Chapter 2: Non-linear equations
Nguyen Thi Minh Tam
ntmtam.vnua@gmail.com
October 24, 2020
1 2.1 Quadratic functions
2 2.2 Revenue, cost and profit
3 2.3 Indices and logarithms
4 2.4 Exponential and Natural logarithm functions
2.1 Quadratic functions
A quadratic function is a function of the form
f (x) = ax 2 + bx + c,
where a, b, c are constants, and a 6= 0.
The graph of a quadratic function is a parabola.
The strategy for sketching the graph of a quadratic function
f (x) = ax 2 + bx + c.
1. Determine the basic shape. The graph has a U shape if
a > 0, and an inverted U shape if a < 0.
2. Determine the y intercept. This is obtained by substituting
x = 0 into the function, which gives y = c.
3. Determine the x intercepts (if any). These are obtained by
solving the quadratic equation
ax 2 + bx + c = 0.
Example 1. Given the supply and demand functions
P = 2QS2 + 10QS + 10
P = −QD2 − 5QD + 52
calculate the equilibrium price and quantity.
2.2 Revenue, cost and profit
The total revenue, denoted by TR, is the amount of money
received by the firm from the sale of its goods.
If Q goods are sold and the price per good is P, then the total
revenue is
TR = PQ
The total cost, denoted by TC, is the amount of money that
the firm has to spend to produce these goods.
The profit function is denoted by π and is defined by
π = TR − TC.
Example 2. Given the demand function
P = 1000 − Q
a) Express TR as a function of Q and hence sketch a graph of
TR against Q.
b) What value of Q maximises total revenue, and what is the
corresponding price?
Fixed costs, FC, are total costs that are independent of output
and include the cost of land, equipment, rent and possibly
skilled labour.
Variable costs, vary with output and include the cost of raw
materials, components, energy and unskilled labour.
If VC denotes the variable cost per unit of output, then the
total variable cost, TVC, in producing Q goods is given by
TVC = (VC)Q
The total cost is
TC = FC + (VC)Q
The average cost function, AC, is defined by
TC FC
AC = = + VC
Q Q
Example 3. Given that fixed costs are 100 and that variable costs
are 2 per unit, express TC and AC as functions of Q. Hence sketch
their graphs.
Note. When the variable cost, VC, is a constant, the graph of the
average cost function is a rectangular hyperbola (L-shaped curve).
Figure: The graph of an average cost function.
Figure: Typical TR and TC graphs sketched on the same diagram when
the demand function is linear and the variable costs are constant.
The two curves intersect at two points, A and B,
corresponding to output levels QA and QB . At these points
the cost and revenue are equal and the firm breaks even.
If Q < QA or Q > QB , then cost exceeds revenue and the
firm makes a loss.
If QA < Q < QB , then revenue exceeds cost and the firm
makes a profit.
Example 4. If fixed costs are 4, variable costs per unit are 1 and
the demand function is
P = 10 − 2Q
obtain an expression for π in terms of Q and hence sketch a graph
of π against Q.
a) For what values of Q does the firm break even?
b) What is the maximum profit?
2.3 Indices and logarithms
Index notation
If M = b n we say that b n is the exponential form of M to
base b.
n is referred to as the index, power or exponent.
If n is a positive integer, then
bn = b × b × . . . × b
b0 = 1
1
b −n =
b√n
n
b 1/n = b = nth root of b
Rule of indices
1. b m × b n = b m+n
bm
2. n = b m−n
b
3. (b m )n = b mn
a n an
4. (ab)n = an b n , = n
b b
The output Q, of any production process depends on factors
of production such as capital K and labour L.
The dependence of Q on K and L may be written as
Q = f (K , L) → production function
A function Q = f (K , L) is said to be homogeneous if there
exists a number n such that:
f (λK , λL) = λn f (K , L)
n is called the degree of homogeneity.
If the degree of homogeneity, n, satisfies:
n < 1, the function is said to display decreasing returns to
scale
n = 1, the function is said to display constant returns to scale
n > 1, the function is said to display increasing returns to
scale.
Example 5. Show that the following production functions are
homogeneous and comment on their returns to scale:
a) Q = 7KL2
b) Q = 50K 1/4 L3/4
Note.
All of the production functions considered so far are of the
form
Q = AK α Lβ ,
where A, α, β are positive constants.
Such functions are called Cobb-Douglas production functions.
They are homogeneous of degree α + β.
Logarithms
If M = b n , then n = logb M (the logarithm of M to base b).
The three rules of logarithms:
1. logb (xy
)= logb x + logb y
x
2. logb = logb x − logb y
y
m
3. logb x = m logb x
2.4 Exponential and Natural logarithm functions
Natural exponential function
The function f (x) = e x , where e = 2.718281828459 . . ., is called
the natural exponential function.
Example 6. The percentage, y , of households possessing
microwave ovens t years after they have been launched is modelled
by
55
y=
1 + 800e −0.3t
1) Find the percentage of households that have microwaves
a) at their launch;
b) after 10 years;
c) after 20 years.
2) What is the market saturation level?
3) Sketch a graph of y against t and hence give a qualitative
description of the growth of microwave ownership over time.
Natural logarithm
If M = e n then n is the natural logarithm of M and we write
n = ln M.
The three rules of logs:
1. ln(xy ) = ln x + ln y
x
2. ln = ln x − ln y
y
3. ln x m = m ln x
Example 7. An economy is forecast to grow continuously so that
the gross national product (GNP), measured in billions of dollars,
after t years is given by
GNP = 80e 0.02t
After how many years is GNP forecast to be $88 billion? What
does the model predict about the value of GNP in the long run?
An economic variable which increases without bound is said to
display unlimited growth.
Exercises
Exercise 1.3, 1.3*: 2, 6ac, 10 (page page 52, 53), 6 (page 54)
Exercise 1.4, 1.4*: 2 (page 45), 2, 4, 7 (page 66)
Exercise 1.5, 1.5*: 3, 4, 8 (page 81), 7 (page 83)
Exercise 1.6: 2, 3, 5, 7 (page 91,92)
Exercise 2.1, 2.1*: 9, 11 (page 137, 138), 11, 12 (page 139)
Exercise 2.2: 9, 11 (page 149)
Exercise 2.3, 2.3*: 5 (page 168-169), 16, 17 (page 171)
Exercise 2.4, 2.4*: 1 (page 182), 5, 7, 1 (page 183), 3 (page 184),
9 (page 185)
