(2.2) Production Function

Question 1

What 2 assumptions do we make in the production function?

Answer 1

• Simple, closed economy
• One consumption good

Question 2

What is capital another way of saying?

Answer 2

Machinery

Question 3

2

What are the inputs of the production function?

Answer 3

• L̄ - Labour
• K̄ - Capital

Question 4

Write the equation for the production function

Answer 4

Y = F(K, L) = Ā K¹ᐟ³ L²ᐟ³

note: unless stated, just use α

Question 5

4

What does Ā represent in the production function?

Answer 5

• Productivity parameter (TFP/tech/Solow residual)
• Effectiveness at combining inputs - ability to produce goods with given resource stocks (K, L)
• Represents unexplained part of production function
• Higher value indicates firm produces more output of good, other things being equal

Question 6

What type of function is the production function?

Answer 6

Cobb-Douglas

Question 7

Given a typical CB production function (i.e. 𝛼 and ‘1-𝛼’ add to 1), what other facts can we say?

Answer 7

• MPK and MPL are both increasing, but dimishing (first derivative > 0)
• MPK and MPL are both convex to origin (second derivative < 0)
• CRS

Question 8

photo

Draw a graph (for A) Production function; B) MPK) when we fix L and vary K

Answer 8

Question 9

In the production function, capital and labour are complements. What can we draw from this?

Answer 9

The marginal product of one factor increases when the other factor increases

Question 10

photo

‘The marginal product of one factor increases when the other factor increases’. Draw a graph for this

Answer 10

Question 11

maths

Prove that MPK is positive

Answer 11

𝜕𝐹/𝜕𝐾
= α𝐾^(𝛼−1) 𝐿^(1−𝛼)
= α 𝐾^(𝛼−1)/𝐿^(𝛼−1)
= α(𝐾/𝐿)^(𝛼−1)>0

Question 12

maths

Prove that MPL is positive

Answer 12

𝜕𝐹/𝜕𝐿
= (1−α)𝐾^𝛼 𝐿^(1−𝛼−1)
= (1−α)𝐾^𝛼 𝐿^(−𝛼)
= (1−α) 𝐾^𝛼/𝐿^𝛼 =(1−𝛼) (𝐾/𝐿)^𝛼>0

Question 13

Describe CRS

Answer 13

• doubling each input exactly doubles output
• F(𝛼K, 𝛼L) = 𝛼F(K,L)
• homogenous function

Question 14

On this graph, what do the tangent lines represent?

Answer 14

gradients represent MPK at a specific K

Question 15

Here is a production function, is it IRS, CRS or DRS? Y = K¹ᐟ³ L²ᐟ³ + B

Answer 15

• Double inputs less than doubles output
• In large K/L, tends to CRS
• Solution: DRS

Question 16

Here is a production function, is it IRS, CRS or DRS? Y = K¹ᐟ³ L²ᐟ³ - B

Answer 16

• Double inputs more than doubles output
• In large K/L, tends to CRS
• Solution: IRS

Question 17

How do you prove IRS/CRS/DRS?

Answer 17

• Double inputs
• Double outputs
• compare

Question 18

How is the wage rate determined?

Answer 18

MPL

Question 19

8 + photo

Assume a production function
Yᵢ = Aᵢ Kᵢ^α Lᵢ^(1−α)
Assume that country A has a higher TFP
Explain the impact on labour

Answer 19

• Compute wage rate = MPL (see photo below)
• MPL higher in Country A
• Workers migrate to Country A in search of higher wages
• Labour input increases in A; declines in B
• via changes in MPL
• As L increases, MPL decreases (and vice versa) -> MPLs become equal
• Process stops when wages are equalised
• No incentive to migrate anymore

Question 20

As a labour input relatively increases for one country, what happens to GDP?

Answer 20

It increases

Question 21

2

As a labour input relatively increases for one country, what happens to capital?

Answer 21

• Total capital stock unchanged
• But capital per worker decreases for increase in labour input (and vice versa)

Question 22

1 + 1(2) + 1(2)

Assume a production function
Yᵢ = Aᵢ Kᵢ^α Lᵢ^(1−α)
Assume that country A has a higher TFP
You know the impact on wages/labour

What happens if A imposes a limit on migration numbers?

Answer 22

• Assume that number of workers that need to move from B to A - until wages equalised - is X
• If cap larger than X
  
  • migration stops when wages equalised
  • maximum number of migrants lower than cap
• If cap smaller than X
  
  • migration does not stop until wages equalise
  • wages remain higher in A

Question 23

1

What prevents wages being equalised between poor and rich countries in the real world?

Answer 23

Tight restrictions to immigration

Question 24

What is the other name for the production function in ‘output per person’

Answer 24

intensive form

Question 25

In the production function, number of workers is equal to...

Answer 25

the number of people

Question 26

Write the output of the production function (intensive form)

Answer 26

y = Y/L = F(K/L, 1) = f(k)

Question 27

# photo Construct the production function in per capita terms (intensive form). Why can we write it like this?

Answer 27

Question 28

# photo What does f(k) look like graphically?

Answer 28

Question 29

# 1 What allows us to write the production function per capita form as Y/L = F(K/L, 1) = f(k)?

Answer 29

CRS | state this at start of answer

Question 30

# 1 When we write something in lower case, what does it denote?

Answer 30

per capita (per worker)

Question 31

# 1 When we put a bar above A, K or L, what does it signal?

Answer 31

Entire stock