বুধবার, ৪ এপ্রিল, ২০১২

Mathematics for Decision Making Lectures



Class Lecture-1+2+3
Mathematics for Decision Making (Math-1202)

Chapter-1: Equilibrium Analysis

Meaning of Equilibrium:

  1. 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.

  1. 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.

  1. 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-

  1. Partial market Equilibrium
  2. General market Equilibrium
  3. Equilibrium in national income or income determination model


  1. Partial market Equilibrium

Definition:

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

    1. 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:

  1. The quantity demanded of the commodity (Qd)
  2. The quantity supplied of the commodity (Qs)
  3. 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
       =5

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
                                                            2a


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
                                                      2
        =    -3±9
                                                      2
         =    -3+9                    (only considering + value)
                                                       2

           =    6
                                                     2
                                          P       =    3

And
         =    -3 - 9                    (only considering - value)
                                                       2

           =    -12
                                                       2
                                          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
       =5

So, the Equilibrium quantity (Qd)=5

Ans:

Equilibrium quantity (Qd)=5
Equilibrium price (P) = 3


  1. General market Equilibrium

Definition:

  1. 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.

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

  1. 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.

  1. 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
                                    2

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

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

(2) Given,

Qd of a commodity = 24-2P
Qs of the same commodity = -5+7P

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

(3) Given,

Qd of a commodity = 51-3P
Qs of the same commodity = 6P-10

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


(4) Given,

Qd of a commodity = 30-2P
Qs of the same commodity = -6+5P

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

(5) Given,

Qd of a commodity = 4-P²
Qs of the same commodity = 4P-1

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

(6) Given,

Qd of a commodity = 3-P²
Qs of the same commodity = 6P-4

Now find the Equilibrium price (P) and quantity demanded of that commodity (Qd)
(7) Given,

Qd of a commodity = 8-P²
Qs of the same commodity = P²-2

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



General market Equilibrium

(8) Find the equilibrium price and quantity demand for the following two goods in two related markets:

Given,



For goods-1,

Qd1 of goods-1 = 10-2P1 +P2
Qs1 Of goods-1 = -2+3P1

For goods-2,

Qd2 of goods-2 = 15+P1 -P2
Qs2 Of goods-2 = -1+2P2



(9) Find the equilibrium price and quantity demand for the following two goods in two related markets:

Given,



For goods-1,

Qd1 of goods-1 = 18-3P1 +P2
Qs1 Of goods-1 = -2+4P1

For goods-2,

Qd2 of goods-2 = 12+P1 -2P2
Qs2 Of goods-2 = -2+3P2



(10) Find the equilibrium price and quantity demand for three complementary goods in three related markets:

Given,



For goods-1,

Qd1 of goods-1 = 23-5P1 +P2+ P3
Qs1 Of goods-1 = -8+6P1





For goods-2,
Qd2 of goods-2 = 15+P1 -3P2+ 2P3
Qs2 Of goods-2 = -11+3P2


For goods-3,

Qd3 of goods-3 = 19+P1 +2P2 - 4P3
Qs3 Of goods-3 = -5+3P1


(11)  Find the equilibrium price and quantity demand for two complementary goods Slacks (S) and Jackets (J) in two related markets:
Given,



For Slacks (S),

QdS = 410-5PS-2PJ
QsS =-60+3PS

For Jackets (J),

QdJ = 295-PS-3PJ
QsJ = -120+2 PJ



(12) Find the equilibrium price and quantity demand for the following two goods X and Y in two related markets:

Given,



For goods-X,

Qdx = 82-3X +Y
Qsx = -5+15X

For goods-Y,

Qdy= 92+2X -4Y
Qsy = -6+32Y



  Equilibrium in national income or income determination model

(13)Find the equilibrium national income Y and consumption C for two sector economy,, where Y= C+I and given,



C=C0 +bY
C0 = 85
b=0.9
I=I0=55

(14)Find the equilibrium national income Y and consumption C for two sector economy,, where Y= C+I and given,



C=C0 +bYd
Yd=Y-T
T=T0 +tY
C0 = 85
b=0.75
t=0.2
T0 = 20
I=I0=30



(15)Find the equilibrium national income Y and consumption C for three sector economy, where Y= C+I +G and given,



C=25 +6Y½
I=I0=16
G =G0 =14

(16)Find the equilibrium national income Y and consumption C for two sector economy,, where Y= C+I and given,



C=C0 +bY
C0 = 65
b=0.6
I=I0 +aY
I0 = 70
a=0.2

(17)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
I=I0
G=G0
X=X0
M=M0











Attention: All of my class lectures are available at: www.islamiceconomicsbd.blogspot.com

Macroeconomics Class Lecture-3

Class Lecture-3
Macroeconomics in Business (Econ-1202)


Rational Expectations Theory

Economic-behavior observation according to which: (1) On average, people can quite correctly predict future conditions and take actions accordingly, even if they do not fully understand the cause-and-effect (causal) relationships underlying the events and their own thinking. Thus, while they do not have perfect foresights, they construct their expectations in a rational manner that, more often than not, turn out to be correct. Any error that creeps in is usually due to random (non-systemic) and unforeseeable causes. (2) In efficient markets with perfect or near perfect information (such as in modern open-market economies) people will anticipate government's actions to stimulate or restrain the economy, and will adjust their response accordingly. For example, if the government attempts to increase the money supply, people will raise their prices and wage demands to compensate for the inflationary impact of the increase. Similarly, during periods of accelerating inflation, they will anticipate stricter credit controls accompanied by high interest rates. Therefore they will attempt to borrow up to their credit capability, thus largely nullifying the controls. This theory was proposed not as a plausible explanation of human behavior, but to serve as a model against which extreme forms of behavior could be compared. It was developed by the US economist Robert Lucas (born 1937) who won the 1955 Nobel Prize for this insight. Not to be confused with rational choice theory. Also called rational expectations hypothesis.

An economic idea that the people in the economy make choices based on their rational outlook, available information and past experiences. The theory suggests that the current expectations in the economy are equivalent to what the future state of the economy will be. This contrasts the idea that government policy influences the decisions of people in the economy.
The idea is that rational expectations of the players in an economy will partially affect what happens to the economy in the future. If a company believes that the price for its product will be higher in the future, it will stop or slow production until the price rises. Because the company weakens supply while demand stays the same, price will increase. In sum, the producer believes that the price will rise in the future, makes a rational decision to slow production and this decision partially affects what happens in the future.

If we think of a stock price. It is common to assume that the price reflects all of the available information about the stock. If there was other information, someone would make money on the poop stock. This is a similar idea to the expectation that the Rational Expectations economist has when looking at economic agents. While acknowledging that the market for stocks has few of the distortions that other markets have, this paradigm allows for pretty good predictions of behavior, largely as a result of the observation that with large numbers the deviations start to cancel out.

 

Business cycle

  1. A predictable long-term pattern of alternating periods of economic growth (recovery) and decline (recession), characterized by changing employment, industrial productivity, and interest rates. also called economic cycle.
  1. The recurring and fluctuating levels of economic activity that an economy experiences over a long period of time. The five stages of the business cycle are growth (expansion), peak, recession (contraction), trough and recovery. At one time, business cycles were thought to be extremely regular, with predictable durations, but today they are widely believed to be irregular, varying in frequency, magnitude and duration
  1. The business cycle describes the phases of growth and decline in an economy. The goal of economic policy is to keep the economy in a healthy growth rate -- fast enough to create jobs for everyone who wants one, but slow enough to avoid inflation. Unfortunately, life is not so simple. Many factors can cause an economy to spin out of control, or settle into depression. The most important, over-riding factor is confidence -- of investors, consumers, businesses and politicians. The economy grows when there is confidence in the future and in policymakers, and does the opposite when confidence drops.

The Stages of the Business Cycle

There are four stages that describe the business cycle. At any point in time you are in one of these stages:
  1. Contraction - When the economy starts slowing down.
  2. Trough - When the economy hits bottom, usually in a recession.
  3. Expansion - When the economy starts growing again.
  4. Peak - When the economy is in a state of "irrational exuberance."

Chapter-2: Circular Flow of Income and National Income Accounting :

Income

Y=C+S
Y= Income                               Or
C=Consumption
S=Saving





Disposable Income

Y=C+S+R-T
Y= Income
C=Consumption
S=Saving
R=Transfer Payments or Subsidy
T=Tax

GDP=Gross Domestic Product

Definition: GDP is the market value of all the final goods and services produced within a country in a given time period.

Market Value: the prices at which each item is traded in market.

Example: if the price of an apple is 10tk, the market value of 20 apple is 10×20=200tk

Final goods and services: A final good or service is an item that is bought by its final user during a specified time period.

Intermediate goods and services: An Intermediate good or service is an item that is produced by one firm, bought by another firm and used as a component of a final good or services.
Example: Computer-Final goods
                  Motherboard- Intermediate goods

Within a country: only goods and services that are produced within a country count as part of that country’s GDP.

In a given time period: GDP measures the value production in a given time period.
Such as:  Yearly-Annual GDP, Quarterly GDP.


GDP: Production of goods and services by Bangladeshi Residents in Bangladesh + Production of goods and services by Foreigner who lives in Bangladesh.


GNP=Gross National Product

Definition: GNP is the market value of all the final goods and services produced by the residents of a country both at home and abroad in a given time period.

GNP: Production of goods and services by Bangladeshi Residents in Bangladesh + Production of goods and services by Bangladeshi Residents in Abroad -Production of goods and services by Foreigner who lives in Bangladesh.

NDP (Net Domestic Product) at market prices: GDP-Depreciation

NNP (Net National Product) at market prices: GNP-Depreciation=National Income


Depreciation: is the decrease in the stock of capital that results from wear &tear and obsolescence, also called capital consumption.

NDP (Net Domestic Product) at factor cost: NDP-Net Indirect Taxes
Net Indirect Taxes=Subsidy-Indirect Taxes.

NNP (Net National Product) at factor cost: NNP- Net Indirect Taxes
=National Income at factor cost


Personal Income: Income+Transper payments

Personal Disposable Income: personal Income+Transper payments-personal taxes

Personal Income: National Income-social security contribution-corporate income taxes-undistributed profits Transfer payments

National Disposable Income: National Income-social security contribution-corporate income taxes-undistributed profits Transfer payments-personal taxes

Disposable Income: C+S


Measurement of National Income:
  1. Expenditure Approach
  2. Income approach
  3. Value added approach
  1. Expenditure Approach
GDP at market price

GDP=Y=C+I+G+X-M
Y= Income/GDP
C=Personal Consumption Expenditure
I=Gross Private Investment
G=Government purchases of goods and services.
X=Export
M=Import


NDP (Net Domestic Product) at market prices: GDP-Depreciation

NDP mp =C+I+G+X-M-Depreciation

NDP (Net Domestic Product) at factor cost: NDP-Net Indirect Taxes
Net Indirect Taxes=Subsidy-Indirect Taxes.

NDP FC= C+I+G+X-M-Depreciation-Net Indirect Taxes

NNP (Net National Product) at market prices: GNP-Depreciation=National Income

NNP (Net National Product) at factor cost: NNP- Net Indirect Taxes
=National Income at factor cost


  1. Income approach

GDP at market price


GDP mp: Compensation of employees or wage +Net interest+ Rent +corporate profit+ proprietor’s income-indirect taxes+subsidy+depreciation

GDP FC: Compensation of employees or wage +Net interest+ Rent +corporate income tax+ undistributed profits+dividends-indirect taxes+subsidy+depreciation