Partial differentiation
(part 1)
Learning objectives
At the end of this lecture, you should be
able to:
• Perform mathematical operations to
find partial differentiations.
• Apply partial differentiations to find
partial elasticities and marginal
functions with two independent
variables.
Function of several
variables
A function of multiple variables
• Most relationships in economics involve more than two variables.
• A function, f, of two variables is a rule that assigns to each incoming
pair of numbers, (𝑥, 𝑦), a uniquely defined outgoing number, 𝑧.
• To evaluate a function two variables, we need to specify the
numerical values of both 𝑥 and 𝑦.
• Example:
• 𝑧 = 𝑓 𝑥, 𝑦 = 𝑥𝑦 + 2𝑦
• Evaluate the function when 𝑥 = 3 and y = 4 gives:
𝑧 = 𝑓(3,4) = 3 ∗ 4 + 2 ∗ 4 = 20
Function of one vs. two independent
variables
Partial derivative
Given the function of two independent variables 𝑧 = 𝑓(𝑥, 𝑦)
The partial derivative of 𝑓 with
respect to 𝑥 is written as:
The partial derivative of 𝑓 with
respect to y is written as:
𝜕𝑧
𝜕𝑓
• 𝑓𝑥 or
or
(read ‘partial df by dx’)
𝜕𝑥
𝜕𝑥
𝜕𝑧
𝜕𝑓
• 𝑓𝑦 or
or
𝜕𝑦
𝜕𝑦
• 𝑓𝑥 is found by differentiating 𝑓
with respect to x, with y held
constant.
• 𝑓𝑦 is found by differentiating 𝑓
with respect to y, with x held
constant.
Exercise
Find expressions for the
first-order partial
derivatives for the function
𝑓 𝑥, 𝑦 = 𝑥 2 𝑦 3 − 10𝑥
Second-order partial derivatives
SECOND-ORDER PARTIAL DERIVATIVES
HOW WE WRITE
Differentiating twice with respect to 𝑥
𝜕2 𝑧
𝜕2 𝑓
𝑓𝑥𝑥 or 2 or 2
𝜕𝑥
𝜕𝑥
Differentiating twice with respect to y
𝜕2 𝑧
𝜕2 𝑓
𝑓𝑦𝑦 or 2 or 2
𝜕𝑦
𝜕𝑦
Differentiating first with respect to x and
then with respect to y
𝜕2 𝑧
𝜕2 𝑓
𝑓𝑦𝑥 or
or
𝜕𝑦𝜕𝑥
𝜕𝑦𝜕𝑥
Differentiating first with respect to y and
then with respect to x
𝜕2 𝑧
𝜕2 𝑓
𝑓𝑥𝑦 or
or
𝜕𝑥𝜕𝑦
𝜕𝑥𝜕𝑦
Exercise
Find expressions for the second-order partial derivatives of the
function:
𝑓 𝑥, 𝑦 = 𝑥 2 𝑦 3 − 10𝑥
Young’s theorem
• If the second partial derivatives of a function are continuous,
differentiating with respect to x then y gives the same
expressions as differentiating with respect to y then x.
• 𝑓𝑦𝑥 = 𝑓𝑥𝑦 𝑜𝑟
𝜕2 𝑓
𝜕2 𝑓
=
𝜕𝑦𝜕𝑥
𝜕𝑥𝜕𝑦
• The theorem can also be applied to functions with more than two
independent variables.
Small increments formulas
DESCRIPTIONS
FORMULAS
𝑥 changes by a small amount ∆𝑥 and 𝑦 is
held fixed
𝜕𝑧
∆𝑧 ≅
∗ ∆𝑥
𝜕𝑥
𝜕𝑧
∆𝑧 ≅
∗ ∆𝑦
𝜕𝑦
𝑦 changes by a small amount ∆y and x is
held fixed
𝑥 changes by a small amount ∆𝑥 and 𝑦
changes by a small amount ∆y
simultaneously
𝜕𝑧
𝜕𝑧
∆𝑧 ≅
∗ ∆𝑥 +
∗ ∆𝑦
𝜕𝑥
𝜕𝑦
𝑥 and 𝑦 both change simultaneously and
∆𝒙 and ∆y tend to zero
𝜕𝑧
𝜕𝑧
𝑑𝑧 =
∗ 𝑑𝑥 +
∗ 𝑑𝑦
𝜕𝑥
𝜕𝑦
𝑑𝑧, 𝑑𝑥, and 𝑑𝑦 are called differentials,
representing the limit of∆𝑧, ∆𝑥, and∆𝑦
Exercises
Given 𝑧 = 𝑥𝑦 − 5𝑥 + 2𝑦
𝜕𝑧
𝜕𝑧
Evaluate and
𝜕𝑥
𝜕𝑦
a. Use the small increments formula to
estimate the change in 𝑧 as 𝑥 decreases from
2 to 1.9 and 𝑦 increases from 6 to 6.1.
b. Confirm your estimate of part (a) by
evaluating 𝑧 at (2, 6) and (1.9, 6.1)
Implicit differentiation
If 𝒇 𝒙, 𝒚 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝒅𝒚
𝒇𝒙
then = −
𝒅𝒙
𝒇𝒚
𝑑𝑦
• Example: Find of the function 𝑓 𝑥, 𝑦 = 𝑦 3 + 2𝑥𝑦 2 − 𝑥
𝑑𝑥
• 𝑓𝑥 = 2𝑦 2 − 1 and 𝑓𝑦 = 3𝑦 2 + 4𝑥𝑦
𝑑𝑦
𝑓𝑥
2𝑦 2 −1
•
=− =− 2
𝑑𝑥
𝑓𝑦
3𝑦 +4𝑥𝑦
Exercises
Use implicit differentiation to
𝑑𝑦
find expressions for given
𝑑𝑥
that:
𝑦 5 − 𝑥𝑦 2 = 10
Review 1
Descriptions
Formulas
𝑥 changes by a small amount ∆𝑥 and 𝑦 is held
fixed
∆𝑧 ≅
𝑦 changes by a small amount ∆y and 𝑥 is held
fixed
𝜕𝑧
∆𝑧 ≅ ∗ ∆𝑦
𝜕𝑦
𝑥 and 𝑦 both change simultaneously
𝜕𝑧
𝜕𝑧
∆𝑧 ≅ ∗ ∆𝑥 + ∗ ∆𝑦
𝜕𝑥
𝜕𝑦
𝑥 and 𝑦 both change simultaneously and ∆𝑥
and ∆y tend to zero
𝑑𝑧 =
Implicit differentiation
If 𝒇 𝒙, 𝒚 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝜕𝑧
∗ ∆𝑥
𝜕𝑥
𝜕𝑧
𝜕𝑧
∗ 𝑑𝑥 + ∗ 𝑑𝑦
𝜕𝑥
𝜕𝑦
𝒅𝒚
𝒇𝒙
then = −
𝒅𝒙
𝒇𝒚
Partial elasticity
and marginal
functions
Elasticity of demand
Suppose that the demand, 𝑄, for a certain good depends on its
price, 𝑃, the price of an alternative good, 𝑃𝐴 , and the income of
consumers 𝑌:
𝑄 = 𝑓(𝑃, 𝑃𝐴 , 𝑌)
for some demand function 𝑓
• Our interest is the responsiveness of demand to changes in any
one of these three variables. This can be measured using
elasticity.
Elasticities formulas
Descriptions
The demand function
The price elasticity of demand (assuming 𝑃𝐴
and 𝑌 are held constant)
The cross-price elasticity demand
Income elasticity of demand
Formulas
𝑄 = 𝑓(𝑃, 𝑃𝐴 , 𝑌)
𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑄 𝑷 𝝏𝑸
𝐸𝑃 =
= ∗
𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑃 𝑸 𝝏𝑷
𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑄 𝑷𝑨 𝝏𝑸
𝐸𝑃𝐴 =
=
∗
𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑃𝐴
𝑸 𝝏𝑷𝑨
EPA < 0: complementary good
EPA > 0: substitutable good
𝐸𝑌 =
𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑄 𝒀 𝝏𝑸
= ∗
𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑌 𝑸 𝝏𝒀
𝐸𝑌 < 0: Inferior good
𝐸𝑌 > 0: Normal good
𝐸𝑌 > 1: Superior good
Substitutable goods vs. complementary goods
Substitutable goods are a
pair of goods that are
alternatives to each other
(Jaques, 2015, p. 80)
Complementary goods are a
pair of goods consumed
together (Jaques, 2015, p.
80)
• Examples: Coke vs. Pepsi;
Grab car vs. Traditional
taxi; etc.
• Examples: printer and
printer cartridges; razors
and razor blades; etc.
Inferior vs. normal good
When income rises
(falls), people
consume less (more)
inferior goods.
When income rises
(falls), people
consume more (less)
normal goods.
Examples: bus
fare, “dumb”
phones, etc.
Examples:
clothes, laptops,
TVs, cars, etc.
Exercises
Given the demand function: 𝑄 = 500 − 3𝑃 − 2𝑃𝐴 +
0.01𝑌
Where 𝑃 = 20, 𝑃𝐴 = 30 and Y = 5000, find:
a. The price elasticity of demand
b. The cross-price elasticity of demand
c. The income elasticity of demand
d. If income rises by 5%, calculate the
corresponding percentage change in demand.
Would this good be classified as inferior, normal
of superior?
Utility
• Utility: the satisfaction gained from consuming a basket of goods
• Suppose that there are two goods, G1 and G2, and that the
consumer buy 𝑥1 items of G1 and 𝑥2 items of G2.
𝑈 = 𝑈(𝑥1 , 𝑥2 )
• Example:
• If 𝑈 3, 7 = 20
and 𝑈 4, 5 = 25
• It means the the consumer derives greater satisfaction from
buying four items of G1 and five items of G2 than from buying
three items of G1 and seven items of G2.
Marginal utility
Marginal utility: The extra satisfaction gained by
consuming 1 extra unit of a good.
The law of diminishing marginal utility:
The increase in utility due to the
consumption of an additional good will
eventually decline.
Marginal utility and Small increments formulas
Descriptions
The rate of change of 𝑈 with respect to 𝑥1 or the
marginal utility of 𝒙𝟏
If 𝑥1 changes by a small amount ∆𝑥1 , and other variables
are held fixed, the change in utility is estimated:
The rate of change of 𝑈 with respect to 𝑥2 or the
marginal utility of 𝒙𝟐
If 𝑥2 changes by a small amount ∆𝑥2 , and other
variables are held fixed, the change in utility is
estimated:
If 𝑥1 and 𝑥2 both change then the net change in 𝑈 is
estimated:
Formulas
𝜕𝑈
𝜕𝑥1
𝜕𝑈
∆𝑈 ≅
∆𝑥1
𝜕𝑥1
𝜕𝑈
𝜕𝑥2
𝜕𝑈
∆𝑈 ≅
∆𝑥2
𝜕𝑥2
𝜕𝑈
𝜕𝑈
∆𝑈 ≅
∆𝑥1 +
∆𝑥2
𝜕𝑥1
𝜕𝑥2
Exercises
An individual’s utility function is given by:
𝑈 = 1000𝑥1 + 450𝑥2 + 5𝑥1 𝑥2 − 2𝑥12 − 𝑥22
Where 𝑥1 is the amount of leisure measured in hours per week and 𝑥2 is
earned income measured in dollars per week.
a. Determine the value of the marginal utilities:
𝜕𝑈
𝜕𝑈
and
when
𝜕𝑥1
𝜕𝑥1
𝑥1 = 138 and 𝑥2 = 500
b. Estimate the change in 𝑈 if the individual works for an extra hour,
which increases earned income by $15 per week.
c. Does the law of diminishing marginal utility hold for this function?
Explain.
Indifference curves
• An indifference curves is defined by:
• 𝑈 𝑥1 , 𝑥2 = 𝑈0
• for some fixed value of 𝑈0
• Point A and B have the same level of
utility (satisfaction level)
• Point C and D have higher level of
utility compared to point A and B
Marginal rate of
commodity substitution
• Marginal rate of commodity
substitution: The increase in 𝑥2
necessary to maintain a constant value
of utility when 𝑥1 decreases by 1 unit.
𝑑𝑥2
∆𝑥2
𝑀𝑅𝐶𝑆 = −
≅−
𝑑𝑥1
∆𝑥1
𝑓𝑥1
𝑑𝑥2
𝜕𝑈/𝜕𝑥1
→ 𝑀𝑅𝐶𝑆 = −
=
=
𝑑𝑥1
𝑓𝑥2
𝜕𝑈/𝜕𝑥2
𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑢𝑡𝑖𝑙𝑖𝑡𝑦 𝑜𝑓𝑥1
∆𝑥2
→ 𝑀𝑅𝐶𝑆 =
≅−
𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑢𝑡𝑖𝑙𝑖𝑡𝑦 𝑜𝑓𝑥2
∆𝑥1
Exercises
Given the utility function: 𝑈 = 1000𝑥1 +
450𝑥2 + 5𝑥1 𝑥2 − 2𝑥12 − 𝑥22
a. Calculate the value of MRCS at the
point (138, 500)
b. Estimate the increase earned in income
required to maintain the current level of
utility if leisure time falls by 2 hours per
week.
Marginal production and Small increments formulas
Descriptions
Output 𝑄, depends on capital, 𝐾, and labor, 𝐿:
Formulas
𝑄 = 𝑓(𝐾, 𝐿)
The marginal product of capital: the rate of change of
output with respect to capital is estimated:
𝜕𝑄
𝜕𝐾
If 𝐾 changes by a small amount ∆𝐾, and other variables
are held fixed, the change in 𝑄 is estimated:
𝜕𝑄
∆𝑄 ≅
∆𝐾
𝜕𝐾
The marginal product of labor: The rate of change of
output with respect to labor
𝜕𝑄
𝜕𝐿
If 𝐾 and 𝐿 both change simultaneously, the net change
in 𝑄 is estimated:
𝜕𝑄
𝜕𝑄
∆𝑄 ≅
∆𝐾 +
∆𝐿
𝜕𝐾
𝜕𝐿
Isoquants and Marginal rate
of technical substitution
• Points on an isoquant represents all possible
combinations of inputs (𝐾, 𝐿) that produce a
constant level of output.
• As capital is reduced, it is necessary to increase
labor to compensate for maintaining the same
production level and vice versa.
• We quantify this exchange of inputs by defining
the marginal rate of technical substitution:
• 𝑀𝑅𝑇𝑆 = −
𝑑𝐾
𝜕𝑄/𝜕𝐿
𝑀𝑃
∆𝐾
=
= 𝐿 ≅−
𝑑𝐿
𝜕𝑄/𝜕𝐾
𝑀𝑃𝐾
∆𝐿
Exercises
Given the production function: 𝑄 =
𝐾 2 + 2𝐿2
a. Write down expressions for the
𝜕𝑄
𝜕𝑄
marginal products and
b.
c.
𝜕𝐾
𝜕𝐿
2𝐿
Show that 𝑀𝑅𝑇𝑆 =
𝐾
𝜕𝑄
𝜕𝑄
Show that 𝐾 + 𝐿 = 2𝑄
𝜕𝐿
𝜕𝐿
Review 2
The marginal utility of 𝒙𝒊 : The rate of change of 𝑈 with
respect to 𝑥𝑖
If 𝑥𝑖 changes by a small amount ∆𝑥𝑖 , and other variables are
held fixed, the change in utility is estimated:
If 𝑥1 and 𝑥2 both change, then the net change in 𝑈 is
estimated:
The marginal product of capital: the rate of change of output
with respect to capital is estimated:
The marginal product of labor: The rate of change of output
with respect to labor
If 𝐾 and 𝐿 both change simultaneously, the net change in 𝑄 is
estimated:
𝜕𝑈
𝜕𝑥𝑖
𝜕𝑈
∆𝑈 ≅
∆𝑥𝑖
𝜕𝑥𝑖
𝜕𝑈
𝜕𝑈
∆𝑈 ≅
∆𝑥1 +
∆𝑥2
𝜕𝑥1
𝜕𝑥2
𝜕𝑄
𝜕𝐾
𝜕𝑄
𝜕𝐿
𝜕𝑄
𝜕𝑄
∆𝑄 ≅
∆𝐾 +
∆𝐿
𝜕𝐾
𝜕𝐿