MODULE No. : 9 : Ordinal Utility Approach

Transcription

1 Subject Paper No and Title Module No and Title Module Tag 2 :Managerial Economics 9 : Ordinal Utility Approach COM_P2_M9

2 TABLE OF CONTENTS 1. Learning Outcomes: Ordinal Utility approach 2. Introduction: Preferences 3.Topic 1: Indifference Curve 3.1.1Notations Assumptions about Preferences Diminishing Marginal Rate of Substitution 3.2 Properties of Indifference Curves 4. Topic Ordinal Utility Analysis 4.1.2Budget Constraint of the consumer Equilibrium of the consumer Graphical Presentation of the equilibrium of the C r Slutsky theorem Normal and Inferior Goods Giffen Goods 4.3 Limitations 5. Summary

3 1. Learning Outcomes: Ordinal Utility Approach After studying this module, you shall be able to Know the Assumptions & Properties of Ordinal Approach Understand consumer equilibrium Identify Understand SlutskyTheorem Evaluate the difference between Normal, Inferior & Giffen Goods

4 2. Introduction: Preferences Preferences are the relationships between bundles. If a consumer chooses bundle (x 1,y 1) when (x2, y 2) is available, then it is natural to say that bundle (x 1, x 2) is preferred to (y 1, y 2) by this consumer. Preferences have to do with the entire bundle of goods, not with individual goods. Consumer is able to order various consumption possibilities. The way in which the consumer ranks the consumption bundles describes the consumer s preferences. 3. Topic 1 : Indifference Curves Notation (x 1, x 2), > (y 1, y 2) means the x-bundle is strictly preferred to the y-bundle. (x 1, x 2) ~ (y 1, y 2) means that the x bundle is regarded as indifferent to the y-bundle. (x 1,x 2) > (y 1, y 2) means the x-bundle is at least as good as the y-bundle. Indifference curves graph the set of bundles that are indifferent weakly preferred to some bundle Assumptions about Preferences Reflexive Any bundle is at least as good as itself. Complete Any two bundles can be compared i.e. a consumer is able to make a choice between any two bundles. If X and Y are any 2 bundles, then consumer can always specify exactly one of the FFfollowing possibilities X>Y, Y>X, X&Y are equally attractive. Transitive If X>Y and Y>Z then X>Z. If bundle X is strictly preferred to bundle Y and bundle Y is strictly preferred to bundle Z, then bundle X is strictly preferred to bundle Z. Rationality The consumer is assumed to be rational. He aims at the maximization of his utility, given his income and market prices. Utility is ordinal Consumer can rank his preferences according to the satisfaction from each basket. Diminishing marginal rate of substitution

5 Preferences are ranked in terms of indifference curves, which are assumed to be convex. This implies that the consumer will be willing to give up lesser amount of good Y for on additional unit of X as one move down the indifference curve. 3.2 Properties of the Indifference Curves An indifference curve has a negative slope which denotes that if the quantity of one commodity Y decreases, the quantity of the other good X must increase for consumer to stay on same level of satisfaction The further away from the origin an Indifference curve lies, the higher the level of utility it denotes, bundles of goods on a higher i.e. are preferred by the rational consumer Indifference Curves do not intersect. If they did, the point of their intersection would imply two different levels of satisfaction, which is impossible The indifference curves are convex to the origin. This implies that the slope of an indifference decreases (in absolute terms) as we move along the curve from the left downwards to the right. 4. Topic 2: Consumer Equilibrium 4.1.1: Ordinal Utility Analysis Such that: u = f(x, y)... (1) u = f(x,y) = k...(1a) u u du =.dy +.dx y x... (2) = MUy.dy + Mu xd x... (2a) Any change doing the indifference curve tears the total utility constant. \ du = MUy.dy + MUx.dx = 0... (3) - dy MUx MRsxy dx = MUy =... (4) Since x increase and y decrease: We often find it useful to refer to the slope of the indifference curve at a particular point. This is known as the Marginal rate of substitution. The name comes from the fact that MRS measures the

6 rate at which the consumer is just willing to substitute one good for another. Suppose we give him an extra unit of X. Then we have to take away from him some amount of good Y that is sufficient to put him back on the indifference curve so that he is just as well off after this substitution of Y for X as he was before. One slight confusing thing about the MRS is that it is typically a negative number. Since the MRS is the numerical measure of the slope of an indifference curve, it will naturally be a negative number. Marginal rate of substitution of y for x is always falling. If consumption of X increases MUx declines and automatically consumption of y declines, so MUy increases the ratio always declines : Budget Constraint of the consumer Suppose that there is some set of goods from which the consumer can choose. In real life a consumer has a number of choices, but we stick to the case of two goods, since it can then depict the consumer s choice graphically. Any utility maximizing consumer faces a budget line. The is endowed with a given income which constraints his behavior. The income constraint in case of two commodities may be written as M = pxqx + pyqy.(5) The Budget Line represents income constraint graphically by the budget line, by solving for q y 1 Px qy = M qx Py - Py (6) Slope of the budget line is the derivative of the above equation which is negative of the ratio of the prices of two goods : Equilibrium Condition Now we will put together the budget set and the theory of preferences in order to examine the optimal choice of consumers. The model of consumers choose the bundles which gives them maximum satisfaction. The consumer is in equilibrium when she maximizes his utility, given his income and the market prices. Two conditions must be fulfilled for the consumer to be in equilibrium. The first condition is that the marginal rate of substitution be equal to the ratio of commodity prices. This is the bundle in the budget set that is on the highest indifference curve. An important feature of this optimal bundle at this choice is where the indifference curve is tangent to the budget line. That is the slope of the indifference curve and the budget line is same at the optimal choice.

7 MRS x,y = MUx = Px.(7) MUy Py This is a necessary but not suffices condition for equilibrium. The second condition is that the indifference curves he convex to the origin. This condition is fulfilled by the axiom of diminishing MRS x,y : Consumer Equilibrium Given the indifference map of the consumer and his budget line, the equilibrium is defined by the part of tangency of the budget line with the highest possible I.C. Here, BC stands for the Budget Constraint, I1, I2 are the indifference curves and X* and Y* are the quantities consumed, respectively, of commodities X and Y, under consumer equilibrium. The slope of BC is Px/Py and the slope of I2 is MUx/MUy = MRSxy. The points that lie along I1 are inferior to I2. The points that lie along I3 are out of bounds for the consumer since they lie beyond the BC. The only point that represents the highest attainable IDC, on I2 is the one that achieves tangency between I2 and BC. This results in maximum satisfaction (the highest attainable IDC combined with the only point on the budget constraint. Figure 1: Consumer Equilibrium 4.2.1: Slutsky theorem

8 The Slutsky's Equation breaks down a change in demand due to price change into the substitution effect and the income effect. The equation takes the form: dx = dhx x dx dpx dpx dm.(8) The term on the left is the change in demand for commodity x, when price of x changes. Here, x is the (Marshallian) demand for a good and px is the price of x. The term h is the Hicksian or the compensated demand. The term dh/dp measures the substitution effect. The term m is the income, and x(dx / dm) measures the income effect. When we read the Slutsky's equation, the term dhx/dpx is the substitution effect. This is because the compensated demand depends on h(px, py, u) which fixes the consumer's utility level, when the consumer's purchasing power remains constant. That is, the IDC remains constant. The term dh/dp only measures the shift in consumption when the price alone changes. On the other hand, the income effect depends on the amount of good the consumer is consuming (x), and the consumer's reaction to an income change dx/dm that comes from the "savings"; or the increase in real income due to a fall in price (of x). Thus, the term X(dx/dm) measures the income effect. Figure 2: Income and Substitution effect for Normal Goods Price Effect

9 A fall in the price of X from P 1 to P 2 results in an increase in the quantity demanded from X 1 to X 2. This is the total price effect which may be split into two separate effects a substitution effect and an income effect. The substitution effect of a price change is always negative (relative to the price; if the price increases, the quantity demanded decreases and vice versa). However, if the sign of the substitution effect is ignored the quantity demand increases, for a fall in price of x. Normally, the income effect is positive. Thus, PE = SE + IE.(9) Substitution Effect The substitution effect is the increase in the quantity bought as the price of the commodity falls, after adjusting income so as to keep the real purchasing power of the consumer the same as before. This adjustment is shown graphically by a non-parallel shift of the new budget line until it becomes tangent to the initial indifference curve. The income effect is caused by a parallel shift of the new budget line in such a way that it is parallel to the compensated budget line, but starts from A, where the original budget line arises from the y-axis. The purpose of the compensating variation is to allow the consumer to remain on the same level of satisfaction as before the price change. The compensated budget line will be tangent to the original indifference curve (I) but to the right to the original tangency point (e 1), at a point (e 1 ), because this line is less sleep than the original one when the price of x falls. The movement from point e 1 to e 1 shows the substitution effect of the price change: the consumer buys more of x now that it is cheaper, substituting x for y. Income Effect The income effect is caused by an increase in the real income due to a fall in the price of x. The additional income which was released due a fall in price of x has to be given back to the consumer, due to which the consumer has in fact a higher purchasing power (from IDC I to IDC II); and, if the commodity is normal, he will spend some of his increased real income on x, thus moving from x1 to x2. Thus, x1 x2 is the income effect of the price change of x : Normal and Inferior goods The income effect of a price change, for normal goods, is positive because as income grows the demand for such goods grows. The price effect x1x2 is the sum of substitution effect x1x1 and the income effect x1 x2. x1x2 = x1x1 + x1 x2.(10) If, however, the commodity is inferior, the income effect of the price change will be negative: as the purchasing power increases, less of x will be bought. Still for most of the inferior goods the substitution effect will more than offset the positive income effect, so that the total price effect will be negative. Thus the negative substitution effect is in most cases adequate for establishing the law of demand.

10 4.2.3: Giffen Goods It is when the income effect is negative and very strong that the law of demand does not hold. This is the case of the Giffen goods, which are inferior and their demand curve has a positive slope. Figure 3: Giffen Goods The original quantity of x is q 11 when the equilibrium is A. When the price ratio falls from m/p2 :m/p 11to m*/p2:m*/p12, the quantity of x rises to q 1*. When the budget line shifts from m*/p2: m*/p12 to m/p2 :m/p 12 falls from q 1*at C to q 12 at B. The net fall in quantity demanded falls from q 11 to q 12. For a fall in price if consumption also falls due to a negative income effect, in such a manner that it outweighs substitution effect, then the total impact of a fall in price of x will that consumption of x will also fall. A fall in price of x leads to a fall in the consumption of that commodity (x). Thus, price effect is positive in the case of Giffen

11 Goods. 4.3: Limitations 1. Axiomatic existence of convex IDC is assumed. 2. It requires a precise ordering of preferences. 3. It has a strong rationality assumption. 5. Summary The optimal choice of the consumer is that bundle in the consumer s budget set that lies on the highest indifference curve. Typically the optimal bundle will be characterized by the condition that the slope of the indifference curve (MRS) will equal the slope of the budget line. A normal good is one for which the demand increase when income increases. An inferior is one for which the demand decreases when income increases. A giffen good is one for which the demand and price is positively related due the negative income effect overcoming a positive substitution. Thus, for a Giffen Good has a positive price effect rather than a negative as in all other commodities. The typical case of a Giffen Good is a diamond whose demand rises when its price increases.