Consumers Equilibrium Utility Analysis-one and two commodity case

Consumer's Equilibrium - Basic

In order to get the fair idea about this topic we need to understand some basic things associated with this topic.

Marginal Utility of a good -: It is always diminishing. Law of diminishing MU will operate. This law states when a consumer consumes more and more amount of a good, he derives less and less satisfaction from the consumption of successive units.

Marginal Utility of Money -: This is assumed to be constant. If a consumer gets satisfaction worth 4 utils from one rupee, he will get satisfaction worth 400 utils from Rs 100/-.

Marginal Utility of a good in terms of Money  (MUx tm)-: This is the ratio of Marginal utility of a good to MU of money. It is just conversion of Marginal utility of a good in to  money. It means how much money is required to buy the given utility.

Suppose Utility of one rupee = 4 utils

Suppose a consumer is consuming a chocolate and getting satisfaction equals to 100 utils. Then utility of how much money is equal to 100 utils. off course  rupees 25/-

MUxtm = MUx/MUm

= 100/4 = 25/-

Where MUx is marginal utility of a good x
MUm is marginal utility of money
MUxtm is marginal utility of a good in terms of money.


Utility per Ruppee -:  It is the ratio of Marginal Utility of a good to price of that good.

  Utility per rupee = MUx/ Px
Where MUx is marginal utility of a good x
Px is price of good x.

Consumer's Equilibrium - Single Commodity Case

A consumer is said to be in equilibrium when he is maximising his satisfaction.

In case of single commodity the consumer is in equilibrium when marginal utility of a good in terms of money becomes equal to the price of that good.


MUxtm= Px          ..........................................          Condition of equilibrium

where MUxtm = MUx/MUm

Suppose MUm is 1 utils and price of commodity is Rs 4.

UNITS	MUx	MUm	MU in terms of money (MUx/MUm)	Price of good x
1	10	1	10	4      Gains
2	6	1	6	4      Gains
3	4	1	4	4       Eq
4	2	1	2	4       Loss

When a consumer is consuming one unit he is getting utility equals to 10 utils which is equal to 10 rupees. But he is spending only 4 rupees. So the consumer is in gains. He will increase his consumption. when he is  consuming second unit he is getting satisfaction equals to rupees 6 but his expenditure on second unit is rupees 4 only. Again he is in gains.He will increase his consumption.

At third unit his satisfaction is equal to rupees 4 and he is spending rupees 4 for this unit. He is in equilibrium. 

At fourth unit he is in loss as price is more than the satisfaction he is getting from fourth unit. he will buy the fourth unit only when the price of fourth unit is reduced to rupees 2.

So   

 MUxtm  > Px   Gains

 MUxtm  = Px   Equilibrium

 MUxtm  < Px   Loss

The condition in single commodity is     MUxtm  = Px    or      MUx/MUm = Px

This can also be explained as 

    MUxtm  = Px    or  MUx/MUm = Px                                           (i)

or

MUx/Px = MUm                                                                              (ii)

The condition may also become -: when the utility per rupee of good x ( ratio of marginal utility of good x to its price) becomes equal to utility of money for that consumer ,the consumer is said to be in equilibrium.

Q.1 Explain consumer's equilibrium in case of single commodity?

Q.2 Given the price of a commodity how does a consumer decide how many units of that good to buy?

Q. 3 Given the utility of money how does a consumer decide how many units of that good to buy?

Q.4 Explain the conditions of consumer's equilibrium in case of single commodity?

Hints 1. As explained above

          2. by using  MUxtm  = Px

          3. by using MUx/Px = MUm

          4. by explaining 

                     MUxtm  > Px   Gains

                     MUxtm  = Px   Equilibrium

                     MUxtm  < Px   Loss

Consumer's Equilibrium - Two Commodity Case

The condition of Consumers Equilibrium in case of single commodity case is-: When marginal utility of a good in terms of money becomes equal to price of that commodity.


Condition in case of good x -: MUxtm = Px

                                        or     MUx/MUm= Px            (i)               As we know MUxtm=MUx/MUm


Condition in case of good y-: : MUytm = Py

                                        or     MUy/MUm= Py             (ii)


Eq (i) can be written as MUx/Px = MUm   and   Eq (ii) can be written as MUy/Py = MUm


comparing both above

       MUx/Px = MUy/Py = MUm

So in case of two commodities the consumer is said to be in equilibrium when the ratio of marginal utilities of both the goods to their respective prices becomes equal to utility of a rupee.

MUx/Px denotes utility per rupee while consuming good x
MUy/Py denotes utility per rupee while consuming good y

We can also say that when utility per rupee from the last rupee spent on  each good becomes equal the consumer attains equilibrium.

If       MUx/Px > MUy/Py  it means utility per rupee from consumption of good x is greater than utility per rupee from consumption of good y.
The consumer should increase the consumption of good x and  reduce the consumption of good y. By doing so the utility per rupee from good x will decrease and utility per rupee from good y will increase and ultimately both become equal. The consumer will attain equilibrium.

If       MUx/Px < MUy/Py  it means utility per rupee from consumption of good x is less than utility per rupee from consumption of good y.
The consumer should increase the consumption of good y and  reduce the consumption of good x. By doing so the utility per rupee from good y will decrease and utility per rupee from good x will increase and ultimately both become equal. The consumer will attain equilibrium.

Hence in case of two commodities the consumer attains equilibrium when
                                          MUx/Px = MUy/Py = MUm

         

   Q.1   Explain consumer's equilibrium in case of two commodities using utility analysis.?
    Q.2  Explain the conditions of consumers equilibrium in case of two commodities using utility analysis?
Q3. A consumer is consuming two goods and he is in equilibrium. Suppose the price of good y increases. How does this affcet his consumption of both the goods. Explain?      

Hints
1. Explained as above
2. Explained as above    
3. when price will increase (means denominator will increase so the value of fraction will decrease) utility per rupee will fall.
at Equilibrium   MUx/Px = MUy/Py
but when price of y increases then MUx/Px > MUy/Py ( as utility per rupee of good y will fall).
             

Budget Set/ Budget Line Indifference curve Analysis

THERE ARE TWO GOODS    GOOD 1 AND GOOD 2

 PRICES OF TWO GOODS ARE    P1 AND P2

 QUANTITIES OF TWO GOODS CAN BE REPRESENTED BY     X1 AND X2

         INCOME OF THE CONSUMER IS M

 ANY COMBINATION OF GOOD 1 AND GOOD 2 IS KNOWN AS BUNDLE.

  (GOOD 1,GOOD 2)  IN THE SAME WAY (1,2) (3,4) (5,4)  (7,8) (4,3) ARE KNOWN AS BUNDLES.

BUDGET SET

   IT COMPRISES OF SETS OF BUNDLES WHICH A CONSUMER CAN PURCHASE FROM HIS INCOME.

    EQUATION FOR BUDGET SET

       P1 X1 + P2 X2 <= M

BUDGET LINE

   IT COMPRISES OF ALL THOSE BUNDLES WHICH COST THE CONSUMERS EXACTLY EQUAL TO HIS INCOME.

   EQUATION FOR BUDGET LINE           P1 X1 + P2 X2  = M

IT CAN BE DRAWN WITH THE HELP OF HORIZONTAL INTERCEPT AND VERTICAL INTERCEPT.

   HOROZONTAL INTERCEPT -: WHEN A CONSUMER SPENDS HIS ENTIRE INCOME ON THE PURCHASE OF GOOD I.

                                                                 M/ P1

   VERTICAL INTERCEPT -: WHEN THE CONSUMER SPENDS HIS ENTIRE INCOME ON THE PURCHASE OF GOOD 2.

                                           

                                                                   M/P2

•          SLOPE OF BUDGET LINE

   BUDGET LINE IS DOWNWARD SLOPPING MEANS IN ORDER TO HAVE MORE OF ONE GOOD A CONSUMER HAS TO GIVE UP SOME UNITS OF OTHER  GOOD .

    SLOPE OF BUDGET LINE DETERMINES BY HOW MUCH AMOUNT GOOD2 IS GIVEN UP IN ORDER TO HAVE ONE MORE UNIT OF GOOD1.

SUPPOSE WE WANT TO INCREASE THE CONSUMPTION OF GOOD 1 BY ONE UNIT WE HAVE TO SPEND P1  AMOUNT MORE AND SO , WE HAVE TO REDUCE CONSUMPTION ON GOOD 2 BY THIS P1 AMOUNT. 

NOW HOW MANY UNITS OF GOOD 2 WE CAN PURCHASE FROM P1 ?

WE CAN PURCHASE         P1 

UNITS FROM   P1.             P2

NOW SUPPOSE PRICE (P1) OF GOOD 1 =Rs 50

 PRICE (P2) OF GOOD 2 =Rs 20

MONOTONIC PREFERENCES

OUT OF GIVEN TWO BUNDLES A CONSUMER WILL ALWAYS PREFER THE BUNDLE WHICH HAS MORE OF AT LEAST ONE GOOD AND THE OTHER GOOD MUST NOT BE DIMINISHING.

  FOR  EXAMPLE

 (4,5)    &     (5,5)

 (4,5)    &     (5,6)

  But in   this case  -:    (4,5)    &     (5,4)   No ?

MARGINAL RATE OF SUBSTITUTION

 IN ORDER TO INCREASE THE CONSUMPTION OF GOOD1 BY ONE UNIT A CONSUMER HAS TO GIVE UP CERTAIN UNITS OF GOOD2 .

                OR

 THE RATE AT WHICH CONSUMER SUBSTITUTE GOOD1 FOR GOOD2 IS KNOWN AS MARGINAL RATE OF SUBSTITUTION.

DIMINISHING MARGINAL RATE OF SUBSTITUTION

 THR RATE AT WHICH CONSUMER SUBSTITUTE ONE UNIT OF GOOD1 WITH GOOD2 DIMINISHES AS HE INCRESES THE CONSUMPTION OF GOOD1.

u GOOD1        GOOD2          MRS

u     1                 20                 -

u     2                 14                6

u     3                  9                 5

u     4                  5                 4  

u     5                  2                 3

AMONG THESE BUNDLES (1,20) (2,14) (3,9)  (4,5)  AND  (5,2)

THE CONSUMER WILL BE INDIFFERENT AS ONE GOOD IS INCREASING BUT OTHER IS DIMINISHING.ALL {EACH OF} THESE BUNDLES (1,20) (2,14) (3,9)  (4,5)  AND  (5,2) GIVES CONSUMER THE SAME LEVEL OF SATISFACTION.

 INDIFFERENCE CURVE REPRESENTS ALL THOSE BUNDLES AMONG WHICH A CONSUMER REMAINS INDIFFERENT AND GETS THE SAME LEVEL OF SATISFACTION.

CONSUMER'S  EQUILIBRIUM

A CONSUMER IS IN EQUILIBRIUM WHERE THE BUDGET LINE IS TANGENT TO THE INDIFFERENCE  CURVE. IT MEANS THE PREFERENCES OF CONSUMER MATCH WITH HIS PURCHASING POWER. 
WHY SO ?

 WHY NOT AT THE POINTS  WHERE THE INDIFFERENCE CURVE INTERSECTS THE BUDGET LINE.

             POINTS A,B & C LIES ON THE SAME INDIFFERENCE CURVE.THIS MEANS ALL THESE POINTS GIVES SAME SATISFACTION TO CONSUMERS.THE EXPENDITURE ON BUNDLE  A AND B IS EQUAL TO THE INCOME AS THEY LIE ON BUDGET LINE. THE EXPENDITURE ON BUNDLE C COSTS LESS THAN INCOME AS THIS POINT LIES BELOW THE BUDGET LINE.

             NOW OUT OF A ,B &C WHAT THE CONSUMER WILL PREFER TO PURCHASE ?

             OFF COURSE C AS IT COSTS LESS THAN A OR B.
BUT THE CONSUMERS WANTS TO SPEND HIS ENTIRE INCOME. SO THE CONSUMER WILL SHIFT TO HIGHER INDIFFERENCE CURVE.
POINT D LIES ON HIGHER INDIFFERENCE CURVE AND IT ALSO COST EQUAL TO INCOME AS IT LIES ON BUDGET LINE .

            WHEN COMPARED WITH A AND B CONSUMER WILL PREFER D AS IT LIES ON HIGHER INDIFFERENCE CURVE.

Budget line tangent to IC curve

Slope of indifference curve = Slope of budget line

MRSxy = Px/Py

So we can say that the consumer is in equilibrium when marginal rate of substitution between two goods become equal to price ratio of those two goods.

Utility analysis

Want satisfying power of a commodity is called  Utility.

When you consume some thing it means you are getting some utility out of it.
To make understanding of the concept' utility is assumed to be measured in terms of numbers.The units of utility is taken as utils. Utility can be measured cardinally.
So if somebody says to you that he has got 80 utils of satisfaction after watching a movie you should understand that he is not joking.

Total Utility is the total psychological satisfaction a consumer obtains after consuming a particular amount of a good.

Marginal utility is net addition to total utility by consuming one more unit of the good.or Utility derived from the consumption of successive units.

Units Total utility Marginal utility
1 ----------- 50 ---------------- 50
2 ----------- 96 ---------------- 46


or
Units Total utility Marginal utility
5 ------------- 120 ----------------- -
6 ------------- 145 ----------------- 25