Consumer Behaviour Module III PDF

Consumer Behaviour Module III Dr Shahid Bashir Senior Lecturer, Mahindra University NET, Ph.D.(Economics), Gold Medalist The Standard Economic Model (SEM) When you walk into a shop or look to make a purchase online, you are confronted with a range of goods that yo...

Consumer Behaviour Module III Dr Shahid Bashir Senior Lecturer, Mahindra University NET, Ph.D.(Economics), Gold Medalist The Standard Economic Model (SEM) When you walk into a shop or look to make a purchase online, you are confronted with a range of goods that you might buy. Of course, because your financial resources are limited, you cannot buy everything you want. The assumption is that you consider the prices of the various goods being offered for sale and buy a bundle of goods that, given your resources, best suits your needs and desires. This model is called the classical theory of consumer behaviour or the standard economic model (SEM) and is fundamentally based on an assumption that humans behave rationally when making consumption choices. 3 The Standard Economic Model (SEM) The SEM provides a theory of consumer choice which provides a complete understanding of demand. It examines the trade-offs that people face in their role as consumers. When making trade-offs, there are assumptions that are made about consumers. These include: Buyers are rational (they do the best they can, given their circumstances). More is preferred to less (monotonicity). Buyers seek to maximise their utility. Consumers act in self-interest and do not consider the utility of others. 4 The Basis of Choice Choices depend on possibilities and preferences. The budget line describes possibilities. The concept of utility describes preferences. 5 Budget Constraint Budget Constraint A budget constraint describes the limits to consumption choices and depends on the consumer’s budget and the prices of goods and services. Budget set: The budget set represents all possible combinations of two or more goods that a consumer can purchase given their income and the prices of those goods. It includes both the affordable combinations (within the budget) and the maximum that can be spent. In general, the budget set can be written as: 𝑷𝑿 𝑿 + 𝑷𝒀 𝒀 ≤ 𝑰 Budget line: The budget line is the boundary of the budget set. It represents all combinations of goods where the consumer is spending their entire income. and the equation of budget line can be written as: 𝑷𝑿 𝑿 + 𝑷𝒀 𝒀 = 𝑰 7 Budget Constraint Let’s look at Tina’s budget line: Suppose, Tina has $4 a day to spend on two goods: bottled water and chewing gum. The price of water (Good X) is $1 a bottle. The price of chewing gum (Good Y) is 50¢ a pack. 8 Changes in budget/income When a consumer’s income/budget decreases, consumption possibilities shrink. Budget line shifts towards left When a consumer’s income/budget increases, consumption possibilities expand. Budget line shifts towards right In both the cases, the slope of the budget line doesn’t change because prices have not changed. 11 Decrease in the price of good X (water) The price of water has decreased from $1 to 50¢ Income $4 Price of gum 50¢ When the price of water falls, the budget line becomes flatter (slope decreases). Recall that slope equals rise over run. 12 Increase in the price of good X (water) The price of water has increased from $1 to $2. Income $4 Price of gum 50¢ When the price of water rises, the budget line becomes steeper (slope decreases). Recall that slope equals rise over run 13 Slope of the budget line Let’s calculate the slope of the initial budget line. When the price of water is $1 a bottle, price of a chewing gum pack is 50¢, and Income $4: Slope equals 8 packs of gum divided by 4 bottles of water. Slope equals 2 packs of gum per bottle. 14 Slope of the budget line Next, calculate the slope of the budget line when water costs 50¢ a bottle. Slope equals 8 packs of gum divided by 8 bottles of water. Slope equals 1 pack of gum per bottle. 15 Slope of the budget line Finally, calculate the slope of the budget line when water costs $2 a bottle. Slope equals 8 packs of gum divided by 2 bottles of water. Slope equals 4 packs of gum per bottle. 16 Slope as an opportunity cost The slope of the budget line is also called the marginal rate of transformation (MRT). You can think of the slope of the budget line as an opportunity cost. The slope tells us how many packs of gum a bottle of water costs. Another name for opportunity cost is relative price, which is the price of one good in terms of another good. A relative price equals the price of one good divided by the price of another good and equals the slope of the budget line. 17 Budget Constraint A Mathematical Treatment Budget Constraint* Consumption bundle** (x1; x2) Prices of the two goods, (p1; p2) Consumer’s income, m Budget constraint/set of the consumer can be written as: p1x1 + p2x2 ≤ m (1) The budget set consists of all bundles that are affordable at the given prices and income. 19 Slope of Budget Line The budget line is the set of bundles that cost exactly m: p1x1 + p2x2 = m (2)* Solving the above equation for x2 we get: 𝒎 𝒑𝟏 𝒙𝟐 = − 𝒙 (3)** 𝒑𝟐 𝒑𝟐 𝟏 𝒎 𝒑𝟏 Where is the vertical intercept and − is 𝒑𝟐 𝒑𝟐 the slope of budget line. Solving the above equation for x1 we get: 𝒎 𝒑𝟐 𝒙𝟏 = − 𝒙 (4) 𝒑𝟏 𝒑𝟏 𝟐 𝒎 Where is the horizontal intercept 𝒑𝟏 20 Interpretation of the Slope of Budget Line The slope of the budget line has a nice economic interpretation. It measures the rate at which the market is willing to substitute good 1 for good 2. Suppose for example that the consumer is going to increase his consumption of good 1 by dx1. How much will his consumption of good 2 have to change in order to satisfy his budget constraint? Let us use dx2 to indicate his change in the consumption of good 2. 21 Interpretation of the Slope of Budget Line Now note that if he satisfies his budget constraint before and after making the change he must satisfy: 𝒑𝟏 𝒙𝟏 + 𝒑𝟐 𝒙𝟐 = 𝒎 (5) and 𝒑𝟏 𝒙𝟏 + 𝒅𝒙𝟏 + 𝒑𝟐 𝒙𝟐 + 𝒅𝒙𝟐 = 𝒎 (6) Subtracting equation (5) from (6) gives 𝒑𝟏 𝒅𝒙𝟏 + 𝒑𝟐 𝒅𝒙𝟐 = 𝟎 (7) This says that the total value of the change in his consumption must be zero. 𝒅𝒙𝟐 Solving for , the rate at which good 2 can be substituted for good 1 while still 𝒅𝒙𝟏 satisfying the budget constraint, gives 𝒅𝒙𝟐 𝒑𝟏 =− (8) 22 𝒅𝒙𝟏 𝒑𝟐 Increase in Consumer’s Income It is easy to see from equation (3 and 4) that an increase in income will increase the vertical and horizontal intercept but not affect the slope of the line. 𝒎 𝒑𝟏 𝒙𝟐 = − 𝒙𝟏 (3) 𝒑𝟐 𝒑𝟐 𝒎 𝒑𝟐 𝒙𝟏 = − 𝒙𝟐 (4) 𝒑𝟏 𝒑𝟏 An increase in income causes a parallel shift outward of the budget line. A decrease in income causes a parallel shift inward of the budget line. 23 Increase in the Price of Good 1 Increase in the price of good 1 while holding price of good 2 and income fixed. This will not change the vertical intercept, but it 𝒑 will make the budget line steeper since 𝟏 𝒑𝟐 will become larger*. 𝒎 𝒑𝟏 𝒙𝟐 = − 𝒙𝟏 (3) 𝒑𝟐 𝒑𝟐 24 Change the prices of good 1 and good 2 at the same time Suppose for example that we double the prices of both goods 1 and 2. In this case both the horizontal and vertical intercepts shift inward by a factor of one-half, and therefore the budget line shifts inward by one-half as well. 𝒎 𝒑𝟏 𝒙𝟐 = − 𝒙𝟏 𝟐𝒑𝟐 𝒑𝟐 𝒎 𝒑𝟐 𝒙𝟏 = − 𝒙𝟐 𝟐𝒑𝟏 𝒑𝟏 Multiplying both prices by two is just like dividing income by 2. 25 Price and income changes together What happens if both prices go up and income goes down? Think about what happens to the horizontal and vertical intercepts. 𝒎 If m decreases and p1 and p2 both increase, then the intercepts and 𝒑𝟏 𝒎 must both decrease. This means that the budget line will shift inward. 𝒑𝟐 What about the slope of the budget line? 𝒑𝟏 If price 2 increases more than price 1, so that decreases (in absolute value), 𝒑𝟐 then the budget line will be flatter. If price 2 increases less than price 1, the budget line will be steeper. 26 Price and income changes together What happens if both prices go down and income goes up? Think about what happens to the horizontal and vertical intercepts. 𝒎 If m increases and p1 and p2 both decrease, then the intercepts and 𝒑𝟏 𝒎 must both increase. This means that the budget line will shift outward. 𝒑𝟐 What about the slope of the budget line? 𝒑𝟏 If price 2 decreases more than price 1, so that increases (in absolute value), 𝒑𝟐 then the budget line will be steeper. If price 2 decreases less than price 1, the budget line will be flatter. 27 Utility What is Utility*? Utility is the benefit or satisfaction that a person gets from the consumption of a good or service. Utility is the want satisfying power of a good. Ordinal Utility***: Satisfaction derived by consuming a product is not measurable but more or less comparable. Ordinal utility theory focuses on how people rank their preferences for goods and services, without assigning numerical values to their satisfaction. Ordinal rankings give us information about the order in which a consumer ranks baskets. For example, a consumer prefers basket A to B. However, an ordinal ranking would not tell us how much more he likes A than B. 29 What is Utility? Cardinal Utility**: Utility is modelled as a quantifiable or cardinal property of the economic goods that a person consumes. Utility is measured numerically in units called utils, which is a Spanish word meaning "useful". Cardinal rankings give us information about the intensity of a consumer’s preferences. With a cardinal ranking, we not only know that he prefers basket A to basket B, but we can also measure the strength of her preference for A over B. We can make a quantitative statement, such as “The consumer likes basket A twice as much as basket B.” A cardinal ranking therefore contains more information than an ordinal ranking. 30 Total and Marginal Utility To use utility to explain and predict choices, we distinguish between: Total utility Marginal utility Total utility is the total benefit that a person gets from the consumption of a good or service. Total utility generally increases as the quantity consumed of a good increases. Marginal utility is the change in total utility that results from a one-unit increase in the quantity of a good consumed. In other words, the additional satisfaction gained by the consumption or use of one more unit of a good or service. MU = ∆TU = TUn – TUn-1 31 Law of Diminishing Marginal Utility* The law of diminishing marginal utility states that as consumption increases, the additional satisfaction or utility gained from consuming each additional unit of a product or service decreases. As more and more of a good is consumed, consuming additional amounts will yield smaller and smaller additions to utility. In other words, the more of any one good consumed in a given period, the less satisfaction (utility) generated by consuming each additional (marginal) unit of the same good. Think about your own marginal utility of the things that you consume. 34 Total Utility and Marginal Utility of Trips to the Club per Week* Consider this numerical example. Trips to Club Total Utility Marginal Utility 1 12 12 Frank loves country music, and a country band plays seven nights a 2 22 10 week at a club near his house. 3 28 6 The table shows how the utility he 4 32 4 derives from the band might change as he goes to the club more 5 34 2 frequently 6 34 0 35 Graphs of Frank’s Total and Marginal Utility 36 Utility-Maximizing Rule The best budget allocation occurs when a person follows the utility-maximizing rule: 1) Allocate the entire available budget. 2) Make the marginal utility per dollar/rupee equal for all goods*. Utility-maximising consumers spread out their expenditures until the following condition holds: 𝑴𝑼𝑿 𝑴𝑼𝒀 = for all goods 𝑷𝑿 𝑷𝒀 where MUX is the marginal utility derived from the last unit of X consumed, MUY is the marginal utility derived from the last unit of Y consumed, PX is the price per unit of X, and PY is the price per unit of Y. 37 Allocation of Fixed Income between Two Alternatives – Maximising Utility* Trips to Club per Week Total Utility Marginal Utility Price Marginal Utility per Dollar (TU) (MU) (P) (MU/P) 1 12 12 $3.00 4.0 2 22 10 3.00 3.3 3 28 6 3.00 2.0 4 32 4 3.00 1.3 5 34 2 3.00 0.7 6 34 0 3.00 0 Basketball Games per Total Utility Marginal Utility Price Marginal Utility per Dollar Week (TU) (MU) (P) (MU/P) 1 21 21 $6.00 3.5 2 33 12 6.00 2.0 3 42 9 6.00 1.5 4 48 6 6.00 1.0 5 51 3 6.00 0.5 6 51 0 6.00 0 38 Preferences Assumptions of Preferences 1. Rationality — Rationality refers to the ability to make decisions based on logical reasoning, typically by selecting the best available option to achieve a specific goal. Consumers make decisions aimed at maximising their utility given their budget constraints. 2. Transitivity — By this we mean that the consumer makes choices that are consistent with each other. A consumer’s preferences over bundles is consistent in the sense that, if the consumer prefers bundle A to bundle B and prefers bundle B to bundle C, the consumer also prefers bundle A to bundle C. If A ≻ B, B ≻ C, then A ≻ C (Transitive preferences) If A ≻ B, B ≻ C, and C ≻ A (Intransitive preferences - cyclicity) 40 Assumptions of Preferences 3. Ordinality— means that utility is ordinal, not cardinal. Consumers can rank different bundles of goods. 4. Completeness—Consumers have the ability to choose among the combinations of goods and services available. When facing a choice between any two bundles of goods, A and B, a consumer can rank them so that one and only one of the following relationships is true: The consumer prefers the A to B, B to A, or is indifferent between the too. 5. Monotonicity (more is better)*—all else being the same, more of a commodity is better than less of it. It means goods yield positive marginal utility. Good—a commodity for which more is preferred to less, at least at some levels of consumption. (MU > 0) Bad—something for which less is preferred to more, such as pollution. (MU < 0) 41 Assumptions of Preferences 6. Consistency: Preferences won’t change. It implies that if a person prefers bundle A over bundle B at one point, he will continue to do so in similar circumstances unless new information changes his perspective. 7. Declining marginal rate of substitution (𝑴𝑹𝑺𝑿𝒀) — the ratio at which a consumer is willing to substitute X for Y, is diminishing as X increases*. 𝜟𝒀 𝑴𝑹𝑺𝑿𝒀 = 𝜟𝑿 42 Indifference curve Indifference curve*—is a set of points, each representing a combination of some amount of good X and some amount of good Y, that all yield the same amount of total utility. Alternatively, Indifference curve is the locus of different combinations of two goods that yield the consumer same level of satisfaction/utility. The consumer depicted here is indifferent between bundles A and B, B and C, and A and C. Because “more is better,” our consumer is unequivocally worse off at A' than at A. 43 Indifference map Indifference map/preference map—a complete set of indifference curves that summarise a consumer’s tastes or preferences. Each consumer has a unique family of Indifference curves. 44 Properties of well-behaved Indifference Curves Higher the indifference curve, Higher the satisfaction- Bundles on indifference curves farther from the origin are preferred to those on indifference curves closer to the origin. An indifference curve goes through every possible bundle. Indifference curves cannot cross. Indifference curves slope downward. Indifference curves are convex to the origin. Indifference curves cannot touch either axis. Indifference curves need not to be parallel to each other. 45 Impossible Indifference Curves (a) Crossing (b) Upward Sloping (c) Thick 46 Impossible Indifference Curves (d) Touching either axis* B, Burritos per semester Z, Pizzas per semester 47 Why IC is convex to the Origin – Declining MRS* Marginal rate of substitution (MRS)—the maximum amount of one good a consumer will sacrifice to obtain one more unit of another good. Marginal rate of substitution of X for Y is 𝜟𝒀 𝑴𝑹𝑺𝑿𝒀 = 𝜟𝑿 where MRS is the slope of the indifference curve. 𝑴𝑼𝑿 ⋅ 𝜟𝑿 = − 𝑴𝑼𝒀 ⋅ 𝜟𝒀 When we divide both sides by 𝑀𝑈𝑌 and by Δ𝑋, we obtain 𝜟𝒀 𝑴𝑼𝑿 𝑴𝑹𝑺𝑿𝒀 = =− 𝜟𝑿 𝑴𝑼𝒀 𝜕𝑼ൗ 𝑴𝑼𝑿 𝜕𝑿 or 𝑴𝑹𝑺𝑿𝒀 = 𝜕𝑼ൗ =− 𝜕𝒀 𝑴𝑼𝒀 48 Curvature of Indifference Curves - Special Cases Perfect substitutes—goods that a consumer is completely indifferent as to which to consume. Perfect complements—goods that a consumer is interested in consuming only in fixed proportions. 50 Curvature of Indifference Curves - Special Cases (a) Perfect Substitutes (b) Perfect Complements 51 Curvature of Indifference Curves - Special Cases (c) Imperfect Substitutes (d) A useless good 52 Curvature of Indifference Curves - Special Cases (e) An economic bad 53 Consumer Choice – Equilibrium Consumers will choose from available combinations of X and Y the one that maximizes utility. In graphic terms, a consumer will move along the budget constraint until he or she is on the highest possible indifference curve. In this case, utility is maximized when our consumer buys X* units of X and Y* units of Y. As long as indifference curves are convex to the origin, utility maximization will take place at that point at which the indifference curve is just tangent to the budget constraint. 54 Consumer Choice – Equilibrium The tangency condition has important implications. Where two curves are tangent, they have the same slope, which implies that the slope of the indifference curve is equal to the slope of the budget constraint at the point of tangency: 𝑴𝑼𝑿 𝑷𝑿 − =− 𝑴𝑼𝒀 𝑷𝒀 By multiplying both sides of this equation by 𝑴𝑼𝒀 and dividing both sides by 𝑷𝑿 , we can rewrite this utility-maximizing rule as: 𝑴𝑼𝑿 𝑴𝑼𝒀 = 𝑷𝑿 𝑷𝒀 This is the same rule derived in our previous discussion without using indifference curves. 55 Numerical Problems Example 1*: Budget Constraint You have an income of $40 to spend on two goods. Good 1 costs $10 per unit, and Good 2 costs $5 per unit. a) Write down your budget equation. b) If you spent all your income on good 1, how much could you buy? c) If you spent all of your income on good 2, how much could you buy? Use blue ink to draw your budget d) Suppose that the price of good 1 falls to $5 while everything else stays the same. Write down your new budget equation. On the graph above, use red ink to draw your new budget line. e) Suppose that the amount you are allowed to spend falls to $30, while the prices of both commodities remain at $5. Write down your budget equation. Use black ink to draw this budget line. 59 Example 2*: Budget Constraint Draw a budget line for each case. a) p1= 1, p2= 1, m= 15 (Use blue ink) b) p1= 1, p2= 2, m= 20 (Use red ink) c) p1= 0, p2= 1, m= 10 (Use black ink) 60 Example 3*: Budget Constraint Sketch the following budget constraints: PX PY Income a. $100 25 $5,000.00 b. 200 125 5,000.00 c. 50 400 2,000.00 d. 40 16 800 e. 3 2 12 f. 0.125 0.75 3 g. 0.75 0.125 3 61 Example 4*: Budget Constraint Suppose the price of X is $5 and the price of Y is $10 and a hypothetical household has $500 to spend per month on goods X and Y. a) Sketch the household budget constraint. b) Assume that the household splits its income equally between X and Y. Show where the household ends up on the budget constraint. c) Suppose the household income doubles to $1,000. Sketch the new budget constraint facing the household. d) Suppose after the change the household spends $200 on Y and $800 on X. Does this imply that X is a normal or an inferior good? What about Y? 62 Example 4*: Solution 63 Example 4.1*: Budget Constraint Kristine is a fan of both action movies and classical concerts. This month she has $480 to spend on the two activities. The original budget constraint is shown in the graph below. Let X represent action movies and Y represent classical concerts. a) What is the equation of the original budget constraint? b) What is the price of a ticket to an action movie and a classical concert? c) Assume a price change occurs, and Kristine now faces the new budget constraint. What is the equation of the new budget constraint? d) With the new budget constraint, what is the price of a ticket to an action movie? A classical concert? 64 Example 4.1*: Budget Constraint- Solution a) The equation for the original budget constraint is 10X + 12Y = $480 𝑰 Hint: Find price of X using horizontal intercept X = and Price of Y by using vertical 𝑷𝑿 𝑰 intercept Y = 𝑷𝒀 b) The price of a ticket to an action movie is $10. The price of a ticket to a classical concert is $12. c) The equation for the new budget constraint is 15X + 12Y = $480. 𝑰 Hint: Find price of X using horizontal intercept X = and Price of Y by using vertical 𝑷𝑿 𝑰 intercept Y = 𝑷𝒀 d) The price of a ticket to an action movie has risen to $15. The price of a ticket to a classical concert is still $12. 65 Example 5*: Utility Maximisation Adrian has $21 to spend on energy drinks and protein bars and wants to maximize his utility on his purchase. Based on the data in the table, how many energy drinks and protein bars should Adrian purchase, and what is his total utility from the purchase? Does the utility maximizing rule hold true for his purchase? Explain. Energy Drinks $3.00 Protein Bars $1.50 Quantity MU TU Quantity MU TU 1 84 84 1 36 36 2 72 156 2 30 66 3 60 216 3 24 90 4 48 264 4 18 108 5 36 300 5 12 120 6 24 324 6 6 126 7 12 336 7 0 126 8 0 336 8 -6 120 66 Example 5*: Utility Maximisation - Solution Energy Drinks $3.00 Protein Bars $1.50 Although, at multiple bundles Quantity MU TU Price of ED MU/P Quantity MU TU Price of ED MU/P MUx MUY 1 84 84 3 28 1 36 36 1.5 24 = Px PY 2 72 156 3 24 2 30 66 1.5 20 3 60 216 3 20 3 24 90 1.5 16 But utility maximising also 4 48 264 3 16 4 18 108 1.5 12 involves entire budget to be 5 36 300 3 12 5 12 120 1.5 8 spent (That is happening only 6 24 324 3 8 6 6 126 1.5 4 at purchasing 5 Energy Drinks 7 12 336 3 4 7 0 126 1.5 0 ($15) and 4 Protein Bars ($6) = 8 0 336 3 0 8 -6 120 1.5 -4 $21) Adrian will purchase 5 energy drinks and 4 protein bars to maximize his utility. His total utility is equal to (300 + 108) = 408. The utility maximizing rule does hold true. The marginal utility of the last energy drink purchased divided by the price of energy drinks is 36/3 = 12. The marginal utility of the last protein bar purchased divided by the price of protein bars is 18/1.50 = 12. So, the marginal utility per dollar spent on the last energy drink and the last protein bar purchased are equal. 67 Example 6*: Utility Maximisation The table shows Darlene’s marginal utility numbers for ice creams and milkshakes. Darlene is trying to decide which item to purchase first, an ice- cream or a milkshake, knowing that she wants to receive the most utility for each dollar she spends. Assuming she has enough money in her budget to purchase either item, which item should she purchase first? Explain. Ice Creams $6 Milkshakes $4 Quandity MU Quantity MU 1 24 1 16 2 16 2 6 3 6 3 2 68 Example 6*: Utility Maximisation- Solution Ice Creams $6 Milkshakes $4 Quantity MU Price MU/P Quantity MU Price MU/P 1 24 6 4.0 1 16 4 4.0 2 16 6 2.7 2 6 4 1.5 3 6 6 1.0 3 2 4 0.5 If Darlene wants to receive maximum utility for each dollar she spends, it does not matter if she buys the first ice cream or the first milkshake, since each one yields the same marginal utility per dollar. Ice cream costs $6, and the marginal utility of the first ice cream is 24, so Darlene will receive a marginal utility per dollar of 24/6 = 4 for the first ice-cream. Milkshake costs $4, and the marginal utility of the first milkshake is 16, so Darlene will receive a marginal utility per dollar of 16/4 = 4 for the first milkshake. Darlene’s satisfaction per dollar spent will be 4 units for either the first ice cream or first milkshake, so it makes no difference which she buys first. 69 Example 7*: Utility Maximisation Suppose that Kendrick has $144 to spend on cigars and brandy each month and that both goods must be purchased whole (no fractional units). Cigars cost $6 each, and brandy costs $30 per bottle. Kendrick’s preferences for cigars and brandy are summarised by the following information: Cigars $6 Brandy $30 Packets per Bottles per TU MU MU/P TU MU MU/P Month Month 1 28 1 150 2 46 2 270 3 62 3 360 4 74 4 420 5 80 5 450 6 84 6 470 7 86 7 480 70 Example 7*: Utility Maximisation – Cont…. a) Fill in the figures for marginal utility and marginal utility per dollar for both cigars and brandy. b) Are these preferences consistent with the law of diminishing marginal utility? Explain briefly. c) Given the budget of $144, what quantity of cigars and what quantity of brandy will maximize Kendrick’s level of satisfaction? Explain briefly. d) Now suppose the price of cigars rises to $8. Which of the columns in the table must be recalculated? Do the required recalculations. e) After the price change, how many cigars and how many bottles of brandy will Kendrick purchase? 71 Example 7*: Utility Maximisation - Solution a) Cigars $6 Brandy $30 Packets per Bottles per TU MU P MU/P TU MU P MU/P Month Month 1 28 28 6 4.67 1 150 150 30 5.00 2 46 18 6 3.00 2 270 120 30 4.00 3 62 16 6 2.67 3 360 90 30 3.00 4 74 12 6 2.00 4 420 60 30 2.00 5 80 6 6 1.00 5 450 30 30 1.00 6 84 4 6 0.67 6 470 20 30 0.67 7 86 2 6 0.33 7 480 10 30 0.33 b) Yes, these figures are consistent with the law of diminishing marginal utility, which states that as the quantity of a good consumed increases, utility also increases, but by less and less for each additional unit. In the tables, the TU figures for both cigars and brandy are increasing, but as more cigars or more brandy are consumed, the MU diminishes. 72 Example 7*: Utility Maximisation - Solution c) Four cigars and Four bottles of brandy. To maximize utility, the individual should allocate income toward those goods with the highest marginal utility per dollar. The first bottle of brandy has a higher marginal utility per dollar than the first cigar, so Kendrick begins by purchasing the first bottle of brandy for $30. The first cigar has a higher marginal utility per dollar than the second bottle of brandy, so now Kendrick should buy a cigar, for a total expenditure of $36. Next, Kendrick should buy the second bottle of brandy, for total spending of $66. This will continue until Kendrick purchases four cigars and four bottles of brandy, for total spending of $144. Cigars $6 Brandy $30 Packets per Bottles per TU MU P MU/P TU MU P MU/P Month Month 1 28 28 6 4.67 1 150 150 30 5.00 2 46 18 6 3.00 2 270 120 30 4.00 3 62 16 6 2.67 3 360 90 30 3.00 4 74 12 6 2.00 4 420 60 30 2.00 5 80 6 6 1.00 5 450 30 30 1.00 6 84 4 6 0.67 6 470 20 30 0.67 7 86 2 6 0.33 7 480 10 30 0.33 73 Example 7*: Utility Maximisation - Solution d) If the price of cigars rises to $8, only the MU/P column for cigars needs to be recalculated: Cigars $8 Packets per month TU MU P MU/P 1 28 28 8 3.50 2 46 18 8 2.25 3 62 16 8 2.00 4 74 12 8 1.50 5 80 6 8 0.75 6 84 4 8 0.50 7 86 2 8 0.25 74 Example 7*: Utility Maximisation - Solution e) Using the same logic as in part (c), Kendrick should purchase three cigars and four bottles of brandy, for a total expenditure of (3 × $8) + (4 × $30) = $144. Cigars Brandy Packets per Bottles per TU MU P MU/P TU MU P MU/P Month Month 1 28 28 8 3.50 1 150 150 30 5.00 2 46 18 8 2.25 2 270 120 30 4.00 3 62 16 8 2.00 3 360 90 30 3.00 4 74 12 8 1.50 4 420 60 30 2.00 5 80 6 8 0.75 5 450 30 30 1.00 6 84 4 8 0.50 6 470 20 30 0.67 7 86 2 8 0.25 7 480 10 30 0.33 75 The Algebra of Consumer’s Equilibrium Utility Function A utility function mathematically represents the level of satisfaction that a consumer enjoys from consuming a bundle of goods. Example: If he consumes bundle 𝑨 = (𝟒𝟎, 𝟑𝟎) and her utility function is 𝒖 𝒙, 𝒚 = 𝟑𝒙 + 𝟓𝒚, her level of utility at bundle 𝑨 is 𝒖 𝟒𝟎, 𝟑𝟎 = 𝟑 × 𝟒𝟎 + 𝟓 × 𝟑𝟎 = 𝟐𝟕𝟎 The utility level from bundle 𝐴 is not as important as the ranking of utilities across bundles: Only the utility ranking matters → “ordinality.” The specific utility level that the consumer reaches with each bundle does not matter → “cardinality.” 77 Cobb-Douglas Utility Function* One commonly used utility function that generates well behaved ICs has the form: 𝒖 𝒙, 𝒚 = 𝑨𝒙𝜶 𝒚𝜷 where 𝐴, 𝛼, and 𝛽 are positive constants. Special cases: 𝜶 + 𝜷 = 𝟏: 𝒖 𝒙, 𝒚 = 𝑨𝒙𝜶 𝒚𝟏−𝜶 𝑨 = 𝟏: 𝒖 𝒙, 𝒚 = 𝒙𝜶 𝒚𝜷 𝑨 = 𝜶 = 𝜷 = 𝟏: 𝒖 𝒙, 𝒚 = 𝒙𝒚 𝑨 = 𝟏, 𝜶 = 𝜷 = 𝟎. 𝟓: 𝒖 𝒙, 𝒚 = 𝒙𝟎.𝟓 𝒚𝟎.𝟓 or 𝒖 𝒙, 𝒚 = 𝒙𝒚 78 Cobb-Douglas Utility Function - Properties Represents well behaved indifference curves. ICs are downward sloping from left to right- MRSxy is negative ICs are convex to the origin- MRSxy declines as x increases. Never touch any axis. 79 Utility Maximisation Problem (UMP) Consumer equilibrium occurs when a consumer maximises his utility subject to his budget constraint. This can be represented using the concepts of an indifference curve (which represents levels of utility) and a budget line (which represents the consumer's income constraint). Let the consumer's utility be represented by a utility function: U= f (x, y) or Utility = U(x, y) where 𝑥 and y are the quantities of two goods consumed. The indifference curve represents combinations of 𝑥 and y that yield the same utility. Subject to the budget constraint* px x + py y = I or I - px x - py y = 0 86 Utility Maximisation Problem (UMP) Constrained maximisation problems are often solved by setting up a Lagrangian function, which in this UMP is: 𝓛 = 𝑼(𝒙, 𝒚) + 𝝀 𝑰 − 𝒑𝒙 𝒙 − 𝒑𝒚 𝒚 (1) where λ represents the Lagrange multiplier, which multiplies the budget line. To solve this problem, we take first-order conditions (partial differentiation) with respect to x, y, and λ, which yields 𝝏𝓛 = 𝑴𝑼𝒙 − 𝝀𝒑𝒙 = 𝟎 (2) 𝝏𝒙 𝝏𝓛 = 𝑴𝑼𝒚 − 𝝀𝒑𝒚 = 𝟎 (3) 𝝏𝒚 𝝏𝓛 = 𝑰 − 𝒑𝒙 𝒙 − 𝒑𝒚 𝒚 = 𝟎 (4) 𝝏𝝀 𝑴𝑼𝒙 Equation (2) can be rearranged to = 𝝀; and, similarly, Equation (3) can be 𝒑𝒙 𝑴𝑼𝒚 𝑴𝑼𝒙 𝑴𝑼𝒚 expressed as = 𝝀. Using these two expression yields =𝝀= 𝒑𝒚 𝒑𝒙 𝒑𝒚 87 Utility Maximisation Problem (UMP) 𝑴𝑼𝒙 𝑴𝑼𝒚 We finally get = 𝒑𝒙 𝒑𝒚 This is the “bang for the buck” coinciding across goods (marginal utility per dollar equal for both the goods). Alternatively, this condition can be expressed as 𝑴𝑼𝒙 𝒑𝒙 = 𝑴𝑼𝒚 𝒑𝒚 which is same as the tangency condition used in the previous analysis. Alternatively, consumer maximises utility at a point where: Slope of indifference curve (MRSxy) = Slope of budget line 88 How to solve a UMP Using Lagrangian Multiplier or Using the following simple rule: 𝑴𝑼𝒙 𝒑𝒙 Step 1: Set the tangency condition as =. Cross-multiply and simplify. 𝑴𝑼𝒚 𝒑𝒚 Step 2: Solve for x or y, and insert the resulting expression into the budget line px x + py y = I and get the value for x or y. Step 3: Lastly, to find the optimal consumption of good x or y, we use the tangency condition as derived in step 1. 90 Example 1* - Interior optimum Consider an individual with a Cobb-Douglas utility function U(x,y)= xy, facing market prices px =$20 and py =$40, and income I =$800. Find his optimal consumption bundle. 91 Example 2* - Interior optimum Consider a variation of example 1 in which the individual now has a Cobb-Douglas utility function u(x, y)=x1/3y2/3, facing market prices px =$10 and py =$20, and income I =$100. Find his optimal consumption bundle. 92