Mathematics for Economics - Lecture notes - Lecture 1 PDF

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Mathematics for economics 2015

Gabriel Leon

Department of Political Economy

[email protected]

Lecture 1

Organisation of Maths Teaching 10 two hour lectures and 5 seminars.

The seminar questions will be posted on Keats - please answer the questions/problems before coming to the seminar. We will go over the answers to these questions during the seminar, and you will bene…t most if you have already tried the questions.

The questions in the midterm and …nal exam will be in the same format as the assignment questions.

The recommended textbook is by Alpha C. Chiang and Kevin Wainwright, “Fundamental Methods of Mathematical Economics”, McGraw-Hill, 2005.

Supplementary textbooks are listed on the module outline

We cover a lot of material fairly quickly, self-study is crucial for this material

Assessment

A&P (10%) Midterm exam on 2 November, during lecture hours (30%): 1 hour long, 5 questions, each worth 20 points. I posted a sample midterm on Keats. Final exam in January (60%): 2 hours long, 10 questions, each worth 10 points. Same format as the midterm, but twice as long.

Topics (by week)

1. Di¤erentiation

2. Optimisation 1

3. Elasticity, continuity, di¤erentiability

4. Optimisation 2

5. Integration

Reading week

6. Midterm

7. Envelope theorem, comparative statics

8. Linear Algebra 1

9. Linear Algebra 2

10. Linear Algebra 3

11. Introduction to Dynamics

Note: indicative, might revise it as we go along.

Most relevant chapters for this week Chiang and Wainwright, chapters 1, 2, 3, 6, 7, 10, 14.

Why Study Maths? It will help you with the material covered in the second and third years

If you decide to study abroad, the economics modules are very mathematical in most destinations

If you think you might want to study economics at the postgraduate level, or you want to read journal articles in economics, maths is essential.

Mathematical skills are a good thing to have.

Although a branch of moral philosophy in the days of Adam Smith, economics has become increasingly mathematical, and is the most quantitative of the social sciences. Of course, this has not been to universal acclaim, even in Smith’s era.

The age of chivalry is gone.

That of sophisters, econo-

mists, and calculators, has succeeded...

-Edmund Burke

(1790)

Mathematics is used by economists to formalise assumptions, logic and to facilitate abstract as well as empirical analysis.

And, while we may not share the opinion of Robert Barro that "Economic theory is mathematical analysis", one advantage of the modern economic approach can easily be seen by considering the debates over what Ricardo, Marx, or even Keynes, actually ‘meant’ in their key writings. By contrast, there is little confusion about the interpretation of the works of Arrow, Debreu and Samuelson.

Oligopoly, a simple and early example

In a competitive market the equilibrium price will equal the marginal cost of production and no agent will make an economic pro…t. Intuition: it is always pro…table for an agent to lower the price (by an in…nitesimally small amount) and capture the whole market, if the price charged by the competitors is higher than the marginal cost of production. But, what happens if there is an oligopoly with only two …rms competing for the market? ‘Duopoly’ The French 19th century mathematician Cournot provided a remarkably simple model of a situation where two …rms compete by setting production volume.

Assuming that the two …rms have the same production costs, with no …xed costs, and a constant marginal cost of

c.

Suppose that demand is a linear function

a  bQ,

Q = qi + qj is the aggregate demanded, and a and b are constants. where

p(Q) = quantity

Then the pro…ts of each …rm are





  = a  b qi + qj qicqi = aqibq2i bqiqj cqi max i qi We can solve this maximisation problem by taking the derivative with respect to

qi

and setting it equal to

(‘First order condition’). Thus, we have

@i = a  2bqi  bqj  c = 0 @qi

0.

Imposing the symmetry implied by the identical production costs of the two …rms and solving for

qi = qj under plausible values of

qi

, we have

ba c

1 (  ) = 3

ab ,

and

c

.

Thus, total production is

qi = 32b (a  c) < 1b (a  c)

2

The latter being total production in perfect competition. Further, we can calculate each …rm’s pro…t

i = j = 91b (a  c)2 > 0

We can(aalso graph each …rm’s best response function c) qj =

b

 2qi

The Cournot model of oligopoly is a staple of microeconomics, and you will probably see it again.

The example illustrates: how to set up a simple optimisation problem, plot graphs, take derivatives and solve simple equations.

Aside: another 19th French century mathematician called Bertrand modelled an oligopoly where …rms set prices, rather than quantities. This changes everything, and the predictions of that model are quite di¤erent.

I will now very quickly review some material that you should already know. If you have forgotten some of this, please review that material on your own and(/or) read the relevant textbooks. All of the maths we used in the Cournot example should be familiar to you.

If not, you will probably …nd this

module quite di¢cult.

You should already know:

 what a coordinate system is  how to plot a linear function  how to plot other common functions such as –

x2

–

x3

–

exponential functions

–

logarithmic functions

Coordinate system

Linear functions

Nonlinear functions

Logarithmic

Exponential

Logistic

Note:

When graphing more complicated functions it is

often helpful to study the …rst and second derivatives in order to …nd maximum, minimum and in‡ection points (if there are any).

You should also know  laws of indices/exponents  laws of logarithms

A brief note If

y = ax then we have an exponential function where and is constant.

x

a is the base

is called the exponent (or index) and

it varies. A typical base in both economics as well as mathematics more generally is the natural number

e, which is approx-

imately equal to 2.718... An exponential function with a positive exponent is typically referred to as a growth function while an exponential function with a negative exponent is often referred to as a decay function. above.

We saw examples of such functions

You should be able to:



solve linear equations



solve quadratic equations



use the quadratic formula

De…nition of a function A function is de…ned as follows:

A function of a real variable

x

with

domain

D

is a rule

that assigns a unique real number to each number

D

.

As

x

x

in

varies over the whole domain, the set of all

possible resulting values

Note however,

f (x)

is called the

range f of

.

that this is a slight oversimpli…cation.

More generally, a function is a rule which, to each element in a set in a set

B

.

A

, associates

one and only one

element

y denote the value of f at x then we can write y = f (x), where x is the independent variable, or argument, of f , and y is the If

f

is a function, and if we let

dependent variable. We de…ne:



Domain

as all the values of the independent variable

for which the formula gives a unique value unless otherwise de…ned.



Range as the set of all values assumes on the domain.

f (x) that the function

Examples of functions

A ‘real world’ example of a function of one variable: suppose a bank charges 2.75% on all purchases abroad, plus a …xed fee of 1.25 pounds per purchase. Denoting the transaction cost of a purchase and price paid (in pounds) and allowing only for positive prices we have, y

p

( ) = 1:25 + 0:0275p

y p

A competitor instead simply charges 2.99%. We have, ( ) = 0:0299p

y p

Assume that an individual is planning to deposit 1000 pounds into an account and make one purchase abroad. Then, the domain for both banks is p ever, the two ranges di¤er.

y

2 (1:25; 28:75]

2 (0; 1000].

How-

For bank one the range is

and for bank two it is y

2 (0; 29:90].

Inverse function

Let f be a function with domain A and range B .

f is

one-to-one if and only if it has an inverse function g with domain B and range A. The function g is given by the following rule:

For each y

2

( )

B , the value g y

( ) = y.

unique number x in A such that f x

( ) = x , y = f (x)

g y

It is common to use the notation f

Note that f

1 6=

1 f (x)

is the

Then,

(x 2 A; y 2 B ) 1

= (f (x))1

for g .

An illustration can be given by the formula that converts degrees centigrade to degrees Fahrenheit. denote the temperature in centigrade and

Let

x

y the tempera-

ture in Fahrenheit. Then we can express the temperature in Fahrenheit as a function of centigrade as follows

9 5

y = x + 32 which gives us the inverse function

5 9

x = (y  32) which in turn expresses the temperature in centigrade as a function of Fahrenheit.

Composite functions y = f (u) and u = g (x), then y is a composite function of x that can be written y = f (g (x)) or, alternatively, y = (f  g )(x). If

Note that functions!

(f  g )(x) and (g  f )(x) are typically di¤erent

For instance, if we have

f (x) = x2 + 3

and

g (x) = 5x

then

(f  g )(x) = (5x)2 + 3 = 25x2 + 3

(g  f )(x) = 5(x2 + 3) = 5x2 + 15

Polynomial functions

Previously we have talked about linear and quadratic functions. These functions are polynomial functions. An n-th degree polynomial function can be written as

( ) = anxn + an1xn1 + ::: + a1x + a0

P x

where the

( )=0

P x

a

represent constants and

an

then the equation has at most

6= 0. n

If we set

unique solu-

tions.

Note:

There is no general (algebraic) solution to poly-

nomial equations of 5 or higher degrees. This result is due to Abel/Ru¢ni.

Increasing and decreasing functions

x1 and x2 belong to an interval I and x2 > x1 and f (x2)  f (x1) then f is increasing in I .

If

Substituting



instead of

decreasing function. Similarly, replacing

 ()



gives us the de…nition of a

with

> ( & < imply strictly so. And, if f 0 = 0 then f is constant in I .

Limits An informal de…nition

f is de…ned for all x near a. (It need not be de…ned at x = a). Then we say that f (x) has the number A as its limit as x tends to a, if f (x) tends to A as x tends to (but is not equal to) a.

Suppose, in general, that a function

We write

or

f (x) = A xlim !a f (x) ! A as x ! a

One-sided limits

From below

f (x) = B 

lim

x!a

or

f (x) ! B as x ! a

or

f (x) ! A as x ! a+

From above

f (x) = A +

lim

x!a

It is a necessary and su¢cient condition for

A that the two one-sided limits of f equal.

at

f (x) = xlim !a

a exist and are

A simple graphical example

Formal de…nition f (x) has limit (or tends to) A as x tends to a, and write lim f (x) = A, if for each number " > 0 x!a there exists a number  > 0 such that (s.t.) j f (x)  A j< " for every x with 0 0)

lim sinx x = 1

x!0

x  1 1+ x =e xlim !1

x1 = 1 lim ln( x!1 x)

L’Hôpital’s Rule Suppose that f

and g are di¤erentiable in an interval

(;  ) that contains a, except possibly at a, and suppose that f (x) and g (x) both tend to zero as x tends to a. If 0 g (x) 6=

0

for all x

then

6= a

in

(;  ),

and if

lim

!a

x

0 (x) g 0 (x)

f

= L,

( ) =L lim x!a g (x) f x

This is true whether L is …nite,

1,

or

1.

Note: The rule also applies to other indeterminate forms such as

1 . 1

A simple example Find the limit to

1.

g(x) 3x2x+5 = f (x) = h(x) 5x2+6x3

when

x!

lim g (x) = 1 and lim h(x) = 1 we have that x!1 f (x) is of an indeterminate form 1 lim 1. x!1 Since

x!1

Hence, we apply L’Hôpital’s rule. We have,

g 0(x) = 6x  1

h0(x) = 10x + 6 ) 6  1x 6 3 6x  1 lim f (x) = lim = = lim = x!1 x!1 10x + 6 x!1 10 + 6 5 10 x and

Using special limits: an example

Let us look at a fairly di¢cult limit

lim x

x!1



1 e x



 1 =?

This looks rather complicated. Is there a way to rewrite the problem so that it looks like one of the special limits?

Recall that there was a special limit

ex  1 lim =1 x!0 x

Now, if we set

lim

1

u!0 u

1 x= u ) u = x1



eu  1

=

then, we have

lim eu

1  eu

!

u ! u e 1 =1 = 1  lim u!0 u u!0

Asymptotes Consider the function:

1 f (x) = 1 + x The …rst thing to note about this function is that it is not de…ned for

x = 0 as 1x

the following limits from below and

0: lim 1 + 1x = 1

x!0

lim 1 + 1x = 1

x!0+

x = 0. We have above as x tends to

is not de…ned for

But what happens as

lim 1 +

x!1

x

tends to in…nity?

1 1 1 1 + = 1 + 0 = 1 = lim = 1 + xlim !1 x x!1 x x

x tends to positive and negative in…nity, f (x) tends to 1. We call y = 1 an (horizontal) asymptote to the graph of f (x) as x tends to 1. Likewise x = 0 is an (vertical) asymptote when x tends to 0.

That is, as

Why do we care about limits?

Economists are often interested in statistical properties that rely on asymptotic properties that involve limits.

We are often interested in di¤erentiability.

If functions are continuous, then this typically guarantees the existence of solutions to optimisation problems.

If functions are di¤erentiable, then we can use …rst and second order conditions to identify and classify potential solutions.

We will return to this in a future lecture

Continuity Continuity is a key concept, and can be de…ned as:

f

is continuous at

f (x) = f (a) x = a if xlim !a

If a function is not continuous at

a then it is discontinu-

ous.

Trivially, we have that This implies that:

lim c = c

x!a

f (x) = c and f (x) = x are

and that

lim x = a.

x!a

continuous everywhere.

As it turns out, as obvious as these observations may seem, they are in fact very helpful when paired with the results on the next slide.

If

f

and

g

are continuous at

 f +g

and

f g

 f g

and

f=g

a,

then:

are continuous at

are continuous at

a

a

 [f (x)]r is continuous at a if [f (x)]r is de…ned (where



r

is any real number)

If

f

I,

then its inverse

is continuous and has an inverse on the interval

f 1

is continuous on

f (I ).

Implications

Basically, the previous slide implies that by combining continuous functions by addition, subtraction, multiplications, division and composition, we can obtain new continuous functions.

Combined with

f (x) = c

and

f (x) = x

being contin-

uous everywhere we also have that polynomials are also continuous.

Also, note that di¤erentiability implies continuity.

Is

y = x + 3 everywhere continuous?

y = x2? y = x3 + 5x2 + 10x  12? y = 1x ?

Di¤erentiation 

what is a derivative



how to compute the tangent



the basic rules of di¤erentiation for a single variable



the sum, product, quotient and chain rules

Many economists think that maximising behaviour lies at the heart of economics. Di¤erentiation in turn lies at the heart of maximisation.

If we denote the derivative as f

f

0 (a)

( )

f x

0 (a)

then we have that:

gives the slope of the tangent to the curve y at the point

(a; f (a))

=

De…ning the derivative more precisely we have that:

The derivative of the function

f 0(a), is given by the formula

f

at point

a,

denoted by

f (a + h)  f (a) h h!0 y = f (x) then alternative ways

f 0(a) = lim Note that if

the derivative are:

dy dx

d...