1-4
Part Chapters 1 Introduction (set theory, algebra)
Part 2 Static Analysis (matrix algebra)
Part 3 Comparative Static Analysis
(derivatives w/ matrix algebra)
Part 4 Optimization Problems
(derivatives w/ constraints w/ matrix
algebra, 2nd order conditions)
Objectivesformathematical
economics
Tounderstandmathematicaleconomicsproblemsby
statingtheunknown,thedataandtheconditions
Toplansolutionstotheseproblemsbyfindinga
connectionbetweenthedataandtheunknown
Tocarryoutyourplansforsolvingmathematical
economicsproblems
Toexaminethesolutionstomathematicaleconomics
problemsforgeneralinsightsintocurrentandfuture
problems(Polya,G.HowtoSolveIt,2nded,1975)
Ch. 1 The Nature of Mathematical
Economics
1.1 Mathematical vs. Nonmathematical
Economics
1.2 Mathematical Economics vs.
Econometrics
1.1 Mathematical vs.
Nonmathematical Economics
Advantage of Language of math in economics
Precise, concise
Draws on math theorems to show the way
Forces declaration of assumptions
Allow treatment of the n-variable case
Language as a form of logic
Logic per se (deduction/induction)
Math as an extension of deductive logic
From reality to rigor; math is a means not and end
Too much rigor and too little reality
1.2 Mathematical Economics vs.
Econometrics
Deduction vs. induction
Deduction: from general to specific
Induction: from specific to general
Weakness of deduction
Correctness of initial assumptions
Weakness of induction
Correctness of final results
Humes paradox:
neither deduction or induction leads to the Truth
so use both: the one is a check on the other
Ch. 2 Economic Models
2.1 Ingredients of a mathematical model
2.2 The real-number system
2.3 The concept of sets
2.4 Relations and functions
2.5 Types of functions
2.6 Functions of two or more ind. Vars
2.7 Levels of generality
2.1 Ingredients of a mathematical
model
Variables, constants, parameters
Equations and identities
2.1 Ingredients of a mathematical
model
A mathematical Economic Model will
consist of:
Variables,
Parameters,
Constants
Equations and Identities
Example: Supply-demand model
Qd = - P
demand equation
Qs = + P
supply equation
Qd = Qs
equilibrium condition
2.2 The real-number system
Real numbers (universal set, continuous,+, -, 0)
Irrational numbers
Those #s that cant be expressed as a ratio, e.g., sqrt. 2, pi,
Rational numbers
Fractions: can be expressed as ratio of integers
Integers: expressed as whole numbers (or ratio of itself to 1)
The number system of a society parallels its
development, e.g, from 1, 2, many to imaginary
numbers, expression of thought (Menninger, 1969)
2.3 The concept of sets
Set notion
Relationships between sets
Operation on sets
Laws of set operations
2.3 The concept of sets
The market or demand supply model above and
its solution can be expressed in set notation as
ordered pairs
D = ( P , Q ) Q = P
S = ( P, Q ) Q = + dP
{
{
}
(
)
}
DS = P, Q
A set {i,,n} is a collection of distinct objects
called elements w/ associated unions ,
intersections , and complements and null .
Exercise 2.3 #2
2.3 Given sets S1 = {2,4,6}, S2 = {7,2,6), S3
= {4,2,6}, S4 = {2,4}, which of the following
statements are true?
(false)
2.3(a) S1 = S2
2.3(b) S1 = R
(false)
2.3(c) 5 S2
(false)
2.3(d) 3 S2
(true)
2.3(e) 4 S4
(false)
2.4 Relations and functions
Ordered Pairs
Relations and Functions
2.4 Relations and functions
A function is a set or ordered pairs with the
property that for (x, y) any x value uniquely
determines a single y value .
More than one x can determine the same
y and still be a function, e.g., curves or
, mpp or cost curves
However if one x determines two or more
ys it is not a function, rather it is called a
relation in that case, e.g., curves or
2.5 Types of functions
Constant functions
Polynomial functions
Rational functions
Non-algebraic functions
Digression on exponents
Property of functions
A function is increasing if
f(xB) f(xA) for xB > xA
A function is strictly increasing if
f(xB) > f(xA)
for xA
xB> A function is decreasing if
f(xB) f(xA) for xB > xA
A function is strictly decreasing if
f(xB) < f(xA)
for xB > xA
2.5 Types of functions
Algebraic functions
C = a0
C = a0+ a1Q
C = a0 + a 1 Q + a 2 Q 2
C = a0 + a 1 Q + a 2 Q 2 + a3 Q 3
C = a/Q
C = aQb
(constant)
(linear)
(quadratic)
(cubic)
(hyperbolic)
(power)
Transcendental (Ch. 10)
C = aQ
lnC = ln(a) +bln(Q)
(Exponential)
(logarithmic)
2.5 Digression on exponents
Rules for exponents
Xn = (X*X*X*...*X) n terms
Rule I:
Xm * Xn = Xm+n
Rule II:
Xm
= X m n
Xn
1
Rule III:
Rule IV:
Rule V:
Rule VI:
Rule VII:
X-n = X n
X0 = 1
nx
X1/n =
(Xm)n = Xmn
Xm * Ym = (XY)m
2.6 Functions of two or more
independent variables
Y = f(X)
one ind. var. (2D)
Y = f(X, Z) two
(3D)
Y = f(W,X,Z) three
(4D) hyperplane
etc.
may be linear, quadratic, cubic, etc.
2.7 Levels of generality
Explicit function
Explicit function w/ numeric perimeters
Y = 10 - .5X + X2
concrete, specific
Implicit function
Explicit function w/ abstract perimeters
Y = a bX
Implicit function
Q = f(N,K)
abstract, general
Part 2: Static Analysis
Ch. 3 Equilibrium Analysis in Economics
Ch. 4 Linear Models and Matrix Algebra
Ch. 5 More linear models and matrix
algebra
Ch. 3 Equilibrium Analysis in
Economics
3.1 The meaning of equilibrium
3.2 Partial market equilibrium linear
model
3.3 PME nonlinear model
3.4 General market equilibrium
3.5 Equilibrium in national income
analysis
3.1 The meaning of equilibrium
Qd
a
Qs =-c+dP
(supply)
( P , Q)
Qd = Qs = Q
Slope of
demand curve
Qd = a bP
(demand
P
O
-c
P
Ceterus _ peribus
Equilibrium: a set of selected interrelated variables
within the model adjusted to a state such that there no
inherent tendency to change.
3.2 Solving a model by going from
the general to the specific
Partial equilibrium model of supply & demand
Qd = Qs =Qe equilibrium condition
Qd = 51 - 3P = a - b(P)
Qs = 6P 10 = -c + d(P)
a bP = -c + dP
(a + c) (b+d)P* = 0
(solve for P)
P* = (a+c)/(b+d) = ?
Q* = (ad-bc)/(b+d) = ?
(linear formula)
3.2a Constructing a linear model of S & D
3.2b Solution by elimination of variables
Partial equilibrium model of supply & demand
Qd = Qs =Qe equilibrium condition
Qd = a - b(P)
(a,b > 0)
Demand
Qs = -c + d(P)
(c,d > 0)
Supply
a - bPe = -c + dPe
a+c
= bPe + dPe
a+c
= Pe(b+d)
Pe = (a+c)/(b+d)
Qd = a-bPe = a-b(a+c)/(b+d) = (ad-bc)/(b+d)
3.3a Quadratic equation vs.
quadratic function
Qd = 4 P2
Qs = 4P -1
Qd = Qs
4-P2 = 4P -1
P2 + 4P -5 = 0
How do you solve for P?
Graphically
Factor
Quadratic equation
3.3 eq.(3.6)
f(Pd):=4-Pd^2;
f(Ps):=4*Ps-1;
soln := {P = -5, Q[d] = -21, Q[s] = -21}, {P = 1, Q[d] = 3, Q[s] = 3}
3.3
f(P):= P +4P-5
2
Qd=4-Pd2
Qs=4Ps-1
Qd=Qs
4-P2=4P-1
P2+4P-5=0
(P+5)(P-1)=0
P={-5, 1}
Qd=4-P2={-21,3}
3.3b The quadratic formula by
completing the square
Take ax2 + bx + c = 0, solve for x in terms a, b, c
x2 + bx/a + c/a = 0
x2 + bx/a + b2/4a2 = b2/4a2 - c/a
(x + b/2a)2 = (b2-4ac)/4a2
x + b/2a = (b2-4ac)/2a
x = (-b (b2-4ac))/2a
quadratic formula
Solve the system of eq. for P and set P = 0 by
moving all terms to one side, then apply
quadratic formula.
3.3b Solving a quadratic system of
equations using the quadratic formula
ax2 + bx + c = 0 P2 + 4P -5 = 0
(
b b 4ac
P1 , P2 =
2a
2
)
1
2
4 (16 (4)(1)(5) )
=
2 1
4 (16 + 20 )
P1 , P2 =
2
1/ 2
1
2
46
=
= {1,5}
2
Cubic
formula:
take
ax3+bx2+
cx+d=0
solve for
x in
terms of
a, b, c, d
3.4 Two-commodity market model
Qd 1 Qs1 = 0
Qd 2 Q s 2 = 0
Qd 1 = a 0 + a1 P1 + a 2 P2
Qs1 = b0 + b1 P1 + b2 P2
Qd 2 = 0 + 1 P1 + 2 P2
Qs 2 = 0 + 1 P1 + 2 P2
General
solution for
P1, P2, Q1,
Q2 for the
twocommodity
linear
model case
>
Summary
Elimination of variable method is ok for v.
simple models only, e.g., three linear or
two linear and one quadratic equation
models
Complexity increases geometrically w/
increase in # equations, e.g., six eq.
Linear model
Need a better way to handle at least the nequation linear models -> matrix algebra!