UNIT 2: BACKGROUND TO DEMAND: THE THEORY OF CONSUMER CHOICE Objectives: Upon completion of this unit students should be able to; Define the concept of utility as basis of demand. Define Marginal utility and diminishing marginal utility. Draw indifference curve and individual demand curve. Distinguish between a cardinal and ordinal utility. Illustrate the law of diminishing marginal utility. State and explain the two conditions for consumer equilibrium in the utility approach. Determine consumer equilibrium given the income of the consumer, the prices of the products and the marginal utilities of the products. Derive an individual demand curve using the utility approach. CONSUMER CHOICE AND DEMAND Utility Theory Utility function: Is an algebraic expression that allows us to rank a consumption bundle by the total utility or satisfaction it provides. It describes the total utility derived from consuming a particular bundle. Consumption bundle: Refer to a particular combination of goods being considered.

Example of Utility function: Total utility U= u(X1, X2, X3,,XN) Where: U = refer to the level of satisfaction you receive from consuming the bundle of goods consisting of X1 ,.., XN U( ) = is the function that maps the bundle of goods into satisfaction. X1 for X n = 1,2,3., n is the quantity of a particular good consumed from a bundle of goods, e.g., X5, May refer to 5 cans. REPRESENTING UTILITY IN MATHEMATICAL TERMS Suppose you have a choice of consuming two different bundles of goods: Bundle 1: consist of 1 large pizza, 2 litre bottle of coke and 20 spicy chicken wings. Bundle 2: consist of 2 doughnuts, 1 litre of coffee and half dozen raising muffins.

For bundle 1; X1= is a large pizza, X2 = 2 litre bottle of coke, X 3= 20 spicy chicken wings. For bundle 2; X1= consist of doughnuts, X2 = 1 litre of coffee, X 3= half dozen raising muffins. Lets assume its early in the morning; Consumption bundle 1 makes you very happy and you call it U 1 . Consumption bundle 2 make you very, very happy and you will call it U 2. Thus, U2>U 1 . Therefore; U 1 = u(large pizza, 2 litre bottle of coke, 20 spicy chicken wings)

U 2 = u(consist of doughnuts,1 litre of coffee, half dozen raising muffins). U( ) is just a function that transform bundles of goods into satisfaction. REPRESENTING UTILITY IN MATHEMATICAL TERMS (CONT.) Utils = Is when utility is broken down into units. Total utility: Is the utility you receive from consuming a particular bundle of goods. Utility can be given a number value e.g. U 1 = 100 and U2 = 200. There are two views of utility: Cardinal utility: The belief that utility can be measured and compared on a unit by unit basis, e.g. Utility measure of 200 is twice as big as a utility measure of 100. Ordinal utility: Where you rank a bundle of goods but you

cannot say how big one bundle is compared to the other. MARGINAL UTILITY AND LAW OF DIMINISHING MARGINAL UTILITY It is appropriate to question how total utility change as greater amount of a particular good is consumed. When change in total utility is associated with change in consumption of a particular commodity it is referred to as Marginal utility. Marginal utility the extra utility a consumer gets from consuming the last unit of a good. Marginal utility of good 1 = total utility in level of consumption of good 1. Thus, MU of Good 1 = U/ X 1

or (U 2 U 1) (X i2 X i1) Where U 2 is the utility received from consuming Xi 2. U 2 = u(Xi 2) U 1 is the utility received from consuming Xi 1. U 1 = u(Xi 1) MU value will always be greater than zero only if we assume that the consumers appetite never becomes totally satiated. MU value will fall (rise) as a goods consumption increases (decreases). MU CALCULATIONS QTY of Hamburgers TU

MU 1 20 - 2 30 10 3 39 9 4 47

8 5 54 7 6 60 6 7 65 5 8

69 4 9 72 3 10 74 2 11 74 0 12

70 -4 What does the table above tell you? When consumption of burgers increases from 2-3, Utility increase by 9 utils. When consumption increases from 8-9, utility increases by only 3 utils. When MU is zero, TU is constant. MU tend to be negative at higher level of hamburger consumption. As Hamburger consumption increases from 11-12, utility decreases by 4 utils. Attempt to draw a Total utility and Marginal utility curve. DERIVING TOTAL AND MARGINAL UTILITY FROM USING AN EQUATION

Suppose your given the utility function for drinking Fanta Grape as: U= u(X), where u(X)= 5x - x 2 . What is you total utility and marginal utility. Derive the Marginal Utility function given the Total Utlity. Amount consumed TU of (X) 1 2 3 4 5 6 MU What have you learned from the Table above? As long as MU is positive, TU will increase. TU follows MU, thus when MU is zero TU is maximized. Thus

demonstrating the law of diminishing MU. The law states that as you consume more units or a particular good during a set time, at some point you MU will decrease as your consumption increases. INDIFFERENCE CURVE Indifference curve is also being referred to as the ISO-utility curve. It is a line that shows all combinations of two goods that give the consumer equal satisfaction. A consumer is indifferent to consumption bundles that yield an equal level of satisfaction or utility. If the two bundles suit his tastes equally well, we say that the consumer is

indifferent between the two bundles. Total Utility is equal at all points along the indifference. FOUR PROPERTIES OF INDIFFERENCE CURVES Higher indifference curves are preferred to lower ones: Indifference curves are downwards sloping: Consumers usually prefer more of something to less of it.

This preference for greater quantity is reflected in the indifference curves. Higher IDs curves represent larger quantities of goods than lower IDs curves. The slope represent the rate at which the consumer is willing to substitute one good for another. Therefore, if the quantity of one good is reduced, the quantity of the other good must increase in order for the consumer to be equally happy. Indifference curve do not cross: . Suppose two IDs curves did cross, as in figure below. At point A is on the same ID curve as point B, the two points will make the consumer equally happy. Point B is on the curve as Point C, these two points will make the consumer equally happy. This implies that point A and C would also make the consumer equally happy even thought point C

has more of both goods. This contradicts our assumption that consumer always prefers more of both goods to less. This is called the axiom of transitivity given any three basket of goods, if A is preferred to B and B to C then A must be preferred to C. Thus IDs curves cannot cross. If the consumer is indifferent between A and B, B and C. She/he must also be indifferent between A and C. But that is impossible. Bundle C is above A, so the consumer should prefer C to A by the more is better property. Therefore because preferences are transitive and more is better than less, ID curves cannot cross. FIGURE 1: INDIFFERENCE CURVE PROPERTIES FOUR PROPERTIES OF INDIFFERENCE CURVES(CONT.) Indifference Curves are bowed inward (Convex to the origin) People are more willing to trade away goods that they have in abundance and less willing to trade away goods of which they have little. These differences in a consumers marginal substitution rates

cause his or her indifference curve to bow inward. DEMONSTRATING AN INDIFFERENCE CURVE Suppose you have 2 goods in your utility function- Sandwiches (S) and Hotdogs (H). To get your indifference curve, you hold your utility constant and look at the different consumption bundles of each good that gives the same utility. Given your utility function as U= u(S,H) = S*H. To derive your indifference curve you keep your utility constant (U). E.g. U=10 what bundles of goods will give you this level of satisfaction. What if U=20? MARGINAL RATE OF SUBSTITUTION (MRS) MRS is the rate at which the consumer is willing to trade one good for another. Or the Maximum amount of one good a consumer will sacrifice to obtain one more unit of another good.

Mathematical Representation of MRS: Y/ X C/ Z Where C is the number of Cola that the consumer must give up to get more Pizzas or vice versa. MRS is the slope of the indifference curve. Moving from A to B, this consumer must give up 6 colas, to obtain more pizzas = 1, so the MRS is -6/1 = -6. That is the slope of the Indifference curve is -6. The negative sign shows that the consumer is willing to give up some of one good to get more of the other. IDs slope downwards. CURVATURE OF INDIFFERENCE CURVES

Must indifference curves be convex to the origin? ID curves do not have to be convex to the origin, but casual observation suggests that most peoples ID curves are convex. When people have a lot of one good, they are willing to give up a relatively large amount of it to get a good of which they have relatively little. However, after that first trade, they are willing to give up less of the first good to get the same amount of the second good.

Figure 3: This consumer is willing to give up three burritos to obtain one more pizza when she is at point a in figure 3.At b, she is willing to trade only two burritos for pizza. At c, she is even willing to trade; she will give up only one burrito for another pizza. This willingness to trade fewer burritos for one more pizza as we move down and to the right along the indifference curve reflects a diminishing marginal rate of substitution: The MRS approaches zero as we move down and to the right along n indifference curve. Figure 3: Indifference Curve Convex to the origin CONCAVE INDIFFERENCE CURVE If indifference curve are concave, the consumer would be willing to give up more burritos to get one more pizza, the fewer the burritos she has. In panel Figure 4., she trades

one pizza for three burritos moving from Bundle c to b , and she trades one pizza for only two burritos moving from b to a, even though her ratio of burritos to pizza is greater. Figure 4: Indifference Curve Concave to the Origin EXTREME EXAMPLE OF INDIFFERENCE CURVES Perfect Substitutes: Goods that a consumer is completely indifferent as to which to consume. Because

Bill cannot taste any difference between Coca Cola and Pepsi, he views them as perfect substitutes. He is indifferent between one can of coke and one can of Pepsi. His indifference curves for the two goods are straight, parallel lines with a slope of -1every where along the curve. The slope of the ID curves of perfect substitutes need not always be -1; it can be any constant rate. For example: Ben knows from reading the labels that Clorox bleach is twice as strong as a generic brand. As a result, Ben is indifferent between one cup of clorox and two cups of the generic bleach. The slope of his indifference curve is -2 where the generic bleach is on the vertical axis. Perfect compliments: Goods that a consumer is interested in consuming only in fixed proportions.

Maureen doesnt like pie by itself or ice cream by itself but loves pie a` la mode: a slice of pie with scoop of vanilla ice cream on top. Her indifference curves have right angles in panel b. If she has only one piece of pie, she gets as much pleasure from it and one scoop of ice cream, Bundle a, as from it and two scoops, Bundle d, or as from it and three scoops, Bundle e. That is, she wont eat the extra scoops because she does not have pieces of pie to go with the ice cream. Therefore, she consumes only bundles like a, b, and c in which pie and ice cream are in equal proportions. EXTREME EXAMPLE OF INDIFFERENCE CURVE Figure 4: Perfect Substitutes Figure 5: Perfect Compliments BUDGET CONSTRAINTS

The budget constraint defines the feasible set of consumption bundles you are able to consume. It is dependent on three things: The prices of the goods in your consumption bundle. The quantity of each good in your consumption bundle. You income. BUDGET CONSTRAINT (CONT.) Assuming that a consumer spend their money on two goods. The budget constraint for two goods is defined as the following: Y = P 1 X 1 + P 2 X 2..(1)

P 1 X 1 is the amount spend on X 1 and P 2 X 2 is the amount spend on X2 . Eguation (1) above is the budget constraint. Its shows that the consumers expenditures on X 1 and X 2 use up his/her entire budget. How many X 1 can this consumer buy? Subtracting P 1 X1 from both sides of equation (1) and dividing both sides by P 1 , we determine the amount of X1 that the consumer can purchase. Y = P 1 X 1 + P 2 X 2 can be rewritten as: X1 = (Y/ P 1 ) (P 2 / P 1 )* X2 .............(2)

Note: Slope of Budget constraint = - P 1 / P 2 Where Slope of the budget line depends on the relative prices. Assume that a consumer faces prices of P 1 = N$1 and P 2 =N$2, so the slope of the budget line is P 1 / P 2 = -1/2 The slope of budget line is called the Marginal Rate of Transformation (MRT): The trade-off the market imposes on the consumer in terms of the amount of one good the consumer must give up to purchase more of the other good: MRT = X 2 / X 1 = - P1 / P 2 Because this consumers MRT = -1/2, He/she can trade an extra X 1 for half X 2 : OR, equivalently , he/she has to give up two X1 to obtain extra X 2 . Thus, intercept on the Y axis will be Y/ P 2 and the x-axis intercept will be Y/ P 1 . EXAMPLE 1:

Suppose you have N$100 to spend on apples and biltong and you know that the price of apples is N$10 and the price of biltong is N$20. Draw your budget curve assuming that apples will be on the Y-axis. The budget equation: N$100=N$20X 1 +N$10X 2 Solving for X 2 = 10 - 2X 1 Solving for the Y intercept: Y/P2=100/10 =10 Solving for the X intercept: Y/P1=100/20 =5 CLASS ACTIVITY 1 QUESTION 1 Consider Garys utility function: u(x, y) = 5xy, where x and y are two goods. 2.1 If the individual consumed 10 units of x and received 250 of utility, how many units of y must the individual consume? 2.2 Would a market basket of x = 15 and y = 3 be preferred to the above combination? Explain. QUESTION 2 Suppose that Bridget spends her income on two goods, food (F) and clothing

(C). Her preferences are represented by the utility function U(F, C) = 10FC. 2.1 What happens to Bridgets utility when the consumption bundle is (10, 5)? 2.2 What happens to Bridgets utility when the consumption bundle is (15, 8)? 2.3 Give any three consumption bundles that result in a utility of 2000, providing the quantity of food exceeds quantity of clothing. SHIFT IN THE BUDGET CONSTRAINT (INCOME) Change in income causes the budget line to shift. Increase in income result in a shift of budget line to the right and a decrease in income would shift the budget line to the left. Suppose your in Example 1 above increase from N$100 to N$160 and the prices of apples and biltong remain the same. What will happen to your budget line? SHIFT IN THE BUDGET CONSTRAINTS (PRICE): Suppose Income remains unchanged but the prices of the goods you are consuming change. Change in a price of any good being

consumed, causes a change in the slope of the line; Price increases; budget gets smaller. Price decreases; budget gets larger. Suppose that the price of Apples increase to N$20 and the price of biltong and income remain the same, what will happen to intercept of Apples? What happens to the budget line if the price of apples increase to N$20 and the price of biltong decrease to N$10? CONSUMER EQUILIBRIUM IN THE UTILITY APPRAOCH To derive consumer equilibrium, the marginal utilities have to be weighted by the prices of the products concerned and consumers available income also taken into account. Consumer is in equilibrium when the weighted marginal utilities are equal for an affordable combinations of goods (When the consumer spend all his/her available income).

When will consumer be in equilibrium based on the utility theory? When weighted Marginal utilities are equal. When all available income is spend. OPTIMIZATION: WHAT THE CONSUMER CHOOSES We now have two pieces to figure out consumer choice: Consumer Preference: Consumers want to get the combination of goods on the highest possible indifference curve. Consumer Budget Constraint: Consumer must also end up on or below his budget constraint. The Consumers Optimal Choices Combining the indifference curve and the budget constraint determines the consumers optimal choice. Consumer optimum occurs at the point where the highest indifference curve and the budget constraint are tangent. The consumer chooses consumption of the two goods so that the

marginal rate of substitution equals the relative price. At the consumers optimum, the consumers valuation of the two goods equals the markets valuation. FIGURE 6: The optimal position of the consumer will be at the highest point on indifference curve where your budget line is tangential to your indifference curve. You can only change to new equilibrium if: Your disposable income change. Relative prices of goods change. Your desire for the two goods changes. At the point of your equilibrium the slope of the indifference curve equal to the slope of the budget line. MRS=MU / MU = P /P OR x y

x y MUY / P Y = MUX / P X Consumer is in equilibrium when marginal utility per dollar spent on good X is equal to the marginal utility per dollar spent on good Y. OPTIMAL MIX: EXAMPLE Jane lives in a dormitory that offers soft drinks and chips for sale in vending machines. Her utility function is U = 3SC (where S is the number of soft drinks per week and C the number of bags of chips per week), so her marginal utility of S is 3C and her marginal utility of C is 3S. Soft drinks are priced at N$0.50 each, chips N$0.25 per bag. 1.1 Write an expression for Jane's marginal rate of substitution between soft drinks and chips if Soft drinks are on the X-axis. 1.2 Use the expression generated in part (1.1) to determine Jane's optimal mix of soft drinks and chips. 1.3 If Jane has $5.00 per week to spend on chips and soft drinks,

how many of each should she purchase per week? CLASS ACTIVITY 2 Suppose Kanu has N$13 to spend on bananas and apples. The price of Bananas is N$2 per kilogram, while the price of apples is N$1 per kilogram. The following table represents kanus total utilities of bananas (TUB) and apples (TUA) respectively. Use the information in the table to answer the following True or False questions: Units TUB TUA 1 52 50 2

92 75 3 120 95 4 140 109 5 154 119

1) 2) 3) 4) 5) When spending his total income of N$13 on bananas and apples, Kanu is in equilibrium if he buys 3 kilogram of bananas and 4 kilograms of apples. When spending his total income of N$13 on bananas and apples, Kanu is in equilibrium if he buys 2 kilogram of bananas and 3 kilograms of apples. When spending his total income of N$13 on bananas and apples Kanu is in equilibrium if he buys 4 kilograms of bananas and 5 kilograms of apples. When spending his total income of N$13 on bananas and apples Kanu is in equilibrium if he buys 5 kilogram of bananas and 5 kilograms of apples. Kanu will be in equilibrium if his total utility is equal to 273 utils.

6) 7) 8) 9) 10) Kanu will be in equilibrium if his total utility is equal to 129 utils. Kanu will be in equilibrium if his total utility is equal to 259 utils. Kanu will be in equilibrium where his total utility is at a maximum, given the prices of bananas and apples and his available income. A consumer is in equilibrium if his or her maginal utility is at a maximum. A consumer is in equilibrium if the marginal utility per dollar is equal for all products he or she can afford to buy. CLASS ACTIVITY 3 QUESTION 1 John consumes two goods, X and Y. The marginal utility of X and the marginal utility of Y satisfy the following equations: MUX = Y and MUY = X. The price of X is N$9, and the price of Y is N$12. 1.1 Write an expression for John's MRS. 1.2 What is the optimal mix between X and Y in John's market basket? QUESTION 2

Eslons utility function is given as U= CS, where C= Chips and S= tin of Stoney per week. Suppose that the Marginal Utility of Chips (MUc) = S and Marginal utility of Stoney (MUs) = C. 2.1 What is her marginal rate of substitution if the Chips are on the vertical axis and Stoney on the horizontal axis? 2.2 Eslons income is N$120, the price of one packet of Chips is N$4, and the price of Stoney is N$2. Derive Eslons budget equation. 2.3 How many Chips and Stoney does he consume to maximize his utility? 2.4 Illustrate your answer in (b) and (c) graphically. What is the slope of the budget line? Also determine his total utility. THANK YOU!!! QUESTIONS???