Math Review

1
Math Review

• A Eco 300
• This review contains some of the math you need
  for this course.However, this material is not
  enough for the math portion of this course, and
  is prepared only to refresh your memory. In
  addition to this review, you should consult a
  standard text book on Differential Calculus.

2
Functional Relations

• Economic concepts are analyzed by functional
  relations.
• e.g. linear, quadratic, exponential etc.
• Example Demand function gt
• Quantity demand is a function of price.
• Q f (P)
• where, P is the price and Q is the quantity.

3
Linear Functional Relation

• General notation Y b m . X
• Y dependent variable
• b intercept
• m slope of the curve
• X independent variable.

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Linear Demand Graph
Y
Y b m.X
m slope
.
b intercept
X
5
Slope of a (linear) function

• The slope shows how steep a line is, and it is
  the first derivative of the curve
• Two things to remember about slope
• its sign.
• its magnitude.

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Slope of a linear function continued
Y

• A negative slope (m lt 0) shows that X and Y are
  inversely related. Or, Y is decreasing in X.
• A positive slope (m gt 0) shows that X and Y
  are positively related. Or, Y is increasing in X.

m lt 0
X
Y
m gt 0
X
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Slope of a linear function continued
Y

• If Slope is zero, m 0, Y is independent of X.
• The magnitude of the slope determines the change
  in Y when X changes by one unit.

X
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Slope of a linear function continued
Y

• Two lines with the same slope must be parallel to
  each other.
• A change in b (the vertical intercept) only
  causes a parallel shift of the curve
• A change in slope, m, causes a non-parallel shift.

m1 m2
m2
m1
X
Y
m1
b2
m1 m2
m2
b1
Y
m1 ? m2
m1
m2
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How to draw a Linear Function
Suppose you have the following function Y 2 -
2X To draw it, find two arbitrary points on the
line. The easiest way is to choose points where
the line meets the axes. First, to find the
vertical intercept put X 0 and compute Y, (Y 2
in this case). Then, put Y 0 (the other
intercept) and compute, X,( X 1in this
case). Since the slope (of a linear function) is
the same, this will always work.
Y
2
1
X
1
10
Calculus Derivatives

• Why a Derivative?
• A derivative tells us about the nature of
  relationship between two variables.
• A derivative can also tell us when a function
  reaches its maximum or minimum.
• A derivative is the slope of a function.

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Derivatives Definition

• Derivative is the change of a dependent variable,
  say Y, due to small changes in the independent
  variable, say X.
• Consider a functional relation Y f (X)
• Derivative of a function f (X) is denoted by
  f(X).
• A function can attain maximum (and minimum) only
  if f(X) 0.

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Derivatives of Polynomials

• Our Typical Problem
• What is the derivative of
• f (X) a0X0 a1X1 a2X2 aNXN ?
• Examples
• f (X) 100 - 5X 10X2 - X3
• f (X) -50 - 15X 8X3 - 5X5
• f (X) 3X - 0.5X2 0.4X5

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Derivatives Rules

• Rule 1 d (a. X N)/ dX N a X N-1
• Examples f (X) 5X gt f (X) 5
• f (X) 4X 3 gt f (X) 12X 2
• f (X) 25 gt f (X) 0
• f (X) 2X 0.5 gt f (X) X -0.5

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Derivatives Rules

• Note The derivative of a constant is always
  zero.
• Examples Y 4, dY/dX 0
• Y 4 4Z dY/dX 0

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Derivatives Rules

• Rule 2 The derivative of sums is the sum
• of the derivatives
• Examples
• f (X) 5X 4X 3 gt f (X) 5 12X 2
• f (X) 25 - 2X 0.5 gt f (X) X -0.5

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Optimization

• Finding the maximum or minimum.
• Examples maximum profit, minimum costs.
• Set f (X) 0 and solve for X.
• Call this X, X. The function takes on a local
  maximum or minimum at X.
• Which is it?
• Second Order Condition (More Derivatives)
• Graph
• Nice functions.

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The Maximum of f(X) Example 1

• Question At what value of X, does f(X) take on
  a maximum when f (X) 50 100X -5X2
• Find f (X). f (X) 100 -10X
• Set f (X) 0. 100 - 10X 0.
• Solve for X. X 10.
• Convince yourself it is a maximum.
• If f (X) lt 0 at X, then Maximum.
• f (X) -10 lt 0 gt Maximum

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Optimization with a constraint

• In some cases we need to find the maximum or
  minimum within a certain range.
• Examples1. What are the maximum amounts of two
  goods a person can buy given that she can afford
  upto a certain limit.
• 2.What are the amounts of Labor and Capital a
  person should use to produce a fixed level of
  output, in order to minimize cost.

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• One of the ways to solve this problem is to use
  Lagrange Method.
• Example Maximize Z f(X,Y) subject to aX cY
  I a,c and I are constants.
• Lagrange Method
• Step 1 gt L f(X,Y) ? (I - aX - cY)
  gt Lagrange Equation
• Step 2 gt FOC-
  i) df(X,Y)/dX -a ?
  ii) df(X,Y)/dY
  -c ?
  iii) df(X,Y)/d ? I - aX - cY
• Step 3 gt Using the above three FOC, solve for X,
  Y and ? .