Economics 352: Intermediate Microeconomics

EC 352: Intermediate Microeconomics, Lecture 3

Economics 352: Intermediate Microeconomics

Notes and Assignment Chapter 3: Preferences and Utility This chapter is an introduction to the underlying theory of how economists think about individuals’ consumption choices. At the root of this are preferences, or how a person feels about different sets of goods and how they choose between them. Based on these preferences, we move to the idea of a utility function, which is a mathematical function that simplifies the choice between different bundles of goods by assigning a number to a bundle of goods and then simply saying that when a person faces a choice between bundles, they will choose the bundle that gives them the highest utility level.

Axioms of Rational Choice We’ll start out with a few basic ideas about rules that should be true for peoples’ decisions between sets of goods. These are fairly basic ideas. 1. Completeness: Any two sets of goods can be compared and chosen between. If these sets of goods have the names A and B, then it must be the case that either: A is preferred to B

AfB

B is preferred to A

ApB

A and B are equally attractive

A~B

So, if you have the choice between, say five gallons of water and two loaves of bread (set A) and three gallons of water and three loaves of bread (set B) it must be the case that you prefer A to B, that you prefer B to A, or that you find them equally attractive. 2. Transitivity: When considering three sets of goods (cleverly named A, B and C) it must be the case that if A is preferred to B ( A f B ) and B is preferred to C ( B f C ), then A will be preferred to C ( A f C ). So, if A f B and B f C , then A f C . It would be weird if this wasn’t true.

3. Continuity: If A is preferred to B ( A f B ) and you have a set of goods that is very similar to A (with a tiny bit more or less of one of the goods, for example) then that bundle must also be preferred to B.

Utility Functions A utility function assigns a number to a set of goods, based on a person’s preferences. So, if you have a set or bundle of goods A, the utility level associated with this bundle is U(A). A bundle of goods that is preferred over another bundle of goods has a higher utility level associated with it. If A f B then U(A) > U(B) The actual numbers from the utility function don’t matter. The only thing that matters is the relative utility levels of the two bundles being compared. The one with the higher utility level is the better bundle, regardless of how big or small the difference between their utility levels might be. To put this differently, imagine that a utility function is based on the quantity of one good that a person has. This good is cleverly named x1. You should think of x1 as your favorite good. One utility function might be that utility is equal to the quantity of that good, or U(x1) = x1

(x1 ≥ 0)

So, if you have five units of the good, x1=5 and U(5)=5. This utility function would be identical to the utility function U(x1) = x12. These two utility functions are identical because, in comparing two bundles, the bundle for which x1 is greater will also be the bundle for which x12 is greater. So, if you have the utility function U(x1) = x1 and are choosing between a bundle for which x1=5 and a bundle for which x1=6, the bundle for which x1=6 would be preferred because it has a higher utility level associated with it (6 versus 5). The ranking is the same with the utility function U(x1) = x12. The bundle for which x1=5 would have the utility level U(5)=25 and the bundle for which for which x1=6 would have the utility level U(6)=36. This utility function would also be identical to such utility functions as: U(x1) = x13 U(x1) = 1000x1

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U(x1) = 500 + x1 because these functions preserve the rankings of U(x1) = x1 and U(x1) = x12. Most of the utility functions we consider will involve more than one good, and these might be written as: U(x1, x2, …., xn)

{The goods are x1, x2 up through xn.)

U(x, y)

{The goods are x and y.)

Alternatively, utility might be expressed as a function of a person’s wealth, with the implication that that wealth is used to buy a set of goods that yields as much utility as possible: U(W)

{W is wealth}

Alternatively, utility might be expressed as a function of the amount of stuff a person consumes, c, and the number of hours of leisure that they are able to enjoy, h, U(c, h)

{c is consumption, h is leisure}

Alternatively, utility might be expressed as a function of the amount of stuff a person consumes this year, c1, and the amount that they consume next year, c2, U(c1, c2)

Utility and Goods More is preferred to less. When choosing between two bundles, if one has a greater quantity of goods in it, that bundle will be preferred to the one with the lesser quantity. For example, if bundle A has one hamburger and six French fries while bundle B has one hamburger and seven French fries, bundle B will be preferred. Put more technically, if bundle A has at least as much of every good as bundle B and more of one good, bundle A will be preferred.

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Alternatively, if bundle A has exactly as much of almost every good as bundle B, but less of one good, bundle B will be preferred. The tricky part is if bundle A has more of some goods and bundle B has more of other goods. In this case, you can’t tell which will be preferred without knowing the exact utility function. In terms of a graph, the idea that more is preferred to less can be shown in a diagram. Starting from the point (x*,y*), the points above and to the right (to the northeast) of (x*,y*) are preferred to (x*,y*). The points below and to the left (to the southwest) of (x*,y*) are worse than (x*,y*). The points to the northwest and southeast may be better than, worse than or equally attractive to (x*,y*), you can’t tell without knowing what the utility function is.

A graph showing the set of points that is better than a specific point and the set of points that is worse than a specific point.

Indifference Curves To the framework of the graph shown above, we will introduce the concept of the indifference curve. An indifference curve is a set of bundles of goods between which a person is indifferent. To put this differently, it represents the combinations of goods that a person finds equally attractive. To put this differently, it represents the combinations or bundles that have the same utility level. Imagine, for example, that a person has the utility function

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U(x, y) = xy This person would be indifferent between the bundles (4,6) and (2,12) because they will both yield a utility level of 24: U(4,6) = 4 x 6 = 24 U(2,12) = 2 x 12 = 24 In terms of an indifference curve, this looks like:

A graph showing two points on an indifference curve. In terms of a choice, if the person was forced to choose between (4,6) and (2,12) they would have a difficult time, because they find the two bundles equally attractive. To extend this idea, you might imagine a different set of bundles that give this person a utility level of U=36. These points constitute a higher indifference curve and a preferred set of bundles:

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A graph showing three points on two different utility curves. To put this more clearly, higher indifference curves are better. That is, higher indifference curves are associated with higher levels of utility. Points on higher indifference curves are preferred to points on lower indifference curves. Now, considering comparing the bundles (4,6) and (3,12). Without knowing the exact utility function, we can’t know which would be preferred. However, because we know that U(x,y) = xy we can tell that (3,12) is preferred to (4,6). Exercise Consider the utility function U(x, y) = xy . Plot out the indifference curves associated with U=64 and U=72. Confirm that the bundle (12,6) is preferred to the point (8,8).

The Marginal Rate of Substitution The slope of an indifference curve is the marginal rate of substitution. This concept has several interpretations. First, it is the rate at which a person is just willing to trade one good for another. Second, it is the slope of the indifference curve. To see that these ideas are equivalent, consider the following indifference curve for the utility function U(x, y) = xy :

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A graph showing three points on the indifference curve 20=xy. This person has a utility level of 20 at the point A (4,5). She would be willing to trade one unit of y for an additional unit of x. This trade leaves her on the same indifference curve at point B and the slope of the indifference curve between these two points is –1. Starting at point B, she would be willing to give up two units of y in order to get five units of x. This would move her to point C (10,5) and leave her with the same level of utility, U=20, that she started with. The slope of the indifference curve between B and C is –2/5. Third, the marginal rate of substitution is the derivative of the indifference curve. In the above case, this is: U( x , y) = xy = 20 20 = 20x −1 y= x − 20 dy = −20 x − 2 = = MRS dx x2 At point A, the slope of the indifference curve is: MRS =

dy − 20 − 20 = = = −1.25 dx 16 42

That is, at point A, she would be willing to give up 1.25 units of y to get 1 unit of x.

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Midway between point A and point B, at (4.472,4.472), the slope of the indifference curve is: MRS =

dy − 20 − 20 = = = −1 2 dx 4.472 20

So, at this point, she would be willing to give up 1 unit of y to get 1 unit of x. At point C, the slope of the indifference curve is: MRS =

dy − 20 − 20 = = = −0.2 dx 10 2 100

So, at point C, she would be willing to give up 0.2 units of y to get 1 unit of x.

The shape of the indifference curve is important. As a person gets more of one good and less of the other, the rate at which they’ll be willing to trade one good for another changes. For example, as the quantity of y a person has decreases, she will demand more and more x to give up one unit of y. In terms of real goods, if you had ten liters of water and one loaf of bread, you might be willing to give up three liters of water to get an additional loaf of bread, so MRS = 3/1 = 3. However, if you had two liters of water and five loaves of bread, you might require four additional loaves of bread in order to give up a liter of water, so MRS = ¼ = 0.25.

Indifference Curves Cannot Cross Indifference curves cannot cross. If two indifference curves did cross, there would be logical inconsistencies that, as the book puts it, would violate the principle of transitivity described at the beginning of this lecture. To put this somewhat differently, consider the following diagram:

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A graph showing the incorrect case of two indifference curves crossing. Points A and B are on the same indifference curve, and points B and C are on the same indifference curve, so this person would be indifferent between points A and B and indifferent between points B and C. This suggests that this person should be indifferent between points A and C. However, point A is to the northeast of point C, so point A represents more of good x and good y, so it must be preferred to point C. This is inconsistent, so it can’t be that indifference curves cross. Instead, indifference curves must be of the form:

A graph showing that higher indifference curves are associated with higher levels of utility.

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Preferences, Convexity and Balance in Consumption A set of point is said to be convex if any line drawn from one point in the set to another point in the set is entirely contained in the set. The set of points preferred to a particular point is convex. In a picture, this looks like:

A graph showing a point on an indifference curve and the set of points that are preferred to that point. The shaded area is the set of bundles that are preferred to the indicated point and is a convex set. Basically, this means that indifference curves have the shape that they do. This carries with it the implication that if a person is indifferent between two bundles of goods, they would actually prefer the average of those to bundles to either of the two. As an example, imagine the utility function U(x,y) = xy. Now, imagine two bundles that both give a utility level of 36, (4,9) and (6,6). Now, imagine that you have the average of these two bundles. That is, the average level of x, which is (4+6)/2 = 5 and the average level of y is (9+6)/2 = 7.5. The resulting utility level from (5,7.5) is U(5,7.5) = 37.5. In terms of a picture, this looks like:

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A graph showing two points on an indifference curve and demonstrating that a point that is midway between those two points will be preferred to either of them. The middle point, (5, 7.5), is slightly above the indifference curve, meaning that it is preferred to either of the two other points, (6, 6) and (4, 9).

The Mathematics of Marginal Rate of Substitution (MRS) and Utility There are a lot of different ways to think about the marginal rate of substitution (MRS). These are presented above. The underlying math is as follows. In its most general form, utility can be expressed as a function of two goods, x and y: U = U(x , y ) If we totally differentiate this function, we get the following:

dU =

∂U ∂U dx + dy ∂x ∂y

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Now, if we think about moving from one point on an indifference curve to another point on an indifference curve, like if we wanted to figure out the slope of the indifference curve, then the level of utility would be held constant. So, there would be no change in the utility level, U, and the change in the utility level, which is dU, would be zero. This gives us: 0=

∂U ∂U dx + dy ∂x ∂y

∂U , or the partial derivative of utility with ∂x respect to x, is the marginal impact that getting a little more of good x has on utility, also known ∂U as the marginal utility of good x. Similarly, , the partial derivative of utility with respect to y, ∂y is the marginal impact that getting a little more of good y has on utility, also known as the marginal utility of good y. These two terms, the marginal utility of x and the marginal utility of y are sometimes written as MUx and MUy. Now, there are a couple of things to know. First,

Second, the slope of the indifference curve is given by the usual term for a derivative, or slope =

dy dx

So, starting with the above equation and solving, we get: ∂U ∂U dx + dy ∂x ∂y ∂U ∂U − dy = dx ∂y ∂x − MU ydy = MU x dx

0=

∂U dy MU x =− = − ∂x ∂U dx MU y ∂y So, the slope of the indifference curve is equal to negative one multiplied by the ratio of marginal utilities. This ratio of marginal utilities is the MRS.

Utility Functions for Some Specific Types of Preferences Here are some particular types of utility functions. Two of these are mentioned because they’re very easy to manipulate and economists use them a lot for just that reason. Two of these are just sort of weird and fun. 12

1. Cobb-Douglas Utility (a useful one) The general form of this utility function is: U ( x , y ) = x α yβ

α > 0, β > 0

The MRS for this utility function is: MU x αx α −1yβ αy α y = = = ⋅ MRS = MU y β x α yβ−1 βx β x You should note that the MRS depends only on the ratio of α over β, not on their actual values. This characteristic is called homotheticity. As the book says, it is often convenient to alter such a utility function so that the ratio of α over β is preserved, but α + β =1. For example, if you started out with U ( x , y) = x 3 y 5

MRS =

3 y ⋅ 5 x

This might be replace with 3 5 U ( x , y) = x 8 y 8

3 y 3 y MRS = 8 ⋅ = ⋅ 5 x 5 x 8

Because the MRSs are identical, the utility functions are basically identical, too.

2. Perfect Substitutes (weird and fun) If two goods are perfect substitutes, then all you really care about is the total number you have. You’re not so worried about the mix of goods. For example, if the two goods are orange pencils (op) and yellow pencils (yp) and you really couldn’t care less what color your pencils are, your utility will depend only on the total number of pencils you have, or: U(op, yp) = op + yp 13

As a second example, if you go out to drink beer with friends, you might consume some combination of eight ounce beers (eb) and sixteen ounce beers (sb), but all your utility depends on is the total volume of beer you consume, or: U(eb, sb) = eb + 2sb That is, a sixteen ounce beer is twice as valuable in terms of your utility as an eight ounce beer. The most general form of this is: U(x , y ) = αx + β y

3. Perfect Complements (weird and fun) This is the opposite of perfect substitutes. With perfect complements, such as left shoes and right shoes, all that matters is which good you have the least of. For example, if you get utility from right shoes (rs) and left shoes (ls), the number of pairs of shoes you have is limited by which you have the least of. So, the utility function here is: U(rs, ls) = min (rs, ls) So, if you have eight right shoes and twelve left shoes, you have eight functioning pairs of shoes and, as written here, U(8,12) = 8. If you have eight right shoes and twenty left shoes, you still have eight functioning pairs of shoes and, again, U(8,20) = 8. More generally, these utility functions are written as: U( x , y) = min(αx , β y) For example, if you had a bunch of cars (c ) and a bunch of tires (t ), you really need to have four tires per car. So, the utility function might be: U(c, t) = min (c, t/4)

or

U(c, t) = min (4c, t)

4. CES Utility (another useful one) Hold on, this is a bit tricky.

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The basic form of a CES utility function is: U(x , y ) =

x δ yδ + δ δ

if δ≤1 and δ≠0

or U(x , y ) = ln x + ln y

if δ=0

Now, this might seem like a weird way to write a utility function, but stick with it for just a minute. First, if δ=1 we have: x1 y1 U(x , y ) = + = x+y 1 1 which is the perfect substitutes utility function. Second, if we have δ=0, it turns out that the first form of the CES function is equivalent to: U(x , y ) = ln x + ln y as indicated above. However, it turns out that this is equivalent to U(x,y) = xy because: U(x , y ) = ln x + ln y

e U (x , y ) = eln x + ln y = eln x eln y = xy This modified version of the original utility function is actually the same as the original utility function because it will preserve the ranking of bundles from the first utility function.

Gosh, Could We Have More Than Two Goods? So, you’re not happy restricting analysis to only two goods? Stepping up to more than two goods isn’t too difficult.

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We start with the utility function: U = U(x1, x 2 , x 3 , x 4 ,..., x n ) We can totally differentiate this to get: dU = MU1dx1 + MU 2dx 2 + MU3dx 3 + MU 4dx 4 ... + MU n dx n Now, the thing about marginal rates of substitution with multiple goods is that the MRSs are really only defined on pairs of goods. That is, you can really only know the rate at which one good would be traded for another good. The concept isn’t defined for more than a pair of goods. For example, you can talk about the rate at which you’d trade hamburgers for cigarettes, the rate at which you’d trade cigarettes for tofu, or the rate at which you’d trade hamburgers for tofu. There’s no way to put them together in terms of a MRS more than two at a time. So, we can think about the marginal rate of substitution between x1and x2, holding quantities of other goods constant. If we hold utility constant, we have dU=0, and if we hold the quantities of other goods constant we have dx3=0, dx4=0 and so on. Putting all of this together we get: dU = MU1dx1 + MU 2dx 2 + MU3dx 3 + MU 4dx 4 ... + MU n dx n 0 = MU1dx1 + MU 2dx 2 + MU3 ⋅ 0 + MU 4 ⋅ 0... + MU n ⋅ 0 0 = MU1dx1 + MU 2dx 2 dx1 MU 2 =− dx 2 MU1 None of this is really different from the two good case. For those of you who like to think about graphs, in the n-good case, the indifference curve will be an n-dimensional plane for which: U(x1, x 2 , x 3 , x 4 ,..., x n ) = k

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Lots of Fun Practice! Here are a lot of practice problems. They are not required. The intention is to give you sufficient opportunity to practice the techniques presented above. You should do these practice problems until you feel you are proficient with the techniques and concepts described above.

Preferences For each of the following utility functions and pairs of bundles, indicate which bundle would be preferred or if the person would be indifferent between them. 1. 2. 3. 4. 5. 6. 7. 8. 9.

U(x,y) = xy U(x,y) = xy U(x,y) = x2y U(x,y) = x + xy + y U(x,y) = 2x + 3y U(x,y) = min (x, y) U(x,y) = min (x, 3y) U(x,y) = x + (y/1.1) U(x,y) = x + lny

(5,8) (5,8) (3,6) (2,8) (3,3) (8,11) (14,5) (1,2) (4,2)

(6,6) (2,20) (2,19) (7,3) (4,2) (28,8) (22,4) (2,1) (5,1)

Plot Some Indifference Curves Plot out the following indifference curves. 1. 2. 3. 4. 5. 6. 7. 8. 9.


U=40 U=1 U=144 U=120 U=24 U=10 U=24 U=100 U=100

Convexity and Stuff For each of following utility functions and pairs of bundles, show that the average of the two bundles is preferred to both of the original bundles. If the average bundle is not preferred, explain why.

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1. 2. 3. 4. 5. 6. 7. 8. 9.


(5,8) (4,10) (4,10) (2,20) (3,6) (1,54) (4,8) (8,4) (4,3) (1,5) (8,12) (28,8) (12,5) (22,4) (10,11) (9,12.1) (4,e) (5,1)

[Note: e ≅ 2.71828.]

Marginal Rates of Substitution Calculate the marginal rate of substitution for each of the following utility functions. 1. 2. 3. 4. 5. 6. 7. 8. 9.

U(x,y) = xy U(x,y) = xy U(x,y) = x2y U(x,y) = x + xy + y U(x,y) = 2x + 3y U(x,y) = x2 y3 U(x,y) = 5x3y4 U(x,y) = x + (y/1.1) U(x,y) = x + lny

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