Solow Growth Model

Question 1

Capital Accumulation Equation

Answer 1

𝐾𝑡+1 = 𝐾𝑡 + 𝐼𝑡 − 𝑑𝐾𝑡
Next year’s capital = this year’s capital + investment - the depreciation rate

Question 2

Change in Capital Stock Equation

Answer 2

∆𝐾𝑡+1≡ 𝐾𝑡+1 − 𝐾𝑡 (—>)* ∆𝐾𝑡+1= 𝐼𝑡 − 𝑑𝐾𝑡

*When Capital Accumulation Equation substituted into 𝐾𝑡

Question 3

𝐾𝑡+1 = 𝐾𝑡 + 𝐼𝑡 − 𝑑𝐾𝑡

Answer 3

The Capital Accumulation Equation

Question 4

What is ∆𝐾𝑡+1= 𝐼𝑡 − 𝑑𝐾𝑡

Answer 4

The Change in Capital Stock Equation

Question 5

Gross Investment and Consumption Equations (In terms of Output)

Answer 5

𝐼𝑡 = 𝑠𝑌𝑡
𝐶𝑡 = (1 − 𝑠)𝑌𝑡

Agents consume some fraction of output and invest the rest

Question 6

𝐼𝑡 = 𝑠𝑌𝑡

Answer 6

Gross Investment Equation (In terms of Output)

Question 7

𝐶𝑡 = (1 − 𝑠)𝑌𝑡

Answer 7

Gross Consumption Equation

Question 8

Labour Equation

Answer 8

𝐿𝑡 = 𝐿

The amount of labour in the economy is given exogenously at a constant level

Question 9

Cobb-Douglas Production Function (L exogenous)

Answer 9

𝑌𝑡 [= 𝐹(𝐾𝑡,𝐿)] = A𝐾𝑡^1/3𝐿^2/3

Question 10

𝑌𝑡 = A𝐾𝑡^1/3𝐿^2/3

Answer 10

Cobb-Douglas Production Function (L exogenous)

Question 11

Equation for MPL (Marginal Product of Labour)

Answer 11

𝑑𝐹(𝐾, 𝐿) / 𝑑𝐿 ≡ 𝑀𝑃𝐿 = 𝑤

𝑤 = wage

First derivative of Cobb-Douglas Production Function with respect to L is MPL

Question 12

𝑑𝐹(𝐾, 𝐿) / 𝑑𝐿 ≡ 𝑀𝑃𝐿 = 𝑤

Answer 12

Equation for MPL

Question 13

Equation for MPK (Marginal Product of Capital)

Answer 13

(𝑑𝐹(𝐾, 𝐿) / 𝑑𝐾) - 𝑑 ≡ 𝑀𝑃𝐾 = r

r = real interest rate = The amount a person can earn by saving one unit of output for a year

First derivative of Cobb-Douglas Production Function with respect to K is MPK

Question 14

(𝑑𝐹(𝐾, 𝐿) / 𝑑𝐾) - 𝑑 ≡ 𝑀𝑃𝐾 = r

Answer 14

Equation for MPK

Question 15

Net Investment Equation

Answer 15

∆𝐾𝑡+1= 𝑠𝑌𝑡 − 𝑑𝐾𝑡
Net investment = Gross investment - Depreciation

Question 16

∆𝐾𝑡+1= 𝑠𝑌𝑡 − 𝑑𝐾𝑡

Answer 16

Net Investment Equation

Question 17

Gross Investment Equation (In terms of the Production Function)

Answer 17

𝑠𝑌𝑡 = 𝑠A𝐾𝑡^1/3𝐿^2/3

Question 18

𝑠𝑌𝑡 = 𝑠A𝐾𝑡^1/3𝐿𝑡^2/3

Answer 18

Gross Investment Equation (In terms of the Production Function)

Question 19

If 𝑠𝑌𝑡 > 𝑑𝐾𝑡, what happens?

Answer 19

Capital stock will increase

Because ∆𝐾𝑡+1= 𝐼𝑡 − 𝑑𝐾𝑡 where 𝐼𝑡 = 𝑠𝑌𝑡

Question 20

What must be true for capital stock to increase?

Answer 20

𝑠𝑌𝑡 > 𝑑𝐾𝑡

Question 21

How can consumption be seen in the Solow diagram?

Answer 21

It is the difference between the output curve and investment and depreciation curves (𝑠𝑌𝑡 and 𝑑𝐾𝑡) at the point where they intercept

Question 22

At what point do the investment and depreciation curves (𝑠𝑌𝑡 and 𝑑𝐾𝑡) intercept?

Answer 22

The steady state - where, in the long run, 𝑠𝑌* = 𝑑𝐾* and ∆𝐾𝑡+1= 0

Question 23

Steady-State Output Equation (Cobb-Douglas)

Answer 23

𝑌* = A𝐾*^1/3𝐿^2/3

Question 24

𝑌* = A𝐾*^1/3𝐿^2/3

Answer 24

Steady-State Output Equation (Cobb-Douglas)

Question 25

Steady-State Level of Capital Equation

Answer 25

𝐾* = ((𝑠A/𝑑)^3/2)𝐿

Sub 𝑠𝑌* = 𝑑𝐾* into the Steady-State Output Equation and rearrange for 𝐾*

An increasing function of:
- the investment rate,
- the size of the workforce
- the productivity of the economy
A decreasing function of:
- the depreciation rate

Question 26

𝐾* = ((𝑠A/𝑑)^3/2)𝐿

Answer 26

Steady-State Level of Capital Equation

Question 27

What is Transition Dynamics?

Answer 27

The process of the economy moving from its initial level of capital to the steady state

Question 28

Steady-State Output Equation (K substituted)

Answer 28

𝑌* = (𝑠/𝑑)^1/2A^3/2𝐿

[All exogenous so all bar]

Question 29

𝑌* = (𝑠/𝑑)^1/2A^3/2𝐿

Answer 29

Steady-State Output Equation (K substituted)

Question 30

Equation for Output-per-Person in Steady-State

Answer 30

𝑦* ≡ 𝑌*/𝐿 = (𝑠/𝑑)^1/2A^3/2

Question 31

𝑦* = (𝑠/𝑑)^1/2A^3/2

Answer 31

Equation for Output-per-Person in Steady-State

Question 32

Why does the economy reach a steady-state?

Answer 32

Investment has diminishing returns - The rate at which production and investment rise is smaller as the capital stock increases… but….
A constant (not diminishing) fraction of the capital stock depreciates each period… so…
Eventually, net investment reaches 0 and output, capital, output per person and consumption per person are all constant

Question 33

What does the Solow model show about long-term growth?

Answer 33

It cannot be fuelled by capital accumulation

Question 34

If the depreciation rate is exogenously shocked to a higher rate, what happens to the depreciation (𝑑𝐾𝑡) and savings (𝑠𝑌𝑡) curves?

Answer 34

• The depreciation curve rotates upwards
• The investment curve remains unchanged

Question 35

What happens to the level of capital and output if the depreciation rate is exogenously shocked to a higher rate?

Answer 35

Steady-state shifts left as depreciation exceeds investment in the short-run, decreasing level of capital (K*)

A leftward shift along the Yt curve decreases Y* This can be shown on a single variable diagram plotted against time.

Question 36

If the savings rate is exogenously shocked to a higher rate, what happens to the depreciation (𝑑𝐾𝑡) and savings (𝑠𝑌𝑡) curves?

Answer 36

• The investment curve shifts/rotates upwards
• The depreciation curve remains unchanged

Question 37

What happens to the level of capital and output if the investment rate is exogenously shocked to a higher rate?

Answer 37

Steady-state shifts right as investment exceeds depreciation in the short-run, increasing level of capital (K*)

A rightward shift along the Yt curve increases Y*. This can be shown on a single variable diagram plotted against time

Question 38

What is s?

Answer 38

The savings rate

Question 39

Different values of s lead to different steady states. How do we know which of these is ‘best’?

Answer 39

The “best” steady state has the highest possible value of consumption

Question 40

Steady-State Consumption Equation

Answer 40

𝐶* = (1 − 𝑠)A𝐾^1/3𝐿^2/3 = (1 − 𝑠)𝑓(𝐾)

In the steady state: 𝐼* = 𝑑𝐾* because Δ𝐾* = 0
So 𝐶* = 𝑌* − 𝐼* = 𝑓(𝐾) − 𝐼
𝐶 = 𝑓(𝐾*) − 𝑑𝐾

Question 41

𝐶* = 𝑓(𝐾) − 𝑑𝐾

Answer 41

Steady-State Consumption Equation

Question 42

How do we notate the steady-state level of capital that maximises consumption?

Answer 42

𝐾*𝑔𝑜𝑙𝑑

Question 43

How do you find 𝐾*𝑔𝑜𝑙𝑑?

Answer 43

The level of 𝐾* at which the gap between the 𝑓(𝐾) and 𝑑𝐾 curves is greatest

Mathematically, where the gradient of 𝑓(𝐾) is equal to the gradient of 𝑑𝐾 (or 𝑓’(𝐾*) = 𝑑)

Question 44

If 𝐾* > 𝐾*𝑔𝑜𝑙𝑑, how should policy makers effect s?

Answer 44

Increasing 𝐶* requires a fall in s.
At all points in the transition period, consumption will be higher, but output and investment will fall

Question 45

If 𝐾* < 𝐾*𝑔𝑜𝑙𝑑, how should policy makers effect s?

Answer 45

Increasing 𝐶* requires a rise in s.
Initially, consumption will fall, but in the long run, consumption, output, and investment will all rise

Question 46

Population and Labour Force Growth Rate Equation

Answer 46

∆𝐿/𝐿 = 𝑛

Where n is exogenous

Question 47

𝑛 = ∆𝐿/𝐿

Answer 47

Population and Labour Force Growth Rate Equation

Question 48

Change in Capital-per-Worker Equation

Answer 48

Δ𝑘𝑡+1 = 𝑠𝑓(𝑘𝑡) − (𝑑+𝑛)𝑘𝑡

Change in capital-per-worker = actual investment - break-even investment

Question 49

Δ𝑘𝑡+1 = 𝑠𝑓(𝑘𝑡) − (𝑑+𝑛)𝑘𝑡

Answer 49

Change in Capital-per-Worker Equation

Question 50

What is break-even investment?

Answer 50

The amount of investment necessary to
keep capital per worker 𝑘𝑡 constant
𝑛𝑘𝑡 - to equip new workers with capital
𝑑𝑘𝑡 - to replace worn out capital

Question 51

What happens to the steady-state level of capital-per-worker with changes in 𝑛?

Answer 51

An increase in 𝑛 causes an increase in break-even investment, leading to a lower steady-state level of capital per worker
An decrease in 𝑛 causes an decrease in break-even investment, leading to a higher steady-state level of capital per worker

Question 52

What are the steady-state growth rates of total capital and output, and per-worker capital and output?

Answer 52

𝑘* - 0
𝑦* - 0
𝐾* - 𝑛
𝑌* - 𝑛

Question 53

How do you construct the growth rate equation for any given equation?

Answer 53

Growth rates operations are one level ‘simpler’ than the operations on
original variables:

1. If 𝑧𝑡 = 𝑥𝑡/𝑦𝑡, then 𝑔𝑧 = 𝑔𝑥 − 𝑔𝑦
2. If 𝑧𝑡 = 𝑥𝑡 × 𝑦𝑡, then 𝑔𝑧 = 𝑔𝑥 + 𝑔𝑦
3. If 𝑧𝑡 = 𝑥𝑡^𝛼, then 𝑔𝑧 = 𝛼 × 𝑔𝑥

Question 54

What is the growth function for the Cobb-Douglas Production Function?
(Growth Rate of GDP Equation)

Answer 54

𝑌𝑡 = A𝐾𝑡^1/3𝐿^2/3
becomes
𝑔𝑌𝑡 = 𝑔𝐴𝑡 + 1/3𝑔𝐾𝑡 + 2/3𝑔𝐿𝑡

Where 𝑔𝐴𝑡 is the growth rate of TFP

Question 55

𝑔𝑌𝑡 = 𝑔𝐴𝑡 + 1/3𝑔𝐾𝑡 + 2/3𝑔𝐿𝑡

Answer 55

Growth Rate of GDP Equation

Question 56

What is TFP?

Answer 56

Total Factor Productivity

Question 57

Positives and negatives of growth accounting

Answer 57

Negative:
- Only reveals the immediate contributors to growth and ignores the deeper issue of what causes those changes

Positive: very useful in studying important economic issues e.g.
- Sources of rapid growth of the newly industrializing countries
- Importance of misallocation of inputs across firms
- Questioning a productivity slowdown or a measurement problem

Question 58

What is convergence theory?

Answer 58

If a country is far below its steady state, it will grow quickly. ‘Poor’ countries should therefore grow quicker than ‘rich’ countries, shrinking the income gap and converging towards a point of equality

Question 59

Why does convergence theory fail?

Answer 59

• Most countries have already reached their steady states
• Most countries are poor not because of bad shocks but because they have
  parameters that yield a lower steady state

Question 60

Strengths of the Solow model

Answer 60

Strengths:
- It provides a framework to determine how rich a country is in the long run
- long run = steady state, pinned down by technology, investment rate etc.
- The principle of transition dynamics can be helpful in understand differences in growth rates across countries

Question 61

Weaknesses of the Solow model

Answer 61

Weaknesses:
- Focusses on investment and capital
- The much more important factor of TFP is left unexplained
- It does not explain differences in investment rates and productivity growth
- The model does not provide a theory of sustained long-run growth

Question 62

What factors might explain very different TFP across countries?

Answer 62

Quality of Institutions:
- Political stability
- Transparency
- Accountability