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vector valued function

1. Basic microeconomics is about the allocation of wealth or expenditure among different physical goods. This gives us relative prices.

2. Basic finance is about the allocation of expenditure across two or more time periods. This gives us the term structure of interest rates.

3. The next step is the allocation of expenditure across (a finite number or a continuum of) states of nature. This gives us rates of return on risky assets, which are random variables.

Then we can try to combine 2 and 3. Finally we can try to combine 1 and 2 and 3. Thus finance is just a subset of micoreconomics. What do consumers do?

They maximise utility given a budget constraint, based on prices and income. What do firms do?

They maximise profits, given technological constraints (and input and output prices). Microeconomics is ultimately the theory of the determination of prices by the interaction of all these decisions: all agents simultaneously maximise their objective functions subject to market clearing conditions.

What is Mathematics?

This section will have all the stuff about logic and proof and so on moved into it.



NOTATION

NOTATION

Throughout the book, x etc. will denote points of Ша for n > l and x etc. will denote points of -ft or of an arbitrary vector or metric space X. X will generally denote a matrix.

Readers should be familiar with the symbols V and 3 and with the expressions such that and subject to and also with their meaning and use, in particular with the importance of presenting the parts of a definition in the correct order and with the process of proving a theorem by arguing from the assumptions to the conclusions. Proof by contradiction and proof by contrapositive are also assumed. There is a book on proofs by Solow which should be referred to here.1 N = jx G KN : Xj > 0,i = 1,...,NJ is used to denote the non-negative or-

thant of 3£N, and = {x eSN : Xj > 0,i = l,..., N} used to denote the positive orthant.

T is the symbol which will be used to denote the transpose of a vector or a matrix.

1 Insert appropriate discussion of all these topics here.



NOTATION



Part I

MATHEMATICS





Chapter 1

LINEAR ALGEBRA

1.1 Introduction

[To be written.]

1.2 Systems of Linear Equations and Matrices

Why are we interested in solving simultaneous equations? We often have to find a point which satisfies more than one equation simultaneously, for example when finding equilibrium price and quantity given supply and demand functions.

To be an equilibrium, the point (Q,P) must lie on both the supply and demand curves.

Now both supply and demand curves can be plotted on the same diagram and the point(s) of intersection will be the equilibrium (equilibria):

solving for equilibrium price and quantity is just one of many examples of the simultaneous equations problem

The ISLM model is another example which we will soon consider at length.

We will usually have many relationships between many economic variables defining equilibrium.

The first approach to simultaneous equations is the equation counting approach:



a rough rule of thumb is that we need the same number of equations as unknowns

this is neither necessary nor sufficient for existence of a unique solution,

- fewer equations than unknowns, unique solution:

x2 + y2 = 0 == x = 0,y = 0

- same number of equations and unknowns but no solution (dependent equations):

x + y = 1 x + y = 2

- more equations than unknowns, unique solution:

x + y = 2 x - 2y + 1 = 0

== x =1, y = 1

Now consider the geometric representation of the simultaneous equation problem, in both the generic and linear cases:

two curves in the coordinate plane can intersect in 0, 1 or more points

two surfaces in 3D coordinate space typically intersect in a curve

three surfaces in 3D coordinate space can intersect in 0, 1 or more points

a more precise theory is needed

There are three types of elementary row operations which can be performed on a system of simultaneous equations without changing the solution(s):

1. Add or subtract a multiple of one equation to or from another equation

2. Multiply a particular equation by a non-zero constant

3. Interchange two equations



Note that each of these operations is reversible (invertible).

Our strategy, roughly equating to Gaussian elimination involves using elementary

row operations to perform the following steps:

1. (a) Eliminate the first variable from all except the first equation

(b) Eliminate the second variable from all except the first two equations

(c) Eliminate the third variable from all except the first three equations

(d) &c.

2. We end up with only one variable in the last equation, which is easily solved.

3. Then we can substitute this solution in the second last equation and solve for the second last variable, and so on.

4. Check your solution!!

Now, let us concentrate on simultaneous linear equations: (2 x 2 EXAMPLE)

x + y = 2 (1.2.1)

2y - x = 7 (1.2.2)

Draw a picture

Use the Gaussian elimination method instead of the following

Solve for x in terms ofy

x = 2 - y x = 2y - 7

Eliminate x

2 - y = 2y - 7

Find y

3y = 9 y = 3

Find x from either equation:

x = 2 - y = 2 - 3 = -l x = 2y - 7 = 6 - 7 = -l



SIMULTANEOUS LINEAR EQUATIONS (3 x 3 EXAMPLE) Consider the general 3D picture ...

Example:

x + 2y + 3z = 6 4x + 5y + 6z = 15 7x + 8y + 10z = 25

(1.2.3) (1.2.4)

(1.2.5)

Solve one equation (1.2.3) for x in terms of y and z:

x = 6 - 2y - 3z

Eliminate x from the other two equations:

4 (6 - 2y - 3z) + 5y + 6z = 15 7(6 - 2y - 3z ) + 8y + 10z = 25

What remains is a 2 x 2 system:

-3y - 6z = 6y - 11 z =

Solve each equation for y:

y = 3 - 2z

17 11

y =---z

y 6 6

Eliminate y:

3 - 2z =---z

6 6

Find z:

= -z

Hence y = 1 and x = 1 .



1.3 Matrix Operations

We motivate the need for matrix algebra by using it as a shorthand for writing systems of linear equations, such as those considered above.

The steps taken to solve simultaneous linear equations involve only the coefficients so we can use the following shorthand to represent the system of equations used in our example:

This is called a matrix, i.e.- a rectangular array of numbers.

We use the concept of the elementary matrix to summarise the elementary row operations carried out in solving the original equations:

(Go through the whole solution step by step again.)

Now the rules are

- Working column by column from left to right, change all the below diagonal elements of the matrix to zeroes

- Working row by row from bottom to top, change the right of diagonal elements to 0 and the diagonal elements to 1

- Read off the solution from the last column.

Or we can reorder the steps to give the Gaussian elimination method: column by column everywhere.

Two n x m matrices can be added and subtracted element by element.

There are three notations for the general 3 x 3 system of simultaneous linear equations:

1. Scalar notation:

Matrix Arithmetic

&3ixi + 32x2 + 33x3

11x1 + 12x2 + 13x3

21x1 + 22x2 + 23x3



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