Math Review for ECON 222 - Queen's Economics...

Math Review for ECON 222Winter Term, 2015

Wenbo Zhu

Queen’s Economic Department

January 13, 2015

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 1 / 37

Outline

1 Exponents

2 Logarithms (Natural logarithms)

3 Growth rates

4 Differentiation

5 Elasticities

6 Examples

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 2 / 37

ExponentsSome graphs

The function: f (x) = ax , a is the “base” and a > 0.

−3 −2 −1 0 1 2 3

05

1015

20

The exponential functions

x

y

●

base = 5base = ebase = 2base = 0.2

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 3 / 37

ExponentsSome basic information about the graph

f (x) = ax has the following properties, regardless the value of a:

the domain of the function is (−∞,∞);

the range of the function is (0,∞);

the function goes through the point (0, 1).

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 4 / 37

ExponentsGeneral Properties of Exponents

an · am = an+m

22 · 23 = 25 = 32(1)

an

am= an−m

23

22= 23−2 = 2

(2)

(an)m = an·m

(22)3 = 22·3 = 64(3)

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 5 / 37

ExponentsQuestion

How about 24 · 42??I first, notice that 2 6= 4, so we cannot apply the equation (1)

above directly ;I however, 4 = 22, so by equation (3), we have

42 = (22)2 = 22·2 = 24;I then by equation (1), 24 · 42 = 24 · 24 = 24+4 = 256.

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 6 / 37

LogarithmsSome graphs

The function: f (x) = loga x , again a is the “base” and a > 0, a 6= 1.

0 2 4 6 8

−4

−3

−2

−1

01

2

The logarithm functions

x

y

●

base = 10base = ebase = 2base = 0.5

*note: in this course, most likely we deal with natural logs, whichmeans the base is “e”.

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 7 / 37

LogarithmsSome basic information about the graph

f (x) = loga x , again regardless the value of a:

the domain of the function is (0,∞);

the range of the function is (−∞,∞);

the function goes through the point (1, 0)

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 8 / 37

LogarithmsThe relation with Exponents: exp ln(a) = a or ln(exp a) = a,∀a > 0

−4 −2 0 2 4

−4

−2

02

4

x

y ●

●

y = e^xy = ln(x)y = x

The exponential and logarithm functions are the inverse of oneanother (i.e. they appear to cancel each other out).

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 9 / 37

LogarithmsUseful Rules for logs

Here are some useful rules for calculation involving logs:

ln(xy) = ln x + ln y (4)

lnx

y= ln x − ln y (5)

ln xp = p ln x (6)

ln 1 = 0 (7)

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 10 / 37

LogarithmsAn example

Please simplify the following expression:

exp[ln(x2)− 2 ln y

]Here is the solution:

exp[ln(x2)− 2 ln y

]= e[ln(x2)−2 ln y]

= e [ln(x2)−ln(y2)]

=e ln(x2)

e ln(y2)

=

(x

y

)2

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 11 / 37

Growth ratesFormulas

Let X and Z be any two variables, not necessarily related by afunction, that are changing over time.

Let ∆XX

and ∆ZZ

represent the growth rates (percentage changes)

I you know that the two-period growth rate of X is Xt+1−Xt

Xt= ∆X

Xt

Rule 1. The growth rate of the product of X and Z equals thegrowth rate of X plus the growth rate of Z .

∆(XZ )

XZ=

∆X

X+

∆Z

Z(8)

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 12 / 37

Growth ratesProof of Rule 1

Proof. Suppose that X increases by ∆X and Z increases by∆Z . Then the absolute increase in the product of X and Z is(X + ∆X )(Z + ∆Z )− XZ , and the growth rate of the productof X and Z is:

∆(XY )

XY=

(Xnew )(Znew )− XZ

XZ

=(X + ∆X )(Z + ∆Z )− XZ

XZ

=XZ − (∆Z )Z + (∆X )X + ∆X∆Z − XZ

XZ

=∆X

X+

∆Z

Z+

∆X∆Z

XZ

The last term can be ignored if both term deviations are small(which they usually are).Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 13 / 37

Growth ratesProof of Rule 1: Graphical

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 14 / 37

Growth ratesRule 2

Rule 2. The growth rate of the ratio of X to Z is the growthrate of X minus the growth rate of Z .

Proof. Let W be the ratio of X to Z , so W = X/Z . ThenX = ZW . By Rule 1, as X equals the product of Z and W , thegrowth rate of X equals the growth rate Z plus the growth rateof W:

∆X

X=

∆Z

Z+

∆W

W∆W

W=

∆X

X− ∆Z

Z

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 15 / 37

Growth rates and logarithms

For small values of x , the following approximation holds :

ln(1 + x) ≈ x

From that property let us show that the growth rate is alsoequal to the difference in logarithms:

Xt+1 − Xt

Xt≈ lnXt+1 − lnXt

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 16 / 37

Growth rates and logarithmsProof

lnXt+1 − lnXt = ln

(Xt+1

Xt

)= ln

(Xt+1 − Xt + Xt

Xt

)= ln

(1 +

Xt+1 − Xt

Xt

)≈ Xt+1 − Xt

Xt

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 17 / 37

DifferentiationSingle-variable Differentiation

Definition:

f ′(x) = limh→0f (x+h)−f (x)

h

There are several ways to denote derivatives

f ′(x) denotes the derivative of f (x) with respect to (wrt) x

f ′x (x , y) denotes the partial derivative of f (x , y) wrt x

you can also denote derivatives with df (x)dx

... and partial derivatives with ∂f (x ,y)∂x

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 18 / 37

DifferentiationRules (with a single function)

Two important derivatives for the course:

f (x) = xa ⇒ f ′(x) = axa−1

f (x) = ln x ⇒ f ′(x) =1

x

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 19 / 37

DifferentiationMore Rules: (with two functions)

Take the derivative wrt each term separately

F (x) = f (x) + g(x) ⇒ F ′(x) = f ′(x) + g ′(x)

F (x) = f (x)− g(x) ⇒ F ′(x) = f ′(x)− g ′(x)

Example:

y = A ⇒ y ′ = 0

y = A + f (x) ⇒ y ′ = f ′(x)

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 20 / 37

DifferentiationMore Rules: (with two functions) Cont’d

Product Rule:

F (x) = f (x)g(x) ⇒ F ′(x) = f ′(x)g(x) + f (x)g ′(x)

Example: y = A · f (x)⇒ y ′ = A · f ′(x)

Quotient Rule:

F (x) =f (x)

g(x)⇒ F ′(x) =

f ′(x) · g(x)− f (x) · g ′(x)

[g(x)]2

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 21 / 37

DifferentiationMore Rules: (with two functions) Cont’d

The Chain Rule: If y is a function of u, and u is a function of x , then

dy

dx=

dy

du

du

dxExample:

y = (ln x)2

dy

dx=

dy

du

du

dx,where y = u2, and u = ln x

= 2 · ln x︸ ︷︷ ︸dy

du

· 1

x︸︷︷︸du

dx

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 22 / 37

DifferentiationMarginal Products with Calculus

Finding the MPK of a Cobb-Douglas production:Y = AKαN1−α

MPK is the change in output for a one unit change in capital(∆Y∆K

when ∆K = 1)

keeping everything else equal.

The MPK is the slope of the tangent line to the productionfunction and is equal to the derivative of the function.

Example: Y = 4K 0.5 Find the MPK when K = 100

MPK =∂Y

∂K= 2K−0.5

=2√100

=2

10=

1

5

Apply the same method to find the marginal product of labour

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 23 / 37

DifferentiationMaximization and Minimization

π(q) = 1000q − cq2, where 1000 > c > 0

to maximize the above objective function, find the partialderivative w.r.t. q and set it equal to zero

π′(q) = 1000− 2cq = 0

q∗ =1000

2c

the procedure for maximization and minimization is the same

a function has been maximized if π′′(q) < 0

why? At the max the slope of the objective function is zero. Anincrease or decrease in q decreases π.

a function has been minimized if π′′(q) > 0

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 24 / 37

DifferentiationSimple Constrained Maximization

π(p, q) = pq − cq where c > 0

subject to p(q) = 1000− 4q

substitute the constraint (p(q)) directly into the π(p, q)objective function and set the partial derivative equal to zero

π(q) = (1000− 4q)q − cq

= 1000q − 4q2 − cq

π′(q) = 1000− c − 8q = 0

q∗ =1000− c

8

the procedure for maximization and minimization is the same

a function has been maximized if π′′(q) < 0

a function has been minimized if π′′(q) > 0

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 25 / 37

ElasticitiesDefinitions

The elasticity of Y with respect to N is defined to be thepercentage change in Y , ∆Y /Y , divided by the percentagechange in N , ∆N/N :

ηY ,N = ∆Y /Y∆N/N

Or if we use derivatives:

ηY ,N = NY∂Y∂N

= ∂ lnY∂ lnN

How so? (Hint: exploit the chain rule):

N

Y

∂Y

∂N=

∂ lnY∂Y

∂Y∂ lnN∂N

∂N=∂ lnY

∂ lnN

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 26 / 37

Example 1

Find the first and second order derivative

f (x) = x2

f ′(x) = 2x

f ′′(x) = 2

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 27 / 37

Example 2

Find the first and second order derivative

f (x) = 3x

f ′(x) = 3

f ′′(x) = 0

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 28 / 37

Example 3

Find the first and second order partial derivative

f (x , y) = yx3

f ′x (x , y) = 3yx2

f ′′x (x , y) = 6yx

*note: you can think of f (x , y) is a production function and y beingthe labour input and x being the capital input.

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 29 / 37

Example 4

Find the first and second order derivative

f (x) = ln(x)

f ′(x) =1

x

f ′′(x) =−1

x2

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 30 / 37

Example 5

f (x) = ex

f ′(x) = ex

why?

y = ex

ln(y) = ln ex

ln(y) = x

1

y

dy

dx= 1

dy

dx= y

dy

dx= ex

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 31 / 37

Example 6

Find the first order derivative

f (x) =ex

x2

f ′(x) =exx2 − ex2x

(x2)2

f ′(x) =ex(x2 − 2x)

x4

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 32 / 37

Example 7

Find the first order derivative

f (x) = ln(x5 − 5x3 − 100

)f ′(x) =

5x4 − 15x2

x5 − 5x3 − 100

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 33 / 37

Example 8

f (x , y) = [αxω + βyω]1ω

∂f (x , y)

∂x=

1

ω· [αxω + βyω]

1ω−1 · αωxω−1

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 34 / 37

Back to growth rate rules (Optional)

Rule 3. Suppose that Y is a variable that is a function of twoother variables X and Z . Then ∆Y

Y= ηY ,X

∆XX

+ ηY ,Z∆ZZ

Proof.: informal. The overall effect on Y is approximately equalto the sum of the individual effects on Y of the change in X andthe change in Z .

Rule 4. The growth rate of X raised to the power a, or X a, is atimes the growth rate of X , growth rate of (X a) = a∆X

X

Proof.: Let Y = X a. Using Rule 3, we find the elasticity of Ywith respect to X equals a.

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 35 / 37

Elasticity of substitution (Optional)

elasticity of substitution is the elasticity of the ratio of twoinputs to a production (or utility) function with respect to theratio of their marginal products (or utilities): σyx = η( y

x ),MRSy,x

We know that the marginal rate of substitution between y and x

is: MRSy ,x =F ′y (x ,y)

F ′x (x ,y)

So if, for example, we want to find σK ,N for the followingCobb-Douglas production function: Y = AKαN1−α

MRSK ,N =MPK

MPN=

AαKα−1N1−α

A(1− α)KαN−α

=α

1− αK

N

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 36 / 37

Elasticity of substitution (Optional) Cont’d

Rearrange the above equation, we get

K

N=

(1− αα

)MRSK ,N .

Then we can apply the formula of σyx and get

σK ,N =MRSK ,N

1−αα

MRSK ,N

· 1− αα

= 1.

Wenbo Zhu (QED) Math Review for ECON 222 January 13, 2015 37 / 37