Linear and Non Linear
Equation for Economics
Dr. Ananda Sabil Hussein
Demand and Supply
Deman and Supply
 𝑄=𝑓 𝑃
 Q = quantity
 P = price
The demand function is the negative
relationship between P and Q which P rises so
Q decreases. The demand equation is:
𝑃 = −𝑎𝑄 + 𝑏
Example
Sketch a graph of demand function
𝑃 = −2𝑄 + 50
Hence, or otherwise, determine the value of
P when Q = 9
Q when P = 10
Market Equilibrium Analysis
(Related to Taxes and Subsidy)
 The demand and supply functions of a good are given by
 𝑃 = −2𝑄 + 50
 𝑃=
1
𝑄
2
+ 25
 Where P, Qd and Qs denote the price, quantity
demanded and quantity supplied respectively.
 Determine the equilibrium price and quantity
 Determine the effect on the market equilibrium if the
government decides to impose a fixed tax of $5 on each
good.
National Income
Determination
 𝐶 = 𝑎𝑌 + 𝑏
 Y= C+S
 C = consumption
 Y = Income
 S = Saving
 Y = C+I+G
 The income that households have to spend on
consumer goods is no longer Y but rather Y – T
(income less tax) is called disposable income Yd.

Given that G = 20 ; I = 35
𝐶 = 0.9𝑌𝑑 + 70
𝑇 = 0.2𝑌 + 25
Calculate the equilibrium level of
national income!
 The simplest non linear function is known as a
quadratic and takes the form
 𝑓 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
 Given the supply and demand functions
 𝑃 = 𝑄𝑠2 + 14𝑄𝑠 + 22
 𝑃 = −𝑄𝐷2 − 10𝑄𝐷 + 150
 Calculate the equilibrium price and quantity.
 Total cost function, TC, relates to the production
costs to the level of output, Q. Total cost consist of
two types elements, fixed cost and variable cost.
 TC = FC + (VC) Q.
 The profit function is denoted by the Greek letter π
and is defined to be the difference between total
revenue, TR, and total cost, TC.

Π = TR – TC
 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.
 For what values of Q does the firm break even?
 What is the maximum profit?
Solution:
We begin by obtaining expressions for the
Total Cost and Total Revenue.
𝑇𝐶 = 𝐹𝐶 + 𝑉𝐶 𝑄 = 4 + 𝑄
𝑇𝑅 = 𝑃 × 𝑄 = 10 − 2𝑄 𝑄 = 10𝑄 − 2𝑄 2
Hence the profit is given by
𝜋 = 𝑇𝑅 − 𝑇𝐶 = 10𝑄 − 2𝑄 2 − 4 + 𝑄
𝜋 = −2𝑄 2 + 11𝑄 − 4
Break even point is illustrated
TR = TC
10𝑄 − 2𝑄2 = 4 + 𝑄
9𝑄 − 4 − 2𝑄2 = 0
2𝑄2 − 9𝑄 + 4 = 0
2𝑄 − 1 𝑄 − 4 = 0
1
𝑄1 =
2
𝑄2 = 4
The number of quantity is even point. Therefore
break even point is happening in the 4 unit of
quantity.
The basic rules of maximum profit is MR = MC
𝑀𝑅 = 𝑇𝑅′ = 10 − 4𝑄
𝑀𝐶 = 𝑇𝐶 ′ = 1
10 − 4𝑄 = 1
9
𝑄 = = 2.25
4
2
𝜋 = −2 2.25 + 9 2.25 − 4 = 6.125
Practice
 Given the quadratic supply and demand functions
 𝑃 = 𝑄2 + 2𝑄 + 12
 𝑃 = −𝑄2 − 4𝑄 + 68
 Determine the equilibrium price and quantity