COMPARATIVE STATICS AND THE CONCEPT OF DERIVATIVE

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COMPARATIVE STATICS AND THE CONCEPT OF DERIVATIVE

• Chapter 6
• Alpha Chiang, Fundamental Methods of Mathematical
  Economics
• 3rd edition

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Nature of Comparative Statics

• concerned with the comparison of different
  equilibrium states that are associated with
  different sets of values of parameters and
  exogenous variables.
• we always start by assuming a given initial
  equilibrium state. Then ask how would the new
  equilibrium compare with old.
• can either be qualitative (direction) or
  quantitative (magnitude)
• problem under consideration is essentially one of
  finding rate of change
• concept of derivative takes significance
  differential calculus

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Rate of Change and the Derivative
Difference quotient
Derivative y f(x) 3x2-4

Notation f(x), f
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Skipped Topics

• The Derivative and Slope of the Curve
• The Concept of Limit
• Digression on Inequalities and Absolute Values
• Limit Theorems
• Continuity and Differentiability of a function

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RULES OF DIFFERENTIATION AND THEIR USE IN
COMPARATIVE STATICS

• Chapter 7
• Alpha Chiang, Fundamental Methods of Mathematical
  Economics
• 3rd edition

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Rules of Differentiation for a Function of One
Variable
Constant Function Rule
Power Function Rule
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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Relationship between Marginal Cost and Average
Cost Functions
Book example
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RULES OF DIFFERENTIATION INVOLVING FUNCTIONS OF
DIFFERENT VARIABLES
Chain Rule If we have a function zf(y) where y
is in turn a function of another variable x, say
yg(x) then the derivative of z with respect to x
is equal to the derivative of z with respect to
y, time the derivative of y with respect to x

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Examples of Chain Rule
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Examples of Chain Rule
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Examples of Chain Rule
Example 4 Given a total revenue function of a
firm Rf(Q) where output Q is a function of
labor input L, or Q g(L), derive the marginal
revenue product of labor

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Inverse Function Rule
If a function y f(x) represents a one-to-one
mapping, i.e. if the function is such that a
different value of x will always yield a
different value of y, the function f will have an
inverse function x f-1(y). This means that a
given value of x yields a unique value of y, but
also a given value of y yields a unique value of
x. The function is said to be monotonically
increasing if
Practical way of ascertaining monotonicity if
the derivative f(x) always adheres to the same
algebraic sign. Examples
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PARTIAL DIFFERENTIATION
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PARTIAL DIFFERENTIATION
Techniques of Partial Differentiation Just hold
(n-1) independent variables constant while
allowing one variable to vary.
Example 1
Example 2
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Applications To Comparative-static Analysis
Market Model
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Applications To Comparative-static Analysis
Market Model
Four partial derivatives
Conclusion
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Applications To Comparative-static Analysis
National Income Model
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Applications To Comparative-static Analysis
National Income Model
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