Ch. 6Comparative Statics and the Concept of Derivative

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Ch. 6 Comparative Statics and the Concept of
Derivative

• 6.1 The Nature of Comparative Statics
• 6.2 Rate of Change and the Derivative
• 6.3 The Derivative and the Slope of a Curve
• 6.4 The Concept of Limit
• 6.5 Digression on Inequalities and Absolute
  Values
• 6.6 Limit Theorems
• 6.7  Continuity and Differentiability of a
  Function

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One Commodity Market Model (2x2 matrix)
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5.6 Application to Market National Income
Models Matrix Inversion (3.5-2, p. 47)


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5.7 Leontief Input-Output Models Structure of an
input-output model
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5.8 Limitations of Static Analysis

• Static analysis solves for the endogenous
  variables for one equilibrium
• Comparative statics show the shifts between
  equilibriums
• Dynamics analysis looks at the attainability and
  stability of the equilibrium

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6.1 The Nature of Comparative Statics

• Comparative statics a study of different
  equilibrium states associated with different sets
  of values of parameters and exogenous variables.
• Begin by assuming an initial equilibrium is given
  
• Examples
• Isolated market model (P0,Q0) (shock)? (P1,Q1)
  
• National income model (Y0, C0) (shock)? (Y1, C1)

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6.1 Shift in Demand
P1
P0
Qd1
Qd0
Q0
Q1
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6.1 Comparative statics

• Static equilibrium analysis
• y f(x)
• Comparative static equilibrium analysis
• y1 - y0 f(x1) - f(x0)
• Where the subscripts 0 and 1 indicate initial
  and subsequent points in time respectively

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6.2 Rate of Change and the DerivativeThe
difference quotient the derivative
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6.1 Comparative statics

• Issues
• Quantitative qualitative of change or
• Magnitude direction
• The rate of change, i.e., the derivative (?Y/ ?G)

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• Macro-economic model(Section 3.5, 1b p. 53)
• Given
• Y C I0 G0
• C a b(Y-T)
• TdtY
• Solving for Y
• Y (a-bd I0 G0)/(1-( b(1-t)))

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Difference Quotient
f(x)
f(x0?x)
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6.2 Difference quotient
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6.2 Difference quotient
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y 3x2 4 (red)?y/?x 6x 3dx x 3,
dx 4, ?y/?x 30 Y 30x 67, secant through
pts (3, f(3), 7, f(7)) (blue)
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6.2 The derivative

• From the previous problem in which y3x2-4
• The difference quotient derivative equal

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Y 3x2 4?Y/?x 6x 3dx x 3, As dx ? 0,
lim ?Y/?x 18Y 18x 31, tangent at point
(3, f(3))
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6.3 The Derivative and the Slope of a Curve

• As the lim of ?x?0, then the f'(x) measures the
  tangent (rise over run) of f(x) at the initial
  point A

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6.4 Concept of limits
Infinitesimals are locations which are not zero,
but which have zero distance from zero.

• The limit (f(x), x?a, direction) function
  attempts to compute the limiting value of f(x) as
  x approaches a from left or right. (eg., N,
  infinity, undefined)
• If q g(v), what value does q approach as v
  approaches N? Answer L
• As v ? N from either side, q ? L. In this case
  both the left-side limit (v less than N) and the
  right side-limit are equal.
• Therefore lim q L

v ? N
v ? N
v?N
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6.4 Evaluation of a limit

• To take a limit, substitute successively smaller
  values that tend to N from both the left and
  right sides since N may not be in the domain of
  the function
• If v is in both the numerator and denominator
  remove it from either depending on the function
• Formal view of the limit concept for a given
  number L, there can always be found a number
  (L-a1) lt L and and other (La2)gtL, where a1 and
  a2 are arbitrary positive numbers. These numbers
  line in the neighborhood of a point on a line.

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6.4 Concept of limits

• If q g(v), what value does q approach as v
  approaches N?
• Answer1 lim q M
• Answer2 lim q M
• In certain cases, only the limit of one side
  needs to be considered. In taking the limit of q
  as v ? ?, for instance, only the left-side
  limit of q is relevant, because v can approach
  ? only from the left.

v ? ?
v ? - ?
v ? -?
V ? ?
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6.4 Concept of a limit

• As v approaches a number N, the limit of qg(v)
  is the number L, if, for every neighborhood of L
  that can be chosen, however small, there can be
  found a corresponding neighborhood of N
  (excluding vN) in the domain of the function
  such that, for every value of v in that
  N-neighborhood, its image lies in the chosen
  L-neighborhood.

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6.4 Concept of a limit

• Given q (2v 5)/(v 1), find the lim q as v ?
  infinity.
• Dividing the numerator by denominator

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6.5 Digression on Inequalities and Absolute
Values Rules of inequalities Absolute values
and inequalities

• Rule I (addition and subtraction)
• a gt b result in a k gt b k
• Rule II (multiplication and division)
• a gt b results in ka gt kb when kgt0
• a gt b results in ka lt kb when klt0
• Rule III (squaring)
• a gt b (b?0) results in a2 gt b2
• /n/ absolute value
• (-n lt /n/ lt n)

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6.5 Digression on Inequalities

• Solve the inequity 1-x lt 3 for x

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6.4 The Concept of LimitLeft-side limit and
right-side limit Graphical illustrations
Evaluation of a limit Formal view of the limit
concept

• Let q??y/?x and v ??x such that q f(v) and

• What value does variable q approach as variable v
  approaches 0?

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6.6 Limit TheoremsTheorems involving a single
functionTheorems involving two functionsLimit
of a polynomial function

• If qavb, then aN b
• If q g(v) b, b
• If q v, then N
• If q vk, then Nk

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6.6 Limit Theorems

• Find lim (1v)/(2 v) as v?0

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6.7 Continuity and Differentiability of a
FunctionContinuity of a function Polynomial
and rational functions Differentiability of a
function

• A continuous function
• When a function qg(v) possesses a limit as v
  tends to the point N in the domain and
• When this limit is also equal to g(N), i.e., the
  value of the function at vN, then the function
  is continuous in N

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6.7 Continuity and Differentiability of a Function

• Requirements for continuity
• N must be in the domain of the function f, qg(v)
• f has a limit as v ? N
• limit equals g(N) in value
• See figure 6.3 p. 134
• function qg(v) includes pt (L, N)not just
  approaches it from left right

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6.7 Continuity and Differentiability of a Function

• Requirements for continuity
• N must be in the domain of the function f, qg(v)
• f has a limit as v ? N
• limit equals g(N) in value
• Check figures in 6.2 p. 130 for continuity
• a b continuous, cd not
• b not differentiable

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6.7 Continuity and Differentiability of a Function

• This rational function is not defined at v 2
  and -2 even though the limit exists as v ? 2 or
  -2. It is discontinuous and therefore does not
  have continuous derivatives, i.e., it is not
  continuous differentiable.

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6.7 Continuity and Differentiability of a Function

• This continuous function is not differentiable at
  x 3 and therefore does not have continuous
  derivatives, i.e., it is not continuously
  differentiable

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6.7 Continuity and differentiability of a function

• For a function to be continuous differentiable
• All points in in domain of f defined
• When the limit concept is applied to the
  difference quotient at x x0 as ?x ? 0 from both
  directions. The continuity condition is necessary
  but not sufficient.
• The differentiability condition (smoothness) is
  both necessary and sufficient for whether f is
  differentiable, i.e., to move from a difference
  quotient to a derivative