# 03_UTILITY ANALYSIS class 12

UTILITY

Utility is a power of goods and services to satisfy a human wants.

UTILIT FUNCTION

Utility function explains the relationship between the utility of the commodity and the units of the commodity consumed.

Ux = f(qx)

TYPES OF UTILITIES

1. Initial Utility

The utility that we obtain from the consumption of first unit of a commodity is called initial utility.

1. Marginal Utility

Marginal utility refers to the net addition to the total utility, when one more unit of the commodity is consumed. It also refers to the utility derived from last unit of the commodity consumed. Marginal utility is slope of total utility curve.

Or                 where  = Change in the units consumed.

1. Total Utility

Total utility means the total satisfaction obtained by the consumer from the consumption of all the units at a given time period.

LAWS OF DIMINISHING MARGINAL UTILITY

According to the law, as more and more units of commodity are consumed,the extra utility that we derive from it goes on declining. Total utility will continue to rise till the point of a consumption when the marginal utility becomes zero. After this point, MU becomes negative which means now the good begin to harm consumer.

Assumption of Law of Diminishing Marginal Utility

1. Utility can be measured in numeric terms.
2. The consumption takes place in the stipulated time period (in continuation).
3. All the consumers are assumed to be rational.
4. Marginal utility of rupee is assumed to be constant (Mu is the Mu per unit of the money income).

 Units consumed MU TU Units Consumed 1 12 12 2 9 21 3 5 26 4 0 26 5 -2 24 Relationship Between Tu and Mu No. of Units consumed MU TU 1 20 20 2 12 32 3 6 38 4 0 38 5 -2 36
1. TU increases with the subsequent consumption of the commodity as long as MU falls but remains positive.
2. TU is maximum when MU is zero.
3. With the further increase in consumption marginal utility of commodity will be negative and Tu starts falling.

Consumer Equilibrium (In case of Single Commodity)

Consumer equilibrium refers to asituation when a consumer gets maximum satisfaction out of the money income he spends on a commodity.

Attainment or Condition

Consumer equilibrium is attained at a point where marginal utility in monetary terms from the units consumed equals the price of the commodity. Utility obtained is the benefit, while price paid is the cost. Consumer compares the benefits and the cost. He will buy the commodity only if the benefits are greater than or atleast equal to cost.

For example :- Marginal

 Units consumed Mu Price Gain Total Gain 0 – – – – 1 6 3 3 3 2 5 3 2 5 3 4 3 1 6 4 3 3 0 6 5 2 3 -1 5

The price of the commodity is ` 3/Unit. Suppose the utility from the first unit is 6utils, the consumer will buy because,benefit is more then cost. On the other hand, consumer will not buy the fifth units because, utility is2utils i.e. less than the price.

The difference between utility and price of the commodity represents the gain to the consumer the total gain from the four units is (3 + 2 + 1) i.e.,  ` 6. If he buys the fifth. Units, the total gain becomes (3 + 2 + 1 + 0 + (-1) = 5. The consumer maximizes the gain, when he buys 4 units.

Thus in a single commodity case, the consumer is at equilibrium, when Mux = Px following are the two conditions of consumer equilibrium.

The consumer  purchases, when Mux>Px. As a rational consumer, he will continue to purchase, so long as MU exceeds the price or greater than price. As the consumption increases. Mux falls due to the law of diminishing Mux. It will get equal to  price which is equilibrium.

1. Total Gain Falls as More is Consumed

It means that consumer should continue to purchase so long as gain is increasing. Thus the purchase of 4 units satisfies both the conditions.

Q.1.   Derive “Law of Demand” with the help of consumer equilibrium in one commodity case?

Ans.  Law of demand states that there is an inverse relationship between the price and the quantity, demanded, others factor remaining constant. It can be derived with the help of cardinal approach of Marshal.

Consumer purchases the good upto a point where,

Mux = Px

When Px falls, it means Mux>Px.

Since Mux>Px it will induce the consumer to buy more units of the good. Infact the consumer must buy more till the point he reach the equilibrium. Thus, it shows that when price falls its demand rise.

 GRAPHIC REPRESENTATION           In the figure. Mu is the marginal utility curve having downward slope indicating that Mu declines with the consumption of successive units. OP is the price which is horizontal line. Consumer equilibrium is           attained at a point E, where MU           curve intersects the price line.

Assumptions of Consumer Equilibrium

1. Consumer is rational
2. Utility can be measured numerically.
3. Income of the consumer and the price of the commodity remains                         fixed.
4. Goods are neither complementary nor substitutes for each other.
5. Tastes and preference of the consumer does not change.

TWO COMMODITY CASE – CONSUMER EQUILIBIRUM

Following conditions must be fulfilled to achieve the consumer        equilibrium :

1. The ratio of the MU to price of x must be equal to the ratio of                        MU andprice of y.

This is known as law of equi marginal utility.

It means the equality of the MU of the last rupee spent on each                           good.

If Mux/Px is greater than Muy/Py. It means that Mu from the last rupee spent on x is greater than Mu of the last rupee spent on y. This induces the consumer to transfer the expenditure from y to x. The consumption of x rise and Mux falls, and that of y rises. This act continues till Mux/Px and Muy/Py are equal.

1. MU of the good falls as more of it is consumed.

This condition is nothing but law of diminishing MU is in operation. It is a condition because if the law is not operating, the law of equi marginal utility will not operate.

For e.g. : A consumer consumes 2 commodities x and y for ` 3/unit and ` 2/unit respectively. It is assumed that`2.

 Units MUx MUy MUx/Px MUy/Py 1 18 16 6 8 2 15 12 5 6 3 9 8 3 4 4 6 4 2 2 5 3 1 1 0.5

Consumer will be at equilibrium, when he consumes 4 units of X and 4 units of Y.

Consumer Preference

Consumer preferences are the consumer’s wants. These preferences are represented graphically through an indifference curve and numerically through in utility analysis.

INDIFFERENCE CURVE

Indifference curve shows combination of two products that provides the same satisfaction to the consumer. The consumer is indifferent between the combinations indicated by any two points on one indifference curve.

For eg.

 Combinations Commodity X Commodity Y 1st 20 1 2nd 14 2 3rd 9 3 4th 5 4 5th 3 5 6th 2 6

The shape of indifference curve is convex because it has decreasing MRS and the total utility on every point on the IC is always constant.

Marginal Rate of Substitution (MRS xy)

MRS is the rate at which consumer is willing to give up one good for the additional unit of another commodity while keeping the same level of satisfaction.

 PRINCIPLE OF DIMINISHING MRS It statics that with every increase in the quantity of commodity x a consumer shall be willing to sacrifice lesser quantity of commodity y. As consumer goes on increasing the consumption of commodity x by one unit the level of sacrifice for y goes on decreasing. Ay1> By2> Cy3> Dy4 > Ey5 Because of this principle only the shape of indifference curve is convex.   It is because of diminishing Marginal utility as.With every increase in the consumptionutility is decreases so the consumer will sacrifice lessor quantity only. INDIFFERENCE MAP It is a collection of indifference curves corresponding to different level of satisfaction. The indifference curve closer to origin indicates lower level of satisfaction while the curve away from the origin indicates higher level of satisfaction.

MONOTONIC PREFERENCES

Indifference map is based on the assumption i.e. more goods more utility. It implies that utility is an increasing function of consumption. As consumption increases total utility also increases along with.

 Properties of Indifference Curves 1.      Indifference curves are downward sloping from left to right :It implies that consumer is willing to give up some units of one commodity to get more units of another commodity so as to stay at the same level of satisfaction. 2.      Indifference curves are convex to the origin :It implies that marginal rate of substitution diminishes along an indifference curve. This implies that as consumer gets more and more of one good, he is willing to give up less and less of another.

3       Indifference curves do not intersect :      Two indifference curves can never intersect. In the diagram, point A and B lie on the same indifference curve IC1 giving same level of satisfaction while point B and C also lie on the same indifference curves IC2 giving same level of satisfaction this implies that combination A and C should also give same level of satisfaction, but this is not possible because combinations C lies on the higher indifference curve than A.

4       Higher indifference curve represents higher level of satisfaction, while lower indifference curve gives lower level of satisfaction because at higher IC, more quantities of both the goods are consumed than at lower IC.

BUDGET SET

Budget set is the list of different combinations of two commodities that a consumer can buy from his income.

For e.g. : Suppose income of consumer = ` 500.

Px = `10 per u nit   Py = `5 per unit

 Quantity of x (units) Quantity of y (units) Total Expenditure (`) 0 100 500 10 80 500 20 60 500 30 40 500 40 20 500 50 0 500

A consumer can spend her income in such a way that expenditure on two commodities is less than or equal to his income.

This is known as budget constraint

[Budget equation]

BUDGET LINE

The graphical representation of the different combination of the goods that a consumer can buy given his money income and the prices of the commodities.

All points on or below the budget line are affordable whereas points above the budget line is not affordable.

 Slope of Budget Line           Slope of the budget line is always downward implying that if consumer wants to increase his consumption, he has to give up some quantity of another commodity. The slope of the budget line is the ratio of the price of the two commodities i.e. Slope of budget line = MRE Budget line is also known as price line because it shows the ratio of the price of two commodities. Shift in the Budget Line           Budget line can be shifted rightwards or leftwards due to any of the two reasons. 1.       Change income 2.       Change in price of commodity Change in Income           With the rise in the income of the consumer, the consumer’s budget line shifts rightwards indicating that now the consumer can purchase more quantities of both the commodities. With the fall in the income of the consumer, the consumer’s budget line shifts leftwards indicating that now the consumer can purchase less. For e.g. Consumer income rises from`500 to` 800. Px =` 10                Py = ` 5 QxÞ50                 Qx = 80 QyÞ100               Qy = 160

 Change in Prices           With the rise in the price of a commodity, the budget line shifts leftwards and slope of the budget line gets changed. For e.g. If the price of x rises from `10 to`20/unit. Now, consumer can purchase only 25 units of x and 100 units of y on spending whole of its income. With the fall in the price of one commodity, the budget line shifts rightward and slope of the budget line also changes, indicating now the consumer can purchase more of x commodity. For e.g. Px changes from `10to `5.

CONSUMER EQUILIBRIUM THROUGH INDIFFERENCE CURVE APPROACH

The consumer attains the equilibrium, when two conditions are fulfilled.

1. MRS must be equal to MRE

MRS Þ MRE

MRS is the rate at which consumers is willing to sacrifice Y to obtain one more unit of x. MRE is the rate at which the consumer actually sacrifice Y to obtain one more unit of x. The consumer will continue to transact so long as MRS ≥ MRE.

As he buys more of x, MRS keeps on declining and at some stage becomes equal to MRE. At this point, consumer stops purchasing more. Thus the consumer reaches the equilibrium.

The two slopes are equal where budget line is tangent to the indifference curve. This is fulfilled at the point E, which is consumer equilibrium.

When MRS > MRE, consumer will increase the consumption.

When MRS < MRE, consumer will decrease the consumption.

The consumer can afford, all the combinations on the budget line. But E is the equilibrium position because combination A and B lie on the lower indifference curve IC1 giving lower level of satisfaction.

The combination R is  providing higher satisfaction but it is not affordable by the consumer.

1. MRS Continuously Falls/IC must be Convex to the Origin

This ensures that of MRS not equal to MRE, the falling MRS will lead to equality again.

This condition means that IC is convex to the origin i.e. consumer is willing to sacrifice lesser and lesser units of one commodity to obtain more and more of another.

 EXERCISE

Multiple Choice Questions

1. The necessary condition for consumer’s equilibrium in case of two commodities is

(a)                                    (b)

(c) Mux=Px                                         (d) none of the above

1. If then to reach the equilibrium position, consumer should

(a) stop buying both the commodities

(b) Buy more of Y and less of X

(c) Buy more of X and less of Y

(d) Buy both the commodities in equal quantities

1. Total utility is __________________ at the point of satiety

(a) zero                                              (b) maximum

(c) minimum                                      (d) none of the above

1. If Mux is 40 and Muy is 30 and price of Y is Rs. 9/unit. What will be the price of X

(a) Rs. 10/unit                                  (b) Rs. 9/unit

(c) Rs. 12/unit                                  (d) Rs. 8/unit

1. Utility differs from

(a) person to person                           (b) time to time

(c) product to product                        (d) All of the above

1. The further the indifference curve from the origin

(a) lower is the satisfaction level        (b) higher is the satisfaction level

(c) same is the satisfaction level         (d) nothing about satisfaction level

1. Slope of budget line is

(a) MRS                                             (b) MRE

(c) MRT                                              (d) MOC

1. MRE refers to

(a) Marginal rate of exchange            (b) Market rate of exchange

(c) Marginal rate of economics           (d) none of the above

1. Principle of diminishing MRS leads to

(a) Downward sloping indifference curves

(b) concave shape of indifference curves

(c) convex shape of indifference curves

(d) none of these

1. Monotonic preferences is the assumption of

(a) indifference curves                        (b) indifference map

(c) budget line                                   (d) consumer equilibrium

1. Marginal utility is the

(a) utility from last unit consumed    (b) utility from each unit consumed

(c) utility from all the units consumed        (d) none of these

1. When Mu is falling and negative

(a) TU is rising                                   (b) TU is maximum

(c) TU is falling                                  (d) None of these

1. Budget line shifts because of

(a) change in income                         (b) change in price of X

(c) change in price of Y                      (d) all of the above

1. Consumer preference are measured through

(a) cardinal utility                              (b) ordinal utility

(c) both (a) and (b)                             (d) none of these

1. Total utility means

(a) utility from last unit of the commodity consumed

(b) utility from each unit of the commodity consumed

(c) utility from all the units of the commodity consumed

(d) none of the these

1. When Mux<Px, consumer will

(a) Increase the consumption            (b) Decrease the consumption

(c) Will not react                                (d) Be at equilibrium

1. Point outside the budget line indicates

(a) Consumer can afford the commodities

(b) Consumer cannot afford the commodities

(c) Income = Expenditure                   (d) None of these

1. Movement on one indifference curve indicates

(a) Higher satisfaction level

(b) Lower satisfaction level

(c) No change in satisfaction level

(d) None of these

1. Point inside the budget line means

(a) Income = Expenditure

(b) Income < Expenditure

(c) Income > Expenditure

(d) MRE = Px/Py

1. Indifference curve is downward slopping

(a) in order to maintain same level of utility

(b) income is limited

(c) both (a) and (b)

(d) none of these

Numerical Questions

1. Price of a chocolate = ` 20 per unit. (Assuming MUR = 1)

Determine equilibrium level of consumption.

Ans. 3 units.

1. A pastry costs Rs. 15 and Smita has consumed 4 pastries. The additional utility derived from 4th pastry is 75 utils. ShoudSmita increase or decrease the consumption of pastry? (Ans. Increase).

1. From the table, calculate, the equilibrium level of consumption.

Given Px=  ` 10 and MUR = 1.

Ans. 6 units.

1. Complete the table
 Units consumed TU MU 1 10 – 2 – 7 3 – 5 4 22 – 5 19 –

1. Compute the consumer equilibrium ® Given Px= ` 12/ unit. (Assume      MUR = 1)
 Units consumed TU 1 20 2 35 3 47 4 51 5 51

1. Explain the situation of consumer equilibrium if Px=`8 /unit. (Assume MUR = 1)
 Units TU 1 42 2 60 3 74 4 85 5 93 6 97 7 99 8 99 9 99 10 92

1. Mr. A has `10 to be spend on X and Y Px = Py=  `2. Determine the extent to which Mr. A should consume each of the commodity to maximize his satisfactory
 Units MUx MUy 1 16 11 2 14 12 3 12 10 4 10 8 5 8 6

Ans. 3 units of X and 2 units of Y.

1. Income of the consumer Þ`22/week, Px = `2, PyÞ`4.

Determine the extent to which consumer would consume to maximize the satisfaction.

 Units Mux Muy 1 20 24 2 18 20 3 16 16 4 12 12 5 8 8 6 6 4

Ans. 5 units of X and 3 units of Y.

Very Short Questions

1. Is the consumer willing to move away from consumer equilibrium point?
2. Why MRS always fall?
3. State the conditions of consumer equilibrium in two commodity case?
4. What do you mean by Mu of one rupee?
5. Why does Tu increases at diminishing rate with the continuous increase in consumption?
6. If one cup of ice-cream gives you a satisfaction of 10 utils and two cups of ice-cream gives you a satisfaction of 25 utils. What is the satisfaction from each cup?
7. Why budget line is a straight line?
8. What is the impact of diminishing MRS on the shape of indifference curve.
9. What does the movement along the indifference curve indicate?
10. What does the movement along the budget line indicate?
11. Why budget line is downward slopping?

1. The income of the consumer is Rs. 40. Price of X and Y are Rs. 5/unit and Rs. 8.unit. Answer the following

(a)      Calculate MRS at consumer equilibrium point.

(b)      Explain whether the combination (3, 4) and (4, 3) will lie on the                          budget line or not? Why?

(c)      Frame the budget equation.

(d)      If consumer is not purchasing any unit of X. How many units of Y                     will be purchased where the point will lie.

(e)      If the following combinations are on the budget line, complete the                       following.

X       Y

A       –        2

B       4        –

C       –        5

1. Consumer is at equilibrium and he consumes two commodities. Explain how the fall in the price of X will raise the demand for X.
2. Consumer is at equilibrium and he consumes one commodity. Explain how rise in the price of X will reduce the demand for X.
3. Explain the law of diminishing MU with the help of example?
4. What is budget line. What is the slope of budget line.
5. What are the reasons of shift in the budget line.
6. Why must MRS be equal to MRE at the point of consumer equilibrium.
7. Frame budget set with Px = Rs. 5/unit and Py = Rs.2/unit and Income = Rs. 50.

True and False

1. If then the consumer should buy more of commodity Y and less of commodity X.
2. MRS remains the same on the indifference curve.
3. Slope of indifference curve is different at different points on the curve.
4. When we add up utility derived from each successive unit, we get total utility.
5. Any consumption beyond the point of satiety leads to disutility.
6. Different points on the indifference curve represent different level of satisfaction.
7. Lower indifference curve represents higher level of satisfaction.
8. Budget line is also known as price line.
9. Marginal utility can never be negative.
10. Two indifference curve can intersect only when they represent same level of satisfaction.