Intermediate microeconomics - Nationalekonomiska …/menu/...Why study intermediate microeconomics? Microeconomics helps us understand human economic behaviour Microfoundationsimportant

IntermediatemicroeconomicsLecture 1: Introduction and Consumer Theory

Varian, chapters 1-5

Who am I?

● Adam Jacobsson

● Director of studies undergraduate and masters

● Research interests– Applied game theory– Environmental inspections and enforcement– Media economics

● Teaching– Microeconomics & game theory

2017-01-18 Adam Jacobsson, Department of Economics 1

Agenda

● Introduction and overview

● Budget constraint

● Preferences

● Utility

● Choice

Adam Jacobsson, Department of Economics2017-01-18 2

Why study intermediate microeconomics?● Microeconomics helps us understand human economic behaviour

● Microfoundations important in modern economics!

● Mostly “same same but different” than NEKI

● Main difference: Now more math but also a deeper understanding of the theory.

● Why do we need more math? After all we’re studying humans, not machines…

● Math helps us to discipline our thinking, formulate testable hypotheses, construct models for

analysis etc – this will make it easier to apply the theory on real world problems.

● Ability to read (scientific) articles.


Aim of the course

The aim of this course is to show how price theory and game theory can be used to explain how different markets work...

…Using the tools of analysis from price and game theory, this course explains how different economic structures arise and work, and how they can be shaped in a socially and economically efficient fashion...


Learning outcomes

Upon completion of this course, the student is expected to be able to:

● Give an account of the central components of modern price and game

theory, and describe how these methods of analysis can be used in

order to explain consumption and production decisions.

● Perform practical calculations of problems that consumers and

producers may encounter, and intuitively explain how and why the

selected calculation method has been used and how this may explain

the results.


Overview of the course

● Topic 1 – Price theory and general equilibrium– Consumer theory– Production theory– Competitive markets

and general equilibrium– “Idealistic” analysis…

2017-01-18 Adam Jacobsson, Department of Economics

Price

Quantity

Supply

Demand

P*

Q*

6

Topic 2 - Uncertainty, Insurance, Investment and Asset Pricing● Now we introduce some complications to the

analysis:– Intertemporal choice – when should we

consume?– Uncertainty – sometimes bad things happen…

How should we deal with this?– Insurance – a way of dealing with uncertainty.– Investment and asset pricing – another way of

dealing with uncertainty – the tradeoff between returns and risk...


Topic 3 - Welfare Theory

● Do we care about the distribution of welfare and what about market failures?– Social welfare

• How should we split ”the pie”?– Externalities

• E.g. environmental consequences of economic decisions…

– Public goods• E.g. military defence, public broadcasting…


Topic 4 - Strategic Interaction and Models of Competition● Are people and organizations passive drones,

just milling about?– Game theory: a theory of strategic interaction– Imperfectly competitive markets (& more

market failures…)


Topic 5 - Asymmetric information, Moral Hazard, Adverse Selection and Contracts

● Oops, people are not only strategic, but also heterogenous and everybody does not know everything…– Asymmetric information– Moral hazard– Adverse selection


Examination

● Written exam at the end of the course.

● Before each exercise, students can hand in solutions to one assignment question (denoted by *),

which will be awarded points by the teachers. If all assignments receive a pass grade, you will

receive a credit for the exam and the associated re-take exam for this semester only. Note that

you cannot save your course credit for later exams. The credit allows you to skip one specified

question on the exam (corresponding to 10% of total exam points). If you do not have the credit,

you then need to solve all questions on the exam for a full score. Solutions are to be handed in by

groups of two or no more than three persons.


Three course components

● Lectures– Introduces theory– the “big picture”

● Group seminars– Apply your theory, practice, practice, practice

● Mathematics lectures– Revise your math skills!

● Math workshops

● And do not forget to read the book!


1. Budgetconstraint

● Theoptimizationprinciple:Anindividualchoosesthebestbundle(combination)ofgoodsthathe/shecanafford.1. Best:accordingtohis/herpreferences2. Canafford:thebudgetconstraint!

● Simplifyingassumption:Thebundleconsistsoftwogoods(good1,good2).● ConsumptionbundleX=(x1,x2),wherex1 andx2 arethequantitiesofgoods1and2

thattheindividualchoosestoconsume– Example:abundleX couldbecomposedof5apples,x1=5,and10oranges,x2=10,

orX=(5,10).Anotherbundle,Y,couldcontain7applesand8oranges,or;Y=(7,8).

•𝒳 isthesetofallconsumptionbundles:X ∈𝒳• PricesP=(p1,p2)


● Individual income: m● The budget constraint:

𝑝F𝑥1 + 𝑝2𝑥2 ≤ 𝑚 (1)● Budget set: set of all X that satisfy the budget

constraint● Budget line: set of all X, for which

𝑝F𝑥1 + 𝑝2𝑥2 = 𝑚 (2)or

𝑥2 =KLM− LF

LM𝑥1 (3)


Solvefor𝑥2!• Divideequ.(2)lhs&rhs by𝑝2 ⟹• LF

LM𝑥1 + 𝑥2 =

KLM

• SubtractLFLM𝑥1 fromlhs&rhs⟹

• Wenowhaveequation(3)

Budget set and budget line


Good2(𝑥2)𝑚𝑝2

𝑚𝑝1

Good1(𝑥1)

Budgetset

Budgetline 𝑥2 =KLM− LF

LM𝑥1

Slope

15

● The slope of the budget line: The opportunity cost of

consuming one more unit of good 1 in terms of good 2.

● Consider the consumption bundle (x1,x2) on the budget

line. Increasing consumption of x1 by 𝑑x1 means we have

to decrease consumption of x2 by 𝑑x2.● Mathematically:

𝑝1 𝑥1 + 𝑑𝑥1 + 𝑝2 𝑥2 + 𝑑𝑥2 =m (4)⟹ 𝑝1𝑥1 + 𝑝1𝑑𝑥1 + 𝑝2𝑥2 + 𝑝2𝑑𝑥2=m⟹ 𝑝1𝑑𝑥1 + 𝑝2𝑑𝑥2 = 𝑚 − 𝑝2𝑥2 − 𝑝1𝑥1

QRSTUVWXYZ[\]SV[

⟹ Y^2Y^F

= − L1LM

(5)


Preferences


Preferences: ordering of consumption bundles.

An individual’s preferences over 𝒳 is a ranking such that:

● 𝑋 ≻ 𝑌 indicates that X is preferred to Y (strict preference),

● 𝑋~𝑌 indicates that X and Y are equivalent (indifference),

● 𝑋 ≽ 𝑌 indicates that X is preferred toY or is equivalent to Y (weak preference).

17

Assumptions about rational preferences

● Completeness. All bundles can be compared, that is, for all X,Y in 𝒳 it is true that either X≻Y,X~Y or X≺Y.

● Reflexivity. Each bundle is at least as good as itself:

𝑋 ≽ 𝑋.

● Transitivity. If 𝑋 ≽ 𝑌 and 𝑌 ≽ 𝑍 then 𝑋 ≽ 𝑍.

Indifference curve: Bundles that the individual is

indifferent between, that is, for all bundles X,Y along an

indifference curve it is true that 𝑋~𝑌.


The indifference curve and the weakly preferred set


Good1(𝑥1)

WeaklypreferredsetofthebundleX

Xx2

x1

Good2(𝑥2)

Indifferencecurve

19

Theorem 1. The indifference curves of an individual with transitive preferences cannot intersect.

Proof. Consider two indifference curves that intersect. Let bundle X lie on one indifference curve and bundle Y on the other and let bundle Z lie at the intersection of the curves. In this case: 𝑋~𝑍and 𝑌~𝑍.Transitivity then implies 𝑋~𝑌.However, the two indifference curves represent different levels of utility. Thus it must be that 𝑋 ≻ 𝑌 or 𝑋 ≺ 𝑌 which contradicts 𝑋~𝑌.


Proof theorem 1.


Good 2

Y

Z

X

Good 1

𝑌~𝑍 and𝑋~𝑍

Bytransitivity:𝑌~𝑋

Butindifferencecurvesrepresentdifferentlevelsofutilitywhereitmustbeeither𝑋 ≻ 𝑌 or𝑋 ≺ 𝑌.Thiscontradicts𝑌~𝑋!QED

21

Well-behaved preferences

(…simplifies the life of an economist…)

The assumption that preferences are well-behaved (=well-behaved indifference curves) is common, although other assumptions can be made.

1. Monotonicity

● Let X=(x1,x2)and Y=(y1,y2).If 𝑥1 > 𝑦1 and 𝑥2 > 𝑦2 (or if 𝑥1 = 𝑦1 and 𝑥2 > 𝑦2), monotonicity implies that 𝑋 ≻ 𝑌.

● That is, more is better (no satiation)!

● Indifference curves then have negative slopes.


2. Convexity

● Let 𝑍 = 𝜆𝑋 + (1 − 𝜆)𝑌, where 𝜆 ∈ 0, 1 .– If 𝑋 ∼ 𝑌, then 𝑍 ≽ 𝑋. – For 𝜆 ∈ 0, 1 ,if 𝑍 ≻ 𝑋 then we have strict

convexity.● That is, averages are preferred to extremes.

● Weakly preferred sets are convex (or strictly convex).


Convexity


Good 2(x2)

X

Z

Y Good 1(x1)

24

𝑍 = 𝜆𝑋 + (1 − 𝜆)𝑌𝜆=1𝜆=0𝜆=0,5• Convex?• Monotone?

3. Utility

Utility is a way to describe an individual’s preferences.

Definition 1. A function is a utility function if it assigns a real number (utility) to every possible consumption bundle (i.e. 𝑢:𝒳 ⟶ ℛ) such that

𝑢(𝑋) > 𝑢(𝑌) ⇔ 𝑋 ≻ 𝑌

The symbol ” ⇔ ”means “if and only if ” (iff)


Utility


X

Y

u(X)>u(Y)

u(X)u(Y)

𝒳:asetofconsumptionbundles

Note:

● According to our assumptions individual preferences imply an ordering of consumption bundles.

● Under these assumptions it is therefore only the rank assigned by different utilities that is important (ordinality).

● Differences in utility levels (cardinality) do not matter under our assumptions.

● Comparability of utility levels and differences in utility between individuals do not matter either.


Theorem 2. If u is a utility function, then every positive monotonic

transformation f of u is a utility function that represents the same preferences.

Proof. If u is a utility function that represents the individual’s preferences,

then

𝑢(𝑋) > 𝑢(𝑌) ⟺ 𝑋 ≻ 𝑌(6)

● If f is a positive monotonic transformation of u (f(u) is always increasing in

u), then

𝑓 𝑢 𝑋 > 𝑓 𝑢 𝑌 ⟺ 𝑢 𝑋 > 𝑢(𝑌) (7)

● Combining (6) and (7), we obtain

𝑓(𝑢(𝑋)) > 𝑓(𝑢(𝑌)) ⟺ 𝑋 ≻ 𝑌(8)

● Hence, f is also a utility function according to Definition 1.


Marginal rate of substitution


Good2(𝑥2)

Good1(𝑥1)

𝑑𝑥2𝑑𝑥1

Theslope=Y^MY^F Indifferencecurve

Marginal rate of substitution (MRS)

The utility function 𝑢(𝑥1, 𝑥2) is differentiated in the point (𝑥1, 𝑥2)and the

result is set to 0 (why?):

𝑑𝑢 =𝜕𝑢(𝑥1, 𝑥2)

𝜕𝑥1𝑑𝑥1 +

𝜕𝑢(𝑥1, 𝑥2)𝜕𝑥2

𝑑𝑥2 = 0

Rearrange:

𝑀𝑅𝑆:= Y^MY^FyYXQR

= −z{z|}z{z|~

The MRS measures how much consumption of good 2 a consumer is

prepared to give up for one more unit of good 1 at a specific consumption

bundle. (MRS is the slope of the indifference curve at a specific point)

Note that the MRS is not affected by a positive monotonic transformation!


4. Choice

● The consumer chooses the most preferred bundle from his/her budget

set (optimization).

● Intuition: move along the budget line until the weakly preferred set

does not overlap the budget set. Or, find the indifference curve with

the highest utility that just touches the budget line…

● With some exceptions, the indifference curve is tangent to the budget

line at the optimal point: MRS = - LFLM

● Exceptions: Corner solutions, kinked indifference curves etc…


The utility maximization problem


Good2(𝑥2)

Good1(𝑥1)

𝑚𝑝2

𝑚𝑝1

u=2u=3u=4

u=5

Note!Thisisanexampleofwellbehavedpreferencesandaninterioroptimum!

Examples where the indifference curve is not tangent to the budget constraint at the optimum


Good2(𝑥2)

Good1(𝑥1)

𝑚𝑝2

𝑚𝑝1

Good2(𝑥2)

Good1(𝑥1)

𝑚𝑝2

𝑚𝑝1

Perfectsubstitutes Perfectcomplements

u=2 u=4u=3 u=5u=2

u=3u=4

u=5

Utility maximization – putting the pieces together (assume well-behaved prefs. and interior solutions)

Maximize utility given the budget constraint.

max^F,^M

𝑢 𝑥1, 𝑥2 𝑠. 𝑡. 𝑝1𝑥1 + 𝑝2𝑥2 = 𝑚 (9)

Set up the Lagrangian:

𝐿 𝑥1, 𝑥2, 𝜆 = 𝑢 𝑥1, 𝑥2 − 𝜆 𝑝1𝑥1 + 𝑝2𝑥2 − 𝑚

Let the optimal solution be denoted by𝑥F∗, 𝑥M∗, 𝜆∗ = (𝑥F∗ 𝑃,𝑚 , 𝑥M∗ 𝑃,𝑚 , 𝜆∗ 𝑃,𝑚 )

(note that P=(p1,p2))

where (𝑥F∗ 𝑃,𝑚 , 𝑥M∗ 𝑃,𝑚 ) are the Marshallian demand

functions.


The first-order conditions (FOC) are given by

��^}

= �X ^}∗,^~∗

�^}− 𝜆∗𝑝1 = 0 (i)

��^~

= �X ^}∗,^~∗

�^~− 𝜆∗𝑝2 = 0 (ii)

��= 𝑝1𝑥F∗ + 𝑝2𝑥M∗ − 𝑚 = 0 (iii)

Solve (i) and (ii) for 𝜆∗:

𝜆∗ =

𝜕𝑢 𝑥F∗, 𝑥M∗𝜕𝑥F𝑝1

=

𝜕𝑢 𝑥F∗, 𝑥M∗𝜕𝑥M𝑝2


At the optimal solution the following is true for the MRS:

𝑀𝑅𝑆 = −

𝜕𝑢 𝑥F∗, 𝑥M∗𝜕𝑥F

𝜕𝑢 𝑥F∗, 𝑥M∗𝜕𝑥M

= −𝑝1𝑝2


The cost minimization problem


Good2

Good1

𝑢�

Amongthebundleson𝑢�,choosethecheapest.

𝑚 = 10𝑚 = 7𝑚 = 5

Cost minimizationMinimize costs to attain a given utility level 𝑢�.

min^F,^M

𝑝1𝑥1 + 𝑝2𝑥2𝑠. 𝑡. 𝑢 𝑥1, 𝑥2 = 𝑢� (10)

Set up the Lagrangian:𝑀 𝑥1, 𝑥2, 𝜇 = 𝑝1𝑥1 + 𝑝2𝑥2 − 𝜇 𝑢 𝑥1, 𝑥2 − 𝑢�

Let the optimal solution be denoted by𝑥F�, 𝑥M�, �̅�� = (𝑥F� 𝑃, 𝑢� , 𝑥M� 𝑃, 𝑢� , 𝜇� 𝑃, 𝑢� )

where (𝑥F� 𝑃, 𝑢� , 𝑥M� 𝑃, 𝑢� ) are the Hicksian demand functions.


The first-order conditions (FOC) are given by

��^}

= 𝑝1 − 𝜇��X ^}�,^~�

�^}= 0 (i)

��^~

= 𝑝2 − 𝜇��X ^}�,^~�

�^~= 0 (ii)

��= − 𝑢 𝑥1, 𝑥2 − 𝑢� = 0 (iii)

Solve (i) and (ii) for 𝜇�:

𝜇� =

𝜕𝑢 𝑥F�, 𝑥M�𝜕𝑥F𝑝1

=

𝜕𝑢 𝑥F�, 𝑥M�𝜕𝑥M𝑝2


At the optimal solution the following is true for the MRS:

𝑀𝑅𝑆 = −

𝜕𝑢 𝑥F�, 𝑥M�𝜕𝑥F

𝜕𝑢 𝑥F�, 𝑥M�𝜕𝑥M

= −𝑝1𝑝2

Same as under the utility maximization problem!

The expenditure function is defined as:𝐸 𝑝1, 𝑝2, 𝑢� : = 𝑝1𝑥F� + 𝑝2𝑥M�


A perspective on preferences

● Preferences indicate what individuals like and utility functions are used to represent different kinds of preferences.

● Common assumption in (traditional) economic analysis: – Homo oeconomicus, i.e. the individual wants

to maximise his/her own material well being.● Is this always so?


● Not necessarily!

● Individuals can also be motivated by altruism, a desire for justice, fairness etc (there is evidence for this!).

● Does this mean we should throw away (traditional) economic models?

● No! But we should be aware of the assumptions we make and how these assumptions match the problem we are analysing.

● Lets look at a recent article:


Evolution and Kantian moralityby Ingela Alger and Jörgen W. Weibull, Games and Economic Behaviour 98, (2016), pp 56-67. (Voluntary reading. This is an advanced scientific text, focus on sections 1, 6 & 7.) For an easy-to-read article about the authors’ reaserach by Thomas Lerner, see Dagens Nyheter, 161228.

● “What kind of preferences should one expect evolution to favour?”, p56.

● Selfish preferences or, perhaps, Kantian preferences?

● Kantian imperative: ”Act only according to that maxim whereby you can, at the same time, will that it should become a universal law.”


● Evolutionary game theoretical model.

● Individuals can play either selfish or more “Kantian” strategies.

● Individuals randomly and repeatedly interact with other individuals

(they have a tendency to be matched with similar individuals).

● The authors show that individuals who play more or less Kantian

strategies tend to better than the more selfish individuals.

● This means that the proportion of “Kantian” individuals in the

population remains large over time.

● That is, evolution favours Kantian behaviour!

● Homo moralis!


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Intermediate microeconomics - Nationalekonomiska …/menu/...Why study intermediate microeconomics? Microeconomics helps us understand human economic behaviour Microfoundationsimportant