Economics 11, Microeconomic Theory
Professor: Maurizio Mazzocco
Office: 8377 Bunche Hall
E-mail: mauriziomazzocco@gmail.com
Office Hour:
Wednesday 4pm-5pm
After each lecture
1
Class Web Site:
• https://moodle2.sscnet.ucla.edu/course/view/19FECON11-1
Text:
• Walter Nicholson, Microeconomic Theory: Basic
Principles and Extensions, 12th Edition.
• The 11th and 10th editions are also fine. You can also buy
older editions. In this case you are responsible for finding
the material covered in class.
2
Course Requirements:
• Students should attend lectures as well as a weekly
discussion sections with a teaching assistant;
• Weekly problem sets;
• 2 midterms and 1 final.
3
Problem Sets:
• Will be assigned approximately once per week and will
be graded;
• Late homework will not be accepted for any reason;
• Each student's lowest homework grade will be dropped;
• You will receive 50% of the points by simply turning in
a complete solution of the problem set;
• The remaining 50% is assigned by grading randomly
one part of each problem set.
4
Discussion Sections:
• The discussion sections start this week;
• Discussion sections will usually focus on
the problem set assigned the previous week;
• They will go over additional exercises;
• TAs will answer question about the material
covered during the lectures that week.
5
Exams:
• Two midterm exams will be given in class on
October 29 and November 21;
• These are the only times these exams will be
offered, i.e. there are no makeup midterms;
• A cumulative final (3-hour) exam will be given on
Friday, December 13, 2019, from 3:00pm to
6:00pm.
6
Grading:
Homework will count 10% of the final grade.
For the remaining 90% two rules will be used:
If the final exam score is below the scores on each
of the two midterms, each exam will count 30%;
For all other students the final exam will count 50%
and the higher of the two midterms will count 40%
(the lower midterm score is dropped).
For any student missing one of the midterms for any
reason, the latter calculation will be used with the one
midterm taken counting 40%.
7
This grading scheme is designed:
• To give students an opportunity to improve their
grades if they do poorly on the first midterm;
• To ensure that a consistent and fair policy is applied
in the unusual case that a student must miss a
midterm;
• To avoid heavy weight on a single poor exam
performance.
8
Is there a curve for the course?
• Yes
• But it depends on the performance of the class
• If I believe that the class as a whole does well in
problem sets, midterms, and final, I will use a
generous curve
9
Economics 89, Honors Seminar
Lecture: Thursday, from 5:30pm.
Once every two weeks, starting October 3, 2019.
Requirements:
Read one paper every two weeks;
Attend the lectures where we will discuss the papers;
Prepare a one-page report discussing the material
covered during the lectures.
10
Economics 89, Honors Seminar
There are only 20 spots in the class
Write one page explaining why you are interested in
the course
Send an email tomorrow morning after 8am to my
gmail account
I will look at the first 40 emails and choose the 20
students with the best explanation
11
Microeconomic Theory
The Course:
• This is the first rigorous course in microeconomic
theory
• One of the main goals is to teach analytical tools
that will be useful in other economic and
business courses
12
Microeconomic Theory
Microeconomics:
• The branch of economics dealing with behavior of
individual decision makers such as consumers
and firms.
13
Everyday Economics
Wake up:
•
Stay in bed
•
Go to class
Breakfast:
•
Cereals and Juice
•
Eggs, bacon and coffee
Economics:
•
Maximizing behavior
14
Your trip to class
You don’t have a car:
•
Bus
•
Cab
You have a car. Have to stop for gas and prices are
higher than last month:
•
I don’t care about prices: full tank
•
I’ll buy half of what I bought last week
•
I’ll take the bus
Economics:
•
Income effect
•
Price elasticity
15
On campus
In class:
•
Doze off
•
Listen to the lecture
After class:
•
Economics 11
•
History
Economics:
•
Maximizing behavior
•
Marginal rate of substitution
•
Price/cost is not only the money you have to pay
16
17
Chapter 2
THE MATHEMATICS OF
OPTIMIZATION
18
The Mathematics of Optimization
• Why do we need to know the mathematics
of optimization?
• Consumers attempt to maximize their
welfare/utility when making decisions.
• Firms attempt to maximizing their profit
when choosing inputs and outputs.
19
Maximization of a Function of
One Variable
• The manager of a firm wishes to
maximize profits:
  f (q)

Maximum profits *
occur at q*
*
 = f(q)
q*
Quantity
20
Maximization of a Function of
One Variable
• If the manager produces less than q*, profits
can be increased by increasing q:
– A change from q1 to q* leads to a rise in 

    1
 *
0
q q  q1
*
*
 = f(q)
1
q1
q*
Quantity
21
Maximization of a Function of
One Variable
• If output is increased beyond q*, profit will
decline
– an increase from q* to q3 leads to a drop in 

  3  

0
*
q q3  q
*
*
 = f(q)
3
q*
q3
Quantity
22
Derivatives
• The derivative of  = f(q) is the limit of
/q for very small changes in q
d df
f ( q  h)  f ( q )

 f ' (q)  lim
h 0
dq dq
h
• The value of this ratio depends on the
value of q
23
Value of a Derivative at a Point
• The evaluation of the derivative at the
point q = q1 can be denoted
d
dq q  q
1
• In our previous example,
d
0
dq q q
1
d
0
dq q q
3
24
Second-Order Derivatives
• The derivative of a derivative is called a
second-order derivative
d (df / dq ) d 2 f
''
 2  f
dq
dq
25
First Order Condition
• For a function of one variable to attain
its maximum value at some point, the
derivative at that point must be zero
df
0
dq q  q *
26
Second Order Conditions
• The first order condition (d/dq) is a
necessary condition for a maximum, but
it is not a sufficient condition

If the profit function was u-shaped,
the first order condition would result
in q* being chosen and  would
be minimized
*
q*
Quantity
27
Second Order Condition
• The second order condition to represent
a maximum is
d 
dq 2
2
 f " ( q ) q  q*  0
q  q*
• The second order condition to represent
a minimum is
d 2
 f " ( q ) q  q*  0
2
dq q  q*
28
Second Order Conditions
f " (q ) q q*  0 the function is concave
• If

Quantity
29
Second Order Conditions
f " (q ) q q*  0 the function is convex
• If

Quantity
30
Functions of Several Variables
• Most goals of individuals depend on
several variables
• In this case we need to find the
maximum and minimum of a function of
several variables:
u  f ( x1 , x2 ,..., xn )
31
Partial Derivatives
• The partial derivative of the function f
with respect to x1 measures how f
changes if we change x1 by a small
amount and we keep all the other
variables constant.
• Treat all the other variables x2,…,xn as
constants and then use the previous
definition of a derivative.
32
Partial Derivatives
• A more formal definition of the partial
derivative is
f
f ( x1  h, x 2 ,..., x n )  f ( x1 , x 2 ,..., x n )
 lim
x1 h0
h
• The partial derivative of u=f (x1,x2,…,xn )
with respect to x1 is denoted by
u
f
or
or f x1 or f1
x1 x1
33
Second-Order Partial Derivatives
• The partial derivative of a partial
derivative is called a second-order
partial derivative
 (f / xi )
2 f

 f ij
x j
xi x j
34
Young’s Theorem
• Under general conditions, the order in
which the second order partial
derivative is computed does not matter
fij  f ji
35
Total Differential
• Consider our utility u = f(x1,x2,…,xn)
• We want to know by how much f changes if
we change all the variables by a small
amount (dx1,dx2,…,dxn )
• The total differential measures this total
effect:
f
f
f
du  df 
dx1 
dx2  ... 
dxn
x1
x2
xn
du  df  f1dx1  f 2 dx2  ...  f n dxn
36
First-Order Conditions
• A necessary condition for a maximum (or
minimum) of the function f(x1,x2,…,xn) is
that du = 0 for any combination of small
changes in the x’s (dx1,dx2,…,dxn )
• The only way for this to be true is if
f1  f2  ...  fn  0
37
First-Order Conditions
• To find a maximum (or minimum) we have to
find the first order conditions:
f/x1 = f1 = 0
f/x2 = f2 = 0
.
.
.
f/xn = fn = 0
38
Second Order Conditions Functions of Two Variables
• The second order conditions for a
maximum are:
• f11 < 0
• f22 < 0
• f11 f22 - f122 > 0
• The second order conditions for a minimum
are:
• f11 > 0
• f22 > 0
• f11 f22 - f122 > 0
39
Constrained Maximization
• Suppose that we wish to find the values
of x1, x2,…, xn that maximize
u = f(x1, x2,…, xn)
subject to a constraint that permits only
certain values of the x’s to be used
g(x1, x2,…, xn) = 0
In our case:
I-(p1x1+p2x2+p3x3…+pnxn)= 0
40
Lagrangian Multiplier Method
• First, set up the following expression:
L = f(x1, x2,…, xn ) + g(x1, x2,…, xn)
• where  is an additional variable called a
Lagrangian multiplier
• L is often called the Lagrangian
• Then apply the method used in the absence
of the constraint to L
41
Lagrangian Multiplier Method
• Find the first-order conditions of the
new objective function L:
L/x1 = f1 + g1 = 0
L/x2 = f2 + g2 = 0
.
.
.
L/xn = fn + gn = 0
L/ = g(x1, x2,…, xn) = 0
42
Lagrangian Multiplier Method
• The first-order conditions can generally
be solved for x1, x2,…, xn and .
• The solution will have three properties:
– the x’s will satisfy the constraint: g(x1, x2,…,
xn) = 0;
– these x’s will make the value of L as large
as possible (as small as possible);
– since the constraint holds, L = f and f is also
as large as possible (as small as possible).
43
Lagrangian Multiplier Method
• The Lagrangian multiplier () has an
economic interpretation
•  measures by how much we can
increase the objective function f if the
constraint is relaxed slightly
• In our case,  represents the increase in
utility/happiness if we increase income
by a small amount.
44
Lagrangian Multiplier Method
• A high value of  indicates that the utility
could be increased substantially by
relaxing the constraint (poor people)
• A low value of  indicates that there is
not much to be gained by relaxing the
constraint (rich people)
• =0 implies that the constraint is not
binding
45
Implicit Function Theorem
• Usually we describe a function as:
y = f(x)
• Sometimes we do not have this explicit
formulation for the function f
• All we have is an equation that describes
how x and y are related:
g(y,x)=0
46
Implicit Function Theorem
• Consider the implicit function:
g(y,x)=0
• The total differential is:
dg=0
gydy + gxdx=0
• If we solve for dy and divide by dx, we get
the implicit derivate:
dy/dx=-gx/gy
• Providing gy≠0
47
Implicit Function Theorem
• The implicit function theorem establishes
the conditions under which we can compute
the implicit derivative of a function.
• In our course we will always assume that
these conditions are satisfied.
48
The Envelope Theorem
• The Envelope Theorem gives us a formula to
compute the derivative of U with respect to a
parameter in the maximization problem , at
the optimum
49
The Envelope Theorem
U   ln c1  (1   ) ln c 2  f ( c1 , c 2 ,  )
• Consider the utility function at the optimum:
U  f ( c1 ( ), c 2 ( ),  )
 f dc 1
 f dc 2
f
dU 
d 
d 
d
 c1 d 
c2 d

dU
 f dc 1  f dc 2  f



d
 c1 d   c 2 d   
50
The Envelope Theorem
• The Envelope Theorem tells us that, at the optimum,
the derivative of U with respect to :
• We only have to consider the direct effect of
utility
on the
• We do not have to consider the indirect effect that
has on the utility through the optimal choice of
consumption
51
Homogeneous Functions
• A function f(x1,x2,…xn) is said to be
homogeneous of degree k if
f(tx1,tx2,…,txn) = tk f(x1,x2,…xn)
– when a function is homogeneous of degree
one, a doubling of all of its arguments
doubles the value of the function itself
– when a function is homogeneous of degree
zero, a doubling of all of its arguments
leaves the value of the function unchanged
52
Homogeneous Functions
• If a function is homogeneous of degree
k, the partial derivatives of the function
will be homogeneous of degree k-1
53
Euler’s Theorem
• Euler’s theorem establishes that, for
homogeneous functions, there is a special
relationship between the values of the
function and the values of its partial
derivatives.
• If a function f(x1,…,xn) is homogeneous of
degree k we have:
kf(x1,…,xn) = x1f1(x1,…,xn) + … + xnfn(x1,…,xn)
54
Euler’s Theorem
• If the function is homogeneous of degree
0:
0 = x1f1(x1,…,xn) + … + xnfn(x1,…,xn)
• If the function is homogeneous of degree
1:
f(x1,…,xn) = x1f1(x1,…,xn) + … + xnfn(x1,…,xn)
55