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Questions and Answers
What happens when effective investment exceeds effective depreciation?
What happens when effective investment exceeds effective depreciation?
In steady state, what condition must hold true for the output per effective worker?
In steady state, what condition must hold true for the output per effective worker?
Which equation represents the steady-state condition for capital per effective worker?
Which equation represents the steady-state condition for capital per effective worker?
What is the relationship between effective investment and effective depreciation when capital per effective worker is decreasing?
What is the relationship between effective investment and effective depreciation when capital per effective worker is decreasing?
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Which factor(s) influence the effective depreciation line in the steady-state equation?
Which factor(s) influence the effective depreciation line in the steady-state equation?
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What happens to steady state output per effective worker after a permanent increase in the savings rate?
What happens to steady state output per effective worker after a permanent increase in the savings rate?
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According to the Solow-Swan model, what is the effect of being far below the steady state on a country's growth rate?
According to the Solow-Swan model, what is the effect of being far below the steady state on a country's growth rate?
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What is the long run growth rate of output per worker determined by, in the Solow-Swan model?
What is the long run growth rate of output per worker determined by, in the Solow-Swan model?
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What does the Convergence Hypothesis suggest about the growth of poor countries compared to rich countries?
What does the Convergence Hypothesis suggest about the growth of poor countries compared to rich countries?
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What does conditional convergence depend on according to the Solow-Swan model?
What does conditional convergence depend on according to the Solow-Swan model?
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What does total factor productivity grow at in the Solow-Swan model?
What does total factor productivity grow at in the Solow-Swan model?
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In the intensive form of the production function, how is output per effective worker defined?
In the intensive form of the production function, how is output per effective worker defined?
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What represents the growth of the labor force in the Solow-Swan model?
What represents the growth of the labor force in the Solow-Swan model?
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What does the Solow residual represent in growth accounting?
What does the Solow residual represent in growth accounting?
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What does the capital accumulation equation indicate?
What does the capital accumulation equation indicate?
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In the equation for the Cobb-Douglas production function, what does the parameter α represent?
In the equation for the Cobb-Douglas production function, what does the parameter α represent?
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Using the formula for productivity growth, which of the following is the correct equation for $gA$?
Using the formula for productivity growth, which of the following is the correct equation for $gA$?
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What does the term $kt$ refer to in the context of the Solow-Swan model?
What does the term $kt$ refer to in the context of the Solow-Swan model?
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What does the term $gA$ signify in the equations discussed?
What does the term $gA$ signify in the equations discussed?
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What was the calculated productivity growth from 1948-2014 according to the given data?
What was the calculated productivity growth from 1948-2014 according to the given data?
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How is the intensive form of the production function typically characterized?
How is the intensive form of the production function typically characterized?
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What key insight can be drawn about the importance of productivity growth relative to capital accumulation based on the example provided?
What key insight can be drawn about the importance of productivity growth relative to capital accumulation based on the example provided?
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What happens to the dynamics of $kt$ in the Solow-Swan model?
What happens to the dynamics of $kt$ in the Solow-Swan model?
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What does the steady-state condition formula sf(k*) represent?
What does the steady-state condition formula sf(k*) represent?
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In the growth accounting formula gY = gA + αgK + (1 − α) gL, what does gA represent?
In the growth accounting formula gY = gA + αgK + (1 − α) gL, what does gA represent?
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What is the primary purpose of dividing the capital accumulation equation Kt+1 − Kt = sYt − δKt by effective labor AtLt?
What is the primary purpose of dividing the capital accumulation equation Kt+1 − Kt = sYt − δKt by effective labor AtLt?
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Which term in the equation (1 + gA)(1 + gL)kt+1 − kt = sf(kt) − δkt represents output per worker?
Which term in the equation (1 + gA)(1 + gL)kt+1 − kt = sf(kt) − δkt represents output per worker?
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What growth rate approximation is used in the equation 1 − (1 + gA)(1 + gL) − δ ≈ −(gA + gL + δ)?
What growth rate approximation is used in the equation 1 − (1 + gA)(1 + gL) − δ ≈ −(gA + gL + δ)?
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In terms of production, what does the natural logarithm transformation ln(Yt) = ln(At) + αln(Kt) + (1 − α)ln(Lt) allow us to analyze?
In terms of production, what does the natural logarithm transformation ln(Yt) = ln(At) + αln(Kt) + (1 − α)ln(Lt) allow us to analyze?
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Which of the following equations correctly corresponds to the concept of capital per effective worker?
Which of the following equations correctly corresponds to the concept of capital per effective worker?
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How does the equation gX = (Xt+1 − Xt) / Xt relate to understanding growth?
How does the equation gX = (Xt+1 − Xt) / Xt relate to understanding growth?
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What do the terms α and (1 − α) represent in the growth accounting formula?
What do the terms α and (1 − α) represent in the growth accounting formula?
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What does the equation Kt+1 = sYt − δKt seek to illustrate in economic modeling?
What does the equation Kt+1 = sYt − δKt seek to illustrate in economic modeling?
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In the steady state, if effective investment equals effective depreciation, then capital per effective worker remains constant.
In the steady state, if effective investment equals effective depreciation, then capital per effective worker remains constant.
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The equation for steady-state capital per effective worker is represented as $sf(k^) = (δ + gA + gL)k^$.
The equation for steady-state capital per effective worker is represented as $sf(k^) = (δ + gA + gL)k^$.
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If effective investment is less than effective depreciation, capital per effective worker increases.
If effective investment is less than effective depreciation, capital per effective worker increases.
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The growth at steady state implies that both capital and output per effective worker grow at different rates than effective labor.
The growth at steady state implies that both capital and output per effective worker grow at different rates than effective labor.
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The output per effective worker in the steady state is represented by the equation $y^* = k^α$.
The output per effective worker in the steady state is represented by the equation $y^* = k^α$.
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An increase in the savings rate shifts the investment per effective worker curve downwards.
An increase in the savings rate shifts the investment per effective worker curve downwards.
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The long run growth rate of output per worker is solely determined by the growth rate of total factor productivity, gA.
The long run growth rate of output per worker is solely determined by the growth rate of total factor productivity, gA.
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The Solow-Swan model predicts unconditional convergence among all countries regardless of their parameters.
The Solow-Swan model predicts unconditional convergence among all countries regardless of their parameters.
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Diminishing returns will cause growth to slow back down to trend over time after an initial increase in the savings rate.
Diminishing returns will cause growth to slow back down to trend over time after an initial increase in the savings rate.
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Countries far below their steady state grow faster than countries close to their steady state according to the Convergence Hypothesis.
Countries far below their steady state grow faster than countries close to their steady state according to the Convergence Hypothesis.
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Total factor productivity in the Solow-Swan model grows at a fixed rate of $gA$ each period.
Total factor productivity in the Solow-Swan model grows at a fixed rate of $gA$ each period.
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The Solow-Swan model suggests that capital accumulation is independent of labor force growth.
The Solow-Swan model suggests that capital accumulation is independent of labor force growth.
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In the intensive form of the production function, output per effective worker can be denoted as $f(kt)$.
In the intensive form of the production function, output per effective worker can be denoted as $f(kt)$.
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A higher savings rate will lead to a decrease in steady-state output per effective worker according to the Solow-Swan model.
A higher savings rate will lead to a decrease in steady-state output per effective worker according to the Solow-Swan model.
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The convergence hypothesis indicates that poorer economies will grow faster than richer ones until they reach similar levels of income.
The convergence hypothesis indicates that poorer economies will grow faster than richer ones until they reach similar levels of income.
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Labor-augmenting productivity refers to productivity that is not affected by changes in the labor force size.
Labor-augmenting productivity refers to productivity that is not affected by changes in the labor force size.
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The equation $K_{t+1} - K_t = sY_t - heta K_t$ summarizes capital accumulation, where $ heta$ represents a depreciation rate.
The equation $K_{t+1} - K_t = sY_t - heta K_t$ summarizes capital accumulation, where $ heta$ represents a depreciation rate.
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In the context of the Solow-Swan model, an increase in total factor productivity $gA$ directly causes the capital per effective worker $kt$ to decrease.
In the context of the Solow-Swan model, an increase in total factor productivity $gA$ directly causes the capital per effective worker $kt$ to decrease.
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In the Cobb-Douglas production function, $Y_t = A_t K_t^{\alpha} L_t^{1-\alpha}$, the parameter $\alpha$ represents the share of labor in output.
In the Cobb-Douglas production function, $Y_t = A_t K_t^{\alpha} L_t^{1-\alpha}$, the parameter $\alpha$ represents the share of labor in output.
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According to the growth accounting equation, productivity growth can be inferred by the formula $gA = gY - \alpha gK - (1 - \alpha)gL$.
According to the growth accounting equation, productivity growth can be inferred by the formula $gA = gY - \alpha gK - (1 - \alpha)gL$.
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The Solow residual refers to the part of economic growth that cannot be explained by increases in capital or labor inputs.
The Solow residual refers to the part of economic growth that cannot be explained by increases in capital or labor inputs.
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In the numerical example from the US economy, the productivity growth over the period 1948-2014 was calculated to be 2.4% per year.
In the numerical example from the US economy, the productivity growth over the period 1948-2014 was calculated to be 2.4% per year.
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Productivity growth accounted for more than half of the overall observed growth during the US economy analysis period from 1948-2014.
Productivity growth accounted for more than half of the overall observed growth during the US economy analysis period from 1948-2014.
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The formula for growth accounting states that $gY = gA + (1 - α)gL$.
The formula for growth accounting states that $gY = gA + (1 - α)gL$.
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The formula $sf(k^) = (δ + gA + gL)k^$ represents the condition for steady-state capital per effective worker.
The formula $sf(k^) = (δ + gA + gL)k^$ represents the condition for steady-state capital per effective worker.
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The approximation $1 - (1 + gA)(1 + gL) - δ ≈ -(gA + gL + δ)$ neglects second order terms.
The approximation $1 - (1 + gA)(1 + gL) - δ ≈ -(gA + gL + δ)$ neglects second order terms.
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In the equation $(1 + gA)(1 + gL)(kt+1 - kt) = sf(kt) - (gA + gL + δ)kt$, the term $(gA + gL + δ)$ represents effective investment.
In the equation $(1 + gA)(1 + gL)(kt+1 - kt) = sf(kt) - (gA + gL + δ)kt$, the term $(gA + gL + δ)$ represents effective investment.
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The growth rate of a variable can be calculated using the formula $gX = \frac{X_{t+1} - X_t}{X_t}$.
The growth rate of a variable can be calculated using the formula $gX = \frac{X_{t+1} - X_t}{X_t}$.
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The growth rate of output per effective worker does not include the contributions from capital accumulation.
The growth rate of output per effective worker does not include the contributions from capital accumulation.
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The natural logarithm transformation allows us to derive the growth accounting equation by relating $ln Y_t$ to $ln A_t$, $ln K_t$, and $ln L_t$.
The natural logarithm transformation allows us to derive the growth accounting equation by relating $ln Y_t$ to $ln A_t$, $ln K_t$, and $ln L_t$.
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The term $gA$ represents the growth of labor in the growth accounting formula.
The term $gA$ represents the growth of labor in the growth accounting formula.
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Effective investment exceeding effective depreciation leads to a decrease in capital per worker.
Effective investment exceeding effective depreciation leads to a decrease in capital per worker.
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The equation $K_{t+1} - K_t = sY_t - δK_t$ illustrates capital accumulation in an economy.
The equation $K_{t+1} - K_t = sY_t - δK_t$ illustrates capital accumulation in an economy.
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What implications arise when effective investment equals effective depreciation in the context of capital per effective worker?
What implications arise when effective investment equals effective depreciation in the context of capital per effective worker?
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Explain the steady-state condition in relation to the output per effective worker in the Solow-Swan model.
Explain the steady-state condition in relation to the output per effective worker in the Solow-Swan model.
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How does effective investment impact capital accumulation according to the Solow-Swan model when it is less than effective depreciation?
How does effective investment impact capital accumulation according to the Solow-Swan model when it is less than effective depreciation?
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What result is observed when effective investment overshoots effective depreciation in the context of capital per worker dynamics?
What result is observed when effective investment overshoots effective depreciation in the context of capital per worker dynamics?
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What happens to the growth rate of output after a permanent increase in the savings rate?
What happens to the growth rate of output after a permanent increase in the savings rate?
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According to the Convergence Hypothesis, what is expected regarding the growth rates of poor and rich countries?
According to the Convergence Hypothesis, what is expected regarding the growth rates of poor and rich countries?
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Discuss the significance of the equation $sf(k^) = (δ + gA + gL)k^$ in the context of steady-state economics.
Discuss the significance of the equation $sf(k^) = (δ + gA + gL)k^$ in the context of steady-state economics.
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What do the terms conditional and unconditional convergence mean in the context of the Solow-Swan model?
What do the terms conditional and unconditional convergence mean in the context of the Solow-Swan model?
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What is the relationship between total factor productivity growth, denoted as $gA$, and the long run growth rate of output per worker?
What is the relationship between total factor productivity growth, denoted as $gA$, and the long run growth rate of output per worker?
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How does an increase in the savings rate affect steady-state capital per effective worker?
How does an increase in the savings rate affect steady-state capital per effective worker?
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What is the significance of growth accounting in the context of the Solow-Swan growth model?
What is the significance of growth accounting in the context of the Solow-Swan growth model?
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How does the Solow-Swan model conceptualize labor-augmenting productivity?
How does the Solow-Swan model conceptualize labor-augmenting productivity?
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Explain the convergence hypothesis as it relates to economic growth.
Explain the convergence hypothesis as it relates to economic growth.
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What role does total factor productivity (gA) play in the Solow-Swan growth model?
What role does total factor productivity (gA) play in the Solow-Swan growth model?
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In the context of the intensive form of the production function, what does the term 'effective worker' refer to?
In the context of the intensive form of the production function, what does the term 'effective worker' refer to?
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How does capital accumulation relate to effective labor growth within the Solow-Swan model?
How does capital accumulation relate to effective labor growth within the Solow-Swan model?
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What happens to the dynamics of capital per effective worker (kt) when effective investment is greater than effective depreciation?
What happens to the dynamics of capital per effective worker (kt) when effective investment is greater than effective depreciation?
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How does an increase in the labor force growth rate (gL) affect steady-state output in the Solow-Swan model?
How does an increase in the labor force growth rate (gL) affect steady-state output in the Solow-Swan model?
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How can one derive productivity growth using the Cobb-Douglas production function, specifically in relation to observed input and output growth?
How can one derive productivity growth using the Cobb-Douglas production function, specifically in relation to observed input and output growth?
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Explain the significance of the Solow residual in growth accounting and its role in understanding economic growth.
Explain the significance of the Solow residual in growth accounting and its role in understanding economic growth.
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What role does the parameter α play in the Cobb-Douglas production function, particularly in terms of factor shares?
What role does the parameter α play in the Cobb-Douglas production function, particularly in terms of factor shares?
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How do the concepts of conditional and unconditional convergence differ in the context of the Solow-Swan model?
How do the concepts of conditional and unconditional convergence differ in the context of the Solow-Swan model?
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What insights can be drawn about the importance of productivity growth relative to capital accumulation based on the numerical example provided?
What insights can be drawn about the importance of productivity growth relative to capital accumulation based on the numerical example provided?
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Explain the steady-state condition represented by the equation $sf(k^) = (δ + gA + gL)k^$ in terms of economic growth.
Explain the steady-state condition represented by the equation $sf(k^) = (δ + gA + gL)k^$ in terms of economic growth.
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How does the growth accounting formula $gY = gA + αgK + (1 − α)gL$ break down the sources of economic growth?
How does the growth accounting formula $gY = gA + αgK + (1 − α)gL$ break down the sources of economic growth?
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In the context of growth accounting, what does the term $gA$ signify?
In the context of growth accounting, what does the term $gA$ signify?
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What is the significance of the term $kt$ in the intensive form of the production function?
What is the significance of the term $kt$ in the intensive form of the production function?
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Describe the implication of having effective investment less than effective depreciation in the Solow-Swan model.
Describe the implication of having effective investment less than effective depreciation in the Solow-Swan model.
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What does the natural logarithm transformation, $ln(Y_t) = ln(A_t) + α ln(K_t) + (1 − α) ln(L_t)$, enable economists to analyze?
What does the natural logarithm transformation, $ln(Y_t) = ln(A_t) + α ln(K_t) + (1 − α) ln(L_t)$, enable economists to analyze?
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How do the terms $ ext{α}$ and $(1 - ext{α})$ function within the growth accounting formula?
How do the terms $ ext{α}$ and $(1 - ext{α})$ function within the growth accounting formula?
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Discuss the role of effective labor growth $(gL)$ in the overall economic growth as presented in the growth accounting formula.
Discuss the role of effective labor growth $(gL)$ in the overall economic growth as presented in the growth accounting formula.
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In the growth rate approximation, how is $gX$ calculated?
In the growth rate approximation, how is $gX$ calculated?
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What does an increase in the savings rate imply for steady-state output per effective worker in the Solow-Swan model?
What does an increase in the savings rate imply for steady-state output per effective worker in the Solow-Swan model?
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If effective investment is less than effective depreciation, capital per effective worker is ______.
If effective investment is less than effective depreciation, capital per effective worker is ______.
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The steady-state condition for capital per effective worker is represented by the equation sf(k*) = (δ + gA + gL)k* which indicates where investment curves ______ depreciation lines.
The steady-state condition for capital per effective worker is represented by the equation sf(k*) = (δ + gA + gL)k* which indicates where investment curves ______ depreciation lines.
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In a steady state, both capital and output per effective worker grow at the same rate as effective ______.
In a steady state, both capital and output per effective worker grow at the same rate as effective ______.
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The output per effective worker in steady state is represented by the equation y* = k*α, where α denotes the ______ parameter.
The output per effective worker in steady state is represented by the equation y* = k*α, where α denotes the ______ parameter.
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If effective investment exceeds effective depreciation, then capital per effective worker is ______.
If effective investment exceeds effective depreciation, then capital per effective worker is ______.
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A country should grow faster when it is far below its steady state and slower when it is close to its ______.
A country should grow faster when it is far below its steady state and slower when it is close to its ______.
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The long run growth rate of output per worker is fully determined by the growth rate gA of total factor ______.
The long run growth rate of output per worker is fully determined by the growth rate gA of total factor ______.
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The growth rate of output per worker minus the growth rate of the labor force equals the growth rate of total factor ______.
The growth rate of output per worker minus the growth rate of the labor force equals the growth rate of total factor ______.
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An increase in the savings rate shifts up the investment per effective worker curve, leading to an increase in steady state capital per effective worker from k∗ to ______.
An increase in the savings rate shifts up the investment per effective worker curve, leading to an increase in steady state capital per effective worker from k∗ to ______.
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The Solow-Swan model predicts that countries with the same parameters will have similar long run output per worker, illustrating the concept of ______ convergence.
The Solow-Swan model predicts that countries with the same parameters will have similar long run output per worker, illustrating the concept of ______ convergence.
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Total factor productivity grows at rate ______ per period.
Total factor productivity grows at rate ______ per period.
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The formula for capital accumulation is given by Kt+1 − Kt = sYt − ______.
The formula for capital accumulation is given by Kt+1 − Kt = sYt − ______.
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The intensive form of the production function relates output per effective worker denoted as ______.
The intensive form of the production function relates output per effective worker denoted as ______.
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Labour force grows at rate ______ per period.
Labour force grows at rate ______ per period.
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In the Cobb-Douglas production function, the term α represents ______.
In the Cobb-Douglas production function, the term α represents ______.
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According to the Solow-Swan model, capital accumulation is determined by the difference between effective investment and ______.
According to the Solow-Swan model, capital accumulation is determined by the difference between effective investment and ______.
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The concept of convergence hypothesis suggests that poorer economies will grow faster than ______.
The concept of convergence hypothesis suggests that poorer economies will grow faster than ______.
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Output per effective worker is represented in the model by the notation ______.
Output per effective worker is represented in the model by the notation ______.
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The formula for productivity growth can be derived as gA = gY - αgK - (1 − α)gL, where gY represents the growth of ______.
The formula for productivity growth can be derived as gA = gY - αgK - (1 − α)gL, where gY represents the growth of ______.
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In a Cobb-Douglas production function, the parameter ______ represents the output elasticity of capital.
In a Cobb-Douglas production function, the parameter ______ represents the output elasticity of capital.
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During the period from 1948 to 2014, the observed growth of labor in the US was denoted as gL, which was ______ % per year.
During the period from 1948 to 2014, the observed growth of labor in the US was denoted as gL, which was ______ % per year.
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According to growth accounting, the ______ residual is a crucial component that reflects productivity changes in the economy.
According to growth accounting, the ______ residual is a crucial component that reflects productivity changes in the economy.
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For the Cobb-Douglas production function, the typical form is Yt = At Ktα L1−α, where At signifies ______ at time t.
For the Cobb-Douglas production function, the typical form is Yt = At Ktα L1−α, where At signifies ______ at time t.
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In growth accounting, the equation gY = gA + αgK + (1 − α) gL indicates the relationship between output growth and various factors, where gY represents the growth of ______.
In growth accounting, the equation gY = gA + αgK + (1 − α) gL indicates the relationship between output growth and various factors, where gY represents the growth of ______.
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The steady-state condition for capital per effective worker is represented by the equation sf(k*) = (δ + gA + gL) ______.
The steady-state condition for capital per effective worker is represented by the equation sf(k*) = (δ + gA + gL) ______.
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The relationship between growth rates of productivity and employment can be represented with the approximation (1 + gA)(1 + gL) ≈ 1 + gA + gL, where gA represents growth in ______.
The relationship between growth rates of productivity and employment can be represented with the approximation (1 + gA)(1 + gL) ≈ 1 + gA + gL, where gA represents growth in ______.
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In the context of the Solow-Swan model, a higher savings rate will lead to a decrease in steady-state output per effective worker according to the ______ model.
In the context of the Solow-Swan model, a higher savings rate will lead to a decrease in steady-state output per effective worker according to the ______ model.
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The capital accumulation equation Kt+1 − Kt = sYt − δKt illustrates how capital interacts with ______ and savings.
The capital accumulation equation Kt+1 − Kt = sYt − δKt illustrates how capital interacts with ______ and savings.
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In terms of growth, the variable kt is defined as capital per effective ______.
In terms of growth, the variable kt is defined as capital per effective ______.
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The Solow residual in growth accounting is used to measure total factor ______.
The Solow residual in growth accounting is used to measure total factor ______.
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When referring to the growth rate of a variable Xt, it is approximated by gX = (Xt+1 − Xt) / ______.
When referring to the growth rate of a variable Xt, it is approximated by gX = (Xt+1 − Xt) / ______.
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The Convergence Hypothesis suggests that poorer economies will grow faster than richer ones until they reach similar levels of ______.
The Convergence Hypothesis suggests that poorer economies will grow faster than richer ones until they reach similar levels of ______.
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In production functions, α represents the share of ______ in the overall production process.
In production functions, α represents the share of ______ in the overall production process.
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Study Notes
The Solow-Swan Growth Model II
- This lecture introduces the Solow-Swan model with productivity and employment growth.
- The model assumes total factor productivity (A) grows at a rate of gA per period and the labor force (L) grows at a rate of gL per period.
- It is more convenient to work with labor-augmenting productivity, where the productivity term is redefined to simplify the math.
- The intensive form of the production function is defined as yt = f (kt) where f (kt) ≡ F(kt, 1) and capital accumulation is given by:
- (1 + gA + gL)(kt+1 - kt) = sf(kt) - ( δ + gA + gL)kt
- The Solow-Swan model diagram remains the same. The dynamics of the model depend on the size of effective investment vs. effective depreciation.
- At steady state, capital and output per effective worker are constant. It follows that capital and output both grow at the same rate as effective labor, which makes up a balanced growth path.
- The long-run growth rate of output per worker is gY - gL = gA, which is fully determined by the growth rate of total factor productivity (gA).
- A permanent increase in the savings rate (s) shifts up investment per effective worker curve along an unchanged depreciation per effective worker line. This results in an increase in steady-state capital and output per effective worker.
- The Solow-Swan model predicts conditional convergence, meaning that countries with the same parameters will have the same long-run output per worker. It does not predict unconditional convergence, where countries would have the same long-run output regardless of parameters.
- Growth accounting is a tool used to decompose observed growth in output into the growth of inputs and "productivity" residual, known as the "Solow residual."
- The growth accounting equation for the Cobb-Douglas production function (Yt = AtKtαL1−α) is gY = gA + αgK + (1 - α)gL, where gA is the growth of total factor productivity.
Steady-State Condition
- The steady-state condition for a model with trend growth is sf(k*) = (δ + gA + gL)k*
Growth Accounting
- Growth accounting allows for measuring the contribution of productivity and input growth to the overall growth of the economy.
- The growth accounting formula is: gY = gA + αgK + (1 - α)gL
Convergence Hypothesis
- The Solow-Swan model predicts that poorer countries will grow faster than richer countries, but only when they have the same parameters such as savings rates, technology growth rates, and depreciation rates. This is known as conditional convergence.
- The model does not predict unconditional convergence, which would mean that all countries will eventually converge to the same level of output.
Productivity Growth and Employment Growth
- The Solow-Swan model is extended to incorporate productivity growth (gA) and employment growth (gL)
- Productivity growth occurs at a rate of gA per period, meaning a 1+gA increase in productivity from one period to the next.
- Employment growth occurs at a rate of gL per period, meaning a 1+gL increase in the labor force from one period to the next.
- More convenient to work with labor-augmenting productivity
- This redefines the productivity term in the production function, making it easier to analyze.
Solow-Swan Diagram Revisited
- The dynamics of the Solow-Swan model with growth are qualitatively similar to the basic model.
- The key difference is the introduction of effective investment and effective depreciation, which include the rate of productivity and employment growth:
- When effective investment exceeds effective depreciation, capital per effective worker (kt) increases.
- When effective investment is less than effective depreciation, capital per effective worker decreases.
- When effective investment equals effective depreciation, capital per effective worker remains constant.
Steady State
- The steady state represents an equilibrium in the Solow-Swan model, where capital per effective worker, output per effective worker, and consumption per effective worker are constant.
- The steady state condition equation is sf(k*) = (δ + gA + gL)k*, where:
- s is the savings rate
- f(k*) is the intensive form of the production function
- δ is the depreciation rate
- gA is the productivity growth rate
- gL is the employment growth rate
- Steady-state output and consumption per effective worker are determined as:
- y* = f(k*)
- c* = (1-s)y*
Balanced Growth
- The steady state implies that capital and output grow at the same rate as effective labor, leading to a balanced growth path.
- The long-run growth rate of output per worker is determined by the productivity growth rate:
- gY - gL = gA
Convergence Hypothesis
- The Solow-Swan model predicts conditional convergence, which means that countries with similar parameters (s, δ, gA, gL) will converge to the same long-run level of output per worker.
- It does not predict unconditional convergence, which would mean all countries converge to the same level of output per worker regardless of their parameters.
Growth Accounting
- This is a tool for decomposing observed growth in output into growth in inputs (capital and labor) and a residual term attributed to productivity.
- Using the Cobb-Douglas production function:
- gY = gA + αgK + (1-α)gL
- gA = gY - αgK - (1-α)gL
- Growth accounting allows us to infer productivity growth from observed output and input growth.
Factors Shares
- α in the Cobb-Douglas production function represents the share of output going to capital.
- Capital's share of output can be calculated using the formula α = (MPK * K) / Y, where MPK is the marginal product of capital.
- The share of output going to labor is (1-α).
Conclusion
- The Solow-Swan model provides a framework for understanding long-run economic growth, emphasizing the roles of productivity, savings, and technology.
- The model predicts conditional convergence, suggesting that countries with similar characteristics should converge to the same long-run growth path.
- Growth accounting allows us to decompose observed economic growth into its contributing factors.
- The model highlights the importance of policies aimed at promoting productivity growth, savings, and technological advancement.
Solow-Swan Growth Model II
- The Solow-Swan model incorporates productivity growth (gA) and employment growth (gL) to account for real-world output growth.
- Productivity: Total factor productivity (TFP) grows at a rate of gA per period, meaning that the efficiency of inputs increases over time.
- Employment: The labor force grows at a rate of gL per period.
- Labor-Augmenting Productivity: The model simplifies by using labor-augmenting productivity, where the productivity term is redefined to make calculations easier.
- Intensive Form of the Production Function: The model can be written in terms of output, capital, and labor per effective worker, assuming constant returns to scale.
- Capital Accumulation: The capital accumulation equation reflects growth in productivity and employment, showing how the change in capital per effective worker depends on investment and depreciation.
- Steady-State: In a steady-state, capital per effective worker remains constant, with investment per effective worker equaling depreciation per effective worker.
- Balanced Growth: The model predicts balanced growth, where capital and output grow at the same rate as effective labor (gA + gL) in the long run.
- Long Run Growth: The long-run growth rate of output per worker is determined by the growth rate of total factor productivity (gA).
- Convergence Hypothesis: The Solow-Swan model predicts conditional convergence: countries with similar parameters will converge to the same level of output per worker in the long run, but not necessarily to the same level of output per worker as other countries with different parameters.
- Growth Accounting: This methodology allows for decomposing observed growth into growth in inputs (capital and labor) and a productivity residual (the Solow residual).
- Cobb-Douglas Production Function: Growth accounting uses the Cobb-Douglas production function to infer productivity from observed input and output growth.
- Numerical Example: By applying growth accounting to historical US data, one can determine the contribution of productivity growth and capital accumulation to overall economic expansion.
Solow-Swan Model II
- The Solow-Swan Model is extended to incorporate productivity growth and employment growth.
Productivity Growth and Employment Growth
- Productivity growth is introduced as a rate of growth in total factor productivity, denoted as gA.
- Employment growth is introduced as a rate of growth in the labor force, denoted as gL.
Labor-Augmenting Productivity
- The model uses labor-augmenting productivity, where productivity is redefined to simplify mathematical calculations.
- Output is represented in terms of effective workers, denoted by Yt / (At Lt).
Intensive Form of the Production Function
- The production function is expressed in terms of effective labor, allowing for simpler analysis.
- The intensive form of the production function, yt = f(kt), represents output per effective worker.
Capital Accumulation with Trend Growth
- The capital accumulation equation is adjusted to account for productivity and employment growth: (1+gA+gL)(kt+1-kt) = sf(kt) - (δ+gA+gL)kt
Steady-State with Trend Growth
- The steady-state level of capital per effective worker (k*) is determined by the point where investment per effective worker equals effective depreciation.
- k* is calculated by the equation: sf(k*) = (δ + gA + gL) k*
Balanced Growth Path
- The model predicts a balanced growth path, where capital and output grow at the same rate as effective labor.
- The steady-state growth rate of output per worker is determined by the growth rate of total factor productivity: gY - gL = gA.
Permanent Increase in Savings Rate
- An increase in the savings rate shifts the investment per effective worker curve upwards, leading to an increase in the steady-state level of capital and output per effective worker.
Convergence Hypothesis
- The Solow-Swan Model predicts conditional convergence: countries with similar parameters converge to the same steady-state level of output per worker.
- Unconditional convergence, where all countries converge to the same level regardless of parameters, is not predicted.
Growth Accounting
- Growth accounting decomposes observed output growth into contributions from input growth (capital and labor) and a productivity residual, often called the Solow residual.
- Using the Cobb-Douglas production function, the growth rate of output can be represented as: gY = gA + αgK + (1 - α)gL.
Numerical Example of Growth Accounting
- The US economy from 1948 to 2014 experienced a growth rate of output per worker of 2.4% per year.
- Assuming a capital share of 1/3, growth accounting suggests productivity growth contributed more than half of the observed output growth.
Convergence Hypothesis in the Data
- Empirically, conditional convergence is observed across countries, indicating that countries with similar parameters tend to converge to similar levels of output per worker over time.
Conclusion
- The Solow-Swan model extended to incorporate trend growth provides insights into the long-run growth path of economies and factors influencing convergence.
- Growth accounting offers a framework for analyzing the relative importance of input growth and productivity improvements on economic growth.
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Explore the intricacies of the Solow-Swan growth model with a focus on productivity and employment growth. This quiz delves into labor-augmenting productivity and the dynamics of capital accumulation. Test your understanding of steady state, effective investment, and balanced growth paths.
