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Quiz 2 :

Preferences and Utility

Quiz 2 :

Preferences and Utility

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Graph a typical indifference curve for the following utility functions, and determine whether they have convex indifference curves (i.e., whether the MRS declines as x increases). img
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img img img img img img img img img img img img img img img img

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In footnote 7 we showed that for a utility function for two goods to have a strictly diminishing MRS (i.e., to be strictly quasi-concave), the following condition must hold: img Use this condition to check the convexity of the indifference curves for each of the utility functions in Problem 3.1. Describe the precise relationship between diminishing marginal utility and quasi-concavity for each case. Reference: Problem 3.1 Graph a typical indifference curve for the following utility functions, and determine whether they have convex indifference curves (i.e., whether the MRS declines as x increases). img
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(a)
For a utility function to have strictly diminishing marginal rate of substitution, it should be strictly quasi-concave. A function is strictly quasi-concave, if the following condition holds:
img ...... (1)
The partial differentials for the utility function are given below:
img img img img img img img Putting in the values in the following equation
img ...... (2)
We get,
img As the inequality (1),
img does not hold, the function is not strictly quasi-concave and hence does not have strictly diminishing rate of substitution.
Both the marginal utilities U x and U y are positive and are constant as reproduced below.
img img As both the marginal utilities are constant, value of equation (2),
img will be zero.
Hence, quasi-concavity condition shown in inequality (1) will not hold. This is true in general for all cases in which marginal utilities of both the goods are constant and are not influenced by the amount of consumption of the other good.
(b)
Let us set utility equal to 10. Then the equation for the curve is
img To simplify the function, while leaving the ordering unchanged, we can square it so the indifference curve is represent by
img Using the above equation, at utility equal to 10, we get the following value of the partial differentials:
img img img img img img img Putting in the above values in equation (2),
img We get,
img Except for when x and/or y are equal to 0, resulting in utility equal to 0, x and y will always be positive quantities.
Thus,
img Hence, strictly quasi-concavity inequality (1),
img holds and the utility function has strictly diminishing marginal rate of substitution.
The marginal utilities for both the goods are reproduced below.
img img Since these are partial differentials (i.e. holding the quantity of the good fixed), the marginal utility of each good is constant in terms of the other good and does not exhibit diminishing marginal utility.
However, as was shown above, the function is strictly quasi-concave. This is because diminishing marginal utility for each good is not a necessary condition for diminishing MRS and consequently strict quasi-concavity.
This can be seen below.
img From the above equation, it can be seen that, as x increases, the marginal rate of substitution will decrease.
Hence, it is necessary that the ratio of the marginal utilities decrease for strict quasi-concavity, irrespective of whether each good has diminish marginal utility or not.
(c)
For this utility function, we get the following values:
img img img img img img img Substituting the above values in equation (2),
img We get,
img As x is always positive,
img Hence, inequality (1)
img holds and the utility function has strictly diminishing MRS.
The marginal utilities of both goods are reproduced below.
img img It can be seen from the above equations that while good x has diminish marginal utility, marginal utility for good y is constant at 1.As has been shown above, the function is strictly quasi-concave. This is because; diminishing marginal utility for each good is not a necessary condition for diminishing MRS and consequently, strict quasi-concavity and only the marginal rate of substitution should decrease.
This can be seen below.
img From the above equation, it can be seen that, as x increases, the marginal rate of substitution will decrease.
Hence, it is necessary that the ratio of the marginal utilities decrease for strict quasi-concavity, irrespective of whether each good has diminish marginal utility or not.
(d)
Let us arbitrarily set utility equal to 4. Then the equation for the curve is
img To simplify the function, while leaving the ordering unchanged, we can square both sides so the indifference curve is represent by
img The values for the partial differential of the above function are as follows:
img img img img img img img Putting in the above values in equation (2),
img We get,
img For the function to be strictly quasi-concave,
img The above inequality holds only when:
img As x and y will always be positive quantities, the above inequality will hold true only when
img However, looking at the original utility function reproduced below:
img We see that for utility to be defined,
img As the quantities are consumed in positive amounts, this also means that
img This is exact opposite to the condition for quasi-concavity i.e.
x y
Hence, for all the defined values of utility, the utility function is not strictly quasi-concave.
The marginal utilities of both the goods are reproduced below.
img img As can be seen from the above equations, while good x exhibits increasing marginal utility, good y exhibits decreasing marginal utility. Above, we found that the function is not strictly quasi-concave in the defined values of utility in which y x.
This is because when y x, the increasing marginal utility of good x overpowers the diminishing marginal utility of good y. The net result is that the curve does not exhibit diminishing MRS (i.e. it is not strictly quasi-concave).
(e)
The values for the differential of the utility function are as follows:
img img img img img img img Substituting the above values in equation (2),
img We get,
img img The above equation would always be negative.
Hence, inequality (1)
img holds and the function is strictly quasi-concave.
The marginal utilities of the two goods are reproduced below.
img img They both exhibit diminishing marginal utility. As shown above, the function is strictly quasi-concave and thus had diminishing marginal rate of substitution. However, this result does not hold in general.
Diminishing marginal utility of both goods is not a sufficient condition for diminishing marginal rate of substitution. This is because it is their ratio that matters, and not the absolute values.

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Consider the following utility functions: img Show that each of these has a diminishing MRS but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude?
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a.
Marginal utility is addition in total utility by consuming one more unit of a commodity.
The utility function is given below:
img The marginal utilities of good x and y for the utility function are given below:
img img As marginal utility of x is y and marginal utility of y is x and they both need to be kept constant at the derivative is calculated of other good. Therefore, it shows the constancy of marginal utilities of x and y.
The marginal rate of substitution (MRS) is the units of one good an individual likes to give up for one additional unit of other good.
The marginal rate of substitution is the ratio of marginal utility of one good to marginal utility of other good.
The formula for the MRS is given below:
img Substituting in the values for the marginal utilities the value of MRS is calculated below:
img img As x and y are the commodities and their consumption can't be negative and there is a negative sign with second derivative therefore, marginal rate of substitution must be decreasing with increase in x
b.
The utility function is given below:
img The marginal utilities of good y and x for the utility function are given below:
img img As both x and y is the consumption and consumption can't be negative. Therefore, both of these marginal utilities are increasing with y and x. Hence, this function shows increasing marginal utilities.
Substituting in the values of the marginal utilities, the value of MRS can be calculated as shown below:
img img As x and y are quantities of goods and this can't be negative, MRS decreases with increase in x. Hence, this utility function shows decreasing MRS.
c.
The utility function is given below:
img The marginal utilities of good x and y for the utility function are given below:
img As the value of x increases, there will be a fall in marginal utility of x. Hence, the goods x shows decreasing marginal utility.
img As the value of y increases, there will be a fall in marginal utility of y. Hence, the goods y shows decreasing marginal utility.
Substituting the values for the marginal utilities, the value of MRS can be calculated as shown below:
img img x and y both are quantities of good and they can't be negative. Therefore, increase in x result in fall in MRS. Hence, this utility function has diminishing MRS.
All the three utility functions of commodity x and y show decreasing MRS.

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As we saw in Figure 3.5, one way to show convexity of indifference curves is to show that, for any two points (x 1 , y 1 ) and ( x 2 , y 2 ) on an indifference curve that promises U = k, the utility associated with the point img is at least as great as k. Use this approach to discuss the convexity of the indifference curves for the following three functions. Be sure to graph your results. a. U(x, y) = min(x, y). b. U(x, y) = max(x, y). c. U(x, y) = x + y. Reference: Figure 3.5 img
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The Phillie Phanatic (PP) always eats his ballpark franks in a special way; he uses a foot-long hot dog together with precisely half a bun, 1 ounce of mustard, and 2 ounces of pickle relish. His utility is a function only of these four items, and any extra amount of a single item without the other constituents is worthless. a. What form does PP's utility function for these four goods have? b. How might we simplify matters by considering PP's utility to be a function of only one good? What is that good? c. Suppose foot-long hot dogs cost $1.00 each, buns cost $0.50 each, mustard costs $0.05 per ounce, and pickle relish costs $0.15 per ounce. How much does the good defined in part (b) cost? d. If the price of foot-long hot dogs increases by 50 percent (to $1.50 each), what is the percentage increase in the price of the good? e. How would a 50 percent increase in the price of a bun affect the price of the good? Why is your answer different from part (d)? f. If the government wanted to raise $1.00 by taxing the goods that PP buys, how should it spread this tax over the four goods so as to minimize the utility cost to PP?
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Many advertising slogans seem to be asserting something about people's preferences. How would you capture the following slogans with a mathematical utility function? a. Promise margarine is just as good as butter. b. Things go better with Coke. c. You can't eat just one Pringle's potato chip. d. Krispy Kreme glazed doughnuts are just better than Dunkin' Donuts. e. Miller Brewing advises us to drink (beer) ''responsibly.'' [What would ''irresponsible'' drinking be?]
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a. A consumer is willing to trade 3 units of x for 1 unit of y when she has 6 units of x and 5 units of y. She is also willing to trade in 6 units of x for 2 units of y when she has 12 units of x and 3 units of y. She is indifferent between bundle (6, 5) and bundle (12, 3). What is the utility function for goods x and y? Hint: What is the shape of the indifference curve? b. A consumer is willing to trade 4 units of x for 1 unit of y when she is consuming bundle (8, 1). She is also willing to trade in 1 unit of x for 2 units of y when she is consuming bundle (4, 4). She is indifferent between these two bundles. Assuming that the utility function is Cobb-Douglas of the form img are positive constants, what is the utility function for this consumer? c. Was there a redundancy of information in part (b)? If yes, how much is the minimum amount of information required in that question to derive the utility function?
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Find utility functions given each of the following indifference curves [defined by img img
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Initial endowments Suppose that a person has initial amounts of the two goods that provide utility to him or her. These initial amounts are given by img a. Graph these initial amounts on this person's indifference curve map. b. If this person can trade x for y (or vice versa) with other people, what kinds of trades would he or she voluntarily make? What kinds would not be made? How do these trades relate to this person's MRS at the point img c. Suppose this person is relatively happy with the initial amounts in his or her possession and will only consider trades that increase utility by at least amount k. How would you illustrate this on the indifference curve map?
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Cobb-Douglas utility Example 3.3 shows that the MRS for the Cobb-Douglas function img is given by img a. Does this result depend on whether img Does this sum have any relevance to the theory of choice? b. For commodity bundles for which y ¼ x, how does the MRS depend on the values of img Develop an intuitive explanation of why, if img . Illustrate your argument with a graph. c. Suppose an individual obtains utility only from amounts of x and y that exceed minimal subsistence levels given by x 0 , y 0. In this case, img Is this function homothetic? (For a further discussion, see the Extensions to Chapter 4.)
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Independent marginal utilities Two goods have independent marginal utilities if img Show that if we assume diminishing marginal utility for each good, then any utility function with independent marginal utilities will have a diminishing MRS. Provide an example to show that the converse of this statement is not true.
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CES utility a. Show that the CES function img is homothetic. How does the MRS depend on the ratio y/x? b. Show that your results from part (a) agree with our discussion of the cases ? = 1 (perfect substitutes) and ? = 0 (Cobb-Douglas). c. Show that the MRS is strictly diminishing for all values of ? 1. d. Show that if x = y, the MRS for this function depends only on the relative sizes of ? and ?. e. Calculate the MRS for this function when y/x = 0.9 and y/x = 1.1 for the two cases ? = 0.5 and ? = -1. What do you conclude about the extent to which the MRS changes in the vicinity of x = y? How would you interpret this geometrically?
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The quasi-linear function Consider the function U(x, y) = x + ln y. This is a function that is used relatively frequently in economic modeling as it has some useful properties. a. Find the MRS of the function. Now, interpret the result. b. Confirm that the function is quasi-concave. c. Find the equation for an indifference curve for this function. d. Compare the marginal utility of x and y. How do you interpret these functions? How might consumers choose between x and y as they try to increase their utility by, for example, consuming more when their income increases? (We will look at this ''income effect'' in detail in the Chapter 5 problems.) e. Considering how the utility changes as the quantities of the two goods increase, describe some situations where this function might be useful.
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Preference relations The formal study of preferences uses a general vector notation. A bundle of n commodities is denoted by the vector img is defined over all potential bundles. The statement img bundle x 1 is preferred to bundle x 2. Indifference between two such bundles is denoted by img The preference relation is ''complete'' if for any two bundles the individual is able to state either img The relation is ''transitive'' if img Finally, a preference relation is ''continuous'' if for any bundle y such that img , any bundle suitably close to y will also be preferred to x. Using these definitions, discuss whether each of the following preference relations is complete, transitive, and continuous. a. Summation preferences: This preference relation assumes one can indeed add apples and oranges. Specifically img b. Lexicographic preferences: In this case the preference relation is organized as a dictionary: If img (regardless of the amounts of the other n - 1 goods). If img (regardless of the amounts of the other n - 2 goods). The lexicographic preference relation then continues in this way throughout the entire list of goods. c. Preferences with satiation: For this preference relation there is assumed to be a consumption bundle (x * ) that provides complete ''bliss.'' The ranking of all other bundles is determined by how close they are to x *. That is, img if and only if img
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In a 1992 article David G. Luenberger introduced what he termed the benefit function as a way of incorporating some degree of cardinal measurement into utility theory.11 The author asks us to specify a certain elementary consumption bundle and then measure how many replications of this bundle would need to be provided to an individual to raise his or her utility level to a particular target. Suppose there are only two goods and that the utility target is given by img . Suppose also that the elementary consumption bundle is given by (x 0 , y 0 ). Then the value of the benefit function, img , is that value of a for which img a. Suppose utility is given by img Calculate the benefit function for x 0 = y 0 = 1. b. Using the utility function from part (a), calculate the benefit function for img Explain why your results differ from those in part (a). c. The benefit function can also be defined when an individual has initial endowments of the two goods. If these initial endowments are given by img is given by that value of ? which satisfies the equation img In this situation the ''benefit'' can be either positive img or negative (when img Develop a graphical description of these two possibilities, and explain how the nature of the elementary bundle may affect the benefit calculation. d. Consider two possible initial endowments, img Explain both graphically and intuitively why img (Note: This shows that the benefit function is concave in the initial endowments.)
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