Islamic Economics: Mathematics for Decision Making Lectures

Class Lecture-1+2+3

Mathematics for Decision Making (Math-1202)

Chapter-1: Equilibrium Analysis

Meaning of Equilibrium:

Equilibrium is a constellation of selected interrelated variables so adjusted to one another that no inherent tendency to change prevails in the model which they constitute.

Equilibrium in Demand and Supply analysis occurs when Qd=Qs.by equating the supply and demand functions, the equilibrium price and quantity can be determined.

The state in which market supply and demand balances each other and, as a result, prices become stable. Generally, when there is too much supply for goods or services, the price goes down, which results in higher demand. The balancing effect of supply and demand results in a state of equilibrium.

Types of Equilibrium:

There are three types of Equilibrium. Such as-

Partial market Equilibrium
General market Equilibrium
Equilibrium in national income or income determination model

Partial market Equilibrium

Definition:

Partial market Equilibrium refers to the situation where Equilibrium position is analyzed when only one commodity (isolated market) is being considered.

A partial equilibrium is one which is based on only a restricted range of data, a standard example is price of a single product, the prices of all other products being held fixed during the analysis.

There are three variables in the model as following:

The quantity demanded of the commodity (Qd)
The quantity supplied of the commodity (Qs)
The price (P)

Example:

(1)Given,

Qd of Apple = -5+3p

Qs of Apple =10-2p

Now find the Equilibrium price (P) and quantity demanded of Apple (Qd)

Ans:

We know that,

Equilibrium=

Qd=Qs

-5+3p=10-2p

or, 3p+2p=10+5

or, 5p=15

or, p=15/5

or, p=3

Equilibrium price (P) of Apple = 3

Now finding Equilibrium quantity demanded of Apple (Qd),

Substituting the value of P into Demand equation of Apple (Qd), we get,

Qd = -5+3p

= -5+ (3×3)

= -5+ 9

= 4

So, Equilibrium quantity demanded of Apple (Qd) =4

Ans: Equilibrium price (P) of Apple = 3

Equilibrium quantity demanded of Apple (Qd) =4

Example:

(2)

Demand and supply equation of a certain commodity are Qd =14-3p and

Qs =P²-4. Find the equilibrium price (P) and quantity (Qd).

(Solve by factoring)

Ans:

We know that,

Equilibrium=

Qd=Qs

14-3p= P²-4

Or, 14-3p – (P²-4)=0

Or, 14-3p – P²+4 =0

Or, 18-3P- P²=0

Or, - P²-3p+18=0

Or, - (P²+3p-18)=0

Or, P²+3p-18=0

We can solve it either by factoring or by quadratic formula; here we will apply factoring,

Or, P²+6p-3p-18=0

Or, P (P+6)-3(p+6) =0

Or, (P+6) (p-3) =0

Now set,

(P+6) =0 and (p-3) =0

Or, P = -6 and P = 3

Since, negative price is not meaningful. So we take only positive value.

Equilibrium price (P) = 3

Now, finding the Equilibrium quantity (Qd),

Qd =14-3p

=14-(3×3)

=14-9

So, the Equilibrium quantity (Qd)=5

Ans:

Equilibrium quantity (Qd)=5

Equilibrium price (P) = 3

Example:

(2)Demand and supply equation of a certain commodity are Qd =14-3p and

Qs =P²-4. Find the equilibrium price (P) and quantity (Qd).

(Solve by quadratic formula)

Ans:

We know that,

Equilibrium=

Qd=Qs

14-3p= P²-4

Or, 14-3p – (P²-4)=0

Or, 14-3p – P²+4 =0

Or, 18-3P- P²=0

Or, - P²-3p+18=0

Or, - (P²+3p-18)=0

Or, P²+3p-18=0

We can solve it either by factoring or by quadratic formula; here we will quadratic formula,

AX²+bX+C=0

Xe = -b±√b²-4ac

Here,

P²+3p-18=0

Let,

a=1

b=3

X=P

C= -18

Now substituting the above values in the formula, we get,

P = -3±√3²-4.1.-18

2.1

= -3±√9+72

2.1

= -3±√81

= -3±9

= -3+9 (only considering + value)

= 6

P = 3

And

= -3 - 9 (only considering - value)

= -12

P = - 6

Since, negative price is not meaningful. So we take only positive value.

Equilibrium price (P) = 3

Now, finding the Equilibrium quantity (Qd),

Qd =14-3p

=14-(3×3)

=14-9

So, the Equilibrium quantity (Qd)=5

Ans:

Equilibrium quantity (Qd)=5

Equilibrium price (P) = 3

General market Equilibrium

Definition:

General market Equilibrium refers to the situation where Equilibrium position is analyzed when more than one commodity or two or more commodities are being considered.

When several interdependent commodities are simultaneously considered in equilibrium analysis that is called General market Equilibrium

General equilibrium theory studies supply and demand fundamentals in an economy with multiple markets, with the objective of proving that all prices are at equilibrium. The theory analyzes the mechanism by which the choices of economic agents are coordinated across all markets.

General equilibrium theory is distinguished from partial equilibrium theory by the fact that it attempts to look at several markets simultaneously rather than a single market in isolation.

Example:

Find the equilibrium price and quantity demand for two substitute goods Beef (B) and Mutton (M) in two related markets:

Given,

For Beef,

Qd of Beef = 82-3PB+PM

Qs Of Beef =-5+15 PB

For Mutton,

Qd of Mutton = 92+2PB-4PM

Qs Of Mutton = -6+32 PM

Ans:

For Beef,

We know that,

Equilibrium=

Qd=Qs

Or, Qd- Qs=0

82-3PB+PM=-5+15 PB

or, 82-3PB+PM - (-5+15 PB)

or, 82-3PB+PM + 5-15 PB = 0

or, 87-18 PB+PM = 0

or, -18 PB+PM = -87

or, PM = -87 + 18 PB

or, PM = 18 PB-87

Similarly For Mutton,

We know that,

Equilibrium=

Qd=Qs

Or, Qd- Qs=0

92+2PB-4PM = -6+32 PM

or, 92+2PB-4PM – (-6+32 PM)

or, 92+2PB-4PM +6-32 PM = 0

or, 98+2PB-36 PM = 0

or, 2PB-36 PM = - 98

or, 2PB = - 98+36 PM

or, 2PB = 36 PM – 98

or, PB = 36 PM – 98

or, PB = 18 PM – 49

Now finding PM,

PM = 18 PB-87

= 18 (18 PM – 49)-87 (substituting the value of PB)

= 324 PM – 882-87

PM - 324 PM = - 969

- 323 PM = - 969

PM = 969

323

PM = 3

So equilibrium price of Mutton PM = 3

Now finding PB,

PB = 18 PM – 49

= (18× 3)– 49

= 54 – 49

= 5

So equilibrium price of Beef PB = 5

Now finding Qd of Beef

QdB = 82-3PB+PM

= 82-3(5)+3 (substituting the value of PB & PM )

= 82-15+3

= 85-15

= 70

SO equilibrium Quantity demand of Beef QdB = 70

Now finding Qd of Mutton,

QdM= 92+2PB-4PM

= 92+2(5)-(4×3)

= 92+10-12

= 90

SO equilibrium Quantity demand of Mutton QdM = 90

Ans:

Equilibrium price of Mutton PM = 3

Equilibrium price of Beef PB = 5

Equilibrium Quantity demand of Beef QdB = 70

Equilibrium Quantity demand of Mutton QdM = 90

3. Equilibrium in national income or income determination model

Equilibrium in national income defines as the level of national income where economy’s total national income equals to total national expenditures.

i.e.

Total national income = total national expenditures

Y= C+I+G+X-M

Where,

Y= total national Income

C=Personal Consumption Expenditure

I=Gross Private Investment

G=Government purchases of goods and services or Government expenditures

X=Export

M=Import

Example:

Find the equilibrium national income Y and consumption C for four sector economy, where Y= C+I+G+X-M and given,

C=C0 +bYd

Yd=Y-T

T=T0 +tY

C0 = 85

b=0.6

t=0.2

T0 = 20

I=I0=30

G=G0=60

X=X0=80

M=M0=50

Ans:

Y= C+ I+G+X-M

= C0 +bYd + I0+ G0+ X0- M0

= C0 +b(Y-T) + I0+ G0+ X0- M0

= C0 +b(Y-( T0 +tY) + I0+ G0+ X0- M0

= C0 +b(Y- T0 - tY) )+ I0+ G0+ X0- M0

= C0 +bY- bT0 - btY+ I0+ G0+ X0- M0

=85+0.6Y-(0.6×20) – (0.6× 0.2)Y+30+60+80-50

=85+0.6Y-12 – 0.12Y+120

Y =205-12 +0.48Y

Y-0.48Y =193

0.52Y = 193

Y= 193

0.52

Y= 371.153846

SO equilibrium national income Y = 371.153846

Now finding equilibrium consumption C,

C=C0 +bYd

= C0 +b(Y-T)

= C0 +b(Y-( T0 +tY)

= C0 +b(Y- T0 - tY)

= C0 +bY- bT0 - btY

=85+0.6(371.153846)-(0.6×20) – (0.6× 0.2) (371.153846)

=85+222.692308-12 – 44.5384615

=307.692308 – 56.5384615

C=251.153847

SO equilibrium Consumption C=251.153847

Ans:

Equilibrium national income Y = 371.153846

Equilibrium Consumption C=251.153847

Home work from previous questions

Partial market Equilibrium

(1) Given,

Qd of a commodity = 51-3P

Qs of the same commodity = -12+6P