Podcast
Questions and Answers
What does k* represent in the context of the model?
What does k* represent in the context of the model?
- The optimal capital needed for growth
- The level of output in the economy
- The initial level of capital
- The steady state capital (correct)
What determines the long-run growth rate of Y/L in this model?
What determines the long-run growth rate of Y/L in this model?
- Investment rates in the economy
- The level of capital accumulation
- Technological progress g (correct)
- The rate of population growth n
If k is greater than k*, what will happen to the economy over time?
If k is greater than k*, what will happen to the economy over time?
- The economy will stabilize at k
- The economy will grow indefinitely
- The economy will converge to k* (correct)
- The economy will experience a decline
What is the implication of diminishing marginal productivity of capital according to the model?
What is the implication of diminishing marginal productivity of capital according to the model?
What condition is represented when sf(k) is greater than (n + g + δ)k?
What condition is represented when sf(k) is greater than (n + g + δ)k?
What condition is necessary for attaining the maximum level of steady-state consumption according to the golden rule?
What condition is necessary for attaining the maximum level of steady-state consumption according to the golden rule?
What is indicated by a saving rate greater than sGR?
What is indicated by a saving rate greater than sGR?
If s < sGR, what does this imply about future consumption?
If s < sGR, what does this imply about future consumption?
In the context of the golden rule, how is the desirability of certain outcomes assessed?
In the context of the golden rule, how is the desirability of certain outcomes assessed?
What does the equation c* = f(k*) - (n + g + δ)k* represent?
What does the equation c* = f(k*) - (n + g + δ)k* represent?
What does the vertical logarithmic scale in the GDP per capita graph facilitate?
What does the vertical logarithmic scale in the GDP per capita graph facilitate?
Which growth rate is indicated in the GDP per capita graph for the United States?
Which growth rate is indicated in the GDP per capita graph for the United States?
In the context of the data, what is the significance of the year 1929?
In the context of the data, what is the significance of the year 1929?
What is represented by the formula $y(t) = e^{gt} y(0)$ in the graph's context?
What is represented by the formula $y(t) = e^{gt} y(0)$ in the graph's context?
What does 'spliced' data refer to in the context of the GDP statistics?
What does 'spliced' data refer to in the context of the GDP statistics?
Which of the following years does not appear in the GDP per person graph?
Which of the following years does not appear in the GDP per person graph?
What aspect of economic growth is noted to vary among different countries according to the data?
What aspect of economic growth is noted to vary among different countries according to the data?
What does the term 'frontier' refer to in relation to economic growth?
What does the term 'frontier' refer to in relation to economic growth?
What does the equation $L· /L = n$ represent?
What does the equation $L· /L = n$ represent?
In the equation $K· = I − δK$, what does the term $I$ represent?
In the equation $K· = I − δK$, what does the term $I$ represent?
What happens to the investment per unit of effective labor if $s$ increases?
What happens to the investment per unit of effective labor if $s$ increases?
Under what condition does the steady state exist?
Under what condition does the steady state exist?
What does the term $c*$ represent in the equation $c* = f(k*) - (n + g + δ)k*$?
What does the term $c*$ represent in the equation $c* = f(k*) - (n + g + δ)k*$?
What does $sf(k*)$ denote in the steady state equation?
What does $sf(k*)$ denote in the steady state equation?
Which of the following statements is true regarding steady state in this model?
Which of the following statements is true regarding steady state in this model?
What is implied if $k$ is differentiated with respect to $t$?
What is implied if $k$ is differentiated with respect to $t$?
What is the relationship represented by Y = C + S in a closed economy with no government?
What is the relationship represented by Y = C + S in a closed economy with no government?
What do the Inada conditions indicate about the marginal products of capital and labor?
What do the Inada conditions indicate about the marginal products of capital and labor?
Under constant returns to scale, how can the production function be expressed in intensive form?
Under constant returns to scale, how can the production function be expressed in intensive form?
What does the assumption of diminishing marginal productivity imply about the relationship between k and f(k)?
What does the assumption of diminishing marginal productivity imply about the relationship between k and f(k)?
According to the neoclassical production function, what does the parameter α represent?
According to the neoclassical production function, what does the parameter α represent?
How does the production function f(k) behave as k approaches zero?
How does the production function f(k) behave as k approaches zero?
What does it mean if f''(k) < 0 for the production function?
What does it mean if f''(k) < 0 for the production function?
What implication does constant returns to scale (CRS) have for the scaling of inputs?
What implication does constant returns to scale (CRS) have for the scaling of inputs?
What is a crucial assumption according to Solow's definition of economic theory?
What is a crucial assumption according to Solow's definition of economic theory?
In the neoclassical production function, what does the term 'AL' represent?
In the neoclassical production function, what does the term 'AL' represent?
What does the equation wL + rK = Y represent in the context of competitive markets?
What does the equation wL + rK = Y represent in the context of competitive markets?
According to the neoclassical production function, what indicates diminishing returns to capital?
According to the neoclassical production function, what indicates diminishing returns to capital?
What does the term 'conditional convergence' refer to in economic theory?
What does the term 'conditional convergence' refer to in economic theory?
Which component is part of the equation Y = K^α(AL)^(1−α)?
Which component is part of the equation Y = K^α(AL)^(1−α)?
What does the symbol 'f''(k)' signify in economic production functions?
What does the symbol 'f''(k)' signify in economic production functions?
In the neoclassical production function, what does the variable 'K' represent?
In the neoclassical production function, what does the variable 'K' represent?
What implication does it have when inputs are paid their marginal products under competitive markets?
What implication does it have when inputs are paid their marginal products under competitive markets?
Which factor is NOT included in the equation wL + rK = Y?
Which factor is NOT included in the equation wL + rK = Y?
Flashcards
Constant Returns to Scale (CRS)
Constant Returns to Scale (CRS)
A production function where doubling all inputs (e.g., capital and labor) doubles output.
Neoclassical Production Function
Neoclassical Production Function
A production function that shows how much output (Y) can be produced with different combinations of capital (K) and labor (AL).
Diminishing Marginal Product
Diminishing Marginal Product
As you add more of one input (e.g., capital), while holding other inputs constant, the increase in output gets smaller and smaller.
Inada Conditions
Inada Conditions
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Capital Share (α)
Capital Share (α)
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Intensive Form Production Function
Intensive Form Production Function
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Marginal Product
Marginal Product
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Closed Economy
Closed Economy
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Constant Growth Rate of Population
Constant Growth Rate of Population
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Constant Growth Rate of Productivity
Constant Growth Rate of Productivity
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Break-Even Investment
Break-Even Investment
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Steady State (Balanced Growth Path)
Steady State (Balanced Growth Path)
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Gross Investment per Unit of Effective Labour
Gross Investment per Unit of Effective Labour
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Depreciation Rate
Depreciation Rate
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Steady-State Capital-Output Ratio (k*)
Steady-State Capital-Output Ratio (k*)
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Capital-Output Ratio (k)
Capital-Output Ratio (k)
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Steady State Capital (k*)
Steady State Capital (k*)
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Steady-State Output (y*)
Steady-State Output (y*)
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Diminishing Marginal Productivity of Capital
Diminishing Marginal Productivity of Capital
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Exogenous Technological Progress (g)
Exogenous Technological Progress (g)
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Convergence to Steady State
Convergence to Steady State
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Long-run GDP per capita growth
Long-run GDP per capita growth
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Logarithmic Scale
Logarithmic Scale
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Chained 2009 dollars
Chained 2009 dollars
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GDP per capita
GDP per capita
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Growth rate (g)
Growth rate (g)
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Frontier countries
Frontier countries
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Economic growth spread
Economic growth spread
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Growth patterns
Growth patterns
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Golden Rule saving rate (sGR)
Golden Rule saving rate (sGR)
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Golden Rule capital stock (kGR)
Golden Rule capital stock (kGR)
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Steady-State Condition
Steady-State Condition
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Dynamic Inefficiency
Dynamic Inefficiency
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Steady-State Consumption (c*)
Steady-State Consumption (c*)
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Solow's Crucial Assumptions
Solow's Crucial Assumptions
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Neoclassical Production Function
Neoclassical Production Function
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Conditional Convergence
Conditional Convergence
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Steady State (Balanced Growth Path)
Steady State (Balanced Growth Path)
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Capital-Output Ratio (k)
Capital-Output Ratio (k)
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Steady-State Capital (k*)
Steady-State Capital (k*)
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Convergence
Convergence
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Exogenous Technological Progress (g)
Exogenous Technological Progress (g)
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Capital’s share (α)
Capital’s share (α)
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Diminishing Marginal Product of Capital
Diminishing Marginal Product of Capital
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Study Notes
References
- Solow-Swan Model notes are based on various resources: Economic Growth (2nd edition), MIT Press (Barro & Sala i Martin, 2004)
- The Facts of Economic Growth (Jones, 2016), Handbook of Macroeconomics, 2A
- Advanced Macroeconomics (5th edition), McGraw Hill (Romer, 2018)
Outline
- Introduction
- Solow-Swan model assumptions
- Dynamics of k
- Steady state
- Level effects
- Golden rule
- Convergence
- Micro-foundations
- Appendix
Introduction
- Figure 26 displays the lack of worldwide convergence in GDP per capita between 1960-2011
The Solow-Swan Model
- Very influential theory of economic growth, simple dynamic model.
- Current income equals current production, based on current capital and labour endowments, and technology.
- Part of current income is saved and invested into additional capital
- Tomorrow's income results from tomorrow's capital and labour, and state of technology.
- Explains how saving, investment, technology, and technical progress affect economic growth.
- Provides empirical predictions to compare with data.
Assumptions
- Agents sell 1 unit of labour services, own and rent capital.
- Income used for consumption and saving (investment in additional capital).
- One final good, output Y, with constant price P = 1.
- Firms hire labor and rent capital using a production function Y = F(K, AL) (AL is "effective" labor, A is labour-augmenting productivity).
- Markets are perfectly competitive.
- Constant returns to scale (CRS).
Assumptions: Neoclassical Production Function
- Constant returns to scale (CRS).
- Positive, diminishing marginal products.
- Inada conditions: lim Fk = ∞ as K approaches 0, lim Fk = 0 as K approaches ∞. Same for FL.
- Production function in intensive form: f(k) = F(K, AL)/AL = F(K/AL, 1). k = K/AL (capital per effective labor), y = Y/AL = f(k) (output per effective labor).
Assumptions: Diminishing Marginal Productivity
- f'(k) > 0: an increase in k leads to an increase in f(k).
- f"(k) < 0: the higher k, the smaller the increases in f(k) when k rises, leading to decreasing returns to capital.
Assumptions: Dynamic Behavior
- Variables are functions of time (t). Dynamic behavior of variables like labor (L), productivity (A), and capital (K) is assumed, accounting for factors like population growth (n), technological progress (g), and depreciation (δ).
Dynamics of k
- k̇ = sf(k) - (n + g + δ)k.
- sf(k) = gross investment per unit of effective labor
- (n + g + δ)k = break-even investment per unit of effective labor
Steady State
- sf(k*) = (n + g + δ)k*. Unique steady state with k* > 0.
- y* = f(k*). Steady state where all variables grow at constant rates.
- k = 0 is ignored.
- K/L and Y/L grow at constant rate g+n.
Steady State & Phase Diagram
- k < k* --> sf(k) > (n+g+δ)k (investment more than break-even).
- k > k* --> sf(k) < (n+g+δ)k (investment less than break-even).
Steady State
- The economy converges to a unique steady state from any starting point k(0) > 0
Golden Rule
- sf(kGR) = (n+g+δ)kGR
- CGR= max{f(k) - (n+g+δ)k} where CGR is maximum consumption.
- This represents the optimal capital stock for maximum long-run consumption.
Golden Rule
- c* = f(k*) - (n+g+δ)k* The optimal saving rate leads to max consumption.
- If S is below SGR, the economy tends to have dynamic inefficiency or over-saving.
- If S is above SGR, it leads to higher initial consumption and lower future consumption.
Absolute Convergence
- identical parameter values lead to the same k*.
- Countries with lower initial capital stocks grow faster toward the common steady state.
Conditional Convergence
- Differences in parameter values lead to different k*.
- Lower initial capital stocks don't necessarily grow faster.
Convergence
- Absolute convergence: converging to same steady state.
- Conditional convergence: differing parameter values could prevent convergence.
Micro-foundations
- Production with side effects (pollution). Is this relevant in the Solow model? What should the government do?
- Model predicts the effects of (1) Income tax reduction on saving rates and (2) News on future productivity levels on today's economic variables.
- Limitations: lacks welfare evaluation and behavioral responses to environment, policies, etc.
- Models like Overlapping Generations and Ramsey address these challenges.
Appendix: Do Differences in Growth Rates Matter?
- Examples showing that even small differences in growth rates lead to large differences in income after years.
Appendix: Lucas on Growth
- Discusses the importance of considering factors like actions governments can take to grow countries like India and Indonesia. The welfare implications are also noted.
Appendix: Solow on Economic Theory
- Theory depends on assumptions that are not always true.
- The art of theorizing is making assumptions that make final results not strongly dependent on the assumptions.
- Crucial assumptions should be realistic.
Appendix: Neoclassical Production Function
- From definitions: Y = F(K, AL), AL = f(k), KF(K, AL) = f'(k) > 0, K^2F(K, AL) = f"(k) < 0.
- Under CRS and competitive markets, inputs are paid marginal products and factor payments exhaust output (wL + rK = Y).
Appendix: Conditional Convergence
- Shows graph of convergence of personal income across Japanese prefectures against the log of 1930 per capita income for growth rates from 1930-1990.
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