Chapter 1 Theory of Demand Ali Mazyaki, Ph.D. Institute for - - PowerPoint PPT Presentation

Microeconomics I

Chapter 1

Theory of Demand

Ali Mazyaki, Ph.D.

Institute for Management and Planning Studies (IMPS)

Agenda

1- Preference and Choice

1-1- Choice set 1-2- Preference 1-3- Rationality 1-4- Utility function

2- Consumer choice 2-1- Commodities 2-2- The consumption set 2-3- Competitive Budgets 2-4- Demand function 2-5- Comparative statics 2-6- The weak axiom of revealed preferences 3- Classical demand theory 3-1- Basics 3-2- Preference and utility 3-3- The utility maximization problem 4- Aggregate demand

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1- Preference and Choiceحیجرت و باختنا

• "Consumption" is the genuine of any economic modeling.

 Precise definition and practical theorizing of this concept is of upmost importance in economics.  Use mathematics: Formalizing the concept using mathematical instruments has several advantages: 1. It provides us a precise language that bans any misinterpretation. This is an invaluable usage of mathematics in economics letting us talk in a flawless understandable way.

– However, one should be aware of the important fact that we do not study mathematics here and all the mathematical formulas have to be interpreted and understood through some logical constructs.

2. Formalizing our understanding is very useful for verification of our

• ideas. To criticize economic conjectures one need to be clear and the

language we normally use in our everyday life is not quite suitable for achieving this clarity.

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1- Preference and Choice

• How do individuals choose from a set of opportunities?
• What can we conclude from observed choices?
• Objective: formulation of a theory that may be applied to a host of

conceivable choice problems.

• Consider a "set of possible alternatives” and call it Consumption Set (Choice

Set) X:

– X is a representation of all alternatives that a consumer may conceive

– Primitive characteristic of the individual: preference relations that summarize his tastes – We impose rationality axioms on preferences and then analyze what this implies for choosing element(s) out of X

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1-1- Choice set

1- Preference and Choice

• To formalize this concept we use preference relation ≽

which is a binary relation on the set of alternatives X:

• Having the above definition we may define:
• Read ≽: "at least as good as"

ای “یبوخ هب لقادح“

• Read ≻: "strictly preferred to“

ای “زا تسا رتهب ًادیکا“

• Read ∼: "is indifferent with“ ای “اب تسا توافت یب“

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1-2- Preference

≽⊂ 𝑌 × 𝑌. 𝑦 ≻ 𝑧 ⇔ 𝑦 ≽ 𝑧 𝑐𝑣𝑢 𝑜𝑝𝑢 𝑧 ≽ 𝑦 𝑦 ∼ 𝑧 ⇔ 𝑦 ≽ 𝑧 𝑏𝑜𝑒 𝑧 ≽ 𝑦

1- Preference and Choice

Now we may define one of the first fundamental definitions in economics which is normally referred to as rationality. However, in some textbooks (Jehle & Reny) it is assumed as the standard assumption of preferences.

• The preference relation ≽ is called rational if it is

complete and transitive:

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1-3- Rationality

لماک ندوب یلاقتنا ندوب

• 1. 𝐷𝑝𝑛𝑞𝑚𝑓𝑢𝑓𝑜𝑓𝑡𝑡: ∀𝑦, 𝑧 ∈ 𝑌 𝑓𝑗𝑢ℎ𝑓𝑠 𝑦 ≽ 𝑧 𝑝𝑠 𝑧 ≽ 𝑦
• 2. 𝑈𝑠𝑏𝑜𝑡𝑗𝑢𝑗𝑤𝑗𝑢𝑧: ∀𝑦, 𝑧, 𝑨 ∈ 𝑌 𝑗𝑔 𝑦 ≽ 𝑧 𝑏𝑜𝑒 𝑧 ≽ 𝑨 ⇒ 𝑦 ≽ 𝑨

1- Preference and Choice

Exercise: Show that if the weak preference relation ≽ of a consumer is rational, then: i. ≻ is irreflexive and transitive

• ii. ∼ is reflexive, transitive and symmetric
• iii. If 𝑦 ≻ 𝑧 ≽ 𝑨 ⇒ 𝑦 ≻ 𝑨

Note that ≻ and ∼ are not rational (why?)

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1-3- Rationality

1- Preference and Choice

 It eliminates the lack of ability to compare

• Comparing alternatives can be difficult if we have

little experience with them (e.g. climate change)

• We neglect the (time) costs of comparing

alternatives

• Cost of being rational may make “being rational”

irrational!; in fact, we set “the cost of thinking” zero which is not very troublesome

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1-3-1- Discussion of basic rationality assumptions

Completeness:

1- Preference and Choice

 It is useful to make the decisions independent from

• ther factors

 Intensity of preferences may be depicted by defining many alternatives

• Problem of “just perceptible differences”:
• Agent may be indifferent between just perceptible

differences of colors for painting a room.

• However, as we repeat this the agent may prefer starting to

final color

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1-3-1- Discussion of basic rationality assumptions

Transitivity:

1- Preference and Choice

The 2002 Nobel laureates Daniel Kahneman (together with Vernon L. Smith) integrated insights from psychological research into economic science, especially concerning human judgment and decision-making under uncertainty. Kahneman and Tversky (1984) show that framing is very important specially when outcomes are uncertain.

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1-3-1- Discussion of basic rationality assumptions

Framing:

1- Preference and Choice

• Problem of framing (manner of presenting alternatives matters for

choice)

• Prices in store 1: €125 for stereo and €15 for calculator
• Salesman tells you that one of them costs €5 less in store 2, which is

located 20 minutes away

• In experiments, fraction that would travel to other store is much higher,

if discount is on calculator

• by contrast, the same individuals express indifference to the following

question

• Because of a stock out you must travel to the other store to get the

two items, but you will receive €5 off on either item as

• compensation. Do you care on which item the rebate is given?
• This violates transitivity

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1-3-1- Discussion of basic rationality assumptions

Framing:

1- Preference and Choice

• x: travel to other store and €5 discount on calculator
• y: travel to other store and €5 discount on stereo
• z: buy both items at first store
• first two choices reveal: 𝑦 ≻ 𝑧 𝑏𝑜𝑒 𝑨 ≻ 𝑧
• third choice reveals: 𝑦 ∼ 𝑧
• but: maybe we have misspecified the choice alternatives
• individuals do also care about making good bargains, often

understood as price reductions in %

• perception for first two choices: discount on individual

product

• perception for third choice: discount on bundle of goods

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1-3-1- Discussion of basic rationality assumptions

Framing:

1- Preference and Choice

• We often take households as the primitive of our analysis
• preferences of mom: ≻𝐵
• preferences of dad: ≻𝐶
• preferences of child: ≻𝐷
• Majority-rule votes produces cyclical household preferences (i.e.

Condorcet Paradox): 𝑦 ≻𝐵 𝑧 ≻𝐵 𝑨 𝑧 ≻𝐶 𝑨 ≻𝐶 𝑦 𝑨 ≻𝐷 𝑦 ≻𝐷 𝑧

• Check that in the majority voting x is strictly preferred to y and y is

strictly preferred to z. While z is strictly preferred to x which is a violation of transitivity.

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1-3-1- Discussion of basic rationality assumptions

Aggregation of preferences :

1- Preference and Choice

• changes in taste
• x: smoke 1 cigarette a day
• y: abstinence (initial situation)
• z: heavy smoking
• Preferences in initial situation: 𝑦 ≻ 𝑧 ≻ 𝑨
• Once the individual has started smoking, preferences change to:

𝑨 ≻ 𝑦 ≻ 𝑧

• “Change-of-taste” models are important for analyzing addictive behavior in

“behavioral economics” (see for example O'Donoghue, T. and M. Rabin 2001 who define several selves for an individual, one of them is rational and the other is not!)

• Heidhues Kőszegi (2010), using the same “bounded rationality” with the

notion of “naiveté”, model “over borrowing” as a reason for financial crisis.

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1-3-1- Discussion of basic rationality assumptions

Bounded rationality:

1- Preference and Choice

• Describing a given preference relation, it is useful to define a utility

function which assigns a numeric value to each alternative, somehow that in comparing to alternatives the alternative with bigger value is the preferred alternative. Definition: A function 𝑣: 𝑌 → 𝑆 is a utility function representing the weak preference relation if, for all x , y ∈ X: 𝑣(𝑦) ≥ 𝑣(𝑧) ⟺ 𝑦 ≽ 𝑧

• u(x) is not unique: Let f be a strictly increasing function f: 𝑆 → 𝑆,

then 𝑤 𝑦 = 𝑔 𝑣(𝑦) is a new utility function representing the same preferences as u(x).

• Note that the utility function is "ordinal" and not "cardinal". In fact,

utility function orders the alternatives and does not show how different they are.

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1-4- Utility function

1- Preference and Choice

Rationality is a necessary condition that a given weak reference relation can be represented by a utility function. i.e. a preference relation can be presented by a utility function only if it is rational.  Proof: Completeness: Since u(.) is a real-valued function defined on X, it must be that for any x , y ∈ X, either u(x) ≥ u(y) or u(y) ≥ u( x). If u(.) represents the preference relation, then either 𝑧 ≽ 𝑦 or 𝑦 ≽ 𝑧 , which implies completeness. Transitivity: Assume 𝑧 ≽ 𝑦, 𝑦 ≽ 𝑨. Then, a utility function representing the preference relation must have u(y) ≥ u(x) and u(x) ≥ u(z), which requires 𝑦 ≽ 𝑨 .

• But rationality is not a sufficient condition i.e. there are some rational

preferences that no utility function may represent them.

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1-4- Utility function

2- Consumer choice

• Consumer choice: decision theory when

individuals face given market prices.

• Commodities: goods and services available in an

economy.

– In principle many distinctions possible, e.g. commodities consumed

• at different time points
• in different states of nature (e.g. umbrella with/without rain)

should, in principle, be viewed as different commodities

– The extent to which aggregation across time, space, or … may be appropriate depends on:

• the specific economic question under consideration,
• and the economic data to which the model is being applied

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2- Consumer choice

• Commodity bundle may be described as a vector

𝑦 = 𝑦1 ⋮ 𝑦𝑀 ∈ ℛ𝑀

• With a total of L commodities
• x is then a point in the L-dimensional commodity space of real

numbers (Negative entries will often represent net outflows)

• Consumption bundle may be described with a commodity bundle.
• Notation: in this lecture, x always represents the above commodity

vector, while xi is a number that denotes the consumption of commodity i

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2-1- Commodities

2- Consumer choice

• Subset of the commodity space. Limitations may result

from physical or institutional restrictions.

• the following figures contain 4 examples of

consumption sets with physical constraint

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2-2- Consumption set

X

24 Leisure Hours Bread

𝑦1 𝑦2

1 2 3

X

• fig. 1: consumption ≤ 24 for leisure
• fig. 2: consumption of good 2 only in

nonnegative integer Amounts

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Bread in New York at Noon

Bread in Washington at Noon

X

Slices of Brown Bread Slices of White Bread

X

4 4

2- Consumer choice

• fig. 3: consumption of one good may

make consumption of another good impossible

• fig. 4: there is a minimum consumption

level, e.g. needed to survive

• One important physical restriction may be that consumption
• f commodities must be nonnegative (this is the case in all 4
• f the above examples)
• the set of all nonnegative bundles of commodities:

𝑌 = 𝑆+

𝑀 = 𝑦 ∈ 𝑆𝑀: 𝑦𝑚 ≥ 0 𝑔𝑝𝑠 𝑚 = 1, … , 𝑀

• This is a convex set: if x and x’ are an element of the set 𝑆+

𝑀

, then the bundle 𝑦" = 𝛽𝑦 + (1 − 𝛽)𝑦′ is also an element of this set for any 𝛽 ∈ 0,1 .

• In the following, we will usually take 𝑆+

𝑀as the consumption

set ⁻ note: aggregation may help to convexify the consumption set, e.g. bread consumed over a longer period in fig. 3.

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2- Consumer choice

• consumption choices are also limited by economic constraints, i.e.

whether a consumer can afford a consumption bundle.

• assumption 1: commodities are traded at prices:

𝑄 = 𝑞1 ⋮ 𝑞𝑚 ∈ 𝑆𝑀 Which are publicly quoted.

• Notation: in this lecture, p always represents the above price vector,

while 𝑞𝑗 is a number that denotes the price of commodity I.

• usually we assume 𝑞𝑗 >0 for all i.
• but, in principle we may have 𝑞𝑗 < 0, e.g. for “bads” (e.g. pollution)
• assumption 2: consumers are price-takers.
• w: a consumer’s wealth level, i.e. a number (usually assumed to be

strictly positive)

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2- Consumer choice

2-3- Competitive budgets

• The Walrasian (or competitive) budget set:

𝐶𝑞,𝑥 = 𝑦 ∈ 𝑆+

𝑀: 𝑞. 𝑦 ≤ 𝑥

=all consumption bundles that are affordable.

• Notation: a dot · between two vectors always represents

the inner product of these two vectors. For example 𝑞. 𝑦, is the number 𝑞. 𝑦 = 𝑞1 ⋮ 𝑞𝑀 . 𝑦1 ⋮ 𝑦𝑀 = 𝑞1𝑦1 + ⋯ + 𝑞𝑀𝑦𝑀

• 𝑞. 𝑦 has the same meaning as 𝑞𝑈𝑦, where 𝑞𝑈is the

transpose of the column vector p

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2- Consumer choice

2-3- Competitive budgets

• Consumer problem is to choose her preferred bundle x

from this budget set.

• The set 𝑦 ∈ 𝑆+

𝑀: 𝑞. 𝑦 = 𝑥 of just affordable bundles is

called budget hyper plane.

• If L=2 it is called the budget line.
• The Walrasian budget set is convex.
• Let 𝑦" = 𝛽𝑦 + (1 − 𝛽)𝑦′. If x and x’ are elements of

the budget set (i.e. if 𝑞. 𝑦 ≤ 𝑥 and 𝑞. 𝑦′ ≤ 𝑥), then: 𝑞. 𝑦" = 𝛽(𝑞. 𝑦) + (1 − 𝛽)(𝑞. 𝑦′) ≤ 𝑥

• And x” is also element of the budget set, i.e. 𝑦" ∈ 𝐶𝑞,𝑥

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2- Consumer choice

2-3- Competitive budgets

• The Walrasian budget set and the effect of

price changes in R2.

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2- Consumer choice

2-3- Competitive budgets

𝑦1 𝑦2 𝑦1 𝑦2

𝐶𝑞,𝑥 𝑥 𝑞2

𝑥 𝑞1

𝑦 ∈ 𝑆+

𝑀: 𝑞. 𝑦 = 𝑥

Slope=−

𝑞1 𝑞2

𝐶𝑞 ,𝑥, 𝑞 = 𝑞1, 𝑞 2 𝑥ℎ𝑗𝑢 𝑞 2 > 𝑞2

• 𝑦 = 𝑦(𝑞, 𝑥); market Ordinary or Walrasian

Demand.

(Demand may not necessarily be single valued but this is assumed in the following.)

• Basic assumptions:

 Homogeneity of degree zero:

𝑦 𝑙𝑞, 𝑙𝑥 = 𝑙0𝑦 𝑞, 𝑥 = 𝑦(𝑞, 𝑥) For any p, w, k>0.

 𝑦(𝑞, 𝑥) satisfies Walras’s law. i.e. for every 𝑞 ≫ 0 and 𝑥 > 0, we have that p. 𝑦 𝑞, 𝑥 = 𝑥.

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2- Consumer choice

2-4- Demand Functions

• Digression: Walrasian demand and choice structure.
• 𝔺𝕩, 𝑦(. ) is a choice structure because:
• Family of Walrasian budget set: 𝔺𝕩 = 𝐶𝑞,𝑥: 𝑞 ≫ 0, 𝑥 > 0
• By homogeneity of degree zero, 𝑦(𝑞, 𝑥) depends only on consumers

budget sets.

• Remark: 𝔺𝕩, 𝑦(. ) does not include all two-and three- element subsets of

X.

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2- Consumer choice

2-4- Demand Functions

• Wealth (income) effect
• the consumer’s Engel function: demand as a function of

wealth for given prices 𝑦(𝑞 , 𝑥)

• wealth/income expansion path: its image in the

commodity space 𝑆+

𝑀 (see figure on next slide).

• 𝜖𝑦𝐽(𝑞,𝑥)

𝜖𝑥

:wealth/income effect for the l-th commodity.

• commodity l is normal if

𝜖𝑦𝐽(𝑞,𝑥) 𝜖𝑥

≥ 0

• commodity l is inferior if

𝜖𝑦𝐽(𝑞,𝑥) 𝜖𝑥

< 0

• we say that demand is normal if every commodity is

normal at all (p , w)

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2- Consumer choice

2-5- Comparative Statics

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2- Consumer choice

wealth effects in matrix notation: 𝐸𝑥𝑦 𝑞, 𝑦 = 𝜖𝑦1(𝑞, 𝑥) 𝜖𝑥 ⋮ 𝜖𝑦𝑀(𝑞, 𝑥) 𝜖𝑥

2-5- Comparative Statics

• (Ordinary) price effect:

𝜖𝑦𝑗(𝑞,𝑥) 𝜖𝑞𝑙

• price effects in matrix form:

𝐸𝑞𝑦 𝑞, 𝑥 = 𝜖𝑦1(𝑞, 𝑥) 𝜖𝑞1 … 𝜖𝑦1(𝑞, 𝑥) 𝜖𝑞𝑀 ⋮ ⋱ ⋮ 𝜖𝑦𝑀(𝑞, 𝑥) 𝜖𝑞1 … 𝜖𝑦𝑀(𝑞, 𝑥) 𝜖𝑞𝑀

• offer curve: demand in 𝑆+

2 as we range over all possible

values of 𝑞2 (see figures on next slide).

• Commodity i is a Giffen good at (p,w) if

𝜖𝑦𝑗(𝑞,𝑥) 𝜖𝑞𝑗

> 0

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2- Consumer choice

2-5- Comparative Statics

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• Some implications of Walras’ law for demand
• 1. By Walras’ Law, 𝑞. 𝑦(𝑞, 𝑥) = 𝑥.

Differentiation w.r.t the price of good k yields: 𝑞𝑚. 𝜖𝑦𝑚(𝑞,𝑥)

𝜖𝑞𝑙 𝑀 𝑚=1

+ 𝑦𝑙 𝑞, 𝑥 = 0

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indirect effects due to demand changes of all goods direct effect of price increase

• n expenditures at given

demand or good k

intuition: total expenditures cannot change in response to a change in prices.

2- Consumer choice

• 2. By Walras’ Law, 𝑞. 𝑦(𝑞, 𝑥) = 𝑥.

Differentiation w.r.t. wealth w yields: 𝑞𝑚.

𝜖𝑦𝑚(𝑞,𝑥) 𝜖𝑥 𝑀 𝑚=1

= 1

• Intuition: Total expenditure must change by an

amount equal to the wealth change

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2- Consumer choice

• in chapter 3 we study preference-based approach.
• comparison with this section allows us to tell how much

more structure is imposed by preference-based approach in comparison to weak axiom (in conjunction with our assumptions that x(p,w) is single-valued, homogeneous of degree 0, and satisfies Walras’ law).

• Definition (MWG 2.F.1)
• The Walrasian demand function x(p,w) satisfies the weak

axiom of revealed preferences if the following property holds for any pricewealth situations (p,w) and (p’,w’):

𝑗𝑔 𝑞. 𝑦 𝑞′, 𝑥′ ≤ 𝑥 𝑏𝑜𝑒 𝑦 𝑞′, 𝑥′ ≠ 𝑦 𝑞, 𝑥 , 𝑢ℎ𝑓𝑜 𝑞′. 𝑦 𝑞, 𝑥 > 𝑥′

• 𝑞. 𝑦 𝑞′, 𝑥′ ≤ 𝑥 𝑏𝑜𝑒 𝑦 𝑞′, 𝑥′ ≠ 𝑦 𝑞, 𝑥 𝑗𝑛𝑞𝑚𝑧 𝑦 𝑞, 𝑥 was

chosen when 𝑦 𝑞′, 𝑥′ was available.

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2-6- Weak Axiom of Revealed Preferences

2- Consumer choice

• ⇒ consumer “revealed” a preference for

x(p,w) over x(p’,w’).

• Consistency “weak axiom” requires that the

preferred bundle was not available when consumer chose x(p’,w’), i.e. p’.x(p,w)<w’.

• following slide shows some examples:
• x* = x(p*,w*), x’ = x(p’,w’), and x* ≠ x.
• remember our assumptions that x(p,w) is

single-valued.

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2-6- Weak Axiom of Revealed Preferences

2- Consumer choice

• Compatible with the weak axiom of revealed preferences?

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2-6- Weak Axiom of Revealed Preferences

2- Consumer choice

• Before we elaborate on the law of demand, an

additional concept shall be introduced.

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2-6- Weak Axiom of Revealed Preferences

2- Consumer choice

References:

• Gravelle, Rees, 2004, Microeconomics. 3rd ed., Harlow et al.
• Heidhues, P., and B. Kőszegi (2010): "Exploiting Naïvete about Self-Control in the

Credit Market“, American Economic Review, 100(5), pp. 2279-2303.

• Jehle & Reny (2001): “Advanced Microeconomic Theory”, Financial times

Prentice Hall, Pearson.

• Mas-Colell, A., Whinston, M., Green, J. (1995): Microeconomic Theory. Oxford

University Press: New York, Oxford.

• O'Donoghue, T. and M. Rabin (2001): “Choice and Procrastination”, The Quarterly

Journal of Economics, 116(1), pp. 121-160.

• Varian, Hal R. (1992): “Microeconomic Analysis”, London: W. W. Norton & Co.

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