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Ch. 6Comparative Statics and the Concept of Derivative

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Title: Ch. 6Comparative Statics and the Concept of Derivative


1
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

2
One Commodity Market Model (2x2 matrix)
3
5.6 Application to Market National Income
Models Matrix Inversion (3.5-2, p. 47)


4
5.7 Leontief Input-Output Models Structure of an
input-output model
5
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

6
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)

7
6.1 Shift in Demand
P1
P0
Qd1
Qd0
Q0
Q1
8
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

9
6.2 Rate of Change and the DerivativeThe
difference quotient the derivative
10
6.1 Comparative statics
  • Issues
  • Quantitative qualitative of change or
  • Magnitude direction
  • The rate of change, i.e., the derivative (?Y/ ?G)

11
  • 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)))

12
(No Transcript)
13
Difference Quotient
f(x)
f(x0?x)
14
6.2 Difference quotient
15
6.2 Difference quotient
16
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)
17
6.2 The derivative
  • From the previous problem in which y3x2-4
  • The difference quotient derivative equal

18
Y 3x2 4?Y/?x 6x 3dx x 3, As dx ? 0,
lim ?Y/?x 18Y 18x 31, tangent at point
(3, f(3))
19
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

20
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
21
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.

22
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 ? ?
23
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.

24
6.4 Concept of a limit
  • Given q (2v 5)/(v 1), find the lim q as v ?
    infinity.
  • Dividing the numerator by denominator

25
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)

26
6.5 Digression on Inequalities
  • Solve the inequity 1-x lt 3 for x

27
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?

28
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

29
6.6 Limit Theorems
  • Find lim (1v)/(2 v) as v?0

30
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

31
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

32
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

33
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.

34
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

35
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
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