Partial differentiation

1
Lecture 3

• Partial differentiation

2
Introduction.1

• So far we have focused upon the simplest case of
  functions yf(x), i.e. functions with a single
  independent variable x e.g., y2x23x10
• Clearly, in economics most functional
  relationships involve more than two variables.
  For instance, we previously saw a case where
  output Qf(L). But as we know the stylized
  production function in economics is one involving
  more than one input, e.g. Qf(L,K). Hence, in
  this lecture we shall review how to apply the
  techniques of differential calculus to such
  multivariate functions. We begin by examining the
  topic of partial differentiation.

3
Introduction2

• Let us consider functions of the general form
  zf(x,y), where the dependent variable is z and
  there are two independent variables x and y.
  Examples include z3x2-9y and ze2x3y. Clearly,
  the two dimentional space would not be
  appropriate for graphical representation of such
  functions. We move to 3 and higher dimentional
  spaces.

z
J
y0
z0
x
K
x0
y
4
Introduction3

• While the multidimentionality makes graphical
  treatment of functions with more than one
  independent variable difficult, most problems in
  economics (as we shall see with the examples of
  utility and production functions) allow us to
  assume that one of the variables under
  consideration is constant at a point in time.
• This assumption reduces the graphical
  representation to one in a two-dimentional space.

5
Partial differentiation . 1

• The arithmetic counterpart is the method of
  partial differentiation.
• Consider the following multivariate function with
  n independent variables zf(x1, x2, x3. xn)
• The assumption of independence implies that
  each xi can vary by itself without affecting the
  others. For example, a change in the value of x1,
  ? x1 while x2, x3. xn remain constant (i.e. ?
  x20, ? x30. ? xn0) will produce a
  corresponding change in z (?z).
• If we take the limit of the rate of change
  of z w.r.t x1 (i.e. ?z/ ? x1 )
• as the change in x becomes very small, the
  limit will constitute the partial derivative of z
  with respect to x1. This partial derivative is
  symbolized by fx1 or dz/dx1.

6
Partial derivatives 2

• Techniques of partial differentiation
• To partially differentiate a multivariate
  function we allow only one variable to vary,
  while all others remain independent.
• e.g. in zf(x1, x2) we have to treat x2 as
  constant.

In general, if zf(x1, x2, x3. xn), then the
partial derivative of y w.r.t xi is given by

7
Partial derivatives .. 3

• Find the first-order partial derivatives of

8
Second order partial derivative..1

• We already saw that function zf(x,y) can give
  rise to two first -order partial derivatives fx,
  and fy
• By differentiating fx, and fy w.r.t. x and
  y, we can obtain four second -order partial
  derivatives.

9
Second- order partial derivatives.2

• The partial derivatives fxy and fyx are called
  the cross partial derivatives, because they
  measure the rate of change of the first-order
  partial derivative with respect to the other
  variable.
• Youngs theorem implies that as long as the two
  cross-partial derivatives are both continuous,
  they will be identical fxy fyx

10
Second-order partial derivatives3

• Find the first and second order partial
  derivatives of

11
Total differential..1

• Univariate case
• We know that f(x)dy/dx ?dyf(x).dx
• Example find dy for yx3.
• dy f(x)dx(3x2)dx

12
Total differential..2

• Multivariate case
• Given zf(x,y) the total differential of z
  is given by
• Example find dz for z2x2y

13
Total derivative (differentiation of a function
of a function)1

• Consider a function zf(x, y), where yg(x). x
  exerts a double effect on z
• - Direct via f
• - Indirect via its own effect on y
• x z
• y

g
f
14
Total derivative (differentiation of a function
of a function).2

• Example Find the total derivative dz/dx of
  zf(x, y)xy3, when yg(x)lnx

15
Economic applications 1 The Utility Function

• The neoclassical assumptions w.r.t. the utility
  function are that it is (i) smooth and
  continuous, (ii) there is non-satiation, (iii)
  the function is characterised by a decreasing
  marginal rate of substitution (MRS). All of these
  determine the shape of the indifference curve.

y
z
y1
z1
y0
z0
U200
x0
x1
U100
x
16
Economic applications 1 The Utility Function
cntnd.

• UU(x,y) the indifference curve is defined as
  the locus of combinations of x and y which give a
  constant level of U.
• Total differentiating U we get
  dUUxdxUydy0,
• hence dy/dx -Ux/Uy

y
Slope of tangentdy/dx
x
17
Economic applications 1 The Utility Function
cntnd.

• Note that the marginal utility tells us how much
  our satisfaction increases if we raise the
  consumption of one good keeping the other
  constant, i.e.,
• MU1Ux1?U/ ? x1U(x1? x1, x2)-U(x1, x2)/
  ? x1
• It satisfies the law of diminishing marginal
  utility ?2U/ ? X2lt0, ? 2U/ ? Y2lt0

XX2
U
XX1
Y
18
Example
19
Economic applications 2 The production function

• The case of the production function is analogical
  to that of the utility function. Namely, the
  isoquant is negatively sloped and characterised
  by a decreasing marginal rate of substitution

K
Q
K1
Q1
K0
Q0
Q100
L0
L1
Q60
L
20
Economic applications 2 The production function
cntnd.

• The production function gives us the combinations
  of inputs which gives us a certain level of
  output.
• MP1?Q/ ? x1Q(x1? x1, x2)-Q(x1, x2)/ ? x1
• dQMPLdLMPKdK0
• MRTSdK/dL -MPL/MPK

K
L
21
Economic applications 2 The production function
cntnd.

• The short term production function is
  characterised buy the law of diminishing marginal
  productivity of labour
• Example Find the marginal rate of technical
  substitution and show that the law of diminishing
  marginal productivity holds if the production
  function has the following form Q(L,K)K0.5L0.5

22
Economic application 3 The link between APL and
MPL
Q
Qf(K0,L)
L
MPL,APL
MPL
APL
L
23
Economic application 4 partial elasticity of
demand

• Example Given the demand function for good A
• QA-100-20PA10PB40Y, where QA represents
  the quantity demanded of good A, PA represents
  the price of good A , PB represents the price of
  good B and Y represents the income of the
  consumer.
• (i) Find the own price elasticity of demand
• (ii) Find the cross-price elasticity of
  demand
• (iii) Is good B substitutable or
  complementary to A?
• (iv) Find the income elasticity of demand

24
Solution

• QA-100-20PA10PB40Y

25
More problems

• Example 1 Find the MRS of utility function
  UX1/2Y1/3
• Example 2 Find the MRTS of production function
  Q2LK?L
• Example 3 For production function Q300L2-L4
  find the point at which the APL is maximized and
  comment on its link between MPL.
• Example 4 For demand function QPaY2/P, where
  Pa10, Y2, P4, find the income elasticity of
  demand.