Solow Growth Model Flashcards
Capital Accumulation Equation
πΎπ‘+1 = πΎπ‘ + πΌπ‘ β ππΎπ‘
Next yearβs capital = this yearβs capital + investment - the depreciation rate
Change in Capital Stock Equation
βπΎπ‘+1β‘ πΎπ‘+1 β πΎπ‘ (β>)* βπΎπ‘+1= πΌπ‘ β ππΎπ‘
*When Capital Accumulation Equation substituted into πΎπ‘
πΎπ‘+1 = πΎπ‘ + πΌπ‘ β ππΎπ‘
The Capital Accumulation Equation
What is βπΎπ‘+1= πΌπ‘ β ππΎπ‘
The Change in Capital Stock Equation
Gross Investment and Consumption Equations (In terms of Output)
πΌπ‘ = π ππ‘
πΆπ‘ = (1 β π )ππ‘
Agents consume some fraction of output and invest the rest
πΌπ‘ = π ππ‘
Gross Investment Equation (In terms of Output)
πΆπ‘ = (1 β π )ππ‘
Gross Consumption Equation
Labour Equation
πΏπ‘ = πΏ
The amount of labour in the economy is given exogenously at a constant level
Cobb-Douglas Production Function (L exogenous)
ππ‘ [= πΉ(πΎπ‘,πΏ)] = AπΎπ‘^1/3πΏ^2/3
ππ‘ = AπΎπ‘^1/3πΏ^2/3
Cobb-Douglas Production Function (L exogenous)
Equation for MPL (Marginal Product of Labour)
ππΉ(πΎ, πΏ) / ππΏ β‘ πππΏ = π€
π€ = wage
First derivative of Cobb-Douglas Production Function with respect to L is MPL
ππΉ(πΎ, πΏ) / ππΏ β‘ πππΏ = π€
Equation for MPL
Equation for MPK (Marginal Product of Capital)
(ππΉ(πΎ, πΏ) / ππΎ) - π β‘ πππΎ = 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
(ππΉ(πΎ, πΏ) / ππΎ) - π β‘ πππΎ = r
Equation for MPK
Net Investment Equation
βπΎπ‘+1= π ππ‘ β ππΎπ‘
Net investment = Gross investment - Depreciation
βπΎπ‘+1= π ππ‘ β ππΎπ‘
Net Investment Equation
Gross Investment Equation (In terms of the Production Function)
π ππ‘ = π AπΎπ‘^1/3πΏ^2/3
π ππ‘ = π AπΎπ‘^1/3πΏπ‘^2/3
Gross Investment Equation (In terms of the Production Function)
If π ππ‘ > ππΎπ‘, what happens?
Capital stock will increase
Because βπΎπ‘+1= πΌπ‘ β ππΎπ‘ where πΌπ‘ = π ππ‘
What must be true for capital stock to increase?
π ππ‘ > ππΎπ‘
How can consumption be seen in the Solow diagram?
It is the difference between the output curve and investment and depreciation curves (π ππ‘ and ππΎπ‘) at the point where they intercept
At what point do the investment and depreciation curves (π ππ‘ and ππΎπ‘) intercept?
The steady state - where, in the long run, π π* = ππΎ* and βπΎπ‘+1= 0
Steady-State Output Equation (Cobb-Douglas)
π* = AπΎ*^1/3πΏ^2/3
π* = AπΎ*^1/3πΏ^2/3
Steady-State Output Equation (Cobb-Douglas)
Steady-State Level of Capital Equation
πΎ* = ((π A/π)^3/2)πΏ
Sub π π* = ππΎ* into the Steady-State Output Equation and rearrange for πΎ*
An increasing function of:
- the investment rate,
- the size of the workforce
- the productivity of the economy
A decreasing function of:
- the depreciation rate
πΎ* = ((π A/π)^3/2)πΏ
Steady-State Level of Capital Equation
What is Transition Dynamics?
The process of the economy moving from its initial level of capital to the steady state
Steady-State Output Equation (K substituted)
π* = (π /π)^1/2A^3/2πΏ
[All exogenous so all bar]
π* = (π /π)^1/2A^3/2πΏ
Steady-State Output Equation (K substituted)
Equation for Output-per-Person in Steady-State
π¦* β‘ π*/πΏ = (π /π)^1/2A^3/2
π¦* = (π /π)^1/2A^3/2
Equation for Output-per-Person in Steady-State
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 increasesβ¦ butβ¦.
A constant (not diminishing) fraction of the capital stock depreciates each periodβ¦ soβ¦
Eventually, net investment reaches 0 and output, capital, output per person and consumption per person are all constant
What does the Solow model show about long-term growth?
It cannot be fuelled by capital accumulation
If the depreciation rate is exogenously shocked to a higher rate, what happens to the depreciation (ππΎπ‘) and savings (π ππ‘) curves?
- The depreciation curve rotates upwards
- The investment curve remains unchanged
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.
If the savings rate is exogenously shocked to a higher rate, what happens to the depreciation (ππΎπ‘) and savings (π ππ‘) curves?
- The investment curve shifts/rotates upwards
- The depreciation curve remains unchanged
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
What is s?
The savings rate
Different values of s lead to different steady states. How do we know which of these is βbestβ?
The βbestβ steady state has the highest possible value of consumption
Steady-State Consumption Equation
πΆ* = (1 β π )AπΎ^1/3πΏ^2/3 = (1 β π )π(πΎ)
In the steady state: πΌ* = ππΎ* because ΞπΎ* = 0
So πΆ* = π* β πΌ* = π(πΎ) β πΌ
πΆ = π(πΎ*) β ππΎ
πΆ* = π(πΎ) β ππΎ
Steady-State Consumption Equation
How do we notate the steady-state level of capital that maximises consumption?
πΎ*ππππ
How do you find πΎ*ππππ?
The level of πΎ* at which the gap between the π(πΎ) and ππΎ curves is greatest
Mathematically, where the gradient of π(πΎ) is equal to the gradient of ππΎ (or πβ(πΎ*) = π)
If πΎ* > πΎ*ππππ, how should policy makers effect s?
Increasing πΆ* requires a fall in s.
At all points in the transition period, consumption will be higher, but output and investment will fall
If πΎ* < πΎ*ππππ, how should policy makers effect s?
Increasing πΆ* requires a rise in s.
Initially, consumption will fall, but in the long run, consumption, output, and investment will all rise
Population and Labour Force Growth Rate Equation
βπΏ/πΏ = π
Where n is exogenous
π = βπΏ/πΏ
Population and Labour Force Growth Rate Equation
Change in Capital-per-Worker Equation
Ξππ‘+1 = π π(ππ‘) β (π+π)ππ‘
Change in capital-per-worker = actual investment - break-even investment
Ξππ‘+1 = π π(ππ‘) β (π+π)ππ‘
Change in Capital-per-Worker Equation
What is break-even investment?
The amount of investment necessary to
keep capital per worker ππ‘ constant
πππ‘ - to equip new workers with capital
πππ‘ - to replace worn out capital
What happens to the steady-state level of capital-per-worker with changes in π?
An increase in π causes an increase in break-even investment, leading to a lower steady-state level of capital per worker
An decrease in π causes an decrease in break-even investment, leading to a higher steady-state level of capital per worker
What are the steady-state growth rates of total capital and output, and per-worker capital and output?
π* - 0
π¦* - 0
πΎ* - π
π* - π
How do you construct the growth rate equation for any given equation?
Growth rates operations are one level βsimplerβ than the operations on
original variables:
- If π§π‘ = π₯π‘/π¦π‘, then ππ§ = ππ₯ β ππ¦
- If π§π‘ = π₯π‘ Γ π¦π‘, then ππ§ = ππ₯ + ππ¦
- If π§π‘ = π₯π‘^πΌ, then ππ§ = πΌ Γ ππ₯
What is the growth function for the Cobb-Douglas Production Function?
(Growth Rate of GDP Equation)
ππ‘ = AπΎπ‘^1/3πΏ^2/3
becomes
πππ‘ = ππ΄π‘ + 1/3ππΎπ‘ + 2/3ππΏπ‘
Where ππ΄π‘ is the growth rate of TFP
πππ‘ = ππ΄π‘ + 1/3ππΎπ‘ + 2/3ππΏπ‘
Growth Rate of GDP Equation
What is TFP?
Total Factor Productivity
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
What is convergence theory?
If a country is far below its steady state, it will grow quickly. βPoorβ countries should therefore grow quicker than βrichβ countries, shrinking the income gap and converging towards a point of equality
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
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
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
What factors might explain very different TFP across countries?
Quality of Institutions:
- Political stability
- Transparency
- Accountability
