Price Theory and Appications

Price Theory and Applications

Ennio Emanuele Piano

December 2, 2015

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Chapter 1

Introduction

1.1 The Task of Economics

Economics is the study of the allocation of scarce resources among competingends, when the object of the allocation is the maximization of such ends. Theallocation of scarce resources among competing end is often referred to as theeconomic problem. The two fundamental notions, in identifying an economicproblem, are scarcity and competing ends. Scarcity is pervasive in humanaffairs. It is the direct and unavoidable result of human nature and the insatia-bility of human wants. If resources were superabundant, and not scarce, thanall human wants could be satisfied and no problem of allocation, that is, nocompetition for such resources, would emerge.

Scarcity in resources implies that there must be a process through whichthese resources have to be allocated. In general, the two alternative when choos-ing how to allocate resources are cooperation and conflict. Throughout history,mankind has experienced a variety of systems for the allocation of resources:

• Involuntary transfer of goods: Goods are allocated through violentmeans. Theft and some government policies have in common the fact thatthey result in a transfer of rights that often differ from what would emergeif only voluntary transfers were to happen.

• Voluntary transfer of goods: Gifts and market exchanges are twocategories of voluntary transfer of goods and services.

• Command: organizations often allocate resources through command in-stead of violence or exchange.

Economics, as a positive science, does not tell you which is the best methodfor the allocation of resources. In the case of a boxing match, the allocationof the winner’s prize is often, though not always, accomplished through theuse of violence, and when exchange between the parties involved is employed,the purpose of the match itself is compromised. Families, on the other hand,

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4 CHAPTER 1. INTRODUCTION

often use a mix of gifts and command for the allocation of resources, whilefirms and other organization choose command to avoid to incur in the hightransaction costs associated with the use of the price mechanism. Finally, mosteconomies rely on voluntary transfer of goods for the allocation of resourcesamong producers as well as consumers.

It is important to understand that scarcity is not the same thing as a short-age. Scarcity is pervasive. Whenever we observe a price, we know that thepriced good is scarce. Sometimes, scarce goods are not priced because of avariety of reasons, including government interventions and moral taboos. Anon-scarce resource has no price, since nobody would pay for something thatis freely available to him at any moment and everywhere. A shortage, on theother hand, consists in the failure of the price system to bring demand to meetsupply. A shortage emerges when the price is ”wrong”, and consumers wouldlike to consume a quantity of the good that is larger than the quantity that sell-ers are willing to give up for that price. Shortages are often due to governmentintervention in setting the price below its equilibrium level, that is, imposing aprice ceiling, or alternatively by an error on the part of the seller. In the lattercase, though, we can expect the shortage to last far less than in the former one,since the price mechanism will incentivize the seller to adjust the price in orderto increase profits.

1.2 The Scope of Economics

Economics as the study of the formation of prices in the market system focuseson four forms of behavior:

1. Production: Production consists in all those processes that change thecharacteristics of resources in order to enhance their capacity to satisfy in-dividual wants. Physical and chemical transformations are therefore to beconsidered as production processes, as well as changes in the geographicalcharacteristics of the resources.

2. Consumption: Consumption consists in the destruction of goods for thepurpose of satisfying the wants of the individual. Economists refer to thisprocess as utility maximization. Utility maximization is not limited tomaterialistic wants, but include every and each good, service, etc. thatthe individual sees the consumption of which as an improvement relativeto its current position, where the evaluation of such an improvement areentirely subjective.

3. Exchange: An exchange is a transfer of property rights over the servicesof particular goods. Exchanges can be voluntary or not. In the market,though, all exchanges are always assumed to be voluntary. For an ex-change to take place, the two parties must value the good offered by theother party more than the good offered by himself.

1.3. THE METHODOLOGY OF ECONOMICS 5

4. Specialization: Specialization occurs whenever some individuals or groupproduces more than he expect to consume, for the purpose of exchangingit for a different goods. Specialization tend to increase the productivityof all parties involved, as well as their ability to attain an higher level ofutility.

Another important notion in the study of markets, and especially in theanalysis of the behavior of consumers and firms, is that of cost. The cost of anindividual’s choice is here defined as the best among all forsaken opportunitiesthat the agent faced at the moment of the choice itself. Thus, cost is alwaysopportunity cost. Unfortunately, the economic notion of cost is often confusedwith the layman notion of cost as the price one has to pay for a good or ser-vice. This confusion is highly misleading. In evaluating courses of actions, it isirrelevant what the absolute monetary cost of taking one action is. What arerelevant are the net benefits associated with all courses of action (independentof the budget constraint of the individual, of course). An action that costs 10,but gives a utility of 1000, will be preferred to one that costs 1, but gives onlya utility of 200.

1.3 The Methodology of Economics

Economics is a positive science, meaning that it only answers questions aboutwhat is, what was, and what will be. It does not answers questions about whatshould be. Economics tries to formulate general laws of causation (if A thenB), and in doing so it focuses only on few of the myriad of features of humannature and of social life. Economics relies on a precise understanding of someuniversal traits of human nature. Individuals are seen as rational utilitymaximizers, meaning that their behavior must be interpreted as always directedtoward the satisfaction of some subjective end.

The unit of analysis of the study of markets is always the individual. Theindividual in question is, as I said, assumed to be a utility maximizer, to havesubjective preferences, and to be able to rank different goods according to hissubjective preference. From these, it follows that the individual allocates hisown budget among services and goods according to a cost benefit analysis thatdeals with marginal units. He will always spend the next dollar to the goodor services the next unit of which is going to give him the most satisfaction.Since he has multiple wants to satisfy, all goods and services have a diminishingmarginal utility, meaning that as more units of the same goods are consumed,the satisfaction that he is going to derive from the next unit is going to be lowerand lower, that is, he will shift to consume units of a different good as soon asthe first on it of the latter gives him more satisfaction then the next unit of theformer. The individual is also assumed to have not just one, but a multiplicityof wants, and that he acts under circumstances of scarcity.

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1.4 The Postulates of Consumer Theory

1. Each person desires many goods and has many goals.

2. For each person, some goods are scarce.

3. Each person is willing to forsake some of a good to get more of othergoods. All values are relative.

4. The more of a good one has, the larger the total personal use value, butthe lower the marginal personal value of a unit.

5. Tastes are subjective.

6. People are innovative but consistent in their choices.

Chapter 2

Equilibria and Optima

2.1 Introduction

Price theory analyses its subject matters by using the tween notions of equilib-rium and optimum. Equilibrium theorizing asks questions about what would bethe final state of rest of individuals acting rationally and in their self interest.To do so, it focuses on those points at which supply and demand intercept,that is, on the equilibrium points. Optimization is the science of how to get tothe best possible outcome given the circumstances. Optimization focuses on thecomparison between marginal magnitudes, such as marginal costs and benefits.

2.2 Equilibrium Analysis

The supply and demand schedule constitutes the most basic use of the notion ofequilibrium in the study of market phenomena. Supply and demand analysis,though simple, is based on a sophisticated understanding of the behavior ofconsumers on the one hand, and of firms on the other. From the economictheory of behavior there emerge some laws, which are the foundations of supplyand demand analysis. These are the two fundamental laws of demand and thelaw of supply.

2.2.1 The First Law of Demand

The first law of demand says that, for any given good Xi, there always is a pricep1 > p such that the individual will buy a quantity q1 < q. Similarly, there isalways a price p2 < p such that the individual will buy a quantity q2 > q.

The law of demand tells us that all demand curves slope downward, meaningthat as the price of a unit of a particular good increases, consumers would onlyby fewer and fewer units of that same good. The law of supply, on the otherhand, tells us that as the price of a unit of a good increases, more and more

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8 CHAPTER 2. EQUILIBRIA AND OPTIMA

firms would like to enter the market and the total quantity of the good offeredwill increase, thus making the supply curve slope upward.

With the potential exception of Giffen and Veblen goods, where the demandcurve might go up for a little bit at relatively high prices, the supply and demandcurves intersect at one point, referred to as the equilibrium point. Demand andsupply equilibrate when, at the price p∗, the sum of the quantities demandedby all consumers at that price equals the sum of the quantity supplied. Pricetheorists often uses the method of comparative statics to analyze how changes inthe data of the market affect the equilibrium price and quantity. These changescan originate within or outside the model. Changes within the model include: i.changes in prices or quantities of substitute goods, as well as in complementarygoods, which affect the demand curve; ii. changes in prices and quantities ofgoods ”related in supply” (as is the case with cattle beef and hides), which affectthe supply curve; iii. changes in income, which also affect the demand curve.Changes outside the model include: changes in technology and resources, whichaffect the supply curve, changes in tastes, that affect the demand curve, butalso changes in the legal and regulatory environment, which affect both supplyand demand.

In doing comparative statics, one must always distinguish between move-ments of the curve itself and movements along the curve, or in other words, thedifference between quantity demanded and quantity supplied on the one hand,and demand and supply on the other. Quantity demanded is the particularvalue assumed by qd at a given price. Demand is a function representing allquantity demanded at all prices. Thus, qd might change without the demandschedule changing, and it might (in some extreme cases) remain the same eventhough the entire demand schedule has shifted. Thus, an increase in demand isa movement of the demand curve itself, an outward curve in this case, while anincrease in quantity demanded is a movement along the demand curve. Move-ments of the curve affect p* and q* in the following ways:

• an increase in demand, with supply staying constant, brings about p’greater than p* and q’ greater than q*

• an increase in supply, with demand staying constant, brings about p’smaller than p* and q’ greater than q*

• an increase in both supply and demand brings about an a new equilibriumprice which is either greater or smaller than the previous one, accordingto which movements is the larger one, and a q’ greater than q*

2.2.2 Supply and Demand

Both supply and demand intercept the ordinate line P. The intersection of de-mand and P is called choke price of demand. The choke price of demandindicates the situation in which the price is so high that no consumer in themarket would like to buy units of the good. The intersection between the supplycurve and P is called choke price of supply. At the choke price of supply,

2.2. EQUILIBRIUM ANALYSIS 9

the price is so low that, although many consumer would be willing to buy it, nofirm would like to supply even one unit of the good.

The most simple mathematical expression of supply and demand curves isthe linear one:

P = a− b ∗Q (2.1)

2.1 is the linear equation for the demand curve, where ’a’ is the value of thechoke price of demand and ’-b’ is the slope (the sign of which is, as I discussabove, negative) of the curve.

P = c+ d ∗Q (2.2)

2.2 is the linear equation for the supply curve, where ’c’ is the choke price ofsupply and ’d’ is the slope (the sign of which is positive) of the curve. Atequilibrium, by definition, the quantity demanded and the quantity suppliedmust be equal, and in the absence of intervention by a third party, also theprices paid by consumers and those received by suppliers must also be equal.Thus, combining the two we obtain the following system of equations:

P = a− b ∗Q (2.3)

P = c+ d ∗Q (2.4)

which, once solved, gives the following solutions:

P = [(a ∗ d) + (b ∗ c)]/(b+ d) (2.5)

P = (a− c)/(b+ d) (2.6)

In a national market, the total supply curve may be the result of the sumof a domestic supply curve and of an import supply curve. To find the totalsupply curve, we must express the two supply functions in terms of Q, and thenadd them. The equilibrium price and quantity would then be found by theintersection of the demand curve and the newly obtained total demand curve.

Third party interventions can influence the equilibrium of a market in avariety of ways. The most simple case is that of state intervention thoughtaxation. States have two methods for taxing consumption: though unit taxes,and through percentage taxes. A unit tax consists in a fixed amount T , whichcan either be levied on suppliers or consumers. In the presence of a unit tax,the price paid by consumers is equal to the price received by the suppliers plusthe value of T , or, that is the same, the price received by the suppliers is equalto the price paid by the consumers minus the amount of the tax:

Pd = Ps+ T (2.7)

The effect of a unit tax can be interpreted as shifting the demand curve inward,which reduces the equilibrium quantity from Q∗ to Q′, while at the same timeproducing state revenues equal to the new equilibrium quantity times the valueof the unit tax:

R = Q′ ∗ T (2.8)


The effect of a proportionate tax is slightly different. Under a proportionalsystem of taxation on consumption, the price received by seller is equal to thatpaid by the consumer times the difference between one and the value of theproportionate tax t:

Ps = Pd ∗ (1− t) (2.9)

Unlike the unit tax, the proportionate tax dos not only shifts the demand curveinward, but it also changes (i.e., increases) its slope.

Another way for governments to interfere with the dynamics of the marketis by imposing price floors and price ceilings. A price ceiling occurs when thegovernment imposes a maximum mandatory price to buyers and sellers. If thisprice is above the equilibrium price, the effect of the policy is null. If, onthe other hand, the price is below the equilibrium price, the effect would beto reduce the quantity supplied, and therefore, the quantity exchanged in themarket. In fact, for a price p′ less than p∗, buyers would be willing to buy moreunits of the good, but at the same time sellers would only like to sell fewerunits. A price floor is the exact opposite of a price ceiling. The governmentimposes a price floor when it considers the equilibrium price ”too low”. Thenew minimum price would therefore be above the equilibrium one, once againsprovoking a contraction of the quantity exchanged. In the presence of a pricefloor, sellers would be willing to sell more units of the good, but their hopeswould be frustrated by the fact that buyers now want fewer units.

2.3 Optimization Analysis

Optimization consists in ”finding the action that leads to the best outcome”.Price theory focuses on two generalized types of optimization: i. Utility max-imization (by consumers or households); ii. Profits maximization (by firms).Price theory finds the solution to these optimization problems through marginalanalysis. Marginal analysis consists in the comparison at the margin between twomagnitudes. In the case of utility maximization, these magnitudes are benefitson the one hand and costs on the other. In the case of profits maximization,these magnitudes are revenues on the one hand and costs on the other.

Marginal analysis uses three types of magnitudes: Total magnitudes; Av-erage Magnitudes; Marginal magnitudes. These magnitudes are reciprocallyrelated. Let us take for example the case of a firm. A firm is assumed to careabout profits, and would therefore do whatever is needed to maximize them.Profits will be maximized when the production of the marginal unit of the goodis expected to costs as much as it is expected to generate in revenues. Sincethere would be no profits in doing so, the firm will not produce that unit. Forthe sake of argument, let us assume that the firm does in fact produce beyondthe quantity at which the marginal unit produced generates zero profits. Thenext unit would then cost more than it is expected to spur in revenues, whichmeans that the firm would incur in a net loss compared to before it producedthis unit. Thus, a profit maximizing firm would not produce it. The same rea-soning applies if a firm were to stop production before reaching the MR = MC

2.3. OPTIMIZATION ANALYSIS 11

condition. In this case, by producing the next unit, the firm would receivefrom it revenues greater than costs. A profit maximizing firm would thereforecontinue production.

The algebraic relation between Total, Average, and Marginal magnitudes isas follows. An average is the ration between the total magnitude (i.e., TotalRevenue) and quantity. A marginal magnitude is the ratio between an increasein the total magnitude (i.e. Total Costs) and the increase in quantity. ThusMarginal Cost (MC) is equal to the increase in total costs (∆C) divided by theincrease in quantity (∆Q). For very small changes in quantity, the relationshipbetween marginal magnitudes can be expressed using calculus:

MC =dC

dQ(2.10)

When only discrete choices can be made, the optimal total magnitude is foundwhen the smallest upward movement brings about MR less than MC and thesmallest downward movement brings about MR greater than MC.

When, on the other hand, infinitesimally small units can be used, the relationbetween T, A, and M has also algebraic and geometrical meanings. The marginalmagnitude is in fact the derivative of T in terms of Q, which means that it isalso the slope of the T curve. A, on the other hand, is the slope of a ray theorigin to the T curve. These relation gives us some more informations about thebehavior of the T, A, and M curves: i. When T rises, M is always positive; ii.When T falls, M is always negative; iii. When T is at its maximum or minimum,M is zero; iv. When M is lower than A, A always falls; v. When M is greaterthan A, A always rise; vi. A and M intersect when A is at its minimum.


Chapter 3

Exchange, TransactionCosts, and Money

3.1 Pure Exchange

The simplest model of a market economy is the two players two goods Edgeworthbox. The Edgeworth box is a rectangle the dimensions of which are determinedby the total quantity of each product that are in the market. In the pure exchangeversion of this model (which excludes the possibility of production), the lengthof the height of the rectangle (Y ) is determined by the sum of the sum of thequantity of the good y owned by A (ya) and that owned by B (yb):

Y = ya + yb (3.1)

The width of the rectangle, on the other hand, is determined by the total quan-tity of the good X in the economy. It is therefore the sum of the quantity of Xowned by A and that owned by B:

X = xa + xb (3.2)

Both A and B are assumed to have well-behaved indifference curves, with theirindifference maps covering the entire surface of the Edgeworth box. Beforeexchange takes place, the reciprocal distribution of good X and Y is identifiedby the point E, which is also a point in which the indifference curves of Aand B intersect. The two indifference curves intersecting at point E delimit anarea called Region of Mutual Advantage. This name expresses the fact that eachindividual would be better off if she could move to any point above the indifferencecurve going through E, and that voluntary exchanges would only occur when botindividuals were made better off, which means that no exchange can result at apoint outside the area delimited by the two curves. Thus, if allowed to exchangequantities of the two goods, the two will end up at point T, which is also thepoint at which two new indifference curves intersect. This process of exchange

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14 CHAPTER 3. EXCHANGE, TRANSACTION COSTS, AND MONEY

continues until the two reach a point at which the indifference curves do notintersect, but are tangent to each other (graphically, at the tangency point,the Region of Mutual Advantage would disappear). Within the Edgeworth box,there is an infinite number of tangency point between A’s and B’s indifferencecurves. These points form a curve called Contract Curve. Algebraically thepoint at which A and B will rest, that is, the final equilibrium point, can bedetermined by solving the following system of equations:

Px ∗Xa + Py ∗ Ya = Px ∗ xa + Py ∗ ya (3.3)

Px ∗Xb + Py ∗ Yb = Px ∗ xb + Py ∗ yb (3.4)

MRSCa = Px/Py = MRSCb (3.5)

Where 3.3 is A’s budget line, 3.4 is B’s budget line, and 3.5 is expresses theidentity solution for a utility maximization problem (the slope of the indifferencecurve equals the slope of the budget line). Equation 3.5 also shows that there isonly one pair of relative prices such that the two indifference curves are tangent,for any given initial allocation of X and Y .

3.2 Supply and Demand in Pure Exchange

To better understand the results of the analysis of the previous section, wemust focus on the behavior of each individual separately. Each individual canbe seen as having two kinds of demand and two kinds of supply. A person’sFull Demand is the quantity that she ends up with at the end of the exchangeprocess. A person’s Transaction Demand is the quantity acquired during theprocess, or, in other words, the difference between her Full Demand and herInitial Endowment. A person’s Full Supply corresponds, in the absence ofproduction, to her Initial Endowment. Her Transaction Supply, on theother hand, is the quantity that she ends up selling in the exchange process, or,in other words, the difference between her Full Supply and her Full Demand :

• Full Demand: xi

• Full Supply: xi

• Transaction Demand: xti = xi − xi

• Transaction Supply: −xti = xi − xi

The price ratio at which xti and −xti both equal zero is called ’AutarkyPrice Ratio’. As the name suggests, at this price at the individual prefer isinitial endowment to all the alternatives, and will therefore restrain from anyexchange whatsoever.

The sum of all individual Full Demands is called Market Full Demand, andthe same applies also to Market Full Supply, Market Transaction Demand, andMarket Transaction Supply :

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• Market Full Demand: X

• Market Full Supply: barX

• Market Transaction Demand: Xt=X-barX

• Market Transaction Supply: -Xt=barX-X

3.3 Exchange and Production

In a market economy, people do not just exchange goods from their initialendowment, but also use their labor (and resources) to produce goods thatcan then exchange for other goods in the marketplace. Each individual has aProduction Possibility Curve of the form:

Q(x, y) = axc + byd1 (3.6)

An example of ProductionPossibilityFunction is:

Q(x, y) = x2 + y2 − 16 (3.7)

In a Robinson Crusoe economy (that is, an economy in which exchange is ruledout), the individual maximizes his utility by producing the bundle goods result-ing from the tangency point between her Production Possibility Curve and herIndifference Curve. In other words, her Production Optimum (R) is also herConsumption Optimum (C). Algebraically, this Robinson’s Optimum is foundby equating her Marginal Rate of Substitution in Consumption to her MarginalRate of Transformation:

MRTS = MRT (3.8)

that is, by equating the first order derivative of the Production Possibility Func-tion to the first order derivative of the Indifference Curve.

By creating the opportunity for trade, the market economy separates pro-duction from consumption. Each individual will produce such as to maximizeher budget line, given the prevailing prices in the market2, by selling a positivequantity of a good, and then use the profits from this sale to buy the optimalquantity of the other good. Individual Full Supply and, thus, Market Full Supplyare a function of the prevailing price for the good in the market.

This discussion highlights the two main advantages of trade: i. Trade al-lows for a better allocation of existing goods; and ii. Trade allowsfor specialization in production. Both these advantages guarantee a bettersituation for everyone involved.

1For simplicity, we are assuming an economy with only two goods2I am assuming a perfectly competitive market.


3.4 The Costs of Exchange

Real markets are seldom perfect. Among the many sources of market imperfec-tion are:

1. Imperfect Communication (Ignorance)

2. Nonunique Prices

3. Transaction Costs

Transaction Costs do not include the costs of physical transfer of commodities.Indeed, the former only exists in a market economy, while the latter will alwaysalso exist in a command economy, that is, in a context where exchange is ruledout. Transaction costs are, as the name suggests, the costs associated with car-rying out a transaction: the costs of communicating and disclosing information;the costs of negotiating and enforcing a contract; and so forth. Transactioncosts can either be proportional or lump-sum. Proportional transaction costshave the same effect of a per unit tax in that the price paid by the buyer exceedsthe price received by the seller by the amount of the cost:

p+x = p−x + g (3.9)

where g is the value of the Per-unit Trading Fee. In the presence of proportionaltransaction costs ”the quantity exchanged [...] is necessarily smaller than the’ideal’ equilibrium amount” (Hirshleifer et al., 2005: 434). Transaction costsaffect the behavior of individuals in production and exchange by creating a gapbetween sellers’ price and buyers’ price. As this gap increases, each individualfinds herself closer and closer to the autarky price. Indeed, for the market as

3.5. THE TRANSACTION COSTS THEORY OF MONEY 17

a whole, for large enough transaction costs, the market can disappear entirely,which geometrically means that the demand and supply curves intersect in thenegative quadrant.

Lump-sum transaction costs are the reason why buyers and sellers hold in-ventories. In the presence of a lump-sum transaction cost, consumer may findpreferable to incur in some inventory costs, and, instead of buying the goodjust before consumption, store it and allocate consumption throughout the timeinterval T . As T ends, each consumer will buy some quantity of the good onceagain and consume it throughout T2, and so forth. So, given a lump-sum cost F ,an individual full consumption flow x, and a production flow x lower than x, theindividual will maintain her consumption flow constant by making a purchaseof size L at any T :

x = x+ (L/T ) (3.10)

The value of lump-sum transaction costs is given by the sum of the ’entry fee’(the lump-sum payment proper) and the cost of keeping an inventory:

V = F/T + hTL/2 (3.11)

Where F/T is the transaction fee per unit of time and hTL/2 is the average costof the inventory. Equation 3.11 shows that as T increases, the average lump-sum cost per unit consumed decreases, while the inventory costs increase. Aconsequence of lump-sum transaction costs is that they transform the autarkyprice of the transaction costs-free world in an interval. Unlike proportionaltransaction costs, though, lump-sum transaction costs do not generate a pricegap between what is paid by consumers and what is received by sellers.

3.5 The Transaction Costs Theory of Money

Money is an institutions that reduces the costs of transacting. It does so byperforming two functions: i. Medium of exchange; and ii. Temporary store ofvalue. Money emerged of exchange in order to reduce the search costs associatedwith finding a counterpart who does not only sells a good for which we havea positive transaction demand, but also has a positive transaction demand forthe good for which we have a positive transaction supply. As a store of value,on the other hand, money solved the problem of inter temporal consumption:a seller may store the money received for a payment in order to allow himselfto consume it in the future. In order to become the prevalent money in aneconomy, a commodity should therefore be relatively cheap to produce and tostore, must have a relatively stable value, and be difficult to counterfeit.

3.6 Auctions

Every market economy is characterized by a variety of exchange mechanisms: i.bilateral exchange; ii. broker markets (which can take the form of geographicaland chronological arbitrage); and iii. Auctions. Auctions emerge to overcome


the problem of asymmetrical information between buyers and sellers. Unlike inmonopoly markets, in an auction the seller does not know the buyers’ reservationprice (or willingness to pay). There are four types of auctions:

1. English or Ascending − Price

2. Dutch or Descending − Price

3. First− Price

4. Second− Price

English and Dutch auctions are done according to the ’public outcry’ method, inwhich each bidder publicly announce her bid. First- and Second-Price auctionsare follow the ’Sealed-Bid’ method. In the English auction, each bidder’s beststrategy is to bid until her own reservation price is reached. The winner willtherefore pay only one unit more than the second bidder’s reservation price.Although they follow a different method, Second-Price auctions are analyticallysimilar, with the winner paying the bid made by the second highest bidder,which exactly corresponds with the latter’s reservation price.s In a sealed-bid,First-Price reservation price, on the other hand, the best strategy is alwaysto bid half your reservation price. Once agains, the Dutch auction, althoughfollowing a different method, arrives has the same optimal strategy: biddingonly half your reservation price.

Chapter 4

The Theory of ConsumerBehavior I: Utility andPreference

4.1 Introduction

The theory of consumer behavior deals with the problem of optimization in con-sumption, and is therefore an application of optimization theory, which meansthat, in its analysis, it makes use of marginal magnitudes such as Marginal Util-ity and Marginal Cost. The two fundamental concepts of Consumer Theory areInstrumental Rationality and Utility Maximization.

4.2 The Laws of Preference

The theory of consumer behavior builds on some fundamental axioms:

1. The Axiom of Comparison: Consumers are able to compare any two bas-kets of goods. For any pair of such bundles A and B, a consumer musteither prefer A to B, prefer B to A, or be indifferent between the two.

2. The Axiom of Transitivity: Transitivity requires that, if a consumer prefersA to B, and B to C, then it must be that she prefers A to C.

3. The Axion of Completeness: Any possible baskets within the consumptionset is either more preferred to another or is less preferred.

The complete ordering of all possible bundles in the reach of the consumercan be expressed by a preference function.The bundles that are ordered by thepreference function can contain three types of commodities:

• Goods: A commodity is called a good whenever more of it is preferredto less, that is, whenever the marginal utility derived from it is positive.

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20CHAPTER 4. THE THEORYOF CONSUMER BEHAVIOR I: UTILITY AND PREFERENCE

• Neuter Commodities: A commodity is called neuter whenever the con-sumer’s utility is not influenced by either an increase nor a decrease in itsquantity, that is, the marginal utility derived from it is zero.

• Bads: A commodity is called a bad whenever less of it is preferred tomore, that is, whenever the marginal utility derived from it is negative.

This categorization is not intrinsic in the commodity itself, but rather dependson consumer preferences. Indeed, the same commodity can be a good, a neutercommodity, and a bad not for different consumers, but it can also be so for asingle consumer. I can prefer more of a commodity (which is thus a good), butonly up to a point, beyond which I am indifferent to an increase in its quantity,but as it increases even more, I would actually like to have less of it, whichmeans that it has become a bad to me.

4.3 Utility and Preference

Price theory assumes that consumers, in the market place, direct their behaviortoward the maximization of their own utility, where utility can be interpretedas a cardinal magnitude having an exact, measurable value, or as an ordinalmagnitude, where we can compare the utility drawn from a good and compareit to that of another good, but never say exactly what is the difference betweenthe two utilities. The latter is called cardinal utility theory, while the latter (themore popular among price theorists) is called ordinal utility theory. In the lattercase ”Utility is the variable whose relative magnitude indicates the direction ofpreference” (Hirshleifer and Hirshleifer, 2005: 73).

When economists say that individuals maximize their utility, they mean that,given the alternatives they face, they will choose the one that will give themmore utility, not that there is a maximum total utility that can be achieved.Cardinal utility theory, on the other hand, assumes that utility can indeed bemeasured (in utils, for example). Geometrically, utility maximization can beexpressed thorough a total utility function and a marginal utility function. Thetotal utility function is positively sloped, which means that utility in consump-tion is an increasing function of quantity consumed, and that marginal utility isalways positive. The marginal utility function, on the other hand, is a decreasingfunction of quantity consumed, meaning that its slope is always negative, a char-acteristic which is referred to as ’diminishing marginal utility’. Algebraically,MU is the slope of the rate of change of ’total utility’:

dU/dC > 0 (4.1)

d2U/dC2 > 0 (4.2)

Equation 4.1 tells us that the slope of the Utility function is increasing in C, andthat MU of a good is always positive. Equation 4.2 tells us that the slope of theMarginal Utility function (the second derivative of the total utility function) isnegative, meaning that MU diminishes as the quantity consumed increases.

4.4. CHARACTERISTICS OF INDIFFERENCE CURVE 21

The utility maximization problem, in its cardinal interpretation, can be rep-resented graphically by three-dimensional Utility Hill. Each point on the surfaceof the hill is associated with a total utility the value of which is determined bythe height of the hill at that point. All points of the hill at the same distancefrom the Cartesian plane are all associated with the same utility. These pointsform the so called curves of total utility, or indifference curves. By maintainingthe indifference curves, but getting rid of the tridimensional hill, economists ar-rive at a graphical representation of the ordinal theory of consumer preferences.

4.4 Characteristics of Indifference Curve

Let us build the utility map for an individual, where each point of the maprepresent a bundle with varying quantities of two goods, X and Y . Then, theindifference curves for this individual will have the following characteristics:

1. Indifference Curves are negatively sloped (otherwise, they would violatethe ’more is better’ assumption)

2. Indifference Curves never intersect (otherwise they would violate the tran-sitivity axiom)

3. There are an infinite number of Indifference Curves (an indifference curvecan, in theory, always be draw in between two indifference curves)

4. Indifference Curves are convex to the origin.1

4.5 Goods and Bads

An indifference map needs not represent the relative preference between bundlesof goods, but also between bundles containing a good and a bad. For example,portfolio preference theory tells us that as the riskiness of an investment in-creases, an individual must be compensated with an increase in mean return inorder to be kept along the same indifference curve. This means that as the quan-tity of the bad (riskiness) increases, the good (mean return) must also increase,and the indifference curves assume a positive slope.

4.5.1 Charity as an application of consumer theory

A charitable person is one who has a positive marginal utility for another per-son’s income (empirically, is fair to say that it will be so only as long as thelatter’s income is lower than her own). An uncharitable person is one whoseindifference curves are straight lines parallel to ordinate. Finally, a malevolentperson will have a positive slope, as the income of the other person is a bad

1According to Hirshleifer et al (2005: 81), this can only be demonstrated if we adopt thecardinals interpretation of utility theory. In fact, I think this conclusion can be drawn by theassumption of diminishing marginal utility.

22CHAPTER 4. THE THEORYOF CONSUMER BEHAVIOR I: UTILITY AND PREFERENCE

from the point of view of the individual, and its increase must be compensateby an increase in the latter’s income to keep it on the same utility function.

Chapter 5

The Theory of ConsumerBehavior II: Consumptionand Demand

5.1 The Geometry of Consumer Choice

The geometrical representation of the optimal consumer choice consists of twoelements: the consumer’s indifference map, and her budge constraint. Thebudget constraint represents the upward limit on the combinations of X and Ythe consumer can achieve for a given income I:

Px ∗ x+ Py ∗ y = I (5.1)

Equation 5.1 is called budget function. By zeroing one of the two variables, it ispossible to identify the intercepts of the function with the horizontal and verticalaxes, which represent the maximum quantity of either good if the consumer wereto spend her entire income on it.

Geometrically, the optimum choice of the consumer is represented by the tan-gency point between the budget constraint and the higher attainable indifferencecurve. In fact, all the other point on the budget constraint would be points ofintersection between it and another, inferior, indifference curve, while none ofthe superior indifference curve ever touch the budget constraint, meaning that,although they could bring about a higher utility, they are not attainable.

23

24CHAPTER 5. THE THEORYOF CONSUMER BEHAVIOR II: CONSUMPTION ANDDEMAND

The slope of the budget is found as follows:

Px ∗x+Py ∗y = I−− > Py ∗y = I−Px ∗x−− > y = I/Py−(Px/Py)∗x (5.2)

where the first element of the right-hand side of the equation is the verticalintercept and the second element is the slope of the line.

The geometry of the optimal bundle offers also a justification for the as-sumption of convexity for the indifference curves. If indifference curves wereconcave, in fact, for any possible budget line, the higher tangency point willalways be a corner solution, meaning that no individual will ever choose to di-versify consumption, no matter her income nor the characteristics of the goodsinvolved. Convex indifference curves, on the other hand, allow both for interiorand corner solutions.

5.2. COMPLEMENTS AND SUBSTITUTES 25

The algebraic understanding of the optimal choice of the consumer variesaccording to whether one choose the cardinal or the ordinal approach to utilitytheory. If one chooses the cardinal theory of individual utility, than an interiorsolution must be consistent with the Consumption Balance Equality :

MUx/Px = MUy/Py (5.3)

Equation 5.3 tells us that, according to cardinal utility theory, the optima bundlewould be such only when the marginal utility derived by the last dollar spent inbuying the good X is equal to the marginal utility derived from the last dollarspent in Y . A corner solution, on the other hand, must be consistent with theConsumption Balance Inequality:

MUx(whenx = 0)/Px < MUy(wheny > 0)/Py (5.4)

If one, on the other hand, chooses the ordinal approach to utility theory, then hecannot use marginal utility as a meaningful notion, since in ordinal utility theory,there is no way to compare two utilities against each other. Marginal utilityis therefore substitutes by the notion of Marginal Rate of Substitution inConsumption, that is, the ratio at which a person is just willing to substitute asmall amount of X for a small amount of Y by keeping the consumer indifferent:

MRSc = −dy/dx|u (5.5)

An interior solution must therefore be consistent with the Substitution BalanceEquation:

MRSc = Px/Py (5.6)

Equation 5.6 tells us that the indifference curve would be tangent to the budgetline when the MRSc equals the absolute value of the slope of the budget line,that is, the ratio between the price of X and the price of Y . For a cornersolution, on the other hand, the choice must be consistent with the SubstitutionBalance Inequality :

MRSc < Px/Py (5.7)

When the entire market is in equilibrium, the price ratio Px/Py is an objectivefact that emerges out of the subjective preferences of all agents, and this priceratio will also be equal to the MRSc of all those who buy a positive quantity ofall goods. The MRSc can also be expressed for one good by using the notionof Marginal Value (MV) or Marginal Willingness to Pay. MV x is obtained byexpressing the MRSc taking Y as the numberer and Py as the price numberer:

MVx = dY/dX = Px/1 (5.8)

5.2 Complements and Substitutes

The consumption choice of the individual depends on three factors: i. Herpreferences; ii. her Income; and, finally, iii. relative prices. The particular char-acteristics of a good may also affect the consumer’s preference for the optima


quantity of the other good. It is easy to imagine, for example, two goods fromwhich an individual can derive a greater utility when consumed in conjunction(i.e., a right and a left shoes). An opposite case will be that of two goods thatwill never be consumed in conjunction, for example, because they satisfy exactlythe same need (i.e., two right shoes). The former are called ComplementaryGoods: Two goods are said to be complementary when a large change in theprice ratio causes only a very small change in the consumption choice. Strictcomplementarity requires that the two goods are consumed always and exclu-sively in fixed proportions. Geometrically, the indifference curve picturing twoperfect complementary goods will be right-angled. The opposite case is thatof Substitute Goods: Two goods are said to be substitute when a very smallchange in the price ratio brings about a large change in the preferred bundle.Perfect substitutes are depicted through a negatively sloped, straight indiffer-ence curve.

5.3 The Consumer Response to Changing Op-portunities

If individual preferences were to remain constant, then only external factors,that is, only changes in opportunities, could account for a change in consumerchoices. Two changes in opportunities can be imagined: i. A change in income;and ii. A change in relative prices. The effect of changes in income is expressedgraphically by the Income Expansion Path (IEP). The IEP represents allthe preferred bundles for a consumer when her income changes and the relativeprices remain the same (different price ratios will produce different IEPs). Thefunction for the income expansion path is given by the Substitution BalanceEquality:

MRSc = Px/Py (5.9)

5.3. THE CONSUMER RESPONSE TO CHANGING OPPORTUNITIES 27

The behavior of the IEP can tell us something about the particular goodsdepicted by the indifference curves. If, for example, the IEP is positively sloped,meaning that as income increases, the consumption of both goods also increases,then both goods are Superior Goods. If, on the other hand, the IEP is nega-tively sloped, meaning that as income increases the consumption of at least onegood decreases, then the good for which consumption has decreased is calledInferior Good, while the good for which consumption has increased more thanthe increase in income is called Ultrasuperior.

The relationship between income and consumption can also be representedfor each good taken separately by the Engel Curve. An Engel Curve has incomeon the horizontal axis and the quantity of the good of interest on the verticalaxis. For superior good, the Engle Curve will always have a positive slope, whilefor inferior goods, the slope will be positive up to a threshold after which it willstart declining, that is, have a negative slope.


The Price Expansion Path does what the IEP did for the relationship betweenincome and quantity for that between price and quantity. The PEP takes incomeas fixed (indeed, there are as many PEPs as there are levels of income) and variesthe price ratio (that is, not the nominal values of the two prices, but the ratiobetween the varying price of X taking that of Y constant). The function forthe PEP is given by the combination of the Substitution Balance Equation andthe Budget Function:

MVx = Px (5.10)

I = Px ∗ x+ Py ∗ y (5.11)

dY/dX = Px (5.12)

y = I/Py − (dY/dX)/Py ∗ x (5.13)

5.3. THE CONSUMER RESPONSE TO CHANGING OPPORTUNITIES 29

The Price Expansion Path has four properties:

1. Utility increases as Px decreases.

2. A negatively sloped PEP indicates that, as Px decreases, the consumerchooses to consumer more of X and less of Y ; a positively sloped PEPindicates that, as Px decreases, the consumer chooses more of both X andY .

3. The PEP can never go above the straight line Y=I/Py, since the consumercannot buy more of Y when Px is declining than when Px is zero.

4. The Pep always goes rightwards, but in the case of a Giffen good, in whichcase the PEP will curl leftward for some values of Px.

A Giffen good is an expectation to the Law of Demand. A commodity isidentified as a Giffen good when an increase in its price brings about an increasein the quantity consumed. For this to be possible, though, the good must be aninferior one, and it must occupy a large share of the budget of the consumers.

The same data used to draw the PEP can also be used to build a demandcurve. The individual demand curve will have Px on the vertical axis and Q/ton the horizontal one, and the same is true for the market demand curve, whichis the horizontal (that is, in terms of Q) sum of all individual demand curves.


5.4 Income and Substitution Effects of a PriceChange

When the relative prices of two or more goods change, while nominal income iskept constant, this has an effect on the consumption behavior of the individual.This effect can be separated into two components:

1. Pure Substitution Effect: Measures the effect of the change in the priceratio by not considering the effect on real income of the price change. Thisis done by observing what the consumer would have chosen if, after theprice had change, she would have been forced to choose the new bundlealong the same indifference curve of the original one.

2. Income Effect: the residual effect once the pure substitution effect hasbeen identified. It is the effect provoked by the change in real income dueto the change in relative prices.

5.5. FROM INDIVIDUAL TO MARKET DEMAND 31

The pure substitution effect is characterized by the fact that it has alwaysthe opposite direction of the price change: when Px increases, X* decreases,when Px decreases, X* increases. Geometrically, since the indifference curveis negatively sloped, when Px decreases, the budget line takes a flatter slope,which means that the tangency point will be to the right of the original one. Onthe other hand, the income effect can have the same and the opposite directionof the change in price. When the change is in the opposite direction, than Xis a normal good, but when it follows the same direction then the good is aninferior one, and if the effect is so large that the new optimal quantity of X islower than the original one, the good in a Giffen good.

5.5 From Individual to Market Demand

Market demand is the horizontal sum of all individual demands in the market,expressed as a function of price:

X =

n∑i=1

xi (5.14)

5.6 An Application: Subsidy Versus Voucher

A subsidy corresponds to a change in the relative prices faced by the recipients,while the voucher represents a change in income that can be spent in one ofthe two goods. Geometrically, thus, a subsidy is exactly the same of a relativeprice change, while the voucher is more similar to an increase in income, wherethe new income line is parallel to the previous one, but keeps the same verticalintercept, which is connected to the new budget line through a horizontal seg-ment. Voucher and subsidies vary in their effect especially for goods that were


not consumed before the introduction of the new policy. In such cases, in fact,subsidies tend to have little if any effect (since the original solution was a cornerone), while a voucher does at least increases the quantity purchased of the goodof interest.

Chapter 6

The Theory of ConsumerBehavior III: Applicationsand Extensions

6.1 The Engel Curve and Income Elasticity ofDemand

The Engle Curve shows the relationship between income and the consumptionof a good. The slope of the Engle Curve is dX/dI can be used to measureconsumers’ responsiveness to changes in income. One problem with this mea-surement is that it is very sensitive to the unit of measurement. To avoid theseshortcomings, the Engle Curve can be substituted by the notion of Elasticity.Elasticity eliminates the problem of sensitivity to the unit of measurement bymeasuring changes in percentage terms:

εx =∆X/X

∆I/I=

∆X/∆I

X/I= ∆X/∆I ∗ I/X

Any nonlinear Engle curve will have a unitary income elasticity in the neighborof the tangency point with a straight line from the origin. Income elasticitycan have values equal, greater than, or smaller than one. We can distinguishbetween: i. Point elasticity (the income elasticity at one point of the EngleCurve); ii. Arc Elasticity (the average of all point elasticities along an arc ofthe Curve).

”A person’s income elasticities over all commodities consumed are connectedby an important condition: The Weighted average of an individual’s incomeelasticities equals one, where the weights are the proportions of the budget

33

34CHAPTER 6. THE THEORYOF CONSUMER BEHAVIOR III: APPLICATIONS AND EXTENSIONS

spent on each commodity” (Hirshleifer and Hirshleifer, 2005: 131):

kx = Px ∗ x/I (6.1)

ky = Py ∗ y/I (6.2)

The weighted average of all income elasticities cannot exceed unity (nor can, inthe unhampered market and for utility maximizing individuals, be lower thanone), since one cannot increase it expenses for the purchase of X and Y by morethan the increase in her income:

1 = kx ∗ εx + ky ∗ εy (6.3)

1 = (dX/dI) ∗ (I/X) ∗ (Px ∗X/I) + (dY/dI) ∗ (I/Y ) ∗ (Py ∗ Y/I) (6.4)

1 = (dX/dI) ∗ Px + (dY/dI) ∗ Py (6.5)

1 = 1/dI ∗ (dX ∗ Px + dY ∗ Py) (6.6)

dI = (dX ∗ Px + dY ∗ Py) (6.7)

6.2 The Demand Curve and the Price Elasticityof Demand

The Demand Curve depicts the responsiveness of consumption to changes inprices. From the point of view of the seller, the demand schedule does notrepresent a good indicator of his total revenues. This is because increases inthe price of the good sold need not, and often do not, result in increases in therevenue of the seller. This sis because as price increases, buyers can respond bybuying less of the good. If the increase in revenues due to the increase in price isoffset by the decrease in quantity demanded, the net result will be a decrease inthe revenues of the seller. When the net effect is negative, the demand scheduleis said to be elastic at the original combination of Q and P. When the neteffect is zero (total revenue remains universe) the demand is said to be unitelastic. When the net effect is an increase in total revenues, the demand issaid to be inelastic. As in the case of the Engel curve, though, the DemandCurve also has a measurement problem due to the unit of measurement. Toovercome these measurement issues, economists use the notion of price elasticityof demand, which measures the responsiveness of consumption to change inprices in percentage terms:

ηx = (∆X/X)/(∆Px/Px) = ∆X/∆Px ∗ Px/X (6.8)

Since the relationship between price and quantities, as expressed by the demandfunction, is usually inverse, dX/dPx is usually negative, η is also negative. Thedemand for X is:

• Elastic, when ηx < −1

• Unit Elastics, when ηx = −1

6.2. THE DEMANDCURVE AND THE PRICE ELASTICITY OF DEMAND35

• Inelastic, when ηx > −1

Different price elasticities have different effect on the total spending on a par-ticular good, Ex:

Ex = Px ∗ x (6.9)

For ηx < −1, that is, when demand is elastic, a decrease in the relative pricePx brings about an increase in total spending on X. Correspondingly, whenηx >1, when demand is inelastic, a decrease in Px provokes a decrease in totalexpenditure on X. To better identify the relationship between price changeand expenditure on X, it is possible to use the notion of Marginal Expenditure,MEx:

MR =dTR

dX(6.10)

MR =d(Px ∗X)

dX(6.11)

MR =dPx ∗X + dX ∗ Px

dX(6.12)

MR =dPx ∗XdX

+dX ∗ PxdX

(6.13)

MR =dPxdX∗X + Px (6.14)

MR = Px(dPxdX∗ dXdPx

+ 1) (6.15)

ηx =dX

dPx∗ PxX

(6.16)

MR = Px(1

ηx+ 1) (6.17)

ηx =Px

MR− Px(6.18)

MR will be positive for elastic demand, zero for unit elastic demand, andnegative for inelastic demand.

6.2.1 The Second Law of Demand

According to the second law of demand, a demand schedule is always moreelastic in the long run than in the short run: ”The longer the time allowedto adjust amount demanded in response to a price change, the greater is thechange in amount demanded, that is, the greater the elasticity” (Alchian andAllen: 28). This greater elasticity in the long run is caused by a process putin motion by any price change. Buyers and sellers alike will respond to a pricechange by trying to find ways to adjust and, for example, buying (or producing,in the case of the sellers) substitute to the good the price of which increased.


6.3 The Cross-Elasticity of Demand

The Cross-Elasticity of Demand is the measure of how a change in the price ofY affects the quantity consumed of the good X:

ηxy =dx/x

dPy/Py= dx/dPy ∗ Py/x (6.19)

When two goods are complements in consumption, an increase in the price ofeither of the two causes a decrease in the consumption of the other. Since thetwo changes have opposite signs, the cross-elasticity will be negative: ηxy < 0When two goods are substitute in consumption, an increase in the price of Yprovokes an increase in the consumption of X. Since the two changes have thesame sign, the cross-elasticity will be positive: ηxy > 0

6.4 Fitting the Demand Curve

The notion of price elasticity can be used to fit econometric data into a statisticaldemand curve for the market as a whole. The elasticity of market demand thusbecomes:

ηx =dX/X

dPx/Px=

Px/X

dPx/dX(6.20)

Where X is the total quantity of X consumed in the market.In fitting the historical data into statistical demand curves, econometricians

usually face an alternative between choosing a Constant Slope demand curveor a Constant Elasticity demand curve. A Constant Slope Demand Curve isa straight line with negative slope. Straight lines have, by definition, constantslope. Price elasticity along a constant slope demand curve varies at each point.At high level of prices, for example, elasticity is larger than at low prices. Thisis because elasticity is measured in percentage terms and a price change is likelyto affect consumption more at high prices that at low prices. At any point ofa constant slope demand curve, price elasticity is equal to the ratio betweenthe slope of a ray from the origin passing through the point of interest, andthe slope of the curve. At all points in the upper interval of the demand curve,the slope of the ray will be greater, in absolute value, than the slope of thecurve. Since, though, the slope of the curve is negative, ηx < −1, meaning thatdemand is elastic. At the lower arc of the demand curve, on the other hand, theslope of the curve is steeper than that of the ray from the origin, meaning thatthe ratio will be, in absolute value, less than one, and, since the slope of thedemand curve is negative, ηx > −1, meaning that demand is inelastic. Demandwill be unit elastic when the slope of the demand curve is exactly the reversefraction of the slope of the ray, that is, at the midpoint of the linear demandcurve. For a non-linear demand curve, the same reasoning applies, with theonly modification that, what matters is not the slope of the demand curve, butrather the slope of the tangent of the demand curve at the point of interest.

6.4. FITTING THE DEMAND CURVE 37

The general function for a linear demand curve, that is, a constant slopedemand curve, is as follows:

X = A+BPx (6.21)

Where A is a constant representing the horizontal intercept, and B is the slopeof the curve, which in normal cases assumes a negative value. The function canbe generalized even more by taking other variables that affect demand for Xsuch as income, price elasticity, the price of substitutes and that of complements,and so forth:

X = A+BPx + CI +DPy + EPz... (6.22)

Where C is the slope of the Income Expansion Path, which is positive whenX is a superior good and negative when is an inferior one, D is the influencethe quantity purchased of a complementary good, which is always positive, andE is the effect of the quantity purchased of a substitute good, which is alwaysnegative.

The general function of a Constant Elasticity Demand Curve is as follows:

X = aP bx (6.23)

Where a is the (varying) slope of the demand curve, and b is the constantelasticity:

ηx =dX/X

dPx/Px=dX

dPx∗ PxX

(6.24)

X = aP bx (6.25)

dX

Px∗ Pxx

= abP (xb− 1) ∗ Px

x= ab

P bxPx∗ Pxx

= abP bxx

(6.26)

ηx = baP bxaP bx

= b (6.27)

Equation 6.21 can also be written in logarithmic form (which can be repre-sented graphically as a straight line):

logX = log a+ b logPx (6.28)

The general function can be extended to take other factors into consideration:

x = aP bx ∗ Ic ∗ P dy ∗ P ez (6.29)

Which, once again, can be expressed in logarithmic terms as:

log x = log a+ b logPx + c log I + d logPy + e logPz (6.30)


6.5 Determinants of Responsiveness of Demandto Price

There are three main determinants of price responsiveness, or price elasticity,on the part of the consumer:

1. Availability of substitutes: the closer the substitute, the greater the elas-ticity, since the consumer can shift to the substitute good as the priceincreases. When there are not close substitutes to a good, on the otherhand, the consumer cannot easily shift to another good when the price ofthe former increases (substitution effect)

2. Luxuries vs Necessities: The demand for luxuries tend to be more elasticthen that for necessities (income effect)

3. High priced vs Low priced goods: low priced goods tend to be less elasticthan high priced goods, especially when we assumed constant slope forthe demand curve

These determinants can help us understand the flaw in the ”importance” fallacy,according to which the less important a good is, in terms of its weight on thetotal expenditure of the consumer, the less the latter will respond to change inprices. This argument is often used to explain the little elasticity of demandfor goods such as salt, the expenditure on which is very small across the entirespectrum of income distribution. But this phenomenon can be explained morerigorously by appealing to the three determinants of demand responsiveness toprice: i. Salt has no close substitutes; ii. salt is a necessity; 3. salt is low priced.Indeed, it is easy to show that two goods, one accounting for the vast majorityof consumers’ income, and the other for only a small fraction of it, can bothhave the same (unitary) elasticity.

6.5.1 The effects of a price increase

An increase in prices has three effects on the behavior of individuals:

1. Endowment effect: If the individual owns units of the good, his endow-ment is now more valuable and his budget constraint moves upward, thusincreasing his own demand for the good.

2. Substitution effect: The price change increases the price of x relative to y,thus making the individual substitute (at the margine) y for x.

3. Income effect: the increase in price reduces the purchasing power of theindividual, and can force him to buy less of both x and y.

6.6. MULTIPLE CONSTRAINTS: RATIONING 39

6.6 Multiple Constraints: Rationing

Rationing constitutes an artificial limitation on the consumption of one or moregoods. It is a popular policy in times of war and is often adopted by socialisticinspired government around the world. There are two types of rationing:

1. Coupon Rationing

2. Point Rationing

A coupon rationing works by limiting the maximum amount of X that anindividual, or an household, can purchase during a given period of time. Acoupon rationing is binding only when the rationed quantity, Rx, is lower thanthe quantity that would have been purchased in the absence of the rationing.

When the rationing is binding, the consumer will choose to consume as much asis allowed of the rationed good and spend the rest of her income on the non ra-tioned good. In the case in which both goods (assuming a two goods economy)are rationed, and both rationing are binding, than the consumer will consumethe maximum allowed quantity of both. The rationale of point rationing is tomodify the relative prices of the rationed goods and, at the same time, restrictthe ability of wealthy individuals to consume more than everyone else. Pointrationing works by introducing a parallel, artificial price ratio in terms of points.Each good is now priced both in money and in points, and each household isgiven a fixed amount of such points N . Now, the choice of the consumer isconstrained by two functions:

• the budget function: I = Px ∗ x+ Py ∗ y

• the point function: N = px ∗ x+ py ∗ y

Under these constraints, three scenarios are possible. In the first scenario, thepoint constraint is not binding, while the budget constraint is, thus allowing


the individual to consume the same bundle as before. In the second scenario,the point constraint is binding (the function lies entirely below the budget func-tion), meaning that the consumption choice will be entirely determine by theartificial price ratio. Finally, in the third scenario, both functions are binding.

Chapter 7

The Firm

7.1 The Firm: Entrepreneurship, Ownership, andManagement

Firms are economic organizations the purpose of which is the production ofgoods and services. In general, firms emerge out of a decentralized marketbecause of the following ”frictions”: i. Transaction Costs; ii. Coordination inproduction.

Transaction costs, that is, the costs associated with transacting, increase thecost of producing a good in a decentralized manner. In order to produce theconsumer good, each producer of the intermediate goods will have to contractwith everyone else. This kind of multilateral contracting is very costly, sinceeach contact must be negotiated and enforced. The firm create the possibility ofbilateral contracting in which the responsible for the firm negotiate with everycontributor to the final product, including workers, suppliers, and so forth.

Furthermore, the firm are created to take advantage of coordination in pro-duction: there simply are some production processes that can only be carriedout by teams of people, and not by the cooperation of independently workingindividuals.

The firm has three fundamental features:

1. Management: The manager is somebody who speaks and acts in the nameof the firm, which gives him or her some discretion over the direction ofthe productive process.

2. Ownership: The owners are those who have invested their money in con-trolling a certain share of the firm. Owners are the residual claimants ofthe firm’s revenues after all costs have been paid. Ownership and man-agement need not, and often do not, coincide,

3. Entrepreneurship: The entrepreneur is the decision maker, the individualresponsible for deciding what actions to take under conditions of uncer-tainty.

41

42 CHAPTER 7. THE FIRM

Firms often take the form of corporations, which allow them to act in aregime of limited liability. Limited liability means that the owner of a share ofthe firm is only liable for his or her share of the firm. Corporations are alsosubject to transferable shares, meaning that the owner of a share can sell it tosomeone else.

All firms have to overcome two problems likely to obstruct their pursuitof profits: i. The Need for Monitoring Management; ii. Distribution of Risk.Sometimes, the owner of the firm cannot perfectly monitor the behavior of themanager. In industries were the monitoring costs are relatively high, we willtend to observe a coincidence of ownership and management. The easier it isto monitor the quality of an input, the more likely it is for management andownership of the asset to be separated, the more difficult it is to monitor thesequalities, the more likely it is that those inputs are going to be owned by themanagers.

Many corporations diversify risk by producing two types of instruments forcorporate capitalization: i. Corporate bonds, ii. Corporate shares. Corporatebonds are for risk-averse individuals, those who do not want to be involved inthe ownership of the firm. Corporate share are for those who are more risk-prone, and are willing to expose themselves to the ”hazards of doing business”.These instruments allow for the specialization in risk-taking.

7.1.1 The Profit-Maximizing Firm

Economists assume that firms are profit maximizers. The economic notion ofprofits, though, is different from the one used in accounting. In price theory,profits are the difference between the firm’s revenue and costs, where costs arereally opportunity costs, and not accounting costs. Economists define oppor-tunity costs as the best among the forgone uses of the resources employed bythe firms. Thus, to calculate the costs of doing business, an economist wouldtake into account the foregone opportunity of earning a salary by working as anemployee for a different firms.

Many have criticized the profit maximizing assumption for firms in whichmanagement and ownership do not coincide. The critics claim that, when thisis the case, monitoring by owners is difficult, which allows managers to pursuetheir own self-interest, rather than that of the firm. This would in turn leadthe firm to fail t maximize profits. For example, a manger may be interestedin ’making a name’ by increasing the performance of the firm in the short run,but depleting the long run profits, thus endangering the position of the owners.This is particularly likely when ownership is diffused, with small shareholdersleft without power.

There are, though, mechanism through which owners, even small ones, canmonitor, although not perfectly, the behavior of managers. For example, theymight decide to sell their share, which lowers the value of those owned by thelarger ones and affects the public perception of the management of the firm. Thediscretionary power of managers is thus limited by the possibility of changes inthe ownership structure. If a firm fails to maximize profits because of bad man-

7.1. THE FIRM: ENTREPRENEURSHIP, OWNERSHIP, ANDMANAGEMENT43

agerial decisions, someone could buy enough shares and put himself (or someonemore capable) in the role of manager and gain the previously unexploited profitopportunity. The more a manager acts against the interest of the firm, then, themore profits are left on the sidewalk, and the more likely it is that people fromoutside the firm will be willing to invest in it by buying shares and changingthe management of the firm.

7.1.2 Production with two or more factors

Within the theory of the market process, firms’ function is the production ofgoods. Production is a process of transformation of inputs (the factors of pro-duction) into outputs (the products). Inputs can be raw materials like woodand iron, or produced factors of production like machinery and human capital.Similarly, a firm can produce consumption goods (like sliced bread) or factorsof production (like machinery) that are going to be used by other firms.

Classical economists used to distinguish between three general categories offactors of production: 1. Labor; 2. Capital; 3. Land. Modern economists addto this list Entrepreneurship and Capital Goods. The services of these factorsare a flow concept.

Land is often defined as ”the original and indestructible power of the soil”.Thus, any service that the land is capable of providing that is the result ofsome human effort cannot be accounted for as services of the land, but ratheras capital goods. It must be noted that, unlike some classical economists likeMalthus thought, land is not fixed in supply.

Capital is the ”stored-up provision for the production of future goods derivedfrom past efforts or abstinence.”

7.1.3 The Law of Diminishing Returns

Keeping one factor of production fixed, and given that such a factor is ”limit-ing” (that is, it is an important factor in the production process of the firm),then eventually the increase in the quantities of all other factors will lead todiminishing returns in production. This means that, all but the fixed factorsincrease, the firm will be able to produce fewer and fewer units of the good.

For example, let us assume that we have a firm using only two factors ofproduction, Land and Labor, in order to produce some output X. Let’s assumethat the quantity of Land that the firm can use os fixed form some reason. Thefirm then adds labor to the fixed land. Initially, the total product ofof laborincreases at an increasing rate, but after a while, adding more labor, thoughincreasing productivity, does so at a diminishing rate. Eventually, the firm willarrive at a point such that adding more labor does not increase the quantityof good produced, and even affects it negatively. In terms of output per unitof labor, the marginal product of labor is first increasing, then decreasing, andfinally zero and even negative. The average product of labor is increasing as longas output is increasing (that is, as long as MP¿AP) and then is decreasing. The


firm will always choose to produce output in the region where average productis diminishing, but the marginal product is not negative.

The law of diminishing marginal returns is ”much misunderstood” (Vickrey,153) tells us that the prerequisite for the existence of a competitive equilibriumwill always be satisfied, somewhere. I

7.2 The Optimum of the Firm Under Pure com-petition

Under pure competition, each firm contributes to market supply only by a neg-ligible amount, and has therefore no influence over the market price. Pure com-petition is the combination of pliopoly, that is, the possibility for firms outsidethe market to enter it as soon as they think they can make economic profits, andpolyploy, that is, that many firms operate in the market at any given moment.

As I discuss above, firms are assumed to maximize profits:

Π = R− C (7.1)

R = P ∗Q (7.2)

C = AC ∗Q (7.3)

where Π is economic profits, R is total revenue, and C is total cost.Each firm’s contribution to market supply is determined by the behavior of

their cost curves, which varies according to whether the firm is acting in theshort run or the long run. In the short run, total costs can be distinguishedbetween fixed costs and variable costs (or direct costs). In the short run, thefirm is assumed to have some inputs, a for example the size of the plant, fixed,meaning that the costs it incurred in paying for it do not affect the choice of howmuch to produce. Total costs curves, in the short run, have two characteristics:

1. At low levels of output, costs rise at a declining rate as quantity increases,thanks to economies of scale.

2. At high levels of output, costs rise at an increasing rate, according tot helaw of demising returns.

The optimal strategy for a competitive firm is to produce until the point atwhich the revenue from selling the next unit equals the cost of producing it:

MR = MC (7.4)

To a competitive firm, though, marginal revenue is equal to price, since thedemand it faces is irresponsive to increases in quantity:

MR = P = MC (7.5)

Where MR = dRdQ and AR = R

Q . On the cost side, MC = dCQ and AC = C

Q . Wecan now rewrite the profit equality as:

Π = R− C = (AR ∗Q)− (AC ∗Q) = (AR−AC) ∗Q (7.6)

7.3. COST FUNCTIONS 45

A firm will therefore produce the quantity at which P=MC. This only applywhen the intersection between the MC curve and the AVC curves intersect ator above the P curve. In fact, if it were otherwise, it would mean that the firmis losing money for each unit it produces. Note that the firm will still produceeven if it is making negative profits, since the revenues are less than the totalcosts when fixed costs are taken into account. In the long run, thus, the firmwould simply not enter in the market when the price is lower than the averagetotal cost. But in the short run, since fixed costs are bygone, production willoccur as long as the average variable cost is not below the price.

7.2.1 Division of Output among Plants

Let us assume that a firm owns two plants, A and B, with different cost curves.Then , the firm will produce in each plant such that the MC in plant A equalsthe market price and the MC in plant B also equals the market price. The totalquantity supplied by the firm will be the sum of the quantities produced in eachplant.

7.3 Cost Functions

Short-run and long-run differ in that, the longer the run, the more inputs arevariable rather than fixed. There is therefore a continuum of short-runs betweenthe condition in which nothing is variable, and the extreme condition in whicheverything is variable. In price theory, the short-run is usually assumed to be


such that some inputs, usually capital, is fixed, while labor is variable. Thelong-run is seen as the condition in which both capital (the plant) and labor areboth variable. In the long run, thus, the total costs curve passes through theorigin, since the firm can always decide to produce zero at zero cost. Thanksto the law of diminishing returns, the TC curve will increase at a decreasingrate at low levels of output, and increase at an increasing rate for high levels ofoutput. The LRTC curve is the lower envelope of all the SRTC curves for anyfixed level of output. The LRTC curve represents e points at which, for eachlevel of fixed costs, it is least costly to produce.

The Long Run Average Cost Curve is the lower envelope of all Short RunAverage Cost Curves. It is tangent to them to the left of their minima whenincreasing returns are in place (that is, when total costs increase at a diminishingrate) and to the right when the law of diminishing returns enters into action, thatis, when costs increase at an increasing rate. The LRAC curve has its minimumat the tangency point with the SRAC curve for which the combination of fixedand variable costs produce the global minimum value of unit cost.

Note that the LRAC curve is tangent to all SRAC curves and always liesbelow them. If this was not so, the LRAC will be greater, for the same quantity,than the SRAC, which is impossible, since in the LR the firm can always shiftto the same combination of fixed and variable costs of the SRAC curve.

Similarly, the Long Run Marginal Cost Curve represents the points of in-terception between all the SRMC curves and the vertical lines that connect thetangency point between LRAC and all the SRAC curves. The LRMC curve cutsthe LRAC curve at the latter’s minimum, which is also the point at which thefirm will produce in the long run. The particular characteristics of this point isthat, at it, LRMC, SRMC, LRAC, and SRAC all touch:

SRMC = LRMC = SRAC = LRAC (7.7)

In real markets, the notion of long run is often different from the one used here.Firms cannot always exit a market as easily as the theory would suggest. Firmsusually encore in fixed costs even while not producing, simply because it is lesscostly than the alternative of selling its plan and machinery every time it has tostop production. This is because transaction costs are positive and because firmsuse specialized resources in their productive process, resources that are thereforeof little value for other firms, even those operating in the same market. Thus,although it would be possible to sell or rent these resources when production isstopped, the cost of doing so is greater than the opportunity cost, which meansthat the firm is better off by keeping them for itself. If transaction costs werezero and all resources were perfectly homogeneous (meaning that they could beused in any production process without any costly modification), then the verynotion of short run will be meaningless, since firms will be able to modify alltheir inputs at any moment in time.

7.3. COST FUNCTIONS 47

7.3.1 Rising Costs and Diminishing Returns

In the model of a pure competitive market in production, the usual assumptionis that Marginal and Average Costs both eventually increase as the quantityproduced increases. Modifications to these assumptions lead to different re-sults. For example, let us assume that MC were falling throughout the caretsgraph. Then, the MC curve will never intersect the MR=P curve from below,meaning that a competitive optimum cannot be found. If, on the other hand,AC falls throughout then a competitive market is also infeasible, since a firmwill in the long run expand its plant indefinitely cutting out all competitorsfrom the market. Thus, falling ACs are said to give life to ’natural monopolies’.


In reality, AC and MC do eventually start to increase, thanks to the Law of Di-minishing Returns. This law tells us that, keeping one input fixed, there isan optimum level of the other input such that the quantity producedis maximized. If this were not the case, it would be possible to increase thelevel of one output to produce an indefinitely large level of output. The Lawwould not apply if all inputs were really variable, since one firm could alwaysavoid an increase in MC and AC by modifying the combination of inputs. Theresult would be flat MC and AC curves and equal to each other. But since notall inputs can be varied together, the law does apply in most circumstances.

7.4 An Application: Peak Versus Off-Peak Op-erations

Let us assume a firm operating in a competitive market in which there are twopossible situations: Peak Demand and Off-Peak demand. The Peak demand isgreater than the Off-Peak one which requires an increase in variable and fixedinputs, and, therefore, in variable and fixed costs. The firm has therefore twonew types of costs:

• Common Costs: Common costs are the costs that the firm must paid inorder to maximize profits both at peak and off-peak.

• Separable Costs: The costs incurred to serve only one market and not theother.

For conveniency, let us assume that in the short run Marginal Common Costs(MCC) and Average Common Costs (ACC) are constant at the value M, mean-ing that geometrically the two curves are horizontal lines. The Marginal Sepa-rable Cost and the Average Separable Cost are assumed to behave in the usualmanner. Then, there are two optimal strategy in the long run:

1. Stable Peak Solution: The MCC must be assumed to have been incurredto meet the larger peak demand, meaning that the only relevant marginalmagnitude is MSC. Thus, Off-Peak, the optimal quantity to produce isfound by equating MSC to the off-peak price Po. On Peak, on the otherhand, MCC matter, thus the optimal strategy is to equal the sum of MCCand MSC to the peak price Pp. Thus, the firm will produce two differentoutputs, qo and qp.

MCo = MSC = Po (7.8)

MCp = MCC +MSC = Ppqo < qp (7.9)

2. Shifting-Peak Solution: The Stable Peak solution only holds when Pp −Po > MCC. When MCC > Pp − Po, another strategy is optimal. Sincethe increase in price cannot pay for the increase in cost due to the Peak

7.4. AN APPLICATION: PEAK VERSUS OFF-PEAK OPERATIONS 49

Demand, the firm must produce the same output in both circumstances.This optimal output is found by equating the sum of the MSC on peak,the MSC off peak, and the MCC to the sum of the prices off and on peak:

MC = MCC +MSCp +MSCo = Po + Ppqp = qo (7.10)


Chapter 8

Market Equilibrium in aCompetitive Industry

8.1 The Supply Function

In a competitive market, in the short run, the optimal strategy for a firm isto produce output until the marginal cost equals the marginal revenue, which,because demand is perfectly elastic, is in this case equal to the market price.The firm, in the short run, will continue producing as long as the market priceis above the average variable cost. On the other hand, in the long run, it willonly stay in the market for a price above the average total cost.

The industry supply curve is the horizontal summation (that is, expressedin terms of Q(p)) of all individual firm supplies. The problem is that this istrue only in a world in which a change in production does not affect the inputmarkets. When we allow for such changes, which are indeed likely to happenin real world markets, than the actual market supply curve must take theminto account. Let assume for example that a change in tastes drive the marketdemand upwards, that is, consumers are willing to pay more at each quantityof the good. The new equilibrium price will not be found by moving alongthe old supply curve. This is because the increase in demand has provokedall firms to produce more of the good, which has in turn increased the priceof the factors of production and, therefore, the cost associated with it. Thenew supply curve will thus be above the old one, and the new equilibrium willbe at the intersection between this and the new demand. The market supplycurve is therefore steeper than those of the individual firms but also of all thoseassociated with different demand curves.

The effect of a change in demand on the supply curve is called Input-PriceEffect. The I-PE tells us that the response of the industry as a whole to anincrease in price is likely to be smaller than it would have been if no such effectexisted. This effect influences the elasticity of supply, that is, the measure of

51

52CHAPTER 8. MARKET EQUILIBRIUM IN A COMPETITIVE INDUSTRY

how much a change in price affects the change in quantity supplied:

κ ≡ ∆Q/Q

∆P/P≡ ∆Q

∆P.P

Q(8.1)

Unlike the price elasticity of demand, the price elasticity of supply tend to bepositive. Because of the IPE, the industry supply elasticity is less elastic thanthe individual supply curve. The individual supply curve corresponds to thefirm’s MC curve above the shut-down decision point (that is, the intersectionbetween MC and AVC in the short run, and between MC and AC in the longrun), and the segment 0PC (where PC is the price level below which the firmwill shutdown) on the vertical axis.

In the long run, the elasticity of supply is greater than in the short run. Thisis because of a combination of factors:

1. The the LRMC is flatter than the SRMC.

2. The Entry−Exit Effect: In the long run, firms are free to enter and exitthe market, which means that an increase in price will lead new firms tostart production. The quantity produced will be therefore greater becauseof the response of firms already in the market and of the new ones.

Within an industry, it is possible for producers to incur in externalitiesprovoked by each firm’s reaction to a price change. These externalities arecalled External Economies, and can be of two types:

8.2. FIRM SURVIVAL AND THE ZERO-PROFITS THEOREM 53

1. Pecuniary External Economies: The Input-Price Effect is an exampleof pecuniary externalities, as is the long run effect of a change in demandon the equilibrium price, since new firms, by entering the market, lower theequilibrium price thus cutting economic profits for everyone. Pecuniaryexternal economies tend to be negative.

2. Technological External Economies: Ideas, and their application toproduction, are an example of Technological External Economies. In thecomputer industry, for example, technological economies have the effect ofreducing the cost of production even though demand has increased, thuscounterbalancing the Input-Price Effect. Technological economies can,potentially, be positive or negative.

When external economies are positive, the industry supply curve will be flat-ter than any individual firm’s supply. For very large positive economies, likea strong technological external economy, the curve might also be negativelysloped, meaning that, a change in the demand curve causes quantity and priceto decrease at the same time (an example: the computer industry).

8.2 Firm Survival and the Zero-Profits Theorem

Survival in the long run requires the firm to have non-negative economic profits.In a competitive industry, though, the long run profits in the industry will be zero.If the industry is open to new firms to enter, as soon as price is above AC, newfirms will enter the market to take advantage of the opportunity, rising the priceof input factors and re-establishing the zero-profit condition.

8.3 The Benefits of Exchange: Consumer andProducer Surplus

The Fundamental Theorem of Exchange tells us that voluntary trade is alwaysmutually beneficial. The benefits of exchange cannot be measured directly interms of the utility of consumers and producers, but we can measure theirwillingness to pay and their reservation price, and, through these measures,have an indication of the benefits of trade.

The Producers’ surplus is the shaded area contained by, on the left, the Yaxis, to the right and below by the supply curve, and above by a horizontal linefrom the intersection of supply and demand to the Y axis. The Consumers’surplus is contained by the Y axis on the left, the demand curve on the rightand above, and the line between the intersection and the Y axis below.The mostvalued a good and the smaller the price, the larger the surplus. The least valueda good, and the higher the price, the smaller the consumer surplus.

These measures can help us assess the effects of, for example, changes intechnology which reduce the cost of production or increase the quality of agood. Under both circumstances, the change shifts one of the two curves (supply


downward the former and demand upward the latter), which means that the areacontained by them (the sum of consumers’ and producers’ surpluses) is largerafter the introduction of the new technology.

8.4 Transaction Taxes and other Hindrances toTrade

It follows from the fundamental theorem of exchange that, the more peopleenter into trade, the larger the benefits. All hindrances to trade will, therefore,reduce these benefits. Taxes are one example of such hindrances. A geometricalas well as algebraic analysis will show that taxes do in fact reduce consumers’and producers’ surplus. Some of this surplus will enter the state treasury andmay finance public goods which also produce some benefits to consumers, buttaxes also always generate a deadweight loss, that is, some surplus is not betransferred but simply disappears. The larger the transaction tax, the larger thedeadweight loss.

Supply quotas are another example. Unlike taxes, though, quotas do notgenerate tax revenues. Quotas simply redistribute some consumer surplus toproducers, and produce a deadweight loss. Consumers are always made worseoff by quotas, while producers, although they lose some surplus as a result ofthe deadweight loss, gain even more thanks to the redistributed surplus from theconsumers to themselves. Many countries have import quotas on some importedgoods, which are a way to redistribute surplus from domestic consumers todomestic and foreign producers.

A third example of wealth destructing hindrances to trade are price ceilings.A price ceiling is a maximum price introduced, at least in theory, in the interestof consumers. In fact, price ceilings have a destructive effect in that they pre-vent the price mechanism to fulfill its function of allocating resources to theirmost valued by spreading informations about profit opportunities. Price ceilingsprevent adjustments of supply to changes in demand, technology, and resources.

Let us assume, for example, that a city council decides to introduce a priceceiling on housing by fixing P to its level at the moment of the adoption ofthe measure. Then, after a little time, an external shock causes the supplyof houses to fall. If the absence of the price ceiling, the price of housing willrise sharply in the immediate run, then decline in the short run thanks to anincrease in quantity supplied by existing providers of housing, and decline evenmore, maybe even to its initial level, in the long run. In the presence of theceiling, though, no price adjustment is possible in the short run, meaning thatonly those sellers the preservation price of which is lower than the mandatedprice will sell their houses.

Furthermore, in the immediate run, some of the houses may end up in thehands of people who do not value them the most, since buyers are preventedfrom bid for the scarce resource. In the long run, though, the effect of a priceceiling is to create a shortage: quantity demanded, at the mandated price,

8.4. TRANSACTION TAXES AND OTHER HINDRANCES TO TRADE 55

will be far larger than the quantity that suppliers are willing to supply at thesame price, which means that some people who would be willing to pay themarket price are prevented from doing so. No shortage is possible whensupply and demand are allowed to adjust to market conditions. Finally,price ceilings have also the feature of generating wasteful practices which area deadweight loss for society. Absent the price system as a mechanism forallocation of resource, other mechanisms, such as waiting lines, will emerge.Unlike a direct payment at the equilibrium price, the loss for the waiting buyeris not being transferred to the seller, and is therefore a net loss for the economyas a whole.

.


Chapter 9

Monopolies, Cartels, andNetworks

Economists refer to a market as being monopolistic when only one firm (themonopolist) operates in it. A monopoly can emerge in only two ways:

1. Natural Monopoly: A natural monopoly emerges when one or more firmsoperating in the same market experience declining Average Costs through-out the relevant section of the schedule. Since the firm could always pro-duce more at lower AC, only one firm can be in the market at equilibrium.

2. Artificial Monopoly: An artificial monopoly is the result of governmentintervention in securing that only one firm operates in the market.

A third possibility is that of monopolistic competition, that is, competitionbetween many firms that, although they operate in the same market for thesame good, they all have some market power and can therefore influence priceat the margin. To avoid confusion, we can call legally protected monopolisticmarkets as ”closed monopolies” and those that emerge because of underlyingeconomic reasons ”open monopolists”.

9.1 The Monopolist’s Profit-Maximizing Opti-mum

A firm operating under pure competition will do its best by producing the quan-tity of output such that the marginal cost equals the price. In the competitivemarket, the single firm has no way of influencing the price, and therefore theMR is always equal to the price, no matter how much it produces. But in thenon perfectly competitive market, the monopolist does have the power to influ-ence the price, since if faces a downward sloping demand curve, which means

57

58 CHAPTER 9. MONOPOLIES, CARTELS, AND NETWORKS

that marginal revenues declines as quantity produced increases.

MR = MC (9.1)

MR ≡ d(P ∗Q)

dQ≡ (P ∗ dQ

dQ) + (Q ∗ dP

dQ) ≡ P +Q ∗ ∆P

∆Q(9.2)

MC = P +Q ∗ ∆P

∆Q(9.3)

The profit-maximizing strategy for a monopolist can also be expressed in termsof price elasticity of demand, η:

η ≡ ∆Q/Q

∆P/P≡ ∆Q

∆P∗ PQ≡ ∆Q

∆P ∗Q∗ P (9.4)

MR ≡ P +Q ∗ dP/dQ ≡ P ∗ (1 +Q/P ∗ dP/dQ) ≡ P ∗ (1 +1

η) (9.5)

The monopolist firm will never produce in territory where η > −1, sincewhere demand is inelastic a change in price leads to a decrease in total rev-enues. On the other hand, in the territory of elastic demand, the monopolistcan produce more, thus indirectly reducing price, but at the same time increaserevenue.

The differences in market conditions between competitive and monopolisticindustry leads to different equilibria. The monopolistic equilibrium will alwayshave lower output and higher price than under pure competition.

A monopoly is restrained in its strategy by two factors: i. the price elasticityof demand; ii. the potential competition of firms from outside the industry.

9.2. MONOPOLY AND ECONOMIC EFFICIENCY 59

Competition, in particular, can constrain the strategy of monopolist even morethan elasticity of demand since the monopolist cannot maintain its position if itwere to produce at a quantity at a price that is larger than the AC of the leastcostly potential competitor.

9.1.1 Two-level optimization under monopoly

The publishing industry is a case of monopolistic market in which there emergesa two-level optimization problem. Each publisher is the sole seller of a particulartitle, over which it has market power. The author, on his part, will want tomaximize her royalties. But the two would choose two different quantities atwhich they maximize profits. The publisher would like to have a higher prize,which means fewer books sold, while authors would like a combination in whichprice is smaller and quantity is larger. If the author could pick her own preferredroyalty rate, knowing that it would be taken into account by the publisher, it willchoose it such that maximizes her profits. But the publisher could offer a lumpsum payment larger than the royalties and then set the price that maximize itsown profits, making both parties better off.

9.1.2 Monopolist with Competitive Fringe

Some markets are characterized by a particular internal organization in whichone of the firm operating in it has market power, while other smaller firms oper-ate in it but are price takers. The monopolist and the fringe firms have differentstrategies. The monopolist first have to subtract the horizontal summation ofthe fringe firms from market demand and then produces the quantity such thatthe marginal revenue of the remaining demand equals its marginal cost. Thefringe firms each produce the output such that its own MC equals the priceresolution from the monopolist’s choice.

9.2 Monopoly and Economic Efficiency

Let us assume we are observing a competitive market with an indefinite numberof small firms. Each of these firms will produce until MC=P, and the final resultwill be an equilibrium that maximizes consumers’ and producers’ benefits. Nowassume that the firms merge creating a giant monopolist, which can now choosethe output such that MC=MR¡P. This will have two effects on consumers’ andtotal surplus. First, the monopolist will be able to transfer to itself some con-sumer surplus. Second, a share of total surplus will be lost on a deadweight loss.As long as its loss is counterbalanced by the share of consumer surplus trans-ferred to itself, the monopolist will be better off, but consumer are definitelyworse off. The loss in total surplus is not the only efficiency loss of monopoly.In the case of government mandated monopoly, in fact, multiple firms competefor the privilege, which generates a competition between rent seekers, the effectof which is a waste of resources.


9.3 Regulation of Monopoly

As a matter of principle, the regulation of monopoly is aimed at forcing mo-nopolists to the zero long-run economic profits condition. The problem withregulation is that it can be wrong, in the sense that it can choose a price thatis not optimal and generates an even larger efficiency loss. Indeed, if the chosencombination of P and Q is such that the AC curve is raising at the intersec-tion point, the MC will be larger than the price fixed, meaning that the firm islosing money for each unit produced beyond the intersection between MC andP, which also means that the firms will not be able to stay in the market inthe long run. Thus, the possibility of such an efficiency loss must be comparedto the efficiency loss associated with an unregulated monopoly, before decidingwhich of the two is the preferable alternative.

Another possibility os the government regulating a natural monopoly, whichhas declining AC over the relevant section of the schedule. Under such condi-tions, the government is likely to choose a too high price, which means that themonopolist would be able to produce even more at a lower unit cost.

As I mentioned above, government regulation is not the only force that coun-terbalance the arbitrary power of the monopolist. Consumers’ price elasticityof demand as well as potential competitors from outside the market limit therange of price that the monopolist can choose autonomously. The influence ofcompetitions from outside the market, which is referred to as ”Competition forthe Field”, is particularly important since, for small differences in the AC curvebetween the monopolist and outside firms, the monopolist will choose a pricethat is close to the one that would emerge under pure competition.

9.4 Monopolistic Price Discrimination

The monopolist can, just by adjusting the output produced, reallocate to itselfa share of consumers’ surplus. But there are ways for it to appropriate an evenlarger share, though a strategy called price-discrimination.

Economists have identifies three types of price-discrimination:

1. Market Segmentation (Third Degree Price Discrimination): The monopo-list sells the same good at different prices to different markets, or segmentsof markets. Market segmentation is only possible when customers in thehigh elasticity market cannot resell to those in the low price elasticitymarket, that is, arbitrate between low-price and high-price markets. Theoptimal strategy for a monopolist facing a fragmented market is to setMC equal to the marginal revenue function of each market, meaning thatthe marginal revenues of each market must be equal to each other:

MC = MR1 = MR2 (9.6)

MC = P1(1 +1

η1) = P2(1 +

1

η2) (9.7)

9.5. CARTELS 61

Thus, the monopolist will choose a higher price for the least elastic market,and a lower price for the most elastic one. Market segmentation need notbe geographical. Firms use other strategies, like fragmenting the marketaccording to age, willingness to pay, and so forth.

2. Block Pricing (Second Degree Price Discrimination): Block pricing con-sists in asking to the same customer declining prices for the further unitsof the same good. Block pricing has a fundamental shortcoming: it is verybad at discriminate between different buyers with different willingness topay.

3. Perfect Discrimination (First Degree Price Discrimination): Under perfectdiscrimination, the monopolist asks different prices for each unit of thegood to the same buyer, according to the latter’s declining marginal utility.The monopolist is therefore able to transfer to itself the entire consumersurplus. Perfect price discrimination is different from the other two formsin that it is efficient from the point of view of society, since every buyer thatwants to buy a unit of the good above its marginal cost is accommodated.There is not, in this case, any monopolistic distortion.

9.5 Cartels

A cartel is an alliance between firms operating in the same market the aim ofwhich is to operate as a monopolist, thus making economic profits and allocatingthem among the members of the cartel. The cartel assigns to each firm a quotaof total output that, if respected by every one firm, will maximize the profits ofthe members.

The problem with cartels is that they are very unstable. For one thing, car-tels only work if they are able to co-opt a very large share of all firms operatingin the industry, and at the same time restrain entry into the market. Further-more, even firms that are part of the cartel have a strong incentive to chisel,that is to start producing more of the good, since it can increase its profits bydoing so. If all firms were to behave in the same way, the cartel will soon breakdown and competitions is restored.

9.6 Network Externalities

Sometimes, a particular technology will monopolize the market because of whateconomists call network effect. A good is said to have network effects when theutility that consumers can expect to derive from it increases as the number ofits consumers increase. Thus, the demand for such goods behave differentlyfrom the demand for other goods. To build the demand curve for a networkgood, we first need to draw demand curves for any possible number of peoplethat can be expected to also buy the good. There will be a demand curve forx expected buyers, one for x + 1 expected buyers and so forth. The market


demand corresponds to the conjunction of all those points of the individualdemands curve in which the number of buyers is exactly its expected value.

Chapter 10

Competition Among theFew: Oligopoly andStrategic Behavior

The notion of oligopoly indicates a situation in which the industry is not dom-inated by a single firm (monopoly) nor are there an indefinitely large numberof firms (polypody and pliopoly). A market is said to be an oligopoly when thenumber of firms is so small that they can behave in a strategic way.

10.1 Strategic Behavior: The Theory of Games

Game Theory is the mathematical study of strategic behavior. A game is aninteraction between one or two players the actions of which have repercussionson those of all others. Through the application of the rationality and self-interestassumption, economists can predict the result of the game based on the patternsof the individual payoffs and the protocol (the rules of the game). Individualpayoffs matter because they are what determine the preferred strategy of self-interested players. Games of strategy can have harmony of interests betweenthe players (coordination games) or divergence of interests, as is the case in thepopular prisoners’s dilemma.

Public goods are an example of game in which individual interests’ diverge.A public good is characterized by non-excludability, meaning that consumptionis open to anyone, and non-rivalry, meaning that one’s consumption does notaffect the consumption of others. The problem with public goods is that it paysfor any individual to contribute only when her contribution is essential for theprovision of the good. When everyone contributes, the individual knows thatshe would be able to consume it no matter if she does or not, and is thereforetempted to free-ride. If everyone behaves in the same way, though, the publicgood would not be provided in the first place.

63

64CHAPTER 10. COMPETITION AMONGTHE FEW: OLIGOPOLY AND STRATEGIC BEHAVIOR

In game theory, a solution is a prediction about which state will emerge as anequilibrium. The solution is often a function of the payoffs and the protocol ofthe game. For example, it often matter whether the game is plaid simultaneouslyor in sequence, where simultaneity is not intended in its chronological meaning,but simply means that none of the player knows what the other players aregoing to play. In a sequential play, whenever the first player chooses based onits prediction of what the other players’ rational response will be, the result scalled subgame perfect equilibrium.

For simplicity, games are often depicted as if the players where behavingwithin a symmetrical situation, meaning that they have the same informationand face the same payoffs. In real world situations, though, it is often thecase that the interacting individuals are in fact behaving under conditions ofasymmetry. For example, they might face different payoff schedules, or possessdifferent informations, and so forth. In sequential games, the second playerfor example posses the information about the first player’s behavior, while thelatter did not know what the second player would have plaid.

In a simultaneous game, players choose their preferred strategy withoutknowing what others are playing, based only on their knowledge about thepayoff schedule. When a player has a preferred strategy regardless of whatother players do, this is called a dominant strategy. The intersection betweenthe dominant strategies among all players is called dominant equilibrium. Whennot all players have dominant strategies, there could still be an equilibrium. Anequilibrium will emerge when non e of the players can increase her own positionby changing strategy keeping other players’ strategy constant. Such an equilib-rium is called Nash equilibrium. Although not all Nash equilibria are dominantequilibria, all dominant equilibria are also Nash equilibria.

Sometimes, when players are only allowed to play pure strategies, a gamemay not have any Nash Equilibrium. But when mixed strategies are allowed,than all games have a Nash equilibrium. The aim of mixed strategies is to makeother players’ expected payoffs equally preferred.

10.2 Duopoly: Identical Products

A duopoly is the simplest form of oligopoly, in which only two firms operate inthe same market. Duopoly has only four solutions:

1. Collusive solution: the duopolists form a cartel and operate as a monop-olist in order to maximize their combined profits.

2. Competitive solution: the duopolist behave as price takers, thus producingthe level of output such that MC=P.

3. Cournot Solution: the Cournot solution consists in a competition overquantity produced. The Cournot solution is the result of a simultaneousgame in which each firm operate as if it was the monopolist of the resid-ual demand once the supply of the other producer is taken into account.

10.3. DUOPOLY: DIFFERENTIATE PRODUCTS 65

The solution of the game if found by solving simultaneously the ReactionFunction of the two firms:

D : P = a− b ∗Q (10.1)

Q = q1 + q2 (10.2)

MC1 = MC2 = 0 (10.3)

MaxΠ1 = P ∗ q1 = a ∗ q1 − b ∗ q21 − q1 ∗ q2 (10.4)

MaxΠ2 = P ∗ q2 = a ∗ q2 − b ∗ q22 − q1 ∗ q2 (10.5)

a− b ∗ 2 ∗ q1 − q2 = 0 (10.6)

a− b ∗ 2 ∗ q2 − q1 = 0 (10.7)

q1 =a− q2

2 ∗ b(10.8)

q2 =a− q1

2 ∗ b(10.9)

Where 10.8 and 10.9 are respectively the Response Function of firm 1and firm 2. The solution to the Cournot game is found simply by solvingsimultaneously the two functions.

4. Bertnrand Solution: in the Bertrand solution, the two firms compete overprice instead of quantity, which drives prices toward the competitive so-lution.

10.3 Duopoly: Differentiate Products

So far I have depicted the situation of two duopolists competing for the marketof the same, identical product. Here we analyze a market in which the twofirms sell two different goods, the difference between them being measured byan index of similarity, s, the value of which can go from 0, meaning that the twogoods are entirely different and thus hat the two firms are each the monopolistover their own markets, to 1, at which value the two firms sell two identicalgoods, that is the scenario I addressed above. The new demand functions facedby the two firm will therefore be as follow:

D1 : P1 = a− bq1 − sq2 (10.10)

D2 : P2 = c− d ∗ q2 − s ∗ q1 (10.11)

MR1 = a− 2 ∗ b ∗ q1 − s ∗ q2 (10.12)

MR2 = c− 2 ∗ d ∗ q1 − s ∗ q1 (10.13)

MC1 = MC2 = 0 (10.14)

q1 =a− s ∗ q2

2 ∗ b(10.15)

q2 =c− s ∗ q1

2 ∗ d(10.16)

66CHAPTER 10. COMPETITION AMONGTHE FEW: OLIGOPOLY AND STRATEGIC BEHAVIOR

The final equilibrium will thus depend on the index of similarity: the closers is to zero, the closer the solution will be similar to that of two monopolistsoperating in different markets, the closer it is to one, the closers the solutionwill be to that of a simple Cournot game.

10.4 Oligopoly, Collusion, and Numbers

The historical evidence suggests that, although it is an inferior solution to theCournot one, price competition between the oligopolistic firms. Indeed, pricesin oligopolistic markets tend to be remarkably stable. This has led economiststo believe that oligopolists have come up with a strategy that forces each one ofthem to keep the same price and quantity constant. This strategy is known as”Kinked Demand Curve”. The kinked demand curve is the result of the followingstrategy: as soon a sone firm decides to sell at a lower price than expected, allother firms do the same, thus reducing the slope of the demand faced beyondthe previous price. The demand curve being kinked, the MC curve intersect theMR curve, which is the one that determines the optimal quantity and price ofthe oligopolist, in a vertical jump, that is, a vertical segment, meaning that thefirm cannot increase its revenue by decreasing price at that point.

10.4.1 The Generalized Response Function

The analytics developed for a duopolist can be generalized for an oligopoly ofany number of firms. Given the market demand function P = a − bQ, whereQ = Σqi, then the equilibrium qi, Q, and P are found as follows:

Q ≡ q1 + q2 + q3 + ...+ qn (10.17)

Q ≡ q1 +Q(n− 1) ≡ q1 + (n1)qo (10.18)

P = a− b(q1 + (n− 1)qo) (10.19)

MR1 = a− 2bq1 − b(n− 1)qo (10.20)

For (10.21)

MC1 = MCo = 0 (10.22)

a− 2bq1 − b(n− 1)qo = 0 (10.23)

since (10.24)

q1 = qo (10.25)

a− 2bq1 − b(n− 1)q1 = 0 (10.26)

a = (2b+ bn− b)q1 (10.27)

q1 =a

bn+ b(10.28)

Q ≡ nq1 ≡an

bn+ b(10.29)

P = a− b( an

b(n+ 1)) = a− an

n+ 1=an+ a− an

n+ 1=

a

n+ 1(10.30)

Chapter 11

The Demand for FactorServices

11.1 Production and Factor Employment witha Single Variable Input

A firm’s production possibility frontier is determined by its production function,the technological relationship between the quantity and ratio of the input factorsand the quantity of output: q = Φ(a, b, c, ...) For simplicity, we can assumethat output is a function of the quantity just one of the many inputs, those ofremaining ones being kept fixed. Then, the production function for one inputfactor becomes q = q(a).

The Law of Diminishing Returns applies exactly to this scenario. Let usassume that a firm that needs two factors, a and b, to produce output q, andthat the quantity of b is fixed. The firm’s total returns, marginal returns, andaverage returns can now all be expressed in terms of input a. According tothe Law of Diminishing Returns, keeping all other factors fixed, the increase ininput a will first cause an increase in the quantity of output, this increase willfirst have increasing rate, reach a maximum, and eventually decline. Similarly,when marginal product is lower than average product, the latter will also startdeclining. Total product will also arrive at its apex, after marginal product andaverage product, after which the increase in factor a produces a decrease in thequantity of output produced.

A firm has only two ways to produce a resource. It can own the factors thatare needed, or it can hire them on the factor market. For simplicity of analysis,let’s assume that the firm always hire all factors in the factor market. The firm’stotal cost function will thus depend on the quantity of each factor as well astheir hire prices:

C ≡ haa+ hbb+ hcc+ ... (11.1)

67

68 CHAPTER 11. THE DEMAND FOR FACTOR SERVICES

Since, by assumption, we are keeping all input factors but a as fixed, equation11.1 can be rewritten as:

C ≡ F + V ≡ F + haa (11.2)

Thanks to the production function, we can express total cost as a function ofoutput produced :

C = F + haa ≡ F + haa(q) (11.3)

For (11.4)

q = αaβ (11.5)

a =q(1/β)

α(11.6)

Thus, the total cost function becomes a function of the hire-price and of theinverse of the production function (that is, input as a function of output).

From the total cost function, it is now possible to derive the Marginal Costfunction and the Average Cost function expressed in term of of quantity ofoutput:

MC ≡ ∆C

∆q≡ (11.7)

≡ ha∆a

∆q≡ ha

∆q/∆a≡ (11.8)

≡ hampa

≡MC (11.9)

where mpa is the Marginal Product of the input factor a. As equation 11.9suggests, than, Marginal Cost is an function of the marginal product of theinput factor. MC increases as the marginal product decreases, and decreases asthe marginal product increases.

AV C ≡ V

q≡ haa

q≡ (11.10)

≡ haq/a≡ haapa

(11.11)

AV C ≡ haapa

(11.12)

Where apa is the average product of the input factor a. AVC increases as apadecreases, and vice versa.

AC ≡ C

q≡ F + V

q≡ (11.13)

≡ F + haa

q≡ F

q+haa

q≡ F

q+

haapa

(11.14)

AC ≡ F

q+

haapa

(11.15)

11.1. PRODUCTION AND FACTOR EMPLOYMENTWITH A SINGLE VARIABLE INPUT69

After having expressed the cost functions in term of output, we can nowderive the demand for the single input factor. Geometrically, the demand forthe input factor a corresponds to the downward sloping section of the vmpacurve, that is, the Value of the Marginal Product (or Marginal Value Product):vmpa ≡ Pmpa, that is, the Marginal Value Product of s is the product of themarginal product and the price of output.

The optimal choice for the firm is to hire a up to the point at which themvpa just equals the mfca (the Marginal Factor Cost of a), which, if the factormarket is a competitive one on the buyers’ side, will be equal to afca (AverageFactor Cost) and to the hire-price of a ha:

mvpa = mfca = afca = ha (11.16)

Equation 11.16 represents the Factor Employment Condition for a CompetitiveFirm. The firm will continue hire the input fact until the cost of hiring it justequal the marginal revenue of employing it in production. If that was not thecase, the firm will not being maximizing its profits, since it could hire somemore of the input factor and make a profit from the next unit of output.

If the firm is a monopolist in the product market (that is, it faces a downwardsloping demand curve), the Factor Employment Condition must be generalizedas:

mrpa = ha (11.17)

Where mrpa is the Marginal Revenue Product, that is, the Revenue that thefirm can make by using the next unit of input factor a1:

mrpa = MR ∗mpa (11.18)

The logical relationship between the firm’s input and output decision can bealgebraically demonstrated:

MC ≡ hampa

(11.19)

dividing both sides by MR

MC

MR≡ hampa ∗MR

(11.20)

since (11.21)

mpa ∗MR ≡ mrpa (11.22)

MC

MR≡ hamrpa

(11.23)

Since the firms equality condition for profit maximization in the product marketis MR = MC it must also be that, at the same output quantity, ha = mrpa.

1The Marginal Value Product and the Marginal Revenue Product differ because the mo-nopolist faces a downward sloping, and not perfectly horizontal, demand curve


11.2 Production and Factor Employment withSeveral Variable Inputs

When referring to the choice of a firm regarding factor employment with morethan one inputs, it is easier to think in terms of two input factors, usuallylabor and capital. Geometrically, the choice of the firm in the factor marketresembles that of the utility-maximizing consumer. As consumers’ choice canbe seen as a ”utility hill”, the firms’ choice can be seen as a ”output hill”. In twodimensions, the choice is represented by a family of curves called ”isoquants”,each representing the infinite combinations of inputs a and b that can be usedto produce the same quantity of output q′, each iso quant being associated witha different level of output.

The Output Hill can be used to illustrate geometrically two properties ofthe production process: i. the Law of Diminishing Returns; and ii. Returnsto Scale. The law of diminishing marginal returns tells us that, keeping oneinput fact fixed, any further increase in the employment of the variable factorwill increase output at a diminishing rate. Returns to scale, on the other hand,refer to the effect of a combined increase in both input factors (in a two-inputfactor world), and the effect of such an increase on output. If the increase inoutput is greater than the increase in the increase in the quantity employed ofthe two factors, the firms is said to have increasing returns to scale. When theincrease in output is smaller than the increase in inputs, the firm is said to havedecreasing returns to scale. When the two changes are of the same magnitude,the firm is said to have constant returns to scale.

11.2. PRODUCTION AND FACTOR EMPLOYMENTWITH SEVERAL VARIABLE INPUTS71

Algebraically, economists often use a function of the form q = κaαbβ toexpress a production function, where κ is expresses the general technologicalprocess of the industry. This function is called Cobb-Douglas, and has the


property of making it easier to see whether a firm has increasing, decreasing, orconstant returns to scale:

q = κaαbβ (11.24)

κ(σa)α(σb)β ≡ κσα+βaαbβ (11.25)

σα+β ∗ (κaαbβ) ≡ σα+βq (11.26)

For α + β > 1, σα+β > σ, meaning that the increase in total output is greaterthan the increase in input. ”Under constant returns to scale, if a factor’s hire-price equals its Marginal Product, then each exponent also equals the fractionalshare of total output going to that factor” (Hirshleifer and Hirshleifer, 2005:354).

The optimal ratio between the two inputs of a firm is given by the tan-gency condition between the budget line of the firm and its isoquants. Unlikea consumer, though, a firm has (given perfectly working capital markets) thepossibility of expanding its own budget line, meaning that it will have not one,but an entire family of budget lines. The tangency points between a firm’s bud-get lines and its isoquants produces a Scale Expansion Path (SEP), the bestcombination of inputs at any level of cost. Algebraically, the points of the ScaleExpansion Path are given by the Factor Balance Equation:

MRSq =hahb

(11.27)

Where MRSq is the Marginal Rate of Substitution in Production (also knownas Marginal Rate of Technical Substitution), that is, the amount of input Bthat the firm needs to hire in order to just compensate a small an infinitesimalquantity of input A, or the slope of the isoquants; and ha

hbis the slope of the

budget lines, or isocosts. Since mpa = ∆q∆a and mpb = ∆q

∆b :

MRSq ≡ −∆b

∆a(11.28)

mpampb

≡ ∆q/∆a

∆q/∆b≡ (11.29)

≡ ∆q

∆a

∆b

∆q≡ ∆b

∆a(11.30)

MRSq ≡mpampb

(11.31)

11.2. PRODUCTION AND FACTOR EMPLOYMENTWITH SEVERAL VARIABLE INPUTS73

We can now rearrange the Factor Balance Equation as follows:

MRSq =hahb

(11.32)

MRSq ≡∆b

∆a≡ mpampb

(11.33)

mpampb

=hahb

(11.34)

mpaha

=mpbhb

(11.35)

Thus rearranged, the Factor Balance Equation simply tells us that the marginalproducts per dollar of all inputs,at the optimal solution, must be equal to eachother. If that was not the case, the firm could produce even more at the samecost by shifting from employing the more costly input to the least costly one.

The Factor Balance Equation can as well be expressed as ha

mpa= hb

mpb. From

equation 11.19, we know that ha

mpa= MC. Thus:

MC =hampa

=hbmpb

(11.36)

At the optimal solution, then:

MC

MR=

haMR ∗mpa

=hb

MR ∗mpb≡ hamrpa

=hbmrpb

(11.37)

Since the optimal strategy is to set MR=MC, it must be also that ha = mrpaand hb = mrpb, which is the Factor Employment Condition for an optimalchoice in the factor market.

As in consumption, in production as well facots are usually not independentto each other. It is often the case that two facts are complementary in produc-tion, meaning that an increase in the quantity employed of one of them increasethe marginal product of the other. Two factors can also be anticomplementary,when the increase in the use of one of them decreases the marginal produc-tivity of the other. When they have no effect on the marginal productivity ofeach other, they are said to be independent. Algebraically, complementarityor anticomplementarity are found by taking the second cross derivative of theproduction function:

q = q(a, b) (11.38)

∂2q

∂a∂b(11.39)

When equation 11.39 is greater than zero, the two input factors are complemen-tary. When it is less than zero, the two are anti-complementary.

With more than one inputs, the firm’s demand for an input is not simplythe downward sloping section of the vmp curve, or, more correctly, it is only so


when the two inputs are independent to each other. When the two are com-plementaries or anticomplementaries, the matter is slightly more complicated.This is because when the hire-price of a factor falls, the firm will buy more ofit, but since the two are complements, the firm will also like to buy more of theother good the mp of which has in the meantime being increased. This processis called reverberation, and can go back and forth between the two input factorsfor a while, put to a limit such that, again, the mrpa = ha and the mrpb = hb.This means that, the firm’s demand curve for an input factor is flatter thanit would have been if it and all other factors had been independent. Since thedemand for a factor must be downward sloping throughout, no Giffen effect ispossible for input factors,

11.3 The Industry’s Demand for Inputs

The industry demand for a factor is usually not simply the horizontal summationof all firms’ individual demand curves. This is so because of two complications.

1. The first complication is the product-price effect. When the hire-price ofan input factor decrease, all firms will like to expand production, whichmeans that the new equilibrium prize in the product market will be lowerthan it would have been if only one firm had adjusted to the change. Thechange in the product market affects the equilibrium in the factor marketthrough the output price. Since the output price determines the vmp of thefactor of interest, a change in the hire price of the fact indirectly lowers thevmp of the factor by lowering the equilibrium price. The industry demandis therefore steeper, that is, less responsive to a change in the hire-price,than the demand of each firm.

2. The second complication is the entry-exit effect. An increase in the hiring-price of an input factor might cause some firm to go out of business,shifting the demand in the factor market to the left. Likewise, when thehiring price decreases, more firms will enter in the market, thus shiftingthe demand curve in the factor market to the right. Thus, the industrydemand curve is flatter than each individual demand curve.

11.4 Monopsony in the Factor Market

A monopsonist firm is the sole buyer of an input factor. This implies that thequantity it buys actually affects the factor hire price. Geometrically, the firmfaces an upward sloping supply curve. The optimal choice for a monopsonist firmis therefore not to buy the quantity if input such that mrpa = ha, but rather thequantity such that the marginal factor cost curve intersect the marginal revenueproduct curve. Like a monopolist in the product market, the monopsonistchooses the quantity to purchase, but the price is determined by the supply

11.5. MINIMUM-WAGE LAWS 75

curve, which corresponds to the average factor cost function afca:

mfca =∆Ca∆a

≡ ha∆a

∆a+a∆ha∆a

≡ ha + a∆ha∆a

(11.40)

An the Factor Employment Condition for a Monopsonist is:

mrpa = mfca (11.41)

11.5 Minimum-Wage Laws

When assessing the effect of minimum-wage legislation, economists face a choicebetween two models, the responses of which can be very different. If the com-petitive model is employed, all minimum wage laws are deemed to generate botha disemployment effect (that is, at the mandated wage, the equilibrium quantityof labor exchanged is lower than at the market price equilibrium) and a unem-ployment effect (meaning that at the mandated price, more people would like towork than employers are willing to hire). When the monopsony model is used,the effect of minimum wage laws is mixed, with some people gaining from itand others being net losers. Low skilled workers are the ones who are the mostnegatively affected, while medium skilled workers might end up with greater em-ployment and higher wage, but is also likely to produce some unemployment.High skill workers are seldom affected by minimum wage laws.


Chapter 12

Resource Supply and theFactor Market Equilibrium

12.1 The Optimum of the Resource Owner

In equilibrium, all income must ultimately come from resources that have beenoffered for hire in the factor market. Resource owners have only two ways toemply their resources: i. Retain them for reservation uses; and ii. offer themin the factor market. One’s labor, or more accurately, one’s labor services anttime are among such resources. Each individual has an initial endowment oftime of 24 hours per day. This time can be employed in its reservation use asleisure time or it can be offered to firms as labor:

Rtot = L+Rleisure (12.1)

(12.2)

The optimal choice of the individual concern the amount of time to allocateto each of these uses. This choice is a function of the resource owner’s ownpreferences and of her income. The latter, in fact, determines the opportunityset of the resource owner, that is, all the attainable combinations of leisureand disposable income. The budget function will thus have the form Itot =Iendowment+ hLL, where the individual disposes of an initial endowment andthe remaining income is determined by the wage offered in the labor marketand the hours of labor she offers:

Rtot = L+Rleisure (12.3)

ltot = Iendowment+ hLL (12.4)

substituting (12.5)

Itot = Iendowment+ hL(Rtot−Rleisure) (12.6)

(12.7)

77

78CHAPTER 12. RESOURCE SUPPLY AND THE FACTORMARKET EQUILIBRIUM

Geometrically, therefore, the budget line is a negatively sloped straight line,which intersects the vertical axis at the point in which income is maximized,that is, when the individual chooses to employ all its time as labor, and avertical line that goes through R = 24, at a point that corresponds to the initialendowment, that is, when she offers none of her time as labor. The slope of thebudget line corresponds to the negative of the hire-price for the resource, thatis, wage.

The optimal choice of the consumer will thus be at the tangency point be-tween its indifference curves (an indifference curves which is function of incomeand leisure) and the budget line. Algebraically, therefore, the optimal choicerequires the Marginal Rate os Substitution in Resource Supply to be equal tothe absolute value of the hire-price of the resource:

MRSR =∆I

∆R|u (12.8)

MRSR = hl (12.9)

MURMUI

= hl (12.10)

MUR = hLMUI (12.11)

As the initial endowment Iendowment increases, the optimal choice of theindividual will vary. Geometrically, this relationship between income endow-ment and labor supplied can be expressed through the Income Expansion Path(IEP). For the usual well-behaved indifference curves, the individual will chooseto supply less and less labor as her initial endowment increases.

Similarly, as the hire-price for the resource, wage, in this case, varies, theoptimal choice of the individual also varies. Geometrically, the choice of theindividuals can be shown by the Wage Expansion Path (WEP). This curvehas the particular feature of being backward bending. Initially, for low valuesof hL, the response of the individual will be that of expanding her supply oflabor. After a threshold, though, an increase in wage provokes a decrease inthe quantity of labor supplied in the market. This is because changes in themarket wage generate both a substitution and an income effect. The substitutioneffect is that of making the resource owner wanting to provide more labor. Thiseffect can though be counterbalanced by an income effect: when wage rises, theresource owner now has more income for any level of labor supplied, and istherefore tempted to forgo some income in order to use the increased income inorder to enjoy more leisure.

The backward bending of the WAP is reflected in the individual labor supplycurve. Unlike most supply curves, the supply curve for labor slopes negativelybeyond some value of W.

This model can be used to predict the effect of the introduction of welfareguarantees. Welfare guarantees are payments to all those individuals who cannotreach a certain level of income. The problem with such guarantees is that theytend to push individuals toward a corner solution in which they supply zerohours of labor.

12.2. PERSONNEL ECONOMICS: MANAGERIAL APPLICATIONS OF EMPLOYMENT THEORY79

12.2 Personnel Economics: Managerial Appli-cations of Employment Theory

Quantity, expressed in terms of hours of work, is not the only relevant feature oflabor. Effort is equally, if not more, important. The problem, with effort, is thatit is very costly to observe, meaning that employers will find it costly to monitortheir employees and pay them the right wage to incentivize effort. The relativecost of monitoring effort and the quality of the product are the main causes offirms’ choice between two payment methods: i. Payment by the Hour; and ii.Payment by the Piece. Payment by the hour is the most common method, andmakes no distinction among workers based on effort, with the result that effortis relatively disincentivised. Payment by the piece is, on the other hand, moreeffort sensitive. The least effort averse employees will like to produce more unitsof output, thus increasing their hour wages. The problem with piece paymentsis that they incentivize workers to produce more but often at the expense ofquality. When quality is easily observed, though, by the piece payment is oftenrelatively more efficient. Thus, we will observe more by the piece payment whenquality monitoring is cheaper.

A third dimension of labor is quality. Usually, the more human capital anindividual has accumulated, the more productive would one unit of her labor be.Employers must therefore find a way to pay differential wages to high qualityand low quality workers. When mechanisms of this sort are too costly, the resultis a Pooling Equilibrium, in which high quality and low quality workers are paidthe same wage. If, on the other hand, the firm is able to find a mechanism to sortthe one type from the other, the result is a Sorting or Separating Equilibrium.A sorting equilibrium can be the result of: i. signaling, when the potentialemployees are the ones sorting themselves out (e.g, getting a degree at a goodcollege); or of ii. screening, when the employer comes up with a way to sortthem out (for example, by requiring candidates to do a test for assessing theirabilities).

The application of the signaling model to biology has produced the theoryof the handicap principle, according to which an animal which is able to survivenotwithstanding an evolutionary handicap (as is the case with a peacock’s tail)indirectly signals that it is stronger or more intelligent than others, thus makingits genes more preferable by its potential partners.

12.3 Factor Market Equilibrium

The factor market equilibrium is determined by the intersection between marketdemand and market supply. Unlike the individual supply curve, the marketsupply curve (which is the horizontal summation of all individual supply curves)needs not be backward bending. Since individual preferences vary a lot, andsince some individual will likely supply more resource as the hire-price increaseat any given level in the relevant section of the demand-supply schedule, themarket supply curve will actually have the usual positive slope.


Demand and supply are each influenced by a combination of factors. Con-cerning demand, these are:

• Technology (which determines the marginal product and complementarityof factors)

• Demand for the final product (which influences the equilibrium price inthe product market and therefore the mrp of the resource)

• Supply of complementary resources

The factors influencing the supply side are:

• Preferences of the resource owners

• Initial endowment of the resource owners

• Demography quantity of resource owners and concentration of resourceownership

• Social Forces and Legislation

12.3.1 Sources of Growing Wage Inequality

The historical evidence seems to suggest that, since the 1980s, the US popu-lation has seen an increase in wage inequality between the top 10 percent andthe bottom 10 percent. Economists have suggested a variety of causes as theresponsible for this process:

1. International competition (competition from low skilled workers has drivendown remuneration for the bottom of the income distribution)

2. Technological change (low skilled workers have been substituted by ma-chinery, while high skill ones have seen their mrp increase)

3. Immigration

4. Weakened unionization

5. Winner-take-all markets

6. Increasing Opportunity Paradox

12.4 The Functional Distribution of Income

Classical economists used to divide resources into Land, Capital, and Labor.According to them, economics consisted in the study of the forces that determinethe distribution of income toward each of these resources, where their returnswere called respectively Rent, Interest, and Wage. Modern economists refutethe analytical element of this distinction, since the laws for the remuneration

12.5. ECONOMIC RENT 81

of resources is the same regardless of their physical characteristics. Moreover,modern economics has questioned the very distinction between such resources,especially when this is based on physical differences. For example, if capital isindeed defined as the produced factor of production, then much of what makesone’s labor productive (such as education) is itself capital (human capital). Atthe same time, land, which included all natural productive powers of the soil,and is assumed to be inelastic in supply, cannot be used to define actual land,the supply of which is at least in part a function of its hire-price.

Capital in particular has been the subject of much discussion in moderneconomics. Capital is often used to refer to two different notions: Real Capital,that is the capital goods in their physical features, and Capital value, that is themarket evaluation of capital goods. According to the law of equal returns, allresources must ultimately receive an equal return, otherwise more firms mightdemand more of the more remunerative one, thus rising its relative price untilthe rates of returns are equilibrated. The rate of return (ROR) is the ratiobetween annual net earnings and capital value:

RORa ≡Za + o∆Pa

Pa(12.12)

Where Za is the annual net earnings, to which it must be added the anticipatedchange in value of the resource ∆Pa, and Pa is the capital value.

12.5 Economic Rent

Economists define economic rent as the differential between the reservation priceof the resource and the actual market price that emerges in the factor market.Economic rent thus corresponds to the producers’ surplus in the product market.


Chapter 13

The Economics of Time

13.1 Present Versus Future

People don’t only exchange present goods for other present goods, but oftenexchange present goods against future ones. The rate of interest 1 + r is thepremium on the value of current goods relative to future goods. Geometrically,the rate of interest is the slope of the intertemporal budget line of the consumerchoice, and algebraically it’s the ratio between the price of present consumptionand that of the future consumption:

P0

P1≡ 1 + r (13.1)

Mathematically, r can be any positive number, but it can also be negativeas long as it is greater than minus one. Indeed, if that was the case, 1 + r wouldbe negative, meaning that one of the prices of the goods involved must alsobe negative, contradicting the assumption that both items are in fact economicgoods.

13.2 Consumption Choices Over Time: Pure Ex-change

Time allows the individual to perform three different but interconnected actions:

1. Borrowing

2. Lending

3. Investing

In a pure exchange economy, no net investment can take place, since there isno production. The consumption choice of the individual is therefore deter-mined in the same way as in the atemporal choice. The optimum combination

83

84 CHAPTER 13. THE ECONOMICS OF TIME

of present and future consumption is found at the tangency point between theinter temporal budget line and the individual’s indifference curve. Let’s assumethe individual in question has an initial endowment of q0 in T0 and of q1 inT1, and that P0 is the numeraire, so that P1 = P0

1+r = 11+r . The budget line

limits the combinations of present and future consumption attainable. At oneextreme, the individual can choose to consume her entire inter temporal budgetin present consumption meaning that she would consume all her endowment ofpresent goods and will also exchange all her future endowment for some quan-tity of present goods, the value of which is determined by the rate of interest.Algebraically, the maximum of present consumption would therefore be equalto q0 + q1

1+r . At the other extreme, if the consumer were to consume only futuregoods, the maximum she could achieve would be given by the sum of the futureendowment and the future value of the present endowment: q1 + (1 + r) ∗ q0,which means that the intercept between the vertical axis and the budget line isgreater than that between the horizontal axis and the budget line.

When the tangency line between the budget line and the indifference curveis such that the optimal quantity of future good is larger than the future en-dowment q1, and the optimal quantity consumed of present goods is smallerthan the initial endowment q0, than the individual is choosing to forego presentconsumption against future consumption, that is, she is a net lender. In theopposite case, when consumption of present good is larger than its initial en-dowment and the consumption of future good is smaller than it, the individualis scarifying future consumption for present one, meaning that she is a net bor-rower. When the tangency point exactly coincides with the initial endowment,the consumer funds herself at ”Polonius Point” (Varian, Ch. 9), the point atwhich she is neither a lender nor a borrower.

The optimal choice of the individual is determined by three factors:

1. His Initial Endowment

2. The Market’s Rate of Interest

3. Individual Preferences

Changes in the rate of interest affect the individual choice as follows. Keepingeverything else equal, an increase in the rate of interest corresponds to a doubleprice change, that is, an increase in today’s price of consumption and a decreasein the price of future consumption. Geometrically, as the interest rate increases,the budget line becomes steeper. similarly, as the rate of interest decreases,the opportunity cost of present consumption is diminished, which geometricallymeans that the budget line becomes flatter.

13.3 Production and Consumption over Time:Saving and Investment

For simplicity, so far I have assumed that no investment were possible, meaningthat today’s foregone consumption could only increase tomorrow’s consumption

13.4. INVESTMENT DECISIONS AND PROJECT ANALYSIS 85

through appreciation at the market’s rate of interest. Production, though, allowsfor an even greater increase in future consumption, since present endowment canbe used to increase the future one (although not indefinitely, thanks to the law ofdiminishing returns). When production is introduced, the individual will havea production optimum and a consumption optimum. First, the individual willchoose the production optimum that maximizes her budget given the prevailingrate of interest. Then, she will find the optimal consumption bundle at thetangency point between the new budget line and the highest indifference curve.

At equilibrium, the supply of saving equals the demand for investment, andsimilarly borrowing equals lending.

13.4 Investment Decisions and Project Analysis

According to the Separation Theorem a person’s production optimum positionQ∗ is entirely independent of his or her personal preferences. The separationtheorem tells us that a maximizing individual will produce such that her budgetis maximized, and that the decision of how much to produce will be determinedentirely by external forces (the relative prices of the goods produced) and notby her preferences about such goods.

The assessment of a project in two periods is based on the present value ofthe flows z0 and z1, which is given by the identity V0 ≡ z0 + z1

1+r1. According

to the present value rule, the individual should always choose to pursue projectswith positive present value, and forego projects with negative present values.This is only true when all projects are independent. When the available projectsare mutually exclusive, the present value rule tells us that we should choose theproject with the greater present value, given that this is positive. More generally,the individual should choose the combination of projects that maximize presentvalue.

Algebraically, the present value of a flow zt in time t is given by the followingequation:

V0 ≡zt

(1 + r)t(13.2)

Where r is the prevailing rate of interest throughout all periods. The presentvalue of n cashflows zi is given by the following identity:

V0 ≡ z0 +z1

1 + r+

z2

(1 + r)2+

z3

(1 + r)3+ ...+

zn(1 + r)n

(13.3)

A particular case is that of the so called ”Consul”, that is, a form of bond thatpays a certain sum of money forever. The present value of a Consul is found as


follows:

V0 ≡z

1 + r+

z

(1 + r)2+

z

(1 + r)3+ ...+

z

(1 + r)n(13.4)

V0 ≡1

1 + r∗ (z +

z

1 + r+

z

(1 + r)2+

z

(1 + r)3+ ...+

z

(1 + r)n) (13.5)

V0 ≡1

1 + r∗ (z + V0) (13.6)

V0 ≡z

1 + r+

V0

1 + r(13.7)

(1 + r) ∗ V0 ≡ z + V0 (13.8)

V0 ∗ r ≡ z (13.9)

V0 ≡z

r(13.10)

An alternative to the present value rule is the ”Rate or Return (ROR) Rule”.According to this rule, a firm should always adopt a project when the ROR ispositive ρ > 0. The rate of return is the discount rate such that the presentvalue of any project is equal to zero:

V0 = 0 (13.11)

0 ≡ z0 +z1

1 + ρ+

z2

(1 + ρ)2+ ...+

zn(1 + ρ)n

(13.12)

When ρ > r, the project should be adopted. The problem with the ROR rule isthat it is not a reliable guide for investment decisions, especially when multiple,incompatible alternatives are presented to the firm. It might in fact be the casethat the project with the highest rate of return is not also the one with the largerpresent value. When this is the case, the firm should always choose the one withthe largest present value, regardless of the ROR.

13.5 Real Interest Rate and Monetary Interest:Allowing for Inflation

In real world economies, there is not just one interest rate, but rather a multitudeof them. Distinguishing between them is very important, especially in the case ofreal and nominal (or monetary) rates of interest. The monetary rate of interestis the premium on current money over future money, that is, r

′is the extra

amount of future money that must be offered in exchange for current money:

1 + r′ ≡ −∆m1

∆m0(13.13)

The nominal and the real rate of interest determine the price level, P . The pricelevel is equal to the ratio between the quantity of money and the quantity of

13.6. THE MULTIPLICITY OF INTEREST RATES 87

tradable goods and services:

P0 =m0

c0(13.14)

P1 =m1

c1(13.15)

(13.16)

The monetary or nominal rate of interest is equal to the sum of the real rate ofinterest, the rate of inflation a1, and the cross product of the two:

r′1 = r1 + a1 + a1r1 (13.17)

The effect of an increase in the monetary base is not univocal, but dependson the expectation of the consumers. If consumers expect the increase in themonetary base to be only temporary, they would be relatively more willing totrade present money for future money, thus decreasing the equilibrium rate ofinterest. But if they interpret the expansion as a signal of even larger ones inthe future, they might rather exchange future money for present money, thusincreasing the equilibrium rate of interest.

13.6 The Multiplicity of Interest Rates

There are three main causes of the presence of multiple interest rates in theeconomy:

1. Risk: Different plans have different levels of risk, which means that themore risky plans must offer a compensation.

2. Transaction Costs: Transaction Costs also generate higher rates of interest

3. Terms: The longer the term of a plan, the higher the interest rate ceterisparibus. This is due to two factors: the first is that the future is riskierthan the present, and risk must be compensated for; the second concernsflexibility. Plans that take longer to produce returns are less flexible, andpeople have a positive preference for flexibility. This lack of flexibility istherefore compensated by higher interest rates.

13.7 The Fundamental of Investment, Saving,and Interest

The equilibrium interest rat between the supply of saving and the demand forborrowing is determined by a variety of factors, which accounted for the factthat, through time and across places, this has varied hugely. Among thesefactors, the most important are:


1. Time-Preference: Individuals end to prefer today’s consumption to tomor-row’s consumption, ceteris paribus

2. Time-Endowment: The allocation of one’s endowment across the differentperiods influence her choice between being a net borrower and a net lender

3. Time Productivity: The more time productive a plan, the more investmentwe will observe, and the lower present consumption will be.

4. Degree of Isolation: Insulated areas with little access to broad financialmarkets have interest rates far from the global average

5. Risk

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Price Theory and Appications