EC270: Microeconomic Theory I.

Chapter 2Mathematics for Microeconomics

Dr. Logan McLeod, PhD

Wilfrid Laurier University, School of Business & Economics

September 4, 2014

Functions

Function: describe the relationship between input and outputvariablesI For each input (independent) variable x , a function

assigns a unique number to the output (dependent)variable y according to some rule

y = 2xy = x2

I we may want to indicate y depends x without a specificalgebraic relationship:

y = f (x)

I Frequently, y depends on several variables (x1, x2, . . . , xn)I We write y = f (x1, x2, . . . , xn)

Graphs

Graph: depicts the behaviour of afunction pictorially

I x is usually on the horizontalaxis

I y is depicted on the verticalaxis

However. . .

I in economics, it is commonto graph functions with theindependent variable on thevertical axis and thedependent variable on thehorizontal axis

I e.g. demand functions

Figure: (A.1) Graphs of functions

Properties of Functions

Continuous function: can be drawn without lifting a pencilfrom the paperI there are no jumps in a continuous function

Smooth function: has no kinks or corners

Monotonic function: always increases or always decreasesI a positive monotonic function always increases as x

increasesI a negative monotonic function always decreases as x

increases

Inverse FunctionsRecall:I A function assigns a unique number y for each xI A monotonic function is always increasing or always decreasingI Thus, a monotonic function will have a unique value of x associated

with each value of yInverse function: a function assigns a unique number x to each yI For example, y = 2x :

y = 2x x = y2

I inverse function exists: a unique value of x is associated witheach value of y

I What about y = x2

y = x2 x = y

I inverse function does not exist: not a unique value of x associatedwith each value of y

Equations and IdentitiesEquations asks when a function is equal to some particular number.

Equation Solution2x = 8 x = 4x2 = 9 x = 3 or x = 3

f (x) = 0 x = x

Solution: a value of x satisfying the equation

Identity: a relationship between variables that holds for all values ofthe variables

(x + y)2 x2 + 2xy + y22(x + 1) 2x + 2

I The symbol means that the left-hand side and the right-handside are equal for all values of the variables

Linear Functions

Linear function: a function of the form y = mx + bI where m and b are constants

They can also be expressed implicitly: ax + cy = dI we would often solve for y as a function of x , to convert

this to the standard form

y =dc a

cx

Changes and Rates of Change

The change in x: x

I A change in x from x1 to x2 is x = x2 x1

Marginal change: very small changes in x

I we are generally interested in only very small changes in x

Rate of change: the ratio of two changes

I assume y = f (x)

I then the rate of change of y with respect to x is

yx

=f (x + x) f (x)

x

Changes and Rates of Change

yx

=f (x + x) f (x)

xNote:

I for a linear function (y = mx + b) has a constant rate of changeof y with respect to x :

yx

=[b + m(x + x)] [b + mx ]

x=

mxx

= m

I for a non-linear function, the rate of change will depend on thevalue of x . For example y = x2:

yx

=[(x + x)2] [x2]

x=

x2 + 2xx + (x)2 x2x

= 2x +x

Slopes and Intercepts

Slope of the function is the rate of change of y as x changesI we commonly interpret the rate of change of a function

graphically as the slope of the function

For example:

y = 53

x + 5

I vertical intercept: thevalue of y when x = 0(which is y = 5)

I horizontal intercept:the value of x wheny = 0 (which is x = 3)

I slope is 53 40 1 2 3

6

0

1

2

3

4

5

X Axis

Y A

xis

Slopes and Intecepts

if a linear function has the standard form y = mx + b, then

vertical intercept bhorizontal intercept bmslope m

if a linear function has the form: ax1 + cx2 = d , then

vertical intercept dchorizontal intercept daslope ac

Slopes and InterceptsA nonlinear function has the property that its slope changes as x changes

I A tangent to a function at some point x is a linear function that has thesame slope

Example:I If y whenever x , then

y will always have thesame sign as x (slopewill be positive)

I If y () wheneverx (), then y and xwill have opposite signs(slope will be negative)

Figure: (A.2B) slope of y = x2 at x = 1

Absolute Values and LogarithmsAbsolute value of a number |x | is a function f (x) defined by:

f (x) =

(x if x 0x if x < 0

Logarithm (or log) of x describes an inverse function of the exponentialfunction f (x) = ax : y = log x or y = log(x)I a logarithm of base a is an inverse function of: f (x) = ax

I the exponent to which base a must be raised to give x .I if f (x) = ax , the loga(x) is the exponent to which base a must be raised

to give xI the natural log of x is ln(x), which has base e (i.e., ex )

Properties of the logarithm:

ln(xy) = ln(x) + ln(y) for all positive numbers x and y

ln

xy

= ln(x) ln(y) for all positive numbers x and y

ln(e) = 1 e = 2.7183 . . .

ln(xy ) = y ln(x)

Derivatives

Derivatives

Derivative is the limit of the rate of change of y with respect tox as the change in x goes to zeroI gives precise meaning to the phrase the rate of change of

y with respect to x for small changes in xI for y = f (x), the derivative (f (x)) is:

df (x)dx

= limx0

f (x + x) f (x)x

Derivatives of Linear FunctionsI Recall, the rate of change of y = mx + b is a constant (m)I Thus, if f (x) = mx + b then

df (x)dx

= m

DerivativesDerivatives of Non-Linear FunctionsI recall the rate of change of y with respect to x will usually depend on xI for example: y = x2 y

x = 2x + xI the derivative of y with respect to x will be a function of x :

df (x)dx

= limx0

2x + x = 2x

Useful derivatives to know:

Family f (x) df (x)dxConstant c 0

Power xc cxc1

Exponential ex ex

Logarithmic ln x 1x

The Power Rule

Assume: f (x) = cx, where c and are constantsI then

df (x)dx

= cx1

I multiply x by its exponent and subtract one from theexponent you began with to find the derivative.

Example: if f (x) = x3, what is df (x)dx ?

The Product Rule

Assume: g(x) and h(x) are both functions of xI define f (x) = g(x)h(x) (i.e., the product of two functions)I then

df (x)dx

= g(x)dh(x)

dx+ h(x)

dg(x)dx

I the first times the derivative of the second, plus thesecond times the derivative of the first

Example: if g(x) = x2 and h(x) = x2 + 3, what is df (x)dx ?

The Chain RuleComposite function:I given two functions y = g(x) and z = h(y)I the composite function is f (x) = h(g(x))

Chain Rule: the derivative of a composite function, f (x), with respect to x isdf (x)

dx=

dh(y)dy

dg(x)dx

Example:

AssumeThen

g(x) = x2 = y

h(y) = 2y + 3

f (x) = 2x2 + 3

dg(x)dx

= 2x

dh(y)dy

= 2

df (x)dx

= 2 2x = 4x

Second DerivativesSecond derivative of a function:I the derivative of the derivative of that function

if y = f (x)

then the 1st derivative isdf (x)

dxor f (x)

then the 2nd derivative isd2f (x)

dx2or f (x)

I measures the curvature of a function

2nd derivative implies f (x) isf (x) < 0 concave near that point (slope is decreasing)f (x) > 0 convex near that point (slope is increasing)f (x) = 0 flat near that point (possible inflection point)

Partial Derivatives

Assume z = f (x , y)

Partial derivative of f (x , y) with respect to x is just thederivative of the function with respect to x , holding y fixed:

f (x , y)x

= limx0

f (x + x , y) f (x , y)x

Similarly, the partial derivative with respect to x2

f (x , y)y

= limy0

f (x , y + y) f (x , y)y

Partial derivatives have exactly the same properties as ordinaryderivatives

Example: Partial Derivatives

Assume a Cobb-Douglas function:

f (x , y) = xy1

Partial Derivative, with respect to xI Use the power rule to get f (x,y)

x :

f (x , y)x

= x1y1 = y

x

1

Second Partial Derivative, with respect to x

2f (x , y)x2

= ( 1)x2y1

Cross Partial Derivative, of f (x,y)x with respect to y

2f (x , y)xy

= (1 )x1y

Youngs Theorem

Youngs Theorem: the order in which partial differentiation isconducted to evaluate second-order partial derivatives does notmatter.

fij = fji

for any pair of variables xi , xj .

Example: f (x1, x2) = x21 x32

f1 = 2x1x32 f2 = 3x21 x

22

f12 = 6x1x22 f21 = 6x1x22

Total Differentiation

Totally differentiating a function:I Tells us the total change in a function from a combined

change in x and y .I Describes movement along a curve.

df (x , y) =f (x , y)x

dx +f (x , y)y

dy

Interpretation of Total Differentiation:

I The partial derivatives(f (x ,y)x and

f (x ,y)y

)indicate the

rate of change in the x and x directionsI dx and dy are the changes in x and y

Optimization

Optimization - One Variable

Optimization refers to the process of finding the largest value(maximum) or the smallest value (minimum) a function(y = f (x)) can take.

Maximum: f (x) achieves a maximum at x iff (x) f (x) for all xI It can be shown that if f (x) is a smooth function that

achieves its maximum value at x, then:

First-order condition:df (x)

dx= 0

(the slope of f (x) is flat at x )

Second-order condition:d2f (x)

dx2 0

(f (x) is concave near x)

Optimization - One Variable

Minimum: f (x) achieves a minimum at x if f (x) f (x) for all xI It can be shown that if f (x) is a smooth function that achieves its

maximum value at x, then:

First-order condition:df (x)

dx= 0

Second-order condition:d2f (x)

dx2 0

(f (x) is convex near x)

Optimization - Multiple Variables

Multivariate Case: if y = f (x1, x2) is a smooth function that achievesa maximum or minimum at some point (x1 , x

2 ), then we must satisfy:

f (x1 , x2 )

x1 0

f (x1 , x2 )

x2 0

I These are referred to as the first-order conditions.

The Envelope TheoremEnvelope Theorem: concerns how the optimal value for a particularfunction changes when a parameter of the function changesI provides a shortcut to calculate the effect of changing a

parameter on the value of a functionI e.g., the effects of changing the market price of a commodity will

have on an individuals purchases

Example: y = x2 + axI Function represents an inverted parabolaI Optimal values of x (x) depend on the parameter a

Value of a Value of x Value of y

0 0 01 12

14

2 1 13 32

94

4 2 45 52

254

6 3 9

The Envelope Theorem - Direct Approach

y = x2 + axFirst: Calculate the slope

dydx

= 2x + a = 0which implies:

x =a2

Second: substitute x into the original function

y = (x)2 + ax

= (a

2

)2+ a

(a2

)=

a2

4

The Envelope Theorem - ShortcutEnvelope Theorem states: for small changes in a, dyda can becomputed by holding x constant at its optimal value

(x = a2

)and

simply calculating ya from the objective function directly

First:ya

= x

Second: evaluate at x:ya

x=a/2

=a2

Note: this is the same results obtained earlier

Envelope Theorem: states the change in the optimal value of afunction (with respect to a parameter) can be found by partiallydifferentiating the objective function while holding x constant at itsoptimal value

dy

da=ya{x = x(a)}

Constrained Optimization

Constrained Optimization

Constrained Optimization: finding a maximum or minimum ofsome function over a restricted values for (x1, x2)I The notation:

maxx1,x2

f (x1, x2)

such that g(x1, x2) = c.

I objective function: f (x1, x2)I constraint function: g(x1, x2) = cI Interpretation:

I find x1 and x2 such that f (x

1 , x2 ) f (x1, x2) for all values of

x1 and x2 that satisfy the equation g(x1, x2) = c.

Solving Constrained Optimization

Assume a linear constraint function:

g(x1, x2) = p1x1 + p2x2 = c

Constrained Maximization Problem

maxx1,x2

f (x1, x2)

such that p1x1 + p2x2 = c.

There are two ways to solve this problem:1. Direct Substitution2. Lagrange Method

Direct Substitution

1. Solve the constraint for one variable:

x2(x1) =cp2 p1

p2x1

2. Substitute x2(x1) into the objective function:

f (x1,cp2 p1

p2x1)

3. Solve the unconstrained maximization problem:

maxx1

f (x1,cp2 p1

p2x1)

(F .O.C.)f (x1, x2(x1))

x1+f (x1, x2(x1))

x2x2x1

= 0

Lagrange Method

Solves the constrained maximization problem by using Lagrangemultipliers.

I define an auxiliary function known as the Lagrangian:

L = f (x1, x2) (p1x1 + p2x2 c)

I the new variable () is the Lagrange multiplier

The Lagrange method says that an optimal choice (x1 , x2 ) must

satisfy three first-order conditions

Lx1

=f (x1 , x

2 )

x1 p1 = 0 (1)

Lx2

=f (x1 , x

2 )

x2 p2 = 0 (2)

L

= p1x1 + p2x2 c = 0 (3)

EC270: Microeconomic Theory I.

Chapter 2Mathematics for Microeconomics

Dr. Logan McLeod, PhD

Wilfrid Laurier University, School of Business & Economics

September 4, 2014