topic 7: partial differentiation - prof. carol newman ... · topic 7: partial differentiation...

Topic 7: Partial DifferentiationReading: Jacques: Chapter 5, Section 5.1-5.2

1. Functions of several variables

2. Partial Differentiation

3. Implicit differentiation3. Implicit differentiation

4. Application I: Elasticity

Application II: Production Functions

Application III: Utility

Functions of several variables

• More realistic in economics to assume an economic variable is a function of a number of different factors:

– Demand may depend on the price of the good and the income level of the consumer

Qd =f(P,Y)

– Output of a firm depends on inputs into the production – Output of a firm depends on inputs into the production process like capital and labour

Q =f(K,L)

Functions of several variables

• Sketching functions of two variables y=f(x,z) :

– Sketch this function in 3-dimensional space or plot relationship between 2 variables for constant values of the third

– For example consider a linear function form:

y =a+bx+czy =a+bx+cz

For different values of z we can represent the relationship between x and y

Functions of several variables

y

z0

z1

x

a+cz0

a+cz1

x0 x1

Functions of several variables

• Consider a non-linear function form: Y=XααααZββββ

0<αααα< 1 & 0<ββββ< 1

(e.g. Production Function)

Y

z =1

z >1

• For different values of z we can represent the relationship between x and y

X x0 x1

Differentiating functions of several variables - Partial Differentiation

• Differentiating functions of one variable y = f(x):

( )xfdx

dy'=

• Differentiating functions of 2 variables y = f(x,z):

Functions of one variable - one first order derivative

xfx

y =∂∂

zfz

y =∂∂

Functions of two variables - two first order derivatives

Differentiate with respect to x holding z constant

Differentiate with respect to z holding x constant

• Examples:

czbxay ++=

Differentiating functions of several variables - Partial Differentiation

325 zxy −=

βα zxy =

Interpretation of partial derivatives

• Functions of one variable:

Change in y as a result of a small change in x:

( ) : is the rate of change of with respect to dy

y f x y xdx

=

dyy x

dx∆ ∆=

• Functions of several variables:

( ),

Holding z fixed, if x changes by a small amount the change in y is given by:

y f x z

yy x

x∆ ∆

=

∂=∂

• Functions of several variables:

If x and z change simultaneouslyy y

y x zx z

∆ ∆ ∆∂ ∂≈ +∂ ∂

This is the small increments formula

• Example:

Evaluate and at the point (1,3)

Small increments formula

3 3y x z z x= −

y

x

∂∂

y

z

∂∂

Hence estimate the change in y when x increases from 1 to

1.1 and z decreases from 3 to 2.8 simultaneously.

x∂ z∂

Implicit Differentiation

Small increments formula:

Since dy = 0 then we can rearrange to find:

y ydy dx dz

x z

∂ ∂= +∂ ∂

x

z

fdz

dx f= −

( ),y f x z c= =

In general if we have the function f(x,y) we can find:

x

y

fdy

dx f= −

Example: Find where dy

dx3 22 5y xy x+ − =

Partial Differentiation

• Second order derivative of a function of 1 variable y=f(x):

( )xfdx

yd''2

2

=

• Second order derivatives of a function of 2 vars y=f(x,z):

fy =∂2

Functions of one variable - one second order derivative

y =∂2

xxfx

y =∂∂

2

zzfz

y =∂∂

2

2

Functions of two variables - four second order derivatives

xzfzx

y =∂∂

∂2

zxfxz

y =∂∂

∂2

Note: In most applications these are equal

• Examples: Find second order partial derivatives

325 zxy −=

Differentiating functions of several variables - Partial Differentiation

xfx

yx 10==

∂∂ 23zf

z

yz −==

∂∂

5 zxy −= xfx x 10==

∂ z z∂

zxy 2= xzfx

yx 2==

∂∂ 2xf

z

yz ==

∂∂

Application I: Elasticityquantity demanded as a function of 1 variable

• Price Elasticity of Demand

Pricein Change alProportion

Demandin Change alProportion=Ed

Q

P

dP

dQEd .=

QdP

• Income Elasticity of Demand

Income in Change alProportion

Demandin Change alProportion=YE

Q

Y

dY

dQEY .=

Application I: Elasticityquantity demanded as a function of several

variables

Own Price Elasticity of Demand

Demandin Change alProportion

( )YPPfQd A,,=

Pricein Change alProportion

Demandin Change alProportion=PE

Q

P

P

QEP .

∂∂=

negative for a downward sloping demand curve

Application I: Elasticityquantity demanded as a function of several

variables

Cross Price Elasticity of Demand

Demandin Change alProportion

( )YPPfQd A,,=

A Good of Pricein Change alProportion

Demandin Change alProportion=PAE

Q

P

P

QE A

APA .

∂∂=

negative for complements, positive for substitutes

Application I: Elasticityquantity demanded as a function of several

variables

Income Elasticity of Demand

Demandin Change alProportion

( )YPPfQd A,,=

Income in Change alProportion

Demandin Change alProportion=YE

Q

Y

Y

QEY .

∂∂=

positive for normal goods, negative for inferior goods

• Example:

Given the demand function:

Application I: Elasticityquantity demanded as a function of several

variables

YPPQd A 1.02100 ++−=find the

(i) price elasticity of demand

(ii) cross-price elasticity of demand

(iii) income elasticity of demand

where:

1000 and 12 ,10 === YPP A

• Marginal Products: The first derivatives of the p.f.

Application II: Production Functions

KMPK

Q =∂∂

The Marginal Product of Capital

LMPL

Q =∂∂ The Marginal Product of Labour

Q = f(K,L)

LMPL

=∂

• Returns to individual inputs: Second-order derivatives

2

2

K

Q

∂∂

2

2

L

Q

∂∂

Negative: Diminishing returns to capitalPositive: Increasing returns to capitalZero: Constant returns to capital

Negative: Diminishing returns to labourPositive: Increasing returns to labourZero: Constant returns to labour

• Elasticity of output with respect to capital

Application II: Production Functions

Q

KMP

Q

K

K

QE KK .. =

∂∂=

Proportional change in output as a result of a proportional change in Capital

• Elasticity of output with respect to labour

Q

LMP

Q

L

L

QE LL .. =

∂∂=

Proportional change in output as a result of a proportional change in Labour

• Marginal Rate of Technical Substitution

Application II: Production Functions

K

L

MP

MP

KQ

LQMRTS =

∂∂∂∂=

The rate at which one input can be substituted for another The rate at which one input can be substituted for another holding output constant

Derivation of MRTS:

Application II: Production Functions

Isoquant: All possible combinations of K and L that produce a constant level of Q (Illustrate)

Figure 5.6

Derivation of MRTS:

Isoquant: All possible combinations of K and L that produce a constant level of Q (Illustrate)

– Downward sloping with slope given by dK/dL

– Rate of substitution between K and L will be

Application II: Production Functions

Using the production function and implicit differentiation an

dK dL−

Q QdQ dL dK

L K

∂ ∂+∂ ∂

≃

Using the production function and implicit differentiation an expression can be found as follows:

Q = f(K,L)

Small increments formula:

dQ = 0 (constant) along isoquant so rearrange:

dK Q L dK Q L

dL Q K dL Q K

∂ ∂ ∂ ∂= − =∂ ∂ ∂ ∂

and -

• Example 1:

Consider the Cobb-Douglas production function:

Application II: Production Functions

5.06.0100 LKQ =

Find an expression for:

(i) the marginal rate of technical substitution

(ii) the elasticity of output with respect to labour

(iii) the elasticity of output with respect to capital

• Example 2:

Show that the function:

Application II: Production Functions

412110 LKQ =is homogenous and comment on the degree of homogeneity.homogeneity.

Comment on

(i) the returns to labour

(ii) the returns to capital

Application III: Utility

• Consumers must make many choices

For example

– Choice between which goods to buy

– Choice between working and leisure time

• The satisfaction that they derive from a particular option we measure in terms of utility

• Suppose there are two goods x1 and x2. The utility function

relates the quantities of these goods to the consumers levels of satisfaction using a utility function:

( )1 2,U f x x=

Application III: Utility

• Marginal utility associated with x1 (x2) gives the change in utility as a result of a one unit change in the quantity of x1

(x2) consumed:

11

UMU

x

∂=∂ 2

2

UMU

x

∂=∂

• When both x1 and x2 change simultaneously we can use the

small increments formula to determine the overall change in utility:utility:

1 21 2

U UU x x

x x∆ ∆ ∆∂ ∂≈ +

∂ ∂

• We would expect the marginal utility to be positive given that we would expect utility to increase with each extra unit of the good consumed

• We would expect the second derivative to be negative given the law of diminishing marginal utility :

2 2

2 21 2

0 0U U

x x

∂ ∂< <∂ ∂

and

Application III: Utility

• Example: Given the utility function:

0.25 0.751 2U x x=

determine the value of the marginal utilities when x1=100 and x2=200.

Estimate the change in utility when x =100 decreases to 99 Estimate the change in utility when x1=100 decreases to 99 and x2 increases to 201.

Does this utility function display diminishing marginal utility?

Indifference curves:

Displays all possible combinations of x1 and x2 that produce a

constant level of Utility (Illustrate)

Application III: Utility

Indifference curves:

Displays all possible combinations of x1 and x2 that produce a

constant level of Utility (Illustrate)

– Downward sloping with slope given by dx2 / dx1

– Rate of substitution between x1 and x2 will be

Application III: Utility

This is known as the Marginal Rate of Commodity Substitution

2 1dx dx−

1 21 2

U UdU dx dx

x x

∂ ∂+∂ ∂≃

This is known as the Marginal Rate of Commodity Substitution

Derivation:

Given the utility function:Small increments formula:

dU = 0 (constant) along indifference curve so rearrange:

2 1 2 1

1 2 1 2

dx Q x dx Q x

dx Q x dx Q x

∂ ∂ ∂ ∂= − =∂ ∂ ∂ ∂

and -

( )1 2,U f x x=

• Example:

Given the utility function:

Application III: Utility

0.5 0.51 2U x x=

find a general expression for MRCS in terms of x1 and x2

Calculate the particular value of MRCS for the indifference curve that passes through (300,500)

Hence estimate the increase in x2 required to maintain the current level of utility when x1 increases by 3 units.