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
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
9.3 Solution of equations: Elimination Methods
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
A square matrix is a matrix in which the number of rows and columns are equal.
The number ad - bc is called the determinant of a square matrix
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.
is called the unit matrix if it satisfies
A 2 x 2 matrix I =
AI = A
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,
1. Find the amount of an ordinary annuity of 10 yearly payments of $1,800 that earn interest at 10% per
year, compounded annually.
2. Find the amount of an ord
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
1. Find the nth term of the arithmetic progression that has the given values of a = 7, d = 5, and n = 9.
2. Find the nth term of the arithmetic progression that has the given values of a = 3.5, d = 0.3
Mathematics of Finance
Amortization and Sinking Funds
Arithmetic and Geometric
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
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
Functions of several variables
7.1 Partial Differentiation
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
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
The exponential function to base a is
y = ax
f ( x) = a x
x is called the index, power or exponent, this is the variable part of the function
bx = b b L b
Rules of Exponential
b0 = 1
b = n
4.3 Logarithmic Functions
y = bx
logb y = x
and x is called the logarithm of y to base b.
(a) log3 9
(b) log 4 2
(d) log10 100
Rules of Logarithms
logb ( xy ) = logb x + logb y
log b = log b x log
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
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
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.
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
6.6 Further Differentiation and Applications
If y = e x then
If y = ln( x) then
Determine the first, second and third derivatives of the following functions:
(a) C = 100 1 e
(b) TC = 10 + ln Q
Rule 7 The
Annuity is amount of equal (mostly) payments made at regular intervals
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