1. Macroeconomic Theory
The Solow Growth Model

2. Introduction
The economy will move toward a stable steady – state equilibrium.
In the steady – state equilibrium, there can be permanent economic growth only if there is technological progress.
An economy will transition toward a higher steady state if there is an increase in its rate of saving or a decrease in its rate of population growth.
An economy will experience higher permanent economic growth if there is an increase in its rate of labor – enhancing technological progress.

3. Assumptions Inputs and Output:
Four variables include: output (Y), capital (K), labor (L) and “knowledge” or the “effectiveness of labor” (A).
Production function: Y(t) = F ( K(t) , A(t) L(t) )
where t denotes time.
Features of the production function:
i)Output changes over time only if the inputs into production change.
ii ) AL is referred to as effective labor, and technological progress that enters in this fashion is known as labor – augmenting.
iii ) The ratio of capital to output (K/Y) being constant.

4. Assumptions Concerning the Production Function
Constant-returns-to-scale
If returns to scale were decreasing, there wouldn’t be much growth at all
If returns to scale were increasing, we wouldn’t only need to observe growth, not explain it.
Surprisingly, constant-returns-to-scale shows up in most empirical estimates and therefore appears to be a reasonable approximation.

5. Assumptions Concerning the Production Function
Constant-returns-to-scale implies that:
Y(t)=F[cK(t),cA(t)L(t)]=cF[K(t),A(t)L(t)] for all c ≥ 0
where c denotes any positive constant.
This means that we can also use intensive form by letting c=1/AL for any t. Then:
F(K/AL,1)=1/AL F ( K , AL )
F ( K , AL )/AL = Y/AL
Y/AL=F{K/AL,1}
y=f(k)
This means that output per unit of effective labor is a function of capital per effective unit of labor

6. Assumptions Concerning the Production Function
Assumptions underlying the capital per unit of effective labor :
f’(k)>0, for all k ⇒ MPk>0
f ”(k) < 0 for all k ⇒ MPk is positive but declines as capital per unit of effective labor rises.

7. Assumptions Concerning the Production Function
The Inada conditions a b)
f(0) = 0 ⇒ No output can be produced without capital.
Indefinitely high level of output per worker is associated with indefinitely large ratio of k.

8. Assumptions Concerning the Production Function
The Cobb-Douglas function is the simplest one to satisfy the assumptions
Y=KαAL(1- α), where 0 < α < 1
Characteristics of Cobb – Douglas production function:
i. It exhibits constant returns to scale as the Solow
model requires.

9. Assumptions Concerning the Production Function
ii. It can be conveniently written in terms of capital per unit of effective labor
Assume that the initial levels of capital, labor and knowledge are given.

10. Assumptions Concerning the Production Function
Labor and knowledge grow at constant rates:
and
where n and g are exogenous and positive parameters.
The growth rate of labor is /L(t) = n
The growth rate of knowledge is /A(t)=g

11. The Evolution of the Inputs Into Production
Capital does not grow on its own
It grows because we save some of our output and reinvest
It depreciate
A simple proportional saving function:
S=sY where 0 < s < 1
Investment is simply the rate of increase of the capital stock i.e. one unit of output devoted to investment yields one unit of new capital.

12. The Evolution of the Inputs Into Production
Given that investment is identically equal to saving:
= S= sY
Now assume existing capital depreciates at constant rate:
= sY(t) – δk(t)

13. The Dynamics of the Model
The Dynamics of k
Since k=K/AL
Then using the chain rule:
while
= sY(t) – δk(t)

14. The Dynamics of k
Using the fact that Y/AL = y = f(k)
sf(k)-(δ+n+g)k
Above equation is the fundamental equation of the Solow model. It states that the rate of change of the capital stock per unit of effective labor is the difference between actual investment sf(k) and breakeven investment (δ+ n + g)k .

15. At k=0, f ’ (k) is large, and thus the s f(k) line is steeper than (n+ g + δ)k the line. Thus, for small values of k, actual investment is larger than breakeven investment.
As k become large, , f ’ (k)
fall towards zero and actual
Investment fall below break
even investment. With actual
Investment line flatter than
Breakeven investment, the two
Lines cross at k* where actual
Investment and Breakeven investment are equal.

16. When k < k*, actual investment > breakeven investment, and so is positive ⇒ k is rising.
is negative ⇒ k is falling.
When k = k*, actual investment
=breakeven investment, and so is zero ⇒ k is constant.
Regardless of where k starts, it converges to k*.

17. The Balanced Growth Path
The Solow model implies that, whatever the initial values of all the variables, the economy moves steadily towards a balanced growth path.
In balanced growth path each variable of the model is growing at a constant rate.
Labor is growing at rate n
Technology is growing at rate g
Effective labor is growing at rate n+g
Capital is growing at rate n+g
Output is growing at rate n+g
Capital per worker is growing at rate g
output per worker is growing at rate g

18. The impact of a change in the saving rate
The rate of saving and the steady state
The long – run balanced rate of growth of a neoclassical economy is the constant exogenous rate of growth of both the labor force, n, and knowledge, g, and is entirely independent of the proportion of income saved.

19. The Impact on Output
An increase in the propensity to save, s, shifts the actual investment line upward, and so k* rises.
Initially at k*old, the increase
in propensity to save raises
savings per worker from A to
D because at this level actual
investment exceeds break
even investment. This mean
more resources are being
devoted to investment
than are needed to hold k
constant and hence, is positive. An increase in capital is produced from D to E.
The extra output creates additional income and expands savings per worker further from E to F. The extra saving in turn produces an increase in capital from F to G. These movements continue until a new equilibrium position k* is reached at B at which point k remains constant.

20. The Impact on Output
A permanent increase in the saving rate produces a temporary increase in the growth rate of output per worker.
When k is constant , growth rate of Y/L is g.
When k is increasing, growth rate of Y/L is g+n.
When k reaches the new value of k*, growth rate of Y/L again return to g.
Thus, a change in saving rate has a level effect but not a growth effect.

21. The Impact on Consumption
Real people care about consumption, not output
It turns out that the effect on consumption of an increased savings rate is ambiguous
We do know that initially consumption has to drop – the savings have to come from consumption initially
But, later, as output rises, it isn’t clear if that foregone consumption will be replaced

22. The Impact on Consumption
In the long run, consumption depends on whether f ’(k*), is more or less than n+ g+ δ.
Let c* denotes consumption per unit of effective labor on the balanced growth path.
The steady – state level of per capita consumption:
c*= (1-s) f(k*)= f(k*) - sf(k*)
On the balanced growth path, sf(k*) = (n+ g + δ)k
c*= f(k*) - (n+ g + δ)k
where k*= k* (s, n , g, δ)
∂c*/ ∂s = f ’[k* (s, n , g, δ)]- (n+ g + δ) ∂ k* (s, n , g, δ)/ ∂s

23. The Impact on Consumption
When f ’ (k*)˂ (n+ g + δ), the additional output from the increased capital is not
enough to maintain the
capital stock at its higher
level. Thus, an increase in
the saving rate lowers
consumption even when
the economy has reached
the new balanced growth
path. In other words, consumption must fall to maintain the higher capital stock.

24. The Impact on Consumption
When f ’ (k*)˃(n+ g + δ),there is more than enough additional output to
maintain k at its higher
level. Thus, an increase
in the saving rate raises
consumption in the
long run.

25. The Impact on Consumption
When f ’ (k*)=(n+ g + δ), 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. In term
of human welfare, the
steady state with the higher level of consumption per worker is known as the Golden Rule level of per worker output and capital.

26. Quantitative Implications

27. The Effect On Output In the Long-run
The long-run effect of a rise in saving on output:
y*= f (k*) )
∂y*/∂s=f’(k*)[∂k*(s, n , g, δ) /∂s)] 2)
Next, find ∂k*/ ∂s
sf(k*)=(n+g+)k*
Take the derivative of both sides with respect to s:
sf’(k*)(∂k*/∂s)+f(k*)=(n+g+)(∂k*/∂s)
f(k*)= (n+g+)(∂k*/∂s)- sf’(k*)(∂k*/∂s)
=(∂k*/∂s)[(n+g+)- sf’(k*)]
(∂k*/∂s) = f(k*)/ (n+g+)- sf’(k*)
∂y*/∂s=f’(k*) f(k*)/ (n+g+)- sf’(k*) 3)

28. The Effect On Output In the Long-run
convert (3) to an elasticity by multiplying both sides by s/y*=s/f(k*)
Using the fact that sf(k*)=(n+g+)k* to substitute for s: =

29. The Effect On Output In the Long-run
Define k*f’(k*)}/f(k*)=k, this is the elasticity of output with respect to capital at the steady state. Now:
[s/y*][∂y*/∂s]= k (k*) /[1- k (k*) ]
If markets are competitive, capital earns its marginal product, the share of total income that goes to capital on the balance growth path is k (k*) .

30. The Effect On Output In the Long-run
In most counters, the share of income paid to capital
k(k*) is about one- third, thus
[s/y*][∂y*/∂s]=0.3333/0.6667=0.5⇒ the elasticity of output with respect to the saving rate.
so it says that if s increases by 10%, from20% of output to 22%, then steady state income will increase by 5%.
Significant changes in saving have only moderate effects on the level of output on the balanced growth path.

31. Convergence and the Solow model
A) Absolute convergence
It assumer that poor economies tend to grow faster per capita than rich economies without conditioning on any other characteristics of economies.
Consider a group of closed economies that are structurally similar in the sense that they have the same values of s, n, g , δ and also have the same production function.
Assume that the only difference among the economies is the initial quantity of capital per person.
Regions or countries with lower starting values of k(0) have higher per capital growth rates and tend to catch up or converge to these with higher capital - labor ratios. Thus all the economies have the same steady-state values k* and y*.

32. Convergence and the Solow model
sf(k)-(δ+ n+ g)k
Divide both sides by k: (t)/k = sf(k)/k - (δ + n + g)
Sf(k)/k is a downward- sloping curve which asymptotes to infinity at k = 0 and approaches zero as k tends to infinity. (δ + n + g) is a horizontal line.
The vertical distance between
the Sf(k)/k and (δ + n + g) line
is equal the growth rate of capital
per person.
Since Sf(k)/k and (δ + n + g) lines
intersect once only, the steady
- state capital - labor ratio k*.

33. When k is relatively low, the average product of capital, f(k)k is relatively high. The actual investment per unit of capital, sf(k)/k is also high. Consequently, the growth rate of capital is also high.
If the economy starts with k(0) > k*, then the growth rate of k is negative, and k falls over time. The growth rate increases and approaches zero as k approaches k*.

34. B) Conditional convergence
The Solow model predicts conditional convergence. Each economy converge to its own steady state and that the speed of this convergence relates inversely to the distance from the steady state.
It is true empirically that countries with higher level of real per capital GDP tend to have higher saving rate, i.e. s poor < s rich and hence k*poor < k*rich
The rich economy would
be predicted to grow faster
per capita than the poor
economy. A poor country
may grow at a slower rate
than a rich one.

35. Sources of variation in output per worker in the Solow model
The Solow model identities two possible sources of variation in output per work: differences in capital per worker ( K/L) and differences in the effectiveness of labor (A).
Capital per worker
In fact, variations in the accumulation of physical capital do not account for a significant part of either worldwide economic growth or cross - country income differences due to
The required differences in capital are too large when trying to account for large differences in income on the basis of differences in capital.
Assume that output per worker in the United States today is ten times larger than it was a hundred years ago.

36. Sources of variation in output per worker in the Solow model
Let capital share k = 1/3 a tenfold difference in output per worker requires a difference of a factor of = 1000.
Indeed, the U.S. capital stock per worker is roughly ten times larger than it was a hundred years ago, note thousand times larger.
Kaldor said that in most of the major industrialized countries over the past century, the growth rates of labor, capital and output are each roughly constant. The growth rates of output and capital are larger than the growth rate of labor so that output per worker and capital per worker are rising. Although capital - output ratios vary across countries, the variation is not great.
In sum, differences in capital per worker are smaller than those needed to account for the differences in output per worker.

37. Attributing differences in output to differences in capital without differences in the effectiveness of labor implies large variation in the rate of return on capital.
Equation implies that the elasticity of the marginal product of capital with respect to output is – 1-a/a.
Let a= 1/3 , a tenfold difference in output per worker arising from differences in capital per worker implies a hundredfold difference in the MP(k) i.e.

38. There is no evidence of such larger the difference in rate of return
There is no evidence of such larger the difference in rate of return. If rates of return were larger by a factor of ten or hundred in poor countries than in rich counties, there would be many incentives to invest in poor countries. Empirically, there are not the facts.
B) Effectiveness of labor
Attributing differences in standards of living to differences in the effectiveness of labor does not require large differences in capital or in rates of return.
Along a balanced growth path, capital is growing at the same rate as output and the MPK is constant.
The Solow model identities the behavior of the effectiveness of labor as the driving force of growth. However, the growth of the effectiveness of labor is regarded as exogenous.

39. The Solow model does not identify what the “effectiveness of labor” is
The Solow model does not identify what the “effectiveness of labor” is. It is just a factor other than labor and capital that affects output.
Interpretations of the effectiveness of labor are : knowledge, the education and skills of labor force, the strength of property rights, the quality of infrastructure, cultural attitudes toward entrepreneurship and work.

40. Empirical Applications
Growth Accounting
Growth accounting is the name given to using a growth model to figure out how big technology is
Start with output
Subtract the contribution of labor
Subtract the contribution of capital
What’s left is a residual called the “Solow residual”
Is this technology?
Is it close to what technology is?

41. Growth Accounting Take a total derivative of the production function
dY/dt=(∂Y/∂K)(dK/dt)+(∂Y/∂L)(dL/dt)+(∂Y/∂A)(dA/dt)
Divide both side by Y, to get:
(dY/dt)/Y=(∂Y/∂K)(dK/dt)/Y+(∂Y/∂L)(dL/dt)/Y+(∂Y/∂A)(dA/dt)

42. Growth Accounting Rewrite the terms on the right hand side:
(dY/dt)/Y=(∂Y/∂K)[(dK/dt)/K](K/Y)+(∂Y/∂L)[(dL/dt)/L](L/Y)+(∂Y/∂A)[(dA/dt)/A](A/Y)
The first and last parts of the first two terms on the RHS are now elasticities of output w.r.t. K and L:
(dY/dt)/Y=αK[(dK/dt)/K]+αL[(dL/dt)/L]+(∂Y/∂A)[(dA/dt)/A](A/Y)

43. Growth Accounting In the previous equation:
The two elasticities are measurable with real data
The last term on the right is what is left over after subtracting capital and labor from output – a residual
This captures the effect of technology
This is known as a Solow residual
The problem with interpreting the Solow residual as technology is that it is really everything but labor and capital

44. Savings and Investment
These are only the same in a closed economy
In the real world, where economies are open
Countries that increase savings rates are pushing their return on capital lower.
In the absence of barriers to capital mobility, an increase in the savings rate in one country should translate into higher investment in those country where return to capital are higher. If the return to capital is equalized across countries, an increase in the savings rate in one country should be spread uniformly across all countries.
So, we shouldn’t see much correlation between savings and investment if the Solow model is correct

45. Savings and Investment
Feldstein and Horioka ‘80 show this isn’t true
Possible explanations:
1. Barriers to capital mobility;
2. Some variables may affect both savings and investment (e.g. tax policy);
3. Governments may implement policies that increase the correlation of savings and investment in order to avoid large trade deficits/surpluses.

46. Extension: Land and Natural Resources
Cobb-Douglas production function augmented to include land and resources:
where R stands for natural resources, T is land, and a, B, γ ˃0 and a + B+ γ ˃ 1. Dynamics of inputs of production function:

47. Extension: Land and Natural Resources
Taking logs of Y (t ) :
Differentiating w.r.t t, we get growth rates:
Substituting growth rate of A,L,T and R
Since on balance growth path gy=gk, we can solve for gy

48. Extension: Land and Natural Resources
The growth rate of output per worker is
This can be either positive or negative. Technological progress encourages growth of output per worker, but falling land and resources per worker limit growth.

49. Extension: Land and Natural Resources
Suppose land and resources grew at the same rate as labor, n. Then, replicating the above derivation:

50. Extension: Land and Natural Resources
Therefore, the limitation that land and natural resources impose on growth of output per worker is
Thus, the greater are the shares of land and natural resources, the lower is growth. Nordhaus (1992) estimates that this drag accounts for (0.24%) of annual growth.

51. Dr. Cavalcanti -- Economic Growth
Michaelmas Term 2010
Conclusion
Solow model: Simple and tractable framework, which allows us to discuss capital accumulation and the implications of technological progress.
It shows us that if there is no technological progress there will be no sustained growth.
Generate per capita output growth, but only exogenously.
Capital accumulation: determined by the saving rate, the depreciation rate and the rate of population growth. All are exogenous.
In steady state output per unit of effective labor remains constant, output per worker depends only on technological growth, and that total output depends on population and technological growth.

52. Research and Development Models
New Growth Theory
Research and Development Models

53. Framework and Assumptions
We’ll model technology as the output of a research and development industry
So the economy produces two things: consumption goods and ideas
Major simplification
We’re going to go back to the idea that the saving rate is constant, and not optimally determined
Continuous time

54. Specifics Same four endogenous variables: K, L, A and Y Two sectors
For simplicity, there is no depreciation
Two sectors
Goods
Ideas
Labor and capital are split between the two sectors
An individual can only work in one sector
Technology is not split
Ideas can be used anywhere

55. Specifics Goods production function Ideas production function
Y(t)=[(1-aK)K(t)][(1-aL)A(t)L(t)](1-)
The shares of labor and capital devoted to goods production, aK and aL, are exogenous.
Ideas production function
dA/dt=B[aKK(t)]β[aLL(t)]γA(t)θ
Note that this is production of new ideas (old ideas do not have to be replaced)
This function allows for decreasing, constant or increasing returns to scale

56. Specifics
Returns-to-scale restrictions are made to capture scalability of production processes
Constant-returns-to-scale makes sense for most physical processes
It isn’t clear what makes sense for the production of ideas, so β+γ+θ isn’t restricted to equal 1

57. Specifics
The key to understanding the model is the similarities and differences between output, capital and technology
New output is produced every period, but it is all used in that period (its “depreciation” = 100%)
New technology is produced every period, but it never depreciates, so it accumulates like capital
Capital is output diverted from consumption
Technology is produced instead of consumption

58. The Dynamics of Knowledge Accumulation
For simplicity, assume =β=0 and there is no capital
The production functions are then
Y(t)=(1-aL)A(t)L(t)
dA/dt=B[aLL(t)]γA(t)θ
The growth rate of technology is then:
gA(t)=(dA/dt)/A=B[aLL(t)]γA(t)θ-1

59. The Dynamics of Knowledge Accumulation
Multiply through by gA to get:
(dgA/dt) =nγgA+(gA)2(θ-1)
The behavior of this quadratic will depend on the size of θ
The growth of technology is endogenous, so new growth models are sometimes called endogenous growth models

60. Case 1: θ<1 The steady state is where:
Here, θ reflect the effect of existing technology on the creation of new ideas
θ<1 implies that limited contribution of the existing technology in the production of new technology
The steady state is where:
(dgA/dt) =nγgA+(gA)2(θ-1)=0,
Or gA* = γn/(1-θ)
So the rate of growth of technology converges to a positive value

61. Case 1: θ<1 This is a model of endogenous growth
Output growth occurs because of technological growth
Technology grows because new ideas are created and old ideas don’t depreciate
It’s kind of odd that the growth of technology depends on population growth
This seems truer at larger scales, but not at smaller ones
Technological growth does not depend on the proportion of people working in research and development

62. Case 2: θ>1
In this case, there is no steady-state growth rate of technology
The growth rate accelerates because the production of new knowledge rises more than proportionally with the existing stock.
This implies that the overall economy never reaches a steady-state either
This is implausible, but shouldn’t be completely dismissed. Growth rates of developed countries have been inching up over the decades.

63. Case 3: θ=1
Production of new knowledge is proportional to existing stock
The model simplifies to:
gA(t)=B[aLL(t)ent]γ
Growth of ideas depends on population
Growth of ideas depends on the proportion of the population working in research and development
(dgA/dt) =nγgA
Growth of ideas is accelerating when population growth is positive
Growth of ideas stops when population growth stops
These models are simple and plausible

64. Thank you