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Solow Growth Model Flashcards

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1
Q

Capital Accumulation Equation

A

𝐾𝑑+1 = 𝐾𝑑 + 𝐼𝑑 βˆ’ 𝑑𝐾𝑑
Next year’s capital = this year’s capital + investment - the depreciation rate

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2
Q

Change in Capital Stock Equation

A

βˆ†πΎπ‘‘+1≑ 𝐾𝑑+1 βˆ’ 𝐾𝑑 (β€”>)* βˆ†πΎπ‘‘+1= 𝐼𝑑 βˆ’ 𝑑𝐾𝑑

*When Capital Accumulation Equation substituted into 𝐾𝑑

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3
Q

𝐾𝑑+1 = 𝐾𝑑 + 𝐼𝑑 βˆ’ 𝑑𝐾𝑑

A

The Capital Accumulation Equation

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4
Q

What is βˆ†πΎπ‘‘+1= 𝐼𝑑 βˆ’ 𝑑𝐾𝑑

A

The Change in Capital Stock Equation

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5
Q

Gross Investment and Consumption Equations (In terms of Output)

A

𝐼𝑑 = π‘ π‘Œπ‘‘
𝐢𝑑 = (1 βˆ’ 𝑠)π‘Œπ‘‘

Agents consume some fraction of output and invest the rest

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6
Q

𝐼𝑑 = π‘ π‘Œπ‘‘

A

Gross Investment Equation (In terms of Output)

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7
Q

𝐢𝑑 = (1 βˆ’ 𝑠)π‘Œπ‘‘

A

Gross Consumption Equation

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8
Q

Labour Equation

A

𝐿𝑑 = 𝐿

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

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9
Q

Cobb-Douglas Production Function (L exogenous)

A

π‘Œπ‘‘ [= 𝐹(𝐾𝑑,𝐿)] = A𝐾𝑑^1/3𝐿^2/3

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10
Q

π‘Œπ‘‘ = A𝐾𝑑^1/3𝐿^2/3

A

Cobb-Douglas Production Function (L exogenous)

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11
Q

Equation for MPL (Marginal Product of Labour)

A

𝑑𝐹(𝐾, 𝐿) / 𝑑𝐿 ≑ 𝑀𝑃𝐿 = 𝑀

𝑀 = wage

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

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12
Q

𝑑𝐹(𝐾, 𝐿) / 𝑑𝐿 ≑ 𝑀𝑃𝐿 = 𝑀

A

Equation for MPL

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13
Q

Equation for MPK (Marginal Product of Capital)

A

(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

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14
Q

(𝑑𝐹(𝐾, 𝐿) / 𝑑𝐾) - 𝑑 ≑ 𝑀𝑃𝐾 = r

A

Equation for MPK

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15
Q

Net Investment Equation

A

βˆ†πΎπ‘‘+1= π‘ π‘Œπ‘‘ βˆ’ 𝑑𝐾𝑑
Net investment = Gross investment - Depreciation

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16
Q

βˆ†πΎπ‘‘+1= π‘ π‘Œπ‘‘ βˆ’ 𝑑𝐾𝑑

A

Net Investment Equation

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17
Q

Gross Investment Equation (In terms of the Production Function)

A

π‘ π‘Œπ‘‘ = 𝑠A𝐾𝑑^1/3𝐿^2/3

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18
Q

π‘ π‘Œπ‘‘ = 𝑠A𝐾𝑑^1/3𝐿𝑑^2/3

A

Gross Investment Equation (In terms of the Production Function)

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19
Q

If π‘ π‘Œπ‘‘ > 𝑑𝐾𝑑, what happens?

A

Capital stock will increase

Because βˆ†πΎπ‘‘+1= 𝐼𝑑 βˆ’ 𝑑𝐾𝑑 where 𝐼𝑑 = π‘ π‘Œπ‘‘

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20
Q

What must be true for capital stock to increase?

A

π‘ π‘Œπ‘‘ > 𝑑𝐾𝑑

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21
Q

How can consumption be seen in the Solow diagram?

A

It is the difference between the output curve and investment and depreciation curves (π‘ π‘Œπ‘‘ and 𝑑𝐾𝑑) at the point where they intercept

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22
Q

At what point do the investment and depreciation curves (π‘ π‘Œπ‘‘ and 𝑑𝐾𝑑) intercept?

A

The steady state - where, in the long run, π‘ π‘Œ* = 𝑑𝐾* and βˆ†πΎπ‘‘+1= 0

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23
Q

Steady-State Output Equation (Cobb-Douglas)

A

π‘Œ* = A𝐾*^1/3𝐿^2/3

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24
Q

π‘Œ* = A𝐾*^1/3𝐿^2/3

A

Steady-State Output Equation (Cobb-Douglas)

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25
Steady-State Level of Capital Equation
𝐾* = ((𝑠A/𝑑)^3/2)𝐿 Sub π‘ π‘Œ* = 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
26
𝐾* = ((𝑠A/𝑑)^3/2)𝐿
Steady-State Level of Capital Equation
27
What is Transition Dynamics?
The process of the economy moving from its initial level of capital to the steady state
28
Steady-State Output Equation (K substituted)
π‘Œ* = (𝑠/𝑑)^1/2A^3/2𝐿 [All exogenous so all bar]
29
π‘Œ* = (𝑠/𝑑)^1/2A^3/2𝐿
Steady-State Output Equation (K substituted)
30
Equation for Output-per-Person in Steady-State
𝑦* ≑ π‘Œ*/𝐿 = (𝑠/𝑑)^1/2A^3/2
31
𝑦* = (𝑠/𝑑)^1/2A^3/2
Equation for Output-per-Person in Steady-State
32
Why does the economy reach a steady-state?
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
33
What does the Solow model show about long-term growth?
It cannot be fuelled by capital accumulation
34
If the depreciation rate is exogenously shocked to a higher rate, what happens to the depreciation (𝑑𝐾𝑑) and savings (π‘ π‘Œπ‘‘) curves?
- The depreciation curve rotates upwards - The investment curve remains unchanged
35
What happens to the level of capital and output if the depreciation rate is exogenously shocked to a higher rate?
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.
36
If the savings rate is exogenously shocked to a higher rate, what happens to the depreciation (𝑑𝐾𝑑) and savings (π‘ π‘Œπ‘‘) curves?
- The investment curve shifts/rotates upwards - The depreciation curve remains unchanged
37
What happens to the level of capital and output if the investment rate is exogenously shocked to a higher rate?
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
38
What is s?
The savings rate
39
Different values of s lead to different steady states. How do we know which of these is 'best'?
The β€œbest” steady state has the highest possible value of consumption
40
Steady-State Consumption Equation
𝐢* = (1 βˆ’ 𝑠)A𝐾*^1/3𝐿^2/3 = (1 βˆ’ 𝑠)𝑓(𝐾*) In the steady state: 𝐼* = 𝑑𝐾* because Δ𝐾* = 0 So 𝐢* = π‘Œ* βˆ’ 𝐼* = 𝑓(𝐾*) βˆ’ 𝐼* 𝐢 = 𝑓(𝐾*) βˆ’ 𝑑𝐾
41
𝐢* = 𝑓(𝐾*) βˆ’ 𝑑𝐾*
Steady-State Consumption Equation
42
How do we notate the steady-state level of capital that maximises consumption?
𝐾*π‘”π‘œπ‘™π‘‘
43
How do you find 𝐾*π‘”π‘œπ‘™π‘‘?
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)
44
If 𝐾* > 𝐾*π‘”π‘œπ‘™π‘‘, how should policy makers effect s?
Increasing 𝐢* requires a fall in s. At all points in the transition period, consumption will be higher, but output and investment will fall
45
If 𝐾* < 𝐾*π‘”π‘œπ‘™π‘‘, how should policy makers effect s?
Increasing 𝐢* requires a rise in s. Initially, consumption will fall, but in the long run, consumption, output, and investment will all rise
46
Population and Labour Force Growth Rate Equation
βˆ†πΏ/𝐿 = 𝑛 Where n is exogenous
47
𝑛 = βˆ†πΏ/𝐿
Population and Labour Force Growth Rate Equation
48
Change in Capital-per-Worker Equation
Ξ”π‘˜π‘‘+1 = 𝑠𝑓(π‘˜π‘‘) βˆ’ (𝑑+𝑛)π‘˜π‘‘ Change in capital-per-worker = actual investment - break-even investment
49
Ξ”π‘˜π‘‘+1 = 𝑠𝑓(π‘˜π‘‘) βˆ’ (𝑑+𝑛)π‘˜π‘‘
Change in Capital-per-Worker Equation
50
What is break-even investment?
The amount of investment necessary to keep capital per worker π‘˜π‘‘ constant π‘›π‘˜π‘‘ - to equip new workers with capital π‘‘π‘˜π‘‘ - to replace worn out capital
51
What happens to the steady-state level of capital-per-worker with changes in 𝑛?
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
52
What are the steady-state growth rates of total capital and output, and per-worker capital and output?
π‘˜* - 0 𝑦* - 0 𝐾* - 𝑛 π‘Œ* - 𝑛
53
How do you construct the growth rate equation for any given equation?
Growth rates operations are one level 'simpler' than the operations on original variables: 1. If 𝑧𝑑 = π‘₯𝑑/𝑦𝑑, then 𝑔𝑧 = 𝑔π‘₯ βˆ’ 𝑔𝑦 2. If 𝑧𝑑 = π‘₯𝑑 Γ— 𝑦𝑑, then 𝑔𝑧 = 𝑔π‘₯ + 𝑔𝑦 3. If 𝑧𝑑 = π‘₯𝑑^𝛼, then 𝑔𝑧 = 𝛼 Γ— 𝑔π‘₯
54
What is the growth function for the Cobb-Douglas Production Function? (Growth Rate of GDP Equation)
π‘Œπ‘‘ = A𝐾𝑑^1/3𝐿^2/3 becomes π‘”π‘Œπ‘‘ = 𝑔𝐴𝑑 + 1/3𝑔𝐾𝑑 + 2/3𝑔𝐿𝑑 Where 𝑔𝐴𝑑 is the growth rate of TFP
55
π‘”π‘Œπ‘‘ = 𝑔𝐴𝑑 + 1/3𝑔𝐾𝑑 + 2/3𝑔𝐿𝑑
Growth Rate of GDP Equation
56
What is TFP?
Total Factor Productivity
57
Positives and negatives of growth accounting
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
58
What is convergence theory?
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
59
Why does convergence theory fail?
- 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
60
Strengths of the Solow model
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
61
Weaknesses of the Solow model
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
62
What factors might explain very different TFP across countries?
Quality of Institutions: - Political stability - Transparency - Accountability

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