Intermediate Microeconomic Theory
Intermediate Microeconomic Theory ECON 3130
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Dr. Shawn Emmerich
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This 80 page Class Notes was uploaded by Dr. Shawn Emmerich on Saturday September 26, 2015. The Class Notes belongs to ECON 3130 at Cornell University taught by A. Guerdjikova in Fall. Since its upload, it has received 70 views. For similar materials see /class/214349/econ-3130-cornell-university in Economcs at Cornell University.
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Date Created: 09/26/15
Notation consumption good labor price of consumption good wage other income hgtlt total amount of time The household39s problem Choose X I f so that uXf is maximized under the constraints Am Guerdykma What is the relevant decision making unit 0 We can distinguish rms according to their i business production units 1quot organizational units iv organizational forms o The organizational form of the rm eg i Sole proprietorship if Partnership iii Corporation determines the allocation of property rights and of the decisionmaking competence O The decision makers are often groups and not single persons 9 Is it possible to indicate a common interest of the group as a whole 0 Moreover decisionmaking is often delegatedto managers 0 The problem of moral hazard arises Managers often make decisions in their own interest instead of in the interest of the owners of the firm MW tlw vw i Do the owners of the firm have an interest in common 0 If there are sufficiently many markets each consumption bundle can be evaluated by the market prices 6 Hence if the owners of the rm prefer more consumption to less they should agree that the rm should maximize the value of the produced consumption bundle F m and Pm dumn quot quotiWHTi Profit maximization with a single input v and a single output y Choose y v so that My v I W 7 WV is maximized under the constraints 1 V 0andv20 y y S 2 F WW and Phi ducmnn iiui iim n First order conditions Marginal product Real wage MP ik v p Production function yquot f v Solution vquot v p W factor demand yquot y p W supply of consumption goods 71 7T P W profit function Am Guerdykma l I Pro t maxim ization ME Example Production function Marginal product Am Guerdykava Example Interior solutions f v is everywhere differentiable hence the foe must hold I i if y 2W Solution vquot v p W 52 factor demand yquot y p W 2 consumption good supply 7tquot 7T P W FW profit function Are there corner solution V v 00 is optimal if w i gt MP 0 P lt gt However MP 0 00 hence there are no corner solutions Am Guerdykma Example Production function f v min aw a9 Marginal product MPltvgtS 539 Am GuerdJiKmra Casel W ilta p vV yaV 7Tp7 Case H w p v e 09 yg 8V E 0 av 71 0 Example Are there corner solutions is optimal if Then Am Guerdykma Solution 7 if 5 gt a vquot v p W 0 7 if a factor demand 0 if 5 lt a a9 if i gt a y y P W 0 a9 if a supply if lt a avpiw if gta 7139quot 7TP W 0 if i a profit function 0 if g lt a Am Guerdykma 0 Production plan y v y1ym v1vn Outputs Inputs 0 Technology set the set of all production plans which are technologically feasible Y The production function indicates for each input vector v v1vn the maximal quantity of output y y1ym o If the firm produces a single output y the production function can be written explicitly y g fv fv1vn o If the firm produces multiple outputs ylym the production function can be written only implicitly Fy1ym v1vn 2 0 How does the output change if the quantity of a single output changes The marginal product of an input fv1 Av1v2 7fv1 v2 7 MP1V1VQ Av 1 i 3fv1v2 i 3V1 I f vv Av if vv WNW 12 2 12 A V2 3f v1 V2 avg and F39mducmn Fummn T u new mm Wt By how much should we increase the input of factor 2 if we want to save 1 unit of factor 1 and leave the output unchanged Marginal rate of technical substitution AVQ 7E1 MP1V1VQ MP2 v1 w MRTS v1 V2 Equ iiibrium and Pm FQI WH W Hum An economy with a single household and a single firm Household 0 Preferences H X l 7 I o Endowment X 07 gt 0 Firm 0 Production function y lt f V 0 Pro t function o The household owns the rm and receives its total pro t 0 Both the household and the rm are pricetakers on the market for labor and on the market for the consumption good 0 The household and the rm make their decisions independently of each other Le a the household neglects the effect of its supply for labor and demand for consumption on the firm s profit a the firm neglects the effect of its demand for labor and supply of consumption on the household s utility Equ hbrmm and Pro THquot H39ww E Lgx Uwaj quot Solution X X P W 7 demand for consumption good Ip w n supply of labor Equ hbrmm and Pro y S f v y 2 0 v 2 0 Solution y y p W supply of consumption good vquot v p W demand for labor 71 7139 p W profit functions Equihbrium and mdmum ciii iiiiiiriwii w Equilibrium prices p Wquot and an equilibrium allocation X yquot lquot vquot such that The household maximizes The rm maximizes its utility at p Wquot 7139quot its pro t at p Wquot XXPW7I yyPW llpw7 i vvpw 7194 Pikyik 7 Wikvik o the market for the consumption good clears XP W y P w o the market for labor clears lp Wquot v p W Equilibrium ar 39 If all markets but one are equilibrated then the last market must be in equilibrium as well The budget restriction of the consumer implies PlXP Wl JP Wl leP W 712 M E 0 for all p W Hence if X p W y p W or equivalently p X p W 7y p W 0 then W v p W 7 l p W 0 or equivalently MP W P W a Guerdjiknva For all A gt 0 if all prices are multiplied by a factor of A o the decision of the firm remains unchanged y AP AW v Ap AW 0 the pro t of the firm is multiplied by A y P W v P W 7139 Ap AW A7 p W o the decision of the household remains unchanged X Ap AW A7 Lp AW A7 Xp w 7139 p W 7139 Hence it is possible to normalize prices Am Guerdjiknva Pm Narmahzatmn 9 Price index ll ilwiiiu nu 0 Under which conditions is a market equilibrium Paretoefficient o Is it possible to obtain Paretooptimal allocations through decentralized markets 0 What criteria can be used to choose among Paretooptimal allocations djimiiliilgm v HM warm it First Welfare Theorem If the preferences of the consumers satisfy non satiation then every equilibrium allocation is Paretooptimal l I Paretooptimality of the market equilibrium in an exchange economy X X8 08 ik YB k i 7 VA quot YB XA 0A Xg IC Ax A YB Am GuerdJikma l I Paretooptimality of the market equilibrium in a RobinsonCrusoe economy Second Welfare Theorem If the preferences of the consumers satisfy a non satiation o convexity and the technology sets of the rms are convex then for any Paretooptimal allocation there exist 0 a price system and 0 a distribution of the initial endowments such that the price system and the Pareto optimal allocation constitute a market equilibrium the Market Equilibrium With 0 nonconvex technology sets eg a increasing returns to scale a fixed costs 9 nonconvex preferences eg Maya is indifferent between a week in Colorado and a week in Hawaii however she prefers both to three days in Colorado and four days in Hawaii a Paretooptimal allocation may not be implementable as a market equilibrium l I A Paretooptimal allocation which cannot be decentralized gtltgt Welfare functions represent social norms and values Example Welfare functions 0 Utilitarian welfare functions WuA uB uA uB 9 Rawl s welfare function W uA uB min um uB 9 Bargaining solution WOW quot3 A DA B DB 0 The welfare functions take into account only final outcomes a They neglect i the decisionmaking procedure if the initial set of alternatives Hi d quot Minn quotiiz ivi Average costs AC T Marginal costs MC Remember 0 For those values of y for which average costs are increasing MC gt AC 36y gt Cy39 3y y o For those values of y for which average costs are decreasing AC gt MC Cy gt 36y y 3y 0 Linear cost functions AC MC const 9 Standard cost functions a MC are decreasing for low output levels and increasing at high input levels a AC are decreasing for low output levels and increasing at high input levels 0 Both MC and AC have a minimum a At the minimum of AClIC Assumption 0 In the short run the input of one of the factors cannot be changed a 92 is the quantity of the factor which cannot be changed in the shortrun quotm if39ii i i39ii 39 39 Cost minimization problem For a given output level 7 and a given quantity of factor 2 92 choose v1 so that c W1 V1 MV2 is minimized under the constraints f v1 V2 V1 i i O quotlt V2 V2 iT39i39iwiiMiiii39i ii The solution of the cost minimization problem determines i the conditional factor demand V1 V1W1W2772 if the cost function 5W1W2772 Ji aiii iid Additional constraints such as v2 92 cannot improve the result of the cost minimization problem ie 6W1 W27 V2 2 CW1 W The longrun costfunction is the boundary of the lower hull of all shortrun costfunctionsThe longrun average costfunction is the boundary of the lower hull of all shortrun average costfunctions In the short run we differentiate between fixed costs a n d variable costs of production short run marginal costs SMC short run average costs SAC average variable costs AVC average fixed costs AFC Cm ivnmrmzamn Example A CobbDouglas production function l l f v1 V2 va23 The longrun factor demand and cost function CW1 WY 2 W1W2 quotlt S i Niw 91W1 W2 7 W1 A 7 W 7 V2W1W2Y I The shortrun cost minimization problem min W1V1 W292 st V1 y V120 lt 5 i Solution i Conditional demand for factor 1 y23 91W1W2772 if Shortrun cost function 3 r Y2 7 CWL W21y1V2 W1W2V2 2 We obtain short run marginal costs SMC short run average costs SAC average variable costs AVC average xed costs AFC 2 3y VQ WQVQ W1 T 2 3Q W1 72 W2 72 y v u H E 3 Mr F M w The reduced optimization problem of the firm Choose y so that 7 PY CW1 my is maximized under the constraints 7 2 CW1 W20 y 2 0 First order condition Price Marginal costs 7 36W1W2y P i 3 y f the F n n Mutiny n i The supply of consumption good y ik y ypW1w2 if The pro t function 94 7T 7TPW1 W2 Short ru n cost a na lysis p Am Guerdykma 000 0 under Fik buying car home insurance saving for retirement deciding on a production plan for your firm participating in a joint venture investing in RampD hiring a manager 0 We model risk by assuming uncertainty about the outcome of a specific action eg o a given production plan x can lead to different outputs Y1 i YQuYn depending on market conditions weather political situation etc o a given portfolio of assets 9 can lead to different returns r1 muff depending on the state of the economy interest rates performance of individual firms etc 0 We assume that economic agents know the probability with which each of these outcomes occurs eg a production plan x leads to output of y1 150 with probability and to an output of y2 300 with probability a portfolio 9 has a return of r1 50 with probability a return r2 10 with probability and a return r3 5 with probability 0 In general we assume that each alternative X is associated with a lotte ry y1 7T12 712mm 71 0 Suppose that alternative X leads to a lottery 01 711 712 7r o The expected value of this lottery is E ly1 7T1y2 7T2yn 7 y17T1 y27f2 yn7fn o Uncertainty about the output 1 2 7 300 7 3J2 17T2 3 E lty1 150 71391 1 2 150 300 250 o Uncertainty about the return of a portfolio 1 1 7 o 7 o 7 o 7 E Km 7 500 711 710y2 100 712 2y3 50 713 5 1 1 2 50E10 5g12 Am GuerdJiKmra 2 under Fik a Decision makers who use the expected value to evaluate a lottery are called riskneutral Elyl 4739L391 4 1 3 E 16397I7 0717 4 KY1 1 4J2 2 4 o In general people are averse towards risk eg they strictly prefer Ely147 114 1 3 E Kn 167I1 Zy207r21gtl 4 0 Different attitudes towards risk can be modelled by first applying a utility function to outcomes 110 and then taking the expected utility of a lottery EUly1 7T1y2 7T2yn 7Ll uy1 711 u y2 712 uyn 7 0 Depending on the curvature of the utility function uy we distinguish between 0 riskaverse decision makers u y is a concave function 0 riskneutral decision makers uy is a linear function a riskloving decision makers uy is a convex function Ecmmrmc Decmm under Ruk Am Guerdykma under Ruk Ly1167r1 EUL M16 u0 Am Guerdykma under Ruk LY1167T1 EUL M16 u0 m under Fik 1 H39ii all De nition The certainty equivalent of a lottery L Y1 7T1yn 7139 CE is the decision makers willingness to pay for the lottery 7 7 07 EUly17T1yn7fnln1uy17rnuyn o For a riskneutral decision maker CE Ey1711yn7fnl Y17T1 Yn7Tn o For a riskaverse decision maker CE lt Ey1 7T1yn 7m y17I1 l Jnnn o For a riskloving decision maker CE gt Ey1 7T1yn 7m y17I1 l Jnnn 1 3 LltY1167T11Y207T2Zgt O For a riskaverse decision maker uy W 3mm 1 Eu 16 7 0 7 KY1 7T1 4J2 JD 4 CE 1since u1 EU L 0 For a riskloving decision maker u y y2 1 3 EU lty1 l6 71391 Zy2 0 712 CE 8 since u8 82 EUL 1 716264 4 Am GuerdJikma De nition The riskpremium is the decision makers willingness to pay in order to receive the expected value of the lottery for sure instead of the uncertain payment of the lottery RP Ey1 711yn w 7 CE 0 For a riskneutral decision maker Ey171391yn7rn CE RP E y1i my 7m 7 CE 0 O For a riskaverse decision maker Ey171391yn7rn gt CE RP Ey171391yn7fni 7 CE gt 0 0 For a riskloving decision maker Ey171391yn7rn lt CE RP Ey1711yn7rn 7 CE lt 0 Am anemia l 3 l E 16 7 0 7 164 KY1 7T1 4J2 7T2 4 4 O For a riskaverse decision maker uy W CE 1 1 3 RPElty1167f11y207121gt7CE4713gt0 o For a riskloving decision maker u y y2 CE 8 RP EltY1167T1y207f2 gtCE47874 Am GuerdJikma W CE lt EH R PEL7CEgt0 1 132 Tum 27116 340 u y y 235 lt0 l llimiqzi 0 Note that the curvature of the utility function u y matters for choices among lotteries o In general a monotone transformation can change the curvature of a function eg My y f quoty y g UM W 0 Therefore u y is unique up to transformations which do not change its curvature auyb with a gt 0 P Spitg 5P P5PD t and XDPD5P5 Am Guerdykma p5pDliT and XDpDSp5 De nition The elasticity of a variable measures the percentage change of its value as a result of an increase of the value of another variable by 1 g L g X where AX respectively Ay denote the change of X respectively y Elasticity Example Elasticity of demand Suppose that the elasticity of demand for gas is 70 3 Then a 5 increase of gas price leads to a l 5 decrease of demand for gas 703 y x 703005 70015 a Guerdjikpva o The elasticity of a function f X y is given as fltl7flt 7 f I 7 f Elasticity M X H X 7 X f X X 0 If f is differentiable at X lel m M dX X HX X 7X the elasticity of f at point X can be written as Elasticity 700 X X Am Guerdjikcva 0 Tra nsaction costs a organized markets a money as a medium of exchange a legal system 9 Competition 0 number of market participants suppliers and customers 0 entrance barriers Remark Transaction costs can lead to a market break down PDPst Market and lnLlLuLlon 1quot l ell39uwiv nli Example Market for lemons 0 There are two types of used cars i highquality h if lemons I with qualities 7 and q qh gt 67 o The relative frequency of highquality cars is 7I0lt7Ilt1 The M A potential buyer reasons in the following way i if p 2 71 then Qaverage 7th JV 7 7139 ql if if p lt qh then Qaverage ql Tllzl l ul o If the seller asks a price p 2 71 the buyer will reject the offer since he expects the quality to be lower that the price he pays Qaverage 7th l 7 7139 ql lt qh S P 0 If the seller asks a price q lt p lt 71 the buyer will reject the offer since he is sure that the quality of the car is lower than the price he pays Qaverage ql lt P o If the seller asks a price p 3 q the buyer will accept the offer but he is sure that the car is of low quality Qaverage ql 2 P 0 Only lemons are traded and in equilibrium pquot ql