Week 5: Consumer Choice
Julio receives utility from consuming food (F) and clothing (C) as given by the utility function U(F,C)=FC. In addition, the price of food is $2 per unit, the price of clothing is $9 per unit, and Julio's weekly income is $50. What is Julio's marginal rate of substitution of food for clothing when utility is maximized? Explain. (Assuming food is on the horizontal axis.) Julio's marginal rate of substitution equals
0.22, which is the price of food divided by the price of clothing.
Bob views apples and oranges as perfect substitutes in his consumption, and MRS = 1 for all combinations of the two goods in his indifference map. Suppose the price of apples is $2 per pound, the price of oranges is $3 per pound, and Bob's budget is $30 per week. What is Bob's utility-maximizing choice between these two goods?
15 pounds of apples and no oranges
Utility Functions
A *utility function* is a formula that assigns each bundle a number to represent the amount of satisfaction provided by that bundle. For example, suppose Phil's utility function for food (F) and shelter (S) is *U(F,S)=FS.* If Phil consumes 4 lb/wk of food and 1 sq yds/wk of shelter, his satisfaction is U(4, 1)=4×1=4 "utils" per week. Utility functions make the ranking of bundles easy. - For (2, 2), Phil's utility is also 4 utils per week. (4, 1) ~ (2, 2). (Food is on the horizontal axis.) - For (3, 1), his utility is 3 utils per week. (4, 1) ~ (2, 2) ≻ (3, 1).
Constructing MRS from Marginal Utilities
Consider a small movement along an indifference curve. Utility along an indifference curve remains constant. Hence, MU_FΔF = MU_s ΔS MU_FΔF is the gained utility from having more food. MU_s ΔS is the lost utility from having less shelter. [slide 14, 3.4]
Why Marginal Utility is Useful
In the indifference curve framework, the best attainable bundle is the bundle on the budget constraint that lies on the highest indifference curve. The tangency condition: MRS= PF /PS. (Food on the horizontal axis.) - the two absolute values of the slopes have to be the same Analogously, the best attainable bundle in the utility-function framework is the bundle on the budget constraint that provides the highest level of utility. Question: what is the analogous condition tangency condition - This is where the concept of marginal utility becomes useful.
Alfred derives utility from consuming iced tea and lemonade. For the bundle he currently consumes, the marginal utility he receives from iced tea is 16 utils, and the marginal utility he receives from lemonade is 8 utils. Instead of consuming this bundle, Alfred should A. buy more iced tea and less lemonade. B. buy less iced tea and lemonade. C. buy more lemonade and less iced tea. D. buy more iced tea and lemonade. E. None of the above is necessarily correct.
E
Ordinal Utility
Just as preference ordering is ordinal, utility is also ordinal. The fact that 𝑈_3 has a level of utility of 100 and 𝑈_2 has a level of 50 does not mean that the bundles on 𝑈_3 generates twice as much satisfaction as those on 𝑈_2. We do not know by how much one is preferred to the other. - cannot say from this information that it is better or worse than the other bundle. Implication: 𝑈 = 4FS represents exactly the same rankings as 𝑈 = 𝐹S. (In fact, given any increasing function 𝑉, 𝑈=𝐹S and 𝑉=𝑉(𝑈)=𝑉(𝐹S) give the preference ordering.) Simple examples of monotone transformation: divide, add to, or subtract from
If Px = Py, then when the consumer maximizes utility,
MUx must be equal to MUy
Satisfaction from consumption is maximized when
Marginal benefit equal marginal cost
Perfect Substitutes
Mattingly is a caffeinated-cola drinker who spends his entire soft drink budget on Coca-Cola and Jolt cola and cares only about total caffeine content. If Jolt has twice the caffeine of Coke, and if Jolt costs $1/pint and Coke costs $0.75/pint, how will Mattingly spend his soft drink budget of $15/wk? MRS < |slope of budget| always, hence a corner solution on the horizontal axis. Intuition: he cares only about total caffeine content, and Jolt provides more caffeine per dollar than Coke does. [Slide 13, 3.3]
Based on his preferences, Bill is willing to trade 6 movie tickets for 1 ticket to a basketball game. If movie tickets cost $12 each and a ticket to the basketball game costs $78, should Bill trade movie tickets for basketball tickets? Why or why not?
Not trade movie tickets for basketball tickets because his marginal utility per dollar spent on movie tickets is greater than his marginal utility per dollar spent on basketball tickets.
Paul consumes only books and DVDs. At his current consumption bundle, his marginal utility from DVDs is 13 and from books is 3. Each DVD costs $10, and each book costs $1. Is he maximizing his utility? Explain. Let MUB be the marginal utility of books, MUD be the marginal utility from DVDs, PB be the price of books, PD be the price of DVDs, and MRS be the marginal rate of substitution.
Paul is not maximizing his utility because MUb / Pb > MUd / Pd *If he is not, how can he increase his utility while keeping his total expenditure constant?* Paul could increase utility while keeping total expenditures constant by consuming more *books* and fewer *DVDs*
Corner Solutions
Sometimes there may be no point of tangency — the MRS may be everywhere greater, or less, than the slope of the budget constraint. In this case we get a corner solution, *Corner solution:* in a choice between two goods, a case in which the consumer does not consume one of the goods. Graphically, the task is still to find the highest affordable indifference curve. I1 and I2 are not the best decision [Slide 12, 3.3] M = $100/wk, Pf = $10/lb, Ps = $5/sq yd. Flatter than budget line: prefers food relatively more At A the MRS is less than the absolute value of the slope of the budget constraint. Diminishing MRS implies that MRS and slope never reach equality. For example MRS = 0.25 at A (benefit of an additional sq yd of shelter). Slope = 0.5 at A. (cost of an additional sq yd of shelter.) Want less shelter, but the amount of shelter already reached zero. *Note that we can have tangency and a corner solution at the same time. Example: MRS = 0.5 at A.
If Marginal Utility per Dollar is Unequal
Suppose that the marginal utility of the last dollar John spends on food is greater than the marginal utility of the last dollar he spends on shelter. For example, suppose the prices of food and shelter are $1/lb and $2/sq yd, respectively, and that the corresponding marginal utilities are 6 and 4. Show that John cannot possibly be maximizing his utility. If John bought 1 sq yd/wk less shelter, he would save $2/wk and would lose 4 utils. But this would enable him to buy 2 lb/wk more food, which would add 12 utils, for a net gain of 8 utils. In general, spending more on the good with higher marginal utility per dollar can increase the consumer's utility.
Marginal utility measures
The additional satisfaction from consuming one more unit of a good.
Graph a Utility Function (in a Two-Good World)
The graph of a two-argument utility function is a three-dimensional utility surface. Indifference curve are like contour lines on the 3-D utility surface projected onto the X-Y plane. Setting the utility function to a constant is like slicing the utility surface by a horizontal plane. Explain two-argument, how the surface is drawn by using a few bundles as examples such as N (10, 2.5, 25), L (5, 5, 25), K (10, 1, 10), J (2, 5, 10). Note that this surface does not correspond to U=XY. *Contour Line:* a line on a map joining points of equal height above or below sea level. An excellent 3D illustration: https://www.youtube.com/watch?v=ghvJGAUTgHc [3.4 SLIDE 7]
Consumers in Georgia pay twice as much for avacados as they do for peaches. However, avacados and peaches are the same price in California. If consumers in both states maximize utility, will the marginal rate of substitution of peaches for avacados be the same for consumers in both states? If not, which will be higher?
The marginal rate of substitution of peaches for avocados will be *higher in California*
Marginal Utility (Continuous Version)
The rate at which the total utility changes with consumption of the good. It is the effect of an infinitesimal increase in the consumption of the good on the total utility (rather than that of a one-unit increase). The continuous version definition will be useful when we have more general utility functions such as U=√FS.
Where Food Stamps and Cash Grants Yield The Same Outcome
[3.3, slide 17] Since the best affordable bundle K locates on AD where the two new budget constraints overlaps, the outcome is the same. Intuition: those who are in need of food find the food stamps as good as cash. [3.3, slide 18]: Since the best affordable bundle L under the cash grant is on ED which is outside the budget constraint FDA associated with the food stamp program, the outcomes are different. Intuition: those who are not in urgent need of food find cash more attractive
Using Utility Function to Construct an Indifference Map
[Slide 6, 3.4] This figure shows four indifference curves (with 1, 2, 3 and 4 utils, respectively) associated with Phil's utility function U(F,S) = FS. For example, for the indifference curve U = 1, the combinations of bundles for which FS = 1, the corresponding function is *S = 1/F* More generally, for any U0, we solve FS=U0 to get *S = U0/F* Utility functions and indifferent maps are equivalent ways to represent consumers' preferences in that for each indifference map, we can find a corresponding utility function (not unique) to represent the map. Conversely, given a utility function, we can also construct an indifference map.
What is the difference between ordinal utility and cardinal utility? Ordinal utility refers to
a ranking of market baskets in order of most to least preferred, while cardinal utility indicates how much one market basket is preferred to another.
Julio receives utility from consuming food (F) and clothing (C) as given by the utility function U(F,C)=FC. In addition, the price of food is $2 per unit, the price of clothing is $9 per unit, and Julio's weekly income is $50. Suppose instead that Julio is consuming a bundle with more food and less clothing than his utility maximizing bundle. This bundle is also on the budget line. Would this marginal rate of substitution of food for clothing be greater than or less than your answer above? Explain. If Julio is instead consuming a bundle with more food and less clothing than his utility maximizing bundle, then
his marginal rate of substitution will be *less than* 0.22 because he will be consuming a bundle that is *to the right of* his satisfaction maximizing bundle.
Marginal Utility Illustrated
https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-1/v/derivative-as-a-concept
When the optimal point on an indifference curve and budget line diagram is a corner solution,
the marginal rate of substitution usually does not equal the ratio of prices for the two goods
Pencils sell for 10 cents and pens sell for 50 cents. Suppose Jack, whose preferences satisfy all of the basic assumptions, buys 5 pens and one pencil each semester. With this consumption bundle, his MRS of pencils for pens is 3 (assuming pencil is on the horizontal axis). Which of the following is true?
Jack could increase his utility by buying more pencils and fewer pens.
When Joe maximizes utility, he finds that his MRS of X for Y is greater than Px/Py. It is most likely that
Joe is not consuming good Y.
Sue views hot dogs and hot dog buns as perfect complements in her consumption, and the corners of her indifference curves follow the 45-degree line. Suppose the price of hot dogs is $5 per package (8 hot dogs), the price of buns is $3 per package (8 hot dog buns), and Sue's budget is $48 per month. What is her optimal choice under this scenario?
6 packages of hot dogs and 6 packages of buns
Marginal Utility (Discrete Version)
The change in the total utility from consuming an additional unit of the good, given fixed the amount of the other good. In a world of more than two goods, we would have to fix the amount of every other good.
2 lemonades for each popcorn.
The price of lemonade is $0.50; the price of popcorn is $1.00. If Fred has maximized his utility by purchasing lemonade and popcorn, his marginal rate of substitution will be
When Tangency Exists - Interior Solutions
For indifference maps for which a tangency point exists, as the previous figure, the best affordable bundle will always lie at the point of tangency. For example, the marginal rate of substitution at F is exactly the same as the absolute value of the slope of the budget constraint *MRS = Ps / Pf* Intuition: measuring everything by food - RHS = the opportunity cost of shelter in terms of food - LHS = the benefit of shelter in terms of food - If LHS ≠ RHS, then it would always be possible for him to purchase a better bundle.
Cardinal Utility
Opposite than Ordinal bundles because absolute value matters In the nineteenth century, economists commonly assumed that people could make statements like "bundle A is 6.43 times as good as bundle B". They assume the satisfaction provided by any bundle can be assigned a numerical, or cardinal, value by a utility function. However, we have no way of telling whether a person gets twice as much satisfaction from one bundle as from another. Nor do we know whether one person gets twice as much satisfaction as another from consuming the same bundle.
The Best Affordable Bundle Explained
The best affordable bundle must lie on the budget line. A result of more-is-better - The more-is-better assumption implies that the best affordable bundle must lie on the budget constraint, not inside it. (Any bundle inside the budget constraint would be less pre-ferred than one just slightly to the northeast, which would also be affordable.) Implies that the best affordable bundle cannot lie on an indifference curve that lies partly inside the budget constraint. Hence: The best affordable bundle lie on an indifference curve that intersects the budget constraint only once. Intuition: keep moving to higher and higher indifference curves until he reaches the highest one that is still affordable - The choice of bundle F makes perfect sense on intuitive grounds. The consumer's goal, after all, is to reach the highest indifference curve he can, given his budget constraint. His strategy is to keep moving to higher and higher indifference curves until he reaches the highest one that is still affordable.
At the optimal point on an indifference curve and budget line diagram (assuming an interior solution)
The consumer spends his or her entire budget on the two goods. The optimal indifference curve is tangent to the budget line. The marginal rate of substitution between the two goods equals the ratio of their prices.
Best Affordable Bundle
The most preferred bundle of those that are affordable. The *indifference map* tells the how the bundles are ranked in terms of preference, the *budget constraint* tells which bundles are affordable, and the consumer's task is to put the two together to chooses the best affordable bundle. - How the consumer uses the two to find the best affordable bundle We now have the tools we need to determine how the consumer should allocate his income between two goods. The indifference map tells us how the various bundles are ranked in order of preference. The budget constraint, in turn, tells us which bundles are affordable. The consumer's task is to put the two together and to choose the most preferred or best affordable bundle. We need not suppose that consumers think explicitly about budget constraints and indifference maps when deciding what to buy. It is sufficient to assume that people make decisions as if they were thinking in these terms, just as expert pool players choose between shots as if they knew all the relevant laws of Newtonian physics. M = $100/wk, Pf=$10/lb, Ps = $5/sq yd. Consider the 5 marked bundles. - G is the most preferred but not affordable. - A, D are affordable but no the best affordable. - F is the best affordable bundle. [SLIDE 6, 3.3]
The Tangency Condition and the Equal Marginal-Utility-per-Dollar Condition
The tangency condition MRS=PF / PS in terms of marginal utilities is (food on the horizontal axis) (MU_F) / (MU_S) = P_F / P_S If we cross-multiply terms in the equation above, we get an equivalent condition that has a very straightforward intuitive interpretation: (MU_F) / P_F =(MU_S) / P_S In words, the *marginal utility per dollar* must be the same for all goods at the optimal bundle.
Explain why the assumption of cardinal utility is not needed in order to rank consumer choices. Cardinal utility is not needed in order to rank consumer choices because economists
can instead use ordinal utility to show how consumers rank different baskets.