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