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EECM 3714

Lecture 10: Unit 10

Returns to scale, homogeneity and partial elasticities
Renshaw, Ch
...
g
...

• Self-check: is 𝑞 =

𝛼𝐾𝛽

+ 1−𝛼

𝐿𝛽

1

ൗ𝛽

homogeneous?

EXAMPLE
Suppose that a firm’s production function is 𝑞 = 10𝐾 0
...
8
...


Is this production function homogeneous? b
...
Does this production function exhibit decreasing returns to capital and labour (i
...

diminishing marginal productivity)?

SOLUTION
• Homogeneity:
• 10 𝜆𝐾

0
...
8

= 10𝜆0
...
3 𝜆0
...
8 = 10𝜆1
...
3 𝐿0
...
1 𝑞

• This function is homogeneous: 𝑟 = 1
...
1 ⟹ 1
...
3 + 0
...
1 > 1 → increasing RTS

• Decreasing returns to labour:
•

𝜕2 𝑞
𝜕𝐿2

= −1
...
3 𝐿−1
...
1𝐾 −1
...
8

• This is negative for all values of K and L 𝐾 ≥ 0, 𝐿 ≥ 0

EXAMPLE 2
Suppose 𝑞 = 10[0
...
5 + 0
...
5 ]0
...

Does this function exhibit increasing, decreasing or constant returns to scale?
Solution:

• First, check if function is homogenous
...
5 𝜆𝐾

0
...
5 𝜆𝐿

0
...
5 =

• ⟹ 10 𝜆0
...
5𝐾 0
...
5𝐿0
...
5 𝜆0
...
5 + 0
...
5 × 𝐿0
...
5
0
...
25 × 0
...
5 + 0
...
5

• Function is homogenous
• Degree of homogeneity is 𝑟 = 0
...
25 < 1, function exhibits decreasing returns to scale

0
...
25 𝑞

PARTIAL ELASTICITIES
• Recall that if 𝑦 = 𝑓(𝑥), then 𝑒 𝑦 =

𝑑𝑦
𝑑𝑥

×

𝑥
𝑦

• Now suppose that 𝑧 = 𝑓(𝑥, 𝑦)
...
r
...
x and y are:
𝜕𝑧

𝑥

𝜕𝑧

𝑦

• 𝑒𝑥𝑧 = 𝜕𝑥 × 𝑧 ; and 𝑒𝑦𝑧 = 𝜕𝑦 × 𝑧

• Note that these two elasticities can also be written as
divided by average (w
...
t
...
e
...
r
...
x)

÷ , i
...
marginal function (w
...
t
...
r
...
y)

• PARTIAL DEMAND ELASTICITIES

• Suppose that demand is 𝑞 𝑑 = 𝑓(𝑝, 𝑝𝑧 , 𝑦), where 𝑝𝑧 , 𝑦 are the price of another product and the
income of consumers
• The other partial demand elasticities are:
• price elasticity of demand, 𝑒 𝑝
• cross-price elasticity of demand, 𝑒 𝑧
• income elasticity of demand, 𝑒 𝑦

• Do examples 17
...
6

PRICE, CROSS PRICE AND INCOME ELASTICITY OF DEMAND
Price elasticity of demand is:

• 𝑒𝑝 =

𝜕𝑞
𝜕𝑝

×

𝑝
𝑞

Income elasticity of demand is:
• 𝑒𝑦 =

𝜕𝑞
𝜕𝑦

×

𝑦
𝑞

Interpretation:

Interpretation:

• If 𝑒 𝑝 > 1 → price elastic demand

• If 𝑒 𝑦 > 0, the good is a normal good

• If 𝑒 𝑝 < 1 → price inelastic demand
• If 𝑒 𝑝 = 1 → unit elastic demand
Cross-price elasticity of demand is:
• 𝑒𝑧 =

𝜕𝑞
𝜕𝑝𝑧

×

𝑝𝑧
𝑞

Interpretation:
• If 𝑒 𝑧 < 0, the two products are complements
• If 𝑒 𝑧 > 0, the two products are substitutes

• If 0 < 𝑒 𝑦 < 1, the good is a necessity
• If 𝑒 𝑦 > 1, the good is a luxury

• If 𝑒 𝑦 < 0, the good is an inferior good

EXAMPLE 1
Suppose that the demand function is 𝑞𝑑 = 1000 − 5𝑝 − 𝑝𝑧2 + 0
...

Find and interpret the
• Price elasticity of demand
• Cross-price elasticity of demand
• Income elasticity of demand

SOLUTION, 1
𝑞 𝑑 = 1000 − 5𝑝 − 𝑝𝑧2 + 0
...

𝑝 = 15; 𝑝𝑧 = 20; 𝑦 = 100
...
005 100
• 𝑒𝑝 =

𝜕𝑞
𝜕𝑝

𝑝
𝑞

× = −5 ×

15
5525

3

= 5525

= −0
...
014 = 0
...
145

• −0
...
015𝑦 2 ×

100
5525

= 0
...
715 > 0 → the product is a normal good;
• 2
...
715

EXAMPLE 2
Suppose 𝑞𝑑 = 10 + 5Τ𝑝 + 2 ln 𝑦 − 2𝑒 0
...
Also suppose that 𝑝 = 2, 𝑦 = 100 and 𝑝𝑧 = 3
...
Find and interpret:
• The price elasticity of demand

• The income elasticity of demand
• The cross-price elasticity of demand

SOLUTION
• 𝑞 𝑑 = 10 + 5Τ𝑝 + 2 ln 𝑦 − 2𝑒 0
...
1×3 = 11
...
2094
𝜕𝑝
𝑞
11
...
8788
• 𝑒 𝑝 = −0
...
2094 < 1 →demand is price

𝑒𝑝

𝜕𝑞 𝑑
𝑦
2
100
2
=
× 𝑑= ×
=
=
𝜕𝑦
𝑞
𝑦
11
...
9394
• 𝑒 𝑦 = 0
...
1675

• 𝑒 𝑦 = 0
...
6𝑒 0
...
1𝑝
𝑧
=
× 𝑑 = −0
...
0678
𝜕𝑝𝑧
𝑞
11
...
9394
𝑒 𝑝𝑧 = −0
...

𝐴𝑃𝐿

This

𝜕𝑞

𝐾

𝑀𝑃𝐾

...
r
...
labour) = 𝑒 𝐿 = × =
𝜕𝐿
𝑞
shows the (percentage) change in production due to a 1% change in labour input
...
r
...
capital) = 𝑒 𝐾 = × =
𝜕𝐾
𝑞
shows the (percentage) change in production due to a 1% change in capital input
...
3 𝐿0
...

a
...


Solution
• 𝑒𝐿 =
• 𝑀𝑃𝐿
𝐿

• 𝑒 =

𝑀𝑃𝐿
𝐴𝑃𝐿
𝜕𝑞
= 𝜕𝐿

𝑀𝑃𝐿
𝐴𝑃𝐿

𝑞

= 8𝐾 0
...
2 and 𝐴𝑃𝐿 = 𝐿 = 10𝐾 0
...
2

=

8𝐾0
...
2
10𝐾0
...
2

= 0
...
8%
...
7 𝐿0
...
7 𝐿0
...
7 𝐿0
...
7𝐿0
...
3

• This means that if capital input increases by 1%, production will increase by 0
...
(note that for CobbDouglas, this is equal to 𝛼)

PARTIAL ELASTICITIES AND LOGS
• Recall from ch

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