EC2104 Lecture 1

Quantitative Methods for Economic AnalysisEC2104

Lecturer: Chua Yeow Hwee

Lecture 111th August 2015

John Maynard Keynes

• The master-economist must possess a rare combination ofgifts .... He must be mathematician, historian, statesman,philosopher – in some degree. He must understand symbolsand speak in words. He must contemplate the particular, interms of the general, and touch abstract and concrete in thesame flight of thought. He must study the present in the lightof the past for the purposes of the future. No part of man’snature or his institutions must be entirely outside his regard.He must be purposeful and disinterested in a simultaneousmood, as aloof and incorruptible as an artist, yet sometimesas near to earth as a politician.

Outline

Readings: SH, Chapters 4, 5.1-5.4

• Models in Economic Analysis

• Logic

• Functions of one variable

• Common functional forms

• Shifting Graphs

• Composite Functions

• Inverse Functions

Models in Economic Analysis

• Real world is complex.

• An economic model is a simplified analytical framework toanalyze important factors and relationships.

• The structure of a model is described by equations.

• Equations relate a number of variables to model key economicrelationships.

• Working with equations, the aim is to derive a set of logicalconclusions that follow from the assumptions.

• This course is designed to give you the tools to make thoseconclusions.

• An equation is a mathematical statement setting twoalgebraic expressions equal to one another.

• Definitional equations: Sets up an identity between twoalternative expressions that have exactly the same meaning.

π ≡ R − C

• Behavioral equations: Specifies the manner in which a variablebehaves in response to changes in other variables.

C = 75 + 10Q

U =Sleep

HW

• Conditional equations: States a requirement to be satisfied.For example, the standard market model requires theequilibrium condition, which describes the pre-requisite forthere to be an equilibrium in the market:

Quantity demanded = Quantity supplied

Simple Logic

• P is a sufficient condition for Q ( If P, then Q)

• P is a necessary condition for Q ( If Q, then P)

• P is a necessary and sufficient condition for Q ( P if and onlyif Q)

• Example: If x is a square, then it is a rectangle. However, if xis a rectangle, it need not be a square!

Functions of One Variable

• If y depends on the value of x , we might be able to say that yis a function of x , or

y = f (x)

• Here, f is the function. The actual symbol for the function isnot important, but f and g are commonly used for genericfunctions. f (x) is the value that the function generates whengiven input x .

• x is the independent variable or the argument, and y is thedependent variable. Again, the symbols are unimportant(e.g. Q = D(P), Cost = C (Output), R = R(x))

• Every value of x must generate a unique value of y = f (x) forf to be a valid function. However, there can be multiplevalues of x that generate the same y .

Functions of One Variable• Example: f (x) = x2

-1

1

3

5

7

9

-4 -3 -2 -1 0 1 2 3 4

f(x)

x

Domain and Range of a Function

• The domain of function f is the set of values of the argumentx for which the function is defined.

• The range of function f is the set of possible resulting valuesof f (x).

Formal Definition of a Function

A function f of a variable x with domain D is a rule that assigns aunique number f (x) to each x in D.

A Note on Graphing Functions• In economics, we generally use Cartesian coordinates, aka xy

or xyz coordinates.• In most graphs, the independent variable goes on the x-axis

and the dependent variable goes on the y -axis.• In price-quantity graphs in economics, the reverse is true.

Price, the independent variable, is always on the y -axis.• Example: Demand functions. D(p) = 6− 2p

0

1

2

3

4

5

6

7

0 1 2 3 4

Quantity

Price 0

1

2

3

4

0 1 2 3 4 5 6 7

Price

Quantity

Common Functions

Here are some types of functions frequently encountered ineconomics:

• Polynomial function: f (x) = anxn + an−1x

n−1 + ...+ a1x + a0where n is a non-negative integer, and the coefficients ai areconstants

• Linear function: f (x) = ax + b where a and b are constants• Quadratic function: f (x) = ax2 + bx + c where a, b and c are

constants

• Rational function: f (x) = g(x)h(x) where g(x) and h(x) are

polynomials and h(x) 6= 0.

• Power function: f (x) = ax r where a and r are constants, andx > 0

• Exponential function: f (x) = Cax , where C and a are positiveconstants.

• Logarithmic function: f (x) = loga x , where a is a positiveconstant.

Polynomial Functions

General form for a polynomial function:

f (x) = a0 + a1x + a2x2 + ...+ anx

n

where n is a non-negative integer and a0, a1, a2, ..., an areconstants.

• Terminology:• The superscript indicators of the powers of x are called

exponents. The highest power, n, is called the degree of thepolynomial.

• Depending on the degree n, we have the following subclasses:• Constant function (n = 0): f (x) = a0• Linear function (n = 1): f (x) = a0 + a1x• Quadratic function (n = 2): f (x) = a0 + a1x + a2x

2

• Cubic function (n = 3): f (x) = a0 + a1x + a2x2 + a3x

3

Linear Function

We often deal with linear relationships in economics.

y = f (x) = ax + b

• Slope defined as 4y4x .

• Take 2 points that lie on the line: (x1, y1) and (x2, y2).

• Since they lie on the line, these are the same as (x1, ax1 + b)and (x2, ax2 + b).

• Slope is then

(ax2 + b)− (ax1 + b)

x2 − x1=

a(x2 − x1)

x2 − x1= a

• If x = 0, then y = f (0) = a ·0 +b = b, so b is the y -intercept.

• Graph is a straight line with slope a and y -intercept b.

Linear Functions

Here are some useful formulas:

• Point-slope formula of a straight line: The equation of thestraight line passing through (x1, y1) with slope a is:

y − y1 = a(x − x1)

• Point-point formula of a straight line: The equation of thestraight line passing through (x1, y1) and (x2, y2), wherex1 6= x2, is:

y − y1 =y2 − y1x2 − x1

(x − x1)

Example: Market ModelSuppose the market demand and supply are given by the followingbehavioral equations.Demand: qD = 10− 2p, Supply: qS = −5 + 3p

• Market equilibrium requires the following:

qD = qS

10− 2p = −5 + 3p ⇒ p = 3

When p = 3, qD = 4, qS = 4.• Alternatively, we can write the equilibrium condition as:

qD = q = qS

Substituting this into the behavioral equations, we get thefollowing system of two equations:

q = 10− 2p

q = −5 + 3p

We can solve these simultaneously to obtain p = 3, q = 4.

Graphical Solution

Demand: qD = 10− 2p, Supply: qS = −5 + 3p

0

1

2

3

4

5

0 1 2 3 4 5 6 7 8 9 10

Price

Quantity

qD = 10 - 2p

qS = -5 + 3p

Quadratic FunctionsQuadratic functions are polynomial functions of degree 2:

f (x) = a + bx + cx2

Example: Solve 2x2 + x − 6 = 0 (find the roots). Draw the graphof the function y = f (x) = 2x2 + x − 6.

• Solution 1: By factoring:

2x2 + x − 6 = 0

(2x − 3)(x + 2) = 0⇒ x =3

2, x = −2

• Solution 2: Using the quadratic formula. The quadratic

formula states that for ax2 + bx + c = 0, x = −b±√b2−4ac2a as

long as b2− 4ac ≥ 0 and a 6= 0. Plugging in a = 2, b = 1 andc = −6 into the equation, we get:

x =−1±

√12 − 4(2)(−6)

2 · 2⇒ x =

−1±√

49

4

which simplifies to x = 32 , x = −2.

How to graph a quadratic function

• Take the roots of the function. In example, the roots of2x2 + x − 6 are, as we found, x = −2 and x = 3

2 .

• Quadratic functions have parabolas as graphs. Find the vertexof the parabola:

• For a quadratic function in the form ax2 + bx + c , the vertexoccurs at x = − b

2a . In the example, this is x = − 12·2 = − 1

4 .• Plug this x value into the function to find the y -coordinate of

the vertex. Example:

y = 2x2 + x − 6 = 2

(−1

4

)2

+

(−1

4

)− 6 = −6

1

8

Graph of 2x2 + x − 6

-7

-6

-5

-4

-3

-2

-1

0

1

-3 -2 -1 0 1 2 3

y

x

(-2, 0) (3/2, 0)

(-1/4, -6 1/8)

Rational Functions

A rational function is a function R(x) = P(x)Q(x) that can be

expressed as the ratio of two polynomials P(x) and Q(x).

• This function is defined for all x where Q(x) 6= 0.

• The rational function R(x) is proper if the degree of P(x) isless than the degree of Q(x). It is improper if the degree ofP(x) is greater than the degree of Q(x).

Example: One of the simplest types of rational function is

R(x) =ax + b

cx + d(c 6= 0)

The graph of R(x) is a hyperbola.

• When a = d = 0, the function has the form f (x) = kx where

k = b/c . Example: Graph of y = 5x .

Power Functions

A function f is a power function if it is defined by the formula

y = f (x) = ax r for all x > 0

for some fixed constants a and r .Three classes (for a > 0):

• r > 1: function is increasing and convex.

• 0 < r < 1: function is increasing and concave.

• r < 0: function is decreasing and convex.

Graphs of Power Functions (a = 1, r > 1)

0

1

2

3

0 1 2

y

x

x3x4 x2

Graphs of Power Functions (a = 1, 0 < r < 1)

0

1

2

0 1 2 3

y

x

x1/4x1/3

x1/2

Graphs of Power Functions (a = 1, r < 0)

0

1

2

3

4

0 1 2 3

y

x

x-1/2

x-1

x-2

Exponential Functions

A quantity y that increases (or decreases) by a fixed factor when xincreases by 1 is said to increase (or decrease) exponentially. If thefixed factor is a > 0, then y is an exponential function f of x where

y = f (x) = Cax for all x ≥ 0

• To see this, if f (x) = Cax , then

f (x + 1) = Cax+1 = Cax · a1 = af (x)⇒ f (x + 1)

f (x)= a

• If a > 1 then f (x) is increasing and if 0 < a < 1, then f (x) isdecreasing.

• Exponential functions with x = t, or time, are used instudying economic growth, population growth, continuouslyaccumulated interest, radioactive decay, etc.

The Natural Exponential Function

The most common base (a in Cax) of an exponential function isthe irrational number e ≈ 2.71828.

• The corresponding exponential function f (x) = ex is calledthe natural exponential function. This can also be written asf (x) = exp(x).

• Many formulas in calculus become much simpler when e isused as the base for exponential functions.

Graph of y = ex and y = 5ex

0

1

2

3

4

5

6

7

8

9

10

-5 -4 -3 -2 -1 0 1 2 3

y

x

ex5ex

Example: Compound Interest

A savings account of K that increases by p% interest each yearwill have increased after t years to K (1 + p

100)t . Why?

Notice that at t = 1, the amount of money is K (1 + p100). At

t = 2, this becomes (1 + p100)× K (1 + p

100) = K (1 + p100)2. At

time t = T , the total amount of money is K (1 + p100)T .

Logarithmic Functions

• In some cases, we are interested in questions that involvesolving equations of the form ax = b for x . For example:ex = 4 or ax = 2.

• In these equations, the unknown x occurs as an exponent.Therefore, we introduce the logarithm as the inverseoperation of exponentiation.

• If ax = b, take the logarithm of base a on both sides of theequation:

loga ax = loga b

By definition, loga ax ≡ x , so x = loga b.

• The order of the two operations can be changed as well, soaloga x = x

• We also have a special name for loge : the natural logarithm,and we use ln rather than loge (e.g. ln x).

Graph of y = ln x

-2

-1

0

1

2

0 1 2 3 4

y

x

Some Rules of Exponents

Assuming a, b > 0; a, b 6= 1; and x and y are real numbers:

• a0 = 1, a1 = a

• ax+y = axay

• ax−y = ax

ay

• axy = (ax)y

• ax · bx = (ab)x

Some Rules of Logarithms

Assuming a, x , y > 0; n any real number, and a 6= 1:

• loga 1 = 0, loga a = 1, loga ax = x , aloga x = x

• loga xy = loga x + loga y

• logaxy = loga x − loga y

• loga xn = n loga x

• logan√x = 1

n loga x

For the natural logarithmic function ln, just replace loga with lnand a with e in the above equations.

Shifting Graphs

If you know the graph of f (x), you also know the graphs off (x) + c , f (x + c), cf (x), and f (−x).

• If y = f (x) is replaced with y = f (x) + c , the graph is movedup by c units if c > 0 (down if c is negative).

• If y = f (x) is replaced with y = f (x + c), the graph is movedleft by c units if c > 0 (right if c is negative).

• If y = f (x) is replaced with y = cf (x), the graph is stretchedvertically if c > 1 and compressed vertically if 0 < c < 1 (Forc < 0, also reflected about the x-axis).

• If y = f (x) is replaced with y = f (−x), the graph is reflectedabout the y -axis.

Shifting y = f (x) =√x

-2

-1

0

1

2

3

4

5

6

-1 0 1 2 3 4 5 6 7 8 9

f(x)

x

�

� � 2

� � 1

Shifting Graphs







-1

0

1

2

3

-4 -3 -2 -1 0 1 2 3 4 5 6

f(x)

x

�

� � 3

� � 1

Shifting Graphs







-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-1 0 1 2 3 4 5 6 7 8 9

f(x)

x

0.5 �

�

�0.5 �

�2 �

2 �

Shifting Graphs







-1

0

1

2

3

-5 -4 -3 -2 -1 0 1 2 3 4 5

f(x)

x

��

Composite Functions

• Suppose that f and g are two functions such that the rangeof f is contained in the domain of g . We define thecomposite function g ◦ f :

g ◦ f (x) = g(f (x))

• g(f (x)) is not the product of g and f (i.e. g(x) · f (x) = g · f ).

• g(f (x)) 6= f (g(x))

• Example: If f (x) = x2 and g(x) = 2 + 5x , theng ◦ f (x) = 2 + 5x2, and f ◦ g(x) = (2 + 5x)2 = 4 + 20x + 25x2

Inverse Functions

• We usually treat quantity demanded as function of price:Q = D(P). This is the consumer’s perspective.

• Producers choose a quantity to produce, which determines theprice according to the demand function.

• We need the inverse of the demand function to representthis: P = D−1(Q).

• In the original demand function, simply solve for P in terms ofQ.

• By definition, D(D−1(Q)) = Q and D−1(D(P)) = P.

• Note: f −1(x) 6= 1f (x)

Formal Definition of Inverse Function

Let f be a function with domain A and range B. If and only if f isone-to-one, it has an inverse function g with domain B and rangeA.

The function g is given by the following rule:For each y in B, the value g(y) is the unique number x in A suchthat f(x) =y.

g(y) = x iff y = f(x)

Graphing Inverse Functions

To graph f −1(x), reflect the graph of f (x) about the line y = x :

f (x) = 2x−1x+3

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

y

x

y = f(x)

y = f-1(x)

y = x

Graphing Inverse Functions

To graph f −1(x), reflect the graph of f (x) about the line y = x :

f (x) = 3√x + 2

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

y

x

y = x

y = g(x)

y = g-1(x)

EC2104 Lecture 1

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