This document summarizes the Solow growth model. It begins by outlining the basic building blocks of the model, including a production function, constant returns to scale, exogenous population growth, capital depreciation, savings rate, and technological progress. It then describes the mechanics of the model, showing how capital stock evolves over time and reaches a steady state. It derives expressions for the steady state capital stock and output in the special Cobb-Douglas case. It also discusses the concept of the "Golden Rule" savings rate that maximizes steady state consumption. Finally, it extends the model to include market forces and derives expressions for factor prices and income shares in a competitive market framework.
Taking progressivity research to European CommissionGRAPE
Social security is essentially about insurance. First, it gives insurance against mortality risk by providing annualization. Second, it provides partial insurance against low-income realization by providing intra-cohort redistribution. However, such redistribution is costly because it distorts labor supply incentives. When the link between social security contribution and future benefits becomes weaker, we treat contributions more and more as taxes, not as implicit savings.
With rising longevity, the social security system in many countries is bound to be put under unprecedented fiscal strain. Therefore, some changes appear imperative. Reforms proposed in the literature usually involve linking pensions to individual contributions, thus improving efficiency at the expense of the insurance loss.
In this paper, we propose a novel way of reforming social security. Our reform consists of two elements. First, we replace the redistributive defined benefit payout scheme with a defined contribution payout scheme, which links individual contributions to individual benefits. It raises efficiency as it reduces labor market distortions associated with contribution rates. Second, we propose to accompany this social security reform with adjustments in the progressiveness of labor taxation. Specifically, we increase progression in income taxes.
Thus, we partially replace the redistribution otherwise provided by social security with the one provided within the tax system.
We show that more redistribution during the working periods can fully or partially compensate for the redistribution during retirement. Given the efficiency gains, privatization of social security accompanied by increased labor tax progression can improve welfare. We show that the scope for this improvement crucially depends on the response of labor supply to the social security reform.
Jump-growth model for predator-prey dynamicsgustavdelius
Talk at the workshop "Stochastic Population Dynamics and Applications in Spatial Ecology" at the ICMS, Edinburgh, June 2009
Abstract: We use a stochastic model to describe the main process determining the size spectrum of organisms in marine ecosystems: larger organisms preying on smaller organisms and growing in size. Instead of the spatial location of the organisms we model their weight, but the techniques are the same. The feeding interaction is non-local in weight-space, determined by a feeding kernel expressing the preference for a certain predator/prey weight ratio. We treat the model both as an individual-based model (stochastic process on configuration space) and as a population model and compare the approaches. The deterministic equation derived from our stochastic model turns out to be a modification of the McKendrick-von Förster equation that has traditionally been used to model size spectra. The steady state is found to be given by a power law weight distribution, in agreement with observation, but we also observe travelling-wave solutions.
Challenges in predicting weather and climate extremesIC3Climate
Presentation from the Kick-off Meeting "Seasonal to Decadal Forecast towards Climate Services: Joint Kickoff Meetings" for ECOMS, EUPORIAS, NACLIM and SPECS FP7 projects
Agents gradually learn the structure of the economy.
Learning model delivers a sizeable recession in 2008-2010,
...whereas RE model predicts a counterfactual expansion.
In a medium scale model learning still matters.
Taking progressivity research to European CommissionGRAPE
Social security is essentially about insurance. First, it gives insurance against mortality risk by providing annualization. Second, it provides partial insurance against low-income realization by providing intra-cohort redistribution. However, such redistribution is costly because it distorts labor supply incentives. When the link between social security contribution and future benefits becomes weaker, we treat contributions more and more as taxes, not as implicit savings.
With rising longevity, the social security system in many countries is bound to be put under unprecedented fiscal strain. Therefore, some changes appear imperative. Reforms proposed in the literature usually involve linking pensions to individual contributions, thus improving efficiency at the expense of the insurance loss.
In this paper, we propose a novel way of reforming social security. Our reform consists of two elements. First, we replace the redistributive defined benefit payout scheme with a defined contribution payout scheme, which links individual contributions to individual benefits. It raises efficiency as it reduces labor market distortions associated with contribution rates. Second, we propose to accompany this social security reform with adjustments in the progressiveness of labor taxation. Specifically, we increase progression in income taxes.
Thus, we partially replace the redistribution otherwise provided by social security with the one provided within the tax system.
We show that more redistribution during the working periods can fully or partially compensate for the redistribution during retirement. Given the efficiency gains, privatization of social security accompanied by increased labor tax progression can improve welfare. We show that the scope for this improvement crucially depends on the response of labor supply to the social security reform.
Jump-growth model for predator-prey dynamicsgustavdelius
Talk at the workshop "Stochastic Population Dynamics and Applications in Spatial Ecology" at the ICMS, Edinburgh, June 2009
Abstract: We use a stochastic model to describe the main process determining the size spectrum of organisms in marine ecosystems: larger organisms preying on smaller organisms and growing in size. Instead of the spatial location of the organisms we model their weight, but the techniques are the same. The feeding interaction is non-local in weight-space, determined by a feeding kernel expressing the preference for a certain predator/prey weight ratio. We treat the model both as an individual-based model (stochastic process on configuration space) and as a population model and compare the approaches. The deterministic equation derived from our stochastic model turns out to be a modification of the McKendrick-von Förster equation that has traditionally been used to model size spectra. The steady state is found to be given by a power law weight distribution, in agreement with observation, but we also observe travelling-wave solutions.
Challenges in predicting weather and climate extremesIC3Climate
Presentation from the Kick-off Meeting "Seasonal to Decadal Forecast towards Climate Services: Joint Kickoff Meetings" for ECOMS, EUPORIAS, NACLIM and SPECS FP7 projects
Agents gradually learn the structure of the economy.
Learning model delivers a sizeable recession in 2008-2010,
...whereas RE model predicts a counterfactual expansion.
In a medium scale model learning still matters.
We explore the application of optimal control techniques in agent-based macroeconomics. We specifically discussed the Ramsey-Cass-Koopman (savings), Barro (public finance), and Ellis-Fender (corruption) models. Model discussions lifted from Sala-i-Martin's lecture notes on economic growth. Some formulations were taken from lectures of Prof. Emmanuel de Dios and Prof. Rolando Danao of UP School of Economics. All errors mine.
This slide set is a work in progress and is embedded in my Principles of Finance course that I teach to computer scientists and engineers.
http://awesome.weebly.com/
PARAMETERS STATIC DATA STREAMED DATA
NATURE
Stable and unchanging over time
Dynamic and continuously generated in real-time
SOURCE
Typically stored in databases, files, or spreadsheets
Often originates from sensors, social media feeds, financial markets, or any source with constant updates
UPDATES
Occur at specific intervals, not continuously
Can be high velocity and high volume, requiring scalable storage and processing
EXAMPLES
Customer information, product catalogs, historical sales data
Sensor data (temperature readings, GPS coordinates), social media feeds (posts, tweets), financial transactions
ANALYSIS
Well-suited for in-depth analysis using sophisticated tools
Focused on real-time insights, often involving extracting patterns and trends from the latest data
The key difference between static and streamed data lies in their characteristics and how they are handled.
All three fields - statistics, data mining, and machine learning - are concerned with extracting knowledge from data, but they differ in their goals and approaches
PARAMETERS STATISTICS DATA MINING MACHINE LEARNING
FOCUS
Summarizing, describing, and drawing conclusions from data
Uncovering hidden patterns and relationships within large datasets Developing algorithms that can learn from data by itself
GOAL
Understand the characteristics of a population based on a sample
Extract insights that can inform decision-making
Build models that can make predictions or classifications on new, unseen data
DATA
Often works with smaller, well-defined datasets. These datasets typically follow a specific structure and are subjected to rigorous cleaning and preparation before analysis
Deals with large and potentially messy datasets. Data can be structured, semi-structured, or unstructured, requiring cleaning and preprocessing before applying mining techniques Primarily utilizes large datasets to train the algorithms. Data quality and pre-processing are crucial for building effective models
APPLICATIONS
Scientific research, market research, experimental studies Fraud detection, Revealing behavioral patterns, Bitcoin mining (Blockchain technology)
Image recognition, spam filtering, stock price prediction, recommendation systems
EXAMPLES
Scientific research: Analyzing experimental data to draw conclusions and support hypotheses.
Market research: Understanding customer demographics, preferences, and market trends Fraud detection: Analyzing financial transactions to identify patterns indicative of fraudulent activity.
Risk management: Assessing potential risks associated with loan applications or insurance claims
Image recognition: Facial recognition in social media or security applications.
Spam filtering: Identifying and filtering unwanted emails.
Stock price prediction: Analyzing market trends and historical data to forecast future stock prices (with inherent limitations)
We explore the application of optimal control techniques in agent-based macroeconomics. We specifically discussed the Ramsey-Cass-Koopman (savings), Barro (public finance), and Ellis-Fender (corruption) models. Model discussions lifted from Sala-i-Martin's lecture notes on economic growth. Some formulations were taken from lectures of Prof. Emmanuel de Dios and Prof. Rolando Danao of UP School of Economics. All errors mine.
This slide set is a work in progress and is embedded in my Principles of Finance course that I teach to computer scientists and engineers.
http://awesome.weebly.com/
PARAMETERS STATIC DATA STREAMED DATA
NATURE
Stable and unchanging over time
Dynamic and continuously generated in real-time
SOURCE
Typically stored in databases, files, or spreadsheets
Often originates from sensors, social media feeds, financial markets, or any source with constant updates
UPDATES
Occur at specific intervals, not continuously
Can be high velocity and high volume, requiring scalable storage and processing
EXAMPLES
Customer information, product catalogs, historical sales data
Sensor data (temperature readings, GPS coordinates), social media feeds (posts, tweets), financial transactions
ANALYSIS
Well-suited for in-depth analysis using sophisticated tools
Focused on real-time insights, often involving extracting patterns and trends from the latest data
The key difference between static and streamed data lies in their characteristics and how they are handled.
All three fields - statistics, data mining, and machine learning - are concerned with extracting knowledge from data, but they differ in their goals and approaches
PARAMETERS STATISTICS DATA MINING MACHINE LEARNING
FOCUS
Summarizing, describing, and drawing conclusions from data
Uncovering hidden patterns and relationships within large datasets Developing algorithms that can learn from data by itself
GOAL
Understand the characteristics of a population based on a sample
Extract insights that can inform decision-making
Build models that can make predictions or classifications on new, unseen data
DATA
Often works with smaller, well-defined datasets. These datasets typically follow a specific structure and are subjected to rigorous cleaning and preparation before analysis
Deals with large and potentially messy datasets. Data can be structured, semi-structured, or unstructured, requiring cleaning and preprocessing before applying mining techniques Primarily utilizes large datasets to train the algorithms. Data quality and pre-processing are crucial for building effective models
APPLICATIONS
Scientific research, market research, experimental studies Fraud detection, Revealing behavioral patterns, Bitcoin mining (Blockchain technology)
Image recognition, spam filtering, stock price prediction, recommendation systems
EXAMPLES
Scientific research: Analyzing experimental data to draw conclusions and support hypotheses.
Market research: Understanding customer demographics, preferences, and market trends Fraud detection: Analyzing financial transactions to identify patterns indicative of fraudulent activity.
Risk management: Assessing potential risks associated with loan applications or insurance claims
Image recognition: Facial recognition in social media or security applications.
Spam filtering: Identifying and filtering unwanted emails.
Stock price prediction: Analyzing market trends and historical data to forecast future stock prices (with inherent limitations)
ALINA MATSENKO AND AJAY MISHRA THE THEORY AND THESTORY OF EVERYONE AND EVERYTHING ALINA MATSENKO AJAY MISHRA REAL LIFE NOT LIE BUT LOFE AND LOVES STORIE OF REALA ND LIVINGA ND DEADPEOLEBUT YES THER
Fr 1987-01-16 1987 resgister 67 type and the prime numbers 1987 which is gaussian prime steven speilebrg heello or shalom is same s priveta dn namaste, but shalom has word om
how to sell pi coins on Bitmart crypto exchangeDOT TECH
Yes. Pi network coins can be exchanged but not on bitmart exchange. Because pi network is still in the enclosed mainnet. The only way pioneers are able to trade pi coins is by reselling the pi coins to pi verified merchants.
A verified merchant is someone who buys pi network coins and resell it to exchanges looking forward to hold till mainnet launch.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
The European Unemployment Puzzle: implications from population agingGRAPE
We study the link between the evolving age structure of the working population and unemployment. We build a large new Keynesian OLG model with a realistic age structure, labor market frictions, sticky prices, and aggregate shocks. Once calibrated to the European economy, we quantify the extent to which demographic changes over the last three decades have contributed to the decline of the unemployment rate. Our findings yield important implications for the future evolution of unemployment given the anticipated further aging of the working population in Europe. We also quantify the implications for optimal monetary policy: lowering inflation volatility becomes less costly in terms of GDP and unemployment volatility, which hints that optimal monetary policy may be more hawkish in an aging society. Finally, our results also propose a partial reversal of the European-US unemployment puzzle due to the fact that the share of young workers is expected to remain robust in the US.
when will pi network coin be available on crypto exchange.DOT TECH
There is no set date for when Pi coins will enter the market.
However, the developers are working hard to get them released as soon as possible.
Once they are available, users will be able to exchange other cryptocurrencies for Pi coins on designated exchanges.
But for now the only way to sell your pi coins is through verified pi vendor.
Here is the telegram contact of my personal pi vendor
@Pi_vendor_247
Yes of course, you can easily start mining pi network coin today and sell to legit pi vendors in the United States.
Here the telegram contact of my personal vendor.
@Pi_vendor_247
#pi network #pi coins #legit #passive income
#US
The Evolution of Non-Banking Financial Companies (NBFCs) in India: Challenges...beulahfernandes8
Role in Financial System
NBFCs are critical in bridging the financial inclusion gap.
They provide specialized financial services that cater to segments often neglected by traditional banks.
Economic Impact
NBFCs contribute significantly to India's GDP.
They support sectors like micro, small, and medium enterprises (MSMEs), housing finance, and personal loans.
how can i use my minded pi coins I need some funds.DOT TECH
If you are interested in selling your pi coins, i have a verified pi merchant, who buys pi coins and resell them to exchanges looking forward to hold till mainnet launch.
Because the core team has announced that pi network will not be doing any pre-sale. The only way exchanges like huobi, bitmart and hotbit can get pi is by buying from miners.
Now a merchant stands in between these exchanges and the miners. As a link to make transactions smooth. Because right now in the enclosed mainnet you can't sell pi coins your self. You need the help of a merchant,
i will leave the telegram contact of my personal pi merchant below. 👇 I and my friends has traded more than 3000pi coins with him successfully.
@Pi_vendor_247
how to sell pi coins effectively (from 50 - 100k pi)DOT TECH
Anywhere in the world, including Africa, America, and Europe, you can sell Pi Network Coins online and receive cash through online payment options.
Pi has not yet been launched on any exchange because we are currently using the confined Mainnet. The planned launch date for Pi is June 28, 2026.
Reselling to investors who want to hold until the mainnet launch in 2026 is currently the sole way to sell.
Consequently, right now. All you need to do is select the right pi network provider.
Who is a pi merchant?
An individual who buys coins from miners on the pi network and resells them to investors hoping to hang onto them until the mainnet is launched is known as a pi merchant.
debuts.
I'll provide you the Telegram username
@Pi_vendor_247
how to sell pi coins in all Africa Countries.DOT TECH
Yes. You can sell your pi network for other cryptocurrencies like Bitcoin, usdt , Ethereum and other currencies And this is done easily with the help from a pi merchant.
What is a pi merchant ?
Since pi is not launched yet in any exchange. The only way you can sell right now is through merchants.
A verified Pi merchant is someone who buys pi network coins from miners and resell them to investors looking forward to hold massive quantities of pi coins before mainnet launch in 2026.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
Seminar: Gender Board Diversity through Ownership NetworksGRAPE
Seminar on gender diversity spillovers through ownership networks at FAME|GRAPE. Presenting novel research. Studies in economics and management using econometrics methods.
how to sell pi coins at high rate quickly.DOT TECH
Where can I sell my pi coins at a high rate.
Pi is not launched yet on any exchange. But one can easily sell his or her pi coins to investors who want to hold pi till mainnet launch.
This means crypto whales want to hold pi. And you can get a good rate for selling pi to them. I will leave the telegram contact of my personal pi vendor below.
A vendor is someone who buys from a miner and resell it to a holder or crypto whale.
Here is the telegram contact of my vendor:
@Pi_vendor_247
what is the future of Pi Network currency.DOT TECH
The future of the Pi cryptocurrency is uncertain, and its success will depend on several factors. Pi is a relatively new cryptocurrency that aims to be user-friendly and accessible to a wide audience. Here are a few key considerations for its future:
Message: @Pi_vendor_247 on telegram if u want to sell PI COINS.
1. Mainnet Launch: As of my last knowledge update in January 2022, Pi was still in the testnet phase. Its success will depend on a successful transition to a mainnet, where actual transactions can take place.
2. User Adoption: Pi's success will be closely tied to user adoption. The more users who join the network and actively participate, the stronger the ecosystem can become.
3. Utility and Use Cases: For a cryptocurrency to thrive, it must offer utility and practical use cases. The Pi team has talked about various applications, including peer-to-peer transactions, smart contracts, and more. The development and implementation of these features will be essential.
4. Regulatory Environment: The regulatory environment for cryptocurrencies is evolving globally. How Pi navigates and complies with regulations in various jurisdictions will significantly impact its future.
5. Technology Development: The Pi network must continue to develop and improve its technology, security, and scalability to compete with established cryptocurrencies.
6. Community Engagement: The Pi community plays a critical role in its future. Engaged users can help build trust and grow the network.
7. Monetization and Sustainability: The Pi team's monetization strategy, such as fees, partnerships, or other revenue sources, will affect its long-term sustainability.
It's essential to approach Pi or any new cryptocurrency with caution and conduct due diligence. Cryptocurrency investments involve risks, and potential rewards can be uncertain. The success and future of Pi will depend on the collective efforts of its team, community, and the broader cryptocurrency market dynamics. It's advisable to stay updated on Pi's development and follow any updates from the official Pi Network website or announcements from the team.
1. Economic Growth
Spring 2013
1 The Solow growth model
Basic building blocks of the model
• A production function
Yt = F (Kt, Lt, At)
– This is a hugely important concept
– Once we assume this, then we are saying that any growth has to be the result of more
capital, more people or better technology!
• Constant returns to scale
F (λKt, λLt) = λF (Kt, Lt)
• Often we’ll look at a special case: Cobb-Douglas production function with labour-augmenting
technology
F (K, L) = Kα
(AL)1−α
• Sometimes people formulate the function as
F (K, L) = AKα
L1−α
(i.e. “neutral” rather than “labour-augmenting” technical change). With a Cobb-Douglas
functional form, it doesn’t make much difference.
• Exogenous population growth
Lt = (1 + n) Lt−1
This is actually a really important assumption
• A constant rate of capital depreciation: δ
1
2. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
• An exogenous savings rate
St = sYt
Ct = (1 − s) Yt
• A closed economy, so savings equals investment
It = St
• Exogenous technological progress
At = (1 + g) At−1
Mechanics of the model
• Suppose that there is no technological progress:
– How much capital will the economy accumulate?
– Will the economy grow? How much? For how long?
• Assume g = 0 for now and normalize A = 1
• Express production function in per capita terms
yt ≡
Yt
Lt
=
1
Lt
F (Kt, Lt)
= F
Kt
Lt
, 1
≡ f (kt)
where kt ≡ Kt
Lt
• Note that we use “per-capita” and “per-worker” interchangeably, but workforce
population
can vary over
time and across countries
• The capital stock evolves according to
Kt+1 = (1 − δ) Kt + It
= (1 − δ) Kt + sYt
∆Kt+1 = −δKt + sYt
2
3. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
• In per-capita terms
∆kt+1 ≡ kt+1 − kt
=
(1 − δ) Kt + sYt
Lt+1
− kt
=
(1 − δ) Kt + sYt
Lt
Lt
Lt+1
− kt
= [(1 − δ) kt + syt]
1
1 + n
− kt
=
syt − (δ + n) kt
1 + n
=
sf (kt) − (δ + n) kt
1 + n
• Interpretation
• Graph: sf (kt) and (δ + n) kt
• The steady state and convergence
• No long-term growth! (GDP grows, GDP per capita does not)
• Growth during transition
• Examples:
– Increase in the savings rate
– Increase in the rate of population growth
– A one-time improvement in technology
Steady state with Cobb-Douglas
• For Cobb-Douglas case we can compute steady-state capital and output explicitly
• Production function is
Yt = Kα
t L1−α
t
yt =
Kα
t L1−α
t
Lt
= kα
t
3
4. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
• The steady state is defined by
ss : ∆kt+1 = 0
so
skα
ss − (δ + n) kss = 0
s
δ + n
= k1−α
ss
kss =
s
δ + n
1
1−α
and
yss =
s
δ + n
α
1−α
so we get an expression for steady state GDP per capita in terms of parameters
The Golden Rule
Question How much should society be saving?
Answer According to one possible criterion known as the Golden Rule, society should have the
“Golden Rule” savings rate. If this sounds a bit tautological it’s because it is. It becomes more
concrete once we describe what the Golden Rule criterion is.
What do we mean by “should”? There are different possible criteria one could use to define
what should be done. The Golden Rule criterion is a very loose interpretation of the moral principle
“one should treat others as one would like others to treat oneself”. Applied to the question of the
savings rate, it can be thought to mean that societies should save in such a way as to maximize
the level of consumption in the steady state. Whether this is a good interpretation of the moral
principle is more of a literary question than an economic one, but let’s accept it for now. One
justification for this objective is that if you were going to be born into a society that is and will
remain in steady state, the Golden Rule society will be the one where you achieve the highest
utility.
Steady state consumption If the economy is at a steady state, consumption will be
css = (1 − s) yss
css depends on s in two ways:
4
5. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
• Directly: the more you save, the less you consume
• Indirectly: the more you save, the higher the steady state capital stock, the higher the output
out of which you can consume
Cobb-Douglas case For the special case of a Cobb-Douglas production function
yss =
s
δ + n
α
1−α
Here we can see the indirect effect: Higher s means higher yss. Therefore steady state consumption
is
css (s) = (1 − s)
s
δ + n
α
1−α
so we can find the maximum by taking a first order condition:
−
s
δ + n
α
1−α
+ (1 − s)
α
1 − α
s
δ + n
α
1−α
−1
1
δ + n
= 0
−1 + (1 − s)
α
1 − α
s
δ + n
−1
1
δ + n
= 0
1 − s
s
α
1 − α
= 1
s = α
General Case Beyond the Cobb-Douglas case, a more general optimality condition for the
Golden-Rule-optimal savings rate comes from the following reasoning.
css (s) = (1 − s) f (kss (s)) (1)
FOC:
−f (kss) + (1 − s) f (kss (s)) ·
∂kss (s)
∂s
= 0 (2)
Now use the steady state condition:
sf (kss) = (δ + n) kss (3)
5
6. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
to compute ∂kss
∂s
:
sf (kss (s)) = (δ + n) kss (s)
f (kss (s)) + sf (kss (s)) ·
∂kss (s)
∂s
= (δ + n)
∂kss (s)
∂s
∂kss (s)
∂s
=
−f (kss (s))
sf (kss (s)) − (δ + n)
(4)
Replace (4) into the FOC (2):
−f (kss) + (1 − s) f (kss (s)) ·
−f (kss (s))
sf (kss (s)) − (δ + n)
= 0
−
f (kss)
sf (kss (s)) − δ
[sf (kss (s)) − (δ + n) + (1 − s) f (kss (s))] = 0
f (kss (s)) = (δ + n) (5)
Let’s go over what this means because it’s not (just) a bunch of maths. We start from equation (1),
which says how much we consume in steady state. This depends on the savings rate directly and
indirectly through the effect of s on kss. We then take first order conditions to find an optimum
and come up with (2). This says that the direct effect, which is negative, is just proportional to
output: the higher the output level, the more we reduce consumption when we increase savings
rates. The indirect effect depends on
1. how much output would increase if we increase the capital stock (that’s why f (kss (s))
appears in the expression)
2. how much more capital we would have if we saved more (that’s why ∂kss(s)
∂s
appears in the
expression)
3. how much of the extra output would we in fact be consuming (that’s why 1 − s appears in
the expression)
This is not the end of it, because we still don’t know how much more capital we are going to have
if we increase the savings rate: we just have the expression ∂kss(s)
∂s
and we need to solve for that.
That’s where we use the fact that in steady state sf (kss) = (δ + n) kss and take derivatives on
both sides to get to (4). We then replace this in the first order conditions and get to (5).
Interpretation of the first order condition Equation (5) has a neat interpretation. Suppose
a society raised its level of savings in such a way that the steady state capital stock were higher.
Would that society have higher consumption? The answer depends on comparing f (kss) against
δ + n. Why? In a steady state, an economy will be saving/investing just enough to make up
for depreciation and population growth. That is what makes a steady state steady! In order to
6
7. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
increase the steady state capital stock by a little bit (call this ∆k) , the economy would have to
increase the absolute amount of investment by just enough to make up for the depreciation and
population growth for this extra capital, every period, i.e. invest an extra (δ + n) ∆k. How much
extra output would society get out of this extra capital? f (kss) · ∆k. When is it the case that
this extra output is enough to cover the required extra investment and have a little extra left over
to consume? Whenever
f (kss) ∆k > (δ + n) ∆k
⇔ f (kss) > δ + n
Therefore it makes sense, according to the Golden Rule, to increase s (and therefore kss) if and
only if
f (kss) > δ + n
The explains condition (5)
• For the Cobb-Douglas case: compute the marginal product of capital in a steady state
f (kss) = αkα−1
ss
= α
s
δ + n
α−1
1−α
= α
δ + n
s
• In order for extra savings to increase steady state consumption, we need
MPK > δ + n
⇔ s < α
Example: economic growth in the USSR in the 1930s
Markets
• So far, “engineering” approach
• Now suppose there is a market for labour and for capital services
– Note metaphor of firms renting capital from households
– Distinction between profits and return on capital
• Questions:
7
8. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
– How many workers will firms want to hire?
– How much capital will firms want to use?
– What ensures that what the firms want coincides with what is actually available?
– What will be the price of labour (the wage)?
– What will be the (rental) price of capital?
• Firms:
max
Ki,Li
F (Ki, Li) − wLi − rK
Ki
FOC:
FK (Ki, Li) − rK
= 0
FL (Ki, Li) − w = 0
• Graphical illustration
• In the Cobb-Douglas case:
αKα−1
i L1−α
i = rK
(1 − α) Kα
i L−α
i = w
• Taking a ratio
1 − α
α
Ki
Li
=
w
rK
Ki
Li
=
w
rK
α
1 − α
(6)
so all firms use the same ratio of capital and labour
• If workers are expensive relative to capital, firms use more capital per worker (and vice-versa)
• (6) implies that they all must have a capital-labour ratio that equals that aggregate, i.e.
Ki
Li
=
K
L
• This lets us find out the factor prices
rK
= αKα−1
L1−α
= αkα−1
w = (1 − α) Kα
L−α
= (1 − α) kα
8
9. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
• Discuss how market-clearing makes these prices come about
• Capital deepening (i.e. increases in K
L
) increases wages and depresses the rental rate of
capital
• We can also compute the total compensation of all workers
wL = (1 − α) Kα
L−α
· L = (1 − α) Kα
L1−α
= (1 − α) Y
and the total capital-income of capital-owners
rK
K = αKα−1
L1−α
· K = αKα
L1−α
= αY
• Constant factor shares. See graph. Is this still true?
• Factor income sums up to total output
• No pure profits for firms (profits = capital income)
• Interest rates:
– If you lend to someone else, tomorrow you get
1 + rt+1
– If you build capital and rent it out, tomorrow you get the rental rate plus your depre-
ciated capital
(1 − δ) + rK
t+1
– Indifference requires
1 + rt+1 = 1 − δ + rK
t+1
rt+1 = rK
t+1 − δ
= FK − δ
• We’ll talk more about the condition
rt+1 = FK − δ
when we talk about investment
9
10. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
Technological progress
• “Steady state”: no tech progress ⇒ no growth
• Now re-introduce technological progress. Focus on Cobb-Douglas case with labor-augmenting
technology
Yt = Kα
t (AtLt)1−α
At+1 = (1 + g) At
• Define “efficiency units of labour”
˜L = AL
and output and capital per “efficiency unit”
˜yt ≡
Yt
˜Lt
=
Kα
t (AtLt)1−α
AtLt
=
Kt
AtLt
α
AtLt
AtLt
1−α
=
Kt
AtLt
α
≡ ˜kα
t ≡ f ˜kt
• ˜k is the amount of capital per efficiency unit of labour.
• The growth rate of ˜L is given by
˜Lt+1
˜Lt
− 1 =
At+1Lt+1
AtLt
− 1
=
At (1 + g) Lt (1 + n)
AtLt
− 1
= (1 + g) (1 + n) − 1
= g + n + gn
≈ g + n
• Rate of “capital-per-efficiency unit of labour” accumulation computed the same way as “cap-
10
11. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
ital per worker” accumulation
∆˜kt+1 =
sf ˜kt − (δ + n + g) ˜kt
1 + n + g
• Same logic for computing steady state in Cobb-Douglas case:
˜kss =
s
δ + n + g
1
1−α
and
˜yss =
s
δ + n + g
α
1−α
• But “steady state” is not so steady! There is growth!
• The key is increasing number of “efficiency units” per worker
• Since everything is constant “per efficiency unit”, everything grows at the same rate: “balanced
growth”
Putting numbers on the parameters
• Parameters:
– α, δ, s, n, g
• Data from NIPA
• Savings rate: s ∈ [0.15, 0.20]
– What about savings=investment? See graph
– Investment by government
– Set s ≈ 0.2
• Share of labour in GDP (from previous graph): 1 − α ≈ 0.65 , so α ≈ 0.35
• Rate of population growth n ≈ 0.01
• Rate of technological progress: g ≈ 0.02.
– How do we know this? According to model, this will be the rate of growth of GDP per
capita when we have balanced growth
11
12. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
• Depreciation: δ ≈ 0.06
– This is hard to measure, because it’s different for different kinds of capital. BEA uses
– Buildings δ ≈ 0.02
– Equipment δ ≈ 0.15
– Computers δ ≈ 0.3
• Capital-output ratio in the model
K
Y
=
s
δ + n + g
≈ 2.22
• Interest rates in the model
r = FK − δ
Compute FK:
FK = αKα−1
(AL)1−α
= α
K
AL
α−1
= α
Y
K
so
r = α
Y
K
− δ
= α
δ + n + g
s
− δ ≈ 9.8%
• Higher than the rates we typically observe
– Mismeasurement?
– Role of risk?
Growth Accounting
• How much growth do we attribute to technological progress, capital accumulation and pop-
ulation growth?
• Start from production function:
Yt = F (Kt, Lt, At)
12
13. ECON 52, Winter 2012 2 EMPIRICAL EVIDENCE ON GROWTH
• Use chain rule:
˙Yt = YKt
˙Kt + YLt
˙Lt + YAt
˙At
• Notation:
˙Xt ≡
dXt
dt
• Divide by Yt to express in percentage terms:
gY ≡
˙Yt
Yt
=
YKt
Yt
˙Kt +
YLt
Yt
˙Lt +
YAt
Yt
˙At
=
YKt Kt
Yt
˙Kt
Kt
+
YLt Lt
Yt
˙Lt
Lt
+
YAt At
Yt
˙At
At
(assume Cobb-Douglas production function)
= αgK + (1 − α) gL + (1 − α) gA
= capital share × % growth of capital
+ labour share × % growth of labour force
+ Solow residual
• Growth of Singapore and Hong Kong (table from Williamson textbook page 231)
2 Empirical evidence on growth
• One view:
– Technology A is the same in all countries
– Countries are not at steady states
– Some countries have more capital than others
– Growth consists of countries approaching the steady state
• This hypothesis is decisively rejected by the evidence!
Convergence
• Compute the growth rate of GDP-per-efficiency-unit of labour (see exercise)
g˜y = αg˜k
13
14. ECON 52, Winter 2012 2 EMPIRICAL EVIDENCE ON GROWTH
• Recall formula for growth in capital-stock-per-effective-unit-of-labour
∆˜kt+1 =
sf ˜kt − (δ + n + g) ˜kt
1 + n + g
• Reexpress in terms of growth rate rather than aboslute change
g˜k =
s
f(˜kt)
˜kt
− (δ + n + g)
1 + n + g
• Replacing:
g˜y = α
s
f(˜kt)
˜kt
− (δ + n + g)
1 + n + g
• If production function is Cobb-Douglas
f ˜k = ˜kα
f ˜k
˜k
= ˜kα−1
= f ˜k
α−1
α
= ˜y
α−1
α
• Replacing
g˜y = α
s˜y
α−1
α − (δ + n + g)
1 + n + g
• Conclusion: countries with lower GDP-per-effective-worker should grow faster!
• If we assume technology is the same in all countries, countries with lower GDP-per-capita
should grow faster!
• By how much? We want to say something like “if country A is x% poorer than country B, it
should grow z% faster”
• Use Taylor approximation
g˜y ≈ gss +
∂g˜y
∂ log ˜y ˜y=˜yss
[log ˜y − log ˜yss]
=
∂g˜y
∂ log ˜y ˜y=˜yss
[log ˜y − log ˜yss] (7)
14
15. ECON 52, Winter 2012 2 EMPIRICAL EVIDENCE ON GROWTH
• Now compute the derivative:
g˜y =
α
1 + n + g
s˜y
α−1
α − (δ + n + g)
=
α
1 + n + g
s exp
α − 1
α
log ˜y − (δ + n + g)
∂g˜y
∂ log ˜y
=
α
1 + n + g
s exp
α − 1
α
log ˜y
α − 1
α
=
α − 1
1 + n + g
s˜y
α−1
α
• In steady state we know that
s˜y
α−1
α = (δ + n + g)
so evaluating the derivative at the steady state leads to
∂g˜y
∂ log ˜y ˜y=˜yss
=
α − 1
1 + n + g
(δ + n + g)
so replacing in (7):
g˜y ≈
α − 1
1 + n + g
(δ + n + g) [log ˜y − log ˜yss]
• This formula tells us how much growth varies when a country is away from steady state
• By extension, also how much growth varies across different countries at different distances
to steady state.
• Putting numbers on parameters
α − 1
1 + n + g
(δ + n + g) ≈ −5.7%
• Countries should converge approximately 5.7% of the way to steady state every year
Evidence on Convergence
• If we plot growth rates against log(GDP per capita), we should find a slope of that magnitude
• Convergence graphs
– Full sample: slope=0
– OECD: slope= -0.11
15
16. ECON 52, Winter 2012 2 EMPIRICAL EVIDENCE ON GROWTH
– W. Europe slope = -0.13
– US states
– Other regions
• Evidence on convergence is mixed at best
• “Conditional” convergence: maybe countries converge to different points
Direct measurement
• Measure the capital stock for each country
• Assume technology is the same in all countries
• Predict that GDP per capita in country i will be
yPredicted
i = kα
i
• Compare this prediction to actual measured GDP per capita
• (see graph)
• Prediction is far off and in a systematic way: based on their capital levels alone, poor countries
should not be so poor
• Maybe the problem is measuring the capital stock?
Evidence from interest rates
• Suppose we don’t trust our measurement of the capital stock
• Ask instead:
– what should k be to account for cross-country differences in GDP?
– if that were the level of k, what interest rates should we observe? are those reasonable?
y = kα
yA
yB
=
kA
kB
α
yA
yB
1
α
=
kA
kB
16
17. ECON 52, Winter 20123 WHAT DOES TFP DEPEND ON? WHERE DOES GROWTH IN TFP COME FROM?
• See chart
• Implications for interest rates:
r = FK − δ
= αkα−1
− δ
= α
Y
K
− δ
• For the US,
Y
K
≈ 2.22
α ≈ 0.35
δ ≈ 0.06
⇒ r ≈ 9.8%
• Replacing the implied Y
K
levels of other countries (see chart)
• Extreme capital scarcity implies extremely high marginal product of capital!
Conclusion
• Differences in capital levels are not the full explanation of differences in GDP across countries
• Policy implications:
– Poor countries will not catch up by rising investment ONLY
– (But investment that brings in new technologies can be important)
3 What does TFP depend on? Where does growth in TFP
come from?
1. Research and development
2. Technological catch-up for countries away from technological frontier
3. “Learning by doing”. Learning by doing what exactly?
4. “Human capital”. Workers not all the same
• How much does this matter? Hall & Jones (1999)
17
18. ECON 52, Winter 20123 WHAT DOES TFP DEPEND ON? WHERE DOES GROWTH IN TFP COME FROM?
• Data on wages by education levels to see how much “human capital” you get per year
of education
• Data on education levels across countries to see how much total human capital different
countries have
• Data: education differences matter, physical capital accumulation matters but a large
chunk is “unexplained” (see graph)
5. Geography: Landlocked and tropical poorer that coastal and temperate (see graph)
• Sachs (2001) proposes explanations. See also Diamond (1997): Guns, Germs and Steel.
(a) Crop yields different by climate.
• But agriculture is not a large fraction of GDP in rich countries
(b) Tropical diseases reduce productivity
• Temperate-zone diseases are/have become less burdensome
(c) Availability of energy resources (especially coal in the beginning of industrialization)
• These factors amplified by
(a) Technologies that don’t transfer well across climates
(b) Demography
(c) Political power
6. Political/social institutions
• Strong correlation between GDP and measures of political transparency, respect for
property rights, rule of law, social trust, etc.
• See graph
• Causality?
• Weaker evidence on more detailed policies
7. Misallocation
• Barriers to entry / expansion
– World bank project on costs of starting a business
• Restrictions on FDI
• Monopolies
18
19. ECON 52, Winter 20123 WHAT DOES TFP DEPEND ON? WHERE DOES GROWTH IN TFP COME FROM?
• Different taxes/subsidies for different firms
• Imperfect capital markets
• Imperfect contract enforcement
– Bloom et al. (2013): the main determinant of firm size in India is the number of
male family members of owners, not the quality of management.
• Labour market regulations
• Discrimination
– Hseih et al. (2013): 1960: 94 percent of doctors and lawyers were white men. 2008:
62 percent. If the difference is due to reduced discrimination, improved allocation
of talent could account for 15-20% of US economic growth.
19