More on consumer theory: predicting and explaining the choices we make
draft Oct 28, 2015
Choices depend on preferences and and constraints
Wrt constraints, and quoting from a course evaluation
"You seem like a nice guy but you are sooooooooo
boring. I would not hang out with you even if you the
only person in my budget set."
In my defense, this person understands the di¤erence between his preferences
and his constraints, and managed to learn in an extreme state of boredom.
In previous lectures we discussed the budget constraint and preferences.
Here we discuss preferences in more detail
The relevant chapter is 10, and, in particular, its appendix.
1
1
Economists assume individuals have preferences: they can rank bundles of goods such
that
Bundle j is preferred to bundle i, or
Bundle i is preferred to bundle j, or
The individual is indi¤erent between bundles i and j.
And this is true for any pair of bundles i and j
Economists often represent preferences with something called a utility function.1
A utility function is simply a mathematical function that assigns a number
to each bundle such that is assigns a higher number to bundle k than to bundle
v if bundle k is preferred to bundle v, and assign bundles h and t the same
number if the individual is indi¤erent between these to bundles.
The function is chosen such that u(k) > u(v) if bundle k is preferred to
bundle v, and u(t) = u(h) if the individual is indi¤erent between the two bundles.
Your u(:) is di¤erent from mine u(:).
One has preferences and a mathematical function is one way to represent
those preference, there are other ways as well.2 For example one could write
down, in a notebook a description of each possible bundle and the ranking of
that bundle.
Economists do no assume people get utility: utility is a way to represent
preferences.
We call these numbers produced by this function, utils or utility. If a bundle
m has a higher utility number than bundle w this simply means that bundle m
is preferred to bundle w.
1 Utility
is a concept that confuses many people. Many people want to make more of
utlity than is there. Economists do not assume that people get utility per sec; it is only a
mathematical metric we use to rank bundles.
2 Note that if one can …nd a mathematical function that ranks bundles in the same order
that George ranks them, then there are an in…nite number of other mathematical functions
that will also rank the bundles in the same order.
2
Economists assume your preferences are your preferences- they are what they
are. If you like to play with your dog, and you would give up everything else
to play with your dog, those are your preferences, which we economists take as
given.
If your preference is to drink motor oil and stick pins in your eyeballs, so be
it if that is what you prefer.
However, economists might want to in‡uence (restrain or encourage) your
choices if your choices directly in‡uence others in a sign…cant way.
Psychologists, on the other hand, might ask why you have the preferences
you have, and consider some preference abnormal, even if they do not lead to
behaviors with external e¤ects.3
3 Actually, many of them would ask whether people have preferences at all in the economic
sense of the word.
3
1.1
KW, and I as well, will proceed assuming there are
only two goods in play
KW assumes the two commodities are restaurant meals and rooms.
they assume u = (r; m). KW choose a matematical function of r and m that
would produce a graph that looks like this one.
4
For my example, my two goods will be aquarium …sh and dog biscuits . I
have a big aquarium that will hold many …sh. My dog, So…e, likes dog biscuits.
My trust fund provides me with a small amount of money each month that I
can only spend on …sh or dog biscuits, So…e has no money.
All of our other needs are fu…lled by care packages sent by relatives, so all
of my money can be spent on …sh or biscuits.
So…e always prefers more dog biscuits to less dog biscuits, and I care about
So…e
Ceteris paribus, I would always prefer to have more …sh in my aquiarium.
5
So…e swimming in my large aquarium
1.1.1
We need to assume a speci…c mathetmatical form for my utility
6
function
1.1.2
I choose u = f :8 d:2 , hopefully it represents my preferences. I
made …sh more important than dog biscuits.
A digression about graphs and mathematical functions. A graph is a visual image of a
speci…c mathematica function. That is, if we know the mathematical function I have
su¢ cient information to graph it, and if one has the graph one can work backwards
and …gure out what mathematical function would produce that graph. As you advance
to higher math courses or higher economics courses, the mathematical function of
interest will be speci…ed. It will then possibly be graphed so the student can visualize
the function. Mostly in Econ 2010 and in the book, the graph is shown without
making explicit the mathematical function that generated that graph. Remember
that underlying the graph, there is a mathematical function.
I have assumed a utility function such that, if I am currently consuming
equal quantities of both, I would prefer an additional …sh to an additional dog
biscuits.
5
4
3
2
5
1
4
3
f
2
1 0
00
1
2
3
4
5
d
My assumed utility for …sh and dog biscuits
uitlity is on the vertical axis.4
4 Note that the utility keeps rising as I consume more of each good, but I can’t draw a
graph that goes on forever.
7
Imagine all the bundles between which I am indi¤erent. How would you
indentify such bundles? GRAB A SWORD?
2
Indi¤erence curves identify all those bundles
that achieve the exact same utility.
Returning to the utility function in KW
Said another way, not using the concept of utility: An indi¤erence curve
represents all of the bundles between which you are indi¤erent)5
5 The existence of indi¤erence curves do not depend on the concept of utility; they depend
only on the individual having preferences (a ranking of consumption bundles).
8
An example indi¤erence curve in KW
9
The individual is indi¤erent between living in a three-room house and eating
out 30 times a month, and 6 rooms and 15 meals out.
If this guy is currently living in three rooms and eating out 30 times he
would be willing to give up 15 meals to get to move to a 6 room place. That
is, his wtp (willingness-to-pay) for three more rooms is 15, measured in forgone
meals out. (Said another way: if he currently lives in 6 rooms and eats out 15
times a month, he would give up three of those rooms to get to eat out 15 more
times a month.)
If this guy is currently living in three rooms and eating out 30 times and he
gave up less than (more than) 15 meals to get to move to a 6 room place he
would be better o¤ (worse o¤), where better o¤ (worse o¤) means he moves to
a higher (lower) indi¤erence curve..
10
2.1
The indi¤erence curves that correspond to my utility
function for …sh and biscuits
2.2
fish
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
biscuits
Indi¤. curvs : Blk u = 1, Red u = 2, Blu u = 3
For example I am indi¤erent between 1 biscuit with 2:4 …sh, and 5 biscuits with
1:6 …sh: each of these bundles generates 2 utils.
Notice how my indi¤erence curves become ‡atter when there are relatively
more biscuits than …sh in the bundle (steeper as there are relatively more …sh
in the bundle). It does not have to be this way, but often is this way.
11
Now let’s look at my utility indi¤erence curve for u = 2 in more detail
fish
4.0
3.5
3.0
2.5
2.0
1.5
0
1
2
3
4
5
6
7
biscuits
Indi¤erence curve for u = 2
Which indicates, as noted above, that I am indi¤erent between 1 biscuits with
2:4 …sh, and 5 biscuits with 1:6 …sh.
Said another way, to increase So…e’s biscuit consumption from 1 to 5 (an
increase of 4 biscuits) I would be willing to give up :8 …sh (2:4 1:6)
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2.3
Let’s do it again but for two smaller changes in the
number of biscuits
fish
4.0
3.5
3.0
2.5
2.0
1.5
0
1
2
3
4
Indi¤erence curve for u = 2
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5
6
7
biscuits
If we currently have 1 biscuit and 2:4 …sh, my wtp to get another biscuit is
:4 …sh. If currently 5 biscuits and 1:6 …sh, my wtp get another biscuit for So…e
is only :1 …sh.
So, for me, the more biscuit-intensive my bundle, relative to …sh, the fewer
…sh I’m will to go without to get another biscuit.
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2.4
Consider the slope of the indi¤erence curve.
Now I redraw the indi¤erence curve for u = 2, adding two tangent lines: one
at b = 1 and one at b = 5. These tangent lines represent the slope of the
indi¤erence curve at b = 1 and at b = 5. Notice that the tangent is steeper (has
a more negative slope) at b = 1. That is, for this indi¤erence curve, the slope
declines in negative value as b increases
fish
4.0
3.5
3.0
2.5
2.0
1.5
0
1
2
3
4
5
6
7
biscuits
Indi¤erence curve for u = 2
At dog biscuits = 1 the slope is :6, and at dog biscuits equal = 5 the slope
is :08. Put loosely, when I am consuming 1 biscuit, I would give up :6 …sh to
get another biscuit, but when we are consuming 5 biscuits I would only give up
:08 …sh to get another biscuit.6
6 The answer are slightly di¤erent from above because, the slope is declines as one increases
biscuits from 1 to 2, and from 5 to 6.
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2.5
wtp (willingness to pay) and M RS (marginal rate of
substitution
The negative of the slope of the indi¤erence curve, at a speci…c amount of dog
biscuits, is, approximately, how many …sh I am willing to give up to get one
more dog biscuit
Said another way, it is my willing-to-pay, wtp, for an additional biscuit in
terms of forgone …sh.
Looking back at my indi¤erence curve for u = 2 , if I am currently consuming 1 dog biscuit I am willing to pay approximately :4 …sh (change my …sh
consumption by :4) to get one more biscuit. Alternatively, if I am currently
consuming 5 biscuits I am only wtp :1 …sh to get on more biscuit.
fish
4.0
3.5
3.0
2.5
2.0
1.5
0
1
2
3
4
5
6
7
biscuits
Indi¤erence curve for u = 2
As the relative amount of biscuits in my bundle increases, my wtp to get more
biscuits, in terms of forgone …sh, declines. This re‡ects diminishing marginal
utility in the consumption of both biscuits and …sh.
If the marginal utility of both goods were constants, independent of the
amount of the good consumed, the indi¤erence curve would be a straight line
(have a constant negative slope).
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2.5.1
What is my wtp for additional …sh in terms of forgone biscuits?
If my wtp for an additional biscuit is :5 …sh, then my wtp for an additional …sh
1
is :5
= 2 biscuits, as it must. (one is the inverse of the other.)
Another name for wtp for an additional biscuits is the marginal rate of substif ish
jholding utility constant
tution of biscuits in place of (for) …sh, M RSbf = 44biscuits
what you will pay to get another biscuit, in terms of …sh.
Another name for wtp for an additional …sh is the marginal rate of substitution of …sh in place of (for) biscuits, M RSf b = 44biscuits
f ish jholding utility constant
what you will pay to get another …sh, in terms of biscuits
f ish
M RSbf = 44biscuits
jholding utility constant is the negative of the slope of
the indi¤erence curves, so a positive number if the indi¤erence curve in negatively sloped. (Note that indi¤erence curves for "goods" are negatively sloped.)
fish
4.0
3.5
3.0
2.5
2.0
1.5
0
1
2
3
4
5
6
7
biscuits
Indi¤erence curve for u = 2
We call it the marginal rate of substitution because it re‡ects the rate at
which the individual is willing to substitute one good for another.
17
Another way to think of M RSbf , the way KW present it, is M RSbf =
change in utility from having one m ore biscuit
change in utility from having one m ore …sh
change in utility from having one m ore …sh
the way KW present it, M RSf b = change
in utility from having one m ore biscuit
So, if, at my current consumption levels for …sh and biscuits, I would get
1 more util from an additional …sh and :5 utils from an additional biscuit the
in utility from having one m ore biscuit
M RSbf = change
= :5
change in utility from having one m ore …sh
1 = :5
so M RSbf =
4 f ish
4 biscuits jholding utility constant
=
change in utility from having one m ore biscuit
change in utility from having one m ore …sh
While there are two ways of describing the marginal rate of substitution, I
…nd my way more intuitive. And, my way does not require the concept of a
marginal utility.
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3
An important aside: consider a world of two
commodities: goods and pollution, where goods
are good and pollution is bad.
Remember that I am an environmental economist, so like environmental examples.
Draw a representative individiual’s indi¤erence curves with goods on the
horizontal axis and pollution on the vertical axis.
Assume everyone has the same preferences.
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T.A.s are making up problems where one commoditiy is a good and one is
a bad.
The following are three indi¤erence curves, each has a positive slope (these
are not three di¤erent utility functions).
pollution
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
goods
Three indi¤erence curves
In what direction is utility increasing?
Remember that uility is constant along each line because each is an indifference curve. Utility is higher along the black line than along the red line and
higher along the red line than along the blue line. Utility increases as one moves
to the southeast (more good and less pollution).7
Try to visualize a utility function when one commodity is a good and one is
a bad.
7 For those mathematically inclined, p = ln(1 + g) for the black line, p = 2 ln(1 = g) for the
red line, and p = 3 ln(1 + g) for the blue line. I chose these functional forms simply because
they have the shape and position I was looking for.
20
9
8
7
6
U
5
4
3
2
4
1
5
4
3
2
1
0
2
00
good
Utility as a function of a good and a bad
What if they were both bads?
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bad
dog sh__t
10
5
0
0 0
pollution
4
2
6
8
10
-2
-4
util
-6
-8
-10
An assumed utility for dog sh_t and pollution
Think about wtp and M RS when one of the commodities is a good and one
is a bad. E.g. what is your wtp for another un it of pollution? and what is your
wtp to get rid of a unit of pollution? Are these amount positive or negative,
and what determines their magnitude?
Your wtp to get more of the good is the additional amount of pollution you
would accept.
Your wtp to get another unit of pollution is how much more goods you would
have to be given to accept that additional unit of pollution, so in it is actualy
your wta (willingness-to-accept) additional pollution, your wtp is negative, so a
wta.
22
Imagine a poor individual, for example one in China. on the blue indi¤erence
curve, in contrast to a rich Boulderite on the black one.
pollution
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
goods
Three indi¤erence curves
I am assuming they both have the same preferences (same indi¤erence map)
How much would the Chinese guy pay, in terms of fewer goods to decrease
pollution from 1:75 to 1:25 units (a decrease of :5 units of pollution)?
The two green horizontal lines on the graph: they will help us to answer this
question.
The top horizontal green line is at 1:75 units of pollution
The bottom horizontal green line is at 1:25 units of pollution
The Chinese guy is currently consuming 1:75 units of pollution and :8 units
of goods. He would give up approx. :3 goods (a decrease from :8 to :5 goods)
to reduce pollution from 1:75 to 1:25
How much would the Boulder guy pay, in terms of fewer goods, to decrease
pollution from 1:75 to 1:25 units? Approx 2:25 units (a decrease from approx
4:75 goods to approx. 2:5 goods).8
8 1:25 = ln(1 + g), Solution is: 2: 490 3
1:75 = ln(1 + g), Solution is: 4: 754 6
1:25 = 3 ln(1 + g), Solution is: 0:516 90
1:75 = 3 ln(1 + g), Solution is: 0:792
23
Why such a big di¤erence between what they would pay?
Di¤erent preferences?
No, we assumed that have the same preferences, the same indi¤erence maps.
The di¤erence in wtp for reduced pollution is much higher in Boulder because
real income is much higher (much higher initial utility level).
This result has important environmental implications.
It basically implies that it is more e¢ cient to locate pollution-intensive industries in poor neighorhoods/countries.
This is because if the neighborhood is poor, their wtp for less pollution is
low relative to the rich neighborhood.
Put simply, poor people relative to rich people, care more about goods than
they care about pollution.
If it is a choice between feeding the kids by working in a polluting factory, or
no job and starving kids, most people would choose the polluting factory, even
if would kill them in twenty years. I, on the other hand, would happily have my
taxes raised (consume fewer goods) if it substantially reduced global warming.
Many people do not like this result, that, ceteris paribus, e¢ ciency dictates
that pollution-intensive industries be located on poor countries/neighborhoods.
Why?
They do not think it is equitable/fair.
A term you often hear is "environmental justice" It is the case that many factories in the U.S. (much heavy industry has already moved abroad) are located
in poor neighborhoods, often poor black neighborhoods.
When arguing that it is not fair keep in many that many of the residents
would rather have a job and the pollution than no job and a cleaner environment. Also keep in mind that, ceteris paribus, rents are cheaper in dirtier
neighborhoods.
Will all this in mind, the equity argument, put simply, is that it is unfair to
have an income distribution such that people at one end of the spectrum can
a¤ord to live in a clean environment and people at the other end of the spectrum
have to live in a dirty environment.
Note that if everyone had the same preferences and the same income, we
would all, ceteris paribus, experience the same amount of pollution
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