7 Indifference Curve Analysis-I

QUADRANT-I

INSTRUCTIONAL OBJECTIVES

At the end of this chapter, learner should be able to understand:

1. Meaning of Cardinal Utility and Ordinal Utility

2. Indifference Curves

Meaning of Indifference Curve

Marginal Rate of Substitution

Indifference Map

Properties of Indifference Curves

3. Budget Line

Meaning of Budget Constraint

Meaning of Budget Line

Slope of Budget line

Shifts in Budget line

4. Consumer Equilibrium with Indifference Curve Analysis

First Order condition for consumer Equilibrium

Second Order Condition for consumer Equilibrium

(Keyword: Utility, Indifference curves, Marginal Rate of Substitution, Budget Constraint, Budget Line, and Consumer Equilibrium.)

“An Indifference curve represents all combinations of two commodities that provided the same level of satisfaction to a person. That person is therefore indifferent among the combinations represented by the points on the curve”-H.L. Varian

Introduction

The concept of Utility is at centre stage of an economic analysis. Utility refers to satisfaction which a consumer attains after the consumption of goods and services. There are two types of utility; one is total utility and other is marginal utility. Total utility means sum of the utilities which a consumer derives after the consumption of goods and services. Marginal utility means change in total utility after the consumption of one more unit of a good. In the Economic Literature, two types of Utility Analysis are much talked about

1) Cardinal Utility Analysis and 2) Ordinal Utility Analysis. Cardinal Utility analysis propounds that Utility is measurable and can be quantified in cardinal number like 1, 2, 3 etc. Under Cardinal utility analysis utility is measured in units called Utils. For example when a consumer consumes two units of Y good, he can quantify the utility he derives in Utils terms say he consumer gets satisfaction or utility worth 10 Utils from the consumption of First unit of Y good and 20 Utils from second unit . In total he gets satisfaction/utility worth 30 Utils by consuming two units of Y good. So value of total Utility is 30 Utils. Similarly here marginal utility is equal to 10 Utils, as the change in total utility is 10 when one more unit of Y is consumed.
On the other hand Ordinal Utility Analysis propounds that utility is a psychic entity, which cannot be measured in cardinal numbers. According to Ordinal Utility hypothesis, consumer can rank his/her preferences. For example, when a consumer consumes two units of Y good, he can tell that the satisfaction he has attained from the consumption of first unit was higher than, lower than or same as that of satisfaction which he derives from consuming second unit of Y good. Indifference curves are one of the important tools of Ordinal Utility Analysis which are used to under about consumer behaviour and later on to understand the concept of consumer equilibrium. In the following sections we shall be studying about Indifference curves, their meaning, indifference map and the various properties of Indifference curves.
The concept of Indifference Curve (IC) Analysis holds a pertinent place in Economic Theory. The technique of IC was initially invented by Edgeworth, one of the famous classical economists. Edgeworth used this technique only to analyse the possibility of exchange between two persons and did not use it for explaining consumer’s demand. It was later J.R. Hicks and R.G.D. Allen who propounded the idea of IC approach based on Ordinal Utility Analysis to explain consumer behaviour. In 1939, J.R. Hicks in his book ‘Value and Capital’ presented the Indifference theory of consumer demand.

Indifference Curve

Indifference curve is a curve which shows the combination of two goods which a consumer consumes and which gives him the same level of utility/satisfaction. It means consumer is indifferent between various combinations of two goods , as they all give him same level of satisfaction.

Example: Let us assume that a consumer consumes two goods X and Y. There are number of combinations of Good X and Y available to him from which he can choose from. The table 1 and figure 1 shows the derivation of the indifference curve.

Table 1: Indifference Curve Schedule

Figure 1 illustrates, that units of Good X are measured horizontally and units of Good Y are measured vertically. At combination A , consumers consumes 1 unit of X and 20 Units of Y , at combination B, the combination is 2X and 16 Y at combination C , it is 3X and 13Y , similarly at combination D and E , consumer consumes 4X and 11Y and 5X and 10Y. When all the Combinations i.e. A, B, C, D, E are joined a convex curve IC1 is formed, which shows that the consumer is indifferent toward any combination as he derives the same level of satisfaction ( ) whether he consumes combination A or D of good X and Y. This curve is called Indifference Curve. One thing is worth mentioning, that when consumer moves away from A to D combination, he consumes more units of X good and less and less unit of Y , that means he substitutes Y for X. The rate at which he substitutes one good for another is called marginal rate of substitution, which is discussed later. The concept of IC is based on certain set of assumptions, which are explained as below.

Assumptions of Indifference Curve Analysis

Indifference curve analysis is based upon following assumptions.

1) Completeness: This assumption is about the completeness of the ordering of preferences. For example in our previous example , in combination A and B of X and Y good , consumer is able to give rank to these combinations, say A is preferred to B or B combination is preferred to A or it could be that consumer is totally indifferent between combination A and B.

2) Transitivity: Preferences are assumed to be transitive. Let there be three combinations of two goods X and Y; combination A, B and C. If consumer prefers combination A over combination B and combination B over combination C, then according to transitivity assumption combination A will be preferred over combination C.

3) Non satiety: According to this assumption wants are non-satiable means which can be satisfied as one want is fulfilled another one crops out and more of any good is always desirable over lesser amount of good. More is always better than less.

Indifference Map

Indifference map represent the entire family of Indifference curves plotted in one diagram. This map represents the complete description of consumer preferences for all combinations of two good which the consumer consumes.

Figure 2

In figure 2, Good X is measured in horizontal axis and Good Y on vertical axis. Three Indifference curves labelled U1 , U2 and U3 are plotted in the figure. Each Indifference curve represent the various combination of Good X and Good Y which gives the consumer same level of satisfaction. Indifference curve U3 represents consumption of more of good X and good Y , than the consumption levels of good X and good Y at indifference curve U1 . It also highlights that the higher Indifference curve represent higher level of satisfaction.

Marginal rate of Substitution

Marginal rate of Substitution is one the major pillars on which the hypothesis of Indifference curves stands. Marginal rate of substitution of X to Y is the maximum amount of Y good which the consumer is ready to sacrifice or pay for one additional unit of X good holding utility constant.
MRSX for Y = ℎ

ℎ Or=∆ ∆

Table 2. Marginal rate of substitution between good X and good Y

In table 2, As consumer moves from Combination A to combination B , He sacrifices 4 units of Y for 1 additional unit of X, utility remain constant. So here Marginal rate of substitution of X for Y is 4:1 which means he is ready to sacrifice 4 units of Y good for 1 additional unit of X good. Similarly as he moves from combination B to C to D and to E , This rate become 3:1, 2:1 and 1:1. In case of Indifference curve , this marginal rate of substitution(MRS) diminishes as explained in Figure 3 which is based on table 2.

In figures 3, X good is measured in X-axis ad Y good is measured on Y-axis. Figure illustrate that for obtaining one additional of X good consumer is ready to sacrifice the unit of good Y, but he sacrifices lesser and lesser unit of Y good. It can be seen that when consumer moves from combination A to combination B, his consumption of X good increases from 1X to 2X but on other hand the consumption of Y is reduced from 20Y to 16Y this is because to maintain same level of utility. Marginal rate of substitution in this case fall or diminishes from 4:1 to 3:1 to 2:1 to 1:1. This diminishing marginal rate of substitution exhibits that consumer is willing to sacrifice lesser and lesser of Y units for one additional unit of X good. This marginal rate of substitution diminishes because good X and Y are imperfect substitutes of each other. MRS of X for Y is the ratio of marginal Utilities of X and Y.

Slope of Indifference Curve:

Marginal rate of Substitution gives the slope of Indifference curve , and because MRS diminishes do slope is represented as

MRSX for Y = − ∆ ∆

Properties of Indifference Curves

Indifference curves has negative slope: Indifference curve should have negative slope or it slopes downward to the right. This property highlights that the combination of two goods which consumer consumes should be substitute of each other, so that when the consumption of one good is increased, the consumption of other good should fall so that level of satisfaction remains same.
Indifference curves are convex to origin: This property of Indifference curves is based on the law of diminishing marginal rate of substitution. It postulates that marginal rate of substitution diminishes as we move left to right on a Indifference curve or consumer will sacrifice less and less units of say Y good for attaining one additional unit of say good X. This is also because good X and Y are imperfect substitute of each other.
Two Indifferences curves cannot intersect each other: Any two indifference curves cannot intersect each other at a point as it violates the transitivity assumption of Indifference curves. This is explained in figure 4.
Figure 4

In figure 4, Good X and Good Y are taken in horizontal axis and vertical axis respectively. Two Indifference curves U1 and U2 shows various combinations of X and Y for which consumer is indifferent and utility level are constant. There are two combination each labelled on indifference U1 and U2 . Combination Z and T are on indifference curve U1 and combination Z and P on Indifference curve U2 . These two indifference curves intersect each other at point Z. Because of this intersection, contradictory results are derived. In Indifference curve U1 , At combination Z we have 1X and 6Y and at combination T , we have 3X and 3Y. Similarly In Indifference curve U2 , there are two combination Z and P , at combination Z , we have 1X and 6Y and at combination P , there are 3X and 5Y. According to definition of Indifference curve, the various combination of two goods on a indifference curve gives consumer same level of satisfaction . In above figure Z is common combination in both Indifference curves so if Z and T combination on Indifference curve U1 gives same satisfaction then similarly Z and P on Indifference curve U2 also give same satisfaction then combination P and T should also give same level of satisfaction, but which is not true as in combination P consumer consumes more of Y( 5Y) than at T ( 3Y). So combination P gives more satisfaction than T. So two indifference curves cannot intersect each other.

Higher Indifference curve symbolises higher level of utility: This property of Indifference curve highlights that the higher Indifference curve give the consumer higher level of satisfaction. The underlying principle is that higher Indifference curve will have those combinations which represent higher level of satisfaction as on higher indifference he will have more of both goods. This is explained in following figure.

Figure 5

In figure 5, Three indifference curves are plotted as U1 , U2 and U3 . On these indifference curves, three different combinations namely A, B and C are shown. A combination on indifference curve U1 exhibits 1X and 4Y, combination B on Indifference curve U2 represent 2X and 5Y and combination C on U3 represent 3X and 6Y. These combination shows that combination C is on the highest Indifference i.e U3 and it represents highest combination of X and Y out of three combinations which are available. Similarly all combination on U3 will represent higher combination of X and Y or more of good X and good Y which in turn means more of satisfaction and utility than combination of X and Y on Indifference curve U1 and U2 .

Shape of Indifference curves in case of Good which are 1) Perfect Substitute and 2) Perfect Complement

1) Perfect Substitute: Two goods are said to be perfect substitute of each other, if the marginal rate of substitution is constant. It means that at various combinations of two goods , consumer is willing to sacrifice same units of one good for having one additional unit of other good .IC of perfect substitute is straight line. Figure 6, Explain this.

FIGURE 6

Units of good X are shown in X axis and units of Y are given on Y axis. Combination A represent 1X and 8Y, B combination has 2X and 7Y similarly at combination C and D , 3X and 6Y and 4X and 5Y are consumed by consumer . When all these combination are joined a straight line Indifference curve is obtained which shows the combination various units of X and Y which gives consumer same level of satisfaction. It also exhibits that Marginal rate of substitution is constant.

2) Perfect Complementary goods: Complementary goods are those goods which are jointly consumed together in fixed proportion. Example left shoe and right shoe, consumer have to buy both shoes in some fixed proportion to fulfil a want. In the case of perfect complementary good the Indifference curve is ‘ L’ shape, which represent that at specific point infinite units of other goods are required to get an additional unit of a good. It also show that a point some fixed proportion of both goods are required and increase in the quantity of good X without an increase in the quantity of Y and vice versa leaves the consumer at same level of utility. The case is represented in Figure 7

FIGURE 7

Consumer Equilibrium using Indifference Curve approach

Equilibrium refers to situation from where there is no motion or change. Equilibrium word is taken from Latin language meaning ‘acquus’ meaning equal and ‘libra’ means balance, so it refers to equal balance. In Economic consumer equilibrium refers to a point at which consumer attains the maximum possible utility from the purchase of the goods and services, given income and price constraints. After attaining this position he has no tendency to move/change from this situation. In the Economics literature, there are two approaches to attain consumer equilibrium. One is Cardinal Utility Approach and other is Ordinal Utility Approach also called as Indifference curve approach. The following section explains how consumer equilibrium can be attained with help of Indifference curve approach. Under Indifference curve approach, consumer attains equilibrium when he attains maximum satisfaction given his income constraint. Maximum satisfaction or level of satisfaction which consumer derives by consuming goods is mapped through Indifference curve which have been discussed in above sections. Income constraint is explained by Budget line or Price line. In the following sections we will discuss about budget constraint, budget line, shifts in budget line and conditions for attaining consumer equilibrium using ordinal utility approach.

Budgetary constraint

Budget constraint is the limit on the consumption of goods and services that a consumer can afford or buy with his income. In a two commodity (X and Y) model, budget constraint can be written as:

Px. Qx + PY. QY = MY

Px is price of X commodity, Qx is quantity of X, PY is price of Y Commodity , QY is quantity of Y, MY is the money income of the consumer. Further solving above equation, we can find the value of QX and QY as follow.

= −

If QY is O i.e consumer does not consume Y good, then =

If Qx is O i.e consumer does not consume X good, then =

When the above equations are translated into diagram, we get a line called Budget line or Price line.

Budge Line: Budget line can be defined as a line which shows all the combination of two goods which a consumer can purchase by spending his entire money income and with given prices of two goods. Budget line or Price line is explained with the help of following schedule and diagram in Table number 3 and figure number 8

Table 3: Budget Line schedule

In the above table and diagram, it is highlighted that price of good X is Rs. 10 per unit and price of good Y is Rs. 20 per unit. The money income of income is Rs. 100. A, B , C and D are various combinations of good X and good Y which the consumer can purchase by spending his entire money income i.e Rs 100 and at given prices of X and Y. Figure 8 shows that at combination A , consumer consumes OX (zero unit of X) and 5Y. Similarly at combination B it is 4X and 3Y , at C combination it is 6X and 2Y and at D point he consumes 10X and 0Y (zero unit of Y). Joining all the points gives a line called Budget line. This diagram also shows that consumer can buy less and save some part of his money income, but he cannot buy more than his budget. The area under the Budget line is also called area of feasible purchase, and the outside area or area on the right hand side of line is called non feasibility area. The Combination A and D are the extreme situation, where consumer purchases either of a commodity only, but which is ideally not preferred as it is against basic tenant of budget line. The budget line has negative slope, which reveals that if consumer want to purchase more of one good than he has to purchase less of other good.

Slope of Budget Line

As mentioned in previously that budget line has negative slope, and it is equal to price ratio of two goods.

Budget line or Price line primarily depend upon Income of the consumer and the prices of the goods and services, which the consumer purchases. Since both income and prices are capable of change, these changes in income and prices also affect budget line. The various factors leading to shifts in budget line can be explained as follow:

Shifts in Budget line

The shift in budget line can be explained in following ways.

1) Change in consumer’s money income, when prices are constant

2) Change in the prices of goods or services , when money income is constant

Change in consumer’s income, when prices are constant: When the income of consumer changes i.e. increase or decrease. Budget line shifts outside and inside but keeping the prices of both good as constant. The effect of income change on budget line is explained in figure 9

Figure 9

Figure 9, shows that initially BL is the budget line, when income of the consumer increases budget line shifts outward and become MN and similarly when the income decreases budget line shifts inward and become CD, keeping the prices constant in both the cases. In above case, there is parallel shift in the budget line this is because only both intercept changes but the price ratio remains same. The same effect will take place if the prices of both goods changes proportionately.

Changes in the prices of goods and services, when money income is constant: If the prices of goods or services changes, budget line will swivel . The following figure explains the change.

Figure 10

Initially BL is the budget line, when the price of good X falls, keeping the price of Y and money income constant, budget line will swivel and become BL1 , similarly is price of good X opposite will happen. Similarly if the price of good Y falls, keeping the price of X and money income constant , budget line will become B1 L.

Consumer Equilibrium with ordinal approach

Now as we are equipped with the necessary tools for understanding consumer equilibrium, in the following section we will study the conditions for consumer equilibrium and how consumer equilibrium is attained with the ordinal utility approach. According to this approach, if consumer equilibrium is to be attained then two conditions must be satisfied.

Condition 1: Indifference curve should be tangent to budget line

Condition 2: At the point of tangency, Indifference curve must be convex to the origin The following paragraph explain the rationale and importance of these two conditions

According to condition number 1 which is also called necessary or first order condition, says that in order to attain consumer equilibrium, Indifference curve should be tangent to budget line or the slope of IC must be equal to slope of budget line. Algebraically

Slope of Indifference curve = Slope of budget line

MRSXY = −

= −

Condition number 1 is explained in Figure number 11.

Figure number 11

In the above figure, BL is the budget line and U1 , U2 and U3 are three indifference curves. Indifference curve number U3 lies in non-feasible area or it is beyond the budget of the consumer . combination M and N of Indifference curve U1 lies on Budget line BL which indicate that these combinations are in feasible space of budget or consumer can afford to buy them, but consumer will not prefer these any of these combination as the combination E of indifference curve U2 also lies on budget line as consumer will prefer combination E over M and N as it represent higher level of satisfaction and IC, U2 is higher than IC U1 . So consumer is attains maximum satisfaction /utility at point E, where the Indifference curve U2 is tangent to budget line BL and here the slope of IC is equal to slope of budget line BL.

Condition Number 2

According to second condition, at the point of tangency between IC and Budget line, C must be convex or the marginal rate of substitution should diminish. If this condition is not fulfilled than the equilibrium formed will not be stable one. Second order condition is explained in figure 12

Figure 12

In figure 12, BL is budget line and initially U1 is the IC and the first order condition is fulfilled at point J , where slope of IC is equal to slope of budget line but here second condition is violated, Ic is concave at point of tangency . Now this equilibrium is not stable as consumer can easily move from point J to point T or point R by moving along budget line BL , and doing this he can reach higher indifference curves either U2 or U3 . So in order to achieve consumer equilibrium, both conditions should be satisfied simultaneously

Summary

Indifference curve is used to analyse the consumer behaviour and conditions under which consumer equilibrium can be achieved. Earlier cardinal utility was more popular method to analyse consumer behaviour. However, the problem of measuring of utility lead to use of ordinal approach, moreover general understanding about consumer behaviour can be generalised using ordinal approach and Indifference curve analysis further help us to understand consumer equilibrium in much wider perspective.

Sources for Further Reading on Indifference curve analysis.

1. Francis Y. Edgeworth , Mahematical Physics , London, C.R. Paul &Co., 1881

2. Hicks, John, and R. G. D. Allen.(1934), “ A reconsideration of the theory of Value”, Economia ,Feb-May, 1934

3. A.W. Stonier and D.C. Hague, A textbook of Economic Theory, London, Longman, 1972

4. Hicks, John R., Value and Capital , 2nd Edition (Oxford University Press, 1946)

5. Maddala ,G.S and Ellen.,Miller., Micro Economics Theory and Practice, Tata McGraw Hill, New Delhi

6. Dwivedi , D.N., Microeconomics Theory and Application , Pearson Education , New Delhi.

7. Pindyck, Robert, D.L , Rubinfeld,., and P.L.Mehta, Microeconomics Sixth Edition, Pearson Education

8. Koutsoyiannis, A. (1990), Modern Microeconomics, Macmillan.

9. Varian H. (2000), Microeconomic Analysis, W.W. Norton, New York.