The Solow-Swan Growth Model II

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Questions and Answers

What happens when effective investment exceeds effective depreciation?

Capital per effective worker increases. (correct)

Investment remains constant.

Capital per effective worker remains unchanged.

Capital per effective worker decreases.

In steady state, what condition must hold true for the output per effective worker?

It is equal to zero.

It remains constant. (correct)

It decreases over time.

It grows at a faster rate than capital per effective worker.

Which equation represents the steady-state condition for capital per effective worker?

sf(k∗) = gA + gL

y∗ = k∗α + (1 − s)y∗

k∗ = (δ + gA + gL)sf(k∗)

sf(k∗) = (δ + gA + gL)k∗ (correct)

What is the relationship between effective investment and effective depreciation when capital per effective worker is decreasing?

Effective investment is less than effective depreciation. Signup and view all the answers

Which factor(s) influence the effective depreciation line in the steady-state equation?

Technological growth and labor growth rates. Signup and view all the answers

What happens to steady state output per effective worker after a permanent increase in the savings rate?

It increases to y ∗′. Signup and view all the answers

According to the Solow-Swan model, what is the effect of being far below the steady state on a country's growth rate?

It causes faster growth. Signup and view all the answers

What is the long run growth rate of output per worker determined by, in the Solow-Swan model?

The growth rate of total factor productivity, gA. Signup and view all the answers

What does the Convergence Hypothesis suggest about the growth of poor countries compared to rich countries?

They should grow faster in the long term. Signup and view all the answers

What does conditional convergence depend on according to the Solow-Swan model?

The same parameters like s, δ, gA, and gL. Signup and view all the answers

What does total factor productivity grow at in the Solow-Swan model?

A rate $gA$ per period Signup and view all the answers

In the intensive form of the production function, how is output per effective worker defined?

As a function of capital per effective worker Signup and view all the answers

What represents the growth of the labor force in the Solow-Swan model?

It grows at a constant rate $gL$ Signup and view all the answers

What does the Solow residual represent in growth accounting?

Growth attributable to technological progress Signup and view all the answers

What does the capital accumulation equation indicate?

Net capital accumulation is driven by savings and depreciation Signup and view all the answers

In the equation for the Cobb-Douglas production function, what does the parameter α represent?

The elasticity of output with respect to capital Signup and view all the answers

Using the formula for productivity growth, which of the following is the correct equation for $gA$?

$gA = gY - αgK - (1 - α)gL$ Signup and view all the answers

What does the term $kt$ refer to in the context of the Solow-Swan model?

Capital per worker Signup and view all the answers

What does the term $gA$ signify in the equations discussed?

Growth rate of technology Signup and view all the answers

What was the calculated productivity growth from 1948-2014 according to the given data?

1.5% per year Signup and view all the answers

How is the intensive form of the production function typically characterized?

By constant returns to scale Signup and view all the answers

What key insight can be drawn about the importance of productivity growth relative to capital accumulation based on the example provided?

Productivity growth accounted for more than half of the growth. Signup and view all the answers

What happens to the dynamics of $kt$ in the Solow-Swan model?

They depend on effective investment levels Signup and view all the answers

What does the steady-state condition formula sf(k*) represent?

Equilibrium between capital accumulation and economic growth Signup and view all the answers

In the growth accounting formula gY = gA + αgK + (1 − α) gL, what does gA represent?

Growth rate of technology Signup and view all the answers

What is the primary purpose of dividing the capital accumulation equation Kt+1 − Kt = sYt − δKt by effective labor AtLt?

To express capital per effective worker Signup and view all the answers

Which term in the equation (1 + gA)(1 + gL)kt+1 − kt = sf(kt) − δkt represents output per worker?

kt+1 Signup and view all the answers

What growth rate approximation is used in the equation 1 − (1 + gA)(1 + gL) − δ ≈ −(gA + gL + δ)?

Standard growth rate approximation Signup and view all the answers

In terms of production, what does the natural logarithm transformation ln(Yt) = ln(At) + αln(Kt) + (1 − α)ln(Lt) allow us to analyze?

Elasticity of output with respect to inputs Signup and view all the answers

Which of the following equations correctly corresponds to the concept of capital per effective worker?

kt = Kt/AtLt Signup and view all the answers

How does the equation gX = (Xt+1 − Xt) / Xt relate to understanding growth?

It captures growth as a rate of change between periods Signup and view all the answers

What do the terms α and (1 − α) represent in the growth accounting formula?

The elasticity of capital and labor with respect to output Signup and view all the answers

What does the equation Kt+1 = sYt − δKt seek to illustrate in economic modeling?

The dynamics of capital accumulation over time Signup and view all the answers

In the steady state, if effective investment equals effective depreciation, then capital per effective worker remains constant.

True Signup and view all the answers

The equation for steady-state capital per effective worker is represented as $sf(k^) = (δ + gA + gL)k^$.

True Signup and view all the answers

If effective investment is less than effective depreciation, capital per effective worker increases.

False Signup and view all the answers

The growth at steady state implies that both capital and output per effective worker grow at different rates than effective labor.

False Signup and view all the answers

The output per effective worker in the steady state is represented by the equation $y^* = k^α$.

True Signup and view all the answers

An increase in the savings rate shifts the investment per effective worker curve downwards.

False Signup and view all the answers

The long run growth rate of output per worker is solely determined by the growth rate of total factor productivity, gA.

True Signup and view all the answers

The Solow-Swan model predicts unconditional convergence among all countries regardless of their parameters.

False Signup and view all the answers

Diminishing returns will cause growth to slow back down to trend over time after an initial increase in the savings rate.

True Signup and view all the answers

Countries far below their steady state grow faster than countries close to their steady state according to the Convergence Hypothesis.

True Signup and view all the answers

Total factor productivity in the Solow-Swan model grows at a fixed rate of $gA$ each period.

True Signup and view all the answers

The Solow-Swan model suggests that capital accumulation is independent of labor force growth.

False Signup and view all the answers

In the intensive form of the production function, output per effective worker can be denoted as $f(kt)$.

True Signup and view all the answers

A higher savings rate will lead to a decrease in steady-state output per effective worker according to the Solow-Swan model.

False Signup and view all the answers

The convergence hypothesis indicates that poorer economies will grow faster than richer ones until they reach similar levels of income.

True Signup and view all the answers

Labor-augmenting productivity refers to productivity that is not affected by changes in the labor force size.

False Signup and view all the answers

The equation $K_{t+1} - K_t = sY_t - heta K_t$ summarizes capital accumulation, where $ heta$ represents a depreciation rate.

False Signup and view all the answers

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.

False Signup and view all the answers

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.

False Signup and view all the answers

According to the growth accounting equation, productivity growth can be inferred by the formula $gA = gY - \alpha gK - (1 - \alpha)gL$.

True Signup and view all the answers

The Solow residual refers to the part of economic growth that cannot be explained by increases in capital or labor inputs.

True Signup and view all the answers

In the numerical example from the US economy, the productivity growth over the period 1948-2014 was calculated to be 2.4% per year.

False Signup and view all the answers

Productivity growth accounted for more than half of the overall observed growth during the US economy analysis period from 1948-2014.

True Signup and view all the answers

The formula for growth accounting states that $gY = gA + (1 - α)gL$.

False Signup and view all the answers

The formula $sf(k^) = (δ + gA + gL)k^$ represents the condition for steady-state capital per effective worker.

True Signup and view all the answers

The approximation $1 - (1 + gA)(1 + gL) - δ ≈ -(gA + gL + δ)$ neglects second order terms.

True Signup and view all the answers

In the equation $(1 + gA)(1 + gL)(kt+1 - kt) = sf(kt) - (gA + gL + δ)kt$, the term $(gA + gL + δ)$ represents effective investment.

False Signup and view all the answers

The growth rate of a variable can be calculated using the formula $gX = \frac{X_{t+1} - X_t}{X_t}$.

True Signup and view all the answers

The growth rate of output per effective worker does not include the contributions from capital accumulation.

False Signup and view all the answers

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 $ln L_t$.

True Signup and view all the answers

The term $gA$ represents the growth of labor in the growth accounting formula.

False Signup and view all the answers

Effective investment exceeding effective depreciation leads to a decrease in capital per worker.

False Signup and view all the answers

The equation $K_{t+1} - K_t = sY_t - δK_t$ illustrates capital accumulation in an economy.

True Signup and view all the answers

What implications arise when effective investment equals effective depreciation in the context of capital per effective worker?

When effective investment equals effective depreciation, capital per effective worker remains constant, indicating a steady state. Signup and view all the answers

Explain the steady-state condition in relation to the output per effective worker in the Solow-Swan model.

The steady-state condition implies that output per effective worker, represented by $y^* = k^α$, remains constant as the capital per effective worker does not change. Signup and view all the answers

How does effective investment impact capital accumulation according to the Solow-Swan model when it is less than effective depreciation?

If effective investment is less than effective depreciation, capital per effective worker decreases, leading to potential economic contraction. Signup and view all the answers

What result is observed when effective investment overshoots effective depreciation in the context of capital per worker dynamics?

When effective investment exceeds effective depreciation, capital per effective worker increases, reflecting growth in the economy. Signup and view all the answers

What happens to the growth rate of output after a permanent increase in the savings rate?

Initially, growth accelerates above the trend due to increased capital accumulation, but over time it slows back down to trend due to diminishing returns. Signup and view all the answers

According to the Convergence Hypothesis, what is expected regarding the growth rates of poor and rich countries?

Poor countries are expected to grow faster than rich countries, allowing them to catch up over time. Signup and view all the answers

Discuss the significance of the equation $sf(k^) = (δ + gA + gL)k^$ in the context of steady-state economics.

This equation represents the steady-state condition where investment per effective worker equals effective depreciation, determining optimal capital levels. Signup and view all the answers

What do the terms conditional and unconditional convergence mean in the context of the Solow-Swan model?

Conditional convergence occurs based on specific parameters like savings rate and productivity, while unconditional convergence implies all countries will converge regardless of their initial conditions. Signup and view all the answers

What is the relationship between total factor productivity growth, denoted as $gA$, and the long run growth rate of output per worker?

The long run growth rate of output per worker is fully determined by the growth rate of total factor productivity, $gA$. Signup and view all the answers

How does an increase in the savings rate affect steady-state capital per effective worker?

An increase in the savings rate leads to an upward shift in the investment per effective worker curve, resulting in higher steady-state capital per effective worker. Signup and view all the answers

What is the significance of growth accounting in the context of the Solow-Swan growth model?

Growth accounting helps to decompose the contributions of different factors, such as total factor productivity, capital, and labor, to overall economic growth. Signup and view all the answers

How does the Solow-Swan model conceptualize labor-augmenting productivity?

In the Solow-Swan model, labor-augmenting productivity is modeled as a function of technology and the labor force, reflecting that productivity improvements depend on effective labor rather than just labor alone. Signup and view all the answers

Explain the convergence hypothesis as it relates to economic growth.

The convergence hypothesis suggests that poorer economies will experience faster growth rates than richer economies, allowing them to catch up in terms of income levels over time. Signup and view all the answers

What role does total factor productivity (gA) play in the Solow-Swan growth model?

Total factor productivity (gA) represents the rate at which technology and efficiency improvements occur, influencing the overall growth of an economy beyond just capital and labor inputs. Signup and view all the answers

In the context of the intensive form of the production function, what does the term 'effective worker' refer to?

An effective worker refers to the productivity of labor that has been adjusted for technological improvements, modeled as the product of labor quantity and labor-augmenting productivity. Signup and view all the answers

How does capital accumulation relate to effective labor growth within the Solow-Swan model?

In the Solow-Swan model, capital accumulation is influenced by effective labor growth since higher levels of effective labor can lead to greater output and, consequently, increased investment in capital. Signup and view all the answers

What happens to the dynamics of capital per effective worker (kt) when effective investment is greater than effective depreciation?

When effective investment exceeds effective depreciation, the capital per effective worker (kt) increases, leading to greater output per effective worker over time. Signup and view all the answers

How does an increase in the labor force growth rate (gL) affect steady-state output in the Solow-Swan model?

An increase in the labor force growth rate (gL) can lead to lower steady-state output per effective worker due to diminishing returns to capital, as more workers share the existing capital stock. Signup and view all the answers

How can one derive productivity growth using the Cobb-Douglas production function, specifically in relation to observed input and output growth?

Productivity growth can be derived by rearranging the equation $gA = gY - \alpha gK - (1 - \alpha)gL$ to infer growth from observed data. Signup and view all the answers

Explain the significance of the Solow residual in growth accounting and its role in understanding economic growth.

The Solow residual captures the portion of output growth not explained by the growth in inputs, highlighting productivity improvements in the economy. Signup and view all the answers

What role does the parameter α play in the Cobb-Douglas production function, particularly in terms of factor shares?

The parameter α represents the output elasticity of capital, indicating the fraction of total income allocated to capital in the production process. Signup and view all the answers

How do the concepts of conditional and unconditional convergence differ in the context of the Solow-Swan model?

Conditional convergence occurs when countries converge to their own steady states based on specific parameters, while unconditional convergence suggests all economies will converge together regardless of conditions. Signup and view all the answers

What insights can be drawn about the importance of productivity growth relative to capital accumulation based on the numerical example provided?

The example shows that productivity growth of 1.5% accounts for more than half of the total growth of 2.4%, indicating that it is critical for long-term economic growth. Signup and view all the answers

Explain the steady-state condition represented by the equation $sf(k^) = (δ + gA + gL)k^$ in terms of economic growth.

The steady-state condition indicates that effective investment, represented by $sf(k^)$, equals the combined rate of effective depreciation and growth in productivity and labor, $(δ + gA + gL)k^$. Signup and view all the answers

How does the growth accounting formula $gY = gA + αgK + (1 − α)gL$ break down the sources of economic growth?

This formula decomposes growth of output per effective worker, $gY$, into contributions from technology growth ($gA$), capital growth ($gK$), and labor growth ($gL$), weighted by their respective output elasticities. Signup and view all the answers

In the context of growth accounting, what does the term $gA$ signify?

$gA$ represents the growth rate of total factor productivity, which captures improvements in efficiency that are not directly tied to increases in labor or capital. Signup and view all the answers

What is the significance of the term $kt$ in the intensive form of the production function?

$kt$ refers to capital per effective worker, which is essential for analyzing the productivity and growth potential of an economy within the Solow-Swan model. Signup and view all the answers

Describe the implication of having effective investment less than effective depreciation in the Solow-Swan model.

If effective investment is less than effective depreciation, capital per effective worker ($kt$) decreases, leading to a contraction in the economy's productive capacity over time. Signup and view all the answers

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?

This transformation allows economists to linearize the relationship between output and its determinants, facilitating easier comparisons and interpretations of growth rates. Signup and view all the answers

How do the terms $ ext{α}$ and $(1 - ext{α})$ function within the growth accounting formula?

The term $ ext{α}$ represents the output elasticity of capital, while $(1 - ext{α})$ represents the output elasticity of labor, indicating their respective contributions to total production. Signup and view all the answers

Discuss the role of effective labor growth $(gL)$ in the overall economic growth as presented in the growth accounting formula.

Effective labor growth $(gL)$ contributes to the increase in output by enlarging the workforce that can utilize capital and productivity advancements. Signup and view all the answers

In the growth rate approximation, how is $gX$ calculated?

$gX$ is approximated by the formula $gX = rac{X_{t+1} - X_t}{X_t}$, which expresses the change in variable $X$ relative to its value at time $t$. Signup and view all the answers

What does an increase in the savings rate imply for steady-state output per effective worker in the Solow-Swan model?

An increase in the savings rate generally leads to higher steady-state output per effective worker, as more capital is accumulated for production. Signup and view all the answers

If effective investment is less than effective depreciation, capital per effective worker is ______.

decreasing Signup and view all the answers

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.

intersect Signup and view all the answers

In a steady state, both capital and output per effective worker grow at the same rate as effective ______.

labor Signup and view all the answers

The output per effective worker in steady state is represented by the equation y* = k*α, where α denotes the ______ parameter.

output Signup and view all the answers

If effective investment exceeds effective depreciation, then capital per effective worker is ______.

increasing Signup and view all the answers

A country should grow faster when it is far below its steady state and slower when it is close to its ______.

steady state Signup and view all the answers

The long run growth rate of output per worker is fully determined by the growth rate gA of total factor ______.

productivity Signup and view all the answers

The growth rate of output per worker minus the growth rate of the labor force equals the growth rate of total factor ______.

productivity Signup and view all the answers

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 ______.

k∗′ Signup and view all the answers

The Solow-Swan model predicts that countries with the same parameters will have similar long run output per worker, illustrating the concept of ______ convergence.

conditional Signup and view all the answers

Total factor productivity grows at rate ______ per period.

gA Signup and view all the answers

The formula for capital accumulation is given by Kt+1 − Kt = sYt − ______.

δKt Signup and view all the answers

The intensive form of the production function relates output per effective worker denoted as ______.

yt Signup and view all the answers

Labour force grows at rate ______ per period.

gL Signup and view all the answers

In the Cobb-Douglas production function, the term α represents ______.

output elasticity with respect to capital Signup and view all the answers

According to the Solow-Swan model, capital accumulation is determined by the difference between effective investment and ______.

effective depreciation Signup and view all the answers

The concept of convergence hypothesis suggests that poorer economies will grow faster than ______.

richer ones Signup and view all the answers

Output per effective worker is represented in the model by the notation ______.

k Signup and view all the answers

The formula for productivity growth can be derived as gA = gY - αgK - (1 − α)gL, where gY represents the growth of ______.

output Signup and view all the answers

In a Cobb-Douglas production function, the parameter ______ represents the output elasticity of capital.

α Signup and view all the answers

During the period from 1948 to 2014, the observed growth of labor in the US was denoted as gL, which was ______ % per year.

2.4 Signup and view all the answers

According to growth accounting, the ______ residual is a crucial component that reflects productivity changes in the economy.

Solow Signup and view all the answers

For the Cobb-Douglas production function, the typical form is Yt = At Ktα L1−α, where At signifies ______ at time t.

total factor productivity Signup and view all the answers

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 ______.

output Signup and view all the answers

The steady-state condition for capital per effective worker is represented by the equation sf(k*) = (δ + gA + gL) ______.

k* Signup and view all the answers

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 ______.

productivity Signup and view all the answers

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.

Solow-Swan Signup and view all the answers

The capital accumulation equation Kt+1 − Kt = sYt − δKt illustrates how capital interacts with ______ and savings.

output Signup and view all the answers

In terms of growth, the variable kt is defined as capital per effective ______.

worker Signup and view all the answers

The Solow residual in growth accounting is used to measure total factor ______.

productivity Signup and view all the answers

When referring to the growth rate of a variable Xt, it is approximated by gX = (Xt+1 − Xt) / ______.

Xt Signup and view all the answers

The Convergence Hypothesis suggests that poorer economies will grow faster than richer ones until they reach similar levels of ______.

income Signup and view all the answers

In production functions, α represents the share of ______ in the overall production process.

capital Signup and view all the answers

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.