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MATH60013/96011 – Mathematics of Business and Economics
创建日期：2024-05-09 15:34:45
所属分类：数学mathematics
标签： MATH60013
Mathematics of Business and Economics

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MATH60013/96011 – Mathematics of Business and Economics 


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
MATH60013/96011 – Mathematics of Business and Economics Imperial College London
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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.
MATH60013/96011 – Mathematics of Business and Economics Imperial College London
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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%
MATH60013/96011 – Mathematics of Business and Economics Imperial College London
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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)
MATH60013/96011 – Mathematics of Business and Economics Imperial College London
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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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