Solow groth model 2

SOLOW GROWTH MODEL
CHAPTER 4:
Rasheed zedan

Solow growth model
• In 1956 𝑠𝑜𝑙𝑜𝑤 introduce his model of economic growth
• Solow recognize the problem of rigid production
function in harrod-domar model
• Solow dropped the fixed coefficient production
function and replaced it by a neoclassical production
function that allows for more flexibility and
substitution b/w factors of production
• Like harrod –domar model 𝑠𝑜𝑙𝑜𝑤 model was
introduced to analyze industrialized economies
• But has been used to explore economic growth in all
countries including developing countries

Solow growth model
• Capital-output and capital-labor ratios are not fixed but
vary depending on the relative endowment of capital
and labor in the economy
• The 𝑠𝑜𝑙𝑜𝑤 model retains from hard-domar model the
assumption of constant returns to scale
• constant returns to scale : means that doubling the
amount of labor and capital leads to double the
amount of output
• In the 𝑠𝑜𝑙𝑜𝑤 model doubling the amount of capital can
be achieved by using a different combination of capital
and labor (the company can use more capital and less
labor or the opposite)

The basic equations in solow model
• The 𝑠𝑜𝑙𝑜𝑤 model can be understood by using all
key variables of per worker-term (capital per
worker, output per worker)
• We can do this by dividing both sides of equation 4-
1 by L we can get:
• Y/L=F(K/L,1) where
• Y/L= OUTPUT PER WORKER
• K/L=CAPITAL PER WORKER
• In this equation we can conclude that output per
worker is function of capital per worker

The basic equations in 𝒔𝒐𝒍𝒐𝒘 model
• Lower-case latters can represent quantities in per
worker term so
• y= output per worker = (Y/L)
• K= capital per worker=(K/L)
• So the first equation In 𝒔𝒐𝒍𝒐𝒘 model is:
• y=f(k)
• The solow model assume production function with
diminishing returns to capital
• diminishing returns to capital:with a fixed labor supply
increasing machines lead to increase output but the
addition to output from each new machine get smaller
and smaller

Figure 4-3 the production function in
𝒔𝒐𝒍𝒐𝒘 growth model

• In the above figure 4-3
• The slope of the curve decline as the capital stock
increase reflecting the assumption of diminishing
marginal product of capital
• The first equation of the model tells us that capital per
worker is fundamental to growth process
• In turn the second equation focuses on the
determinants of changes in capital per worker
• ∆𝒌 = 𝒔𝒚 − 𝒏 + 𝒅 𝑲
• Where:
• ∆𝒌 =change in capital per worker
• Sy =saving per worker

• This equation is very important because it implies that
the change in capital per worker (∆𝒌) is determined
by three things
• The ∆𝒌 is positively related to saving per worker
.because s is the saving rate and y is the income so
the term sy means saving per worker so as the saving
per worker increase investments will increase and
capital per worker k will increase
• The ∆𝒌 is negatively related to population growth
because growth in population means that labor force
will increase with no increase in investments means
that capital per worker will fall
• Depreciation erodes capital stock so each year the
amount of per worker capital fall by the amount of
depreciation

• Therefore saving increase capital per worker and
depreciation ,population growth decrease it
• If the saving per worker (𝒔𝒚) is greater than the
amount of capital needed to compensate labor and
depreciation (𝒏 + 𝒅)𝒌 then ∆𝒌 is positive number
meaning that capital per worker will increase this
situation is called capital deepening.
• capital deepening: economies in which workers have
access to more machines , computers and other
equipment these economies can produce more output
per worker
• If the amount of saving per worker is exactly equal to
the amount of capital needed to compensate labor and
depreciation this situation is called capital widening

• Capital widening: economies in which the widening of
saving is exactly equal to the expansion of labor force
and depreciation(this situation leads to no change in
capital per worker)
• Note :a country with a high saving rates can deepen
its capital and rapidly expand its capital per worker
providing the base for growth in output
• In hard-domar model saving play a central role in the
growth process
• In 𝑠𝑜𝑙𝑜𝑤 model relation b/w saving and growth is not
linear because of diminishing returns to capital in the
production function
• In addition 𝑠𝑜𝑙𝑜𝑤 model introduce the role of
population growth rate and allows for substitutions
b/w the factors of production

The 𝒔𝒐𝒍𝒐𝒘 diagram

• The 𝑠𝑜𝑙𝑜𝑤 diagram consist of 3 curves
• The first is the production function y=f(k)
• The second curve is the saving function which can
be driven directly by multiplying both side of the
production function by the saving rate we get
sy=s× 𝒇 𝒌 because saving is assumed to be a fixed
fraction of income
• the third curve is the line (n+d)k which is a straight
line through the origin with the slope (n+d). This
line represent the amount if capital needed to
compensate the growth of labor force and
depreciation to keep capital per worker (k) constant

• Note that: the second and third curves are
representation of the two right-hand terms of
equation }4-14{
• The second and third curves intersect at point A
where 𝒌 = 𝒌𝒐 at point A 𝒔𝒚 is exactly equal to ( 𝒏

• Note: to the left of point A (𝒌 = 𝒌𝟏) the amount of saving
is greater than the amount needed to compensate the
labor force and depreciation (capital deepening) and the
economy continue to shift to the right until it reaches the
steady state point (A) in terms of production function
shifting to the right represent an increase in output per
worker from 𝒚𝟐 𝒕𝒐 𝒚𝒐
• To the right of point A. 𝒌 = 𝒌𝟐 the amount of saving 𝒔𝒚 is
smaller than the amount needed to compensate the
growth in labor force and depreciation 𝒏 + 𝒅 the
economy continue to shift to the left until it reaches the
equilibrium at point A . this shift to the left caused output
per worker to decline from 𝒚𝟐 𝒕𝒐 𝒚𝟎

• Point A is the only place in which the amount of
saving 𝒔𝒚 is equal to the amount needed by the
labor force and depreciation 𝒏 + 𝒅 𝒌 so that the
amount of capital per worker 𝒌 remain constant
• Point A is called the steady state in 𝑠𝑜𝑙𝑜𝑤 model
• Output per worker at the steady state (𝒚𝒐) is
mentioned as the steady state, long run, potential
level of output per worker
• It should be noted that all values that remains
constant are expressed in per worker term
• However output per worker is constant the total
output grow at rate 𝒏

Changes in the saving rate and population
growth rate in the 𝒔𝒐𝒍𝒐𝒘 model
• Both harrod-domar and 𝑠𝑜𝑙𝑜𝑤 models put saving at
the core of the growth process
• In the harrod-domar model an increase in the
amount of saving leads directly to an increase in
aggregate output
• As shown in figure 4-5 increasing saving from 𝒔‘𝒕𝒐 𝒔
increasing saving means that saving per worker
increase and the saving function shift from 𝒔𝒚 to be
𝒔‘𝒚 so the amount of saving per worker is greater
than the amount needed to compensate labor and
depreciation (𝒏 + 𝒅)𝒌 so k increase from 𝒌𝒐 𝒕𝒐 𝒌𝟑

Figure 4.5 an increase in saving rate

Changes in the saving rate and population
growth rate in the 𝒔𝒐𝒍𝒐𝒘 model
• And the output per worker increase from 𝒚𝒐 𝒕𝒐 𝒚𝟑 and
the economy shift to a new long run equilibrium at
point B
• Therefore increasing the saving rate means that
investment will increase which will lead to increase per
worker capital
• The 𝑠𝑜𝑙𝑜𝑤 model predicts economies that save more
have higher living standard than those saving less
• The increase in per capita income however is smaller
than for a similar increase in 𝒔 in the harrod-domar
model because the 𝑠𝑜𝑙𝑜𝑤 model has diminishing to
returns in production

Changes in the saving rate and population growth
rate in the 𝒔𝒐𝒍𝒐𝒘 model
• Higher saving also lead to a temporary increase in
the economic growth rate as the steady state shifts
from A to B
• However the increase in the saving rate does not
lead to a permanent increase in the long run rate of
output growth rate which remain at n
• The solow diagram also can be used to evaluate the
impacts of a change in the population growth rate
• An increase in the population growth rate from
𝒏 𝒕𝒐 𝒏‘ rotates the capital widening line to shift
from (𝒏 + 𝒅)𝒌 to (𝒏‘ + 𝒅)𝒌 as shown in fig.4-6

Fig. 4.6: The effect of Population Growth
in the Solow Model

Changes in the saving rate and population growth
rate in the 𝒔𝒐𝒍𝒐𝒘 model
• As a result of increasing the population growth rate
the labor force increase and saving per worker
decline so the capital per worker falls from
𝒌𝒐 𝒕𝒐 𝒌𝟒 and the economy shift to a new steady
state C saving per worker falls from 𝒔𝒚𝒐 𝒕𝒐 𝒔𝒚𝟒
,output per worker also falls from 𝒚𝒐 𝒕𝒐 𝒚𝟒
• Thus an increase in the population growth rate
leads to lower average income in the 𝒔𝒐𝒍𝒐𝒘 model
• With a higher population growth rate ,Y need to
grow faster to keep y constant

Changes in the saving rate and population growth rate in
the 𝒔𝒐𝒍𝒐𝒘 model
• The 𝑠𝑜𝑙𝑜𝑤 growth model suggests that growth
rates differ across countries for two main reasons:
• Two countries with the same current level of
income may experience different growth rates if
one has a higher steady state level of income than
the other
• Two countries with the same long run steady state
level of income may have different growth rates if
they are In a different points in transition to the
steady state

Technological changes in the 𝒔𝒐𝒍𝒐𝒘
model
• The 𝑠𝑜𝑙𝑜𝑤 model is a powerful tool for analyzing
the inter-relationships b/w saving, investment,
population growth, and economic growth
• The technological progress allows output per
worker to continue to grow
• The technological progress is a key driver but not
the only driver of productivity growth
• The technological progress play a central role of
explaining productivity growth

Technological changes in the 𝒔𝒐𝒍𝒐𝒘 model
• To incorporate economy’s ability to produce more
output with the same amount of capital and labor we
modify the original production function by adding new
variable T to represent technological progress as
follows:
• 𝒀 = 𝑭(𝑲, 𝑻 × 𝑳)
• Technology is introduced in such a way that directly
enhance the input of labor as shown T is multiplied by L
this type of Technological changes is referred to as
a labor augmenting
• As the technology increase the efficiency and
productivity of labor increase because the same
amount labor are now able to produce more
output

• Increase in T can result from improvements In
technology or in term of human capital such as
improvements in health, education, or skills of the
work force
• The combined term 𝑻 × 𝑳 Is sometimes referred to
as the amount of effective units of labor
• The expression 𝑻 × 𝑳 measures both the amount
of labor and its efficiency in the production process
• An increase in either T or L lead to an increase in
the amount of effective labor and aggregate output

• An increase in T differ from an increase in L
however , because the rise in income from new
technology does not need to be shared with
additional worker
• Technological changes and productivity growth
allows output per worker to increase
• The usual assumption that technology grow at fixed
rate denoted by 𝜽 so that
∆𝑻
𝑻
= 𝜽
• if technology grows at 1 percent per year , then
each worker will be1 percent more productive each
year

• With the work force growing at n , growth in the
effective supply of labor is equal to 𝒏 + 𝜽
• If the work force grows by 2 percent per year and
the technology grows by 1 percent per year the
effective supply of labor increase by 3 percent
• To show technological change in the 𝑠𝑜𝑙𝑜𝑤 diagram
we need to modify our notations
• Previously we express y and k as output per worker
and capital per worker
• we need now to express these variable in terms of
output and capital per effective worker

• So instead of dividing Y and K by L as previously (to
obtain y and k)we will now divide them by (𝑻 × 𝑳)
• Output per effective worker (𝒚𝒆)=Y/(T× 𝑳)
• Capital per effective worker (𝒌𝒆)=K/(T× 𝑳)
• With these changes the production function can be
written as follows: 𝒚𝒆 = 𝒇(𝒌𝒆)
• Saving per effective worker: 𝒔𝒚𝒆
• Effective labor now growing at the rate (𝒏 + 𝜽)
• The capital accumulation equation can be written
as: ∆𝒌𝒆 = 𝒔𝒚𝒆 − 𝒏 + 𝒅 + 𝜽 𝒌𝒆

• The new term 𝒏 + 𝒅 + 𝜽 𝒌𝒆 is greater than the
original one 𝒏 + 𝒅 𝒌𝒆.indicating that more
capital is needed to keep capital per effective
worker constant

Strengths and weaknesses of the 𝒔𝒐𝒍𝒐𝒘
model
• Strengths:
• It is more powerful tool for understanding the
growth process
• Provide more reasonable flexibility of factors
proportions in the production process.as it replace
the fixed coefficient production function with a
neoclassical one
• It emphasize the important role of factor
accumulation and saving
• Does much better job of describing real world
outcomes than harrod-domar model

Strengths and weaknesses of the 𝒔𝒐𝒍𝒐𝒘 model
• Provide the role of technological changes and
productivity growth in the growth process
• Provide powerful insights in the relationship b/w
saving, investment , population growth and
technological change on the steady-state level output
per worker

Strengths and weaknesses of the 𝒔𝒐𝒍𝒐𝒘 model
• Weaknesses:
• Specifying productivity growth exogenous it do not
spell out exactly how it takes place or how the
growth process itself might affect it
• The model helps focus attention on the more
fundamental influences on the steady state and the
growth rate but it does not provide a full
understanding of the precise pathway through
which these factors influence output and growth

Solow groth model 2

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Solow groth model 2