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Lecture Notes – Intermediate Microeconomics Xu Hu [email protected] Department of Economics, Texas A&M University November 12, 2010

Lecture Notes on Intermediate Microeconomics

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Lecture Notes Intermediate [email protected] of Economics, Texas A&M UniversityNovember12,20102Contents1 Introduction 51.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Demand-supply analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.3 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 ConsumerBehavior 112.1 Preference and Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.1 Marginal Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Indierence Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.3 Marginal Rate of Substitution . . . . . . . . . . . . . . . . . . . . . . . 172.2 Budget Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Utility Maximization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 ProducerBehaviorintheCompetitiveMarket 273.1 Producer Behavior with single input . . . . . . . . . . . . . . . . . . . . . . . 293.1.1 Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.2 Prot Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Producer Behavior with two inputs. . . . . . . . . . . . . . . . . . . . . . . . 383.2.1 Production Technology with two inputs . . . . . . . . . . . . . . . . . 413.2.2 Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Long Run Equilibrium in Competitive Market . . . . . . . . . . . . . . . . . . 4734 CONTENTS4 Monopoly 534.1 Question in front of a monopolist . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Prot Maximization for a monopolist . . . . . . . . . . . . . . . . . . . . . . . 564.2.1 Revenue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.2 Prot Maximizing Condition . . . . . . . . . . . . . . . . . . . . . . . 564.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 Duopoly 615.1 Cournot Duopoly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2 Stackelberg Duopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3 Bertrand Duopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Chapter1Introduction1.1 PrologueThis short article serves as an introduction to the course ECON 323, Intermediate Microe-conomics. In this article,I will present the major topics to be discussed in this course andwill review some basics of the demand-supply analysis.The goal for economics is to understand how economy works. The approach we take is verysimilar to the one used by natural scientists. In order to understand the aggregate behaviorof the economy, we rst examine the working of its components,i.e., various types of market,suchascommoditymarkets, labormarket, capital market, etc.1Afterweobtainagoodunderstandingof howeachmarketworks, weinvestigatehowonemarketislinkedtotheothers. By examining the inter-dependence of dierent markets,we are able to discuss theworking of the whole economy.In this course, we only concentrate on the working of markets, mainly the commodity market.The analysis of how one market is linked to the others is left to the intermediate Macro course.Again, theapproachwetaketounderstandtheworkingof amarketisbyexaminingitscomponents,i.e., peoplewhoparticipateinthemarket. Wegroupthembythefunctionthey play. The most relevant classication for us is buyer and seller, or consumer andproducer. Thus the rst topic is the theory which explains how an individual consumer acts(Part II) and how an individual producer acts(Part III). On the basis of this, we investigate theinteractions between consumers and producers and try to understand how price is determined.The mechanism under which consumers interact with producers depends on the structureofmarket, which also inuences the determination of price.Let me give you an example to illustrate this point.Suppose there are two isolated islands . In island A, there is a lake rich in shes. In island B,there is a eld of corn. Let us call the people who live in island A shermen, and in island Bfarmers. Even though two islands are separated, in between there lies a small island whichhasnoinhabitant. SinceshermeninislandAalsowanttoeatsomecornandfarmersin1SometimethismethodologyiscalledReductionism56 CHAPTER1. INTRODUCTIONisland B want to eat some sh, they come to the small island between them to trade. Thisis the market.If the corn and the sh traded in the market are homogeneous, i.e., without any typesof dierence, suchasquality(Averyunrealisticassumption.), andif thereisalargenumberofshermenandfarmersinthemarketsothatnoonealonehasthepowerto inuence the market price, we think this market is competitive. The device we useto analyze the working of the market here is Demand-Supply analysis. We will reviewthe basics later. What you have not learned is how the demand and supply curves arederived, which will be discussed in the rst topic.Whatifthecornandshtradedinthemarketarenothomogeneous? Forexample,some farmers in island B are able to produce the corn of better quality than the rest.In this case how the market functions?Whatif shermeninislandAactcollusively, i.e., formingamonopolyforshoramonopsony for corn? In this case shermen as a whole act as one person, having thepower to inuence the market price. In part IV and V we will deal with this type ofsituation.1.2 Demand-supplyanalysisIn this section,we will start with an example as an illustration of what demandcurveandsupplycurve are, andthenIwill elaboratealittlebitontheconceptof equilibriumandcomparative static analysis. In the end, we will discuss the concept of elasticity.1.2.1 EquilibriumFrom our two-island story,we have seen that the shermen in island A and the farmers inisland B are willing to trade with each other. The shermen in island A are the buyers forcornandthesuppliersforsh. Now, letusconsiderthemarketforcorn. ThefollowingscheduletellsuswhatthetotalamountofcorndemandedbytheshermeninislandAisunder a given price.Price(Unit: sh per corn) Quantity demanded for corn1 102 83 64 45 26 0This schedule tells us that when farmers in island B ask one sh for each corn the shermenas a whole in island A want to have 10 corns;if the farmers ask 3 shes for each corn,the1.2. DEMAND-SUPPLYANALYSIS 7shermen will only demand 6 corns. We can notice that as the price decreases, the quantitydemanded for corn increases.We can represent this schedule by a linear function, i.e.,Qdc= 12 2Pc, whereQdcis the quantity demanded for corn andPcis the price of corn in terms of sh.We can also draw this schedule into a two-dimensional curve. In the horizontal axis, we useQ to denote the quantity demanded. In the vertical axis, we use Pto denote the price. Eachpair in the table above corresponds to a point in such a two-dimensional plane. We draw aline which transverses all the points, which gives us the demand curve for corn. It should bedownward sloping, which says the higher the price the lower the quantity demanded.Now lets consider the farmers in island B, the suppliers of corn. The following schedule tellsus what the total amount of corn supplied by the farmers in island B is under a given price.Price(Unit: sh per corn) Quantity supplied for corn1 22 43 64 85 106 12ThisscheduletellsusthatwhentheshermeninislandAoeroneshforeachcornthefarmers as a whole in island B are willing to provide 2 corns; if the shermen oer two shesfor each corn,the farmers will increase their supply to 4 corns. We can notice as the pricedecreases, in contrast with the demand curve, the quantity supplied decreases as well. We candraw a line to represent such schedule in a two-dimensional plane as we do for the demandschedule. And the functional representation is,Qsc = 2Pc, whereQdcis the quantity supplied for corn.Now, look at two schedules. We notice that only at price 3 shes per corn, they can reach anagreement in the sense that the quantity demanded is equal to the quantity supplied. Whatif the price is one sh per corn? When the price is 1 sh per corn,the shermen in islandAasawholewanttohave10cornswhileonly2cornsareprovidedbythefarmers. Thisimplies under this price the need of some shermen in island A who want to buy some cornisnotsatised. Whatshouldtheydo? Theycangotonegotiatewiththefarmers. Fromthe demand curve, we see that some of them are willing to oer a higher price for each corn.By oering a favorable term(a higher price), the unsatised shermen will nd some farmersin island B who want to provide more. This process will continue until the price reaches 3shes per corn. What if the price is 5 shes per corn? In this case,the shermen demandless than the farmers are willing to provide. Thus some farmers can not sell out their corns.Byoeringalowerprice, theywillhavemorebuyers. Thisprocesswillcontinueuntilthe8 CHAPTER1. INTRODUCTIONprice reaches 3 shes per corn. Only when the price is exactly 3 shes per corn, everyone issatised.Formally, we call the price as the equilibrium price provided it makes the quantity demandedequal to the quantity supplied. (Warning: I did not say the demand curve is equivalent withthe supply curve.) In the graph, the intersection of the demand curve and the supply curvegives us the equilibrium price and quantity.Equilibrium in general is the situation where no individual has any incentive to change theirdecisions. In this case, equilibrium is the situation when quantity demanded is equal to thequantitysupplied. Mostofeconomistsbelievethattheequilibriumstateisarestingpointsuch that the economy will nally achieve. Whenever the economy is not at the equilibriumstate, there always exists a tendency for the economy to move toward the equilibrium. Thushistorically economists also call the equilibrium state as the stationary state.Take the market for corn we analyzed above for example. Only when the price is 3 shes percorn everyone is satised and no one is willing to deviate from that state. When the price isnot 3 shes per corn, either the buyers will oer a higher price or the suppliers will requesta lower price. In other words, the tendency is present toward the equilibrium price 3 shesper corn.We have mentioned that as long as the equilibrium state is reached,the economy will staythere forever. Does that mean the economy will not change at all after that?How can we usesuch a static method to analyze a changing world?And nobody will believe the existence ofsuch a non-changing world. The world is always changing. So how?1.2.2 ComparativeStaticsBefore answering this question, it is necessary to introduce a pair of concepts exogenousandendogenousvariables. In this world,we have some observations,and they make us curiousabout why they are so. Therefore, theorists are trying to set up models to explain them. Inamodel, therearesomevariableswearetryingtoknowhowtheirequilibriumvaluesaredetermined. We call them endogenous variables. And there are some other variables whichforthepresentpurposewearenottryingtoexplainbutinsteadwhosevaluewetakeasgiven. In our two-island example, even though we did not specify what exogenous variablesare,they actually determine thedemand and supply schedules. For instances,the weatherdetermineshowmanycornthefarmerscancollecteachdaywhichaectshowmanycornthey want to trade regardless of the price. Given the demand schedule and supply schedule,we can determine the equilibrium price and quantity. So you see the price and quantity inour case are the endogenous variables.Come back to the question: How can we use such a static method to analyze the phenomenain a changing world?We rst classify all the changes as changes in exogenous variables. As the exogenous variablesvary, the demand and supply curve mightshift(NOT the change in quantities !), which inthe end determines the new equilibrium price. This type of process is called the ComparativeStatic analysis.Lets see some examples.1.2. DEMAND-SUPPLYANALYSIS 91.2.3 ElasticityInthissection, webrieyreviewauseful concept, elasticity. Herewefocusonthepriceelasticity. Price elasticity of demand (supply) measures the sensitivity of quantity demanded(supplied) in response to the changes in price. For example, in the case of the market for cornfrom our two-island story, you might want to know if the price falls 1% by what percentagethequantitydemandedwill increaseandthequantitysuppliedwill fall. If thequantitydemanded (supplied) increases (falls) to a large degree in the percentage sense,we call thedemand (supply) is fairly sensitive to the price and the demand curve is elastic. Otherwisewe call it inelastic.Formally, the following is the denition, =Q/QP/PRemarks:In most of cases, the price elasticity of demand is negative due to the Law of demand,which says as the price increases, the quantity demanded falls. But it is NOT alwaystrue. We will see some counter examples in Part II.More precise denition for elasticity calls for the use of calculus. See Textbook Page46footnote1.The price elasticity is evaluated at some point. (See Example below for details.)When the absolute value of the elasticity is large, it implies the demand curve or supplycurve at the point where the elasticity is evaluated is elastic.|| = 1 0 || < 1 || > 1Unitary elasticity Inelastic ElasticExample:Price(Unit: sh per corn) Quantity demanded for corn Elasticity2 82/81/2= 0.53 62/61/3= 14 42/41/4= 25 22/21/5= 5Price(Unit: sh per corn) Quantity supplied for corn Elasticity2 42/41/2= 13 62/61/3= 14 82/81/4= 15 102/101/5= 110 CHAPTER1. INTRODUCTIONChapter2ConsumerBehaviorIn this article, we will focus on the theory which explains how consumers make their choices.Lets rst take a look at the big picture of the theory.Infrontof theconsumers, thereareabunchof choicesforthemtopick. Takeacollegestudent for example. He can have a lunch at Subway, Mcdonalds, PizzaHut, or Jins Cafe.He can buy a Honda,toyota,or Fold.... But not all the choices are aordable for him.Thesetof thechoicesthatconsumerscanpickfromisrestrictedbytheresourcestheyown,forexample, money. Wethinktheyhaveapreferenceorderingoverall thechoices. ThepreferenceorderingsayschoiceAisbetterthanchoiceB,choiceCisworsethanchoiceD,etc. According to such preference ordering, they will pick a best aordable choice.Inaword, somehowwebelievethat individuals behaveas if theyweremaximizingthesatisfaction resulted from their actions. Another way of sayingisthat they arecalculatinggains and pains when they are making choices, and choose the best one to maximize the gainand minimize the pain.Asetofquestionsmightarise: whatdoImeanbygainsandpains? materialisticorpsy-chological? Doesthetheorysuggeststhatpeopleonlycareaboutmoney? Howtoexplaingenerous donations and charity activities?Thetheorydoes not sayanythingabout what kinds of gains andpains that peoplearecalculating at all. It can be materialistic or psychological. Neither does the theory suggestthatdierentpeoplehavethesamepreference. Whatkindofpreferencethatpeoplehaveisnotthequestionthateconomistsaretryingtoanswer. Economistsonlyassumethereexistssuchapreferenceorderingbutthespeciccontentislefttobeopensothatitcanaccommodate the variety of tastes among people.ExampleSuppose one sherman in island A on July 3rd has ve shes, and the market price of sh is2 corns per sh. Then we know the aordable choices for him are as follows.1112 CHAPTER2. CONSUMERBEHAVIORFish Corn0 101 82 63 44 25 0Notice that choices like 6 shes, 4 shes and 3 corns, or 11 corns are not aordable. Why?suppose the man wants to have 4 shes and 3 corns. He has ve corns. Eat 4 of 5,only 1sh is left. Since the price of sh is 2 corns per sh, the maximal amount of sh he can haveis 2. Thus 4 shes and 3 corns are not aordable for him.Now suppose he has the following preference ordering over the choices.Desirability Fish CornSuper Best 4 2Second Best 3 4good 5 0OK 2 6Just so so 1 8Worst 0 10Accordingtothetableabove, theoptimal choiceforthemaninislandAistodemand4shes and 2 corns when the price of sh is 2 apples per sh.You may notice that the optimal choice made by the man is essentially dependent on threethings.1. subjective valuation over the choices : this provides a criterion for individuals to decidewhich choice is best for them.2. the market price: this determines aordable choices that individuals can pick.3. the initial endowment(or income) (5 shes in this case.)To address this point, we can think of the following changes. Suppose now, the market priceof sh is 3 corns per sh. The set of all aordable choices for the man in island A is changedto the one in below.Fish Corn0 151 122 93 64 35 02.1. PREFERENCEANDUTILITY 13Since now the man can choose some combinations of sh and corn which are not aordableunder theprevious pricelevel,i.e., 2corns per sh, theoptimal choicehemakes will bedierent from the one he made before. Now consider a dierent change. Suppose the priceof sh is still 2 corns per sh, but the man in island A is endowed with 6 shes. The new setof all aordable choices is as follows.Fish Apple0 121 102 83 64 45 26 0Again, the total resources also aect the set of all aordable choices.In below, we study the preference ordering in more details. And then we move on to discussthe budget constraint which shapes the set of choices aordable for individuals. In the end,we explain how to derive optimal choices.2.1 PreferenceandUtilityLetX=choices, denotes the set of all the possible choices. In this course, we only considertwo-commodity case. Thus, the elements in X are pairs. In our two-island example, there aretwo commodities, sh and corn. In this case, the set of all possible choices contains elementslike, (5 shes, 1 corn), (3 shes, 2 corns), etc. Formally,X = {(a, b) : a, b R+}, whereR+denotes nonnegative real numbers. The rst coordinate indicates the amount of sh and thesecond the amount of corn. For example, 5 shes and 1 corn can represented by (5,1). Thus,we can associate each element inXwith a point in a two-dimensional plane.In the rest of the course, we assume divisibility of the quantity of goods. In other words, anyreal number can denotes certain quantity of one good, even though we know 1/3 of a car isnot a car any more, and can be not sold in the real world. However, we have this assumptionfor the convenience of our analysis. Therefore, X= {(a, b) :a, b R+}, whereR+denotesnonnegative real numbers. Graphically,Xis simply the rst quadrant.Preferenceisdenedover Xwhichspeciesarelationbetweentwopairs. Forexample,(4,2)isbetterthan(3,4), whichsays4shesand2cornscombinationisbetterthanthecombination of 3 shes and 4 corns. Or (3,4) and (5,0) are the same, which means 3 shesand 4 corns are the same with 5 shes. Note, dierent people might have dierent preferenceorderings. For example. One guy might prefer the combination (3,4) to (4,2) while the otherguy might choose the opposite. Thus in essence, the preference order reects peoples tastesand subjective valuation of commodities.There are several axioms on preference.We require all preferences should satisfy following veaxioms.14 CHAPTER2. CONSUMERBEHAVIOR1. completeness All pairs inXare comparable.2. transitivityIfpairAisbetterthanpairB, andpairBisbetterthanpairC, thenpairAisbetterthanpairC. Forexample, forsomeoneif the4shesand2applescombinationisbetterthecombinationof 3shesand4apples, and3shesand4apples combination is better than 5 shes, then the 4 shes and 2 apples combinationis better than 5 shes.3. continuity A technical condition which preference is representable via a real function.(Not required to know. If interested, come to me.)4. monotonicity The more the better. For example, (4,2) is better than (1,1).5. convexity Diversication is desirable. Technically, it says for example, (1,2) is betterthan both (0,4) and (2,0) when (0,4) and (2,0) are the same. In other words, if you areindierent to eat 4 shes and 2 apples, the mixture of them makes you happier. Noticethat 1/2 of (0,4) and 1/2 of (2,0) is (1,2).Withaxiom1-3, wecanprovethereexistsafunctionu: XRsuchthatfunctionupreserves the ordering. We call such function, utility function, which assigns a real numberto each pair, and call such number as the utility from consuming such pair. For example:pairs(sh,apple) utility(5,0) 6(3,4) 10(4,2) 12(4,5) 15... ...Thus, we see u(5,0)=6, u(3,4)=10.WhatdoImeanbyutilityfunctionpreservestheordering?Ifpair(4,2)isbetterthan(3,4), then u(4,2)=12> u(3,4)=10. Formally, if according to the preference ordering pair Ais better than pair B , then u(A) > u(B). Thus the utility actually represents the satisfactionfromconsumingonecertaincombinationofgoods. Ifyoustillremember, Imentionedthetheory somehow suggests that individuals behave as if they were maximizing the satisfactiontheygainfromconsuming. Herewecanseethatmaximizingthesatisfactionisequivalentwithmaximizingtheutility. Inotherwords,pickingthebestchoiceisequivalentwithchoosing the pair which yields the highest utility.Now the question is Does it matter if we change the number but keep the relative relationintact?For example,pairs(sh,apple) utility(5,0) 90(3,4) 100(4,2) 134(4,5) 156... ...2.1. PREFERENCEANDUTILITY 15In this case,you see the utility from consuming 3 shes and 4 apples is 100,dierent fromthepreviouscase. Soyes, wehaveadierentutilityfunction, butwedidnotchangetherelativerelations. Forexample,(4,2)isstillbetterthan(3,4). Soeventhough,twoutilityfunctions might give us two dierent numbers for one pair, as long as they reect the samepreference ordering, they will induce the same behavior. We will come to this again when wediscuss how to derive the optimal choice.2.1.1 MarginalUtilityNow we need to introduce the concept of marginal utility. Here I just give you the denition,and we will use this concept later. Marginal utility means the additional amount of utilityyou can gain from consuming one more unit of one good. For example,pairs(sh,corn) utility Marginal Utility (per sh)(2,4) 4 NA(3,4) 10 6(4,4) 14 4(5,4) 16 2... ...Inthisexample, whentheamountofshisincreasedfrom2to3, theutilityisincreasedfrom 4 to 10. 6 per sh is the marginal utility evaluated at pair(2,4). (Warning: when youcalculate the marginal utility for one good, for example sh, you need to keep the amount ofother goods constant, say corn in our case. When you say marginal utility, you should alwaysspecify which commodity you are talking about and it is evaluated at which point.)Now lets present the mathematical denition of marginal utility.2.1.2 IndierenceCurveNow we are ready to introduce the useful tool for our analysis, i.e., indierence curve. Indif-ference curve collects all the pairs which give the same utility.Example 1: Linear utility functionSuppose the utility function isu(x1, x2) = x1 + x2, where x1denotesthequantityconsumedof commodity1andx2denotesthequantityconsumed of commodity 2. What this function does is for each pair ofx1andx2it gives anumber(which is the utility from consuming such pair) by summing up these two numbers.Lets see some examples to illustrate this.pairs (commodity 1, commodity 2) utility(5,0) 5(3,4) 7(4,5) 9... ...16 CHAPTER2. CONSUMERBEHAVIORFigure 2.1: Example 1: Linear Indiernce CurveNowletslookfortheindierencecurvethepointsonwhichgiveutility10accordingthisutility function.pairs (commodity 1, commodity 2) utility(10,0) 10(5,5) 10(4,6) 10... ...We can draw this as a straight line in a two-dimensional plane. See Figure 1.Example 2Suppose the utility function isu(x1, x2) = x1 x2, where x1denotesthequantityconsumedof commodity1andx2denotesthequantityconsumed of commodity 2.Lets see the pairs which give utility 10.pairs (commodity 1, commodity 2) utility(10,1) 10(5,2) 10(4,2.5) 10... ...Example 3: Leontif Utility FunctionSuppose the utility function isu(x1, x2) = min{x1, x2}, where x1denotesthequantityconsumedof commodity1andx2denotesthequantityconsumed of commodity 2. What this function does is for each pair ofx1andx2it gives theminimum of the two. For example,u(3, 4) = 3, andu(4, 5) = 4.2.1. PREFERENCEANDUTILITY 17Figure 2.2: Example 2Lets see the pairs which give utility 10.pairs (commodity 1, commodity 2) utility(11,10) 10(12,10) 10(13,10) 10... ...(10,11) 10(10,12) 10(10,13) 10... ...Remarks on indierence curve.1. Two indierence curves which have dierent utility level never intersect.2. The indierence curve which has higher utility level will always lies above the one whichhas lower utility level. We have this because we assume the more the better.2.1.3 MarginalRateofSubstitutionNow we are ready to introduce the concept, Marginal Rate of Substitution, (MRS).If oneunitof agoodisgivenup, inordertokeeptheutilitythesameMRSmeanstheamount of the other good that needs to increase to compensate the loss. This implies we can18 CHAPTER2. CONSUMERBEHAVIORFigure 2.3: Example 3: Leontif Indierence Curvecalculate MRS from the indierence curve since along an indierence curve the utility is thesame.ShapeofIndierenceCurveThe next question is the shape of indierence curve. In the rest of the course, most of time,we assume the strict convexity of the preference, therefore, the indierence curve should havethe shape similar to the one we see in Example 2. We have already mentioned the economicmeaning of the convexity of the preference. Now lets see the consequence of this assumptionon the MRS.MarginalRateofSubstitutionandMarginalUtilityNow it is a good place to give you the formula to calculate MRS. My purpose is not to teachyou the math, but to link the concept of marginal utility with MRS.MRS12 =MU1MU2, whereMRS12meansthemarginalrateofsubstitutionofcommodity1withrespecttocommodity 2, and MU1 and MU2 means the marginal utility of commodity 1 and commodity2 respectively. WHY?suppose now you consume one additional unit of commodity 1, how much utility you gain?That isMU1 by the denition of marginal utility. In order to keep the utility the same, youhavetoreducesomeamountofcommodity2. Howmany? FirstyouhavetoreduceMU1this much of utility resulted from consuming one more unit of commodity 1. We know if youreduce one unit of commodity 2, we will loseMU2 this much of utility, and therefore, to lose1 unit of utility, we have to reduce1MU2this amount of commodity 2. Thus to reduceMU1this much of utility, you have to reduce MU11MU2this amount of commodity 2. And wait,theamountofcommodity2thatneedstobereducedifonemoreunitofcommodity1isincreased in order to keep the utility the same, emmmmm...., what is that?Oh that is MRS.:-).2.2. BUDGETCONSTRAINT 19Figure 2.4: The Budget Set2.2 BudgetConstraintIn this section, we learn how to derive the set of all aordable choices for an individual, andthesetiscalledthebudgetset. Therearetwothingsexogenousforthem, thepricesandtotal resources.Lets rst take a look at the general setup for deriving the budget set. Suppose the person weare considering has incomeIin terms of dollars. And there are commodities in front of him.We use X1 and X2 to denote the amount of commodity 1 and commodity 2 respectively. Weuse P1 and P2 to denote the price of commodity 1 and commodity 2 in terms of dollars. It isnot hard to realize that total use can not exceed the total resources. That says money spenton commodity 1 and commodity 2 can not exceed the total income, I. Thus, we should have,P1 X1 + P2 X2 IThis inequality restricts the choices ofX1 andX2. Lets represent this in a two-dimensionalplane. Recall, wehaveassumedthatthemorethebetter, whichimpliestheindividualswill always choose the combination on the boundary of the budget set, i.e., the budget line.(see Figure 4)It is the good point to introduce the concept of relativeprice. The relative price of com-modity 1 in terms of commodity 2 is the maximal amount of commodity 2 you can have if20 CHAPTER2. CONSUMERBEHAVIORyou give up one unit of commodity 1. In our general setup, the relative price of commodity1 in terms of commodity 2 isP1P2. Why?If you give up one unit of commodity 1, then we save P1dollar, and you can use this many ofdollars to buy commodity. We know with 1 dollar, we can buy1P2this amount of commodity2. And then withP1dollar, we can buyP1 1P2this amount of commodity 2. By denitionthis is relative price of commodity 1 in terms of commodity 2. Graphically what isP1P2?Thatis the absolute value of the slope of the budget line!!Now I want to talk about the changes in the budget set. As I mentioned before, in generalthere are two things which aect the budget set, prices and income. We are considering fourtypes of changes.2.2.1 Applications1. Intertemporal substitutionLets imagine the following situation. When you are just born, you are thinking you mightearn some money when you are young, sayI, and you would earn nothing when you are old.Andyounoticethereisaprogramwhichhelpsyousavesomemoney. Andthisprogrampromises to pay you an interest when you are old for each dollar you invest into it when youare young.We useR to denote the gross interest rate, which means, if you invested 1 dollarwhen you were young, it will return youR dollar including the principal plus the interests.Suppose there is only one good to consume, and we use Cy to denote the quantity of the goodthat you consume when you were young andCo to denote the quantity of the good that youconsume when you are old. And the price of this good is 1 $ per unit and remains the samethroughout your life time. So the situation looks like as follows,young oldIncome($) I 0consumption(quantity) CyCoprice of consumption goods($ per unit) 1 1The question is to write down the life-time budget constraint you are facing. When you areyoung, there are two options for you, spend some money on the consumption and save somemoney. LetsuseStodenotesavingexpressedintermsofdollars. Therefore, themoneyspent on the consumption plus the saving should not exceed the total incomeI, i.e.,1 Cy + S I,whererecallthepriceofthegoodis1$perunit. Whenyouareold,thetotalresourcesavailable to you are saving times the gross interest rate, i.e.,R S. Therefore, we see1 Co R S, whichsaysthemoneyspentonconsumptionwhenweareoldcannotexceedthesavingplus interest earnings. Now do some algebra, we combine two inequalities, and we get,Cy +CoR I2.2. BUDGETCONSTRAINT 21Basically, there are two things in the individuals mind who are making such intertemporaldecision, how much to consume today and how much to consume tomorrow. Denitely theirdecisiondependsonhowimpatienttheyarefordelayingtheconsumption. IfIamtellingyou now Ill give you a brand new car tomorrow, you probably will be quite excited. Whatif I am telling you yes Ill give you a new car, but 20 years from now?I bet you will be lessexcited. On the other hand, their decision also depends on how much they will lose if theychoose to consume earlier, since if they postpone their consumption and save them, they willalways earn some interests later. And this is captured in the gross interest rate. When theinterest rate is high, they will lose more if they choose to consume earlier. So in this sense,the price of consuming today in terms of consuming tomorrow is the gross interest rate. Weoften call this the relative price of consuming today to consuming tomorrow. According tothe denition of relative price, why so?Suppose you give up one unit of consumption today,howmuchyousave? 1 1 = 1dollar. Andyoucaninvestthisamountofmoneyintotheprogram which returns you 1 R dollars when you were old. And how many consumptiongoodsyoucanbuybyusingthisamountofmoney? ThatisR1. Thustherelativepriceofconsuming today to consuming tomorrow should beR, which is the gross interest rate.2. Consumption and LeisureSupposenowyouareinthefollowingsituation. Weallknoweachdaythereare24hours.LetT= 24. And supposew denotes the wage rate, $ per hour. We assume there is only onegood to consume and the price of it is 1 $ per unit. And we useCto denote the quantityof the good that you consume andL to denote the hours you choose to have a rest or haveafun,butnotwork. Sothequestionistowritedownthebudgetconstraint. Nowwecaneasily see the money you earn, that is, (T L) w, the hours you work times the wage rate.And the money spent on consumption should not exceed the money you earn. Therefore, wehave,C (T L) wDo some algebra, we see thatC + L w T wAgain. What is the price of leisure in terms of dollar? Yes the wage rate. If you choose tosleep at home for a hour, you are actually giving up the opportunity to work for a hour whichearns you 1 w dollar.Now lets have some complication.1. supposeif theworkinghoursexceed1/3of T, thatis8hours, youwill begivenaone-time bonus sayB.2. suppose if you decide to work outside the regular working hours, that is 8, you are givenextra dollars, say, for every hour you work beyond the regular hours.3. suppose the worker union forces the congress to pass law which forbids the citizens towork longer than 8 hours each day.22 CHAPTER2. CONSUMERBEHAVIORFigure 2.5: Example 22.3. UTILITYMAXIMIZATION 23Figure 2.6: Utility Maximization: Graphical Presentation2.3 UtilityMaximizationInthesectionof preferenceandutility, I havementionedthat thetheorysuggests thatindividuals behaveas if theyweremaximizingthesatisfactionresultedfromconsuming,andtheutilityfunctioncapturesthesatisfactionindividualsgainfromconsumingcertaincommoditypairs, like3shesand4corns, or4shesand2corns. Sogiventhesetofallpossiblechoices, theindividual will chooseapairwhichmaximizetheutilityfunction. Inother words, the individual will choose a pair which has the highest utility among all thosepairs within the budget set. Our goal is to know under certain prices and income what theindividuals optimal choice will be.We rst take a look at the graphical presentation and then I will give some applications. Inthe end, I give the mathematical presentation.(which is denitely not required but good toknow.)We have already know the absolute value of the slope of indierence curve evaluated at somepointistheMRSevaluatedatthispoint. Fromthegraph, weobservethatifthepointisthe optimal choice, the MRS evaluated at this point, i.e., the absolute value of the slope oftheindierencecurve, isequal toabsolutevalueof theslopeof thebudgetline, whichistherelativeprice. Thisequalityhasquitealotofeconomicstolearn. RecalltheMRSofcommodity 1 in terms of commodity 2 is equal toMU1MU2. And the relative price isP1P2. Then24 CHAPTER2. CONSUMERBEHAVIORwe see,MU1MU2=P1P2Do some algebra, we see that,MU1P1=MU2P2What is this?If you spent 1 $ on commodity 1, we can have how many commodity 1, that is,1P1, if you have one additional commodity 1, how much additional utility you can gain, thatis,1P1MU1. In a word, the termMU1P1means if you spend 1 $ on commodity 1, the amountof utility you can have. Suppose,MU1P1=MU2P2, let s assume thatMU1P1>MU2P2. what will youdo? Move 1 $ from consuming commodity 2,and use it to consume commodity 1,becausethat additional dollar will gain your more utility. Utill when you will stop move from one tothe other. utill they are equal.ExampleNow, lets see an example. Consider a consumer s choice of sh and corn. Suppose his utilityfunction isu(f, c) = f cAccording to his utility function, the marginal utility of sh MUfcan be calculated by usingthe following formula,MUf= cAndMUc = fAnd thus,MRSfc the marginal rate of substitution of sh with respect to corn is,MRSfc =cfNoticethisutilitydoesnothavethepropertyofdecreasingmarginalutilitybutthepropertyofdiminishingmarginalrateofsubstitution.Suppose the consumer has 8 dollars, and the price of sh is 2 $ per sh and the price of cornis 1$ per corn. The following choices are on the budget line.pairs(sh,corn) utility MUfMUcMUfPf|MUcPcMRSfc(4,0) 0 0 4 0 4 0(3.5, 1) 3.5 1 3.5 0.5 3.513.5(3, 2) 6 2 3 1 3 2/3(2.5, 3) 7.5 3 2.5 1.5 2.5 6/5(2, 4) 8 4 2 2 2 2(1.5, 5) 7.5 5 1.5 2.5 1.5 10/3(1, 6) 6 6 1 3 1 6(0.5,7) 3.5 7 0.5 3.5 0.5 14(0,8) 0 8 0 4 0 Now lets have some applications. Back to our example2 (Consumption and Leisure) in theBudget Constrain section.2.3. UTILITYMAXIMIZATION 25Figure 2.7: Utility Maximization: Applications26 CHAPTER2. CONSUMERBEHAVIORChapter3ProducerBehaviorintheCompetitiveMarketIn this lecture, we will see a simple model which attempts to understand producer behaviorin the competitive market.Lets rst take a look at the big picture of the model. We dene a producer as an entity ofa production technology which can transform certain amounts of inputs into certain amountof output. Take a construction company for example. The company needs some constructionworkers rst; second it also needs some materials like concrete, woods, iron, etc; third, someconstructiondeviceslikedrills, arealsoneeded. All theseareinputsfortheconstructioncompany. Andtheoutputwillbebuildings. Wewilluseafunctiontodescribethequan-titative relation between inputs and outputs, which simply says a certain amount of inputscan produce a certain amount output. We call this production technology. In the model weare considering,the production technology is exogenous for producers. In other words,theproducers will take the technology as given. And the activities like R&D are excluded fromthe model. We are interested in the questionhow producers choose the quantity of outputgivenaproductiontechnologyandthemarketpricesofitsoutputsandinputs. Wethinkthe producers choose the quantity of output so as to maximize its prot. In other words, wethink the prot maximization is the objective that rms are trying to achieve. Is that true?...ExampleLets consider an Island where people want to produce baskets. Suppose in order to producebaskets, they only need labor as input, and he is given the following production technology.Labor (hours) Basket (quantities)1 22 53 74 85 8.5... ...2728 CHAPTER3. PRODUCERBEHAVIORINTHECOMPETITIVEMARKETThe table above simply says: if the man works one hour he can make 1 basket; if he workstwo hours, 5 baskets will made. It is often convenient to write down the inverse relation ofproduction which says in order to produce one unit of output the minimal amount of inputis needed. Suppose the inverse relation is as follows,Basket (quantities) Labor1 50 min2 one hour3 one hour and 15 min4 one hour and 35 min5 two hours... ...The table above says: if the man wants to make 1 basket he needs to work 50 minutes; if theman wants to make 2 baskets he needs to work one hour.Now suppose the market price of labor is 1$ per hour and the market price of basket is 0.4 $per basket. The question for us is how much to produce?Now rst lets calculate the cost ofproduction. Since the only input is labor, and we know in order to produce a certain amountof baskets, how many hours are needed, and then we can calculate the market value of thelabor used to produce such amount of baskets.Basket (quantities) Labor Total Cost (dollars)1 50 min 5/62 one hour 13 one hour and 15 min 1.254 one hour and 35 min 1.595 two hours 2... ... ...The table above simply says that if 1 basket is produced, one hour of labor is used and it hasthe worth of 5/6 dollars; if two baskets are produced, two hours of labor, worth 1$ are used.At the sametimewecan calculateif 1basked is sold,what therevenuewill be. Since themarket price of basket is 0.4$ per unit, we know,Basket (quantities) Revenue (dollars)1 0.42 0.83 1.24 1.65 2... ...Everyone knows that the prot is just the dierence between revenue and cost. Then we seethat,3.1. PRODUCERBEHAVIORWITHSINGLEINPUT 29Basket (quantities) Revenue ($) Cost($) Prot($)1 0.4 5/6 -0.432 0.8 1 -0.23 1.2 1.25 -0.054 1.6 1.59 0.015 2 2 0From the information we know so far, the man in island C will choose to produce 4 baskets,which gives the highest prot.This example illustrates how we will proceed in later sections. First we study the productiontechnology from which we can derive the cost function for the producer. And then we moveon to characterize the prot-maximizing choice.Now one more problem is left before we move on. In the title of this lecture, you see a term Competitive Market. What do I mean by that?How that inuences our analysis?. Actuallyit is a big assumption. We will discuss this issue later in prot maximization section.3.1 ProducerBehaviorwithsingleinputIn this section, we only consider the case where there is only one input for the production. Wecandescribeaproductiontechnologybasicallyintwoways: writetheproductionfunctiony =f(x) anddrawthecurveinthegraph. Inthis section, weonlygivethegraphicalpresentation.First lets consider the concept of marginal product (or sometimes we call marginal return).Themarginal productof theinputisjusttheadditional amountof outputwhichcanbeproduced if the one more unit of the input is added. Now suppose I give you a productiontechnologypresentedbyacurve. Howcanyoundoutthemarginal product? Takethefollowing curve for example. (Figure 1)Wecanseethattheslopeoftheproductionfunctionisjustthemarginal product. Ifthemarginal product is decreasing as the quantity of input is increasing, we call this productiontechnology has the diminishing marginal return property. Graphically, that means, the slopeoftheproductionfunctionwillbedecreasing. Ifthemarginalproductisincreasingasthequantity of input is increasing, we call this production technology has the increasing marginalreturn property. The economic meaning of diminishing marginal return is that as the quantityof inputusedforproductionisincreasing, itsproductivityisactuallydecreasing, sinceitgenerates less output if one more unit of input is used.Now let me give you all the possible production technology we will consider in this section.See Figure 2.Here, Iwanttointroducetheconceptofinversefunctionofproductionfunction, say, x=f1(y), where x is input and y is output. The meaning of this function is that given a certainlevel of output,y, the minimal amount of input x is needed to produce such amount of output.30 CHAPTER3. PRODUCERBEHAVIORINTHECOMPETITIVEMARKETFigure 3.1: Marginal Product : An exampleFigure 3.2: Production Function3.1. PRODUCERBEHAVIORWITHSINGLEINPUT 31Figure 3.3: Inverse FunctionFigure 3.4: Inverse Function32 CHAPTER3. PRODUCERBEHAVIORINTHECOMPETITIVEMARKETThe next question is: given a production function, how can I draw the inverse function of itin the graph. See Figure 4.3.1.1 CostThere are several types of cost. There are three relevant for us. The important thing aboutthem is that they are all functions of the quantity of output. In other words, they will varyas the quantity of output changes.Total CostTotal cost: asitsnamesuggests, itjustthetotal costassociatedwithproduction. Inourcase, it only comes from the cost of buying input. Without loss of generality, we can assumethe price of input is 1 $ per unit. Therefore, the total cost of production is just 1 times thequantity of input used to produce certain amount of output.Back to our basket-producing example.Basket (quantities) Labor Total Cost (dollars)1 50 min 5/62 one hour 13 one hour and 15 min 1.254 one hour and 35 min 1.595 two hours 2From the example above, we can see that the inverse function of production is used to cal-culate the total cost.Average CostAveragecostmeansinaveragewhatisthecostforproducingoneunitofoutput, i.e., thetotal cost/the number of output.Example:Basket (quantities) Labor Total Cost ($) Average Cost($ per unit)1 50 min 5/6 5/62 one hour 1 0.53 one hour and 15 min 1.25 0.424 one hour and 35 min 1.59 0.39755 two hours 2 0.4Marginal CostMarginal cost means the additional cost if one more unit is produced.Example:3.1. PRODUCERBEHAVIORWITHSINGLEINPUT 33Figure 3.5: Cost: Example1Basket (quantities) Labor Total Cost (dollars) Marginal Cost(Dollars per unit)1 50 min 5/6 NA2 one hour 1 1/63 one hour and 15 min 1.25 0.254 one hour and 35 min 1.59 0.345 two hours 2 0.41Nowthequestionis: graphicallygivenaproductionfunctionhowtogureoutthecostfunction. Andthenhowtocalculatetheaveragecostandmarginalcost. Weconsidertwospecial cases. See Figure 5 and Figure 6. From Figure 6, we see there is a well-known resultbetween average cost and marginal cost. In the presence of xed cost, the average cost reachesits minimum as it is equal to the marginal cost.34 CHAPTER3. PRODUCERBEHAVIORINTHECOMPETITIVEMARKETFigure 3.6: Cost: Example23.1. PRODUCERBEHAVIORWITHSINGLEINPUT 35Figure 3.7: Graphical presentation of Cost3.1.2 ProtMaximizationI have mentioned in this model we assume that the rms are trying to maximize their prot byusing the quantity of output. Now in this section, we want to have a concrete characterizationas we did in the theory of demand. I said in most of cases, the optimal choice by a typicalconsumer who is trying to maximize his/her utility should satisfy the condition (sometime wecall it the marginal condition), i.e., the marginal rate of substitution is equal to the relativeprice. Here we also have such concrete characterization.Before going to derive such prot-maximizing condition, lets represent the prot in the graph.Everyone knows that the prot is just the dierence between revenue and cost. Now lets seewhat the cost should be associated with a certain level of output. see Figure 7.Nowsupposethemarketofpriceoftheoutputisp$perunit. letsseewhattherevenueshould be in the graph when the output isq. see Figure 8.Now in order to nd out the prot, we can combine Figure 7 with Figure 8. See Figure 9.Now we are ready to derive the condition for prot maximization. lets rst state the condi-tion,marginal cost = the price ofoutput, NOTE: we have to pay very attention to the fact that we have assumed the price of inputis one. If the price of input is not one, the condition should bemarginal cost = the relative price ofoutput in terms ofinput. I need to give several remarks on this condition:1. this condition presupposes the existence of maximization. It is possible that the maxi-mum does not exist at all.36 CHAPTER3. PRODUCERBEHAVIORINTHECOMPETITIVEMARKETFigure 3.8: Graphical presentation of revenueFigure 3.9: Graphical presentation of prot3.1. PRODUCERBEHAVIORWITHSINGLEINPUT 372. we know that the marginal cost varies as the quantity of output changes. Therefore wecould nd the quantity which satises this condition. That quantity is the one whichmaximizes the prot.3. There is one more assumption for the statement to be true, i.e., we are considering theproducers in the Competitive Market. What do you mean by Competitive Market?If amarket is perfectly competitive, we mean the producers in such market are price-takers.In other words, they think their action can not inuence the price. This point is verysubtle. You may say well denitely their action can inuence the price; for example, ifone producer cuts its production, the price will go up as long as the demand remains thesame. You are right. They actually can inuence the price but I assume they feel thatthey can not. Is that a bad assumption?Not really, when the number of producers inthis market goes to innity, any single producer has very tiny inuence on the marketprice. Andthereforetheyfeel theycannotinuencetheprice. Forexample, whenyougotoHEB, whenyouarebuying1gallonofmilk, areyouthinkingthatifyoubuyonemoregallon,thepriceofmilkwillmarkupsignicantly? Ithinkyouwontbecause your purchasing takes up only a tiny part of the demand. This idea applies totheproducersideaswell. IfyouareaproducerofbasketinislandC,andtherearemillions of and tons of competitors out there, will you think you can inuence the priceofbasket? Idontthinksobecauseyoursellingonlyformsatinypartofthesupplyof basket. You will see an amazing justication of this assumption later. I can brieymentionithere. Laterwewill discussmonopoly. Denitelythemonopolistisnotaprice-taker, since it is the only producer and thus apparently it can inuence the price.Furthermore, whenwemodelthebehaviorofmonopolist, wethinktheyactuallysettheprice,usuallyhigherthanthepriceincompetitivemarket. Andthenwewillseethe so-called duopoly, where there are two producers. And we can see the market priceinduopolywhichisalsohigherthanthepriceincompetitivemarketbutlowerthanthepricesetbymonopolist. Question: asweareaddingmoreproducersandastheproducers number goes to innity, will the price converges to the price in competitivemarket?The answer is yes. We will see the details later.Now lets give the justication of prot-maximizing condition. As before, we prove by con-tradiction. Suppose it it not true. And supposemarginal cost < the price ofoutputIn such situation, what the producer will do? If the producer decides to produce one moreunit, the cost associated with this one more unit (NOT the total cost) is just the marginalcost. By selling such one more unit, the additional revenue is just the price of output. Andif marginal cost the price ofoutputIn such situation, if the producer reduce production of one unit, it saves the cost of producingthat one unit, i.e., the marginal cost, and at the same time it loses the revenue, i.e., the priceof output. Sincemarginalcost >the price ofoutput, itimpliesthatactuallyproducing38 CHAPTER3. PRODUCERBEHAVIORINTHECOMPETITIVEMARKETFigure 3.10: Graphical presentation of prot maximizationthis unit and selling it does not gain any prot but instead it causes loss. Thus if the prothas been achieved the maximum, the marginal cost should NOT be higher than the price ofoutput otherwise there is a room to save the loss by reducing production of one unit.Now lets show graphically the prot-maximizing condition stated above is true. see Figure10.Now lets see one exception see Figure 11.Now lets have another graphical presentation of the prot maximization. See Figure 12.3.2 ProducerBehaviorwithtwoinputsInthissection, weconsiderthecasewheretheproductiontechnologyrequirestwoinputs.The same as before, we rst specify the production technology, derive the cost function, andthen by prot-maximization hypothesis we nd the optimal quantity of output. As long asweknowthecostfunction, webacktotheanalysiswehavedoneintheprevioussectionwhere there is one single input. There will be no essential dierence between this section andthe previous one in the manner of deriving optimal quantity of output. The dierences takeplaceonlyinderivingthecostfunction. Therefore, weonlytalkabouthowtoderivecostfunctionwhentherearetwoinputs. Andthenweshouldbeabletoknowhowtondtheoptimal quantity of output with two inputs.Lets back to our story: the producer of basket. Now suppose producing baskets needs laborand bamboo as inputs.Suppose the producer has the following technology,3.2. PRODUCERBEHAVIORWITHTWOINPUTS 39Figure 3.11: prot maximization : exceptionFigure 3.12: prot maximization : Another Graphical Presentation40 CHAPTER3. PRODUCERBEHAVIORINTHECOMPETITIVEMARKETlabor(hours) bamboo(quantities) Basket(quantities)30 mins 1 11 hr 1 11 hr 2 22 hr 3 33 hr 2 21 hr 4 2... ... ...I want to use the example to illustrate the idea of deriving cost function when there are twoinputs. Recall, the cost function simply says in order to produce certain amount of outputwhat the total cost should be to produce that amount. Suppose now the producer wants toproduce6baskets. Therststepistondoutallthecombinationsoflaborandbamboosuch that producing 6 units of basket is possible. According to the technology the producerhas, suppose we have the following combinations,labor(hours) bamboo(quantities) Basket(quantities)3 hr 6 64 hr 6 65 hr 6 66 hr 6 63 hr 7 63 hr 8 6... ... ...Now suppose if the price of labor is 1$ per hour as before, and the price of bamboo is 0.5$per unit. Then we can know the total cost of all combinations which are able to produce 6unitsofbasket. Incontrastwiththecasewherethereisonesingleinput, herewehaveabunchofchoicestoproducecertainamountofoutput. Thequestionis: whatistherulefortheproducertopickonecombinationofinputsgiventhequantityofoutput? Hereweassume that the producer will choose a combination which is the cheapest. In other words,theproducerisminimizingthecostwhilehe/herischoosingthecombinationofinputstoproduce certain amount of output.labor(hours) bamboo(quantities) Cost ($)3 hr 6 3+3=64 hr 6 4+3=75 hr 6 5+3=86 hr 6 6+3=93 hr 7 3+3.5=6.53 hr 8 3+4=7... ... ...From the table above, we see that the producer will work 3 hours and use 6 units of bambootoproduce6baskets, andthiscosts6$whichistheminimalamongallotherproductionchoices.3.2. PRODUCERBEHAVIORWITHTWOINPUTS 41Now suppose the producer wants to produce 8 units of basket. In this case, we can also ndout all the combinations which makes this possible, compare them by their costs, and choosethe cheapest one. Finally we know producing 8 units of baskets will cost the producer 8$.labor(hours) bamboo(quantities) basket(quantities) Cost ($)4 hr 8 8 4+4=84 hr 9 8 4+4.5=8.55 hr 8 8 5+4=96 hr 8 8 6+4=104 hr 10 8 4+5=94 hr 11 8 4+5.5=9.5... ... ... ...Thus we see we could associate any level of output with a minimal cost,and then we derivethe cost function. Suppose we have the following cost function for the man in island C.basket(quantities) Cost ($)1 12 23 34 46 68 8... ...Actually, I did NOT make up this table randomly. What we observe is a linear cost functionwhich is resulted from the special technology function we are using implicitly behind the seriesoftableIgaveabove. Ingeneral, iftheproductionfunctionisLeontief, thecostfunctionderived from it is linear. We will get into that later in more details.3.2.1 ProductionTechnologywithtwoinputsSuppose we consider the production technology with two inputs, capital and labor. Suppose,weuseKtodenotecapitalstockandLtodenotelabor. Everyoneknowsthatthemoneypaid for the service provided by capital is called interest and the money paid for the laborservice is called wage. We use r to denoted the interest, $ per unit and w to denote the wagerate, $ per unit.For example, suppose, capital is a machine, if it is used for a hour, the producer needs to pay400$, therefore the interest is 400$ per hour of usage.Suppose we write the production function as follows,q = f(K, L), where q is the quantity of output. Then there are several major concepts I want to introduce.42 CHAPTER3. PRODUCERBEHAVIORINTHECOMPETITIVEMARKETMarginal ProductWe have already seen the denition of marginal product in the case of single input. We canapplythesamedenitionherewithsomecaution. Themarginalproductofcapitalisjustthe additional amount of output produced if one more unit of capital is added. The marginalproduct of labor is just the additional amount of output produced if one more unit of laboris added. Take the construction company for example,Capital labor output Marginal Product of CapitalMachine(Hour of Usage per day) Worker(persons per day) buildings per month buildings per hour of usage per day2 hrs 10 0.5 NA3 hrs 10 0.9 0.44 hrs 10 1.2 0.35 hrs 10 1.4 0.26 hrs 10 1.5 0.1... ... ... ...Notice, whilecalculatingthemarginal productofcapital,theamountoflaborremainsthesame. And we also notice that as the usage of the machine per day is increasing, the marginalproduct of capital is decreasing. Lets take a look at an example of the marginal product oflabor.labor Capital output Marginal Product of LaborWorker(persons per day) Machine(Hour of Usage per day) buildings per month buildings per person per day11 3 hrs 1 NA12 3 hrs 1.5 0.513 3 hrs 1.8 0.314 3 hrs 2 0.215 3 hrs 2.1 0.1... ... ... ...IsoquantIsoquantisacurvewhichcollectsall thecombinationsof inputswhichproducethesameamount of output.Lets look at the example of the construction company. If the construction company needs toproduce one building in a month, the following combinations of capital and labor are possiblechoices,labor(persons per day) machine(hours of usage per day)11 3 hrs10 3.5 hrs9 4.1 hrs8 4.9 hrs7 6 hrs... ...Marginal Rate of Technical SubstitutionThe marginal rate of technical substitution (MRTS) of capital with respect to labor means if3.2. PRODUCERBEHAVIORWITHTWOINPUTS 43Figure 3.13: MTRSthe one unit of capital is reduced in order to produce the same amount of output, the amountof labor needs to be added. The marginal rate of technical substitution of labor with respectto capital means if the one unit of labor is reduced in order to produce the same amount ofoutput, the amount of capital needs to be added.Lets look at the example of the construction company.labor(persons per day) machine(hours of usage per day) MRTS of labor11 3 hrs NA10 3.5 hrs 0.59 4.1 hrs 0.68 4.9 hrs 0.87 6 hrs 1.1... ...What we observe is that as the quantity of labor used for production is decreasing it needsmore extra capital, additional amount of capital to make 1 building within a month for thecompany. In other words, the quantity of labor is increasing, the marginal rate of technicalsubstitution of labor is decreasing. This is called diminishing marginal rate of technical sub-stitution.Now the question is to nd the MRTS from the isoquant.The shape of isoquant, and its consequence on MRTS. see Figure 21.The following is a formula you may nd analogous to the one we have seen about marginalrate of substitution in the theory of demand,MRTSKL =MPKMPLwhere MTRSKL is the marginal rate of technical substitution, MPK and MPL are marginalproduct of capital and labor respectively.44 CHAPTER3. PRODUCERBEHAVIORINTHECOMPETITIVEMARKETFigure 3.14: The shape of IsoquantWhy this is true?The question is left to you to think about.Returns to scaleSuppose the quantities of capital and labor are doubled. The question is: will the quantityof output be doubled?If the output more than doubles, there are increasing returns to scale.If the output less than doubles, there are decreasing returns to scale. If the output exactlydoubles, there are constant returns to scale. We can see that Returnstoscale is the rateatwhichoutputincreasesasinputsareincreasedproportionately. Taketheconstructioncompanyforexample. If thereareincreasingreturnstoscale, weshouldhavefollowingsituations,labor machine percentage increase of inputs buildings percentage increase of output11 3 NA 1 NA22 6 100% 2.5 150%33 9 200% 3.8 280%... ... ... ... ...Constant Returns to scale,labor machine percentage increase of inputs buildings percentage increase of output11 3 NA 1 NA22 6 100% 2 100%33 9 200% 3 200%... ... ... ... ...Decreasing Returns to scale,3.2. PRODUCERBEHAVIORWITHTWOINPUTS 45labor machine percentage increase of inputs buildings percentage increase of output11 3 NA 1 NA22 6 100% 1.8 80%33 9 200% 2.5 150%... ... ... ... ...3.2.2 CostMinimizationOurgoal hereistoderivethecostfunction. Wehavealreadyknownthatgivenacertainlevel of output, there are a bunch of possible combinations of inputs to produce that amountof output and those choices are on the same isoquant. By cost-minimization hypothesis, wethink the producer will choose a combination of inputs which has the lowest total cost. Whatwe are heading for is to nd a condition which can characterize the cost-minimizing choiceof inputs given a certain level of output. And we will show how to nd the cost-minimizingchoice graphically.In order to do that, we need to introduce the concept of Isocost. Isocost is a straight linewhich collects all the combinations of inputs, capital and labor which have the same total cost.For example, the producer of basket. Suppose, the price of labor is 1$ per hour as before, andthe price of bamboo is 0.5$ per unit. And we are now looking for the combinations of laborand bamboo such that the total cost is 6$. It is not hard to check the following combinationscost 6$ in total,labor(hrs) bamboo(quantities) total cost($)0 12 61 10 62 8 63 6 64 4 65 2 66 0 6Denitelytherearemanymoreothers. Fromthetableabove, wecanseetheisocostwiththe total cost 6$ should be a straight line and we draw it in the graph.What if the cost is 7?The the combinations should be as follows,labor(hrs) bamboo(quantities) total cost($)0 14 71 12 72 10 73 8 74 6 75 4 76 2 77 0 746 CHAPTER3. PRODUCERBEHAVIORINTHECOMPETITIVEMARKETFigure 3.15: IsocostFigure 3.16: Cost MinimizationWhat we observe is that the isocost with the cost 7$ is parallel to the isocost with the cost6$. This really is because the prices of inputs remains the same. Now the question is: whatdoes the slope of isocost mean in economic terms?Now we are ready to show how to nd the cost-minimizing choice of inputs graphically, giventhepricesofinputsandthequantityofoutputtobeproduced. Step1Given the prices of inputs, we can draw a series of isocost with dierent costs.Step2Given the quantity of output, we can draw the isoquant, the combinations of inputs on whichare able to produce that amount of output.Step3Looking for an isocost which is the tangent line of the isoquant.see Figure 23.3.3. LONGRUNEQUILIBRIUMINCOMPETITIVEMARKET 47From the graph, we observe a condition, which is also analogous to the one we nd in utilitymaximization.MTRS = Relative PriceWhy this is true?Similar argument.3.3 LongRunEquilibriuminCompetitiveMarketThis section, we nally touch upon some equilibrium concept, but it is a partial equilibriumconcept, in the sense that we only consider one market with all other markets being exogenous.Take the construction company for example, we consider in this case, how the price of building(output) is determined while the prices of inputs, like wage rate or the rent for the machinesare exogenous for our analysis and we are NOT attempting to understand how the prices ofthem are determined.Thequestionwearetryingtoansweris: supposethereisacompetitivemarketweareconsidering, and suppose there are a bunch of producers in this market. Given the demandcurve, we are looking for the equilibrium price and quantity in this market.Assumptions of a perfectly competitive market1. Price Taking: all producers and consumers are price takers.2. Free entry and Exit3. Product Homogeneity : this implies all producers are facing the same cost function.4. The factor industry will not be inuenced by the output industry.Conditions for the long-run equilibrium1. all producers are maximizing prot2. no producer has any incentive to either enter or exit the industryWith the assumption we made above, and from the conditions for the long-run equilibrium,we see that in the equilibrium, there will be zero prot for each producer. Why?If the protis not zero, more producers will be attracted into the industry. And thus by the condition 2,only when the prot is zero, we have the equilibrium conditions satised.We are ready to use the equilibrium conditions to nd the equilibrium price in a competitivemarket.So how can we nd the equilibrium quantity and equilibrium price?1. First we derive the supply curve of a typical producer48 CHAPTER3. PRODUCERBEHAVIORINTHECOMPETITIVEMARKETFigure 3.17: Cost Function for example 1Figure 3.18: Marginal Cost and Average Cost for example 12. then we see what the supply curve of the whole industry should be.3. the intersection of the demand curve and supply curve gives us the equilibrium quantityand equilibrium price.Example 1Supposeweareconsideringthemarketof basketinislandC. Andeachproducerhasthefollowingtypeofcostfunctionwhichislinear. seeFigure24. Nowwearelookingfortheprice level such that for each producer the prot is zero. Since all producers have the samecost function, we consider a representative producer. This cost just tells us that for a typicalproducer producing 3 baskets costs 6$, 4 baskets 8$, 5 baskets 10$, so on and so forth.What do you think the marginal cost and average cost should be in this case?It should bea horizontal line with the vertical coordinate being 2. see Figure 25.Now the question is which supply curve for each producer?Now suppose the price of outputis 3, what is happening?In order to gain more prot, the producer will produce as much as3.3. LONGRUNEQUILIBRIUMINCOMPETITIVEMARKET 49Figure 3.19: Individual Supply curve for example 1Figure 3.20: Supply Curve and equilibrium for example 1possible, that is, innity. And this is true for any price higher than 2. What if the price ofoutput is 1? The produce will not produce anything, since it is losing money. From above,we can the supply curve of an individual producer should be as follows, see Figure 26.Fromherewecanderivethesupplycurveof basketforthemarket. Whenthepriceofbasket is above 2, the quantity supplied will be innity. When the price of basket is below2, thequantitysuppliedwill zero. Whenthepriceisexactlyequal to2, thesupplywillbeanything.(ItdoesNOTmeanitwillbeinnity. Itonlymeansproducerwillbeequallysatised with any level of output simply because all of them bring zero prot.)So the supplycurveshouldbeasfollows. SeeFigure27. Weseeitshouldbeahorizontal line. Thencombiningwith thedemand curve,wend theequilibriumquantity and equilibriumprice.What we can see is that the price is the one which makes a typical producer gain zero protand the quantity is somehow determined by the demand curve. To see this, suppose for somereason, the demand curve shifts up. The equilibrium quantity will increase.Example 2Now lets look at a dierent example. Suppose in this basket industry, all producer have thesomecostfunctionindicatedbyFigure28. Wehavealreadyknownthemarginalcostand50 CHAPTER3. PRODUCERBEHAVIORINTHECOMPETITIVEMARKETFigure 3.21: Cost Function for example 2Figure 3.22: Marginal Cost and Average Cost for example 2average cost for this case. see Figure 29.Now lets see what the supply curve for any individual producer is. Suppose the price we ndin Figure 29 isp. Then we see when the price of output is larger than and equal to p,thequantity produced should make the marginal cost equal to the price. Therefore, the marginalcost above the price p part is the supply curve. When the price is lower than p, the producerwill produce nothing. see Figure 30Now what do you think the supply curve of the whole basket industry should be?When theprice is larger than p, each producer wants to produce something, and more importantly theyare gaining non-zero prot, which attract more and more producers. Since we have assumedfree entry, as long as the prot is not zero, there will be innitely many producers enteringthemarketwhichmakesthetotal supplyof basketsamounttoinnityeventhougheachproducer produces something nite. When the price is lower thanp,no producer will stayin the market. So we conclude that only when the price isp, the total supply will anything.(First, it does NOT mean each producer is indierent with all level of output. Instead each3.3. LONGRUNEQUILIBRIUMINCOMPETITIVEMARKET 51Figure 3.23: Individual Supply curve for example 2Figure 3.24: Supply curve for example 2one of them will produce q. It does suggest that producers are indierent to enter or exit themarket.) Therefore, the supply curve of the whole industry should be as follows, see Figure31.From this two examples, we could see in the competitive market, the supply curve is a hori-zontal line. And the market price is equal to the marginal cost, and the equilibrium quantityis determined by the demand side. And all producers gain zero prot in the equilibrium.You might ask me why the supply curve is a horizontal line, not a upwards sloping curve?Ifyou relax any assumption we made, we will get a upwards sloping curve.52 CHAPTER3. PRODUCERBEHAVIORINTHECOMPETITIVEMARKETChapter4MonopolyIn this lecture, we are trying to understand how monopolists make their production decision.4.1 QuestioninfrontofamonopolistSuppose, thereisamarketwherethereisonlyasinglesupplier. Weareinterestedintwoquestions:1. Will this producer behave in a dierent manner comparing with a typical producer ina competitive market?2. Will the equilibrium price and quantity in such a market be dierent from the ones ina competitive market?Recall the behavioral assumption we made for a typical producer in a competitive market isthatittakesthepriceofoutputandinputsasgiven. Wehavetogiveupthisassumptionwhen we are analyzing the behavior of a monopolist. Since the monopolist by denition isthe single supplier, and thus it denitely feels it has the power to inuence the price. So wethink a monopolist will consider the impact of its action (decision of how to produce) on themarket price.Now the next question is : do you think the monopolist will choose the quantity of outputarbitrarily?No. The monopolist will try to maximize the prot as any producer does. Thedierence between the producer in a competitive market and a monopolist is the monopolistwill use its power to inuence the market price of output and thus the revenue.Lets take a look at an example. There is a single producer of basket, who denitely is lonely.And suppose the demand curve of basket in front of this lonely monopolist is described bythe equation,p = 9 qwherepdenotesthepriceandqdenotesthequantity. wecanalsodescribethisdemandschedule in a table as follows.5354 CHAPTER4. MONOPOLYFigure 4.1: Demand CurvePrice of Basket Demand for Basket($ per basket) (Quantities)1 82 73 64 56 37 28 19 0... ...We can also draw the line on the graph. see Figure 1.Now we can express the relation between price and quantity in the other way (see the tablebelow), which is sometime called the inverse demand function. This function simply tells youif the quantity demanded is this much, what the price should be so that the consumers willdemand that much. In the table below, we see that when the quantity is 8, the price shouldbe 1$ per basket so that the consumers will demand 8 baskets.Quantity Demanded for Basket Price of Basket(Quantities) ($ per basket)8 17 26 35 43 62 71 80 9... ...4.1. QUESTIONINFRONTOFAMONOPOLIST 55Now why this inverse demand function is relevant for the lonely monopolist in island C? Thispiece of information is important for the monopolist to know what the revenue will be if thehe produces certain amount of baskets. Suppose, he wants to produce 6 baskets, and then heknows that the price will be 3$ per basket. Because, if the price is lower than 3, the quantitydemanded will be more than 6; and if the price is higher than 3, the quantity demanded willbe less than 6, according to the table we have above. And thus, the revenue for producing 6baskets will be 6 times 3, 18 $. Now lets work for all levels of output.Quantity produced Price of Basket Revenue(Quantities) ($ per basket) ($)8 1 87 2 146 3 185 4 204 5 203 6 182 7 141 8 80 9 0... ... ...Nowthemonopolististryingtocalculatetheprotforproducingalllevelsofoutputandthen he can decide which level of output will generate the highest prot. Suppose the costfunction is linear for the monopolist, and its functional form isc(q) = qwhereq is the quantity. we can describe the cost function in the table below.Quantity produced Total Cost(Quantities) ($)8 87 76 65 54 43 32 21 10 0... ...Now, we combine the revenue and cost.56 CHAPTER4. MONOPOLYQuantity produced Price of Basket Revenue Total Cost Prot(Quantities) ($ per basket) ($) ($) ($)8 1 8 8 07 2 14 7 76 3 18 6 125 4 20 5 154 5 20 4 163 6 18 3 152 7 14 2 121 8 8 1 70 9 0 0 0Then we see the monopolist will produce 4 baskets which generates the highest prot.4.2 ProtMaximizationforamonopolist4.2.1 RevenueIn this lecture, we only consider the following type of demand curve,p(q) = a b qwhere q is the quantity and p is the price,and a, b > 0 constant. And then we see the revenuefunction should beR = p q = p(q) q = (a b q) q = a q b q2We need to introduce the concept marginal revenue.Marginalrevenueisjusttheadditionalrevenuetheproducerwillhaveifonemoreunitofoutput is produced. Precisely it should be rst order derivative of the revenue function, wesee then,MR = a 2 b qNow lets draw the demand curve and marginal revenue in the same graph. see Figure 2.4.2.2 ProtMaximizingConditionThe same as before, we are tryingtondthe conditionwhichcharacterize the prot-maximizing choice. Let me rst state the condition.Marginal Revenue = Marginal CostThenletmeexplainwhythisistrueforprot-maximizingquantityof output. Suppose,itisnottrueandMarginal Revenue>Marginal Cost. Whatwill behappening? Themonopolist will nd it is protable to produce one more unit. What if Marginal Revenue 0 constant. We use q1 to denote the quantity of baskets produced by Brian, andq2 to denote the quantity of baskets produced by Justin. The problem for Brian is to chooseq1 givenq2 to maximize the prot.First, we knowq1 + q2is the total quantity supplied, then we know the market price shouldbep = a (q1 +q2), therefore, the total revenue for Brian givenq2 isR = (a q2) q1q21,then the marginal revenue for Brian isMR = (a q2) 2 q1. Let the marginal cost equalto marginal revenue. We have (a q2) 2 q1 = c. Do some algebra, we see that,q1 =a c2q22We call this function, the best response function by Brian, which is the function of the numberofbasketsproducedbyJustin. WecanalsoderivethebestresponsefunctionbyJustininthe exactly same way, we can have,q2 =a c2q12See Figure 1.Now the question is what the equilibrium concept is. The equilibrium concept we will use iscalled Nash equilibrium by John Nash. But for me the idea has been known among economistsfor a quite long time.64 CHAPTER5. DUOPOLYFigure 5.1: Best Response FunctionFigure 5.2: Equilibrium: Example. . . , thegeneral ideaof equilibrium, refers toacertaintypeof relationshipbetween the plans of dierent members of a society. It refers to, that is, the casewheretheseplansarefullyadjustedtooneanother,sothatitispossibleforallofthemtobecarriedoutbecausetheplansofanyonememberarebasedontheexpectation of such actions on the part of the other members as are contained inthe plans which those others are making at the same time.-By Friedrich A. Hayek The Pure Theory of Capital Midway Reprint 1975, p18.Letstransformthequoteintoourlanguage. BrianandJustinaremakingdecisiononthebasis of their expectationof howmanybaskets theother will produce. Inother words,their action depends on the others action. The equilibrium refers to the situation where alltheir plan can be carried out when their expectation becomes true. For example, lets makea=11andc=2, thenwebacktothenumerical exampleIgaveattheverybeginning.Suppose Justin anticipates that Brian will produce 2 baskets. In gure 2, or by the formulaabove,we see that he will produce 3.5 baskets. We see this plan made by Justin is based onhis expectation of how many baskets Brian will produce. If Brian anticipates that Justin willproduce 3.5, he will produce 2.75 which is not what Justin anticipates. In other words, theirplans are NOT compatible.So how can we nd the equilibrium? The intersection of best response function. Now lets5.1. COURNOTDUOPOLY 65Figure 5.3: Equilibrium: Exampleverify this. In the same example as above, if Justin believes that Brian will produce 3 baskets,then Justin will produce 3 baskets. If Brian knows Justin is producing 3 baskets, he will dowhat Justin anticipates. Then their plans are fully adjusted. See Figure 3.So we can nd the equilibrium by looking at the intersection of two best response functions.Mathematically, we can have two equations and two variables, we can solve them.q1 =a c2q22 q2 =a c2q12we have,q1 = q2 = (a c)/3lets see a numerical example. Demand function: p(q) = 10q, cost function: c(q) = 2q. Theresult we have is each of two producers will produce 8/3 , and market price is 108/38/3 =14/3. Lets compare this with the monopolistic market and competitive market. Monopoly Competitive Market Cournot DuopolyEquilibrium price 6 2 4.6Equilibrium quantity 4 8 5.3Prot for each producer 16 0 6.76Theconclusionisthepriceincournotduopolywillbehigherthanthecompetitivemarketandlowerthanthemonopoly,andthequantitywillbelowerthanthecompetitivemarketand higher than the monopoly.Nowsupposethenumberof producerbecomesthree. Whatistheequilibriumpriceandquantity?What if the number of producer becomes 100?...66 CHAPTER5. DUOPOLY5.2 StackelbergDuopolyThesameasCournotduopoly, Stackelbergduopolyalsoconsidersamarketwithtwopro-ducers. The dierence is in Stackelberg duopoly one producer is the leader in the market andthe other is the follower. In Cournot duopoly, two producers choose the quantity of outputsimultaneously. Buthere, theleaderwill choosethequantityrst, andthenitisthefol-lowers turn to choose. The situation becomes dierent because when the follower is makingthe decision,the leaders decision has become a given condition for him/her in Stackelbergduopoly while in Cournot duopoly two producers also need to consider what their competitorwill do but they only have anticipations of their competitors action.Lets take a look at an example. Suppose there are two producers, Brian and Justin, in themarket of basket. Suppose Brian is the leader and he will choose the quantity of basket rst.Suppose, thedemandcurvetheyarefacingisp(q)=10 q. Andthecostfunctiontheyhave isc(q) = 2q. The real thing between them is : On one hand, since Brian is the leaderand therefore he can take a large portion of demand, and this will force Justin to restrict theproduction otherwise the price will be too low to earn some prot; on the other hand, JustincanalsogiveBriansometroublebyincreasingtheproductionsuchthatbothofthemwillnot earn some prot. For example, Suppose Brian chooses to produce 7 baskets, notice thisis a large portion of demand, since when the total quantity supplied is 8 baskets the marketprice will be equal to the marginal cost. Now how Justin deals with this?If Justin produces1 baskets, the prot he can earn is zero. He can earn some prot only when he produces lessthan 1 baskets. Lets calculate what the best decision for him is in this case.Baskets produced by Justin Baskets produced by Brian Total Supply Market price Revenue Cost Prot0 7 7 3 0 0 00.25 7 7.25 2.75 0.6875 0.5 0.18750.5 7 7.5 2.5 1.25 1 0.250.75 7 7.75 2.25 1.6875 1.5 0.18751 7 8 2 2 2 0Then we see that Justin will produce 0.5 baskets and he will earn 0.25 $ if Brian chooses toproduce 7 basket. On the other hand,Brian will earn 3.5 $. Justin might think he can domuch better. He goes to threaten Brian, saying If you produce more than 4 baskets, I willpunish you by producing more baskets to make you lose money. For example, in the case wejust talked about, if Brian produced 7 baskets, what Justin can do to make Brian crazy is toproduce more than 1 basket, say 2 baskets. In a result, the market price will be 1 and Brianactually loses 7 $ while Justin just loses 2$. One thing here we have to pay attention to isBrian has already made the decision while Justin is making his choice. For Brian , after hemade the choice, it it irreversible. So if Brian is trying to take a large portion of the demand,he is also taking the risk of being punished by Justin.Now the question is will Brian be threatened ?Put it in the other way, will Brian take muchcredit of what Justin is saying?in other words, will Brian believe Justin will what he is saying?5.3. BERTRANDDUOPOLY 67The Answer is NO. What Justin said is called EmptyThreat. The threat is empty becauseJustin will not do what he said. Since we have known if Brian produces 7 baskets, the bestchoiceforJustinistoproduce0.5baskets,not2,orsomethingelse. OfcourseifJustinisemotionally unpredictable person, we can not expect him to do something reasonable. Butthe point is the reasonable choice for Justin in that case is to produce 0.5 baskets Not 2.5.3 BertrandDuopolyIn this section, we consider Bertrand duopoly model named after Joseph Louis Bertrand, aFrench mathematician. In this model, we consider a market of two producers as before. Here,thechoicevariablefortheproducerisNOTthequantityofoutputanymore. Instead, theproducer is choosing the price at which level its product is sold. The demand for its productis dependent on its own price and its competitors price. The goal for the producer is still tomaximize prot.Let s take a look at an example. Suppose there are two producers, Brian and Justin, in themarket of basket. The demand curve they are commonly facing isp(q) = 10 q. Supposethey have the same cost function, c(q) = 2q. Brian and Justin will set their prices simultane-ously. The producer with the lower price will take over the whole demand. If the prices arethesame, twoproducerssharethedemand. Forexample, ifforeachbasketBriancharges2$butJustinchargesonly1$,consumerswillgotobuybasketsproducedbyJustin. Andthe quantity will be 9, according to the demand curve. If both of them charges 2$ for eachbasket, the total quantity demanded will be 8, and each one of them will get a half of 8, thatis,4. Soweseethatthepricesetbyoneproducerisdependentonhisanticipationoftheprice that his competitor charges. Now lets me illustrate how Brian and Justin are makingthe decision on the basis of their anticipation of what the other will do. Suppose Brian thinksJustinwillcharge4$perbasket,anypricelowerthan4willbeagoodchoiceforhim,say3.5. But if Justin knows that Brian will charge 3.5$ for each basket, Justin will cut its price,say 3$ per basket. But if Brian knows Justin will cut the price to 3, he will lower the pricefurther. So this process will continue. Until....??Until the price is zero?No until the protiszero. When? whenthepricereaches2$perbasket. Why? SupposeBrianknowsthatJustin will charge 2$ per basket. Then he has incentive to undercut its competitors price ?NO ! If Brian lowers the price to 1$ per basket, he takes over the whole demand (9 baskets),the revenue he gains is 9$, what about the cost?18$ !!!, Brian is losing money. It is true forany price lower than 2. So if Brian anticipates Justin will charge 2 dollars for each basket,he does not have incentive to undercut the price, will he charge more than 2?To answer thisquestion. Suppose Brian charges 2 dollars also. The prot he gains is zero. If he charges morethan 2 dollars,no consumer will come to him and thus he gains nothing. So he is actuallyindierent to these two choices.Now the question is what the equilibrium should be?The Nash equilibrium here should be:bothBrianandJustincharge2dollarsforeachbasket. Why? IfBrianknowsJustinwillcharge 2$,he is indierent between setting price at 2 and higher. If Brian also charges 2$,Justinwill nothaveincentivetochangehisdecision, still settingpriceat2. Sothisisa68 CHAPTER5. DUOPOLYresting point.Now let me give the formal presentation of the model.Suppose the cost function for both two producers isc(q) = c qI useq1to denote the demand for Brians product, andq2the demand for Justins product.I usep1to denote the price set by Brian, andp2the price set by Justin. So the demand forBrians product is dependent on the price set by Brian and Justin, we haveq1 = a p1 + 1/2 p2ThisdemandfunctionforBriansproductsaysthatthehigherthepricesetbyBrian, thelower the demand for his product; the higher the price set by Justin, the higher the demandfor Brians product. Symmetrically, we have the demand curve for Justins productq2 = a p1 + 1/2 p1There is one thing we need to pay attention to is that for each producer the price set by itscompetitor is exogenous; in other words, each producer takes the price set by its competitoras given.Step1Therststepistoderivethebestresponsefunctionforeachproducer. Letsrstlook at Brians problem. Given the price set by Justin,p2, and facing the demand curve forhisownproductq1=a p1 + 1/2p2, Brianistryingtomaximizetheprotbysettingthepricep1.Now what is the revenue?R = p1 q1 = p1 (a p1 + 1/2p2)What is the cost?C = 2 q1 = c (a p1 + 1/2p2)What is the prot?Profit = R C = p1 (a p1 + 1/2p2) c (a p1 + 1/2p2)Then we haveProfit = (p1 c)(a p1 + 1/2p2)Lets take the derivative of prot function with respect top1, we have(a p1 + 1/2p2) (p1 c)setting it to zero,we can obtain the price which maximizes the prot for Brian whenp2isgiven,p1 =a + c2+p245.3. BERTRANDDUOPOLY 69Figure 5.4: Best Response Function in Bertrand DuopolyThisisthebestresponsefunctionbyBrian. WecanhavethebestresponsefunctionbyJustin by doing the similar thing,p2 =a + c2+p14Lets draw the best response function in the graph. As before, the equilibrium should be theintersection of the best response functions. We can know then, in the equilibrium the priceset by two producers should be the same, that is,2(a+c)3. m should be the intersection of thebest response functions. We can know then, in the equilibrium the price set by two producersshould be the same, that is,2(a+c)3.