Math Guide for Econ 11
These notes are intended as a reference guide for the mathematical tools of calculus that
are used in Econ 11. The following pages provide all the mathematical results, along with
brief explanations, illustrative examples of their use and additional exercises. You should
have seen some, if not most of this material in your calculus course, especially the parts that
are related to single-variable calculus. Some of the concepts may be new, but these will be
discussed in detail, when they are introduced during the lectures; these notes are intended
to serve you as a quick reference as you go along with the course. The
f
rst two TA sessions
will review the material in this handout in detail.
1 Introduction
Consider the following two examples:
1. A
f
rm that has to decide how much to produce of a certain good. The market price
for the
f
rm’s product is
p
, and it costs the
f
rm
C
(
q
)=
1
2
q
2
to produce a quantity
q
of the
good. The
f
rm wants to maximize its pro
f
ts, i.e. sales revenue minus production costs, and
can choose the quantity
q
that it produces. With a sales revenue
pq
,wecanwr
itethe
f
rm’s
pro
f
ts as
F
(
q
)=
pq
−
1
2
q
2
,andthe
f
rm’s decision problem consists in
f
nding the quantity
q
that maximizes
F
(
q
)
.
2. A student has to prepare for an exam the next day. She has
12
hours remaining before
the exam. For each additional hour of studying, she expects to raise her grade by
2
points
(out of a 100), but she also realizes that her concentration level during the exam depends on
how much she sleeps. If she sleeps
7
hours, she is able to work with full concentration. If she
sleeps less than that, her concentration goes down. If she didn’t study any more, and slept
exactly
y
hours before the exam, she expects a grade of
11 +
y
(14
−
y
)
(for y=7, this gives
a grade of 60; for y=6, this gives a grade of 59, for y=5, this gives a grade of 56, and so on).
Therefore, if the student sleeps
y
hours and studies
x
hours until the exam, her grade (as a
function of
x
and
y
)is
F
(
x, y
)=11+
y
(14
−
y
)+2
x
. The student wants to allocate her
time optimally between sleeping and studying so as to maximize her expected grade. But
her choices of
x
and
y
have to satisfy the additional constraint that
12
≥
x
+
y
,i
.e
. the
amount of time that the student studies plus the amount that she sleeps cannot exceed the
12 hours remaining before the exam.
These two examples are representative of the
decision problems
that are studied in Mi-
croeconomics, and that we are concerned with in much of this course. Formally, we consider
problems in which a
decision-maker
(i.e. a consumer, a household, a
f
rm, etc. depending
on the exact context), has to maximize an
objective function
F
(
x
)
with respect to some
choice variable
x
.I
nt
h