3.2 Equilibrium and Break-even
Equilibrium in the goods and labour markets
1. Goods Market Equilibrium
Goods market equilibrium occurs when the quantity supplied
(Qs ) by producers
(Qd ) by consumers are equal. Equivalently,
market equilibrium occurs when
Chapter 9
9.2 Matrices
Suppose that a firm produces three types of good, G1, G2 and G3, which it sells to
two customers, C1 and C2. The monthly sales for these goods are given in Table.
Monthly sales for goods
Sold
to C1
customer
C2
G1
7
1
G2
3
5
G3
4
6
D
9.3 Solution of equations: Elimination Methods
Gaussian Elimination
In its simplest form, Gaussian elimination is a technique for solving a system of n
linear equations in n unknowns by systematically adding multiples of equations to
other equations in su
9.4 Determinants
Definition
A square matrix is a matrix in which the number of rows and columns are equal.
Definition
a b
A=
The number ad - bc is called the determinant of a square matrix
and is
c d
written as
det(A )
or
A
ab
or
cd
Example
Find the de
9.5 Matrix inversion
For simplicity we concentrate on 2 x 2 and 3 x 3 matrices, although the ideas and
techniques apply more generally to n x n matrices of any size.
Definition
1 0
is called the unit matrix if it satisfies
0 1
A 2 x 2 matrix I =
AI = A
Section 4.1
MULTIPLE CHOICE
1. Find the simple interest on a $400 investment made for 5 years at an interest rate of 7%/year. What is
the accumulated amount?
a. The simple interest is $140,
the accumulated amount is $540.
b. The simple interest is $115,
t
Section 4.2
MULTIPLE CHOICE
1. Find the amount of an ordinary annuity of 10 yearly payments of $1,800 that earn interest at 10% per
year, compounded annually.
a.
b.
c.
d.
$4,668.74
$28,687.36
$87,798.04
$3,600.00
ANS: B
PTS: 1
2. Find the amount of an ord
Section 4.3
MULTIPLE CHOICE
1. Find the periodic payment R required to amortize a loan of P dollars over t years with interest earned
at the rate of r%/year compounded m times a year. Round your answer to the nearest cent.
P = 120,000, r = 7, t = 8, m = 2
Section 4.4
MULTIPLE CHOICE
1. Find the nth term of the arithmetic progression that has the given values of a = 7, d = 5, and n = 9.
a.
b.
c.
d.
ANS: B
PTS: 1
2. Find the nth term of the arithmetic progression that has the given values of a = 3.5, d = 0.3
4
Mathematics of Finance
Compound Interest
Annuities
Amortization and Sinking Funds
Arithmetic and Geometric
Progressions
4.1
Compound Interest
Simple Interest Formulas
Simple interest is the interest that is
computed on the original principal only.
5.4 Unconstrained optimization
For functions of two variables
z = f(x, y)
the optimum points are found by solving the simultaneous equations
f x ( x, y ) = 0 and f y ( x, y ) = 0
We can classify a optimum point (a, b) as
(1) a minimum (Figure (a) if all t
7.2 Applications of Partial Differentiation
Production function
A production function describes the output, Q, as a function of labour, L and
capital, K. That is
Q = f(L,K)
A production function that is widely used in economics is the Cobb-Douglas
product
Chapter 7
Functions of several variables
7.1 Partial Differentiation
Definition
A function, f, of two variables is written in the general form as
z = f(x, y)
where x and y are the independent variables and z is the dependent variable, that is
its value de
Chapter 4
Non-linear Functions and Applications
4.1 Quadratic, Cubic and Other Polynomial Functions
If the demand and supply graphs are curved and, in these circumstances, it is essential
to model them using more complicated functions.
The simplest non-li
4.2 The exponential functions
Definition
The exponential function to base a is
y = ax
f ( x) = a x
or
x is called the index, power or exponent, this is the variable part of the function
bx = b b L b
Rules of Exponential
1.
b0 = 1
2.
bn =
1
bn
1
n
3.
b = n
4.3 Logarithmic Functions
Definition
If
y = bx
then
logb y = x
and x is called the logarithm of y to base b.
Example
Evaluate
(a) log3 9
(b) log 4 2
1
log 7
(c)
7
(d) log10 100
Rules of Logarithms
1.
logb ( xy ) = logb x + logb y
2.
x
log b = log b x log
Chapter 6
Differentiation and Applications
6.1 Slope of a curve and Differentiation
In section 2.1 the slope of a straight line was defined to be the change in height
(y ) per unit increase in the horizontal distance (x ) . It was also noted that the
slop
6.2 Applications of Differentiation
Marginal functions, Average Functions
In this section we concentrate on three main areas which illustrate differentiation
applicability in economics:
revenue and cost
production
consumption and savings.
Revenue and c
6.3 Optimization for functions of one variable
In this section we show how to locate maximum and minimum points of any
economic function. Look at the graph sketched in Figure. Points B, C, D, E, F and G
are referred to as turning points of the function. A
6.4 Economic Applications of Maximum and Minimum Points
The task of finding the maximum and minimum values of a function is referred to as
optimization. This is an important topic in mathematical economics.
Example
The demand function for a good is given
6.5 Curvature and Other Applications
Second derivative and curvature
The first and second derivatives (y' and y") have already been used to find maximum
and minimum points of various functions. In this section, both y' and y" will be used to
describe curv
6.6 Further Differentiation and Applications
dy
= ex
dx
Rule 5
If y = e x then
Rule 6
If y = ln( x) then
dy 1
=
dx x
Example
Determine the first, second and third derivatives of the following functions:
Y
(a) C = 100 1 e
(b) TC = 10 + ln Q
(
)
Rule 7 The
4.2
Annuities
Annuity is amount of equal (mostly) payments made at regular intervals
of time.
So there is NO PRINCIPAL.
Example:- Monthly deposits into a savings account.
Monthly home mortgage payments.
Book Pg 215- Q1-26 (odd)
Future Value of an Annuity