Mathematics for Microeconomics

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Mathematics for Microeconomics

• Economics 102
• 2nd Sem, 2009-2010
• Tirso Paris, UPLB

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Derivatives
y
Derivative Slope at a point a
b
?y
a
?x
x
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Rate of Change and the Derivative
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Function of One Variable
Constant Function Rule
Power Function Rule
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Function of One Variable
Generalized Power Function Rule
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Rules Of Differentiation Involving Two Or More
Functions Of The Same Variable
Sum-Difference Rule
Product Rule
Quotient Rule
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Product Rule
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Quotient Rule
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Chain Rule
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Derivatives of logarithm and exponentials
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Partial Derivatives
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Functions of Several Variables

• Most goals of economic agents depend on several
  variables
• trade-offs must be made
• The dependence of one variable (y) on a series of
  other variables (x1,x2,,xn) is denoted by

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Partial Derivatives

• The partial derivative of y with respect to x1 is
  denoted by

• It is understood that in calculating the partial
  derivative, all of the other xs are held constant

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Partial Derivatives

• A more formal definition of the partial
  derivative is

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Partial derivatives
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Partial derivatives
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Calculating Partial Derivatives
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Calculating Partial Derivatives
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Partial Derivatives

• Partial derivatives are the mathematical
  expression of the ceteris paribus assumption
• show how changes in one variable affect some
  outcome when other influences are held constant

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Partial Derivatives

• We must be concerned with how variables are
  measured
• if q represents the quantity of gasoline demanded
  (measured in billions of gallons) and p
  represents the price in dollars per gallon, then
  ?q/?p will measure the change in demand (in
  billiions of gallons per year) for a dollar per
  gallon change in price

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Differentials
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Total Differential
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Elasticity of Demand

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Elasticity and logarithms
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The Mathematics of Optimization

• Many economic theories begin with the assumption
  that an economic agent is seeking to find the
  optimal value of some function
• consumers seek to maximize utility
• firms seek to maximize profit
• This chapter introduces the mathematics common to
  these problems

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Optimum Values and Extreme Values

• Economics is by and large a science of choice.
• common criterion of choice among alternative in
  economic is the goal of maximizing something or
  minimizing something
• optimization means quest for the best
• maximum and minimum as mathematical concepts
  collective term is extremum.
• Optimization
• delineate an objective function
• independent variables are choice variables
• eg. p is the object of maximization and Q is the
  choice variable.

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Maximization of a Function of One Variable

• Simple example Manager of a firm wishes to
  maximize profits

?
Maximum profits of ? occur at q
? f(q)
Quantity
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Maximization of a Function of One Variable

• The manager will likely try to vary q to see
  where the maximum profit occurs
• an increase from q1 to q2 leads to a rise in ?

?
?
? f(q)
?1
Quantity
q
q1
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Maximization of a Function of One Variable

• If output is increased beyond q, profit will
  decline
• an increase from q to q3 leads to a drop in ?

?
?
? f(q)
Quantity
q
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Derivatives

• The derivative of ? f(q) is the limit of ??/?q
  for very small changes in q

• The value of this ratio depends on the value of q1

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Value of a Derivative at a Point

• The evaluation of the derivative at the point q
  q1 can be denoted

• In our previous example,

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First Order Condition for a Maximum

• For a function of one variable to attain its
  maximum value at some point, the derivative at
  that point must be zero

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Second Order Conditions

• The first order condition (d?/dq) is a necessary
  condition for a maximum, but it is not a
  sufficient condition

?
If the profit function was u-shaped, the first
order condition would result in q being chosen
and ? would be minimized
Quantity
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Second Order Conditions

• This must mean that, in order for q to be the
  optimum,

and

• Therefore, at q, d?/dq must be decreasing

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Second Derivatives

• The derivative of a derivative is called a second
  derivative
• The second derivative can be denoted by

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Second Order Condition

• The second order condition to represent a (local)
  maximum is

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Relative Maximum and Minimum Necessary
condition

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Second Derivative Test
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Example
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A Problem On Profit Maximization
.
5 pts each