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# ECON 352D1 Lecture Notes - Lecture 1: Partial Derivative, Twill, FaradExam

Department
Economics
Course Code
ECON 352D1
Professor
Study Guide
Final

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Econ 352 2018
1
0. Some mathematical tools
1. Differential equations
Definition.
A differential equation is one that involves an unknown function and its derivatives.
Intuitively, it is an equation that relates the change (or rate of change) in some process to
the process itself and other processes that are changing over time. For instance
( ) ( )
34' 0yt y t t++=
(1)
y(t) is the dependent variable and t the independent variable
The solution to (1) is not a point but rather a function
( )
yt
. In general this type of
equations, in which functions are the solutions, are referred to as functional equations.1
A differential equation is said to be autonomous, if the independent variable only
enters the equation through the dependent variable and not by itself or through other
variables. (1) is not an autonomous differential equation.
Genesis of a Differential Equation. Where do they come from?
Widely used in natural sciences and engineering.
Ex. 1 A population of bacteria growing asexually by self-division (binary fission). Suppose
that at time
t
the population contains
x
individuals and that we observe the population for
a short interval of time,
tÎ”
. Let the instantaneous probability of reproduction (division) be
r
. Then we can calculate the increase in population as
which is a differential equation.
1 Notation: Given a function,
(,)yFxt=
, I will denote its partial derivative with respect to time as
âˆ‚
F x
,
t
( )
âˆ‚
t
=âˆ‚
y
âˆ‚
t
=
Ftx
,
t
( )
=
yt
=ï€¦
F x
,
t
( )
=ï€¦
y
=ï€¦
F
and its partial with respect to any variable
x
as
âˆ‚F x,t
( )
âˆ‚x
=âˆ‚y
âˆ‚x
=Fxx,t
( )
=yx=Fx
. Given this notation, the growth rate of a variable (through time) will be
denoted as
Ë†
F
â‰¡
ï€¦
F
F
. Finally I will be using * to refer to equilibrium/optimal values.

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Econ 352 2018
2
Ex. 2 Many radioactive materials, for instance Carbon-14, decay at a rate proportional to
the amount of it present in other materials. If at time t we have a quantity of Carbon-14,
C(t), then the change in the quantity of Carbon-14 through time is given by,
which is another a differential equation.
Carbon is a component of many materials (wood, bone, hair, pollenâ€¦), carbon is made of 3
isotopes (C12, C13 and C14) which appear in constant proportions in all living organisms.
C14 atoms are always decaying, but as long as the organism is alive, they are always being
replaced by new C14 atoms. Once a living organism dies it stops absorbing new C14
atoms. The radiocarbon dating method measures the C14 concentration of a sample whose
age is not known, and compares it with the concentration of a living organism. Then using
the decay rate of C14 a date for the death of the sample could be estimated, since the
change in proportion of C14 atoms begins when our undated material died.
Ex. 3 A good approximation to evaluate the change in temperature of an object is known as
Newtonâ€™s law of cooling. This law states that the change of temperature of an object
depends on the difference between its temperature and the temperature of the surrounding
environment. We can model this as,
ï€¦
T
(
t
)=âˆ’
ÂµT t
( )
âˆ’
S t
( )
( )
0
Âµ
>
which is another differential equation.
In economics the use of differential equations arises from the explicit modeling of an
additional dimension: time. Most economic processes involve a sequence of actions, where
current decisions affect future opportunities.
Any of you sitting in this classroom, taking this course. This is part of a larger plan
to get a degree in economics. This will probably allow you to get a job in a sector you are
interested in or to proceed into graduate school. As a result, your current choices (and
performance) will have an effect in the opportunities you have in the future and therefore in
A firm maximizes profits by hiring labor,
N
, up to the point that its marginal cost
(the real wage) and its marginal product coincide, this is a static problem. But if the firm is
allowed to modify its productive capacity the problem becomes more complicated because
at any point in time the firm needs to make a decision on how much labor to hire and how
much to invest
( )
( )
It
in capacity
( )
( )
ct
. This will affect the future productive capacity of
the firm and therefore the decision needs to take into account not only current but also
(forecasted) future market conditions. The nature of this problem is dynamic, and the
evolution of the capacity of the firm could be modeled as
( ) ( )
C t rC t=âˆ’
ï€¦

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Econ 352 2018
3
ï€¦
c t
( )
=
I t
( )
âˆ’
Î´c t
( )
where
Î´
is a measure of the depreciation of the capacity of the firm, i.e. the wear and tear
of plant and equipment that results from its use in production.
The solution to this differential equation (given the path of investment and the initial
capacity) will define the whole path of capacity,
( )
ct
, through time.
Solving Differential Equations
When we use calculus we identify the type of problem we are facing, then we apply
a set of standard rules that lead us to the solution.
Problem Ã  Rules Ã  Solution
So if you have a system with two equations on two unknowns (
x
and
y
). You solve the
first equation for
x
and replace the resulting expression in the second equation. As a result
you have a single equation on
y
. You find the value of
y
that satisfies that equation and
you use that value to recover
x
.
Most of the time when we work with differential equations we do not have such a
predetermined set of rules, in fact some of the most interesting differential equations are
still analytically intractable (and need to be solved numerically or characterized
graphically). With differential equations we are going to
Problem Ã  Guess for the solution Ã  Conditions that make the guess a solution
In this context practice and good intuition become crucial. In this class we are going to
solve only very simple differential equations where there is a standard set of rules that is
known to work.
The general form of a first order linear differential equation is
ï€¦
y t
( )
+
u t
( )
y t
( )
=
w t
( )
(2)
we are looking for a function, y(t), that satisfies (2)
CASE I: Constant coefficients
( ) ( )
,ut uwt w==
a) Homogeneous case,
w=0