Solow model Economics question/ DEADLINE in 8 hours
has776
2020-Fall-Econ433-Lecture05-Copy.pdf
71
The Solow Model So Far
In the Solow model so far:
–Worker productivity is held constant.
–income per capita is constant in the
steady state.
Neither point is true in the real world:
‐‐Worker productivity rises frequently
–1904‐2004: U.S. real GDP per person
grew by a factor of 7.6, or 2% per year.
Next: We will allow worker productivity to
change.
Solow Model with Labor Productivity Growth
• We started with the basic Solow model
without labor force growth, without
technological growth.
• We then added population growth.
• We now add worker productivity growth.
The Nature of Technological Progress
• Let’s go back to the Cobb-Douglas Production
Function: Y = A.KαLβ
• There are two ways of thinking of
technological progress here:
• If “A” rises, same values of K and L will increase Y.
this is called a neutral technological progress.
Labor Productivity: h
• What if each worker becomes more efficient over time?
• Higher productivity: Each worker behaves as if the worker is doing h times the work of the previous worker.
• In other words, h = index of human capital. It was implicitly h = 1 before.
• Will consider this issue first.
The Nature of Productivity Growth
• First, note that each worker today is more
efficient than a generic worker in 1750.
• In 1750, farmers used rudimentary agricultural
tools, they used inefficient machines and
equipment.
• One reason why workers today are more efficient
than the workers in 1750 is because they have better
education, better knowledge, better skills, they also
work with different – more efficient – K.
The Nature of Productivity Growth
• One way to capture all this is to assume that each
worker gets a productivity or human capital package
called “h.” If h is 1, then that’s the baseline scenario
– i.e., the productivity package of workers in 1750
had an index value of h = 1.
• As time went on, h started rising. So a worker in, say,
1850 was 10 times as productive as a typical worker
in 1750.
The Nature of Productivity Growth
• This means:
• A worker in 1850 = (10)(worker in 1750) as far as
his/her contribution to GDP was concerned.
• The human capital index h will rise for various
reasons: people get more skilled and educated, the
underlying technology changes, the nature of capital
changes, etc.
Human Capital in the Solow Model
• Thus a new variable: h = labor efficiency (human capital) is introduced directly in the Solow model.
• Assume first that h increases exogenously: human capital
progress is labor-augmenting: it increases labor efficiency at
the exogenous rate g:
• g = ∆h/h
6
Human Capital in the Solow Model
• We now write the production function as:
• Y = F(K, hL)
• where hL = the number of effective workers.
• Interestingly, and counter-intuitively, increases in labor efficiency have the same effect on output
as increases in the labor force.
• Although you are one worker, you are effectively like h times the worker in 1750.
• So if you produce 100 units, and your h is 10, it is as if each 1750 worker has produced 10 units each. 8
A Graphical Intuitive Understanding
Adding the labor productivity growth rate, g, shifts the (δ + n) curve anti-clockwise.
It may appear that the new steady state goes from A to C.
But it really goes from A to B when we take productivity growth of labor into account
The newer equation of motion for k
break-even investment
actual investment
• With population growth, the equation of motion for k is
k = s f(k) − ( + n + g) k
16
The newer equation of motion for k
• But keep in mind that the values of y, k we get from this model, must be multiplied by the labor productivity index: h.
• Also remember that h grows at g percent (Solow’s assumption)
• This new h = old h(1+g)t
Steady State in the Expanded Solow Model
• Thus, formally, this expanded model is very similar to the original model we started with which had no population growth and no Productivity growth.
• In that original model the steady state equation was: k = sf(k) – k = 0;
• Or, s/ = k/f(k)
• The expanded model now becomes (the new steady state condition): s/( + n + g) = k/f(k)
The Steady State Condition in the Solow Model
s/( + n + g) = k/f(k)
This is the final version of the Solow Steady State Condition
Solow model with population growth and lab productivity growth
• The graphs are the same, except that is replaced with + n + g.
• Look at the next slide and confirm the following:
• If s rises, s f(k) shifts out, and the steady state k and y rise.
• If or n or g rises, ( + n + g) k shift up, y and k fall.
Human Capital in the Solow Model
Investment, break-even investment
Capital per
worker, k
( +n +g)k
s f(k)
k*
k = s f(k) − ( +n +g )k
20
Solow Full Model
Symbol Steady-state growth rate
k 0
y 0
yh g
Y = yhL n + g
Working with percentages
• In the next few slides, the multiplication rule and the division rule of percentages are discussed
Working With Percentages
We will use the following result:
Growth rate of Y = gy = ΔY/Y
= (dY/dt)/Y = (1/Y)dY/dt
= (d log Y/dY)(dY/dt)
= d log Y/dt
Working With Percentages • Let Z = XY.
• Taking logs: log Z = log X + log Y
• d log Z /dt = d log X/dt + d log Y/dt
• Using the previous result:
• gZ = gX + gy
• Similarly:
• Let W = X/Y.
• Taking logs: log W = log X - log Y
• Using the previous result:
• gW = gX - gY
Useful Info for Solow Model: Working With Percentages
Consider two variables X and Y. Suppose X grows at a% and Y grows at b%.
Rule 1 Then XY grows approximately at (a + b)%
Rule 2 And (X/Y) grows approximately at (a – b)%
Rule 3 If GDP grows at a%, it takes approximately (70/a) years for the value of X to double. Y(1+r)t = 2Y Whence t ln(1+r) = ln 2 Or, t = ln 2/ln (1+r) = = ln2/r (using Taylor Series Expansion) = 0.70/r
Useful Info: Working with Percentages • Examples: • If GDP grows at 5% and Population grows at 2%,
per capita GDP grows approximately at (5-2)% = 3%.
• GDP will double approximately in 70/5 = 14 years.
• Per Capita GDP will double approximately in 70/3 = 23.3 years.
• Now back to the Solow model. 23
The Steady State
• Recall the definitions of k and y in this expanded version. They have no real life meanings, they are just algebraic variables. Let’s try to understand the meaning of steady state in this context.
• By definition, steady state is where, k and y do not change, i.e., they grow at 0%.
• But since Y/L = h.y, and h grows at g%, Y/L, the real per capita income grows at g% at steady state.
• Since Y = h.y.L, and L grows at n%, Y, the real income of a country grows at g% at steady state.
Solow Full Model
Symbol Steady-state growth rate
k 0
y 0
yh g
Y = yhL n + g
Golden Steady State in the Full Model • Golden Rule for Savings: MPK = δ + n + g.
Is the US saving enough?
From US data: k = 2.5y δk = 0.1y Capital share = MPK*k = 0.3y
Solve for depreciation rate: δ = 0.1y/k = 0.1(1/2.5) = 0.04 Solve for MPK = 0.3y/k = 0.3(1/2.5) = 0.12
Are we saving enough?
• The US grows at about 2% = n + g
• So, δ + n + g = 0.04 + 0.02 = 0.06
• Which is half of US MPK.
• Even if the US is close to steady state, US can certainly do better by saving more.
Solow Bottom-Line
• Steady State GDP of a country grows at n+g percent.
• Steady State Per Capita income of a country grows at g percent.
• Higher saving increases per capita income level.
• Higher n increases steady state growth rate of GDP
More on Productivity Growth in the Solow Model
• The exogenous parameter g plays a crucial role in the
Solow Model.
• But it is exogenous! The model does not explain g.
• What determines g? many papers have been written
about what determines g in the post- Solow era.
• We will look at the role of Productivity in more
detail, next.
The Mysterious “A”: The Solow Residual
• We will use the following result:
• gy = ΔY/Y = (dY/dt)/Y
• = (1/Y)dY/dt
• = (d log Y/dY)(dY/dt)
• = d log Y/dt
Solow Residual • Now consider:
• Y = AKαL1-α
• Take logs on both sides with respect to t:
• d log Y/dt
• = d log A/dt + α dlog K/dt + (1-α) d log L/dt
• Or, gy = gA + αgK + (1-α) gL
• Or, gA = gy - [αgK + (1-α) gL ]
gA: Exogenous reasons for Progress • gA could include:
• “Neutral” technological progress
• Better management techniques
• Social/cultural/religious changes that allow more productivity
• Infrastructural changes that help all industries
• S-curve effects
• Much of technology may be “non-rival.” Improvements may affect all industries.
• General increase in literacy, skills and automation may benefit all products.
• Health improvements may benefit general productivity
Solow’s Basic Results
• Basic Solow Model: – Late starters have higher growth rates
(because f(k) increases at a decreasing rate).
– Eventually, late starters will have growth rates similar to the growth rates of matured countries.
Solow’s Basic Results
– All countries will eventually stagnate unless technology comes to the rescue.
– Higher savings rate in a late-start country will take the country closer to the income level of matured countries.
– High savings rate will take a country to a higher level of GDP in the long run. But the savings rate itself does not affect GDP growth rate.
Algebra of the General Solow Model
Solow Model General Steady State Formulas
• Consider the Cobb-Douglas production function:
• Which can be re-written as
29
Solow Model General Steady State Formulas
• Using the steady state condition we get:
• Let:
• Thus
30
Solow Model Steady State Formulas
• This gives us the steady state capital stock, k* (a “*” shows that it a steady state variable.)
• Or,
Solow Model Steady State Formulas
• Or,
Or,
Solow Model Steady State Formulas
Or, if you substitute for θ,
Solow Model Steady State Formulas
Substituting,
Solow Model Steady State Formulas
• Or,
Given, s, n, δ, A and α, we can find steady state values of per capita GDP and per capita capital stock.
Solow Model Steady State Formulas
• Substituting for θ, the steady state value of per capita income y* becomes:
• This formula will be used in the following example.
36
Richland-Poorland Example • Two countries, Richland and Poorland, are described by
the Solow growth model. They have the same Cobb-
Douglas production function, F(K,L) = A KαL1−α, but with
different quantities of capital and labor. Richland saves 32
percent of its income, while Poorland saves 10 percent.
Richland has population growth of 1 percent per year,
while Poorland has population growth of 3 percent. (The
numbers in this problem are chosen to be approximately
realistic descriptions of rich and poor nations.) Both
nations have technological progress at a rate of 2 percent
per year and depreciation at a rate of 5 percent per year.
a. What is the per worker production function f(k)?
b. Solve for the ratio of Richland's steady-state
income per worker to Poorland’s.
c. If the Cobb-Douglas parameter α takes the
conventional value of about 1/3, how much higher
should income per worker be in Richland
compared to Poorland?
d. Income per worker in Richland is actually 16
times income per worker in Poorland. Can you
explain this fact by changing the value of the
parameter α? What must it be?
e. Can you think of any way of justifying such a
value for this parameter? How else might you
explain the large difference in income between
Richland and Poorland?
Per Worker Capital Stock and Per Workers Income:
y*R / y *
P
= {[ 𝑠𝑅 δ+𝑛𝑅+𝑔
]/ [ 𝑠 𝑃
δ+𝑛𝑃+𝑔 ]}
α/(1-α)
= {[ 0.32 0.05+0.01+0.02
]/ [ 0.10
0.05+0.03+0.02 ]}
α/(1-α)
= 4α/(1-α)
If α = 1/3,
the expression becomes:
41/2 = 2 Richland is twice as rich as Poorland.
If, in reality, Richland is 16 times as rich as Poorland, then
= 4α/(1-α) = 16 Which means: α/(1-α) = 2, or, α = 2/3 Capital has 2/3 weight.
This may mean that human capital is more important or that technology biased toward human capital is more important.
Solow Residual
•Significant Differences •Across Time •Across Countries
Moore’s Law (1965). “Time is the teacher.” The cost of a unit decreases exponentially over time. In his 1965 Electronics paper, “Cramming more components onto integrated circuits,” Intel co-founder Gordon Moore predicted that—over a ten year span—the number of transistors on a chip would double each year. He reportedly revised the doubling-time to two years, and others later reformulated it into the familiar “Computing power doubles every 18 months.” Like the Mars rovers Spirit and Opportunity, Moore’s Law has continued to operate far past its original design horizon.
Wright’s Law (1936). “We learn by doing.” The cost of a unit decreases as a function of the cumulative production.
Moore’s Law, Wright’s Law, Goddard’s Law , Nordhaus Synthesis, Sinclair, Klepper, and Cohen’s synthesis
Goddard’s Law (1982, in IEEE Transactions on Components, Hybrids, and Manufacturing Technology). “Economies of scale.” Unit cost decreases as the scale of production increases.
Nordhaus’s synthesis (2009): “Time and experience,” combines Moore’s and Wright’s formulations to project unit costs as a function of both time and cumulative production.
Sinclair, Klepper, and Cohen’s synthesis (2000): “Experience and scale,” combines Wright and Goddard to predict that unit costs fall as a function of both cumulative production and the rate of production.
Moore’s Law, Wright’s Law, Goddard’s Law , Nordhaus Synthesis, Sinclair, Klepper, and Cohen’s synthesis
We Need a Good Theory of Technological Growth
Scale Economy and Path-Dependency
• Cannot identify comparative advantage with a convex PPC.
If the green line is the PPC
And black line is the price,
The country has comp adv
In which good?
The basic structure of the HO
Theory (increasing cost)
does not hold; so comp adv is not well‐defined!
Downward Sloping Cost Curves
• Downward sloping cost curves come from convex PPCs: If cost reduces as one produces more, how do we know what to produce?
Slope falls as more wine is produced! Wine becomes cheaper!
Learning-by-Doing: Liberty Ships • Liberty Ships built by the United States Maritime Commission in World War II
• "Liberty ship" was the name given to the EC2 type ship designed for "Emergency" construction by the United States Maritime Commission in World War II. Liberty ships were nicknamed "ugly ducklings" by President Franklin Delano Roosevelt.
• The first of the 2,711 Liberty ships was the SS Patrick Henry, launched on Sept. 27, 1941, and built to a standardized, mass produced design. (2,710 ships were completed, as one burned at the dock.) The 250,000 parts were pre-fabricated throughout the country in 250-ton sections and welded together in about 70 days. One Liberty ship, the SS Robert E. Peary was built in four and a half days. A Liberty cost under $2,000,000.
• The Liberty was 441 feet long and 56 feet wide. Her three-cylinder, reciprocating steam engine, fed by two oil-burning boilers produced 2,500 hp and a speed of 11 knots. Her 5 holds could carry over 9,000 tons of cargo, plus airplanes, tanks, and locomotives lashed to its deck. A Liberty could carry 2,840 jeeps, 440 tanks, or 230 million rounds of rifle ammunition.
• http://www.usmm.org/libertyships.html
Liberty Ship
Invention and Diffusion
• Diffusion Over Time
• Diffusion Over Space
69
Imitation Lag and Product Cycle
• Assume that it takes time for new technology to spread
• Imitation lag: the time between product’s introduction in country 1 and appearance of a version of that product produced in country 2
• Demand lag: time between products appearance in country 1 and its acceptance in country 2
• Net lag: imitation – demand lag
• The product cycle theory says that the rich countries will develop the products, export them, then end up importing those same products!
70
The Product Cycle Theory
• The product cycle consists of three stages (new, maturing, standardized)
1. A new product is introduced in a rich country – High‐income demands, labour‐saving production
technique – The firms operate only in the domestic markets and learn
production techniques and consumer responses 2. Maturing product
– Economies of scale start to realize – Demand in other rich countries starts to emerge – Part of the production may be shifted to these countries and
they might even start exporting to the original country
71
The Product Cycle Theory
3. Standardized product
– Product is well know to consumers and producer
– Production may shift to the developing counties
time
Production, consumption (in the developed country)
exports imports
new product stage
maturing product stage
standardized product
stage
Consumption1
Production1
Vernon (1966): International Investment And International Trade in the Product Cycle. Quarterly Journal of Economics 80(2).
The Product Cycle
time
Prodn, consn
t0 t1
New product phase
U.S. production
U.S. consumption
The New Product Phase: An Example
• 1961: Sumlock Comptometer introduces the first desk‐top calculator, the Anita Mk VII
• Cost: about $1,000 • 1965: Texas Instruments begins development of first hand‐held
calculator • 1970: first models were introduced by Busicom, Sharp,
and Sanyo. • Price: $600. • Very few were exported.
The Anita Mk VII
TI’s Cal Tech (1967)
The Busicom LE‐120‐A (1970)
The Maturing Product Phase
• By 1972, Japanese companies had entered
• Price fell to around $400.
• Later in the 1970s, Singapore and Taiwan began to produce, taking advantage of lower labor costs.
• U.S. market share dropped.
TI‐1680 (1977)
TI‐35 II (1984)
Tydlig
The Standardized Product Phase
• By the 1980s, prices of pocket calculators were down to $15 or so (the little ones are almost free now!)
• The Pacific Rim countries became the major players
• The PLC for calculators was complete in 25 years or so.
• Computers have become lot more efficient
Diffusion Over Time: S-Curve Effect
Reason’s for Slow Diffusion
• High Fixed Costs to Transform
• Network Externalities
• QWERTY Effect
Invention and Diffusion
• Slow Geographic Diffusion
Can Agglomeration be Accidental? “Path‐Dependent?”
• Why were the Swiss Watchmakers so good?
• https://www.hautehorlogerie.org/en/encyclopaed ia/history-of-watchmaking/
• Why are there so many quilt‐makers in Georgia?
• http://www.georgiaencyclopedia.org/articles/ar ts-culture/quilt-making
Why are there so many software firms in San Jose?
• Silicon Valley: The region is home to many of the world's largest technology companies including Apple, Google, Facebook, HP, Intel, Cisco, eBay, Adobe, Agilent, Oracle, Yahoo, Netflix, and EA…Silicon Valley continues to be the leading hub for high‐tech innovation and development, accounting for 1/3 of all of the venture capital investment in the United States: Wikipedia.
Accidents
• These puzzles may be explained by fortuitous circumstances and by accidents in history.
• Georgia became the quilt center because one woman, Harriet Powers, made great quilts. So more quilt makers moved in to Georgia…
• Comparative advantage in quilt making was an accidental “path dependent” phenomenon.
Georgia Quilts by Harriet Powers (1837 – 1910)
Path‐dependent comparative advantage
Georgia Quilts by Harriet Powers
Can Comparative Advantage be Accidental?
• Swiss Watches, Georgia Quilts, Software in Silicon Valley: Accidental Traders?
• A combination of economies of scale (internal and external) and Hotelling’s theory about spatial economic competition (beach vendor example) could create a “path‐dependent” comparative advantage and geographical clustering.
Hotelling’s Law
• Hotelling’s Law says that that the producers may congregate in one area.
• Such clustering can not only explain malls and the LA Jewelry District, but also the location of many products.
• That contradicts the HO Theorem.
Adoption and Adaption
• Technology adoption may rise over time.
• Technology adaption may be easier over time.
• Technology may be patented.
• Technology may be inappropriate in another country
• But it is hard to come up with a theory of tech growth which will give us insights into the process.
The Nature of Technological Progress
• In the basic Solow model, per capita income grows at g% which is the rate of growth of h.
• What determines h?
• Note that the determination of “A” and the determination of “h” are similar problems:
• Y = AKα(hL)1- α = A. (h)1- α Kα (L)1- α ≡ A* Kα L1- α
Some attempts to Model Technology: Romer and Others
• We did not say much about A.
• If “A” rises continuously, there will be no steady state either!
• Can “A” rise continuously?
• Solow, Romer and others seem to have gone back to Adam Smith and are asking:
• Can technology rescue us and break the curse of steady state?
Standard Solow/Cobb-Douglas Production Function
The Y = AK Model
• If our knowledge enhanced/technology enhanced production function is linear, we should be very happy indeed.
Linear Production Function? • This economy exhibits unlimited growth (there is no
steady state!) under A Smithian production function
f(k)
sf(k)
(δ + n)k
k
y
48
Romer Model: Implications
Investment, break-even investment
Capital per
worker, k
(+n)k
k*?
44
sY Y = f(k)
We could even have exponentially growing production function!
How Does Technology/Labor Productivity Change?
• Is technology like a fish-pond (and have we already overfished?)
• Is it multiplicative from the last period?
• Is it exponential?
Solow vs. Harrod-Domar
• Harrod-Domar: High Saving leads to high growth rate.
• In the Solow model, high s does not affect growth rates in the long run.
• Note that in Harrod-Domar, “v” is assumed to be constant.
• In the Solow model, v rises as the economy grows.
62
The Romer Model
• The Romer model assumes that societies through R&D create knowledge and such knowledge benefits everyone in the society.
• The knowledge variable enhances the K variable in the Cobb-Douglas production function.
• So the Romer equation is:
• Y = KαL1- αKβ where Kβ is a knowledge enhancing component of Y.
The Romer Model The equation can be written as:
Y = Kα+ β L1- α
A curious feature of this model is that α+ β is not necessarily less than 1 and that exponents add up to more than 1.
This violates the constant returns to scale assumption we mentioned earlier.
We can now have a graph such as this:
Romer Model: Implications
Investment, break-even investment
Capital per
worker, k
(+n)k
k*?
44
sY Y = f(k)
Romer Model • There is no steady state in such models
because the sY curve does not intersect with the ( +n) k line!
• In other words, knowledge creates power - power to break the curse of the steady state.
• The model can also be written as:
• Y = K α+ β
L 1-α = KαKβ L1-α = ARomerK
β L1-α
• The nature of “A” has changed, but it is similar to the Solow Model.
Relation between Adam and Robert
46
A Smithian Model
• Remember Adam Smith? Smith said that technology depends on the size of the market.
• What if technology depends on GDP or Y in a positive way?
• What if h = BY? Where B is a parameter?
• Then (substituting), Y = (BL)(1 – α/ α)K
• Or, Y = AsmithK which is a linear production function between Y and K.
• This produces linear production functions again without steady state (next slide).
Linear Production Function? • This economy exhibits unlimited growth (there is no
steady state!) under A Smithian production function
f(k)
sf(k)
(δ + n)k
k
y
48
Relation between Harrod/Domar and Robert/Romer
Smith-Harrod Domar-Solow-Solow-Romer
• Y = AsmithK
• HD Model: Y = (1/v)K
• Solow Model: y = Akα
• Romer Model: Y = ARomerk α
Thoughts on Tech Growth
• Hypothesis 1: Fish Pond Theory of Technology
– A limited number of technologies are available in space and time. We have already discovered most of them.
56.
Thoughts on Tech Growth
• Hypothesis 2: Multiplicative Speed of Technological Progress
• Example: Number of inventions this century are three times the number of inventions last century
• Hypothesis 3: Combinatorial Speed of Technological Progress
• Example: Number of inventions this century is equal to the possible combinations of any two previous inventions during the last century.
• http://en.wikipedia.org/wiki/Combination
57.
Fish Pond Theory of Technology
• If you get better at fishing as time passes, the fish pond theory may produce the following sequence of new revolutionary technologies:
58
18th
Century 19th
Century 20th
Century 21st
Century 22nd
Century 23rd
Century 24th
Century
4 10 100 110 40 10 0
Multiplicative Speed of Technological Progress
• The multiplicative theory (Hypothesis 2: this century’s invention is three times the invention of the last century) produces:
59
18th
Century 19th
Century 20th
Century 21st
Century 22nd
Century 23rd
Century 24th
Century
4 12 36 108 324 972 2916
The Combinatorial Theory
18th
Century 19th
Century 20th
Century 21st
Century 22nd
Century 23rd Century 24th
Century
4 6 15 105 5460 14,903,070 ?
60
The Combinatorial theory produces: (assuming we start with 4 major discoveries in the 18th century, where k = 2 and we begin with n = 4). Use the formula:
Which hypothesis is correct?
• Which hypothesis is correct? 1, 2 or 3?
• No one can tell in the 21st century:
– All three theories produce near identical results for the 21st century (110, 108, and 105 major discoveries). So we don’t know what will happen in the next century – would the tech innovations explode or stagnate?
61
So Long, Solow • Solow model, as elegant as it is, does not see that a rich
country and a poor country are qualitatively different.
• In the Solow model, the rich and the poor are different because they may have different savings rates, different population growth rates, different technological growth rates and may even have even difference depreciation rates.
• But there is nothing organically different about the countries in the Solow’s vision of the world. Poor countries are younger versions of rich countries.
• There is a large literature that argues the rich and the poor countries are different in their economic frameworks.
Creative Destruction: Schumpeter
• Political issues
• Patents
• Patent Trolls
Creative Destruction: Schumpeter
Schumpeter’s Creative Destruction A Theory of Technological Growth
• We invest a lot on technology.
• Why invest in R&D?
• Does investment in technology increase A or h?
The Role of Technology in Growth: Weil’s Model
Weil’s Model y = A(1- γA)L y = Per Capita GDP A = General Technology Index γA = Fraction of labor force engaged in R&D μ = Price of new invention measured in units of labor
dA/dt = LA μ
dy/dt = dA/dt = γA μ
L
A, and therefore, y, keep growing. No output stagnation.
Effect of Shifting Labor into R&D
Weil’s Two Country Model
y1 = A1(1- γA1)L1 y2 = A2(1- γA2)L2
dA1/dt = γA1 μ 𝑖
L1
μc = c 𝐴1
𝐴2
Cost of Copying for the Follower Country
Steady State in the Two-Country Model
Steady State Tech Growth
dA1/dt = γA1 μ 𝑖
L1 = dA2/dt = γA2 μ 𝑐
L2
Let:
L1 = L2 = L
Then: γA1 μ 𝑖
= γA2 μ 𝑐
μ𝑐 = γA2 γA1
μ𝑖
Effect of an Increase in R&D in the Follower Country on the Steady State
Effect of an Increase in gA2 on Productivity and Output
Tech Transfers: Compatibility
• Tacit Knowledge
• Inappropriate technology
• Lack of demand
• Capital Saving Technology
Neutral Technological Change
Capital-Biased Technological Change
Solow Model: Concluding Comments
• In addition to A, K, L, s, δ, n, g, α, many other things influence economic growth.
• Role of geography, social environment, political environment, corruption, and a host of other things must matter.
• Solow model is a one-sector model – which may not be realistic.
• Solow may have just scratched the surface of growth theory.
Is the Solow Model Appropriate for Very Poor Countries?
• Solow model was a one-sector model. • Arthur Lewis proposed a two-sector model with
characteristics of less developed countries. • Two sectors: agriculture and industry/manufacturing. • In the Lewis model, the traditional agriculture sector is
socially different from the modern industrial sector. • There is disguised unemployment in the agricultural sector
where food is shared. • Disguised Unemployment in agriculture has two similar
connotations: (1) Zero Marginal Product : the last worker does not increase total product at all or, (2) The last worker adds very little (less than his wage) to total product.
Zero Marginal Product is Incompatible with Solow
• In Solow:
Y = AKαL(1-α)
MPK = ∂Y/∂K = αAK(α-1)L(1-α)
MPL = ∂Y/∂L = (1-α)AKαL(1-α-1)
If K and get their marginal products:
K.MPK + L.MPL = KαAK(α-1)L(1-α) + L(1-α)AKαL(1-α-1) = (α + 1 – α) AKαL(1-α)
= Y. GDP is distributed between K and L according to their Marginal products.
Zero Marginal Product is Incompatible with Solow
• If workers are paid their marginal product at zero:
• K.MPK + L.MPL = K.MPK
• Then, who gets the rest of Y? If capitalists get the rest of the Y, not clear why workers work at all.
Street Vendors in Poor Countries
• You see street vendors like this (next slide) in the villages of less developed countries.
• Do they make enough money to support themselves?
Rural Street Vendor
Street Vendors
• Look at what he is selling.
• How much money does he make per day?
• Is it enough to support himself?
• Probably he makes one or two dollars worth of profit/day – not enough to support himself.
• Other family members help him out. May be the parents help. May be the siblings help.
A Different Paradigm
• If other family members consistently help the rural street vendor, the model then will not follow the typical economics models.
• In rational economics, wage = value of your marginal product. If your work adds 5 widgets to a factory’s production, and widgets can be sold for $2/piece, then your wage is (5)(2) = $10 or less.
Poor People Have to Share
• Since marginal product of rural subsistence level workers is very low (may be even zero), Wage should be zero or close to zero.
• Near zero wage means starvation and death. • So what does he do? • His poor siblings, parents and others must share food with him so
that he does not die. • The the poor have to share • They share because they think about their mutual insurance. If the
lucky (relatively) high- wage-earner loses his/her job tomorrow, he/she may need help from the extended family at that time. So the high-wage earner shares his wage.
• So everybody tries to help everybody else in an extended family type setup.
Socially Different Agricultural Sector
• If the extended family members share their incomes, the vendor’s true wage is equal to whatever he earns selling his stuff plus the amount the family provides for his upkeep.
• If he earns less one week, the family probably gives him more that week.
• If he earns more one week, the family probably gives him less that week.
• His subsistence wage is thus institutionally fixed.
• How does the family provide for him? Where does the family get the extra resources?
• If there are L workers in the extended family, all L workers pool their resources and divide their income equally.
Socially Different Agricultural Sector
• Some of these workers have high marginal product, some have low marginal product – but in reality they share their total wage.
• This may not be totally accurate, but something similar to wage sharing happens which ensures that everyone gets a minimum institutionally fixed wage.
• What is the institutionally fixed wage? • Wage = Total product/L = average product of labor or,
APL. • This is different from the modern, professional urban
manufacturing sector where wage = value of marginal product.
Lewis Dual Economy
Model (also Known as
the Labor Surplus Model)
A Village in China
• Models of Economic Growth : Growth Models for Developing Countries.
• 1954. “Economic Development with Unlimited Supplies of Labour.” Manchester School 22 (May): 139–191.
• 1955. The Theory of Economic Growth. London: Allen and Unwin.
The Lewis Dual Economy Model
• Two sectors:
– Sector 1:
• Agricultural, rural, informal, traditional sector providing subsistence level wages and “bad” jobs.
– Sector 2:
• Manufacturing, urban, formal, modern sector providing above-subsistence level wages and “good” jobs.
Recall that production function looks like this (K is fixed here)
4
c
Right Side of the Production Function
• Notice that to the right of c, total product does not rise.
• This means that to the right of c, the marginal product is zero.
• Average product (total product/labor) is still positive to the right of c.
• The next slide shows this.
7
Labor Supply in Industry
• Wages are not allowed to fall below an institutionally fixed level which is probably close to the subsistence level.
• This is the situation in agriculture. • So how does the labor supply curve to industry
look like? • No one works for the industry if industry offers a
wage below the institutionally fixed wage in agriculture.
• Wage rate in industry must not be less than the wage rate at h (next slide).
8
Demand for Labor in Industry
• Demand curve for labor in industry is like any other labor demand curve. Capitalists are hard-nosed and rational. They don’t share anything.
9
10
The Economy Begins from Low Wages
• The equilibrium wage in the previous slide is k.
• In this case, both agriculture and industry pay the institutionally fixed wage.
• There is a lot of disguised unemployment in the agricultural sector e.g., there are people who really contribute nothing to agricultural production (their marginal product is zero).
11
Dynamics of the Lewis Economy
• What happens in the next period?
• Recall that capitalists are the only ones making extra income.
• Hopefully, they reinvest their income.
• They use more machines in the next period.
• As they use more machines, each worker becomes more productive.
The Capitalists Can Invest Their Income
W=P.MPL
Labor
• If they reinvest their income in new machines, what happens?
This triangle shows Capitalists’Income.
W
Labor Demand Curve
12
13
More Capital = Workers More Productive
• Addition of new machines will make each worker more productive than before.
• Because each worker becomes more productive, labor demand curve shifts out in the next period.
Dynamics of the Lewis Economy • As capitalists reinvest, they raise (shift up) the
industrial demand curve continuously.
14
Lewis Turning Point
15
Dynamics of the Lewis Economy
• After some time, the flat part of industrial labor supply curve will no longer be relevant.
• The industrial sector labor demand curve will intersect the industrial sector labor supply curve at the upward sloping part of the industrial labor supply curve. See n or n’.
16
Dynamics of the Lewis Economy
• What happens in the rural sector at that time?
• Rural sector no longer pays the institutionally fixed wage, it starts using the marginal productivity rule – it slowly but surely becomes more capitalistic!
Capital Accumulation and Labor Demand
If the capitalists do not invest domestically due to political instability etc., the TPM would not shift up.
26
27
Roadblocks to the Lewis Development Process
What if the new investment is not labor-biased at all?
What if investment takes place, but such investment is biased toward high tech capital?
(See the next slide).
Lewis Model and Stagnation
28
In this case, labor demand does not rise. The Lewis economy stagnates.
29
Lewis gives us an interesting model of development, but …
Lewis Economic Development Process is Not Guaranteed!
30
Vicious Circle of Poverty
Vicious Cycle of Poverty
Nurkse and Rosenstein Rodan: Balanced Growth
Buy
Sell
Buy
Sell
Sell
Buy Buy
Sector 1
Sector 2
Sector 3
Balanced Growth
• Invest in all sectors since all sectors are related by supply and demand.
• On sector supplies input to the other.
• One sector demands goods from the other.
• The idea has roots in Say’s Law.
Singer/Hirschman: Unbalanced Growth
• There is always resource constraint: balanced growth may be impractical.
• Choose the sector with highest backward and forward linkages and invest in that sector
Steel Could be the Leading Sector
31
End of Growth Theories
• We have now come to the end of survey of growth theories.
• It was hard. But it had to be done because growth is an integral part of economic development.
From Growth to Development • There is a recognition in the literature that development is a
broader concept that is over and above what the Solow model suggests.
• Arthur Lewis has a model with two sectors – one of them (agriculture) is not a capitalist sector.
• Rosenstin-Rodan advocates “Big Push” by the government.
• Nurkse was a proponent of “Balanced Growth.”
• Albert Hirschman was in favor of unbalanced growth.
From Growth to Development
• Here is a brief review of that literature.
• Continue with the audio version after this point.
• The video for this part is a collection of lectures by Professors Tyler Cowen and Alex Tabarrok.
• Simply watch the video for this part.
