School

McGill UniversityDepartment

EconomicsCourse Code

ECON 352D1Professor

Alvarez-CuadradoStudy Guide

FinalThis

**preview**shows pages 1-3. to view the full**15 pages of the document.**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

Î”x t

( )

=rx t

( )

Î”tâ†’Î”x t

( )

Î”t

=rx t

( )

Î”tâ†’0

âŽ¯ â†’âŽ¯âŽ¯ !

x t

( )

=rx t

( )

â†’!

x t

( )

âˆ’rx t

( )

=0

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

your future choices.

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

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