The document summarizes key aspects of the Solow growth model. It explains that the Solow model replaced the fixed production function of the Harrod-Domar model with a neoclassical production function allowing for factor substitution. It presents the basic equations of the Solow model showing that changes in capital per worker are determined by savings, population growth, and depreciation. It illustrates the Solow diagram and how steady state equilibrium is reached. It analyzes how changes in the saving rate and population growth rate impact the model.
The document discusses optimal control and agent-based economic models. It summarizes key concepts from neoclassical production theory including the production function, Cobb-Douglas production function, and capital dynamics. It also discusses the utility function, consumption, savings and investment. The document then provides an overview of concepts in optimal control theory including the Hamiltonian, Pontryagin's maximum principle, and infinite horizon problems. It concludes with an introduction to the Ramsey-Cass-Koopmans model for determining optimal savings.
Chapter 5 The Solow Growth ModelHikaru SaijoUniversitWilheminaRossi174
The document summarizes Chapter 5 of the Solow Growth Model. It introduces the Solow model, which uses a Cobb-Douglas production function to model how capital accumulates over time. It shows graphically and mathematically how the model reaches a steady state where growth stops. While the model does not explain long-term growth, it can be used to understand differences in output across countries and experiment with changes to parameters. The strengths and weaknesses of the Solow model are also discussed.
This document provides an overview of three theories of income distribution:
1) Kalecki's theory argues that the distribution of national income between wages and profits is determined by the degree of monopoly in the economy. A higher degree of monopoly results in a larger share of income going to profits.
2) Kaldor's theory views income as split between wages and profits. The ratio of profits to national income depends on the ratio of investment to national income, given saving propensities.
3) Ricardo's classical theory holds that wages are determined by the wage fund, rents arise from land scarcity, and profits are whatever income is left after paying wages and rents.
The document summarizes concepts related to economic growth calculation and analysis. It explains how to calculate growth rates using formulas that relate the current and future values of income given a growth rate over time. It also discusses how small differences in growth rates, like 1% vs 2%, can lead to large differences in income levels over many years. The document then covers factors that influence economic growth like capital accumulation, productivity increases, savings, investment, and production functions. It introduces the concept of total factor productivity (TFP) in analyzing sources of economic growth.
Es 2 k21-18 returns to scale ppt naveen chouhanNaveenChouhan13
Returns to scale refers to the relationship between changes in total input and the resulting changes in total output. There are three types of returns to scale: increasing, constant, and diminishing. Increasing returns occur when a proportional increase in all inputs results in a more than proportional increase in output. Constant returns occur when output increases proportionally to inputs. Diminishing returns occur when output increases by less than the proportional increase in inputs. The law of returns to scale explains how output responds to proportional changes in all inputs in the long run.
Production analysis by Neeraj Bhandari ( Surkhet.Nepal )Neeraj Bhandari
The document provides information on production functions and related concepts:
- A production function describes the technological relationship between inputs and outputs. It represents the maximum output attainable from various combinations of inputs.
- Inputs can be fixed or variable. The short run is when some inputs are fixed, while the long run allows variation in all inputs.
- Isoquants represent combinations of inputs that produce equal output. They have properties like being negatively sloped, non-intersecting, and convex to the origin.
- Laws of production include diminishing marginal returns and variable proportions. Returns to scale can be increasing, constant, or decreasing in the long run depending on output and input changes.
The document summarizes key aspects of the Solow growth model. It explains that the Solow model replaced the fixed production function of the Harrod-Domar model with a neoclassical production function allowing for factor substitution. It presents the basic equations of the Solow model showing that changes in capital per worker are determined by savings, population growth, and depreciation. It illustrates the Solow diagram and how steady state equilibrium is reached. It analyzes how changes in the saving rate and population growth rate impact the model.
The document discusses optimal control and agent-based economic models. It summarizes key concepts from neoclassical production theory including the production function, Cobb-Douglas production function, and capital dynamics. It also discusses the utility function, consumption, savings and investment. The document then provides an overview of concepts in optimal control theory including the Hamiltonian, Pontryagin's maximum principle, and infinite horizon problems. It concludes with an introduction to the Ramsey-Cass-Koopmans model for determining optimal savings.
Chapter 5 The Solow Growth ModelHikaru SaijoUniversitWilheminaRossi174
The document summarizes Chapter 5 of the Solow Growth Model. It introduces the Solow model, which uses a Cobb-Douglas production function to model how capital accumulates over time. It shows graphically and mathematically how the model reaches a steady state where growth stops. While the model does not explain long-term growth, it can be used to understand differences in output across countries and experiment with changes to parameters. The strengths and weaknesses of the Solow model are also discussed.
This document provides an overview of three theories of income distribution:
1) Kalecki's theory argues that the distribution of national income between wages and profits is determined by the degree of monopoly in the economy. A higher degree of monopoly results in a larger share of income going to profits.
2) Kaldor's theory views income as split between wages and profits. The ratio of profits to national income depends on the ratio of investment to national income, given saving propensities.
3) Ricardo's classical theory holds that wages are determined by the wage fund, rents arise from land scarcity, and profits are whatever income is left after paying wages and rents.
The document summarizes concepts related to economic growth calculation and analysis. It explains how to calculate growth rates using formulas that relate the current and future values of income given a growth rate over time. It also discusses how small differences in growth rates, like 1% vs 2%, can lead to large differences in income levels over many years. The document then covers factors that influence economic growth like capital accumulation, productivity increases, savings, investment, and production functions. It introduces the concept of total factor productivity (TFP) in analyzing sources of economic growth.
Es 2 k21-18 returns to scale ppt naveen chouhanNaveenChouhan13
Returns to scale refers to the relationship between changes in total input and the resulting changes in total output. There are three types of returns to scale: increasing, constant, and diminishing. Increasing returns occur when a proportional increase in all inputs results in a more than proportional increase in output. Constant returns occur when output increases proportionally to inputs. Diminishing returns occur when output increases by less than the proportional increase in inputs. The law of returns to scale explains how output responds to proportional changes in all inputs in the long run.
Production analysis by Neeraj Bhandari ( Surkhet.Nepal )Neeraj Bhandari
The document provides information on production functions and related concepts:
- A production function describes the technological relationship between inputs and outputs. It represents the maximum output attainable from various combinations of inputs.
- Inputs can be fixed or variable. The short run is when some inputs are fixed, while the long run allows variation in all inputs.
- Isoquants represent combinations of inputs that produce equal output. They have properties like being negatively sloped, non-intersecting, and convex to the origin.
- Laws of production include diminishing marginal returns and variable proportions. Returns to scale can be increasing, constant, or decreasing in the long run depending on output and input changes.
The document discusses production concepts and cost analysis, including:
- Production functions show the relationship between inputs and outputs. Common types include Cobb-Douglas, CES, and Leontief functions.
- Total, average, and marginal products are defined for analyzing how output changes with variable inputs like labor.
- Short-run and long-run periods are distinguished based on whether inputs are fixed or variable.
- Isoquants and isocost lines are introduced to explain the concept of producer equilibrium between inputs.
cost of production / Chapter 6(pindyck)RAHUL SINHA
topics covered
•Production and firm
•The production function
•Short run versus Long run
•Production with one variable input(Labour)
•Average product
•Marginal product
•The slopes of the production curve
•Law of diminishing marginal returns
•Production with two variable inputs
•Isoquant
•Isoquant Maps
•Diminishing marginal returns
•Substitution among inputs
•Returns to scale
•Describing returns to scale
The Cobb-Douglas production function models the relationship between an output and inputs like labor and capital. It assumes outputs increase with inputs but at a decreasing rate. The formula relates the natural log of output to the natural log of inputs with elasticity coefficients representing the percentage change in output from a 1% change in an input. If the coefficients sum to 1 there are constant returns to scale, less than 1 is decreasing returns, and more than 1 is increasing returns. An example using Taiwan agricultural data from 1958-1972 estimated elasticities of 1.5 for labor and 0.4 for capital, indicating increasing returns to scale.
1. Returns to scale refers to the change in total output resulting from a proportional change in all inputs.
2. There are three types of returns to scale: increasing, constant, and diminishing.
3. Increasing returns to scale occur when a 1% increase in all inputs leads to a more than 1% increase in output. Constant returns mean a proportional change in output, while diminishing returns mean output increases by less than the input increase.
This document provides an overview of production theory, including:
1. It defines production as the transformation of inputs (capital, labor, etc.) into outputs (goods and services) and discusses technical vs. economic efficiency.
2. It introduces the concepts of economic efficiency (lowest cost to produce output) and technological efficiency (cannot increase output without increasing inputs).
3. It explains that production theory applies the principles of constrained optimization, where firms aim to minimize costs or maximize output given constraints. This leads to the same rule for allocating inputs and technology choice.
4. It provides examples and explanations of key production concepts like production functions, production tables, short-run vs. long-run production,
This document provides an overview of production theory, including:
1. It defines production as the transformation of inputs (capital, labor, etc.) into outputs (goods and services) and discusses technical vs. economic efficiency.
2. It introduces the concepts of economic efficiency (lowest cost to produce output) and technological efficiency (cannot increase output without increasing inputs).
3. It explains that production theory applies the principles of constrained optimization, where firms aim to minimize costs or maximize output given constraints. This leads to the same rule for allocating inputs and technology choice.
4. It provides examples and explanations of key production concepts like production functions, production tables, short-run vs. long-run production,
III. work studyprinciples of Ergonomics,Krushna Ktk
This document provides an overview of production theory, including:
1. It defines production as the transformation of inputs (capital, labor, etc.) into outputs (goods and services). Managers aim for both technical and economic efficiency in production.
2. It introduces the concepts of economic efficiency (lowest cost of production) and technological efficiency (cannot increase output without more inputs). Production theory applies constrained optimization to minimize costs or maximize output.
3. It discusses production functions, production tables, short-run vs long-run production, returns to scale, total, average and marginal product, and the law of diminishing returns in the short-run. Isoquants and marginal rate of technical substitution are introduced
1. The document discusses production theory and the concepts of efficiency, production functions, and returns to scale. It provides definitions and examples.
2. Key aspects covered include the difference between technical and economic efficiency, definitions of production functions and how they relate inputs to outputs, concepts of short-run and long-run production, and how returns to scale are classified.
3. Production theory models are presented including isoquants, production tables, and different types of production functions like Cobb-Douglas and CES. Properties like elasticity of substitution and factor intensities are defined.
The document discusses using a vector autoregressive (VAR) model to analyze the relationship between stock returns and credit default swap (CDS) spreads of firms over time. It identifies stock returns and CDS spreads as dependent variables, and volatility, financial leverage, and other firm metrics as explanatory and control variables. The conceptual framework involves estimating two equations using ordinary least squares regression with one-period lagged returns as explanatory variables to avoid autocorrelation.
Theory of production attempts to explain how firms determine optimal input and output levels. It involves fundamental economic principles like the relationship between input and output prices and quantities. A production function is a precise mathematical equation relating total output to amounts of inputs. Common assumptions in production functions include constant technology and full efficiency. The Cobb-Douglas production function models output as a function of capital and labor. Isoquants illustrate combinations of two inputs that produce the same output level, and have properties like being downward sloping and convex to the origin. Marginal rate of technical substitution measures the rate at which one input can substitute for another while maintaining output.
Cost & Production Analysis For MBA Students.pptaviatordevendra
This document discusses key concepts related to production and cost analysis. It defines inputs, fixed and variable inputs, production functions, total product, average product, marginal product, and the law of diminishing marginal returns. It also covers production in the short run and long run, including isoquants, marginal rate of technical substitution, and returns to scale. The document analyzes costs including fixed, variable, and total costs. It discusses average and marginal costs, breakeven analysis, and economies of scale. Finally, it covers long run cost relationships including long run total cost, average cost, and marginal cost curves.
1) The simple growth model contains 5 main equations that relate key economic variables like investment, saving, capital stock, labor force, and output.
2) The aggregate production function shows that total output is a function of capital stock and labor force. As capital stock and labor increase, output will also increase.
3) Saving is determined by the saving rate multiplied by total income. Investment is equal to saving in a closed economy where saving must be used for investment or consumption.
The document discusses key concepts related to production and returns to scale. It can be summarized as follows:
1. Production involves using factors of production like labor, capital, land, and raw materials to transform inputs into outputs. The relationship between inputs and outputs is represented by production functions.
2. In the short run, at least one factor is fixed while others can vary. This relationship is explained by the law of variable proportions, which outlines three stages of production - increasing, constant, and diminishing returns.
3. In the long run, all factors are variable. The behavior of output with changes in all inputs is known as returns to scale and can exhibit increasing, constant, or diminishing returns depending
The document analyzes the production function of Tata Steel using regression analysis and correlation analysis. Regression analysis is used to estimate Tata Steel's production function from historical data collected over 8 years. The regression results show that Tata Steel's production function is capital intensive rather than labor intensive, with capital having a greater impact on production volume than labor. Correlation analysis also indicates production is more related to capital than labor. Therefore, it can be concluded that Tata Steel can increase production more by employing more capital as opposed to labor.
Page 2 of 41
THE MODEL SETUP AND QUESTIONS
GDP (the demand side of the economy) is given simply by
our standard expenditure equation:
Y = C + I + G +NX
For these notes we make the simplifying assumption that
there is no government or exchange of goods and
services with the rest of the world. Hence, G = NX = 0 and
GDP (again, the demand side of the economy) is given
simply by:
Y = C + I.
You might be asked to think about what happens if there
is government and exchange with the rest of the world at
some point though. So you have to fully understand the
model to be able to tweak it, in case and answer those
questions.
We’ll look at an economy with given “structural
characteristics”:
A given production function ==> the Cobb Douglas
production function that we have studied already.
This represents the supply side of the economy.
A given exogenous savings rate for the economy: s
A given population growth rate: n
A given depreciation rate of capital: d
Page 3 of 41
With this info we want to analyze the economy long run
behavior…that’s what growth is all about. We want to try
to understand the evolution of GDP and other
macroeconomic variable with a long time horizon
perspective.
In particular, we want to analyze changes in the economy
over time:
We have seen so far that to affect productivity we need
to understand physical capital and investment so:
– How do these structural characteristics interact
to determine the investment level, and the
evolution of the capital stock?
– How does the evolution of the capital stock
interact with population in determining the
change in production?
– We’ll discuss how these factors determine the
behavior of the economy period after period,
and the implication of this for its long run
evolution.
What are the level of physical capital, output,
investment and consumption in the long run for
a specific economy?
Page 4 of 41
THE EQUATIONS OF THE MODEL
We have 5 basic ingredients (equations) in the Solow
model (yes, you need to memorize those and be able to
work the math out). Thankfully, we have seen 4 of these
5 equations previously at some point during this course
so it is just a matter of putting them together, and
understanding how they interact:
1) The production function: We have seen this equation
concerning the production function already in the slides
for chapter 12. For these notes we will use the Cobb
Douglas production function which, again, you have seen
in details. It has the constant returns to scale property.
Formally:
A is the TFP (or technology).
is physical capital at period t
is labor at period t
0 < < 1 is called the capital share you should know
this already.
1 is called the labor share you should know this
already.
Only 2 factors of productions (K, L) are analyzed jointly
with technology (A) here. This is for simplicity. It is
Page 5 of 41
possible to make the model more complicated and
consider more factor of productions such as human
capital, knowledge capital, organiz.
Final Time series analysis part 2. pptxSHUBHAMMBA3
The document discusses key concepts in time series analysis including stationarity, trend, seasonality, autocorrelation, and partial autocorrelation. It defines stationarity as a time series having constant mean and variance, and that it is important for analysis and accurate predictions. Trend refers to the general tendency of data to increase or decrease over a long period. Seasonality describes regular, predictable changes that occur each calendar year. Autocorrelation measures the correlation between values of a time series at different points in time, while partial autocorrelation measures the direct correlation between two time points excluding intermediate values.
Vicinity Jobs’ data includes more than three million 2023 OJPs and thousands of skills. Most skills appear in less than 0.02% of job postings, so most postings rely on a small subset of commonly used terms, like teamwork.
Laura Adkins-Hackett, Economist, LMIC, and Sukriti Trehan, Data Scientist, LMIC, presented their research exploring trends in the skills listed in OJPs to develop a deeper understanding of in-demand skills. This research project uses pointwise mutual information and other methods to extract more information about common skills from the relationships between skills, occupations and regions.
The document discusses production concepts and cost analysis, including:
- Production functions show the relationship between inputs and outputs. Common types include Cobb-Douglas, CES, and Leontief functions.
- Total, average, and marginal products are defined for analyzing how output changes with variable inputs like labor.
- Short-run and long-run periods are distinguished based on whether inputs are fixed or variable.
- Isoquants and isocost lines are introduced to explain the concept of producer equilibrium between inputs.
cost of production / Chapter 6(pindyck)RAHUL SINHA
topics covered
•Production and firm
•The production function
•Short run versus Long run
•Production with one variable input(Labour)
•Average product
•Marginal product
•The slopes of the production curve
•Law of diminishing marginal returns
•Production with two variable inputs
•Isoquant
•Isoquant Maps
•Diminishing marginal returns
•Substitution among inputs
•Returns to scale
•Describing returns to scale
The Cobb-Douglas production function models the relationship between an output and inputs like labor and capital. It assumes outputs increase with inputs but at a decreasing rate. The formula relates the natural log of output to the natural log of inputs with elasticity coefficients representing the percentage change in output from a 1% change in an input. If the coefficients sum to 1 there are constant returns to scale, less than 1 is decreasing returns, and more than 1 is increasing returns. An example using Taiwan agricultural data from 1958-1972 estimated elasticities of 1.5 for labor and 0.4 for capital, indicating increasing returns to scale.
1. Returns to scale refers to the change in total output resulting from a proportional change in all inputs.
2. There are three types of returns to scale: increasing, constant, and diminishing.
3. Increasing returns to scale occur when a 1% increase in all inputs leads to a more than 1% increase in output. Constant returns mean a proportional change in output, while diminishing returns mean output increases by less than the input increase.
This document provides an overview of production theory, including:
1. It defines production as the transformation of inputs (capital, labor, etc.) into outputs (goods and services) and discusses technical vs. economic efficiency.
2. It introduces the concepts of economic efficiency (lowest cost to produce output) and technological efficiency (cannot increase output without increasing inputs).
3. It explains that production theory applies the principles of constrained optimization, where firms aim to minimize costs or maximize output given constraints. This leads to the same rule for allocating inputs and technology choice.
4. It provides examples and explanations of key production concepts like production functions, production tables, short-run vs. long-run production,
This document provides an overview of production theory, including:
1. It defines production as the transformation of inputs (capital, labor, etc.) into outputs (goods and services) and discusses technical vs. economic efficiency.
2. It introduces the concepts of economic efficiency (lowest cost to produce output) and technological efficiency (cannot increase output without increasing inputs).
3. It explains that production theory applies the principles of constrained optimization, where firms aim to minimize costs or maximize output given constraints. This leads to the same rule for allocating inputs and technology choice.
4. It provides examples and explanations of key production concepts like production functions, production tables, short-run vs. long-run production,
III. work studyprinciples of Ergonomics,Krushna Ktk
This document provides an overview of production theory, including:
1. It defines production as the transformation of inputs (capital, labor, etc.) into outputs (goods and services). Managers aim for both technical and economic efficiency in production.
2. It introduces the concepts of economic efficiency (lowest cost of production) and technological efficiency (cannot increase output without more inputs). Production theory applies constrained optimization to minimize costs or maximize output.
3. It discusses production functions, production tables, short-run vs long-run production, returns to scale, total, average and marginal product, and the law of diminishing returns in the short-run. Isoquants and marginal rate of technical substitution are introduced
1. The document discusses production theory and the concepts of efficiency, production functions, and returns to scale. It provides definitions and examples.
2. Key aspects covered include the difference between technical and economic efficiency, definitions of production functions and how they relate inputs to outputs, concepts of short-run and long-run production, and how returns to scale are classified.
3. Production theory models are presented including isoquants, production tables, and different types of production functions like Cobb-Douglas and CES. Properties like elasticity of substitution and factor intensities are defined.
The document discusses using a vector autoregressive (VAR) model to analyze the relationship between stock returns and credit default swap (CDS) spreads of firms over time. It identifies stock returns and CDS spreads as dependent variables, and volatility, financial leverage, and other firm metrics as explanatory and control variables. The conceptual framework involves estimating two equations using ordinary least squares regression with one-period lagged returns as explanatory variables to avoid autocorrelation.
Theory of production attempts to explain how firms determine optimal input and output levels. It involves fundamental economic principles like the relationship between input and output prices and quantities. A production function is a precise mathematical equation relating total output to amounts of inputs. Common assumptions in production functions include constant technology and full efficiency. The Cobb-Douglas production function models output as a function of capital and labor. Isoquants illustrate combinations of two inputs that produce the same output level, and have properties like being downward sloping and convex to the origin. Marginal rate of technical substitution measures the rate at which one input can substitute for another while maintaining output.
Cost & Production Analysis For MBA Students.pptaviatordevendra
This document discusses key concepts related to production and cost analysis. It defines inputs, fixed and variable inputs, production functions, total product, average product, marginal product, and the law of diminishing marginal returns. It also covers production in the short run and long run, including isoquants, marginal rate of technical substitution, and returns to scale. The document analyzes costs including fixed, variable, and total costs. It discusses average and marginal costs, breakeven analysis, and economies of scale. Finally, it covers long run cost relationships including long run total cost, average cost, and marginal cost curves.
1) The simple growth model contains 5 main equations that relate key economic variables like investment, saving, capital stock, labor force, and output.
2) The aggregate production function shows that total output is a function of capital stock and labor force. As capital stock and labor increase, output will also increase.
3) Saving is determined by the saving rate multiplied by total income. Investment is equal to saving in a closed economy where saving must be used for investment or consumption.
The document discusses key concepts related to production and returns to scale. It can be summarized as follows:
1. Production involves using factors of production like labor, capital, land, and raw materials to transform inputs into outputs. The relationship between inputs and outputs is represented by production functions.
2. In the short run, at least one factor is fixed while others can vary. This relationship is explained by the law of variable proportions, which outlines three stages of production - increasing, constant, and diminishing returns.
3. In the long run, all factors are variable. The behavior of output with changes in all inputs is known as returns to scale and can exhibit increasing, constant, or diminishing returns depending
The document analyzes the production function of Tata Steel using regression analysis and correlation analysis. Regression analysis is used to estimate Tata Steel's production function from historical data collected over 8 years. The regression results show that Tata Steel's production function is capital intensive rather than labor intensive, with capital having a greater impact on production volume than labor. Correlation analysis also indicates production is more related to capital than labor. Therefore, it can be concluded that Tata Steel can increase production more by employing more capital as opposed to labor.
Page 2 of 41
THE MODEL SETUP AND QUESTIONS
GDP (the demand side of the economy) is given simply by
our standard expenditure equation:
Y = C + I + G +NX
For these notes we make the simplifying assumption that
there is no government or exchange of goods and
services with the rest of the world. Hence, G = NX = 0 and
GDP (again, the demand side of the economy) is given
simply by:
Y = C + I.
You might be asked to think about what happens if there
is government and exchange with the rest of the world at
some point though. So you have to fully understand the
model to be able to tweak it, in case and answer those
questions.
We’ll look at an economy with given “structural
characteristics”:
A given production function ==> the Cobb Douglas
production function that we have studied already.
This represents the supply side of the economy.
A given exogenous savings rate for the economy: s
A given population growth rate: n
A given depreciation rate of capital: d
Page 3 of 41
With this info we want to analyze the economy long run
behavior…that’s what growth is all about. We want to try
to understand the evolution of GDP and other
macroeconomic variable with a long time horizon
perspective.
In particular, we want to analyze changes in the economy
over time:
We have seen so far that to affect productivity we need
to understand physical capital and investment so:
– How do these structural characteristics interact
to determine the investment level, and the
evolution of the capital stock?
– How does the evolution of the capital stock
interact with population in determining the
change in production?
– We’ll discuss how these factors determine the
behavior of the economy period after period,
and the implication of this for its long run
evolution.
What are the level of physical capital, output,
investment and consumption in the long run for
a specific economy?
Page 4 of 41
THE EQUATIONS OF THE MODEL
We have 5 basic ingredients (equations) in the Solow
model (yes, you need to memorize those and be able to
work the math out). Thankfully, we have seen 4 of these
5 equations previously at some point during this course
so it is just a matter of putting them together, and
understanding how they interact:
1) The production function: We have seen this equation
concerning the production function already in the slides
for chapter 12. For these notes we will use the Cobb
Douglas production function which, again, you have seen
in details. It has the constant returns to scale property.
Formally:
A is the TFP (or technology).
is physical capital at period t
is labor at period t
0 < < 1 is called the capital share you should know
this already.
1 is called the labor share you should know this
already.
Only 2 factors of productions (K, L) are analyzed jointly
with technology (A) here. This is for simplicity. It is
Page 5 of 41
possible to make the model more complicated and
consider more factor of productions such as human
capital, knowledge capital, organiz.
Final Time series analysis part 2. pptxSHUBHAMMBA3
The document discusses key concepts in time series analysis including stationarity, trend, seasonality, autocorrelation, and partial autocorrelation. It defines stationarity as a time series having constant mean and variance, and that it is important for analysis and accurate predictions. Trend refers to the general tendency of data to increase or decrease over a long period. Seasonality describes regular, predictable changes that occur each calendar year. Autocorrelation measures the correlation between values of a time series at different points in time, while partial autocorrelation measures the direct correlation between two time points excluding intermediate values.
Vicinity Jobs’ data includes more than three million 2023 OJPs and thousands of skills. Most skills appear in less than 0.02% of job postings, so most postings rely on a small subset of commonly used terms, like teamwork.
Laura Adkins-Hackett, Economist, LMIC, and Sukriti Trehan, Data Scientist, LMIC, presented their research exploring trends in the skills listed in OJPs to develop a deeper understanding of in-demand skills. This research project uses pointwise mutual information and other methods to extract more information about common skills from the relationships between skills, occupations and regions.
Solution Manual For Financial Accounting, 8th Canadian Edition 2024, by Libby...Donc Test
Solution Manual For Financial Accounting, 8th Canadian Edition 2024, by Libby, Hodge, Verified Chapters 1 - 13, Complete Newest Version Solution Manual For Financial Accounting, 8th Canadian Edition by Libby, Hodge, Verified Chapters 1 - 13, Complete Newest Version Solution Manual For Financial Accounting 8th Canadian Edition Pdf Chapters Download Stuvia Solution Manual For Financial Accounting 8th Canadian Edition Ebook Download Stuvia Solution Manual For Financial Accounting 8th Canadian Edition Pdf Solution Manual For Financial Accounting 8th Canadian Edition Pdf Download Stuvia Financial Accounting 8th Canadian Edition Pdf Chapters Download Stuvia Financial Accounting 8th Canadian Edition Ebook Download Stuvia Financial Accounting 8th Canadian Edition Pdf Financial Accounting 8th Canadian Edition Pdf Download Stuvia
Independent Study - College of Wooster Research (2023-2024) FDI, Culture, Glo...AntoniaOwensDetwiler
"Does Foreign Direct Investment Negatively Affect Preservation of Culture in the Global South? Case Studies in Thailand and Cambodia."
Do elements of globalization, such as Foreign Direct Investment (FDI), negatively affect the ability of countries in the Global South to preserve their culture? This research aims to answer this question by employing a cross-sectional comparative case study analysis utilizing methods of difference. Thailand and Cambodia are compared as they are in the same region and have a similar culture. The metric of difference between Thailand and Cambodia is their ability to preserve their culture. This ability is operationalized by their respective attitudes towards FDI; Thailand imposes stringent regulations and limitations on FDI while Cambodia does not hesitate to accept most FDI and imposes fewer limitations. The evidence from this study suggests that FDI from globally influential countries with high gross domestic products (GDPs) (e.g. China, U.S.) challenges the ability of countries with lower GDPs (e.g. Cambodia) to protect their culture. Furthermore, the ability, or lack thereof, of the receiving countries to protect their culture is amplified by the existence and implementation of restrictive FDI policies imposed by their governments.
My study abroad in Bali, Indonesia, inspired this research topic as I noticed how globalization is changing the culture of its people. I learned their language and way of life which helped me understand the beauty and importance of cultural preservation. I believe we could all benefit from learning new perspectives as they could help us ideate solutions to contemporary issues and empathize with others.
[4:55 p.m.] Bryan Oates
OJPs are becoming a critical resource for policy-makers and researchers who study the labour market. LMIC continues to work with Vicinity Jobs’ data on OJPs, which can be explored in our Canadian Job Trends Dashboard. Valuable insights have been gained through our analysis of OJP data, including LMIC research lead
Suzanne Spiteri’s recent report on improving the quality and accessibility of job postings to reduce employment barriers for neurodivergent people.
Decoding job postings: Improving accessibility for neurodivergent job seekers
Improving the quality and accessibility of job postings is one way to reduce employment barriers for neurodivergent people.
Economic Risk Factor Update: June 2024 [SlideShare]Commonwealth
May’s reports showed signs of continued economic growth, said Sam Millette, director, fixed income, in his latest Economic Risk Factor Update.
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Independent Study - College of Wooster Research (2023-2024)
Solow completa - updated.pptx
1. ECONOMIC GROWTH: CONTEXT
We take it for granted but its new in terms of human history
While it is common to think about growth today as being somehow natural, even
expected – in fact, if world growth falls from 3.5 to 3.2%, it is perceived as a big crisis –
it is worthwhile to acknowledge that this was not always the case.
Pretty much until the end of the 18th century growth was quite low, if it happened at
all.
In fact, it was so low that people could not see it during their lifetimes
Then, towards the turn of the 18th century, something happened that created
explosive economic growth as the world had never seen before.
2. Source: Campante, F., Sturzenegger, F. y Velasco, A. (1) (2021). Advanced Macroeconomics: An Easy Guide LSE Press
3. Source: Campante, F., Sturzenegger,
F. y Velasco, A. (1) (2021). Advanced
Macroeconomics: An Easy Guide
LSE Press
5. SOLOW GROWTH MODEL -
INTRODUCTION
• Traditional model
• Other models (more advanced) are often best understood
in relation to the Solow model (often called Solow-Swan
model)
• The Solow model has no optimization in it:
• It takes the savings rate as exogenous and constant
• (later we will relax these assumptions and add other
endogenous elements to the model).
6. SOLOW GROWTH MODEL -
ASSUMPTIONS
INPUTS AND OUTPUTS
• Model focuses on 4 variables: Y, K, L and A
• A is knowledge, or the “effectiveness of labour”
• At a point in time, the economy has some inputs of K, L and A – and
these are combined to produce Y:
𝑌 𝑡 = 𝐹 𝐾 𝑡 , 𝐴 𝑡 𝐿(𝑡)
• Labour’s value in output depends on its effectiveness, A.
• t denotes time
• So output changes over time as inputs change
7. SOLOW GROWTH MODEL -
ASSUMPTIONS
INPUTS AND OUTPUTS
• Output obtained from a given stock of K and L can only be
augmented if there is technological progress => A↑
• A x L is multiplicative
• A.L = effective labour
• Technological progress enters into the model as labour
augmenting.
• The implication of this is that the ratio of K/Y eventually “settles
down” in the longer run (we will see this in the phase diagrams):
• It basically makes the analysis simpler
8. SOLOW GROWTH MODEL -
ASSUMPTIONS
ASSUMPTIONS CONCERNING THE PRODUCTION
FUNTION
• Central assumptions relate to the properties of the
production function (and the evolution of inputs).
• We assume Constant Returns to Scale (CRS) in A.L and
K
• That means that doubling K and AL (that is, 2xL with
A fixed) doubles the output
• Generally:
9. SOLOW GROWTH MODEL -
ASSUMPTIONS
ASSUMPTIONS CONCERNING THE PRODUCTION FUNTION
• Why CRS?
• This is really a combination of 2 separate assumptions
1. The Economy is big enough that the gains from specialization have been exhausted (smaller
economies might still gain)
2. Inputs other than K and L and A are relatively unimportant (E.g.: no land etc).
• Assumption of CRS allows us to work with the production function in its
intensive form:
• Setting 𝑐 =
1
𝐴𝐿
:
• 𝑭
𝑲
𝑨𝑳
, 𝟏 =
1
𝐴𝐿
𝐹 𝐾, 𝐴𝐿
10. SOLOW GROWTH MODEL -
ASSUMPTIONS
ASSUMPTIONS CONCERNING THE PRODUCTION
FUNTION
•
1
𝐴𝐿
𝐹 𝐾, 𝐴𝐿 = 𝑭
𝑲
𝑨𝑳
, 𝟏
•
𝑲
𝑨𝑳
is the amount of capital per unit of effective labour
•
𝐹 𝐾,𝐴𝐿
𝐴𝐿
=
𝑌
𝐴𝐿
= the output per unit of effective labour
• If we define 𝓀 =
𝑲
𝑨𝑳
, 𝑦 =
𝒀
𝑨𝑳
, and 𝒇 𝓀 = 𝑭 𝓀, 𝟏
• Then we get the production function in its intensive form:
𝑦 = 𝒇 𝓀
11. SOLOW GROWTH MODEL -
ASSUMPTIONS
ASSUMPTIONS CONCERNING THE PRODUCTION
FUNTION
𝑦 = 𝒇 𝓀
• What is this?
• This is just the OUTPUT PER UNIT OF EFFECTIVE LABOUR
• Output per unit of effective labour is a function of capital
per unit of effective labour
• This makes sense when you think of it.
• It just means that the output that we are capable of
producing is proportional to the ratio of the main inputs
12. SOLOW GROWTH MODEL -
ASSUMPTIONS
ASSUMPTIONS CONCERNING THE PRODUCTION
FUNTION
• Note that these new variables are not of interest in
and of themselves, but they are tools to learn about
the variables that we are interested in
• We will see that the easiest way to analyse the model
is to focus on the behaviour of 𝓀, rather than on K,
and AL
• Example: the behaviour of Y/L (per worker) = A(Y/AL) or
Af(𝓀)
13. SOLOW GROWTH MODEL -
ASSUMPTIONS
ASSUMPTIONS CONCERNING THE PRODUCTION FUNTION
• INTUITION?
• Imagine that the economy can be divided into little mini
economies, AL of them (we can do this because there are CRTS):
ECONOMY (total)
1,
𝐾
𝐴𝐿
1,
𝐾
𝐴𝐿 1,
𝐾
𝐴𝐿
1,
𝐾
𝐴𝐿
Each mini economy
is for one unit of
effective labour
Amount of effective
labour per mini
economy
Amount of capital
per mini economy
Each mini economy produces
1/AL as much as the big economy
So output depends only on the
amount of K per unit of AL, and
nothing more (i.e.: not on the size
of the economy)
15. SOLOW GROWTH MODEL -
ASSUMPTIONS
ASSUMPTIONS CONCERNING THE PRODUCTION FUNTION
MATHEMATICALLY:
• Since 𝐹 𝐾, 𝐴𝐿 = 𝐴𝐿 𝐹 𝐾/𝐴𝐿 the marginal product of capital,
𝜕𝐹 𝐾,𝐴𝐿
𝜕𝐾
:
𝜕𝐹 𝐾, 𝐴𝐿
𝜕𝐾
= 𝐴𝐿. 𝑓
′
𝐾
𝐴𝐿 ×
1
𝐴𝐿
= 𝒇′ 𝓀
• What is the implication?
• The MPK is positive, but is declining in 𝐾/𝐴𝐿 (think of a typical
factory example)
• A typical example of this type of function is a COBB-DOUGLAS
function:
𝐹 𝐾, 𝐴𝐿 = 𝐾∝(𝐴𝐿) 1−∝
Where 0 < α < 1
16. SOLOW GROWTH MODEL - ASSUMPTIONS
ASSUMPTIONS CONCERNING THE PRODUCTION FUNTION
• This gives us CRS:
𝐹 𝑐𝐾 𝑡 , 𝑐𝐴 𝑡 𝐿(𝑡) = 𝑐𝐾 ∝(𝑐𝐴𝐿) 1−∝
= 𝑐𝛼 𝑐1−𝛼𝐾𝛼(𝐴𝐿)1−𝛼
= 𝑐𝐹(𝐾, 𝐴𝐿)
• To get the intensive form, divide both sides by AL:
𝒇 𝓀 = 𝑭
𝑲
𝑨𝑳
, 𝟏
=
𝑲
𝑨𝑳
𝛼
= 𝓀𝛼
𝒇′ 𝓀 = 𝛼𝓀 𝛼−1
(meets assumptions)
17. EVOLUTION OF INPUTS INTO
PRODUCTION
• How do L, A and K evolve?
• The model is set in continuous time; variables are defined at every
point of time
• Initial levels are exogenous (taken as given)
• Labour and Knowledge grow at constant rates:
𝐿 𝑡 = 𝑛𝐿 𝑡
𝐴 𝑡 = 𝑔𝐴 𝑡
• ‘n’ and ‘g’ are exogenous (we cannot choose them in this model)
• And a dot above a variable signifies it’s variance over time: 𝑋 𝑡 =
𝑑𝑋(𝑡)
𝑑𝑡
18. EVOLUTION OF INPUTS INTO
PRODUCTION
• What is a “growth rate”?
• A proportional rate of change:
𝑋 𝑡
𝑋(𝑡)
• Implication?
•
𝐿 𝑡
𝐿 𝑡
= 𝑛 and
𝐴 𝑡
𝐴 𝑡
= 𝑔.
• As they are both constants, we have constant growth rates
• Another important piece of information:
• Growth rate of a variable = rate of change of its natural log
•
𝐿 𝑡
𝐿 𝑡
=
𝑑 ln 𝐿 𝑡
𝑑𝑡
• See Romer for the proof using the chain rule.
19. EVOLUTION OF INPUTS INTO
PRODUCTION
• What is the evolution path of L and A?
• Because the variables’ growth rates are ∆𝑙𝑛:
𝑙𝑛𝐿 𝑡 − 𝑙𝑛𝐿 0 = 𝑛𝑡
𝑙𝑛𝐴 𝑡 − 𝑙𝑛𝐴 0 = 𝑔𝑡
• Applying the exponent to both sides (to get rid of the “ln”’s)
𝑙𝑛𝐿 𝑡 − 𝑙𝑛𝐿 0 = 𝑙𝑛
𝐿 𝑡
𝐿 0
= 𝑛𝑡 → 𝐿 𝑡 = 𝐿 0 𝑒𝑛𝑡
𝑙𝑛𝐴 𝑡 − 𝑙𝑛𝐴 0 = 𝑙𝑛
𝐴 𝑡
𝐴 0
= 𝑔𝑡 → 𝐴 𝑡 = 𝐴 0 𝑒𝑔𝑡
Thus, 𝐿 and 𝐴 grow exponentially
20. EVOLUTION OF INPUTS INTO
PRODUCTION
• Output can be used for consumption or investment.
• In this model, the fraction of output devoted to Investment (through
savings = s) is exogenous and constant
• And one unit of output devoted to investment => one unit of capital.
• In addition, capital depreciates at the rate 𝛿. So:
𝐾 𝑡 = 𝑠𝑌 𝑡 − 𝛿𝐾 𝑡
• Although there are no restrictions placed on n, g, and 𝛿 individually,
their SUM (𝑛 + 𝑔 + 𝛿) is assumed to be positive.
21. A SUMMARY OF THE MODEL
• Very simplified
• Only a single good
• No government
• Fluctuations in employment are ignored
• Production only incorporates 3 inputs
• 𝑠, 𝑛, 𝑔 and 𝛿 are all assumed to be constant
• However, while simple, this is not always a defect
• A model’s job is to provide insights about CERTAIN ASPECTS of the world
• We cannot use a model that is too complicated to understand
• If the simplification does not give INCORRECT answers, then the lack of realization can be a
virtue.
22. THE DYNAMICS OF THE MODEL
• The evolution of two of the three inputs into production (L and A) is
exogenous
• The main analysis of the dynamics of the model is then focused on K as this
is the “DRIVING VARIABLE” in the model.
• THE DYNAMICS OF K
• We will focus on the capital stock per unit of effective labour, 𝓀
• Since 𝓀=K/AL, we can use the quotient rule to find:
• 𝓀 𝑡 =
𝐾(𝑡)
𝐴 𝑡 𝐿(𝑡)
−
𝐾 (𝑡)
[𝐴 𝑡 𝐿 𝑡 ]2 𝐴 𝑡 𝐿 𝑡 + 𝐿 𝑡 𝐴 𝑡
• =
𝐾(𝑡)
𝐴 𝑡 𝐿(𝑡)
−
𝐾 𝑡
𝐴 𝑡 𝐿 𝑡
𝐿 𝑡
𝐿 𝑡
−
𝐾 𝑡
𝐴 𝑡 𝐿 𝑡
𝐴 𝑡
𝐴 𝑡
.
ℎ 𝑥 =
𝑓(𝑥)
𝑔(𝑥)
ℎ′
𝑥 =
𝑔 𝑥 𝑓′
𝑥 − 𝑓 𝑥 𝑔′(𝑥)
𝑔(𝑥)2
23. THE DYNAMICS OF THE MODEL
•
𝐾 𝑡
𝐴 𝑡 𝐿 𝑡
is just 𝓀.
𝐿 𝑡
𝐿 𝑡
is 𝑛 and
𝐴 𝑡
𝐴 𝑡
is 𝑔. And we know that 𝐾 𝑡 = 𝑠𝑌 𝑡 −
𝛿𝐾 𝑡 .
• We can substitute these elements into the equation to get:
• 𝓀 𝑡 =
𝑠𝑌 𝑡 −𝛿𝐾 𝑡
𝐴 𝑡 𝐿(𝑡)
− 𝓀 𝑡 𝑛 − 𝓀 𝑡 𝑔 = 𝑠
𝑌 𝑡
𝐴 𝑡 𝐿 𝑡
− 𝛿𝓀 𝑡 −𝑛𝓀 𝑡 − 𝑔𝓀 𝑡
• And, because we know that
𝑌 𝑡
𝐴 𝑡 𝐿 𝑡
= 𝑦 = 𝑓(𝓀):
𝓴 𝒕 = 𝒔𝒇(𝓴 𝒕 ) − 𝜹𝓴 𝒕 − 𝒏𝓴 𝒕 − 𝒈𝓴 𝒕
24. THE DYNAMICS OF THE MODEL
𝓴 𝒕 = 𝒔𝒇(𝓴 𝒕 ) − 𝒏 𝒕 + 𝒈 𝒕 + 𝜹 𝒕 𝓴
This equation is the key equation of the Solow model
It says that the rate of change of the capital stock per unit of effective labor is the difference
between 2 terms:
• 𝑠𝑓(𝓀 𝑡 ): the fraction of output per unit of effective labour that is saved – this is ACTUAL INVESTMENT
• 𝓀 𝑛 + 𝑔 + 𝛿 : the amount of investment that must be done just to keep 𝓀 at it’s existing level – this is
BREAK EVEN INVESTMENT
There are 2 reasons that some investment is needed to keep 𝓀 from falling:
• Existing capital is depreciating: 𝛿𝓀 𝑡
• The quantity of effective labour is growing: 𝑛𝓀 𝑡 + 𝑔𝓀 𝑡
• To keep 𝓀 constant requires that the growth in population and technology as well as
depreciation is accounted for.
25. THE DYNAMICS OF THE MODEL
In other words, the rate of capital must grow at a rate larger than the sum of
the:
• depreciation rate,
• the population growth rate and
• the rate of growth in technology
In order to maintain the same K/AL ratio which is the key driver of growth in the
model.
When actual investment > break even investment, 𝓀 is rising
When actual investment < break even investment, 𝓀 is falling
In equality, 𝓀 is constant.
26. GRAFICAR:
• 𝒔𝒇 𝓴 𝒕
• 𝒏 𝒕 + 𝒈 𝒕 + 𝜹 𝒕 𝓴
• En el espacio: horizontal
= k,
• vertical = inversión/AL,
Y/AL
•
s, y
k
(n+g+δ)k
27. THE
DYNAMICS
OF THE
MODEL
At f(0)=0, actual and break even
investment are equal at zero.
ACTUAL INVESTMENT:
We know from our assumptions
regarding the production
function that when 𝓀 is close to
zero, f’(𝓀) is very large.
But as 𝓀 increases, the marginal
product of 𝓀 decreases until it
falls towards zero.
28. THE DYNAMICS OF THE MODEL
At some point the slope of the ACTUAL INVESTMENT line falls below that
of the BREAK EVEN INVESTMENT line (which has a constant slope).
So the two lines must cross.
The fact that f’’(𝓀) < 0 implies that the two lines only intersect ONCE for
𝓀 > 0.
We call the level of 𝓀 the point of intersection 𝓀*.
𝓀* is the point at which break even and actual investment are equal and
we are in equilibrium.
29. THE DYNAMICS
OF THE MODEL
This is the phase diagram for 𝓀
What is a phase diagram?
It is when we map the change in
the variable as a function of the
level of the same variable (𝓀 in
terms of 𝓀).
If 𝓀 < 𝓀*, 𝓀 is positive
If 𝓀 > 𝓀*, 𝓀 is negative and capital
per unit of effective labour is
falling.
𝓀 = 𝓀*, 𝓀 is zero.
Thus we have convergence to 𝓀*.
30. DYNAMICS: THE BALANCED GROWTH
PATH
• Since 𝓀 converges to 𝓀*, it is of value to know how the other variables behave when 𝓀= 𝓀*.
• By assumption, L and A are growing at their constant rates of 𝑛 and 𝑔, respectively.
• The capital stock?
• Becuase K = AL𝓀, and 𝓀 is constant at 𝓀*, K is growing at a rate of 𝑛 + 𝑔. (i.e.:
𝐾
𝐾
= 𝑛 + 𝑔)
• So we have K and AL growing at the same rate of 𝑛 + 𝑔.
• And becuase we have constant returns to scale, we know that Y is also growing at 𝑛 + 𝑔.
• What about K/L and Y/L?
• They are growing at rate 𝑔 (the rate of growth of technology).
31. DYNAMICS: THE BALANCED GROWTH
PATH
• The implication is that the economy, regardless of where it starts from,
will end up converging to a point where all variables grow at the same
constant rate.
• This is called the BALANCED GROWTH PATH.
• Importantly, the key implication of this is that on this path, the only way
that we can have an increase in the output per worker is by achieving
technological progress.
32. IMPACT OF A CHANGE IN THE SAVING
RATE
• The most policy relevant parameter in the model is the savings rate.
• Why policy relevant?
• Because in practice, government policies can impact on the savings rate in
many ways, for eg.:
• the division of its budget between consumption and investment,
• it’s funding source for the budget: taxes or borrowing
• How savings and invesment are treated for tax purposes
• All of these policy settings will impact on the fraction of output that is
invested, that is s.
• So it is natural to examine the impact of a change in s on the model.
33. IMPACT OF A CHANGE IN THE SAVING
RATE
• To investigate this, we will first assume that the economy is
initially on a balanced growth path
• The shock will be a permanent increase in the rate of savings.
• We can then investigate how the model behaves when it has
been “tipped off” it’s balanced growth path.
IMPACT ON OUTPUT
• The increase in s shifts the actual investment line upwards, and
so 𝓀* rises.
• But 𝓀 does not immediately jump to the new value of 𝓀*….
34. IMPACT OF A
CHANGE IN THE
SAVING RATE
• When 𝓀= 𝓀*OLD, with the new
savings rate, actual investment
exceeds break even investment:
• 𝑠𝑓(𝓀 𝑡 ) > 𝓀 𝑛 + 𝑔 + 𝛿
• That is, more resources are
devoted towards investment
than are needed just to keep 𝓀
constant.
• 𝓀 is now positive and 𝓀 begins
to rise –it rises until it reaches
𝓀*NEW.
• At this point it will remain
constant again.
35. IMPACT OF A
CHANGE IN THE
SAVING RATE
• These results are summarized in
these 3 panels.
• t0 denotes the time that there is
a jump in the savings rate. By
assumption it remains constant
thereafter.
• Since the jump in s causes
actual investment to exceed
break-even investment by a
strictly positive amount, 𝓀
jumps from zero to a strictly
positive amount.
• 𝓀 rises gradually from the old
value of 𝓀 *to the new value,
and 𝓀 falls gradually back to
zero
36. IMPACT OF A
CHANGE IN THE
SAVING RATE
• What about the behavior of output per worker, Y/L = Af (𝓀)?
• When 𝓀 is constant, Y/L grows at rate g, the growth rate of A.
• When 𝓀 is increasing, Y/L grows both because A is increasing
and because 𝓀 is increasing.
• Thus its growth rate exceeds g.
• When 𝓀 reaches the new value of 𝓀*, however, again only the
growth of A contributes to the growth of Y/L, and so the
growth rate of Y/L returns to g.
• Thus a permanent increase in the saving rate produces a
temporary increase in the growth rate of output per worker:
• 𝓀 is rising for a time, but eventually it increases to the
point where the additional saving is devoted entirely
to maintaining the higher level of 𝓀.
37. IMPACT OF A CHANGE IN THE SAVING
RATE
• In sum, a change in the saving rate has a level effect but
not a growth effect:
• it changes the economy’s balanced growth path, and
thus the level of output per worker at any point in time
• But it does not affect the growth rate of output per
worker on the balanced growth path.
• Indeed, in the Solow model only changes in the rate of
technological progress have growth effects; all other
changes have only level effects.
38. THE IMPACT ON CONSUMPTION
• As a household’s welfare is also dependent on
consumption, we are interested in the behaviour of
consumption as well.
• Consumption may be more important for households
than output.
• Consumption, C, per unit of effective labour (AL) =
𝑓 𝓀 ∗ 1 − 𝑠 .
• This is, C/AL is equal to the amount of capital per unit
of effective labour multiplied by the fraction of output
that is not saved.
• Thus, since s changes discontinuously at t0 and 𝓀 does
not, initially consumption per unit of effective labor
jumps downward.
• Consumption then rises gradually as k rises and s
remains at its higher level.
39. THE IMPACT ON CONSUMPTION
• Whether consumption eventually exceeds its level before the rise in s is not immediately clear.
• Let c∗ denote consumption per unit of effective labor on the balanced growth path.
• c∗ equals output per unit of effective labor, 𝑓 𝓀∗ , minus investment per unit of effective labor,
s𝑓 𝓀∗
• On the balanced growth path, actual investment [s𝑓 𝓀∗ ] equals break-even investment, (𝑛 + 𝑔 +
40. THE IMPACT ON CONSUMPTION
This implies:
𝜕𝑐∗
𝜕𝑠
=
𝜕𝑐∗
𝜕𝓀
.
𝜕𝓀∗
𝜕𝑠
𝜕𝑐∗
𝜕𝑠
= 𝑓′(𝓀∗
𝑠, 𝑛, 𝑔, 𝛿 ) − (𝑛 + 𝑔 + 𝛿) .
𝜕𝓀∗
𝑠, 𝑛, 𝑔, 𝛿
𝜕𝑠
What does this tell us?
• We know that
𝜕𝓀∗ 𝑠,𝑛,𝑔,𝛿
𝜕𝑠
is positive.
So to know whether
𝜕𝑐∗
𝜕𝑠
is positive or not, we need to know whether
𝑓′(𝓀∗
𝑠, 𝑛, 𝑔, 𝛿 ) − (𝑛 + 𝑔 + 𝛿) is positive, or:
𝑓′
𝓀∗
𝑠, 𝑛, 𝑔, 𝛿 > (𝑛 + 𝑔 + 𝛿)
41. THE IMPACT ON CONSUMPTION
Intuitively, when 𝓀 rises, the marginal increase in intensive capital
must be sufficient to compensate the marginal reduction in
intensive capital due to the impacts of 𝑛 + 𝑔 + 𝛿 on the level of
K/AL.
• If f’(𝓴∗
) is less than 𝑛 + 𝑔 + 𝛿, then the additional output from the
increased capital is not enough to maintain both consumption and
capital at their higher levels
• The proportionate level of C/AL must fall to allow for the higher
K/AL.
• If f’(𝓴∗) exceeds 𝑛 + 𝑔 + 𝛿, there is more than enough
additional output to maintain 𝓀 at its higher level, and so
consumption rises.
42. THE IMPACT ON CONSUMPTION
• In reality, 𝑓′(𝓀∗ 𝑠, 𝑛, 𝑔, 𝛿 ) can be greater or less than (𝑛 +
𝑔 + 𝛿).
• We can look at the effects in the diagram.
• The figure in the next slide shows not only (𝑛 + 𝑔 + 𝛿)𝓀
and 𝑠𝑓(𝓀), but also 𝑓(𝓀).
• Since consumption on the balanced growth path equals
output less breakeven investment, 𝑐∗
is the distance
between 𝑓 𝓀 and (𝑛 + 𝑔 + 𝛿)𝓀 at 𝓀 = 𝓀∗
44. The figure shows the determinants of c*
for three different values of s (relating to
three different levels of 𝓴∗
.
A: s is high, and so 𝓀∗ is high and f’(𝓀∗) is
less than 𝑛 + 𝑔 + 𝛿.
As a result, an increase in the saving rate
lowers consumption even when the
economy has reached its new balanced
growth path
s is low, 𝓀∗ is
low, f’(𝓀∗
) is
greater than
𝑛 + 𝑔 + 𝛿, and
an increase in s
raises
consumption in
the long run.
s is at the level that causes f’(𝓀∗
)
to just equal 𝑛 + 𝑔 + 𝛿 —that is,
the tangent of the two lines are
parallel at𝓀 = 𝓀∗
. In this case, a
marginal change in s has no effect
on consumption in the long run,
and consumption is at its
maximum possible level among
balanced growth paths.
45. THE IMPACT ON CONSUMPTION
• This value of 𝓀∗
is known as the golden-rule level of
the capital stock.
• Of course, in the Solow model, where saving is
exogenous, there is no more reason to expect the
capital stock on the balanced growth path to equal
the golden rule level than there is to expect it to equal
any other possible value.
46. IMPLICATIONS OF THE SOLOW MODEL
• The Solow model shows that capital accumulation by itself cannot sustain growth in per capita
income in the long run: diminishing marginal returns.
• At some point the capital stock becomes large enough that a given savings rate can only provide
just enough new capital to replenish ongoing depreciation and increases in labour force.
• Alternatively, if we introduce exogenous technological change (productivity), we can generate long-
run growth in income per capita, but we do not really explain it – we are not explaining those
differences, we are just assuming them!
• As a result, nothing within the model tells you what policy can do about growth in the long run.
• We do learn a lot about growth in the transition to the long run, about differences in income levels,
and how policy can affect those things.
(i) convergence – the model predicts conditional convergence;
(ii) dynamic inefficiency – it is possible to save too much in this model; and
(iii) long-run differences in income – they seem to have a lot to do with differences in productivity.