Analysis and assessment of urban transformation - Notes

Introduction.

This section of the course in “Economic assessment of urban transformation” concerns the first part of the

title: the “economic assessment” of a project, policy or any other kind of public intervention. Two particular

words mentioned in the previous sentence represent the basic concepts that will accompany us through

the following discussion.

The first word is economic. In this course we will consider any problem from the point of view of an

economist, which is different compared with the point of view of a philosopher, of a sociologist or of any

other social scientist. Then, the first question to be answered involves the meaning of this word: what is

economics about?

The British economist Lionel Robbins (1898 - 1984) provided an effective answer. According to him,

economics is the science studying human behavior as a relationship between given ends and scarce means

which have alternative uses. The first point to be noticed is the definition of economics as a science. As any

science, it starts from the observation of the physical world in order to formulate statements of

uniformities and consistencies of relationship between natural phenomena. In our case, the object of the

study (the natural phenomena) concerns human behavior. More precisely, recalling Robbins’ definition,

economics is aimed at explaining and predicting human behavior through the analysis of three main issues:

given ends, scarce means and alternative uses. Try to think at a very simple example. You are in the railway

station, waiting for the train, and you are very thirsty. What you want to do (your given ends) is to drink

something. Close to you there is an automatic distributor. It is filled with many bottles: water and soft

drinks. The scarce means are represented by the few coins in your pocket: you would like to drink a cold

coke but you do not have enough money for that. Then, after having checked the bottom of your bag

looking for some forgotten coins, you will decide to buy a bottle of plain water.

Economists formalize such problems in mathematical terms. In other words, they try to give a functional

form to both preferences and financial constraints. The aim of this procedure is to build a behavioral

model, able to explain and predict individuals’ decisions and attitudes. In particular, microeconomics

focuses on the behavior of the units: a general consumer (for instance the thirsty individual of our example)

or a single firm (producing for example the bottles in the distributor). By contrast, macroeconomics studies

the behavior of the economy as a whole, analyzing the interactions between aggregate measures. A typical

macroeconomic concept is, for instance, the Gross Domestic Product (that is, the value of all the final goods

produced within a country) and its interdependencies with private consumption, defined as the sum of all

consumers’ spending. In the following lines we will restrict our attention to some basic topics in

microeconomics. More in details, we will focus on the consumer problem.

At this point one question comes naturally to mind: why should we be interested in predicting the

economic behavior of consumers, of firms and of all the other economic agents? The very final goal of

economic theory is to provide suggestions and ideas about the best organization of our society. Actually,

there are many economic theories, such as Marxism, Liberalism, Keynesianism, Mercantilism and so on.

Some theories are no longer accepted, but each of them represents a way of looking at the world and at

the interactions among economic agents. Moreover, each theory provides some advices about the most

desirable social organization. For instance, some theories argue in favor of a strong public intervention in

the economy, while some others require competition as a necessary condition. Who is right and who is

wrong? Since economics is not an exact science, such as physics or medicine, answering this question is

almost impossible. However, this topic will be addressed in the second chapter of these notes, concerning

the results of the two fundamental theorems of welfare economics. These theorems (representing two of

the most important results of mainstream economics) provide arguments supporting competition and free

markets. We will review these findings and we will compare the theoretical predictions with the empirical

observation of the real word. Are markets organized as we would expect based on the two theorems’

results? This discussion will lead us to the second core concept of our analysis: the role of the public sector.

Recalling the first line of this introduction, any project or policy will be considered from the public point of

view. Compared with private assessment methods, this perspective is completely different as it accounts

for all the members of a society rather than for a few categories. The last section of these notes is devoted

to a comparison between two families of project assessment methods: Cost-Benefit Analysis (CBA) and

Multicriteria Analysis.

1. Consumer theory.

In the introduction to these notes we said that the objective of economic theory consists in explaining and

predicting economic agents’ behavior. At this stage, we will consider the case of two main actors: the

consumer and the producer. Their problems are very similar. The consumer makes a choice, among many

consumption possibilities, based on his preferences (or tastes, what he likes) and on his budget constraint

(the money he has in his pocket, what he can do). Imagine having 10€ to spend for your dinner. Your choice

will depend on this restriction andon your tastes. For instance, if you dislike Chinese food but you love pizza

you will choose an Italian restaurant. The producer faces a very similar problem. He has to decide what to

produce among many options. The conclusion is based on both what he wants to do (given the available

technology) and on what he can do (determined by the firm’s budget).

The solution of the consumer problem leads to the setting of the so-called demand curve for a certain

commodity. On the other hand, the analysis of the producer’s behavior allows us to draw the supply curve

for the same good. The study of the interactions between demand and supply is one of the primary scopes

of economics. We will handle this topic in chapter 3 and 4, while in this section we will focus on the

consumer problem. At the end of the chapter you will be able to draw, starting from individual preferences,

a demand curve. We will not approach the case of the producer. A few lines will be devoted to this issue, in

order to let you understand the similarities with the consumer’s case and the rationale behind the slope

and the shape of the supply curve.

Before starting, a preliminary remark is necessary. As we mentioned in the introduction, economic theory

provides a mathematical formalization of economic agents’ behavior. In order to be able to generalize its

results, the theory must be based on a set of assumptions, allowingthe economists to formally specify these

behaviors.The first relevant question is the following one: who are the economic agents? At this very first

stage we identify two main characters: consumers (spending their money for buying commodities) and

firms (producing consumption goods and selling them to the consumers).

Start by considering the consumption case. As we said, consumers always have to choose among a set of

alternatives the best one, based on their own preferences. According to our theory, the three following

assumptions hold:

1.individuals always have a well defined notion of what they like;

2.they perfectly know what their objectives are;

3.they make decisions aimed at fulfilling their objectives or, put differently, they always pursue their own

self-interest.

In a nutshell, consumers are said to be rational. In the following paragraphs we will better understand the

meaning of this assumption. The hypothesis about rationality has a counterpart for the producer: firms are

assumed to be profit maximizing. Intuitively, the scope of the firm is to earn the highest profits possible.

Finally, a third assumption involves the place where consumers and producers meet: the market. Markets

are assumed to be highly competitive. Competition means that there are so many agents (buyers and

sellers) in the market that none of them is able to influence the price of the commodities. This assumption

is particularly important, as it will follow us for the whole discussion. For this reason it is better to clarify it

immediately. Consider the outdoor market close to your place. You can find there many peddlers, selling

fruits and vegetable. Suppose you want to buy some apples, and that the price of the apples is 1€ per kilo.

Suppose to go to the first peddler asking him for a discount: what about selling your apples at 0.90€ per

kilo? In a competitive market, the peddler will refuse this trade. There are so many customers in the market

(willing to buy apples at 1€ per kilo) that the single consumer does not have any power to influence the

market price. If you will not buy the apples at 1€ per kilo, someone else will accept this price. The peddler

does not have any incentive to allow for a discount. The same reasoning holds for the production side of

the economy. What if one peddler decides to increase the price of apples to 1.10€ per kilo? Well, he will

simply loose all his customers, since in the market there are so many sellers (willing to sell their apples at

1€ per kilo) that the single peddler cannot influence the market price. His costumers, when facing the new

price of 1.10€, will buy the apples from someone else. In other words, consumers do not have any incentive

to pay the apples more than 1€ per kilo. The term incentive is particularly important in economic theory.

This will be clear in chapter 3, when we will understand that economic policies are nothing but rules aimed

at providing the right incentives (from the perspective of the social planner) to economic agents.

This set of assumptions (consumers’ rationality, profit maximizing firms and competitive markets)defines

the so called basic competitive model. It is a basic model, since it ignores many relevant actors (for instance

the role of the government, which will be analyzed in chapter 3).You have to consider it as a convenient

approximation of what is going on in the real world, but not as an accurate description of it. This model

constitutes the starting point for studying economics. Its peculiar strength consists in allowing us to

formalize economic agents’ behavior and, as a consequence, to generalize our conclusions.

1.1. Scarcity: the consumption set.

As mentioned in the introduction, consumers’ choice is the result of the interaction between what you can

do and what you want to do. In the following lines we will illustrate the consumer problem through a very

simple example. We will consider the story of Jim, a young student, who has to allocate his poor monthly

budget among different consumption options. Our first step concerns what he can do with his money.

Jim usually has lunch at the cafeteria close to the university. After morning lectures he is both hungry and

thirsty. The menu, rather basic, includes just two items: bottles of beer and slices of pizza. In principle, Jim

would like to fill his stomach with tons of pizza and liters of beer. Figure 1 represents the so

calledconsumption set.

Figure 1.1. The consumption set.

The horizontal axis reports the number of slices of pizza, whilst on the vertical axis you find the bottles of

beer. As we said, Jim is very hungry and probably he would choose to eat 15 slices of pizza drinking 15

bottles of beer (point B (15,15)). However this consumption bundle is not feasible, since Jim has to attend

another lecture within one hour. Then, he has not enough time to consume the amount of pizza and beer

represented by the consumption bundle B. Moreover, he does not have enough money for all these

expanses. Maybe, after some careful considerations about his financial condition, Jim will decide to buy just

a couple of slices of pizza and two beers (point A). More formally, the consumption activity is subject to

both time andbudget constraints. In our discussion we will not care about time constraints, while we will

focus on the connections between the available amount of money and the consumer choice. Concerning

figure 1.1, there is still one thing to remember: any bundle in the first quadrant is available for

consumption. For instance, Jim could choose to buy ½ beer and 2.3 slices of pizza, or 4.5 beers and 4.5 units

of pizza (point C). In other words, we are assuming that goods are infinitely divisible.

At this point we have already learnt something. We understood how to represent the consumer problem

when dealing with two commodities (but the same reasoning holds if we raise the number of consumption

goods: simply, it cannot be easily represented on a two-dimensional sheet of paper). Moreover, we noticed

that Jim has not enough money to buy whatever he wants. Hence, the next problem to be solved is the

following one: how can we include in our analysis Jim’s budget constraint? In order to answer this question

we must introduce two new elements: prices and income (or, more in general, wealth). Suppose Jim is

working as a clerk in a bookshop. He works just for a few hours a day, since he has to attend his courses as

well. At the end of the day he gets paid 20€. This amount of money is Jim’s income, and it represents the

sum he can invest in his lunch. Assume that every slice of pizza costs 4€, while the price of a beer is 2€.

Then, given his budget, he can buy a maximum of 5 units of pizza (20/4=5, allocation B in figure 1.2) or 10

bottles of beer (20/2=10, allocation A in figure 1.2). These two solutions are rather “extreme”, in the sense

that Jim would spend all his money exclusively in one single good (beer or pizza). However, he could prefer

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

Slices of pizza

Beer

A

B

C

to buy a mix of the two commodities. If he is very thirsty, for instance, he could decide to buy 2 slices of

pizza and 6 beers. Notice that he would still spend all his money (2∙4 + 6∙2 = 20, allocation C in figure 1.2).

Moreover, point C lies on the line connecting A and B. This line, better defined as the budget constraint (or

budget line), represents the combinations of goods that a consumer can purchase given the current system

of prices and his income, when he spends all his income in the consumption activity.

Figure 1.2. Budget line andopportunity set.

If Jim chooses any bundle on his budget line, he will spend 20€ for the lunch. But he could also decide to

save some money: any point in the blue area, including the allocations below the budget line, is attainable

by Jim. This area, well-known as the opportunity set, comprehends the bundles available for consumption,

given current prices and income. It is bounded by the budget line which, as we have seen, is defined by the

maximum amount of goods can be purchased.

By introducing the budget line we restricted Jim’s possibility set. Now we know that he will not be able to

consume at the same time 4 slices of pizza and 9 beers (point D). Does the budget line tell us anything else

about what Jim can do? Consider the slope of the constraint. Suppose Jim is taking into consideration the

consumption bundle represented by point C, in which he gets 2 units of pizza and 6 beers. He asks himself

how many bottles of beer he should give up in order to have one extra slice of pizza, still spending all his

income in the consumption activity (or, in other words, still lying on the budget line). You can easily see

that if Jim wants to raise the consumption of pizza to 3 units, he must renounce to two beers, moving from

point C to point E in figure 1.2. In point E Jim would eat 3 units of pizza and only 4 beers. The slope of the

budget line is given by the ratio between the rise (the change measured on the vertical axis, i.e. the change

in the consumption of beer) and the run (the change on the horizontal axis, i.e. the change in the

consumption of pizza). In our case, the slope of Jim’s budget line is simply equal to:

4 6 22

3 2 1

Rise

Run

− −= = = −−

D(4,9)

B(5,0)

C(2,6)

A(0,10)

E(3,4)

0 1 2 3 4 5 60

2

4

6

8

10

12

Slices of pizza

Beers

This ratio shows that Jim has to renounce to 2 beers (the numerator, -2) in order to have one more slice of

pizza (the denominator, +1). Pay attention to two peculiar characteristics of the slope of the budget line.

First of all it is always negative: this makes sense, since it means that if you want to increase the

consumption of one good you must give up a certain amount of the other one. Secondly, it is constant. This

means that starting from any point on the budget line (A, C, E and so on), Jim will still have to renounce to

two bottles of beer if he wants to get one more pizza. In other words, the slope of the budget line is always

equal to -2. Also this property is sensitive, because the budget line represents nothing but the relative price

of one good in terms of another. If you must give up 2 bottles of beer in order to have one more pizza, it

means that the price of each slice is twice the price of each bottle. This is exactly what we assumed at the

beginning of our story, when we said that the pizza costs 4€ per slice whilst the price of beer is 2€ per

bottle. If the price of beer increases (for instance to 4€), keeping constant Jim’s income and the price of

pizza, the slope of the budget constraint would change. In this case it would be equal to -1 since, being the

price of the two commodities the same, Jim should renounce just to one beer in order to have one extra

unit of pizza.

Summing up what we reviewed in this paragraph, we observed that Jim cannot buy whatever he wants. His

possibilities are constrained by his income and by the market prices.

1.2. Preferences: utility and consumers’ rationality.

1.2.1. The concept of utility.

In the previous paragraph we showed what Jim can do. Given his poor income, he cannot eat more than 5

slices of pizza and he cannot drink more than 10 beers. At this stage, we can ask ourselves about Jim’s

objectives. What does he want to do? This question is relevant, since if we understand what his objectives

are, we will be able to explain and predict his behavior.

Jim is interested in buying some pizza and beer, but we did not spend many words about his motivations.

We just said that he is thirsty and hungry, and for these reasons he is decided to buy something at the

cafeteria. In this case, the consumption activity is somehow necessary, in the sense that Jim really needs

something to eat and drink. However, people often buy things they do not really need. Every day we

purchase goods and services simply because we like them. Hence, the question is: why do people buy?

The underlying assumption on which is based the whole consumer theory is that

individualsconsumesincethey derive happiness, satisfaction from the consumptionactivity.Economists

rarely talk about happiness or satisfaction, but they usually use a synonym for these terms: utility.

Increasing one’s own utility is the final goal of individuals’ behavior. Since utility depends on consumption,

people spend their money in purchasing things they like. For instance, you bought a new cell phone even if

the old one was still working because you prefer the last model, and you feel happier with your new

gadget.

More formally, we are assuming that utility directly depends on the amount of goods we are consuming. In

other words, utility can be expressed as a function of the commodities consumed. In mathematical terms,

this can be written as

1 2( , )U f x x=

In the expression Uis the total amount of utility, whilst x1 and x2 represent the two consumption goods.

For example, Jim’s utility could be expressed as

( , )U f pizza beer=

Thus far, we clarified one point: individuals buy since consumption increases their utility. Be careful, this

does not mean that the consumption of any good or service has a positive effect on utility. Obviously

consumers buy just what they like.

We can stress one more point, which is really relevant as it represents one of the underlying ideas in

economic theory. As you observed, Jim’s utility exclusively depends on his own possessions. His happiness

derives from the quantity of beer and pizza he consumes. His behavior is aimed at maximizing his own self-

interest. In other words, he is not concerned about other people’s welfare, but just about his own utility.

Box 1.1. What is a function?

In the following lines I quote the definition of function from the book “Fundamental Methods of

Mathematical Economics – 4th

ed.”, by A.C. Chiang and K. Wainwright. This is a useful reference if you

want to refresh your mathematical skills. I added (in italic) some comments to the original text.

Ordered pairs. In writing a set {a,b} we do not care about the order in which the elements a and b appear,

because by definition {a,b} ={b,a}. The pair of elements a andb in this case are unordered pair. When the

order of a and b does carry significance, however, we can write two different pairs denoted by (a,b) and

(b,a), which have the property that (a,b)≠(b,a) unless a=b. For instance, in figure 1.2 (as well as in the other

graphs) we considered a set of ordered pairs, where the first element pertains to the horizontal axis and

the second element is a y value. Hencepoint B (5,0) is different from point (0,5).

Functions. A function is a set of ordered pairs with

the property that any x value (on the horizontal

axis) uniquely determines a y value (on the vertical

axis). Although the definition of function stipulates

a unique y for each x, the converse is not required.

In other words, more than one x value may

legitimately be associated with the same y value.

This possibility is shown in figure B.2.1, where the

values x1and x2 in the x set are both associated with

the same value in the y set by the function y=f(x).

x

y

A B

y=-x²+3x+2

x1 x2

y1

A function is also called a mapping, or transformation; both words connote the action of associating one

thing to another. In the statement y=f(x), the functional notation f may thus be interpreted to mean a rule

by which the set x is “mapped” (“transformed”) in the set y. Thus we may write:

f:x→y

Where the arrow indicates mapping and the letter f symbolically specifies a rule of mapping.

In the function y=f(x), x is referred to as the argument of the function and y is called the value of the

function. The set of all permissible values that x can take in a given context is known as the domain of the

function, which may be a subset of the set of all real numbers.

In economic models, behavioral equations enter as functions (consider for instancePatrick’s utility function

in section 1.2.3). Since most variables in economic models are by their nature restricted to being

nonnegative real numbers (think at the quantity of a certain good, it cannot be negative), their domains

are also so restricted. This is why most geometric representations in economics are drawn only in the first

quadrant.

In the example represented in figure B.2.1 the rule of mapping f is equal to (-x²+3x+2). The domain of the

function is the set of nonnegative numbers (we do not say “positive” numbers since the 0 is included in the

domain). Hence, when choosing the functional form for utility you want to find a rule for transforming your

preferences into a measure of satisfaction. Pay attention to one point: we said that the definition of

function stipulates a unique y for each x. Consider figure B.2.2: does it represent a function?

Figure B.2.2.

You may answer “no”, since x1 is associated with

two different values in the y set, y1 and y2.

The answer is somehow correct, in the sense that in

this case y is not a function of x. That is, we cannot

write y=f(x).

But be careful, as the opposite holds, since x is a

function of y: x=f(y). Then, figure B.2.2 still

represents a function, in which y is the argument

and x is the value. Note that there is a unique x for

each y, as required by the definition of function.

More specifically, the rule of mapping is very similar

to the one graphically represented in figure B.2.1:

x = -y²+3y+2

1.2.2 Some assumptions about preferences.

At this point we have to represent consumer preferences graphically, similarly to what we made in figures

1.1 and 1.2. We said that utility is a function of the amount of goods consumed. But how can we model

these functions? Answering this question requires some assumptions about individuals’ behavior.

As we mentioned in the introduction to this chapter, individuals are assumed to be rational. In a nutshell,

individuals are assumed:

• to have a well defined notion of what they like or dislike;

• to care about quantities (more is better);

x

y

x1

y1

y2

• to make decisions and choices based on their own self-interest (what we anticipated in the

previous paragraph).

These ideas can be more formally expressed by the following definitions:

a. Individual preferences are complete if for any two consumption bundles A and B, either ARB (A is at

least as good as B), or BRA (B is at least as good as A), or both. In this statement we make use of the

symbol R, which means “as least as good as”. For instance, consider point C and E in figure 1.2. If

we write CRE we are saying that Jim either prefers C to E or he is indifferent between C and E, but

not both. Put differently, this assumption states that Jim (or any other individual) is always able to

order his consumption options from the most preferred one to the least preferred one. Even if this

hypothesis may sound obvious, it is not trivial at all. In some cases, it is very hard to define an order

of preferences between alternatives. Do you prefer Star Wars or Schindler’s List? Roberto Baggio or

Giaunluigi Buffon? The extinction of the Bengal tiger or the fall of the leaning tower of Pisa? In such

cases it is almost impossible to give an answer, as we are comparing two different kinds of movies,

two different kinds of football players and two different kinds of potential catastrophic events.

b. Individual preferences are reflexive, if for all A, ARA (A is at least as good as itself). This assumption

is less questionable compared with the previous one: it simply states that Jim likes 2 slices of pizza

and 6 beers (point C) at least as well as 2 slices of pizza and 6 beers.

c. Individual preferences are transitive if ARB and BRC implies that ARC. Come back to figure 1.2. If

Jim prefers D to E (DRE) and E to C (ERC), he will prefer D to C as well (DRC). Intuitively, this

assumption makes sense. However, problematic issues could arise if we consider a set of

consumption bundles very close to each other. The Sorities paradox constitutes a good example.

Suppose you go to breakfast and you find 1,000 cups of coffee. The first one contains one grain of

sugar, the second one contains two grains and so on, until the last cup which has been sweeten

with 1,000 grains of sugar. Imagine you dislike coffee without sugar. Hence, you prefer the last cup

to the first one. However, you are not able to get the difference between cups 999 and 998, cups

998 and 997, and so on until cups 1 and 0. You are indifferent between all these couples of

alternatives. As a consequence, given the transitivity assumption, you are supposed to be

indifferent between cups 1 and 1000 as well. This paradox provides experimental evidence about

the non-universal validity of this assumption.

d. Individual preferences are strongly monotonic if for any two consumption bundles A=(a1, a2) and

B=(b1, b2) if a1 ≤ b1, a2 ≤ b2, and A ≠ B, then B is preferred to A.Economists usually refer to this

hypothesis as to the non-satiation assumption. Roughly speaking, this condition states that more is

better. Consider again figure 1.2. Compared with C, in point E Jim consumes a larger amount of

both pizza (4 slices against 2) and beer (9 bottles against 6). Then, he must prefer the consumption

bundle represented by E with respect to C. More in general, consider figure 1.3. Compared with the

consumption bundle A, all the bundles in the green area contain more units of at least one good.

Then, Jim will prefer any point in this area to A. The opposite holds when we look at the red

portion, whose allocations are characterized by lower amounts of beer and/or pizza with respect to

A. Hence, Jim will prefer A to any of these bundles. What about A and B? At present we are not

able to say anything about that, but we will answer this question very soon.

Figure 1.3. Non-satiation assumption.

Focusing on the non-satiation assumption, you have to pay attention to one important detail: we

are not talking about prices, but just about preferences! From the previous paragraph we already

know that, given Jim’s income and the current system of prices, the point E is outside his

possibilities, defined by his opportunity set. Here we are simply stating that Jim prefers E to C,

without any consideration about his budget constraint. In principle, you will agree that the non-

satiation assumption sounds reasonable. However, also in this case, counterexamples could be

presented. Try to think about the Sorities paradox discussed a few lines above. If you like sugar-

sweetened coffee, you will prefer cup 1,000 to cup 1. In other words, the larger the amount of

sugar the more you will appreciate your coffee. Since sugar has a positive effect on your utility, this

relation is consistent with the non-satiation assumption. But if you are not able to distinguish

between the intermediate couples of cups, given their small differences in terms of sugar, this rule

will not hold across the other options. According to our hypothesis, you should prefer cup 11 to cup

10, cup 178 to cup 177 and so on. In the Sorities paradox this is not true due to the negligible

increase in the level of sugar between the pairs of cups.

1.2.2. Measurability of utility: cardinal utility.

So far, we introduced some concepts and ideas about Jim’s preferences. Summing up, we said that:

• he buys items because the consumption activity has a positive effect on his utility;

0 10 20 30 40 500

10

20

30

40

50

Slices of pizza

Beers

A

B

• he is always able to order (based on his tastes) alternative consumption bundles from the best one

to the worst one;

• potentially, he would like to consume an infinite amount of goods (non-satiation assumption).

The objective of this section is to translate Jim’s preferences from an abstract and indefinite idea to its

graphical representation. Since Jim’s preferences are described by its utility function, we have to express

the functional form of:

1 2( , )U f x x=

In general, the form of the utility function describes the effect of an increase in consumption on happiness.

As a consequence, its mathematical formulation depends on individual tastes.

Consider for instance the following example. Jim’s uncle, Patrick, is a drunkard, spending his nights and

days in a pub. Suppose that our economy is still made up by two goods, bottles of beer and slices of bread.

Patrick has a strong preference for beer rather than for bread. Hence, we can write his utility function as:

21 2 1 2( , ) ( , ) 5U f beer bread f x x x x= = = +

In this expression x1stays for the amount of beer, whilst x2 represents the slices of bread. This notation is

useful in order to save some space but it changes nothing: we could keep on using “beer” and “bread”

rather than x1and x2. We defined the functional form of Patrick’s utility as

21 25U x x= +

because he definitely prefers beer to bread. Suppose that beer and bread has the same price per unit. For

instance, one bottle of beer costs 1€, the same price of each slice of bread. If Patrick has 5€ in his pocket,

he will have to decide, based on his preferences, how to spend his money.

Figure 1.4. Patrick’s utility function.

If he spends his budget entirely in alcoholic drinks, buying five bottles of beer, his utility will be equal to:

2 21 25 5(5) 0 125U x x= + = + =

00 1Beers

3

Bread

5410

5 2

200

20

15

B

A

120

Utility

80

60

40

100

If he decides to devote the whole amount of money to bread, buying five slices, his utility will be:

2 21 25 5(0) 5 5U x x= + = + =

He could also choose to buy some beers and some slices of bread: in that case his utility level will be

included in the interval between 5 and 125 (as an exercise, you can check it by including in the formulation

some different combinations of beer and bread).

Figure 1.4 represents Patrick’s utility function. This is a tridimensional graph since, compared with the

example in figure 1.1, we are dealing with more than two dimensions: the two commodities and the utility

generated by the consumption activity. Utility (on the vertical axis) is a function of the quantity of beer and

bread consumed. You can verify on the graph that when Patrick gets five units of bread and no beers, his

utility is very low (we said that it is equal to five). In the graph, this combination leads to a point on the light

blue area (A). If Patrick spends his entire income on beer, his utility will be much higher, represented by

point B in the red area, the top right corner of the box represented in figure 1.4. Given his tastes, he prefers

to spend all his money on beer.

Box 1.2. Individuals’ rationality: is it a realistic assumption?

If you really want something in this life, you have to work for it. Now quiet, they're about to announce the

lottery numbers!

Homer Simpson

Talking about consumers’ preferences, we claimed that economic theory generally assumes individuals to

be rational. According to these set of hypotheses about human behavior (founding the so called concept of

homo economicus), individuals are assumed to care exclusively about their own self interest (or better,

their own utility) and to be always able to identify among different alternatives the best one, according to

their preferences.

We explained this concept through a very trivial example: if you dislike Chinese food you will avoid it in

favor of pizza or something else. But what if the choice is more complex than that? Are individuals always

able to choose the best option? In their book Nudge, Richard Thaler and Cass Sunstein show that

people are often irrational (though they may not realize it) in choosing behaviors that may not be in

their own best interest. For instance, people eat, drink alcohol and smoke in excess. They do not save

enough for retirement or invest poorly their savings. Thaler and Sunstein compare this impulsive,

clueless human being with the homo economicus portrait, pointing out some of the major sources of

unrealism in economic theory. The book is fully understandable and enjoyable for non-economists.

Other than this book, a broad literature focused on the same issue. A good reference for better

understanding this topic is the paper by AmartyaK.Sen “Rational fools: a critic of behavioral

foundation of economic theory”.

Even if most of these objections are meaningful and justified, you must remember what we said in the

very first lines of this chapter: an economic model is not intended to be an accurate description of the

real world, but simply a convenient approximation.

Obviously we can consider Patrick’s case as a sort of “extreme” example. We said he is a drunkard. Most of

people would prefer a mix of the two goods. In principle, you can give to the utility function any functional

form. Consider for instance another example. Jim goes to the pub with his close friend Paul. The pub is

selling just two drinks: beer and lemonade. Suppose Paul’s favorite drink is panaché, a mixture of equal

amounts of beer and lemonade. Hence, we could write his utility function as:

1 2 1 2( , ) ( , )U f beer lemonade f x x x x= = =

Assume the price of the two goods to be the same, 1€ per bottle. Paul has 6 € in his pocket. This time, if he

decides to spend his entire budget on beer, his utility will be equal to 0:

1 2 6 0 0U x x= = ⋅ =

The same holds if he chooses to buy six bottles of lemonade:

1 2 0 6 0U x x= = ⋅ =

Given his preferences, the best choice consists in buying a combination of the two goods. In particular, the

best solution is to buy an equal amount of beers and bottles of lemonade:

1 2 3 3 9U x x= = ⋅ =

You can verify that, buying different combinations of the two commodities (such as 4 beers and 2

lemonades) leads to a lower level of utility. This makes sense, as we said that panaché is a 50% mixture of

beer and lemonade and since we assumed that Paul prefers panaché to the other two drinks (dry beer and

dry lemonade). If Paul had loved beer and disliked lemonade we would have chosen another functional

form for his utility, maybe similar to Patrick’s one.

So far we interpreted utility as a measurable concept. That is, for every consumption bundle we expressed

utility as a cardinal number. For instance, when Patrick spends his budget entirely on beer, his utility is

equal to 125 units. But can we really consider utility as a measurable concept? Is it a realistic assumption?

From Jeremy Bentham (1748-1832) to Alfred Marshall (1842-1924), many economists considered utility as

a measurable concept. In particular, Marshall developed a theory of demand (that is a theory explaining

consumers’ behaviour) based on this principle and on the so-called diminishing marginal utility law (Carl

Menger, 1840–1921). In the previous sentence there are two ideas you probably do not fully understand.

The first one concerns the term marginal, whilst the second one refers to the diminishing utility law. What

do they mean? We will start from the concept of marginality.

Marginality is one of the most important concepts in economic theory. The most difficult decisions we

make in our life are not whether to do something or not, but whether to do a little more or a little less of

something. For example, you enrolled the university since you want to graduate. You never ask yourself

about whether you should leave your studies and spend the rest of your life taking the sun on your terrace.

This question has no sense, since you know you must study. But probably you ask yourself whether you

should increase (or decrease) the amount of time devoted to your studies and decrease (or increase) the

time spent with your friends.

Box 1.3. The water-diamond paradox.

Adam Smith (1723-1790), often considered as the founding father or modern economic theory, was the

first to note and state the so called water-diamond paradox. In An Inquiry into the Nature and Causes of

the Wealth of Nations (1776) he pointed out that, even though life cannot exist without water and can

easily exist without diamonds, diamonds are, pound for pound, vastly more valuable than water. Smith

was not able to explain this paradox. The marginal-utility theory of value provided a solution. Water in

total is much more valuable than diamonds in total because the first few units of water are necessary for

life itself. But, because water is plentiful and diamonds are scarce, the marginal value of a pound of

diamonds exceeds the marginal value of a pound of water. Put differently, the value of a cup of water is

lower than the value of a cup filled with diamonds, simply because we have plenty of water whilst

worldwide only a small amount of diamonds is available. A marginal change (in this case, an additional

cup) in your water consumption is less valuable than a marginal change in your diamond endowment.

Obviously, if you had to compare the value of the entire amount of water in the world and the value of the

global resources of diamonds, you would recognize that the former is more valuable.

People, consciously or not, think at the trade-offs at the margin in most of their decisions. Hence, Marginal

Utility (MU) is the change in utility caused by a one-unit increase in the consumption of a certain good.

To better understand this concept, recall our example about Patrick’s utility function. For ease of

representation, focus on the relationship between the consumption of beer and utility, without accounting

for bread.

Figure 1.5. Patrick’s utility as a function of the quantity of beer.

Figure 1.5 reports Patrick’s utility as a function of the bottles of beer consumed. Notice that the connection

between consumption and utility is still the one represented in figure 1.4: when Patrick drinks five beers,

his utility is equal to 125 (point D). Here, we want to focus on the concept of marginality. Consider the

allocation A. In this point Patrick consumes one beer, and his utility is equal to 5. If he moves to B, where

he consumes one more bottle of beer, his utility grows to 20. Since we defined MU as the change in utility

generated by a one-unit increase in consumption, we can easily obtain its numerical value:

20 5 15BMU = − =

0 1 2 3 4 5 60

20

40

60

80

100

120

140

Beers

Utility

D

B

A

C

What if we add another bottle of beer to Patrick’s basket? Well, we will move from B to point C,

characterized by a level of utility equal to 45. Hence, the MU will be:

45 20 25CMU = − =

This example is useful for two reasons. First of all since the concepts of marginality and MU should be more

clear now. Moreover, it allows us to make a consideration about the realism of our framework.

Consider the first bottle of beer. The MU associated to the first bottle is equal to 5 (point A). As we showed,

the MU associated with the second beer is three times larger, equal to 15 (point B). The MU of the third

beer is even higher (25, point C). The same applies for the fourth beer, the fifth one and so on. The utility

associated to the last beer consumed (or, better, to the marginal beer) is always higher than the one

associated to the previous one. Then, MU is increasing. Does such evidence make sense? Think at your own

experience. When you enter a restaurant you are very hungry. You ask for a pizza. Since you are hungry,

you appreciate it a lot. You immediately feel better, having filled your empty stomach. Say that you still

have hunger, and then you decide to order a second pizza. You enjoy it, but probably this time the

experience is less satisfactory compared with the previous one. Following the same reasoning, if you decide

to eat a third pizza you are likely to appreciate it a little bit less than the first two ones. In this example, the

MU is decreasing. This is realistic, as it means that the higher the number of pizza consumed, the lower the

pleasure of eating the last one. As mentioned a few lines above, Alfred Marshall developed a theory which

is based on the diminishing marginal utility principle. Hence, according to Marshall’s idea, we can

reconsider our examples, and translate them in more realistic terms.

Figure 1.6 and table 1.1. Jim’s utility as a function of the quantity of pizza.

For instance, try to find a proper formalization of Jim’s preferences for pizza. We said he likes pizza. Hence,

a possible utility function could be expressed by1:

0.51 1( ) ( )U f pizza f x x= = =

1Rememberthat

0.5x x=

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

3.0

Slices of pizza

Utility

A

B

C

Pizza Utility ∆∆∆∆Utility

(MU)

1 1 1

2 1.4142 0.4142

3 1.732 0.3178

4 2 0.268

5 2.236 0.236

This formulation produces the graph reported in figure 1.6. The first pizza has a positive effect on Jim’s

utility. The MU of the first slice is equal to 1 (point A). What if Jim decides to eat a second pizza? In this case

(point B) his utility will be equal to:

0.5 0.51 2 1.4142U x= = =

As a consequence, the corresponding MU is given by:

1.4142 1 0.4142BMU = − =

The same holds for the third marginal increase in consumption (point C):

0.5 0.51 3 1.732 1.732 1.4142 0.3178CU x MU= = = → = − =

As you can easily note, this time MU is decreasing in consumption, since the utility associated to the first

pizza (1) is larger than the one associated with the second one (0.4142), and so on, as summarized in table

1.1.

It is worth noting one particular thing: if the MU is decreasing, the absolute level of utility is always

increasing in consumption. It means that even after 100 slices of pizza the 101st

will have a positive effect

(obviously very small) on Jim’s utility. Does it sound reasonable? Yes, it does, recall the non-satiation

assumption!

1.2.3. Measurability of utility: ordinal utility.

In the previous paragraph we introduced in our model a little bit of realism through the principle of

decreasing marginal utility. However, our story still sounds somehow unrealistic. We assumed utility to be a

measurable concept. Hence, we attached to every consumption bundle a number, representing a measure

of its effect on individual welfare. From table 1.1 you know that one pizza provides Jim with one unit of

utility, two slices with 1.4142 units and so on. Is this representation reasonable? You are certainly able to

say that, for instance, you prefer ice cream to pizza, but could you really attach a cardinal value to your

preferences?

John Hicks (1904-1989) and Roy Allen (1906-1983) moved the same objection to Marshall’s theoretical

framework. They developed a new demand theory, based on ordinal utility. In their model individuals are

not required to be able to attach a numerical value of utility to any consumption bundle, but they are

simply assumed to be able to rank all the options from the best one to the least preferred one. Note that

Hicks and Allen reached the same conclusions of Marshall’s theory, but dropping out from their model the

unrealistic assumption about cardinal utility.They just substituted the diminishing marginal utility law with

the diminishing marginal rate of substitution principle. In a few lines we will understand what it means.

Before proceeding it is better to recall our objective. We said that Jim’s choice (what we want to analyze) is

the result of a combination between his possibilities and his preferences. What Jim can do is represented in

figure 1.2 by his budget constraint. His preferences, what he wants to do, are reproduced in figure 1.6.

However, the two illustrations are not fully comparable. Both graphs report on the horizontal axis the

number of slices of pizza consumed by Jim. But on the vertical axis we plotted two different items: the

bottles of beer (figure 1.2) and Jim’s utility (figure 1.6). In order to solve the problem about Jim’s choice we

have to represent his possibilities and his preferences in the same form. The theory developed by Hicks and

Allen comes into help. In particular, they introduced a new tool for representing individuals’ utility, the so-

called indifference curve.

1.2.4. Indifference curves: definition and basic properties.

In this paragraph we will provide a definition of indifference curve and summarize its basic properties. In

the next sections we will better understand why indifference curves have their particular shape. For the

time being, focus on the meaning of their graphical representation.

An indifference curve shows the various combinations of goods that make a person equally happy. In other

words, along an indifference curve the utility level is constant. Consider figure 1.7.

As usual, Jim has to choose between two goods, pizza and beer. Obviously, he prefers some consumption

bundles to some others: his preferences depend on his tastes. But in some cases he is indifferent between

alternative options.

Figure 1.7. The indifference curve.

For instance, allocations A and B are the same for him. More formally, for Jim A is as good as B and vice

versa. Hence, A and B lie on the same indifference curve. All the bundles on this curve are characterized by

the same utility level. Note that in B Jim is eating a larger amount of pizza than in point A, but at the same

time he is drinking a smaller amount of beer. This trade-off holds whenever we compare any two points on

the indifference curve. Moreover, it is necessary: if point B had a larger amount of both pizza and beer than

A, Jim would not be indifferent between the two allocations. Why? Because of the non-satiation

assumption (more is better). Given this hypothesis, Jim would definitely prefer B to A.

Notice that the graphical representation of preferences in figure 1.8 is fully comparable with the budget

constraint in figure 1.2. In both graphs the two axes show the quantity of the two commodities.

0 2 4 6 8 10 12 14 16 18 20 220

2

4

6

8

10

Slices of pizza

Beers

A

B

Finally, you have to pay attention to one important feature. Indifference curves have nothing to do with

prices. Different combinations of goods along the same curve have different costs. Here we are just talking

about preferences. For instance, assume the price of pizza to be very low, say 1€ per slice. Beer, on the

contrary, is very expensive, each bottle costing 4€. Then, Jim will have to pay about 20€ for the allocation B

(where he eats 8 slices of pizza and drinks around 3 beers) and approximately 28€ for the allocation A

(where he consumes 4 units of pizza and roughly 6 beers). Obviously, being Jim indifferent between point A

and B, given these prices he will choose the cheapest allocation (B). But we will talk about choices in the

next paragraphs, what you have to understand at this stage of the discussion is that along an indifference

curve utility is constant but prices are not!

Clearly, there is not just one indifference curve, as in figure 1.8. For any individual and any set of

consumption goods we can build a map of indifference curves. What is the difference between each curve

and the others? The difference relies on the peculiar level of utility of each curve. Indifference curves close

to the origin of the axes are characterized by lower utility levels than curves far from point (0,0). This rule is

the first one of a set of properties determining the shape of indifference curves.

The shape of indifference curves: property 1. Consumers prefer bundles on higher indifference curves than

allocations on lower indifference curves.

Figure 1.8. Indifference curves: property 1.

Consider figure 1.8, reporting the map of Jim’s indifference curves for pizza and beer. Along each curve

Jim’s utility is constant. But curves far away from the origin are identified by higher levels of utility. For

instance, take into account point A and point B. They lie on two different curves: which one is associated

with higher utility? Recalling the non-satiation assumption, we know that the right answer is “B”, since this

allocation includes a larger amount of both pizza and beer than A.

The shape of indifference curves: property 2. Indifference curves cannot have a positive slope.

Until now we always considered indifference curves with a negative slope.

0 2 4 6 8 10 12 14 16 18 20 220

2

4

6

8

10

Slices of pizza

Beers

A

B

Figure 1.9. Indifference curves: property 2.

In general, indifference curves cannot have a positive slope. In figure 1.9 we divided the first quadrant into

four portions: areas B, C, D and E. Having a negative slope, the indifference curve passes through E and C.

Could it cross sectors B and D? Suppose the answer to be positive. Then, lying on the same indifference

curve, allocation A and X should be characterized by the same utility level. But this is impossible, as the

consumption bundle X is made up by 12 slices of pizza and 6 beers, whilst point A includes just 10 units of

pizza and 5 bottles of beer. X must be preferred to A. Hence, a positive sloped indifference curve would

violate the non-satiation assumption.

The shape of indifference curves: property 3. Indifference curves cannot cross. Consider figure 1.10. It

represents two indifference curves, I1 and I2. Suppose they cross at point A. Given the definition of

indifference curve, we know that the utility in point B is the same than in point A, since they lie on the same

curve.

Figure 1.10. Indifference curves: property 3.

Hence, Jim is indifferent between allocations A and B. Following the same reasoning, the utility in point C is

equal to the utility in point A, as they still lie on the same curve (I2). Then, Jim is indifferent between A and

C as well. As a consequence, recalling the transitivity assumption (hypothesis c, paragraph 1.2.2), Jim has to

0 2 4 6 8 10 12 14 16 18 20 220

2

4

6

8

10

12

Slices of pizza

Beers

E

D C

B

XZA

0 2 4 6 8 10 12 14 16 18 20 220

2

4

6

8

10

Slices of pizza

Beers

A

I1

I2

B

C

be indifferent between C and B. But this is a clear violation of the non-satiation assumption, because the

consumption bundle C includes a larger amount of both pizza and beer.

1.2.5. From utility functions to indifference curves.

In the previous paragraph we provided the definition of indifference curve and we showed that their shape

is bounded by a set of properties based on our assumptions about preferences. We said that this graphical

representation of preferences has been introduced by Hicks and Allen. However, it may appear not

completely clear the link between the utility function and the indifference curve. How can we shift from

one concept to another? To fully understand this point, it is better to take a step back reconsidering Jim’s

problem. At the beginning of our story we said that every day he enters the cafeteria and has to decide

about the allocation of his monetary budget between pizza and beer. In the last sections we also showed

that, if we want to add some realism to our narration, it makes sense to assume decreasing marginal utility.

Try to update our example according to the former principle.

Assume Jim’s preferences to be described by the following function:

0.5 0.31 2 1 2( , ) ( , ) 2U f pizza beer f x x x x= = = +

As you remember from figure 1.6, this rule leads to a utility function characterized by decreasing MU. But

this time we have two commodities, rather than one. Hence, in order to visualize the relation between the

consumption of beer and pizza and utility we need a tridimensional graph.

Figure 1.11. Jim’s utility as a function of beer and pizza (1).

Figure 1.11 shows Jim’s utility as a function of beer and pizza. The vertical axis represents utility, while on

the horizontal ones the quantities of the two goods are plotted. As you can easily verify (by substituting the

numerical values into the utility function) when Jim eats one slice of pizza and drinks no bier, his utility is

equal to 1. When he decides to drink one beer without buying any pizza, his utility is equal to 2.

0 22 46 810 108 6

Beers Pizza

4

2

0

4

6

Utility

0

Now suppose pizza and beer to have the same price. Say they cost 1€ per unit. If Jim has just 2€ in his

pocket, he will consider three options. He can consume two beers (option 1), two slices of pizza (option 2)

or one beer and one slice of pizza (option 3). What about his utility in each case? This question can be easily

answered:

Option 1 0.5 0.3 0.5 0.3

1 1 22 (0) 2(2)U x x 2.46= + = + =

Option 2 0.5 0.3 0.5 0.3

2 1 22 (2) 2(0)U x x 1.4142= + = + =

Option 3 0.5 0.3 0.5 0.3

3 1 22 (1) 2(1)U x x 3= + = + =

The best option is the third one, where Jim’s utility is equal to 3. In general, given this functional form of

utility, Jim prefers a combination of the two goods, rather than having many units of one single commodity.

You can come to the same conclusion by looking at the surface in figure 1.1. It looks like a circus tent: it is

higher in the middle (the intense red part, where Jim consumes a combination of the two goods) than at

the edges, close to the horizontal axis (where his consumption is unbalanced in favor of one commodity).

We will come back to this point very soon, for now just observe and keep in mind this evidence.

Coming back to our initial goal, we said that utility cannot realistically be expressed in cardinal terms.

Hence, we want to shift from a concept of utility expressed by numbers (as 3 or 1.4142, what we

summarized in the previous lines) to indifference curves. Why? Because indifference curves allow us to

represent ordered preferences without assuming utility’s measurability. As reported in the previous

paragraph, an indifference curve is defined as the locus of points representing all the different

combinations of two goods which yield equal level of utility to the consumer.

Figure 1.12. Jim’s utility as a function of beer and pizza (2).

Reconsider figure 1.11. Utility is represented on the vertical axis. We thought at the surface depicted in the

picture as at a circus tent. You could also think at it as if it was a pie without the filling. Imagine you are

cutting very thin, horizontal slices of this cake. Figure 1.12 may help you in visualizing what we mean.

0 22 46 810 108 6

Beers Pizza

4

2

0

4

6

Utility

0

Since you are cutting horizontal slices, every slice represents a fixed amount of utility. The first slice from

the bottom of the pie stays for a utility of 1, the second one stays for a level of utility of 1.1 and so on.

Figure 1.13. Jim’s utility as a function of beer and pizza (3).

In other words, along each slide the utility is constant, but the quantity of the two commodities (what is

represented on the two other axes, the base of the cake) obviously varies. This definition sounds familiar,

doesn’t it?

The graphical representation should appear familiar as well, but we are observing the slices from the wrong

perspective. Imagine to rotate the box depicted in figure 1.12, and to look at it from the top. In this manner

(figure 1.13) the vertical axis (representing utility) is pointed toward your nose, whilst the other two ones

are placed in the traditional fashion (compare for instance figure 1.13 with figure 1.7). The graph in figure

1.13 sheds the final light on our mystery: the slices of cake are nothing but indifference curves. Put more

formally, you should understand how we passed from cardinal utility to ordinal utility. Indifference curves

close to the origin of the axes are worse than those far away from point (0,0), as they are characterized by

small amounts of the two commodities (the light blue area in figure 1.11).

1.2.6. The slope of indifference curves.

Based on what we reviewed in the previous paragraph, you should recognize that indifference curves

represent an acceptable tool for measuring individual preferences. If you are convinced about that, we can

go beyond the pure definition of indifference curve, trying to understand its use in explaining consumers’

choices. In order to do that, we have to deal with a new concept: the Marginal Rate of Substitution (MRS).

We already cited this idea, claiming that its introduction constitutes one of the most important innovations

in Hicks and Allen’s framework compared with Marshall’s theory. We also said that, in the new demand

theory, the principle of diminishing MRS substituted the law of diminishing MU. To better understand the

meaning of this sentence, come back to our story. Jim is dealing with two goods, beer and pizza, and his

0 2 4 6 8 10

10

8

6

Beers4

Pizza0

7

Utility

2

0

utility directly depends on the consumption of both of them. Hence, we have two MU as well, the MU of

beer and the MU of pizza. They represent the utility Jim gets from having one-extra slice of pizza and the

utility he gets from consuming one extra bottle of beer. The two MU were represented in figure 1.11, as the

lines forming the edge of the surface, where Jim was consuming just one kind of good. In other words, they

reproduce the effect of the consumption of a certain commodity (pizza or beer) on utility. We plotted them

separately in a couple of two-dimensional graphs (figure 1.14).

Since along the same indifference curve the level of utility does not change, an important consideration

follows. If Jim wants to add one slice of pizza to his consumption basket, keeping constant his utility (that is,

lying on the same indifference curve) he must give up a certain amount of beer. This situation is

represented in figure 1.15. In point A Jim has 8 slices of pizza and 6.25 bottles of beer. Say that he wants

one more pizza, keeping constant his utility. This means that he desires to stay on the same indifference

curve, but with one extra pizza. Graphically, he wants to move from point A to B. In point B he obtains the

extra pizza (he has 9 slices instead of 8) but he must give up some beer. How much? An amount equal to

the red segment in figure 1.15.

Figure 1.14.Marginal utility of pizza and beer.

The same reasoning can be expressed in more formal terms. Since along an indifference curve the utility

level is constant, the following identity holds:

( ) ( )beer pizzaMU beer MU pizza−∆ = +∆

This identity means that if you want to keep your utility constant (that is, if you want to lie on the same

indifference curve) the increase of utility generated by an increase in the consumption of pizza (the right

part of the expression: MUpizza(+∆pizza)) must be compensated by a decrease of utility generated by a

reduction in the consumption of beer (the left part of the identity: MUbeer(-∆beer)). This is exactly what we

represented in figure 1.15.

The identity above can be rewritten as:

pizza beer

beer pizza

MU

MU

∆= −∆

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

3.0

Slices of pizza

Utility

A

B

C

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

3.0

Beers

Utility

A

B

C

Figure 1.15.The slope of an indifference curve.

This ratio measures the marginal rate of substitution (MRS). Hence, the MRS between pizza and beer

measures how much beer Jim is willing to give up in return of one more pizza, keeping constant his overall

utility.

Once we know the meaning of MRS, we have to understand why Hicks and Allen formulated the principle

of diminishing MRS. What does it mean?

Figure 1.16.The principle of diminishing MRS.

Consider figure 1.16. Jim finds himself in point A, where he consumes 2 slices of pizza and 12.5 bottles of

beer. Suppose he wants to obtain one more pizza without changing his overall utility. How many bottles of

beer should he give up? The answer is very easy: he should move from A to B, where he consumes 3 slices

of pizza (one more than before) and 8.3 bottles of beer. Hence he should give up:

12.5 – 8.3 = 4.2 bottles of beer

This means that, in point A, Jim is prepared to renounce to more than 4 beers in order to have one extra

pizza.

Now suppose Jim to be in point B. This consumption bundle is made up by 8 units of pizza and 3.1 bottles of

beer. Again, what if Jim decides to add one pizza to his consumption basket? We already know that he will

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

2

4

6

8

10

Slices of pizza

Beers

A

B

0 1 2 3 4 5 6 7 8 9 10 110

2

4

6

8

10

12

14

Slices of pizza

Beers

A

B

C D

have to give up some beer. He will move from C to D, where he consumes 9 slices of pizza and 2.8 beers.

Hence, for the additional pizza Jim will have to give up:

3.1 – 2.8 = 0.3 bottles of beer

Try to compute the MRS for the two situations. In the first case (from A to B) we will have:

4.24.2

1pizza

beer

MU

MU

−= − = −

In the second one (from C to D) we will write:

0.30.3

1pizza

beer

MU

MU

−= − = −

As you can see, the absolute value of the MRS in point D is much lower compared with the MRS in point B.

This is reasonable, since in point A Jim has a lot of beers, whilst he has just a few slices of pizza. Then, he

will be willing to sacrifice a significant amount of beer for one extra pizza. In point C the picture is quite

different: Jim has just a few beers and a relevant quantity of pizza (8 slices). As a consequence, he will not

accept to give up the same amount of beers in order to get one more slice of pizza. He will accept to give up

just a smaller number of bottles (0.3).

Figure 1.17.The assumption about convex preferences.

This example explains the rationale behind the law of diminishing MRS: as the consumer has more and

more pizza, he is prepared to forego less and less beer (and vice versa). This justification seems realistic and

it should remind you something we exposed in the previous sections. In our comments to figure 1.11 we

claimed that Jim prefers a combination of pizza and beer rather than a bundle exclusively filled with one

kind of good. For this reason, compared with point B, in point D he is less willing to give up his endowment

of beer. Notice that this behavior depends on a particular assumption about preferences we did not

mention yet. The hypothesis about convex preferences concerns the benefits from the diversification of a

consumption bundle, in order to avoid the preponderance of one good over another. More formally, this

assumption states that, given two bundles A and B making a consumer equally happy, a combination of

these two allocations will be judged at least as good as the original ones. In other words, if a consumer is

Slices of pizza

Beers

A

B

C

I2

I1

1.17.1

Slices of pizza

Beers

A

B

C

I2

I1

1.17.2

indifferent between A(0,2) and B(2,0), for a combination of these two bundles (for instance ½ of A and ½ of

B = C(1,1)) the following relations hold: CRA (C is at least as good as A) and CRB (C is at least as good as B).

It is due to this assumption that the indifference curves have their particular “banana” shape. Consider for

example figure 1.17. In figure 1.17.1 (on the left) the consumer is indifferent between allocations A and B,

as they lie on the same indifference curve, I1. The consumption bundle C, a linear combination of A and B,

lies on a higher indifference curve (I2). Hence, we can write that CRA and CRB (C is at least as good as both

A and B). Now move to the graph on the right. In this case the consumer is still indifferent between points A

and B, lying on the same indifference curve I1. What about a linear combination of these two allocations?

This time the convexity assumption does not hold anymore. We cannot say that C is at least as good as A

and B. Actually, C is strictly worse than A and B, lying on an indifference curve (I2) closer to the origin.

Therefore, if we assume convex preferences indifference curves must have their “banana” shape. Keep in

mind that this rule holds only if we assume convexity of preferences. As we said, this is an acceptable

assumption, due to its realism. If you open an economic textbook 99 times out of 100 you will find

indifference curves shaped in that way. However, it is not a universal rule: in some particular cases you may

find exceptions.

Box 1.4. “Shaken, not stirred!”: James Bond preferences and the shape of indifference curves

In the last paragraph we said that Jim (fig. 1.16) is always ready to trade beer for pizza and, keeping

constant his utility, he will be indifferent between points A, B, C, D and all the other allocations lying on the

same indifference curve. Many times this is a convenient and reasonable assumption, but in some cases

individuals could not be willing to accept any combination of two goods. As a consequence, indifference

curves will not have their traditional “banana” shape. Consider a couple of examples.

As you probably know, agent 007 never drinks dry gin or dry vodka: he always gets a cocktail made of three

parts of gin and one part of vodka. Hence, if you replace vodka for beer and gin for pizza in figure 1.16, you

can easily understand that James Bond will be interested only in those allocations including the two drinks

in the same proportion, 3:1. This situation is represented in figure 1.18 (graph on the left). 007 will be

indifferent between allocation A (1 bottle of vodka and 3 bottles of gin) and allocation A’ (1 bottle of vodka

and 4 bottles of gin), because the extra bottle of gin in bundle A’ cannot be used for preparing cocktails.

What about the shape of indifference curves in this particular case? Figure 18 shows indifference curves

shaped like 90 degrees angles. Why? Consider the first indifference curve (I1) and imagine Bond to be in

point A, where he has 1 bottle of vodka and 3 bottles of gin. According to the non-satiation assumption, an

increase in consumption should generate an increase in utility. But if 007 moves from the consumption

bundle A to the allocation A’ (including 1 bottle of vodka and 4 bottles of gin) his utility does not change. A’

still lies on the indifference curve I1. The same reasoning applies if he moves from A to A’’. The extra bottle

of gin (or vodka) is useless without a corresponding increase in the consumption of vodka (or gin). In order

to increase his utility, Bond needs a proportional increase of both drinks. With this additional quantity of

alcohol he would shift from point A to the consumption bundle B, located on a higher indifference curve.

Figure 1.18.The assumption about convex preferences: some particular cases.

The second case presented in figure 1.18 (the graph on the right) involves another particular situation

characterized by linear indifference curves. Be careful, the lines drawn in the picture are not budget

constraints, even if they have the same shape! In this case we are considering a consumer who has to

choose between two very similar goods: Coca-Cola and Pepsi Cola. The horizontal axis reports the cans of

Coca-Cola consumed by the individual, whilst the vertical axis represents his consumption of Pepsi. Here we

are assuming that the consumer is indifferent between the two goods: he does not care if he drinks Coke or

Pepsi. Consider for instance the first indifference curve, I1. When the individual has 4 Pepsi and no Coke

(point A) he would be willing to give up 1 can of Pepsi in order to obtain one more (actually, his first one)

can of Coke. In this situation his MRS is equal to 1. Consider now the allocation B, where the consumer has

3 cans of Coke and 1 Pepsi. He will be willing to renounce to his last Pepsi in exchange of one more unit of

Coke. Then, his MRS is still equal to 1. As you can see, this example constitutes a particular case in which

preferences are not convex and, as a consequence, the principle of MRS does not hold. Indifference curves

have this particular shape since we are assuming that the individual is completely indifferent between the

two goods. If he had preferred one drink to the other one, we would have drawn indifference curves in the

common format (figure 1.17.1). This example constitutes another violation of the hypothesis of convex

preferences and of the principle of diminishing MRS.

1.3. The choice.

In the introductory part of this chapter we claimed that the choice (what you will do) is a combination of

what you can do and what you want to do. The first two paragraphs have been devoted to the issue about

scarcity and to the one concerning the representability of consumers’ preferences. Now that we solved

both problems, and we are able to reproduce both individuals’ constraints and preferences in a graph, the

final step simply consists in putting them together.

0 1 2 3 4 50

2

4

6

8

10

Vodka

Gin

A

B

C

I1 I2 I3

A'

A''

0 1 2 3 4 5 6 7 8 9 10 110

2

4

6

8

10

Coca Cola

Pepsi

I1 I2 I3

A (0,4)

B (3,1)

Recall figure 1.2, presenting Jim’s budget line and opportunity set. We said that, given his income and the

current system of prices, Jim is able to buy any consumption bundle in the shaded area. However, we also

claimed that the assumption about rational choice means consuming the highest possible combination of

the two goods we can get given our constrained resources (in this case, given our scarce amount of

money). As a consequence, we will ignore the shaded area in figure 1.19 (graph on the left) and we will

focus just on the budget line, since we know that Jim strictly prefers the consumption bundles located at

the limit of his financial possibilities.

Figure 1.19. Opportunity set and budget line.

In the first paragraph, we learnt that the slope of the budget line answers the following question: given the

current prices and my income, how many bottles of beer must I give up in order to obtain one more slice of

pizza? In other words, the slope of the budget constraint is equal to the relative price of one good (in this

case, beer) in terms of the other one (here, pizza). The answer to the previous question, when considering

figure 1.19, is that Jim must sacrifice two bottles of beer in order to get one additional pizza (the slope of

the budget line is -2). Remember, since the budget constraint is a linear function, its slope is constant.

Compared with the budget line, the message conveyed by the indifference curve could sound somehow

similar. Be careful: the formulation is similar but its meaning is completely different! In particular, the slope

of the indifference curve answers the following question: how many bottles of beer am I willing to give up

in order to get one more slice of pizza? Since we assumed convex preferences, the slope of the indifference

curve (that is, the MRS) is not constant.

Now it should be completely clear that the budget constraint returns you some information about

consumers’ possibilities (what they can do given their scarce resources), whilst the indifference curve

describes their preferences (what they would like to do if they were not subject to any monetary

constraint). If this idea is clear in your mind the next step will appear as the natural extension of our

reasoning. At this point we have to put together what we can do and what we want to do, keeping in mind

that our goal is to get the highest level of utility we can reach given our budget constraint.

Consider figure 1.20. Jim has to decide how many slices of pizza and bottles of beer he will buy for lunch.

He is very hungry and thirsty, and he would like to eat 3 slices of pizza and 7 beers (point D). Unfortunately,

0 1 2 3 4 5 60

2

4

6

8

10

12

Slices of pizza

Beers

0 1 2 3 4 5 60

2

4

6

8

10

12

Slices of pizza

Beers

this allocation is outside his opportunity set: he simply does not have enough money for it. He notices that

he could buy 1.5 units of pizza and about 3.3 bottles of beer (point A).

Figure 1.20. Consumer’s choice.

This consumption bundle is affordable, as it is included in the opportunity set. However, Jim easily

understands that the allocation B (made up by almost 2 slices of pizza and 4 beers) would be even better

than A, because it lies on a higher indifference curve. Following the same reasoning, he realizes that C is

better than B. Jim reiterates this reasoning until he reaches the threshold defined by his monetary budget.

This occurs in point E. This allocation lies on the budget constraint: Jim would like to go further, toward

point D, but he cannot due to his scarce amount of money. Then, he will decide to buy 2.5 slices of pizza

and 5 bottles of beer.

The consumption bundle E has an important characteristic. In E the budget constraint is tangent to the

indifference curve. In other words, in point E the budget line and the indifference curve have the same

slope. Hence, in that point Jim’s preferences (described by the slope of the indifference curve) are

compatible with his monetary possibilities (defined by the budget constraint).

Try to express this concept more formally. We know that the slope of the budget line is given by:

The slope of the indifference curve can be written as: pizza

beer

MU

MU−

Then, the condition characterizing point E is: pizza pizza

beer beer

p MU

p MU=

which leads to: pizza beer

pizza beer

MU MU

p p= (1)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.00

2

4

6

8

10

12

Slices of pizza

Beers

AB

C

DE

pizza

beer

p

p−

The last identity means that the consumer will adjust his consumption until he gets to the point where the

marginal utilities of the two goods, per euro spent, are equal. In choosing between two goods, a consumer

will adjust his choices to the point where the marginal utilities are proportional to the prices. When this

condition is met, the consumer’s problem is solved.

This graphical explanation probably convinced you. Nevertheless, a simple numerical example will remove

any doubt about its truth.

Table 1.2.

Slices of pizza Total utility

(pizza)

MUpizza Bottles of beer Total utility

(beer)

MUbeer

1 48 48 1 34 34

2 94 46 2 66 32

3 138 44 3 96 30

4 180 42 4 124 28

5 220 40 5 150 26

6 258 38 6 174 24

7 294 36 7 196 22

8 326 32 8 216 20

9 356 30 9 232 16

10 384 28 10 244 12

Suppose pizza and beer to have the same price: every slice and every bottle cost 2€. Table 1.2 summarizes

Jim’s options. The first column indicates the number of slices of pizza consumed by Jim. The second column

list the total utility provided by the consumption of pizza. As you can see, utility increases with quantity,

consistently with the non-satiation assumption. The third column reports the marginal utility corresponding

to the consumption of each slice of pizza. For instance, the MU of the second slice is represented by the

increase in utility due to the consumption of this additional unit (94 – 48 = 46). The rest of the table

provides the same information, but concerning the consumption of beer. Suppose Jim’s income amounts to

20€. He could decide to devote his entire wealth to the consumption of pizza. He can buy a maximum

amount of ten slices. Is this allocation the best choice? Jim reflects a little bit and he realizes that the

consumption of the last unit of pizza (the tenth slice) produces an increase in his utility equal to 28. If he

does not buy this last slice, he could devote the money saved (2€) to the purchase of the first bottle of

beer. This would be a good deal indeed: the consumption of the first beer produces an increase in utility

equal to 34 (which is higher than 28). Hence, Jim decides to consider the allocation including 9 slices of

pizza and 1 bottle of beer. Is this consumption bundle satisfactory? Again, Jim considers the increase in

utility generated by the last slice of pizza (the ninth one). It is equal to 30. If he gives up this additional slice

for the second bottle of beer he would feel better, as the second beer is associated to a MU of 32. Then,

Jim will decide to shift to the new consumption bundle, where he eats 8 units of pizza and 2 beers. Does he

have any incentive to move from this allocation? The answer is negative, because the eighth slice of pizza

has a MU of 32, while its alternative (the third bottle of beer) is characterized by a lower marginal utility

(30). Jim has no incentive to move from this allocation. Then, he will consume 8 slices of pizza and 2 beers.

We learnt that the peculiarity of such a situation is that the slope of the indifference curve and the slope of

the budget line must assume the same value, leading to the following equilibrium condition:

pizza beer

pizza beer

MU MU

p p=

What if we substitute in this identity the numerical values reported in table 1.2? In the selected allocation

(8 slices of pizza and 2 beers) we have:

32 32

2 2= → the identity (1) is verified.

Table 1.3.

Slices of pizza Total utility

(pizza)

MUpizza Bottles of beer Total utility

(beer)

MUbeer

1 48 48 1 22 22

2 94 46 2 42 20

3 138 44 3 60 18

4 180 42 4 76 16

5 220 40 5 90 14

6 258 38 6 102 12

7 294 36 7 112 10

8 326 32 8 120 8

9 356 30 9 126 6

10 384 28 10 130 4

The same result holds if the prices of the goods differ. For instance, assume the price of each slice of pizza

to be twice the price of each bottle of beer. Every unit of pizza costs 4€, whilst every beer costs 2€. We

already know (figure 1.2) that, given this system of prices, the slope of the budget constraint is equal to -2.

With his income of 20€, Jim can buy a maximum amount of 10 bottles or 5 units of pizza. He considers the

first option, where he allocates all his resources to the consumption of beer. He evaluates the possibility to

renounce to the tenth bottle in order to receive the first two slices of pizza (remember that now with the

money saved he is able to buy two units of pizza). He understands that this trade is profitable: the MU

associated to the last slice (28) is lower than the marginal utility associated to the first two bottles (22 +

20=42). Then, Jim decides that buying 10 units of pizza is not his best choices, and he will prefer to spend

his money in 9 slices and 2 beers. What if he gives up another unit of pizza? He notices that in this case he

would have enough money to buy the third and the fourth beer. Since the MU of the ninth pizza is 30, and

the MU of the alternative option is 34 (18 + 16), Jim decides to change another time his consumption

bundle. At this point he wonders whether he should keep on substituting pizza with beer or not. If he gives

up the eighth slice he will receive the fifth and the sixth bottles of beer. But this time the deal is not

profitable anymore: the utility associated with the last unit of pizza (32) is higher than the utility of the two

supplementary beers (14 + 12 =26). Hence, he will decide to consume a bundle made up by 8 slices of pizza

and 4 bottles of beers. As this option represents the solution of the consumer’s problem, we must verify

the identity:

pizza beer

pizza beer

MU MU

p p=

If we substitute the values of our example we have:

32 16

4 2= →8 8= → the identity (1) is verified.

These simple examples showed that when the marginal utilities of the two commodities are proportional to

the prices, the consumer does not have any incentive to look for another consumption bundle. Are you

convinced about that? Concerning table 1.2 and 1.3 you may move one objection: the examples are built in

order to get the desired results. For example, what if the MU of the fourth beer in table 1.3 was equal to

15? Jim would still decide to consume 8 slices of pizza and 4 beers, but this time the identity would not be

verified anymore:

32 15

4 2≠

This objection makes sense: the example is designed to reach the desired conclusion. However, in the aim

of simplicity, we considered only integer quantities of the two goods. As mentioned in the paragraph 1.1,

Jim is able to consume any amount of pizza and beer. Then, between the options “1 bottle” and “2 bottles”

he can choose among an infinite number of intermediate solutions: “1.01 bottles”, “1.34 bottles”, “1.56

bottles” and so on. The MU associated to each of these intermediate options is always a little bit smaller

than the MU of the previous one. Table 1.4 presents an example. Obviously the same applies to the

consumption of pizza. Hence, by slightly decreasing the quantity of pizza and, at the same time, by

imperceptibly increasing his consumption of beer, Jim will be always able to find an allocation in which the

identity (1) is verified.

Table 1.4.

Bottles of beer Total utility (beer) MUbeer

0 0

0.05 1.2 1.5

0.1 2.95 1.45

0.15 4.38 1.43

… … …

1 22

The absence of incentives to leave the selected consumption bundle (8 slices of pizza and 4 beers) is

graphically represented in figure 1.21.

Figure 1.21.

This picture recalls the example presented in figure 1.20. We said that, in this case, Jim is choosing an

allocation made up by 2.5 slices of pizza and 5 bottles of beer. If he wants one more pizza he could move

from E to E’ (he gives up 2 beers), still lying on the boundary limits determined by his budget constrained.

However, Jim will not accept this deal, as E’ lies on a lower indifference curve compared with E. Jim would

be willing to renounce to the 2 bottles of beer only for a larger amount of pizza, such that he could reach

point E’’. But he cannot reach the allocation E’’, since it is outside his opportunity set.

At this point the solution of the consumer’s problem should be clear. Maybe one last question involves the

location of the selected bundle. Must it always lie more or less in the middle of the budget line? In other

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.00

2

4

6

8

10

12

Slices of pizza

Beers

E'E''

E

words, am I always forced to choose a consumption bundle composed by a mixture of the two

commodities? The answer is negative. As you should remember, by assuming convex preferences we said

that, generally speaking, people prefer a combination of different commodities rather than an allocation

entirely filled with one kind of good. However, we also showed that some exceptions to this rule do exist

(figure 1.18): the shape of the indifference curve depends on consumers’ tastes. For instance, consider an

example involving Jim’s uncle, Patrick the drunkard. He decides to visit his nephew and to have lunch

together. In the cafeteria, he will have to choose between beer and pizza. But we know that, between

these two goods, he has a strong preference for beer.

Figure 1.22.

Hence, we could imagine a set of indifference curves shaped as in figure 1.22. As you can see, assuming a

budget of 20€ and a set of prices equal to the one described in paragraph 1.1, Patrick will choose to devote

his entire income to the consumption of beer, without eating any pizza. This case, represented by point E, is

usually referred as “corner solution”, since it is located at the extreme of the budget constraint.

Box 1.5. The mathematical formalization of the problem.

In the previous paragraphs we analyzed the consumer problem essentially through the aid of graphs and

figures. Surely you realized that every line or curve is nothing but the graphical representation of a

mathematical function. We spent just a few lines talking about functions, when we introduced the concept

of utility function. In the rest of the presentation, we avoided a formal mathematical treatment of the

subject in favor of a more intuitive geometrical approach. However, the most curious among you could

doubt about the reliability of our final results. How did we draw those curves? How can we derive the

coordinates of the selected allocation?

Actually the geometrical problem presented in this firs chapter is nothing but a graphical representation of

the mathematical formalization of the consumer problem.

You already know that, as a first step, we have to formalize the consumer’s preferences:

( )1 2U f x ,x=

0 1 2 3 4 5 60

2

4

6

8

10

Slices of pizza

BeersE

I1

Then, given the prices and his income, we can write the budget constraint: 1 1 2 2y p x p x= +

The quantity demanded by the consumer (that is, the selected consumption bundle) is represented by

the solution of the following maximization problem:

Max1 1 2 2y p x p x= +

Subject to (s.t.) 1 1 2 2y p x p x= +

How can we solve such a problem? I remind you (this is not a compulsory topic) to chapter 12 in

“Fundamental Methods of Mathematical Economics – 4th

ed.”, by A.C. Chiang and K. Wainwright.

Here we will review the mathematical representation of the problem described in figure 1.22. In that

particular case, we decided to represent Patrick’s utility as:

( )1 2 1 2 25U f x ,x x x x= = +

Where x1is the quantity of pizza and x2 is the quantity of beer. His consumption is subject to the following

budget constraint:

1 210 4 2x x= +

assuming that the system of prices and Patrick’s income are the same as in the case described in

paragraph 1.1.

The map of indifference curve has the following equation:

1 2 25U x x x= +

→ 2 1 5U x (x )= +

→ 2 1 5x U / ( x )= +

In the indifference curve plotted in figure 1.22 U is equal to 50. If the level of utility decreases, the

indifference curve moves downwards (in figure B.4.1 utility decreases to 40 and the indifference curve

moves from I1 to I3). The opposite holds if the level of utility increases (in figure B.4.1 utility raises to 60

and the indifference curve moves from I1 to I2).

In order to derive the coordinates of the selected allocation we have to solve an optimization problem in

which our objective function (represented by the utility function) is constrained but the budget line.

As you maybe already know (if it is not the case and you are interested in the topic have a look at the

Chiang and Wainwright book for a detailed explanation), such a problem can be solved by the so called

Lagrange-multiplier method. The essence of this method is to convert a constrained problem into another

form, by incorporating the constraint into a new function (the Lagrangian function). The latter can be

written as:

1 2 2 1 25 10 4 2x x x ( x x )α λ= + + − −

As you can see, this function is nothing but the

objective function plus the constraint multiplied by

an undetermined number λ.

We consider λ as an additional choice variable in α.

The first order conditions (FOC) of the

(unconstrained) maximization problem will consist

of a set of simultaneous equation:

2 2

1

1 12

1 2

0 4 0 4

0 5 2 0 2 5

0 20 4 2 0

x xx

x xx

x x

α λ λ

α λ λ

αλ

∂ = → − = → =∂

∂ = → + − = → = −∂∂ = → − − =

Figure B.4.1.

Verify that if you substitute the expression for x2 and

x1,obtained from the two first equations, into the

last one, you get:

20 4 2 5 2 4 0 5 2( ) ( ) /λ λ λ− − − = → =

If you substitute this value into the expressions for

x2and x1you obtain:

2

1

54 4 10

25

2 5 2 5 02

x

x

λ

λ

= = ⋅ =

= − = ⋅ − =

→ verify that the allocation (10,0) is the solution of the problem, represented by point E in figure 1.22.

Try to change the utility’s functional form and verify how Patrick’s behavior is going to change!

1.4. Different scenarios.

The previous part of the chapter has been devoted to the analysis of the consumer problem. The

perspective was static: we investigated Jim’s choices when he enters the cafeteria with a given amount of

money. At this stage, once that we are able to predict his behavior, we can look at the dynamic aspects of

the problem: what does it happen if some of the conditions change? We will consider two cases: an

increase (or decrease) in Jim’s income and a rise (or a fall) of prices. Through these examples we will learn

how to draw Jim’s demand curve for pizza, beer or any other commodity.

1.4.1. Different scenarios: a change in income.

Jim’s employer is very satisfied with him. Jim works as a clerk in a bookshop and in the last month the

volume of retails significantly increased. For this reason, the boss decides to raise the basic wage of his

employee from 20€ to 40€. The first effect of this change operates on Jim’s budget constraint: now, when

he enters the cafeteria ha has 20 extra Euros to spend.

Given his new budget, and keeping constant the prices of the two commodities (ppizza= 4€, pbeer= 2€), Jim

can buy a maximum of 20 bottles of beer or 10 slices of pizza. The new budget line is represented by the

green line in figure 1.23. Notice that the slope of the curve does not change (it is still equal to -2), as it

depends on the price of one good relative to the other: in this example the prices do not change, then their

ratio is always the same. What if Jim’s income decreases, for instance to 10€? As you probably have already

0 1 2 3 4 5 60

5

10

15

Slices of pizza

Beers

E

I1I2

I3

realized in this case the budget line moves leftwards, since Jim can buy only 2.5 slices of pizza and 5 bottles

of beer (the red line in figure 1.23).

Intuitively, when the income increases the opportunity set widens, whilst the opposite happens as the

wealth decreases. What about Jim’s choices? As usual, what he will do directly depends (alongside with his

possibilities) on his preferences. In general, we can guess that Jim will consume a larger quantity of both

pizza and beer. Figure 1.23 represents this eventuality, where the selected bundle moves from E to E’ to E’’

as Jim’s income increases.

Figure 1.23. The budget line when income changes.

The income expansion path is the locus of the optimal consumption bundles as income varies. In a case like

the one represented in figure 1.23 economists talk about normal goods. A normal good is a good for which

demand increases as the income of the consumer increases and the relative prices remain constant.

Moreover, the share of budget devoted to the consumption of a normal good is always the same. For

instance, assume that Jim’s income is equal to 20 €, and that he spends half of his budget in beer. If his boss

decides to raise his income to 40 € (and if beer is a normal good for Jim), he will devote to the purchase of

beer a sum still equal to one half of his budget (20 €).

Figure 1.23.Jim’s choices as his budget increases.

0 1 2 3 4 5 6 7 8 9 10 110

2.5

5

7.5

10

12.5

15

17.5

20

Slices of pizza

Beers

0 1 2 3 4 5 6 7 8 9 10 110

5

10

15

20

Slices of pizza

Beers

Income expansion path

E'

E'

E''

If the proportional change in the consumption is larger than the change in income, we talk about superior

goods. Finally, if the demand of a certain good decreases as income increases (still keeping constant the

relative prices) we are dealing with an inferior good. Figure 1.24 shows an example of inferior and superior

goods. Potatoes are inferior goods: Jim is consuming less and less of them as income raises. The opposite

holds for the consumption of meat.

Figure 1.24.Inferior and superior goods.

The rationale behind this example is that poor people can consume only cheep food (potatoes, rice, bread).

When they get richer probably they will decide to stop eating their usual food items in favor of something

more tasty and appealing as, for instance, meat. In general, remember that if goods are superior, normal or

inferior depends on individual preferences. Beer is probably a normal good for Jim, but it is a superior good

for his uncle Patrick.

1.4.2. Different scenarios: a change in relative prices.

Another group of dynamic scenarios is represented by those cases in which the relative price of the two

goods changes.

For example, suppose that the price of pizza decreases from 4€ to 2€. Keeping constant Jim’s income (20€)

and the price of beer (2€ per bottle) the budget line would rotate rightwards, as shown in figure 1.25, from

A to B. Now Jim can buy a maximum of 10 bottles of beer or 10 slices of pizza. Notice that the slope of the

budget constraint is not equal to -2 anymore. It is equal to -1, as the ratio between the two prices changed.

Obviously, if the price of pizza increases to 5€ per unit, the budget line would rotate leftwards (from A to C)

and Jim would be able to consume just 4 slices when he decides to devote his entire income to the

consumption of pizza.

As in the previous case, we could wonder what is the effect of such a change on the consumer’s choices.

Figure 1.26 sheds some light on this issue. As you can see, when every slice of pizza costs 4 € (the budget

line in the middle) Jim consumes 2.25 units (point B). If the price of pizza reduces, the budget line rotates to

the right. Here, Jim will buy a larger amount of pizza: 3.2 slices (point C). Finally, if the price of pizza

0 2 4 6 8 10 12 14 16 18 20 220

2

4

6

8

10

Potatoes

Meat

increases the budget constraint will rotate to the left and Jim will purchase a smaller amount of the good:

just 2 slices (allocation A). Intuitively, Jim’s behaviour makes sense: the higher the price of a certain

commodity (keeping constant the income and all the other prices), the lower the quantity purchased.

Starting from figure 1.26 we are finally able to derive Jim’s demand curve for pizza. In correspondence to

each price (in the previous example, 2 €, 4 € and 5 € per slice) we know the amount of pizza demanded by

Jim (respectively 3.2, 2.25 and 2 slices).

Figure 1.25. The effect of an increase in the price of pizza (1).

Figure 1.27 provides a graphical illustration of these alternative scenarios. The horizontal axis, as usual,

describes the quantity of pizza consumed by the individual. Rather than the quantity of the other

consumption good, this time the vertical axis shows the price of pizza.

Figure 1.26. The effect of an increase in the price of pizza (2).

From figure 1.26 we are able to represent the relationship between the quantity of pizza purchased by Jim

and its price. The allocations A, B and C (figure 1.27) correspond to the consumption bundles described in

figure 1.26. The only difference between the two graphs is that in figure 1.27 we completely ignore the

consumption of beer, focusing on the demand of pizza in correspondence of different price levels. The

demand curve can be drawn simply by connecting the allocations identified in figure 1.26. It is worth noting

0 1 2 3 4 5 6 7 8 9 10 110

2

4

6

8

10

12

Slices of pizza

Beers

A BC

0 1 2 3 4 5 6 7 8 9 10 110

2

4

6

8

10

Slices of pizza

Beers

A

B

C

one particular feature of the demand curve: since the higher the price is the lower the quantity purchased,

the demand curve always has a negative slope. The red line in figure 1.27 represents Jim’s demand of pizza.

How can we shift from the individual to the market demand? Since the market is made up by all the

consumers in the economy, the market demand will simply be equal to the sum of all individual demand

curves. This procedure is represented in figure 1.28. Suppose the economy to be made up just by two

consumers: Jim and his friend Sean.

Figure 1.27. The effect of an increase in the price of pizza (2).

In this case, the market demand curve will be equal to the horizontal sum of the two individual curves.

When the price of pizza is equal to 5 € per unit, Jim and Sean purchase respectively 2 and 4 slices. Then, in

correspondence to this price the market demand will be equal to 6 units (2 + 4). Applying the same

reasoning for all the other price levels leads to the derivation of the market demand curve.

Figure 1.28. From individual to market demand.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.00

1

2

3

4

5

6

7

8

9

Slices of pizza

Price of pizza

A

B

C

2

5

4

6=2+4

Market demand

5

Pizza

PPPP

PP

Jim Sean

Pizza

Pizza

1.4.3. Demand and consumer surplus.

The last concept presented in this chapter is particularly important, as it will come back very often in

thorough the next sections. It concerns the demand curve and it can be explained by a simple example.

Suppose Jim to be very hungry. He is desperately looking for a restaurant and, after a long walk, he finds a

cafeteria. He is so hungry that he would be ready to pay 9€ for a slice of pizza. He reads the menu and he

finds out that the price of pizza is 4€ per slice. He orders the first slice and enjoys his meal. Once he has

polished off his portions he feels better: his appetite is quite satisfied. However he would enjoy another

slice of pizza. Of course he is less hungry than before and he would not accept to pay 9€ for it. Suppose he

is willing to pay 5€ for the second pizza. Since the price is lower (4€) he will ask the waiter to bring him

another portion. What can we learn from this story? The main message is that Jim derives a sort of “extra

benefit” from the consumption of the first pizza. The “standard” benefit concerns the increase in his utility

due to the consumption itself. However, he was willing to pay 9€ for that slice and he had to pay just 4€.

Then, we can say that he achieved a sort of “extra” benefit equal to the difference between his willingness

to pay and what he had to pay (13 – 4 = 5€). Economists refer to this amount of money as to the consumer

surplus. Notice that the consumer surplus is positive also for the consumption of the second slice of pizza

(5 – 4 = 1€). This concept can be graphically described through the demand curve.

Figure 1.29.Demand and consumer surplus.

The graph in figure 1.29 reports Jim’s demand of pizza. We said that he is willing to pay 9€ for the first slice,

but it costs just 4€. Translated on the graph, this means that he would be willing to pay the red area but,

given the market price, he has to pay only the blue square. Hence, the consumer surplus is equal to the

difference between what he would be willing to pay (red + blue area) and what he has to pay (the blue

area). In figure 1.29 the Jim’s surplus is equal to the red area.

0 1 2 3 40

2

4

6

8

10

12

14

Slices of pizza

Price of pizza

1.5. The problem of the firm.

In the introduction to this chapter we mentioned that any economy is made up by two main actors: the

consumers and the producers or, better, the firms. The firms produce consumption goods, with the final

objective to sell their output to the consumers. Unfortunately in our course we do not have enough time to

cover this topic with the same details of the consumer case. In the following lines we will provide you with

some basic concepts and intuitions concerning the firm problem, in order to let you understand the

similarities with the framework explored in the previous sections of this chapter.

As in the consumer case the firm2 has to decide (once it has decided the good to produce) the quantity to

produce and the technology to exploit in the production process. Consider for instance the case of a firm

producing T-shirts. It can choose to produce 10,000 shirts, or maybe only 1,000. Moreover, it can decide to

employ a lot of workers, placing its factory in a non-developed country. Or maybe it can decide to produce

its output in a hyper-technological manufacturing plant exploiting a small number of workers. In the

following lines we generalize this problem, in order to understand how the general firm is facing these

issues. Just an aside: what about the selling price of the good? Is it fixed by the firm? The answer is

negative, as least in the basic competitive model, which is the case considered here. Remember we

assumed the markets to be competitive. This means that the single firm (as the single consumer) is not able

to affect the market price.

In the previous part, defining the basic competitive model, we stated that while the consumers are

assumed to be rational, the firm is assumed to be profit maximizing. What does it mean? It means that the

final goal of the firm is to maximize the difference between the total revenue and the total costs. Total

revenue (TR) is defined by the product of the number of units sold (for instance the number of T-shirts) and

the selling price. The total cost (TC) is defined as the costs undertaken by the firm for the workers and for

the investments in machineries, raw materials, etc. As we said, the difference between the total revenue

and total costs represents the profit (∏ ) of the firm. The quantity of output produced by the firm is such

that a further increase (or, better, a marginal increase) in the production cannot generate a corresponding

increase in the firm’s profit. In order to better understand the last sentence we must recall the concept of

marginality, mentioned in the previous section. The marginal cost (MC) is the cost of producing one more

unit of a certain good; it can be written as:

MC TC / Q= ∂ ∂

where Q represents the quantity of output produced. The same reasoning applies to the revenues of the

firm. Hence, the marginal revenue (MR) can be defined as the increase in revenue generated by the sale of

one additional unit of output:

MR TR / Q= ∂ ∂

2 Obviously with this term we mean the entrepreneur holding the firm

In our model the marginal revenue is fixed and equal to the market price of the good. As a consequence,

the marginal profit (MP) is represented by the difference between MR and MC:

MP / Q MR MC= ∂ ∏ ∂ = −

If the revenue generated by the production of an extra-unit of output (MR) is higher than the costs

undertaken for its production (MC), the firm will choose to produce it. Consider for instance the case of our

firm producing T-shirts. If the production costs of the first T-shirt are lower than its selling price the firm will

decide to produce it. Then, it will consider the possibility to produce he second shirt as well, is it a

profitable deal? Again, if the revenue is higher than the cost of producing the second unit of output the

answer is affirmative. The firm keeps on reasoning like this up to the point where the production of the

marginal T-shirt is not profitable anymore. That is, when the MC is higher than the MR and the firm would

lose money. If MC and MR are equal the firm will produce the extra-unit, since it will cover its costs. In

general, the level at which marginal revenue equals marginal cost (and marginal profit is zero) is the one at

which the firm maximizes its profit:

0MR MC− =

or

0/ Q∂∏ ∂ =

Since the MR is fixed (and equal to the market price) the framework for action of the firm is restricted to

the costs. The firm aims at minimizing the costs of producing its output. In other words it wants to employ

the most efficient technology affordable with its budget, i.e. the amount of money devoted to the purchase

of work, machineries, etc.

Consider a firm specialized in the production of bread. It has to invest some money in buying the

machineries needed in this particular kind of production activity and to devote part of the budget to the

workers’ salaries. This problem finds a graphical representation in figure 1.30. On the horizontal axis we

represent the units of labor (L) employed for production (for instance, you can interpret it as the number of

working hours per day). The vertical axis shows the amount of capital (K). Here the term “capital” stays for

the money invested in the machineries and in the other physical equipments. Concerning our simple

example about a firm producing bread, you can think at the capital as the number of bowls needed for

mixing water with flour. Labor and capital are the inputs of the production process, where bread is the final

output. Given a budget of 400€, if both the hourly wage and the price of each bowl are equal to 20€, the

firm’s constraint is represented by the line plotted in figure 1.30.

Figure 30. The problem of the firm: the isocost.

This line is called the isocost, as it represents all the combinations of inputs which cost the same amount of

money (in this case, 400€). The slope of the isocost is constant and it reflects the relative price of the two

inputs. In this case, since K and L have the same price per unit, the slope of the isocost is equal to -1.

Figure 30 should remind you something. Obviously, the reasoning is equivalent to the one followed in the

consumer case, when we introduced the concept of budget constraint. The isocost is nothing but the

counterpart, for the firm, of the consumer’s budget line.

Once the firm has analyzed its financial constraints, estimating the maximum amount of inputs available,

the next step concerns the best use of these scarce resources. Is it better to invest the entire budget in

machineries? Or maybe it is better to employ as many workers as possible? Or perhaps the best solution is

to split the budget in two identical portions? The answer crucially depends on the available technology. The

so-called production function specifies the units of output for all the possible combinations of inputs.

Consider our example about the firm producing bread. Table 1.5 shows the number of loafs of bread

produced with different combinations of inputs. For instance, when the firm decides to employ a worker

for two hours (2 units of labor) and to invest part of its budget in the acquisition of two bowls (2 units of

capital), it will produce 15 loafs. As you have probably already understood, this relation can be expressed in

functional terms. The quantity of output is a function of the quantity of inputs.

Table 1.5. Loafs of bread produced for various combinations of inputs.

Units of labor

Un

its

of

cap

ita

l

1.5 2 3 4 5

1.5 8 12 15 18 20

2 12 15 17 20 22

3 15 17 18 21 23

4 18 20 21 22 24

5 20 22 23 24 25

Hence, we can write:

( , )Q f L K=

0 2 4 6 8 10 12 14 16 18 20 220

5

10

15

20

L

K

Where Q stays for the number of loafs produced while L and K represent respectively the units of labor and

the units of capital employed in the production process. Obviously, an increase in the amount of inputs

used for the production leads to a raise in the production volumes. This can be easily verified considering

the example in table 1.5. The concept of production function is very similar to something we reviewed in

the first part of this chapter: the utility function. As utility can be represented as an increasing function of

the consumption goods, the quantity of output is an increasing function of the quantity of inputs.

Since all good things come in threes, the consumer and the firm problem share one more similarity.

Pay attention to table 1.5. As you can note, the same volume of output can be produced by using different

amounts of inputs. Consider for instance the green cells. When the firm employs 1.5 units of labor and 5

units of capital, the output will be equal to 20 loafs.

Figure 1.31. The problem of the firm: the isoquant.

Call this combination “A”. The same number of loafs can be produced when the firm decides to buy 5 units

of labor and 1.5 units of capital. Call this second combination of inputs “D”. You can see from table 1.5 that

the production is equal to 20 loafs also when the firm employs 2 units of labor and 4 units of capital

(combination “B”) and 4 units of labor and 2 units of capital (combination “C”). What if we plot these

combinations of inputs on a graph? The result is reported in figure 1.31.

The line connecting points A, B, C and D is called the isoquant. The isoquant is the curve showing the

various combinations of the inputs yielding to the same level of output. In other words, along the isoquant

the output is constant, while the volumes of input change. Compared with the line plotted in figure 1.31,

isoquants closer to the origin will be associated with lower volumes of output. Following the same

reasoning, isoquants far from the origin correspond to large volumes of output. The similarity with the

indifference curve is evident: the two curves have the same shape. The slope of the isoquant (the Marginal

Rate of Technical Substitution, MRTS) represents the rate at which one factor can be substituted for

another while holding the level of output constant. As for the MRS, the MRTS is decreasing: as the firm

employs more and more units of capital, it must forego less and less units of labor if it wants to keep the

0 1 2 3 4 5 6 7 8 9 100

5

10

15

L

K

A

C D

Output = 20 loafs of bread

B

output unchanged. It is worth noting that whilst the shape of the indifference curve is somehow

“subjective”, in the sense that it depends on individual preferences, the shape of the isoquant is

“objective”, since it is determined by the available technology. The solution of the problem is determined

by matching the isoquant with the isocost. In the equilibrium, the two curves are characterized by the same

slope.

Even if we will not review it in details, the solution of the problem of the firm leads to the derivation of the

supply curve. The supply curve shows the various amounts of output produced by the firm at different price

levels. More precisely, Figure 1.32 represents an example. It is worth noting that the supply curve always

has a positive slope. This means that if the price raises, the firm will be willing to increase its production.

Figure 1.32. The problem of the firm: the supply curve.

In figure 1.32, when the price of bread is equal to 50 cents per kilo, the firm does not to produce: if the

price is too low, the costs exceed the revenues. When the price increases, the firm will be willing to

produce larger volumes of output. As for the demand curve, the market supply is nothing but the sum of

the quantities supplied, at the same price level, by all the firms in the economy. Putting together supply

and demand leads to the definition of the market equilibrium. The equilibrium is a situation from which

neither the consumers nor the firms have an incentive to move. Consider for instance the market of bread,

represented in figure 1.33. Suppose the economy to be in point C. In C we have an excess of supply: when

the price is equal to 2.7€, consumers’ demand is equal to 5 kilos of bread, whilst the firms are supplying

10.5 kilos. Since firms have to compete among themselves in order to sell their products, the price of bread

will fall to the equilibrium level B (2€ per kilo).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Bread (kg)

Price

Figure 1.33. The market of bread and its price equilibrium.

The same reasoning applies in the case of an excess of demand. If the price is too low (1€ per kilo, the

quantity demanded (11.5 kilos) is much larger than the quantity supplied (2.5€). Sine also consumers

compete among themselves in order to get the few units of bread available on the market, the price of

breed will rapidly increase, up to its equilibrium level (point B).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Bread (kg)

Price

A

B

C

D E

Demand

Supply