Microeconomics: Utility and Demand

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Consumer Behavior: Utility and Demand
Cardinal Utility
Consumer Surplus
Ordinal Utility
Indifference Curves
The Consumer’s Constrained Maximization Problem

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  • Let’s start considering three assumptions which form the basis of consumer theory. Firstly, consumer is unable to change the prices of the goods (P X , P Y ). In this case, coke = X; and popcorn = Y. In other words, consumers are price takers. Secondly, we assume that there is only one time period in this model and the consumer consumer’s income is M, but cannot borrow any more. Given prices, income and individual tastes, the consumer’s goal is to maximize his/her utility U = U (X, Y), where X is units bought of the good X, coke, and Y represents units bought of the good Y, popcorn.
  • The problem is to measure the utility or satisfaction an individual receives from consuming goods. We have two approaches to utility analysis in economic theory. Cardinal utility: assumes that the consumer has the ability to accurately measure the level of utility he/she derives from consuming a particular combination of goods, and assign an number to it. When economists first studied utility (late nineteenth-early twentieth-century), they assumed that utility could be measured in terms of some unit of satisfaction. Ordinal utility: consumers are assumed to rank consumption bundles and choose among them.
  • In developing our theory of consumer behavior, we begin by considering the consumption of a single good. In our example, X represents cans of coca-cola, Y represents cups of popcorn. and the value of this good Y is X = amount of cans of coca-cola Y = amount of cups of popcorn So, we are interested in in the full satisfaction (or total utility, U) resulting from the consumption of different units of coke when the value of the good Y is held constant (e.g. Y = 1; 1 cup of popcorn).
  • We would expect that for most consumers the increase in utility from consuming the first can of coca-cola would be the greatest: for example, 10 utils. We assume that utility is measured in utils. The marginal utility is the satisfaction given by the last unit consumed; in this case, is the same: 10 utils. But, if the individual consumes a second can of coca-cola, combined with one cup of popcorn, we expect an increase in the total utility or satisfaction, but the increase in his/her utility is likely smaller than for the first can; it comes to say that the marginal utility, the satisfaction provided by the last unit, decreases. As additional cans of coca-cola are consumed, we expect that the increases in his utility, MU X , will continue to diminish until he achieves a maximum level of utility, 22 utils, corresponding to 5 cans, because additional units of good X result in decreases in his utility and negative marginal utility.
  • If changes in the independent variable, good X, can be reduced to infinitesimally small amounts, and the other independent variable, good Y, is held constant, then the marginal utility is the partial derivative of U with respect to X.
  • The slope of the utility curve is the marginal utility. In mathematical terms, we can state that the slope of this utility curve, Δ U/ Δ X. Utility rises at a decreasing rate with respect to increases in the consumption levels of good, until S. We can observe that while the successive increments in good X are of equal size, the corresponding increments in utility are diminishing up to X = 5 units. In other words, the slope of this utility curve, the marginal utility, is diminishing up to S. In this point, the slope is 0, or marginal utility is zero, which indicates that the utility curve achieves a maximum. After X=5 utility actually begins to decline with any further increases in X. In other words, the marginal utility, or the slope of the utility curve, is negative.
  • With the data in the previous table, we can draw the marginal utility function. The utility obtained from each additional can of coke falls as the total number rises.
  • We are going to consider a consumer buying two goods: X = cans of coke; Y = cups of popcorn. Both goods have same price per unit of $1.00. Consumer’s monetary income is $5.00. In this case, the goal of the consumer is to select the number of cans of coke and cups of popcorn that will maximize his/her utility, given prices and income. With the first dollar, the individual will consume coke, because the marginal utility is greater than the marginal utility of popcorn. With the second dollar, the consumer can choose either coke or popcorn, because the marginal utility is the same. Imagine that he/she choose Y. With the third dollar the consumer will decide to buy another can of coke. With the forth dollar he/she would choose either coke or popcorn. Imagine he/she chooses popcorn. Finally, the consumer has only one dollar to spend. He/she will buy a new can of coke. So, the purchase will be: three units of good X, and two units of good Y. We see that the marginal utility per dollar of X is 4: MU X /P X The marginal utility per dollar of Y is 4 too: MU Y /P Y This is the condition required for a consumer to be maximizing utility.
  • Now, we are going to apply the previous ideas to develop the concept of consumer surplus
  • Imagine a smoker who smokes a lot. Suppose that we have collected the following information on the basis of an personal interview with him. This smoker would be willing to pay $3.00 for one pack per week. For a second pack per week this smoker would be willing to pay $2.80. And $2.60, $2.40, and $2.20 for successive packs from the third to the fifth per month. Of course, the amount the smoker is prepared to pay reflects the marginal utility he/she gets. The marginal utility we know decreases when the consumer increases the consumption of a good. So, he/she would be willing to pay progressively smaller amounts for each additional unit consumed. However, in practice, the consumer pays the same amount of money for each pack bought. For example, imagine that he/she buys 5 packs, and each pack of cigarettes costs $2.20. As long as he/she would be willing to pay more than the market price for any unit, he/she obtains consumer surplus on in when he/she buys it. For example, for the first pack the smoker would be willing to pay $3.00. But he/she in reality only pays $2.20. So, the consumer surplus are the fifty cents saved.
  • With the data shown in the table we can draw the individual demand for cigarettes. The negative sloped demand curve shows that the smoker would be willing to pay progressively smaller amounts for each additional unit consumed, reflecting diminishing marginal utility of a good with increases in its consumption. We remember that U is sigma MU. So, U is: $13.00. So, the sum of the values that he/she places on each pack gives us the total value that the smoker places on all 5 packs of cigarettes. In this case, $13.00 per week. If the changes in consumption were infinitesimally small, then the total utility would be the area under the demand curve.
  • Therefore, total utility is the amount he/she would be willing to pay if he/she were offered the cigarettes one pack at a time. However, this smoker does not to pay a different price for each pack he/she consumes each week; he/she buys 5 packs paying the same for each one: $2.20. He/she pays: $11.00. Consumer surplus is the sum of the extra valuations placed on each unit above the market price paid for each
  • To develop the indifference theory, let’s assume that there are only two consumption goods, good X and good Y. In our case, let’s suppose that X are cons of ice-cream and Y are cups of cold lemonade). In this approach, consumers are assumed to rank consumption bundles and choose among them. Each consumption bundle contains x units of X and y units of Y
  • To develop the indifference theory, let’s assume that there are only two consumption goods, good X and good Y. In our case, let’s suppose that X are cons of ice-cream and Y are cups of cold lemonade). In this approach, consumers are assumed to rank consumption bundles and choose among them. Each consumption bundle contains x units of X and y units of Y
  • Imagine that the consumer choose the bundle A (8 units of good Y and 2 units of good X). If the consumer wants to get one more unit of ice-cream, receiving the same level of utility (bundle B), he/she should be prepared to give up 4 cups of lemonade. This is the idea of marginal rate of substitution. In this case, the marginal rate of substitution is -4. A negative MRS means that to increase consumption of one product, the consumer is prepared to decrease consumption of a second product. But, how much lemonade would the consumer be prepared to give up to get more units of ice-cream? The MRS would be b an c. We observe (in absolute value) a diminishing marginal rate of substitution moving down to the right along indifference curve. This result reflects, in part, the law of diminishing utility stated earlier.
  • Imagine that the consumer choose the bundle A (8 units of good Y and 2 units of good X). If the consumer wants to get one more unit of ice-cream, receiving the same level of utility (bundle B), he/she should be prepared to give up 4 cups of lemonade. This is the idea of marginal rate of substitution. In this case, the marginal rate of substitution is -4. A negative MRS means that to increase consumption of one product, the consumer is prepared to decrease consumption of a second product. But, how much lemonade would the consumer be prepared to give up to get more units of ice-cream? The MRS would be b an c. We observe (in absolute value) a diminishing marginal rate of substitution moving down to the right along indifference curve. This result reflects, in part, the law of diminishing utility stated earlier.
  • An indifference map consists of a set of indifference curves. All points on a particular curve indicate alternative combinations of X and Y that give the consumer equal utility. The farther the curve is from the origin, the higher is the level of utility it represents. For example, U3 is a higher indifference curve than U2; thus all the points on U3 yield a higher level of utility than do the points on U2 or U1. Because the consumer wishes to maximize utility, he wishes to reach the highest attainable indifference curve. So, consumer prefers U3. The indifference curves illustrate consumer’s tastes. But, may be, the consumer cannot afford this curve. Why? Because along with the tastes we need to consider the prices and the monetary income.
  • P X = $2.00 P Y = $1.00 M = $10.00
  • Here we have different possibilities to the consumer if he/she decides to spend all income: $10 (last column). If the consumer buys only lemonade, the most cups he/she can purchase is 10 units. Possibility a. If the consumer buys only ice-cream, the most cons he/she can buy is 5 units. This is the possibility f. But we can consider consumption bundles that combine lemonade and ice-cream. Possibilities b, c, d, and e. For example, possibility d, a bundle consisting of 4 units of lemonade and 3 units of ice-cream.
  • Microeconomics: Utility and Demand

    1. 1. Microeconomics Consumer Behavior: Utility and Demand Dr. Manuel Salas Velasco
    2. 2. How Do Consumers Make Their Decisions? <ul><li>Consumer is unable to change the prices of the goods (P X , P Y ) </li></ul><ul><li>Consumer’s income: M </li></ul><ul><li>Given prices, income, and individual tastes, the consumer’s goal is to maximize his/her utility U = U (X, Y) </li></ul>Dr. Manuel Salas Velasco The theory of consumer behavior Some assumptions:
    3. 3. How Well Do Economists Measure Utility? <ul><li>Cardinal utility: assumes that the consumer has the ability to accurately measure the level of utility he/she derives from consuming a particular combination of goods, and assign an number to it. </li></ul><ul><li>Ordinal utility: consumers are assumed to rank consumption bundles and choose among them. </li></ul>Dr. Manuel Salas Velasco We have two approaches to utility analysis in economic theory:
    4. 4. Cardinal Utility Consumer Behavior: Utility and Demand Dr. Manuel Salas Velasco
    5. 5. Cardinal Utility Dr. Manuel Salas Velasco Let’s assume that there are only two consumption goods, good X (cans of coca-cola) and good Y (cups of popcorn). The utility function can be expressed as: U = U (X, Y) We are going to focus on the relationship between utility and the consumption levels of only one of the goods: We are interested in the full satisfaction (or total utility, U) resulting from the consumption of different units of coke when the value of the good Y is held constant (e.g. Y = 1; one cup of popcorn).
    6. 6. Total and Marginal Utility Schedules Dr. Manuel Salas Velasco Quantity (X), cans of coca-cola Total Utility (U), utils Marginal Utility (MU X ), utils 0 0 1 10 10 2 16 6 3 20 4 4 22 2 5 22 0 6 20 -2
    7. 7. Dr. Manuel Salas Velasco <ul><li>This law states that as additional units of a good are consumed, while holding the consumption of all other goods constant, the resulting increments in utility will diminish </li></ul>Law of Diminishing Marginal Utility X U MU X 0 0 1 10 10 2 16 6 3 20 4 4 22 2 5 22 0 6 20 -2
    8. 8. Utility Function (U) Dr. Manuel Salas Velasco S <ul><li>The slope of the utility curve is the marginal utility </li></ul><ul><li>Utility rises at a decreasing rate with respect to increases in the consumption levels of good, until S </li></ul><ul><li>After X = 5, utility actually begins to decline with any further increases in X </li></ul>
    9. 9. Marginal Utility Function (MU X ) Dr. Manuel Salas Velasco
    10. 10. The Consumer’s Decision Dr. Manuel Salas Velasco Purchase: U = U (X, Y) Income = 5 dollars P X = P Y = 1 dollar (X, Y, X, Y, X) <ul><li>The consumer will allocate expenditure so that the utility gained from the last dollar spent on each product is equal </li></ul>Coke (X) Popcorn (Y) MU X MU Y 0 1 10 6 2 6 4 3 4 2 4 2 0
    11. 11. Consumer Surplus Consumer Behavior: Utility and Demand Dr. Manuel Salas Velasco
    12. 12. Consumer Surplus Dr. Manuel Salas Velasco Packs of cigarettes (per week) Amount the consumer is prepared to pay ($) Consumer surplus on each pack ($) if cigarettes cost $2.20 per pack 1 3.00 0.80 2 2.80 0.60 3 2.60 0.40 4 2.40 0.20 5 2.20 0.00
    13. 13. Consumer Surplus for an Individual Dr. Manuel Salas Velasco Total utility U = 3.00 + 2.80 + 2.60 + 2.40 + 2.20 U = $13.00 U = integral of the function D over the interval [0, 5] <ul><li>If the changes in consumption were infinitesimally small, then the total utility would be the area under the demand curve up to 5 </li></ul>
    14. 14. Consumer Surplus for an Individual Dr. Manuel Salas Velasco Total utility; U = $13.00 Expenditure Consumer surplus Expenditure = 2.20 x 5 = $11.00 Surplus = 13 - 11 = $2.00 Consumer surplus is the sum of the extra valuations placed on each unit above the market price paid for each
    15. 15. Ordinal Utility Consumer Behavior: Utility and Demand Dr. Manuel Salas Velasco
    16. 16. Ordinal Utility: Indifference Theory <ul><li>The second approach to study the theory of demand assumes that consumer can always say which of two consumption bundles he/she prefers without having to say by how much he/she prefers it: consumers are assumed to rank consumption bundles and choose among them. </li></ul><ul><li>Let’s assume that there are only two consumption goods, X and Y; each consumption bundle contains x units of X and y units of Y: ( x , y ). </li></ul>Dr. Manuel Salas Velasco
    17. 17. A Consumer’s Ordinal Preferences <ul><li>Let’s suppose that: X = cons of ice-cream; Y = cups of cold lemonade. </li></ul><ul><li>Consider three consumption bundles (units per week): </li></ul><ul><ul><li>A (2, 8) </li></ul></ul><ul><ul><li>B (3, 4) </li></ul></ul><ul><ul><li>C (5, 2) </li></ul></ul><ul><li>Suppose that each bundle gives the consumer equal satisfaction or utility: the consumer is indifferent between the three bundles of goods. </li></ul>Dr. Manuel Salas Velasco Alternative bundles giving a consumer equal utility Good Y (cups of cold lemonade) Good X (cons of ice-cream) A 8 2 B 4 3 C 2 5
    18. 18. Indifference Curves: A Way to Describe Preferences Dr. Manuel Salas Velasco A B C <ul><li>An indifference curve shows all combinations of goods that yield the same satisfaction to the consumer </li></ul>U = U (X, Y) The axiom of transitivity
    19. 19. Characteristics of Indifference Curves 1. Indifference curves generally possess negative slopes Dr. Manuel Salas Velasco
    20. 20. The Marginal Rate of Substitution Dr. Manuel Salas Velasco A B C B’ MRS = rate at which a consumer is willing to substitute one good for the other within his/her utility function, while receiving the same level of utility. b c A negative MRS means that to increase consumption of one product, the consumer is prepared to decrease consumption of a second product.
    21. 21. The MRS measures the slope of the indifference curve … Dr. Manuel Salas Velasco A B C U = U (X, Y) If consumption bundles are continuous (or infinitely divisible) then the MRS is a ratio of marginal utilities. Slope of the indifference curve (negative)
    22. 22. Characteristics of Indifference Curves 2. Indifference curves cannot cross Dr. Manuel Salas Velasco
    23. 23. Indifference Curves That Cross Dr. Manuel Salas Velasco A B D indifferent indifferent indifferent X Y A D B This contradicts the assumption that A is preferred to D
    24. 24. 3. The farther the curve is form the origin, the higher is the level of utility it represents. Characteristics of Indifference Curves Dr. Manuel Salas Velasco
    25. 25. An Indifference Curve Map Dr. Manuel Salas Velasco X Y (Units per time period) (Units per time period) <ul><li>Consumer wishes to maximize utility, he wishes to reach the highest attainable indifference curve </li></ul>> >
    26. 26. What Choices Is an Individual Consumer Able to Make? Budget Constraint Dr. Manuel Salas Velasco
    27. 27. The Budget Constraint Dr. Manuel Salas Velasco Good Y (cups of cold lemonade) Good X (cons of ice-cream) Total expend. Price Quantity Expend. Price Quantity Expend. a 1 10 10 2 0 0 10 b 1 8 8 2 1 2 10 c 1 6 6 2 2 4 10 d 1 4 4 2 3 6 10 e 1 2 2 2 4 8 10 f 1 0 0 2 5 10 10
    28. 28. The Budget Line Dr. Manuel Salas Velasco a b c d e f <ul><li>The budget line shows all combinations of products that are available to the consumer given his money income and the prices of the goods that he/she purchases </li></ul>
    29. 29. The Budget Line Dr. Manuel Salas Velasco a b c d e f M = 10; P X = 2; P Y = 1 Point f horizontal intercept Point a vertical intercept Slope: The equation for the budget line: Relative price ratio
    30. 30. The Consumer’s Utility Maximizing Choice Dr. Manuel Salas Velasco E <ul><li>The consumer’s utility is maximized at the point (E) where an indifference curve is tangent to the budget line </li></ul><ul><li>At that point, the consumer’s marginal rate of substitution for the two goods is equal to the relative prices of the two goods </li></ul>The condition for utility maximization
    31. 31. The Consumer’s Constrained Maximization Problem Dr. Manuel Salas Velasco Maximize U = U (X, Y) the objective function P X = price of good X P Y = price of good Y M = consumer’s income Subject to: the constraint P X X + P Y Y = M <ul><li>A basic assumption of the theory of consumer behavior is that rational consumers seek to maximize their total utility, subject to predetermined prices of the goods and their money income </li></ul><ul><li>Formally, we can express this concept of consumer choice as a constrained optimization problem </li></ul>
    32. 32. The Consumer’s Constrained Maximization Problem Dr. Manuel Salas Velasco Maximize: U = U (X, Y) the objective function Subject to: the constraint P X X + P Y Y = M Step 1 . Set up the Lagrangian function: To do so, we first set the constraint function equal to zero: M – P X X – P Y Y = 0 We then multiply this form by lambda to form the Lagrangian function: <ul><li>In order to solve such a problem, we will use the Lagrangian Multiplier Method </li></ul>
    33. 33. The Consumer’s Constrained Maximization Problem Dr. Manuel Salas Velasco Step 2 . Determine the first-order conditions:
    34. 34. The Consumer’s Constrained Maximization Problem Dr. Manuel Salas Velasco Step 3 . Solving for lambda from the two first-order conditions: Therefore: The condition for utility maximization
    35. 35. The Consumer’s Constrained Maximization Problem. Example Dr. Manuel Salas Velasco Maximize: U = X 0.5 Y 0.5 (Cobb-Douglas Indifference curve) Subject to: 4 X + Y = 800 (where: P X = 4; P Y = 1: M = 800) Step 1 . Set up the Lagrangian function:
    36. 36. The Consumer’s Constrained Maximization Problem. Example Dr. Manuel Salas Velasco Step 2 . Determine the first-order conditions: Y = 800 – 4X
    37. 37. The Consumer’s Constrained Maximization Problem. Example Dr. Manuel Salas Velasco Y = 800 – 4X Y = 800 – 4X; 4X = 800 – 4X; 8X = 800; X* = 100 units; Y* = 400 units U = X 0.5 Y 0.5 = (100) 0.5 (400) 0.5 = 200 Lambda = 0.5 (100) 0.5 (400) -0.5 = 0.25 Lambda (lagrangian multiplier) measures the change in utility due to a one dollar change in the consumer’s income ( ∂U/∂M)

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