Notes-Chapter 6-7

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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 z=f(y) where y is in turn a function of another variable x, say y=g(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 R=f(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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