数学代写-MATH60013/96011
时间:2022-05-21
MATH60013/96011 – Mathematics of Business and Economics Imperial College London
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MATH96011 – Mathematics of Business and
Economics

Dr Ioanna Papatsouma
Spring 2022
Introduction

This course aims to provide a broad mathematical introduction to economics and its
application in a business setting. In this introduction, I provide some initial definitions
relevant to the content of the course, in addition to detailing the course structure and
specifying some course objectives.

What is an economy?

An economy is an ecosystem, in which governments, markets, firms and individual
consumers all interact with the aim of enabling the provision of goods and services in return
for payment. Broadly speaking, there are four groups of agents that enable an economy to
function:
- individuals and households, who act as consumers in obtaining goods and
services from producers, and as suppliers in providing labour to producers;

- firms, who provide goods and services to consumers, and who also employ
individuals from the first group; firms also act as consumers for other firms;

- governments and regulatory bodies, who provide oversight, regulation and
intervention in order that their economies function smoothly and in service of
particular goals;

- and trade partners external to the economy, who also interact with these
agents to influence production and consumption within the economy.

The above definition of an economy indicates that economic analysis can be carried out at a
number of different scales. Households and firms come together to trade goods and
services within (conceptual) markets; the analysis of such interactions and the behaviour of
these markets is the principal focus of microeconomic theory. In contrast, the actions of
governments that influence the operation of markets, as well as the interaction of markets
with external trading partners, falls under the heading of macroeconomic analysis.

This course will focus on the interaction of economic agents within a market economy, i.e.
one where production and trade are private enterprises. As such, we consider an ecosystem
whose constituent parts are each controlled by their own decision-making processes; the
study of economics is therefore important to understand how these processes affect one
another and how each agent should operate in order to satisfy their particular goals whilst
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acknowledging the behaviour of other agents and changes to the economic environment in
general.

Course Structure

The structure of this course reflects this division of analyses. In Parts 1 and 2, we focus on
microeconomics, before taking a macroeconomic viewpoint in Part 3.
- In Part 1, we will analyse the aims and objectives of both firms and consumers,
and we will use mathematical arguments to show how these objectives lead to
the observed behaviour in a competitive market environment.
- In Part 2, we will consider how the behaviour of firms and consumers is affected
by the properties of the market in which they operate, and how their behaviour
is affected by changes to the market environment.
- In Part 3, we analyse the macroeconomic environment; in particular, we explore
aggregated concepts of supply and demand, we look at the circular flow of
income and discuss the Gross Domestic Product (GDP).

Syllabus:
Theory of the firm

Profit maximisation for a competitive firm
Cost minimisation. Geometry of costs
Profit maximisation for a non-competitive firm

Theory of the consumer

Consumer preferences and utility maximisation
The Slutsky equation

Levels of competition in a market
Consumers’ and Producers’ surplus
Deadweight loss

Macroeconomic theory

Circular flow of income
Cross Domestic Product
Social welfare and allocation of income

Mathematical Methods:
(Constraint) Optimisation. Quasi-concavity. Preferences relations and orders.

Course Objectives

This course provides an introduction to the fundamental aspects of both microeconomics
and macroeconomics, using a mostly rigorous mathematical approach to both the
exposition and demonstration of these subjects.
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In a business context, this course will provide you with the tools required to analyse the
goals of a firm and the decisions that a firm may make in the context of their particular
market. At the end of the course, you should understand the effects that these decisions
have on the firm itself, on the various connected individuals and firms, and on the economy
as a whole.

Solving problems in this course will require both an economic understanding of the
concepts as well as a sound mathematical derivation. That means that you should be
prepared to come up with mathematical proofs as well as to explain notions in form of
(very) short essays.

Additional Course Information
Lectures: Pre-recorded lectures

Office Hour: Doodle poll

Problem Classes: Bi-weekly; see weekly plan

The problem sheets will be available on Blackboard and the
solutions to the problem sheets will be uploaded after the
problem classes.
If anybody wants to have some feedback on their un-assessed
problem sheets, you can give me your solutions and I will have
a look at it.

Course Rep: You should agree on a course rep in the first week.

Lecture notes: The lecture notes are available on Blackboard. They have gaps
and we will fill these gaps during the lectures.



Textbooks:
All the material used in this module can be found in various textbooks.

Gillespie, A. (2013) Business Economics (2nd Edition). Oxford University Press.
Varian, H. R. (1992) Microeconomic Analysis (3rd Edition). W. W. Norton & Co.
Varian, H. R. (2014) Intermediate Microeconomics (9th Edition). W. W. Norton & Co.

These books can be found (some electronically) in the college library. However, the course
will be self-sufficient.

Assessment: 1 piece of assessed coursework, worth 10% (Probably from 28/02 till 14/03)
1 two-hour final exam, worth 90%
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Part 1 - Microeconomics

Supply and Demand – an introduction

Principal questions to be addressed by economics – what price should we be paying for
goods, what price should a vendor be selling their goods for? Do the different motivations of
the different parties give different answers to this question?

Supply refers to the quantity of a product that a vendor (or vendors) is willing and able to
sell, at a given price in a given period of time.

Correspondingly, demand refers to the quantity of the product that the buyer (or society at
large) is willing and able to purchase at a given price in a given period of time
- Willingness and ability both important, e.g. pint of beer, flat in South Kensington

- ‘…in a given period of time’ also important

The good’s price is not the only determinant
- Demand can also depend on…
• number and price of substitute goods e.g. tea/coffee, bus/tube
• number and price of complementary goods e.g. cars-gas, printers-ink
cartridges
• level and distribution of income, e.g. flat in South Kensington
• consumer’s expectation with regards to the future
• consumer’s tastes and habits, e.g. chocolates, tobacco
- Alternative determinants for supply:
• Changes to the overall cost of production
• Change in prices of inputs to production
• Change in technological capabilities
• Organisational change


- Both supply and demand may also change over time
Demand: e.g. tapes-> CDs -> Minidiscs -> MP3s
Supply: e.g. cheaper production due to tech advances

Law of Demand: Ceteris paribus (everything else being equal), an increase in price will
usually lead to a dropp in demand.
- This is often linked to either the income effect or the substitution effect:
• A rise in price results in a decrease in the consumer’s purchasing power:
their income no longer covers the same quantity of the good in question.
(income effect)
• A rise in a good’s price may result in consumers substituting it for a
similar, less expensive good.
(substitution effect)
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Law of Supply: Ceteris paribus, an increase in price will lead to an increase in supply.
- This is because higher price (per unit) will incentivise greater production. It may
be, e.g., that a manufacturer produces more than one product, and in order to
optimise their profit, they have to split resources according to the revenue
earned by each product. If this revenue split changes, so must the resource split,
and therefore the amounts being produced.

In a market economy, the price of a good is determined according to its supply and demand,
through the price mechanism:
- If supply exceeds demand, price drops as the producers compete to sell the
good. This acts to encourage demand via the Law of Demand. (addressing the
good’s surplus)
- If demand exceeds supply, the price increases as consumers compete with each
other to obtain the good. This acts to incentivise production, as per the Law of
Supply. (addressing the shortage of the good)

This interaction of the price mechanism with the Laws of Supply and Demand means that
changes in both supply and demand will both cause and be caused by changes in the price
of the good. This relationship determines the equilibrium price of the product; where supply
and demand are equal.

Supply and demand curves

In mathematical terms, the demand and supply can be considered as functions and
mapping a price ∈ [0,∞) to some level of demand () or supply (). It depends on the
good of interest if these functions are integer-valued (discrete) or real-valued (continuous).
If they can be inverted, their inverses are referred to as inverse demand and inverse supply.
As such, they map from ℕ or ℝ to [0,∞).
Note that we ignore the fact that prices are also reported in discrete units and treat the
price variable as a continuous quantity.

It is often convenient and revealing to analyse the supply and demand function graphically.
For historic reasons (due to the economist Alfred Marshall) we use the convention that
prices are depicted on the vertical axis and quantities on the horizontal axis. For trained
mathematicians, this praxis is rather counter-intuitive. However, since the convention to do
so is pervasive in the economic literature, we shall stick to it in this course.
The graphs of the supply and demand functions are referred to as supply and demand
curves.

If one is to analyse stylised facts rather than precise quantitative results, one commonly
uses linear functions for supply and demand for the sake of simplicity.






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Examples of supply and demand curves:



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A good with more close subs will have higher |!|
The higher the proportion of income spent on a
good, the greater the elasticity
Analysing the supply and demand curves:
It is of interest to economists to characterise the sensitivity of a product’s supply or demand
to shifts in its price; the measure of such sensitivity is known as the price elasticity of
supply/demand.
Example: a) Change in demand for a pint of bear if price changes from £3 to £4.
b) Change in demand for a car if price changes from £30000 to £30001.
In general, the elasticity of a quantity refers to the relative magnitude of its reaction to a
change in any variables on which it depends.

Consider the demand for a product, which depends on its price:
- if the demand for a product is fairly resilient and robust to price changes, then it
is inelastic: demand changes relatively little for a given change in price
- Conversely, if the demand is particularly sensitive to changes in the price, then it
is said to be elastic.

More rigorously, we denote and define the price elasticity of demand to be
ϵ! = ϵ!() = % (or proportional)change in quantity demanded% (or proportional)change in price

= () /()1/
= () () = ln(exp(ln())) ln

Note that generally ϵ! < 0, as demand is decreasing in .

- For a good with elastic demand at price , if |!()| > 1
- For a good with inelastic demand at price , |!()| < 1
- If |!()| = 1, then the good is said to have unit elastic demand at price .


Determinants of ε!:
- Number and closeness of substitute goods
- Proportion of income spent on the good
- Time period

We define the price elasticity of supply " similarly; this measures the sensitivity of a good’s
supply function to changes in the good’s price.
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In addition, we can consider:
- the income elasticity of demand (supply), which measures the sensitivity of a
good’s demand (supply) function to changes in the consumer’s income; and
- the cross-price elasticity of demand (supply), which measures the sensitivity of
one good’s demand (supply) function to changes in the price of another good.


Demand, Price and Revenue

The revenue generated by a particular good is simply defined as the product of its price and
quantity demanded () = () ⋅ .

Since the demanded quantity of a good is inversely related to the good’s price, an increase
in price will not necessarily increase the revenue generated.

When will an increase in price result in an increase in revenue? How is this linked to the
price elasticity?




Consider the derivative of with respect to :
() = () + () > 0 ⟺ − () () < 1 ⟺ −ϵ!() < 1

So an increase in price will lead to an increase in revenue when demand is inelastic.
When demand for a good is elastic, revenue is increased by decreasing the price.



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Theory of the Firm

Production Functions, Cost, Revenue and Profit

The principle aim of a firm is to turn various inputs, such as raw materials and labour, into
output that can then be sold, ideally for profit. Inputs to the production process are referred
to as factors of production, and are broadly split into four categories:

Labour Land Raw Materials Capital

The first three are self-explanatory; capital requires some explanation:

- Broadly speaking, capital refers to those inputs to production that may be
consumed now, but that will deliver greater overall value to the firm if
consumption is deferred.

- Capital goods are those inputs to production that are themselves produced
goods, and which are durable, such as machinery.

- Capital finance refers to the financial assets of a firm that are themselves used to
generate wealth; it differs from ‘money’ in general in that it is not used to
purchase consumable goods and services.

We start by considering constraints that might be placed upon a firm’s production
capabilities; only certain combinations of input and output quantities may be
technologically feasible, and so these are referred to as technological restraints, and the set
of all inputs and outputs that satisfy such restraints are referred to as the production set.

We will denote the vector of input factor quantities as ∈ ℝ#$% and the vector of output
quantities as ∈ ℝ#$& ; thus the production set is the collection of vectors U, V ∈ℝ#$% × ℝ#$& such that ≤ (), for some production function . The production function,
also known as the technology of the firm, prescribes the maximum level of output for a
given level of input .

For given ∈ ℝ#$& , the set of all points ∈ ℝ#$% , such that (', (, … , %) =
is known as an isoquant; this is the combination of inputs that can produce the given .
Mathematically speaking, this is the pre-image )'({}).

We will mostly consider single-output cases, = 1, but the methodology extends to ≥2. Similarly, we will usually consider ∈ ℝ(, but mostly for illustrative convenience.
When ∈ ℝ#$, the input requirement set is the set of all vectors that produce at least ,
that is )'([,∞)).

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We now consider three examples of production functions that are often used in
microeconomic analysis.



Leontief Technology (aka fixed proportions/perfect complements)
Suppose we have two inputs ' and (: the Leontief production function takes the
form (', () = min(', () , , ≥ 0


Example
- Output: car, inputs: engine, wheels



Perfect Substitutes: (perfect complements)
Suppose, in contrast, we have inputs to production that can be easily substituted for
one another without affecting the level of output.
(', () = ' + (, , ≥ 0


Example
- Output: coffee, input: milk or soya milk



Cobb-Douglas Technology:
Suppose we have two inputs, ' and (: the Cobb-Douglas production function takes
the form (', () = '*(+ , , , ≥ 0















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Properties of production functions / the input requirement sets:

Monotonicity: if some input is increased, then the maximum output will be at least as great
as it was before ∗ ≥ ⟹ ('∗, … , %∗) ≥ (', … , %)
This reflects the property of free disposal: if a company can dispose of an input without it
costing anything, then increasing the inputs available can only be a benefit.

Convexity of the input requirement set: If for some , (', () and (′(, ′() are both in the
input requirement set, so will be their weighted average. This allows a producer to mix
production techniques.
The convexity of the input requirement set is equivalent to the quasi-concavity of the
production function .


Long-run and Short-run

Broadly speaking, in the economic literature, analysis is split into two scenarios, considering
the behaviour of the firm or individual in either the short-run or the long-run. These are
inexact periods of time, defined implicitly by the number of production inputs ', … , % that
may vary within such a timeframe: in the long-run, all inputs may vary, whereas in the short
run, at least one input will be held constant.

The Marginal Product

Question: how much can we increase output by varying the input factors?

Suppose we are operating with an element (', (, ) in the production set of and
we wish to obtain a level of output - > by increasing '…

- The marginal product of factor is defined as .(', () = (', (). , = 1,2

There is a slight issue with the marginal product – it is dependent on the units used to
measure the input quantities and also on the unit of the output quantities.

In order to measure the marginal effect of increasing each input independently of its units,
we turn to the output elasticity with respect to each input.

For a production function : ℝ#$% → ℝ#$, the output elasticity with respect to input . is
defined as the ratio of the relative change in output to the relative change in input: .(', () = (', (). .(', () (= ln (', () ln . = 1,2)

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Substitution

How about if we wish to keep the same level of output, but use less of a particular input
factor? Graphically, this is simply the act of moving along an isoquant.
We are interested in the rate of change of input ( with respect to ', in order to keep the
output constant. This is the Marginal Rate of Technical Substitution (MRTS), or the
technical rate of substitution.
Mathematically speaking, for a fixed ∈ ℝ, we are interested in the derivative of a function /, where / is implicitly defined as
/(') = ( ⟺ (', () =

The implicit function theorem asserts that such a / exists at least locally. Moreover, if is
a '-function, then also / is a '-function.
Then we obtain
U', /(')V =

Differentiation with respect to ' yields
' U', /(')V + ((', /(')) /′(') = 0.

This implies that, assuming ( U', /(')V ≠ 0,
/- (') = −' U', /(')V( U', /(')V.





We define the MRTS of a production function to be
(', () = −'(', ()((', ().

Reasoning: Start with the pair (', () and determine (', () = . Then proceed with the
rationale from above, choosing / .





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Consider the behaviour of the marginal product as a function of ', … , %: since the
production function is nondecreasing, we have that the marginal product is nonnegative,
and it is common to assume that it is nonincreasing. In other words, increasing an input '
from 100 to 101 is likely to result in a smaller increase in production than if we were
increasing ' from 1 to 2. This is known as the law of diminishing marginal productivity,
and holds ceteris paribus. This is also referred to as the law of diminishing returns, not to be
confused with decreasing returns to scale (later).
Similarly, it is common to assume that a firm’s production has diminishing marginal rate of
technical substitution: if we consider substituting factor ( for factor ' (i.e. decreasing '
and increasing ( such that the output is fixed), the larger the value for ' (before
substitution), the smaller the absolute value of the MRTS: in other words, the absolute value
of the slope of the isoquant must decrease as ' increases.


Example: (', () = √'(. Then we get (', () = − 0!0"









Returns to Scale:
We have considered the effects on the production function of increasing individual factors
whilst keeping others fixed, and we have considered the effect of substituting one factor for
another whilst keeping the output level fixed. We now consider the effect on the production
function of scaling all variable input factors by the same constant, keeping the ratio of
inputs fixed.
That means for some production function : ℝ#$% → ℝ#$ and ∈ ℝ#$% you should consider
the behaviour of the function ℝ#$ ∋ → rs.



- The most common scenario is that of constant returns to scale, and this is where
an increase in all inputs results in a proportional increase in the output:
rs = rs ∀ > 0 ∀ ∈ ℝ#$%

This is considered to be the most common scenario as, in most scenarios, the
production process can simply be replicated: if we double all input factors (land,
capital, labour and raw materials) then the production process can simply be
duplicated.

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- In some scenarios, increasing returns to scale may be observed; this is where
rs > rs ∀ > 1 ∀ ∈ ℝ#$%



- Decreasing returns to scale refers to the case where rs < rs ∀ > 1 ∀ ∈ ℝ#$%




Decreasing returns to scale can only happen when one or more inputs cannot be
varied. In reality, all inputs are variable in the long run – so decreasing returns to
scale is a short-run phenomenon


Example – returns to scale of the Cobb-Douglas production function.







In order to fully characterise the scalability of a firm’s production process, we wish to find a
quantitative measure of the returns to scale; we turn again to the use of an elasticity
measure. ℎ() = ()
We are interested in the reaction of to percentage changes in when we start with = 1.
If is differentiable, we can define its elasticity of scale at as
rs = ℎ() ℎ()y12' = rs ()y12' =
rs y12'() = zrs, |rs

That means rs is a local measure of the scale behaviour.
If is continuously differentiable, one can show that
(i) rs ≥ 1 for all ∈ ℝ#$% if and only if rs ≥ rs ∀ ≥ 1 ∀ ∈ ℝ#$% .
(ii) rs ≤ 1 for all ∈ ℝ#$% if and only if rs ≤ rs ∀ ≥ 1 ∀ ∈ ℝ#$% .
(iii) rs = 1 for all ∈ ℝ#$% if and only if rs = rs ∀ > 0 ∀ ∈ ℝ#$% .
The idea of the proof is to consider for ∈ ℝ#$% the function 0: (0,∞) → ℝ, 0() = rsrs.


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Additional potential properties of the production function

- Homogeneity: : ℝ#$% → ℝ is positively homogeneous of degree ∈ ℝ if rs = 3rs ∀ > 0 ∀ ∈ ℝ#$% .
This has obvious links to the returns to scale: e.g. displays increasing returns to
scale if and only if is homogeneous of degree > 1.

- Homotheticity: : ℝ#$% → ℝ is homothetic if there is a positively homogeneous
function ℎ: ℝ#$% → ℝ and a strictly increasing function : ℝ → ℝ such that rs = UℎrsV ∀ ∈ ℝ#$% .


Homogeneous and homothetic production functions are useful modelling scenarios
as they prescribe isoquants that vary simply for differing levels of output. A
particularly useful property of such technologies is that, for each, the corresponding
MRTS is independent of the scale of production.





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Profit and profit maximisation
So far, we have only considered production functions and the influences of different factors
to the production. However, what is the ultimate reason and motivation for a firm to
produce products at all?
Most microeconomic analyses assume that the firm is profit driven – we will too!

The profit of a firm is simply its revenue minus its costs, where all costs of the firm are taken
into account. It is often easy to overlook costs (e.g. labour costs for a self-employed person).

In application, the decision of how to maximise profit comes down to deciding how much
output to produce and at what price, or how much input to buy and at what price. We
therefore construct the profit function in terms of these variables:
(, , ) = rs4 − 4 (, , ) = rs − 4 (single-output case)

We will assume the conditions of a purely competitive market, i.e. where all firms are
assumed to be price-takers (their actions have a negligible effect on the prices).
In such a scenario, one need only determine the quantities of inputs/outputs in order to
maximise profits.

Treating this as an unconstrained optimisation problem then, the first-order conditions for
to be maximised can be found straightforwardly:
∀ = 1,… , ∶ rs. = 0 ⟺ ƒ5 5rs.&52' = .

This first-order condition is often referred to as the fundamental condition of profit
maximization:
- Profit is maximised if marginal revenue is equal to marginal cost
- Also stated as: ‘the value of the marginal product wrt a factor is equal to its price’.
For fixed U, V, this condition is easily solved to provide a necessary condition for
the profit-maximising input vector ∗, which can then be used to establish the profit-
maximising output.

So for the single-input case, the second-order condition for profit maximisation is that --(∗) ≤ 0 (necessary) or --(∗) < 0 (sufficient).


For ∈ ℝ#$% , > 1, the corresponding condition is that the Hessian of ↦ rs4 is
negative semidefinite (necessary) or negative definite (sufficient) at ∗.


Quite often, this (local) concavity of the production function is part of our assumptions.
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There are some caveats with this canonical procedure:

1. The production function might not be differentiable, e.g., in a Leontief technology.
We need to be careful about that.
2. Some of the input variables might be discrete and not continuous. So that means we
need a discrete minimisation here.
3. We might have boundary solutions (usually at 0 meaning that it is optimal not to use
a certain factor in the production at all).
Example: () = and > .
4. A best strategy might not exist (e.g. because the profit function is not bounded).
Example: () = and < .
5. The optimal strategy might exist, but is not unique.
Example: () = and = .













For a specific production function : ℝ#$% → ℝ#$& we introduce the map
∗ ∶ ℝ#$& × ℝ#$% → 2 ℝ#$%

yielding the / an optimal specification of input quantities, given a price ∈ ℝ#$& and prices
for input factors ∈ ℝ#$% . That is
∗ U, V = argmax0∈ ℝ#$% (, , ) = argmax0∈ ℝ#$% rs4 − 4.


In the light of the caveats mentioned above, our standard assumption is that the optimal
input specification exists and is unique. Under this assumption, the values of ∗ are
singletons and we can identify the singletons with their unique element.
Consequently, we can consider ∗ as a function – the factor demand function – being a map ∗ ∶ ℝ#$& × ℝ#$% → ℝ#$% .

Recall that we are in the setting of a competitive firm such that the firms are price takers
and r, s can be considered as exogenous variables.
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Under this assumption we can also define the output supply as the function
∗ ∶ ℝ#$& × ℝ#$% → ℝ#$& U, V ↦ ∗ U, V = …∗ U, V†




Moreover, we can define the profit function
∗ ∶ ℝ#$& × ℝ#$% → ℝ
U, V ↦ ∗ U, V = U∗ U, V , , V = max0∈ ℝ#$% (, , ) = max0∈ ℝ#$% rs4 − 4



Properties:
1. The factor demand function is positively homogeneous of degree 0. ∗ U, V = ∗ U, V ∀ > 0 ∀ U, V ∈ ℝ#$& × ℝ#$%
2. The profit function is positively homogeneous of degree 1.
3. The profit function is non-decreasing in and non-increasing in .
4. The profit function is convex.
5. Under some regularity assumptions, the profit function is continuous.

Proof: 5. Follows with the Berge Maximum Theorem. The other assertions are exercises.

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Hotelling’s Lemma (special case of the Envelope Theorem)

The output supply and factor demand functions can be obtained directly from the
maximised profit function through partial differentiation with respect to the price vector.
Recall ∗ U, V = …∗ U, V†4 − ∗ U, V4
∗ U, V5 = 5 …∗ U, V† +ƒ3&32' ƒ.3 …∗ U, V† .∗ U, V5%.2'− ƒ. .∗ U, V5%.2' = 5 …∗ U, V† +ƒ %.2' ‡ƒ3.3 …∗ U, V†&32' −.ˆ‰ŠŠŠŠŠŠŠŠ‹ŠŠŠŠŠŠŠŠŒ$
.∗ U, V5
= 5 …∗ U, V†

∗ U, V5 = ƒ3&32' ƒ.3 …∗ U, V† .∗ U, V5%.2' − 5∗ U, V −ƒ3&32' 3∗ U, V5 ƒr∗s.∗ .∗5%.2' − 5∗ −ƒ. .∗5%.2'
= − 5∗ U, V +ƒ %.2' ‡ƒ3.3 …∗ U, V†&32' −.ˆ‰ŠŠŠŠŠŠŠŠ‹ŠŠŠŠŠŠŠŠŒ$
.∗ U, V5 = − 5∗ U, V


One convenient corollary of Hotelling’s Lemma is that each row of the maximised profit
function Hessian (wrt ) is equivalent to the gradient of the firm’s supply function for the
corresponding output. This makes it easier to derive some properties of profit-maximizing
firms, such as…

The LeChatelier principle:

The long-run supply response to a change in price is at least as large as the short-run supply
response. Essentially, it states that .∗ U, V. ≥ 0.
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Weak Axiom of Profit Maximization (WAPM):

So far, we have started with a given production function and we have derived some
results about the factor demand function or the output supply function. Now, we turn the
perspective and assume that we can observe a firm’s ‘behaviour’. The Weak Axiom of Profit
Maximization (WAPM) is a necessary condition for the rational, i.e. profit maximizing
behaviour of that company.
In a first step, we can use the WAPM to check if a company is profit maximizing by checking
if the observed dataset satisfies the WAPM.
In a second step, we can even consider some attempts of statistical inference for the
production function using a dataset that satisfies WAPM. The details for the second step will
be the content of some assignment.


Suppose we observe the net output vectors 1 = U1 , −1V ∈ ℝ#$& × ℝ#$% and their
corresponding price vectors 1 = U1 , 1V ∈ ℝ#$& × ℝ#$% for some firm at discrete time
points = 1,… , . Assuming that the firm is acting to maximise profits, we can deduce that
1r1s4 ≥ 1r9s4 ∀, = 1,… , .
Where we index via superscripts as subscripts are reserved for indexing vectors
This is the Weak Axiom of Profit Maximisation.
⟹ 1r1 − 9s4 ≥ 0 −9r1 − 9s4 ≥ 0

Writing WAPM with indices switched yields as a corollary:

r1 − 9sr1 − 9s4 ≥ 0 ∀, = 1,… , .

Some comments are in order:
- Can be related to infinitesimal second order conditions for profit maximization,
derived using the profit function. Note change in sign due to input elements in
being −.

- Constructed using only the definition of profit maximisation, no assumptions
about () have been made

- Valid for any price change, not just infinitesimal price changes.

- WAPM fully characterises a profit-maximising firm, i.e. for all datasets that satisfy
WAPM, a profit-maximizing production function can be found.

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Cost Minimisation

We previously looked at a direct, unconstrained approach to profit maximisation: given
fixed costs for our inputs ∈ ℝ#$% and a fixed price for our output ∈ ℝ, what production
choices lead to maximum profit? In non-competitive markets, however, since the output
price ∈ ℝ is not necessarily fixed, it is useful to split this into two constituent problems:

- For fixed ∈ ℝ#$% , what is the minimum total cost to the firm of producing a
level of output ?
- Given this knowledge, what is the most profitable level of output?

We consider the first part of this problem here.
Note that, for = 1, this is trivial; we therefore assume here that ≥ 2.

For input vector ∈ ℝ#$% with associated prices ∈ ℝ#$% , we are interested in solving the
optimisation problem:

Find argmin0∈:&"({/}) 4

To solve this constrained optimisation problem, we convert it into a Lagrangian problem; we
incorporate the constraint into the objective function, and subsequently treat it as an
unconstrained minimisation.
First, define the Lagrangian, ℒ:
ℒ ∶ ℝ#$% × ℝ → ℝ, ℒ(, ) = 4 − rrs − s


Then, find the first-order conditions for minimising ℒ:
(+1 unknowns, +1 FOCs) ℒ(, ). = . − .rs = 0, = 1,… , (, ) = rs − = 0

Solve these +1 equations for the +1 unknowns ', … , %,
.rs = . , = 1,… ,

rs =

The conditions on . for cost-minimisation look reassuringly similar to those obtained for
competitive profit maximisation; they are not the same however. Here, is simply a dummy
variable, which we must get rid of if we are to make any progress; we cannot solve explicitly
for . in terms of known prices.

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If has a known, differentiable form…

- Use solved condition to find in terms of and .

- Substitute into the constraint and rearrange to find in terms of and .

- Re-substitute into the solved condition to find in terms of and .

Thus, we can determine the level of each input factor in terms of the input factor prices and
the desired level of output; these relationships constitute the conditional factor demand
function, which we denote ∗r, s. Formally, this is a function
∗ ∶ ℝ#$% × ℝ#$ → ℝ#$% , r, s ↦ argmin0∈:&"({/}) 4


The minimum total cost to the firm of producing a level of output with input prices can
subsequently be obtained as the price-weighted conditional factor demand functions:
∗ ∶ ℝ#$% × ℝ#$ → ℝ#$% , ∗r, s = min0∈:&"({/}) 4 = ∗r, s4

Example: Cobb-Douglas











We note that the first-order conditions above can be restated in terms of the marginal rate
of technical substitution:
('∗, (∗) = −'('∗, (∗)(('∗, (∗) = −'(


That means the marginal rate of technical substitution coincides with the economic rate of
substitution.



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Possible problems when finding the conditional factor demand function:

1. The production function might not be differentiable, e.g., in a Leontief technology.
We need to be careful about that.
2. We might have boundary solutions (usually at 0 meaning that it is optimal not to use
a certain factor in the production at all).
3. If the production function is continuous, surjective on [0,∞) and ≫ 0, there is
always a cost minimizing strategy (in contrast to the possible problems with profit
maximization:
The objective function ↦ 4 is continuous.
A continuous function attains a minimum and a maximum over a compact set.
Since is continuous, the pre-image )'({}) is closed.
We can just pick some arbitrary element ′ ∈ )'({}) (if the preimage is non-
empty) and restrict attention to the set
)'({}) ∩ { ∈ ℝ#$% | 4 ≤ ′4}

which is a bounded and closed set (and hence compact).

4. The optimal strategy might exist, but might not be unique.







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Properties of the cost function

The cost function ∗r, s is…
- Nondecreasing in : - ≥ ⇒ ∗r-, s ≥ ∗(, )

- Homogeneous of degree 1 in ∗r, s = ∗r, s ∀ > 0

- Concave in ∗r + (1 − )-, s ≥ ∗r, s + (1 − )∗r-, s ∀ ∈ [0,1]

- Continuous in for ≫ 0
For any $ ≫ 0 ,
∗($, ) exists, and is equal to the limit lim?→?$ ∗r, s.
Proof: Exercise.


Shephard’s Lemma (another application of the Envelope Thm)

We can obtain the conditional factor demand functions from the cost function through
differentiation with respect to the price vector ; this is Shephard’s Lemma: if ∗r, s is
differentiable at r, s, and . > 0 for = 1,… , , then .∗r, s = ∗r, s.
“Alternative” proof:
ie not a function, but a vector
Let - be a cost-minimizing bundle of goods that produces at prices - and define rs = ∗r, s − -4
Since ∗ is minimized for the argument , rs ≤ 0 for all possible .
In particular, we note that r-s = 0.
Since this is the maximum, we must have rs. y?2?' = 0
for = 1,… , .
Thus, ∗r, s. y?2?' = .-
as required.

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Weak Axiom of Cost Minimisation (WACM)

We can make very similar considerations as in the case of the Weak Axiom of Profit
Maximisation. That means the Weak Axiom of Cost Minimisation (WACM) gives a necessary
condition on data to stem from a cost minimising (and thus rationally operating) firm.

Assume we have observations of prices 1 ∈ ℝ#$% , inputs 1 ∈ ℝ#$% and outputs 1 ≥ 0 at
time points = 1,… , . Then the WACM states that
1r1s4 ≤ 1r9s4 ∀, = 1,… , such that ≤ .

This implies as a corollary that

r1 −9sr1 − 9s4 ≤ 0 ∀, = 1,… , such that = .



Note: If all prices but . are held constant, then in order to minimize costs whilst keeping
output constant, the change in . must be in the opposite direction to the change in ..


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Long-run vs. short-run costs

We have considered cost minimisation under the assumption that all of our inputs to
production are allowed to vary in quantity – recall that this is a long-run scenario. In the
short run, at least one factor will remain fixed.
Let , ⊆ {1,… , } be index sets with ∪ = {1, … , } and ∩ = ∅. Here, the set
consists of the indices of variable short-run factors and comprises the indices of the fixed
long-run factors.
Then, for ∈ ℝ#$% we shall write = rC , Ds ∈ ℝ#$C × ℝ#$D .

The level of the fixed factor will influence both the minimised cost, given by the short-run
cost function
9∗ ∶ ℝ#$% × ℝ#$C × ℝ#$ → ℝ
9∗r, C , s = min0(∈ℝ#$( EFGH IHJI :K0),0(M2/ 4
= C C4 + min0(∈ℝ#$( EFGH IHJI :K0),0(M2/D D4

…and the cost-minimising choices of the variable factors, given by the short-run conditional
factor demand functions
9∗ ∶ ℝ#$% × ℝ#$C × ℝ#$ → ℝ#$D

9∗r, C , s = argmin0(∈ℝ#$( EFGH IHJI :K0),0(M2/ 4 = argmin0(∈ℝ#$( EFGH IHJI :K0),0(M2/D D4


Note that
∗r, s ≤ 9∗r, C , s

We cannot establish a similar relation between the short-run conditional factor demand
function and its long-run version.

Consider examples of firms with specific variable and fixed production factors:
- Coal-fired power station
- Hydroelectric power station
- Newspaper
- Somebody folding paper aeroplanes
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Notational convention – ignore
dependence on e.g.
Average and marginal costs

The cost function is used to gain insight into the economic capabilities of the firm; indeed,
much of the firm’s economic behaviour can be gleaned from ∗r, s. It is particularly
important to be able to analyse the behaviour of the cost function as the level of changes,
and so we now define a series of derived quantities for both short-run and long-run
analyses.

Assuming the costs = (C , D) to be fixed, we shall supress the costs in the notation and
we define the short-run average cost () to be the per-unit cost of producing units of
output: () = 9∗r, C , s


Assuming that the firm is cost-minimising, D = 9∗r, C , s
() = C C4‰‹Œ"NCO(/) +D 9
∗r, C , s4‰ŠŠŠŠ‹ŠŠŠŠŒ"NDO(/)

where SAFC = short-run average fixed costs and SAVC = short-run average variable costs.

It is also useful to consider the rate at which a firm’s costs increase (or decrease) with
respect to its output; the short-run marginal cost () is defined as
() = 9∗r, C , s .

In the long-run, we have only variable input factors, i.e. = {1,… , }, = ∅, leading to =D, therefore the long-run average and marginal costs are defined accordingly: () = ∗r, s , sometimes written () () = ∗r, s , sometimes written ()


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Geometry of costs

The shape of the average and marginal cost curves can be illuminating, and indicative of the
economic capabilities of a firm with a given production function. Suppose we are operating
in the short run.

As increases, SAFC will clearly decrease – what will happen to the variable costs?

Law of diminishing marginal productivity (valid for the variable component of the short-run
analysis) states that, as inputs increase, the marginal productivity of each subsequent unit of
input will decrease.

This is because increasing variable inputs are combined with a given quantity of fixed inputs
(e.g. increasing hole-diggers but keeping one shovel)
This implies that as we increase output, the input required to maintain a unit increase in
output will itself increase; thus, the per-unit variable costs will increase as y increases, i.e.
SAVC increases as y increases (note that SAVC may be initially decreasing in y, but law of dim
marg prod means that eventually, it will be increasing)
So, roughly,























SAFC SAVC SAC
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Consider the minimum of the SAC - where does this occur?
() =  9∗r, C , s ¡ = ¢
"∗r, C , s − "∗r, C , s£( () = 0 ⟺ "∗r, C , s = "∗r, C , s




…so we have that, at its local minimum, the average cost curve is intersected by the
marginal cost curve. Similar analysis reveals that:

- () < () ⟺ short-run average costs are decreasing in
- () > () ⟺ short-run average costs are increasing in

The link between the average and marginal costs in the short-run can be further probed;
consider how each behave at = 0?
() = C C 4 ‰Š‹ŠŒ→P
¤¥¦¥§"NCO +D 9∗4r, C , s‰ŠŠŠŠ‹ŠŠŠŠŒ…?
¤¥¥¥¥¦¥¥¥¥§"NDO
Consider the behaviour of the variable costs as → 0: lim/→$D 9∗4r, C , s
= lim/→$⎩⎨
⎧ D 9∗4r, C , s 1«⎭⎬
⎫¤¥¥¥¥¥¥¥¦¥¥¥¥¥¥¥§"SO(/) by l’Hopital’s rule.
(limit of a ratio where both numerator and denominator tend to 0 is equal to the
limit of the ratio of corresponding derivatives)




So, as → 0,
- the short-run average costs explode in the presence of fixed costs…
- but the short-run average variable costs and short-run marginal costs are equal

It shouldn’t be surprising that the link between average and marginal costs is chiefly through
the variable costs…the fixed costs do not contribute to the marginal costs! Indeed, we can
further note:
- The area under the marginal cost curve (MC) will give the total variable costs –
this is a straightforward result of the fundamental theorem of calculus.
- As we saw above with the short-run average and marginal costs,
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• () = () at the local minimum of ()
• () ≷ () ⟺ () is increasing/decreasing in
Two options for SMC(y) and SAVC(y):





We have previously defined the short-run and the long-run according to the ability to vary
the factors of production. We have considered the long-run to be the period of time in
which all factors can be varied. We revisit the notions of fixed and variable factors, and
consider their role in short-run and long-run cost analyses.

Fixed costs are those costs that do not scale with the firm’s output. One cannot influence
the level of production through altering a fixed factor of production. Even if a firm was to
drop all output to = 0, fixed costs would still require payment.

In contrast, variable costs are dependent on the firm’s level of output, as the output is
influenced by changing the variable factors of production.

In the short-run, there are both fixed and variable factors of production, and thus also fixed
and variable costs. In the long-run, some of the fixed factors may be more easily varied, and
so can be used to influence the level of output. Thus, factors that are fixed in the short-run
are often variable in the long-run.

There may well be, however, some costs that are constant with respect to the level of
output even in the long-run, as long as the firm is producing a positive level of output (i.e. is
still in business); these are referred to as quasi-fixed costs, and they correspond to quasi-
fixed factors of production.

To summarise:
In the short-run, we have: Variable Costs + Fixed Costs

In the long-run, we have: Variable Costs + Quasi-fixed Costs

Consider, now, the long-run () and () curves. The existence of quasi-fixed costs
implies that the average and marginal cost curves will have a similar shape in the long run as
in the short-run.

Recall from before: (cf p21) ∗r, s ≤ 9∗r, C , s
We argued that fixed costs in the short-run can be varied, & thus optimised, in the long-run.
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i.e. short-run costs are always greater than or equal to the long-run costs. This still holds,
even in the presence of quasi-fixed costs; factors that are ‘fixed’ in the long-run will also be
fixed in the short-run.

Recall also, however, that ∗r, s = 9∗r, C∗r, s, )

i.e. for each level of output there will be an optimal level of the fixed factor given by its
conditional factor demand.



We conclude that the long-run () curve:
- has a similar shape to ();
- lies below or on the () curve at all points > 0;
- and is tangential to the () at the point ∗ for which C = C∗rC , ∗s.




Note that, as C varies, the point ∗ at which () = () will move. If the fixed
factors can vary continuously, then the curve will trace out the curve; we say that
the curve is the lower envelope of the curves. If C can only be varied discretely,
then this lower envelope will be tangential to each curve at more than one point:


NB: ∗ is not necessarily
the minimising point of (), i.e. doesn’t
necessarily lie tangential
to the at either of
their minima.

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In order to characterise the behaviour of the long-run marginal costs, we note that the
relationships that held between average and marginal costs in the short run will also hold in
the long run:

- () = () ⟺ long-run average costs are at a (local) minimum

- () < () ⟺ long-run average costs are decreasing in

- () > () ⟺ long-run average costs are increasing in

We further note that if the short-run fixed factors C are fixed at their long-run conditional
factor demand for a given output, then
∀ ≥ 0 ∗r, s = 9∗r, C∗r, s, ) ⇒ (∗) = (∗).





The argument is as follows: ∗r, s = 9∗r, C∗r, s, ) holds for any . That means, we
can take the total derivative on both sides. That is
() = ∗r, s = ∗r, s = 9∗r, C∗r, s, ) = C 9∗r, C∗r, s, )‰ŠŠŠŠŠŠ‹ŠŠŠŠŠŠŒ2$ C∗r, s + 9∗r, C∗r, s, )‰ŠŠŠŠŠŠ‹ŠŠŠŠŠŠŒ2"SO(/)

The first summand is 0 since the function C ↦ 9∗r, C , s is minimised by C = C∗r, s.
So the first order condition implies that the partial derivative vanishes at this point.

With the second summand, one also needs to be cautious. This is only the SMC if C =C∗r, s which means that = ∗.

The argument is very similar to the one in the Envelope Theorem.












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Profit maximisation given minimised costs

We have now considered the choices that a firm must make to minimise its costs, given
knowledge of its factor prices and a given level of output . As mentioned previously, we
now consider how a firm should subsequently choose an optimal level of output in order
to maximise profits conditional on minimised costs.

To set up the profit-maximisation question in this conditional framework, we initially
maintain the assumption of perfect competition; we also assume to begin with that we are
operating in the short-run.

Recall that previously, profit maximisation was framed as a question of how much input to
use, and that the output of the firm was specified by the production function . Now, all of
the firm’s technical constraints are implicitly specified by the cost function.

We therefore reformulate our profit maximisation problem: we seek
argmax TU$ ³ − "∗r, C , s´
max TU$ ³ − "∗r, C , s´

First- and second-order conditions for the optimal level of output given minimised costs are
given by: U − "∗r, C , sV = 0 ⇒ = () ()
(( U − "∗r, C , sV ≤ 0 ⇒ ("∗r, C , s( ≥ 0 ()


These conditions suggest that, in order to maximise profits, the output should be such that
the corresponding short-run marginal cost is increasing and equal to the output price .

For a cost-minimising competitive firm, this specifies a relationship between the market-
defined output price and the quantity of output that the firm should provide – this is the
short-run supply curve for an individual firm!

Example 1:
Suppose that a firm’s short-run cost function for a good is specified as
9∗r', (, C , s = 2¶'( ( + rC , Cs
If the market price for the good is £16 and each input costs the firm £4, how many
units of the good should the firm produce in the short run, and what is their
maximised profit if fixed costs are £12?

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: () = 4¶'( = ⇒ ¸ = 4√'( = 1616 = 1 (9∗r', (, C , s( = 4¶'( ≥ 0, ¸ = 1
= ¸ − 9∗r', (, C , s = 16 ⋅ 1 − 2 ⋅ 4 ⋅ 1( − 12 = −4.


In the short-run, i.e. when there are fixed costs, the most profitable position for a firm may
be one that returns negative profit, as fixed costs will always require payment.

Example 2:
Consider the cost function 9∗r, C , s = ''( +(( + (C , C)
What is the maximised profit here, when ' = 2, ( = '(, and = 2? : () = '2 )'( + 2( = ⟹ V( − '( + 1 = 0 ⟹ '( = 1 '( = −1 + √52 ( ) ⟹ ¸ = 1 ¸ = 3 − √52 : ¸ = 1 ⟹ (∗( > 0 ; ¸ = 3 − √52 ⟹ (∗( < 0 ¸ = 1
Max profit is given by ¸ − 2¶¸ − '( ¸( − = − 2 '( −
So, since < 2 '(, even if fixed costs are zero, the firm makes a loss.


















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So, as illustrated in Example 2, in some circumstances it may be preferable for a firm to go
out of business rather than provide > 0. The minimum of the cost function is on the
boundary.

Indeed, we can generalise: it will be preferable to go out of business when (The LHS is the
profit when = 0) −C C4 > − 9∗r, C , s ⇔ 0 > − D D∗rD , s4 ⇔ () = D D∗rD , s4 >
This is known as the shutdown condition; when satisfied, it is preferable for the firm to go
out of business.
So we must refine our definition of the firm’s chosen short-run supply. The competitive
cost-minimising firm should choose a positive level of output such that:
- () = ;
- () is increasing in ;
- and () ≤

If no such > 0 exists for the given , then the firm should set = 0.

These conditions are satisfied by the portion of the curve that is increasing in and
that lies on or above the curve:











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In the long run, we have a very similar story. Neither the first- nor second-order conditions
above explicitly require the costs to be dependent on fixed factors of production; these
translate to the long-run scenario as would be expected. The long-run profit-maximising
supply for a cost-minimising firm is given by such that
- () = ;
- () must be increasing in ;
- () ≤ . Converse of the shutdown condition in the long-run

Once more, if no such > 0 exists for the given , then the firm should choose to go out of
business.

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Profit maximisation for a noncompetitive firm

To contrast, we consider the profit maximisation problem for a cost-minimising monopolist.
Whilst monopolists have more control over output prices than in a competitive market, they
cannot choose price and output independently of one another; they must respect the
market demand for their product. We therefore assume that the monopolist chooses the
amount of output to provide, , and the output price is determined according to the market
demand for this output, i.e. as a function of , ().
The other way is also valid (i.e. choose and consider ()), but this is more convenient.
The function () is the inverse of the market’s demand function and is referred to as the
inverse demand function “facing the firm”; we note that it may be dependent on other
determinants, but assume these to be held constant in our analysis.
(e.g. substitute goods or services from other markets, e.g. transport in London rail/bus/bike)

To maximise profits, we therefore seek argmax/#$ ³() − "∗r, s´
We’ve simply dropped C from the notation as it isn’t of much interest to us in optimising!
First- and second-order conditions for finding a profit-maximising position for a monopolist
facing an inverse demand function are therefore given by
U() − "∗r, sV = 0 ⇒ () + () = () ()
(( U() − "∗r, sV ≤ 0 ⇒ ("∗r, C , s()( ≥ (()()( + 2() ()

We can rearrange the first-order condition: () Á1 + () ()‰ŠŠŠŠ‹ŠŠŠŠŒ'W 'X*(/) = ()
where !() ≔ !(()) is the price elasticity of demand facing the monopolist.

Recall that () = )'(), such that = (()). Taking the derivative with respect to
yields ′() = 1′(()).

We note that, since !() < 0 (demand decreases with increasing price) and () ≥ 0
(costs should not decrease in ) we must have |!()| ≥ 1,

Otherwise violation of sign.
i.e. elastic demand facing the firm, in order to have a profit-maximising position.



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Example:
Consider the monopolist faced with a linear inverse demand () = ' − ( ', ( > 0, 0 ≤ ≤ '/(

and Cobb-Douglas variable costs in the short term: (CD with = = 1) "∗r, C , s = 2¶'(( + rC , Cs.
What is the maximum profit that this monopolist can achieve?

First, check the elasticity: 1!() = () () = − (' − ( < 0 ! < 0 ∴ |!()| ≥ 1 ⇒ '( − 1 ≥ 1 ⇒ ≤ '2(

So any profit-maximising level of output must be below *"(*!.


Now, solve the FOC: (¸) Á1 + 1!(¸)Â = (¸) ⇒ ' − 2(¸ = 4¶'( ¸ ⇒ ¸ = '2( + 4√'( ≤ '2(

Evaluating the SOC verifies that this is a maximum.

Substituting into the linear inverse demand curve gives the resulting price () and
allows us to calculate the maximised profit as '(4(( + 2√'() − rC , Cs

















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We can see from this example that it is also possible for profit-maximising monopolists to
experience losses in the short-run; this is not a phenomenon unique to competitive markets.

The above optimisation assumes that > 0. Just as for competitive firms, however, we note
that the profit-maximising (loss-minimising) position for a monopolist may be to go out of
business, i.e. to set = 0. This happens when the losses incurred by setting output
according to the above first- and second-order conditions are greater than the fixed costs,
i.e. when () > ()


We also note that, as for competitive firms, the extension to the long-run is trivial. For a
cost-minimising monopolist, the long-run profit-maximising output will satisfy the
following conditions:
- ()[1 + !)'()] = ()
- Y!Z∗K?,/MY/! ≥ Y![(/)(Y/!) + 2 Y[(/)Y/
- () ≤ ()

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Theory of the Consumer
We now focus on the theory of the consumer, where we will formalise the notion of
consumer preferences and show how optimal behaviour of the consumer with respect to
their preferences will lead to a specification of the demand function.
In the course of our analysis, we will see a lot of similarities and analogies to the Theory of
the Firm.

Preferences & Utility

We start by considering the goods consumed by a consumer.

Define the consumption bundle for a particular consumer to be the quantities of a collection
of goods that the consumer is willing to consume:
= (', (, … , %) ∈ ℝ#$% .

The set of possible consumption bundles is referred to as the consumption set; this is
usually taken to be some closed and convex set ⊆ ℝ#$% .

Consumers are assumed to have preferences between bundles , - ∈ :

- ≼ - means that the consumer has a preference for bundle ′ over bundle .

- ≺ - means that the consumer has a strict preference for ′ over . ≺ - ⟺ ≼ - ′ ≼

- ∼ - denotes indifference between and -. ∼ - ⟺ ≼ - ′ ≼

Mathematically speaking, the preference relation ≼ is a subset of the Cartesian product × .
We are working under the condition that the preference relation satisfies the three axioms
of a complete weak order on . That is

Completeness ∀ , - ∈ , ≼ - or - ≼

Reflexivity ∀ ∈ , ≼

Transitivity ∀ , -, -- ∈ , if ≼ - and - ≼ --, then ≼ --



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Beware that reflexivity actually follows from completeness.

In addition, the following assumptions are useful but not necessary:

Continuity ∀ ∈ , the sets ³-: - ≼ ´ and ³-: ≼ -´ are closed sets (wrt the Euclidean topology).

Weak / Strong Monotonicity
≤ - ⟹ ≼ - (weak)
(at least as much of everything is as least as good – “greed is good”) ≤ - ≠ - ⟹ ≺ - (strong)


Local nonsatiation

For all ∈ and for all > 0, ∃ - ∈ with Ò- − Ò < such that ≺ -.
(We can always do a little better, even with just small changes to )

(Strict) Convexity

Given , -, -- ∈ with ≼ ′ and ≼ ′′,
≼ - + (1 − )-- for 0 ≤ ≤ 1
...this is convexity.
Strictness….
If - ≠ --, ≺ - + (1 − )-- for 0 < < 1



One can easily verify that strong monotonicity implies local nonsatiation.



Note – we have not yet used the symbols ≽ or ≻; we can use this as would be expected, i.e. ≼ - ⟺ - ≽

but it is no more than a notational convenience.



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How does a consumer decide between bundles in some subset of ? How do we judge the
suitability, or usefulness, of a consumption bundle ? More to the point, how can we, as
economists, model the unobserved preference allocation of consumers? … and do sensible
forecast about consumers’ behaviour?

It is useful to model consumer preferences by a utility function, which we define to be a real
mapping : → ℝ.
We say that represents the preference relation ≼ if
r-s ≤ rs ⟺ - ≼ .

For this reason, (⋅) is sometimes referred to as the ordinal utility.

- If only the ordering imposed by a utility function is relevant, one speaks of an
ordinal utility. If is an ordinal utility, any strictly increasing transformation of
represents the same preferences.
- If one wants to compare different utility differences, say Õr-s − rsÕ Õr-′s − rsÕ,

one speaks of a cardinal utility. Cardinal utilities are in general only preserved by
affine and increasing transformations.



Existence of an (ordinal) utility function:

Suppose a consumption set is imbued with a preference relation that is complete,
transitive, continuous and strongly monotonic. Then there exists a continuous utility function ∶ → ℝ that represents this preference relation.

Note – the assumption of strong monotonicity can be dropped, though the proof is more
complex. This more general case is known as Debreu’s Theorem (1954).

Proof:
Outline:
- We will consider bundles of goods that contain the same amount of each good,
i.e. ‘homogeneous’ bundles;
- We will show that if, for every ∈ , there exists a homogeneous bundle to
which the consumer is indifferent, then the level of the homogeneous bundle
can be taken as an appropriate utility function, i.e. one that preserves the
ordering of ≽;
- We will then show that such a homogeneous bundle exists and is unique.
§ We show uniqueness of the chosen homogeneous bundle
§ This is not the same as uniqueness of the utility function
- Continuity is beyond the scope of the course


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Let = (1,… ,1) be an -length vector of ones.

For any consumption bundle ∈ , suppose there exists some value rs ∈ ℝ such
that rs ∼ .
For , - ∈ ,
rs > r-s ⟹ rs > r-s ⟹ rs ≻ r-s (by strong monotonicity)
⟹ ≻ - (by transitivity)
Similarly rs ≤ r-s ⟹ ≼ -.

So rs maintains the ordering of preferences as required.

To proceed, we seek to prove the existence, uniqueness and continuity of the function rs.
Existence:
- Let = ³ ∈ ℝ ∶ ≽ ´ = ³ ∈ ℝ ∶ ≼ ´;
the required value will be in both sets.
- Umax. .V ≥ ⟹ Umax. .V ≽ by monotonicity, so is nonempty

- 0 ∈ , so is nonempty.

- By continuity of ≽, and are both closed; we consider the lower bound of
and show that it is also the (inclusive) upper bound of .

- Since is nonempty and closed, set ∗ ≔ inf ∈ , and let % = ∗ − '%
• % < ∗ ⇒ % ∉ (as ∗ is a lower bound) ⇒ % ≺ ⇒ % ∈
• Also, % → ∗ as → ∞, and is closed, so ∗ ∈ .

- Since ∗ ∈ and ∗ ∈ , ∗ ∼ , so rs = ∗, and therefore exists.

Uniqueness:

• Suppose ∼ 'rs, and suppose also that ∼ (rs

- By transitivity, 'rs ≽ ≽ (rs ⟹ 'rs ≽ (rs

⟹ 'rs ≥ (rs by strict monotonicity (see below)

- Similarly, 'rs ≤ (rs, so 'rs = (rs.

For strictly monotonic preference relation ≽,
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≥ ⟺Ù/\9,]\*^^/ ≽



Proof of continuity is Debreu’s Theorem (1954) – “Representation of a Preference
Ordering by a Numerical Function”.

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Properties of a utility function
If the underlying preferences are complete, transitive, continuous and (strictly) monotone,
the corresponding utility function will be continuous and (strictly) monotone.
If the preferences are (strictly) convex, the utility function is (strictly) quasi-concave.

Note that a function : → ℝ, where is a convex set, is strictly quasi-concave if for all , ∈ , ≠ , and for all ∈ (0,1) r(1 − ) + s > min{(), ()}.

Substitution in demand

Suppose the availability of good drops, such that . must decrease. In order to preserve
the same level of utility in their overall consumption bundle, consumers will want to
compensate by replacing with a separate good. By how much should the consumer alter 5
such that the utility remains constant?

This is an analogous problem to that of technical substitution of inputs to production on the
supply-side!

Indeed, we define the marginal rate of substitution (MRS) to be the rate of change of good with respect to the change in good :
.,5rs = − rs.rs5 ,
where we also define .() ≔ rs.
to be the marginal utility with respect to good .

The MRS is, of course, the consumer-side analogue to the MRTS (marginal rate of technical
substitution. One can check that the .,5 is indeed invariant under a strictly monotonic
transformation of the utilities.

Just as it is often useful to consider a graphical representation of a firm’s economic and
technological capabilities, it can be useful to graphically represent consumer preferences. As
a demand-side analogue to the isoquant, we define the indifference curve to be a level set
of the utility function:



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Sketch of an indifference curve, note properties of indifference curve –
- Continuous
- Convex
- Marginal rate of substitution – derivative of the indifference curve




Budget Restraints, Utility Maximisation and Demand
A fundamental assumption underlying consumer-side economic analysis is that the
consumer will choose to purchase the most preferred consumption bundle from the set of
all affordable bundles.

Represent the set of all affordable bundles by the budget set: = [,& = Ü ∈ : 4 ≤ Ý ⊆ ,
for a fixed budget , with the vector of prices of goods.

At the heart of consumer choice, then, is the problem of finding the most preferred bundle ∈ .

This is the problem of finding argmax0∈_ rs:

A solution to this problem will exist so long as is continuous and is closed and
bounded…
…this is guaranteed if > 0. i.e. if . > 0, for all ∈ {1, … , }.

Denote the constrained utility-maximising bundle ∗ ∈ :
- ∗ will be independent under a strictly increasing transformation of the utility
function.
- ∗ will, in general, be dependant both on prices and on the budget .
- ∗ is homogeneous of degree zero jointly in prices and budget.
This is because multiplying income and prices by > 0
will not alter , and thus will not alter ∗.
How can we find ∗?
- Kuhn-Tucker as constraints are inequalities
- Lagrangian in case of equality


If we make some reasonable regularity assumptions about the consumer’s preference
ordering ≼, we can simplify our constrained optimisation problem.
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Assume local nonsatiation and suppose ∗ = argmax0∈_ rs:
- If 4∗ < , that means ∗ is in the interior of , then there would exist some ,
close enough to ∗, such that both 4 < and (by nonsatiation) ≻ ∗.
- This would imply that ∗did not maximize (), and so we have a contradiction. ⟹ 4∗ ≮ ⟹ 4∗ = .
This is the ‘budget line’. Note that { ∈ |4∗ = } is the relative boundary in .
We do not consider the boundary in ℝ%.


Thus, since we assume local nonsatiation, we need only seek argmax0∈Y_ rs. We can address
this using the Lagrangian!
max0∈` rs such that 4 =

Some economists call the fact that utilities are maximised only if people spend all their
money Walras’ Law.

Example: Consider the consumer with utility function (', () = '*(')* ∈ (0,1)

ℒ(, ) = rs − U4 −V = '*(')* − ('' + (( − )

First-order conditions for maximising ℒ: ℒ(, ). = 0, = 1,2, ℒ(, ) = 0 ( ℎ ) ℒ(, )' = 0 ⟹ …'(†*)' = ' , ℒ(, )( = 0 ⟹ (1 − ) …'(†* = (

Dividing both equations yields:
1 − …('† = '(


Rearranging and using the budget constraint '' + (( = , ( = 1 − '( … − ((' † ⟹ (∗(', (, ) = (1 − )(
Substituting into the constraint once more gives: '∗(', (, ) = '

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Emphasise use of r. , 5s = −./5 ; this ratio of prices sometimes referred to as the
economic rate of substitution

What could have made our analysis substantially easier?
à A log-transformation of the data!
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A note on second-order conditions for the Lagrangian:

In order for the Lagrangian ℒ to be maximised, we require negative semidefiniteness of the
matrix of second derivatives of ℒ with respect to each of the variables (necessary condition).
This matrix is known as the ‘bordered Hessian’; in the current context, the Hessian refers to
the matrix of second derivatives of the utility function. Requiring the bordered Hessian to be
negative semidefinite is the same as requiring the Hessian to be negative semidefinite,
subject to a linear constraint:

ℎ4∇(r∗sℎ ≤ 0 ∀ ℎ ∈ ℝ% such that ∇r∗s4ℎ = 0
where ∇(r∗s ∶= ¢(r∗s.5 y020∗£.,52'
% .

This is necessary for the fact that the utility function is locally quasi-concave.

That means there is a neighbourhood containing ∗ such that is quasi-concave on .

A sufficient second order condition for local quasi-concavity is that the utility function is
strictly locally quasi-concave. A sufficient condition for that is
ℎ4∇(r∗sℎ < 0 ∀ ℎ ∈ ℝ%, ℎ ≠ 0 such that ∇r∗s4ℎ = 0.

The choice of the consumption bundle that maximises the consumer’s constrained utility
function will be exactly the bundle that the consumer demands; this is unsurprisingly
referred to as the demanded bundle or demand function, ∗ ∶ ℝ#$% × ℝ#$ → ℝ#$% , ∗ U,V = argmax0∈Y_ rs.

Note that this is sometimes referred to as Marshallian demand.

Note that we have discussed the existence of the argmax. However, it is per se not clear
whether the argmax is unique, that means, whether the maximum is attained at a single
point over . This can be guaranteed if the underlying preferences are strictly convex (and
prices are strictly positive).
You will show this in an exercise.

Moreover, the function ∗(,) is homogeneous of degree 0 in (,).

We also note that, faced with a set of goods with prices , the maximum utility achievable
with a given budget is known as the indirect utility function: ∶ ℝ#$% × ℝ#$ → ℝ, U,V = …∗ U,V† = max0∈Y_ ().
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This indirect utility function is itself a quantity of interest, and we note some of its key
properties here:
- Nonincreasing in : - ≥ ⟹ U-, V ≤ U,V
…and nondecreasing in : If the underlying preferences satisfy the local
nonsatiation assumption, then is even strictly increasing in . - ≥ ⟹ U,-V ≥ U,V
- Homogeneous of degree 0 in U,V: U, V = U,V ∀ > 0
- Quasi-convex in : Ü ∈ ℝ#$% : U,V ≤ Ý is a convex set for all ∈ ℝ and for all ≥ 0.
- Continuous at all ≫ 0, > 0.


The indirect utility function is often illustrated using so-called price indifference curves.
These are the level sets of the indirect utility function with a fixed budget . That is
{ ∈ ℝ#$% ∶ U,V = } for some ≥ 0, ∈ ℝ.

They are analogous to the indifference curves of the utility function.






Plot price indifference curves for two good problem.
Note…
- nondecreasing as we move towards the origin
- lower contour sets are convex


Sketch a graph of U, ⋅V.


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A direct consequence of the local nonsatiation assumption of the underlying preferences is
that for fixed , the indirect utility function U, ⋅V is strictly increasing in .
Therefore, U, ⋅V is injective and can be inverted on its image. Denote this image with [ = { U,V ∶ ≥ 0}.

Then we define the expenditure function
U,⋅V ∶ [ → [0,∞), ↦ U, V . . = …, U, V†


The expenditure function provides the minimum level of income required to obtain a given
level of utility at prices . Note that U, V can also be obtained as the solution to the
optimisation problem

Find min0 4 subject to the constraint rs ≥ .

The consumer’s expenditure function is simply the demand-side analogue to a firm’s cost
function, and their properties are also the same: U, V is
- nondecreasing in
- homogeneous of degree 1 in
- concave in
- continuous in for ≫ 0
The aims of firms and consumers are the same here – both wish to minimise their
costs/expenditure such that they can achieve a desired level of output/utility.

The dual quantity to the expenditure function is the Hicksian demand (sometimes referred
to as the compensated demand)
a∗ ∶ ℝ#$% × ℝ → ℝ#$% , U, V ↦ a∗ U, V = argmin0∈b&"([b,P)) 4

Recall that, on the firm side, for a specified level of output, the cost-minimising combination
of production inputs can be found via Shephard’s Lemma. We can also apply this result in
the current scenario, yielding an expression for the expenditure-minimising consumption
bundle in terms of prices and desired utility level: a,.∗ U, V = U, V. .
The Hicksian demand function is formally the same as the conditional factor demand
function on the supply side.

Note that, when we refer to the demand function without qualification, it is assumed to be
the Marshallian demand.
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Unlike the Marshallian demand, the Hicksian demand function is not observable; indeed, it
depends on utility, which is itself unobservable. Nonetheless, under some of the usual
regularity assumptions, the Hicksian and Marshallian demands satisfy the following
identities:
For all > 0, ≥ 0, ∈ [
- …, U,V† = . Since is the max utility possible at , the minimum
expenditure needed to achieve is exactly .
- …, U, V† ≡ . Similarly, since is the min expenditure needed to achieve ,
the max utility possible at is exactly .
- a,.∗ …, U,V† = .∗ U,V. The utility-maximising bundle for budget is exactly the
same as the bundle that minimises expenditure for that maximised utility.
- .∗ …, U, V† = a,.∗ U, V. The expenditure-minimising bundle for desired
utility is exactly the bundle that maximises utility for that minimised
expenditure.


Slutsky’s Equation


For economists, it is important to understand how consumers react to changes in the
economic environment. For instance, we can consider how the optimal choice of
consumption bundle ∗ U,V will change with respect to the price vector . Slutsky’s
Equation states that the total effect of a change in demand of a good when the price of a
good is changed can be decomposed into a substitution effect and an income effect.


- the substitution effect, or the change in compensated demand
- which is the change in the demanded bundle resulting from the change in the
optimal balance of goods; this is the change that we can make to optimise
expenditure whilst keeping utility fixed
- the income effect
- This is the change in the magnitude of the optimally-balanced bundle, due to the
decrease / increase in purchasing power.











Continuous ,.
Preferences satisfy
local nonsatiation
Solutions to the
expenditure
minimisation and
the utility
maximisation
problems both
exist
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Theorem: Under the usual regularity conditions, we have that
5∗ U,V.‰ŠŠ‹ŠŠŒde1*^ \::\Z1 =
a,5∗ …, U,V†.‰ŠŠŠŠŠ‹ŠŠŠŠŠŒ"b+91.1b1.e% \::\Z1 −
5∗ U,V .∗ U,V‰ŠŠŠŠŠ‹ŠŠŠŠŠŒf%Ze&\ \::\Z1


for all ≫ 0, > 0 and for all , ∈ {1, … , }.

Remark: Sometimes, notation can be quite misleading. Especially when considering the
Slutsky equation it is crucial not to mix partial derivatives and total derivatives. A partial
derivative is an operator mapping a (differentiable) function to its derivative which is again a
function. In contrast, a total derivative needs a free variable in an equation and takes the
derivative with respect to this equation.

Example: Consider the function ∶ ℝ( → ℝ, (', () ↦ (', () = 'V + (g. Then
YY0! is again a function, namely the function YY0! ∶ ℝ( → ℝ, (', () ↦ 5(h. Of course,
we can also give other arguments to the function, e.g.
( (, ) = 5h.

On the other hand, the total derivative acts like follows:
dd( (', () = 5(h.

But dd( (, ) = 0
since (, ) does not depend on ( at all. We see that actually, the operator YY0! indicates
that we have to take the partial derivative with respect to the second argument of , where
usually ( stands. But the denomination involving ( can be quite misleading. It would be
better just to indicate the argument with respect to which one takes the derivative. A better
notation is therefore ( instead of YY0!.








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In the light of this discussion, the Slutsky equation takes the form:
.5∗ U,V = .a,5∗ …, U,V† − %W'5∗ U,V .∗ U,V



Proof of Slutsky’s Equation:

Let ≫ 0, > 0 . For any ∈ ℝ we have the identity
a,5∗ U, V = 5∗ …, U, V†.

Therefore, we can take the total derivative with respect to . and obtain
.a,5∗ U, V = . a,5∗ U, V = . 5∗ …, U, V† = .5∗ …, U, V† + %W'5∗ …, U, V† . U, V‰Š‹ŠŒ20,,.∗ i[,bj .

For = U,V, this yields
.a,5∗ …, U,V†= .5∗ â, …, U,V†‰ŠŠŠ‹ŠŠŠŒ2& ã+ %W'5∗ â, …, U,V†‰ŠŠŠ‹ŠŠŠŒ2& ãa,.∗ …, U,V†‰ŠŠŠŠ‹ŠŠŠŠŒ20.∗i[,&j .

Rearranging the quantities yields the assertion. ∎
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It is clear from the Slutsky equation that the income effect plays a major part in determining
how the demand for a set of goods will react to changes in their prices. For firms, consumers
and economists alike, then, it is important to ascertain how the demand of certain goods
will react to changes in consumer budget.

Indeed, economists class goods according to the manner in which they react to changes in
consumer income:
- For normal goods, an increase in income will result in an increase in demand; 5∗ U,V ≥ 0
- For inferior goods, an increase in income will result in a decrease in demand. 5∗ U,V < 0
It is also worth noting the different subclasses of normal goods: suppose we have an
increase in consumer income …
- …for luxury goods, demand will increase more than proportionally to income;
Income elasticity is >1
- …for necessary goods, demand will increase less than proportionally
Income elasticity is positive, but <1
- …and if demand increases proportionally to income, the consumer is said to have
homothetic preferences for the set of goods under consideration.
Income elasticity is =1

Finally, we note that goods can also be classified according to how changes in price impact
their consumer demand:
- For ordinary goods, a decrease in price will lead to an increase in their demand; 5∗ U,V5 ≤ 0

- For Giffen goods, a decrease in price will lead to a decrease in demand 5∗ U,V5 > 0



That means our previously stated law of demand only holds for ordinary goods, but not for
Giffen goods.

What is an example for a Giffen good? Some theoretical considerations can help us finding
necessary conditions for Giffen goods. In particular, the Slutsky equation helps to establish a
relation between ordinary and normal goods on the one hand side, as well as Giffen and
inferior goods on the other side.



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Recall that from Slutsky’s Equation with = :
5∗ U,V5 = a,5∗ …, U,V†5 − 5∗ U,V 5∗ U,V



Since a,5∗ …, U,V† = Y\k[,li[,&jmY[/ and the expenditure function is concave, we obtain
for the substitution effect:
a,5∗ …, U,V†5 = ( …, U,V†r5s( ≤ 0.

Since 5∗ U,V ≥ 0, we have the following implications:
5∗ U,V ≥ 0 ⟹ 5∗ U,V5 ≤ 0.

That means a normal good is always an ordinary good. The contraposition holds too:
5∗ U,V5 > 0 ⟹ 5∗ U,V < 0

A Giffen good is always an inferior good.


These considerations lead to a list of three necessary conditions for a good to be a Giffen
good:

1. It must be an inferior good.

2. The substitution effect must be relatively small – i.e. there is no close substitute
good.

3. A substantial part of income is spent on the respective good, but not all of it.

Example: Staple food in relatively poor countries.
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Part 2 – Markets and Competition

Markets – Demand, Supply, and Equilibrium

We define a market for a good or service to simply be the union of the individuals and firms
that operate on both the supply and demand sides of a potential transaction. There are
many different types of market that we should be aware of, some of which form rich areas
of study themselves. We will continue to focus on markets for goods and services, however
notable other markets include:
- Labour market
- Capital market
- Shares market

The intentions and wishes of each side of a market are, of course, specified through the
demand and supply curves, and we recall from the start of the course that (for a competitive
market) the prices at which goods are sold is settled through the price mechanism.

Recall that the price at which supply equals demand is referred to as the equilibrium price.
Suppose that a market is settled at an equilibrium price $:

- If the demand for a good changes for
some reason, then the demand curve
will shift; demand and supply will no
longer be equal at $.

- An increase in demand, or a decrease in
supply, will lead to an excess in
demand.

- A decrease in demand, or an increase in
supply, will lead to an excess in supply.

- These excesses invoke the price
mechanism, changing the equilibrium
price. The speed at which this happens
will vary between markets.


Excesses in demand or supply can be measured as long as we can express supply and
demand in terms of the good’s price.
- For an individual consumer, demand is measurable through the Marshallian
demand under the assumption that they are maximising their utility.


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- For an individual firm, we have the (short-run or long-run) supply curve (under
profit-maximising assumptions)


We want the market demand and market supply (Also referred to as the industry demand
and industry supply) (i.e. the total demand for a good across the market, and the total
quantity of the good supplied)
These are simply the sum of the respective individual quantities.

Suppose a market for a single good contains consumers and firms. Further, suppose that
consumer has demand given by .∗(,.)
where is the price of the good and . is the budget for consumer
and that the supply curve for firm is specified by 5∗ UV
We specified the supply curve as portions of the SMC/LMC, i.e. costs as a function of output
– we just need to invert this to get output required for a given price
The market demand for the good is then defined as ∗ U,', … ,fV =ƒ.∗ U,.Vf.2'


and the corresponding market supply is defined as ∗ UV =ƒ5∗ UVn52'

Example:
Suppose that the market for bananas contains 1000 utility-maximising consumers
with demand functions .∗ U,.V = . è __( + N(é = . è _])'_] + N]é = 1,… ,1000
also dependent on the price of apples
Further, suppose the banana market comprises two suppliers, with supply curves 5∗ UV = _2 = 1,2
What is the equilibrium price for bananas?

Find the market demand and industry supply: ∗ U,', … ,'$$$V = ƒ . è __( + N(é'$$$.2' = è __( + N(é ƒ .'$$$.2' = è __( + N(é

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∗ UV = '∗(_) + (∗(_) = 3_4

_,$ occurs when the market demand and industry supply are equal: è __( + N(é = 3_4 ⟺ _,$ = ê43 − N(



Consumers’ and Producers’ Surplus – Social Welfare

To analyse the consequences of a change in prices or income – or more generally, a change
in policy – it is useful to have a measure of social welfare. We will see that a handy such
measure is the sum of consumers’ and producers’ surplus. It also gives rise to another
characterization of the equilibrium price and equilibrium quantity, maximising this social
welfare measure.

We put ourselves into the general framework of utility maximising consumers and profit
maximising firms where the utility and production functions satisfy our usual assumptions.
Suppose we have firms with cost functions 5∗(⋅), ∈ {1, … , }, and consumers with
respective utility functions .(⋅), ∈ {1, … , }, and corresponding quantities (i.e. indirect
utility function ., expenditure function ., Marshallian demand .∗, and Hicksian demand a,.∗ as well as profit-maximising output 5∗)


Consider fixed income levels ', … ,f and a price change from (') ∈ ℝ#$% to (() ∈ ℝ#$% .
Suppose that the price change affects only one single product and that w.l.o.g. the product
gets more expensive. To save notation, we will only explicitly denote the variable with a
price change, suppressing all the other ones. So in the specific good, we will consider a price
change from (') > 0 to (() > 0 where we assume without loss of generality that (') <(().
We assume that producers are concerned about their change of profit. So we can measure
the effect of the price change with the quantity
ƒ5∗r(()s − 5∗r(')s = n52' ƒë 5∗() .[(!)[(")
n
52'

Recall that 5∗() = 5∗() − 5∗ U5∗()V. Hence
5∗() = 5∗-() = 5∗() + 5∗-() − 5∗- U5∗()V 5∗-() = 5∗() + 5∗-() … − 5∗- U5∗()V†‰ŠŠŠŠ‹ŠŠŠŠŒ2$ = 5∗()
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Consequently, we get ƒ5∗r(()s − 5∗r(')s = n52' ƒë 5∗() = ë () [(!)[(") .[(!)[(")
n
52'


Consequently, we introduce the producers’ surplus at price í as one part of the measure
for social welfare measure:
(̂) = ë () [o$ .


The consumer side is a bit trickier. Following the utility maximisation rationale of the
lecture, each individual consumer cares about the difference in their individual indirect
utility, that is
.r((), .s − .r('), .s.

However, this approach is problematic since

• Utilities are ordinal and we cannot aggregate them;

• We would like to compare the effect on the consumers with the effect on the
producers.

So we need a monetary measure!



A natural possibility is to consider the difference of the individual expenditure functions,
keeping the initial (indirect) utility fixed. This quantity is known as compensating variation

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r('), ((), .s = . U((), .r('), .sV − . U('), .r('), .sV= . U((), .r('), .sV − . = ë . U, .r('), .sV [(!)[(")= ë a,.∗ U, .r('), .sV [(!)[(")


where we have used Shephard’s Lemma.



What is problematic with this approach?

• Firstly, the indirect utility – and thus, Hicksian demand – is not directly observable.
Only Marshallian demand is observable.


• In line with that argument, we only have aggregated demand in terms of the sum of
Marshallian demand, not of Hicksian demand.


• What is somewhat inelegant is that fact that compensated variation is in general not
(anti-)symmetry, whereas the change in overall profit on the producer side is.
That is, we usually have that
.r((), ('), .s ≠ −.r('), ((), .s





Consequently, we prefer to work with a quantity that relies on Marshallian demand only
and that is (anti)symmetric. To this end, consider
ë .∗(,.)[(!)[(")


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One can show that – under our standard assumptions and if they are normal goods (using
Slutsky’s equation) – for (') < (() it holds that
−.r((), ('), .s ≤ ë .∗(,.)[(!)[(") ≤ .r('), ((), .s.



So we define the consumer surplus at price ̂ as

(̂) = ë ƒ.∗(,.)f.2'P[o = ë (,', …f)P[o .

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Finally, the sum of consumers’ surplus and producers’ surplus, the community surplus, can
be considered as a measure of social welfare.


Changes in the market demand or industry supply of a good will lead to a change in its
equilibrium price. Taxes and subsidies are an interesting example of factors that lead to such
a change.

Indirect Taxes and Equilibrium:

An indirect tax is one that can be passed on to another party. In the context of providing
goods and services, an indirect tax on producers is one that is passed on to consumers. In
general, a tax that is dependent on the quantity of good being produced can be treated as
an indirect tax.
e.g. VAT, …
How much of an indirect tax is passed on to consumers? As much as possible though, in
general, both consumers and producers will bear some brunt!











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The ‘incidence’ or ‘burden’ of the indirect tax on consumers and producers is determined by
the relative price elasticity of supply and demand:
- If demand is more price-elastic than supply, then the producer will bear the
majority of the tax burden
- If supply is more price-elastic than demand, then the producer will pass on the
majority of the burden.





















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Indirect taxes on the production of goods can be imposed in one of two ways:
- the tax may be a fixed amount per unit sold – this is a unit, or specific tax
- or it may be a percentage of the good’s price – this is an ad valorem tax.


Diagram showing the effect of unit and ad valorem taxes
- unit taxes shift the supply curve in parallel
- value-added taxes cause the supply curve to ‘diverge’












In any case, imposing taxes reduces the community surplus. The difference between the
original community surplus and the new community surplus plus the tax revenue is called
deadweight loss. However, if taxes are present, it is crucial that we include the government
into the consideration and computation of the community surplus. That is, in the presence
of taxes, the community surplus is the sum of the producers’ surplus, the consumers’
surplus and the tax revenue.
A deadweight loss occurs when the market price/quantity deviates from the equilibrium
price/quantity. We have seen that taxes can cause a deadweight loss.

But then, why do we have taxes at all?
- To raise money for the government/pay for public services, such as NHS and as
well as investment in public projects, such as roads, rail and housing.
- To account for (negative) externalities that are not accounted for by either of the
parties (consumers and firms).
Examples:
- Harm to health (tobacco, alcohol)
- Harm to the environment (CO2)

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On the other hand, some activities are deemed to have positive externalities (education,
culture, …), which functions as justifications for subsidies.
Subsidies can also cause a deadweight loss. Also in the presence of subsidies, one must
include the government into the consideration and computation. That is, the community
surplus is the sum of producers’ surplus and consumers’ surplus minus the total size of the
subsidy.
Moreover, maximal or minimal prices as well as quantities can cause a deadweight loss.





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Abnormal Profits, Long-run Equilibrium and Productive Efficiency

Consider a competitive firm and suppose it has costs given below, on the right, whilst
operating in the market with industry supply and market demand as given on the left:












The firm’s individual supply curve is obtained under the assumption that the firm is profit-
maximising: by construction, we have that at any point on the firm’s (positive) supply curve,
their marginal revenue will equal their marginal cost.
Suppose the industry supply is given by ' in the short run.


What will happen as we move into the long run?

- Other firms want to enter the market, as it is easy to make money!
- This will drive the supply up (increase the quantity for fixed price), until…
- …marginal cost equals average cost.








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Thus, the long-run equilibrium is the point at which no individual firm makes a profit.
In other words: The shutdown condition we have derived in the previous chapter is binding
in the long run!


But surely companies make long-run profits all the time?!
We are considering different types of profit here! This is where we need to distinguish
between accounting costs and economic costs:
Accounting costs include all financial costs of production
o All paid costs
o Includes fixed and variable costs
o This is what we’ve considered so far



Economic costs are accounting costs, plus opportunity costs
o Opportunity costs measure the foregone benefit of employing the firm’s
resources elsewhere, in the best alternative manner.
o Unit costs for each factor can be defined as economic costs (including
opportunity costs of each factor).
o In the long run the opportunity costs should match the accounting profit.
§ Otherwise firms will want to enter or leave the market.







Firms that exactly cover their economic costs have zero economic profit, but their
accounting profit is equal to their opportunity costs; they are said to be earning a normal
profit.




Firms which cover more than their economic costs are said to be making an abnormal profit.
This will encourage entry into the market by other firms.

Firms that make normal profits are said to be productively efficient – they produce at the
minimum of the average cost curve, when taking opportunity costs into account.

Overall, firms in perfect competition can only make normal profit in the long-run. This is due
to the infinite price-taker firms and the absence of barriers to entry or exit the market.



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Part 3 – Macroeconomics

In the following section, we will consider a number of fundamental concepts central to
macroeconomic analysis.

Macroeconomy concerns country-wide economics and the economic variables that affect all
firms and individuals to different extents. It also looks at the interaction of different (micro)
markets and factors that influence the inter-operation of these markets. Such factors may
include:
- Interest rates
- Exchange rates
- Inflation
- National income
- (Un)Employment
- Taxation

Although macroeconomics covers a much greater scale than microeconomics, there are
some parallels. For example, an important concept in macroeconomic analysis is the idea of
aggregate supply and aggregate demand. Note that these are separate concepts to the
market supply and demand; here, we are talking about the entire economy, not just single
markets!

The circular flow of income

The circular flow of income is a model that analyses how money, goods and services flow
through the economy over a particular time horizon. It originates from the work of the
French economist and physician François Quesnay (1694 – 1774) and is akin to the
circulatory system of the body. Thus, it depicts the interdependence of the various
economic agents. On the other hand, it illustrates the constitution of National Income.

Two sector model


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We can actually see two circuits: One reflects actual physical goods (goods, services, factors
of production/labor). The second is the circuit of money flowing into the opposite direction
than the first one.
For the sake of simplicity, we will only denote one of the two circuits, which will be the one
associated to money.
In order to quantify the flows, we introduce the following variables:
- C: Consumption expenditure
- Y: Income
As the name suggests, the circular flow needs to be in equilibrium in the two sector model
in the sense that = .
It has the interpretation that the households need to earn what they spend, but also spend
what they earn. Another way to consider it is that demand (C) needs to be equal to supply
(Y) or production. Now, one might wonder whether this model is too simplistic – do
households always spend all their money in a certain period? And how can firms grow and
increase their production capacities? We definitely need more agents in our model.

The five sector model
Three additional sectors:
1) Government
2) Financial sector
3) Overseas sector










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Sketch the five sector model.
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a) Leakages

Of course, not all of a household’s income will be spent on domestic goods and services.
- S: Savings (Financial sector)
- T: Taxes (Government sector)
- M: Import spending (Overseas sector)


…these are all called ‘withdrawals’ or ‘leakages’ from the economy; they do not feed back
into demand for domestic output


b) Injections

Additionally there will be ’injections’ into the economy in the form of…
- I: Investment spending by businesses (Financial sector)
- G: Government spending on goods and services (Government sector)
- X: Export spending (Overseas sector)






Injections and leakages such as these will obviously have an impact on the demand for all
goods and services within the economy: leakages will reduce the aggregate demand and
injections will boost it. Indeed, when the injections compensate for the leakages, then
aggregate demand will equal aggregate supply, and the economy will be in equilibrium:
This will be when ( + + ) = ( + +).

Indeed, we can be more specific and define aggregate demand as follows:
= + + + − = +


Our definition of aggregate demand above requires careful consideration of various sources
of demand for the final goods and services being provided to the economy by firms. In
contrast, we can straightforwardly define aggregate supply to be the total value of all final
goods and services provided by firms to the economy.


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Recall that equilibrium in the economy occurs when + + = + +. These
variables can be paired according to the respective sectors:
- Financial sector (, ): In order to make investments, firms obtain the required
financing from financial institutions; these institutions are able to provide the
financing due to the savings of consumers. might not equal – decisions to invest/save are made by different parties.



- Government sector (, ): The government can both inject and withdraw from
the economy, via spending and taxation, respectively. might not equal … governments may choose to run a budget surplus ( > )
or a budget deficit ( < ).



- Overseas (,): Demand for exports and demand for imports are also linked,
but may not necessarily be equal.
Not necessarily balanced trade. (The trade balance is the value of exported goods
and services minus the value of imported goods and services. A positive value
indicates a trade surplus and a negative value indicates a trade deficit.)
We can have a trade surplus ( > ) or a trade deficit ( < ), e.g. China –
trade surplus, UK – trade deficit.



Considering the second pairing in particular, we see that governments can therefore
influence the economy by forcing a mismatch in withdrawals and injections.

One can qualitatively discuss what happens if the economy deviates from the equilibrium.
Suppose that, over a given time period, injections exceed withdrawals:

- There will be an excess in aggregate demand, motivating an increase in
aggregate supply to move towards equilibrium. This is economic growth!

- The resulting increase in aggregate supply may cause firms to increase their
labour supply, leading to a fall in unemployment.

- An increase in demand will also increase prices….this leads to inflation

- Excess in demand will increase imports as consumers buy elsewhere; exports will
decrease due to rising prices.



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Gross Domestic Product (GDP)

How do we exactly measure the overall production of an economy? We also need this
measure to talk about deviations in production, which can be recession or growth.
So we introduce one of the most famous notions of macroeconomics.

The Gross Domestic Product (GDP) measures the nominal gross value of all goods and
services produced in a certain country in a certain period of interest.


Comments:

• The GDP is a monetary / nominal value, not a real value.

• It measures only the gross value which is added in the course of production. à
Avoiding double counting.

• How to measure services provided by the state (e.g. administration, education,
security, defense)? (that don’t have a market price)
One measures it just taking into account the costs of those services.


• The confining quantities are time, but also area – that’s why it’s called domestic. A
(historic) alternative is the Gross National Product (GNP). The difference to GDP is
that GNP measures the gross output of all citizens of a certain nationality
irrespective of their residence.
It was used to measure the overall production foremost until the first half of the 20th
century.


Different ways to calculate the GDP
We start with an example: Consider two firms.
F1 (Steel producer):
- Revenue: £100
- Wages: £50
- Capital: £30
- Profit: £20
F2 (Car producer):
- Revenue: £210
- Steel: £100
- Wages: £70
- Profit: £40
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1) Production Approach: Calculate the gross value added in the domestic production.
This is gross value of output minus intermediate consumption. In our example, this is
= (£100 + £210)‰ŠŠŠŠ‹ŠŠŠŠŒp]e99 l*^b\ − £100‰‹Œ.%1\]&\q.*1\ Ze%9b&[1.e% = £210

This approach reflects the very definition of GDP probably best. Note that the GDP
avoids double counting. A good intuitive justification for this is to imagine that F1
and F2 merged. Then we would not see the intermediate consumption (it would not
be reported to the national statistics office). In macroeconomics, one considers the
entire supply side as one firm such that this perspective makes sense.

How would GDP change if F1 were an overseas steel supplier?
= £210‰‹Œp]e99 l*^b\ − £100‰‹Œ.%1\]&\q.*1\ Ze%9b&[1.e% = £110


2) Expenditure approach: This approach takes the angle that everything that was
produced has to be bought. We can use the circular flow diagram:
= + + + ( −)

In our example, we have
= £210, = = = = 0
= £210.

Again, if F1 were overseas, we had = £100, = = = 0
= £210 − £100 = £110.

3) Income approach: Somebody has to earn the value that has been created. This is
usually income from labour, capital, and taxes, which was generated domestically. In
our example, that is
= (£50 + £70)‰ŠŠŠ‹ŠŠŠŒr*+eb] + (£20 + £30 + £40)‰ŠŠŠŠŠ‹ŠŠŠŠŠŒs\%19, .%1\]\91, q.l.q\%q9 = £120 + £90 = £210

Again, for the alternative scenario with F1 being abroad, we get = £70ùr*+eb] + £40ùs\%19, .%1\]\91, q.l.q\%q9 = £110.
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Criticism concerning GDP as an overall welfare measure

1. Since GDP is a nominal value, price changes (due to inflation) can cause an increase
in GDP (but they do not affect production in real terms – at least primarily).

2. Many services are ignored, e.g., child-rearing, care for elderly people, working in an
honorary capacity (in societies, congregations, sports, …)

3. Externalities are often ignored, e.g., adverse effects to the environment

4. Depreciation is often ignored, e.g., destruction of infrastructure by (natural)
disasters such as storms, floods, but also war. Depreciation can also come from
natural sources.

5. It ignores the benefits of leisure.

GDP is often used as a proxy or an indicator of the overall welfare of a society. To some
extent, this is certainly justified. However, what can happen if decision-makers in politics,
business, and society are mixing up the notion of an indicator with overall welfare itself?














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They might try to cause in increase of GDP without actually improving the overall welfare of
society (or maybe even deteriorating welfare).
1. Inflation increases the GDP since it is a nominal measure.

2. Tendency to commercialise those services such as child-rearing (kindergartens), care for
elderly people (retirement home), or voluntary work.

3. Increase of industry production despite adverse effects to the environment (for example
in the former German Democratic Republic).

4. There might be incentives to cause some depreciation in order to re-build infrastructure
(destroy streets on order to rebuild them). Also in business, there might be an incentive
to build products with a high deterioration rate.

5. People might be pushed into (dependent) work despite their preferences.










Can you think of possibilities how solve these problems in the practical usage of GDP as a
proxy for overall welfare of a society?





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Allocation of income – connections to social welfare

There is an ongoing debate (over the last centuries) how income should be distributed.
Before addressing this normative question, let us first turn to the descriptive side of it. How
can the distribution of income be measured?
Certainly, this is a statistical question. The most informative measure would be to report
each individual’s income (which amounts to reporting the empirical distribution function of
income). However, this is too complex to report. So we are actually looking for some
number that summarises the distribution of income.
Suggestions:
- Mean - Range
- Median - IQR (Interquartile range)
- Some quantiles
- Variance/standard deviation
What is often of interest is a relative measure for dispersion in order to measure how
equally income is distributed. There are several ways how to do this.
One way could be to consider the ratio of standard deviation and mean. However, this
measure is not normalised.

A measure that is most commonly used for this purpose is the Gini coefficient. For a
population of persons with income ', … , % it is defined as

= ∑ Õ. − 5Õ%.,52'2 ∑ .%.2'

Indeed, one can show that the Gini coefficient is
- Positively homogeneous of degree 0 in ', … , %;
- Always between 0 and 1
- 0 if and only if there is complete equality. That is, if and only if ' = ⋯ = %;
- For fixed it is maximal if and only if there is complete inequality. That is, if and
only if there is some ∈ {1,… , } such that 3 = ∑ 5%52' . Indeed, then = − 1 .

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Normative positions
Besides merely describing income and wealth allocation, there is an ongoing debate
between different normative positions. Recall that two of the main drawbacks in this
discussion is that utilities are not directly observable. And moreover, since we are working
with ordinary utilities, there is no way how to meaningfully aggregate them.
Here is an overview of some positions, including possible criticisms.

• Equal distribution: This position asserts that everybody should have the same
income which is akin to communistic theory.
What is problematic is that there are only week (monetary) incentives to contribute
to economic growth (at least if we assume the Homo Economicus).

• Minimax approach: Suppose you were born into a society with uncertainty in which
class you will be born. The most risk averse approach would then be to minimise the
maximally adverse outcome (therefore the name). Equivalently, one would try to
maximise the minimal utility in the population.
Criticism: One needs to compare different utility functions.

• Pareto efficiency:
A Pareto improvement is a change that makes at least one person better off, without
making any other person worse of. An allocation is Pareto efficient if no Pareto
improvement is possible.
Even though this is a widely agreed criterion for income allocation, it is also a rather
week notion of efficiency. E.g. if utilities are strictly increasing in income, also an
allocation is Pareto efficient where one person owns the entire income and the rest
does not have anything at all.

• An alternative perspective is that one should not care about the outcome of the
income allocation, but only about the underlying mechanism. If the allocation
mechanism is fair (if it works according to fair rules and laws), then any resulting
allocation is fair.
This position is probably most akin to a purely capitalistic approach.

In the entire discussion, it is crucial to make a precise distinction between relative and
absolute allocation of income. Bear in mind that a third of a very large cake might still be
better than half of a rather small cake.
However, some studies show that people often rather care about their relative income…


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