O quantity pay the same price 5 . Plug into profit function to find q , * = ( q . q y , 2 - used when it is difficult to identify profit Level 2 or illegal to charge different prices ' Dominant Strategy : an action that ditz ldqz = 8 - q , - 292=0 to groups w/ highest WTP provides a higher payoff regardless qz* , L q - q , ) , z - Block Pricing : firms choose one price Of how an opponent plays → if any for the first few units E another price player has a dominant strategy , a q , = ( 8 - L 8 - q , ) 12 ) 12 for subsequent units dominant strategy solution exists zq , = q - 4 +I q , = 4+ I q , 1. Determine firm 's profit function as . l Strictly ) Dominated Strategy : an function of quantity 2 block a action that always provides a lower 3/2 q , T 4 IT pl of , ) q , t plqz ) ( q z - q , ) - C ( q z ) payoff than another possible action q , = 4 . I = 8/3 2. Take first order condition Wrt both regardless of how the player 's quantity variables opponents play q z = 8/3 3. Solve for quantity variables . Nash Equilibrium : set of strategies . Subscription G unit price EX 4. Plug into market demand to find Such that no player has an incentive profit maximizing prices to unilaterally deviate 9 , = 6 - p S . Plug into profit function to find . Prisoner 's Dilemma : players have a q z =3 - O . 5 p no MC profit level dominant Strategy to cheat , preventing - firms must be able to prevent beneficial cooperation lower demander : q z resale G must have market power , . Coordination Games : multiple Nash p PS increases , CS decreases . more Equilibria exist , each corresponding 6 - units sold than w/o price disc rim , to each player doing what the other more total surplus , W/ more blocks players are doing p - p can approach efficient outcome - " pushing " a coordination game into D . 2 Part Tariff : charge lump sum a ' ' good " equilibrium can be achieved 3- . 's p Q , subscription fee , S , for the right to through costless expectations ; however , buy all ; unit prices , u , are uniform players must believe you are willing to CS z = I ( 6 - p ) ( 3 - . 5 p )1. Calculate consumer surplus as a pay if the equilibrium doesn't occur function of u for the lower - Hawk I Dove Games : multiple Nash IT = 25 t pg . t pqz - C Cq , tqz ) demanding customer . Set this Equilibria exist , each corresponding to equal to s . one player being " strong " and the = L 6 - p ) ( 3- . Sp ) t p ( 6 - D ) 2. Determine profit function as a other being ' ' weak " ( opposite actions ) t p ( 3- . Sp ) - O function of U - To be a hawk , commit irreversibly to IT = 25 t 09 , t Uqz - C Cq , t q z ) the strong position = I 8 t 3 p - P2 3. Take the first order condition - if the game is repeated L both d IT Idp =3 - 213=0Wrt U players are stubborn , consider a 4. Solve for profit - maximizing u p = 3/2 5. Plug U into demand functions to compromise . Steps to solve for the NE of a 2 q , = 6 - I . 5 = 4 . 5 determine profit - maximizing quantities player continuous game 6. Plug into profit function to find qz =3 - . 5 ( I . S ) = 2 . 25 I . For player I , derive player I 's best profit level response LBR ) for each possible S = . 5 ( 6 - I . S ) L 3- . SLI . S ) ) = 5- firms must be able to prevent resale . strategy of player 2 must have market power E identify 1000 of each type → customer WTP types , firms charge 2 . For player 2 . derive player 2 's BR for user fee equal to consumer surplus each possible strategy of player I Max fixed cost ? of lower demander , When demand 3 . Find the set of Strategies that Sim Ulta - 1000 ( g + 1. g ( 4. s ) ) + types are similar firms charge low newly solves the BR functions U and high S , when demand types - Cournot Game : firms produce identical 1000 ( S t I . S L 2 . 25 ) ) = 20125 are different firms charge high u goods , firms commit irreversibly to a and low S , producer doesn't certain quantity level , when qt pt , extract all surplus each firm is profit maximizing . Tie in Sales : in order to buy one . Bertrand game : each firm sets prices ; item , customer must buy another the firm that has the lowest price gets . Group Price Discrimination : price all customers ; if they have identical varies by group , used when difficult prices, they share customers 50150 to price on individual WTP but can determine avg WTP for a group