Neoclassical Growth Models

CHAPTER FOUR
The neo-Keynesian Harrod–Domar model;
The Solow–Swan neoclassical model; and
The Romer–Lucas-inspired endogenous growth models.
1 Macro Economics-II CH-4 WSU By Zegeye Paulos

4.1 The Harrod- Domar Growth
Model Harrod (1939, 1948) and Evsey Domar (1946, 1947) independently
developed theories that relate an economy’s rate of growth to its
capital stock.
 The model assumes
an exogenous rate of labour force growth (n),
a given technology exhibiting fixed factor proportions
i.e. - constant capital–labour ratio, K/L) and
- a fixed capital–output ratio (K/Y).
Assuming a two-sector economy (households and firms),
 we can write the simple national income equation as (4.1):
Yt = Ct + St -----------------------------------------------------4.1
where Yt = GDP, Ct = consumption and St = saving.
 Equilibrium in this simple economy requires (4.2):
It = St ------------------------------------------4.22
Macro Economics-II CH-4 WSU By Zegeye Paulos

Substituting (4.2) into (4.1) yields (4.3):
Yt = Ct + It ----------------------------------------------------4.3
Within the Harrod–Domar framework the growth of real GDP
is assumed to be proportional to
 The share of investment spending (I) in GDP and
 For an economy to grow, net additions to the capital
stock are required.
The capital stock over time is given in equation as
Kt+1 = (1− δ)Kt + It ---------------------------------4.4
where δ is the rate of depreciation of the capital stock.
The relationship between
 the size of the total capital stock (K) and total GDP (Y)
is known as the capital–output ratio (K/Y = v) and is assumed fixed.

 Given that we have defined v = K/Y, it also follows that v = ΔK/ΔY
 Assume that total saving is some proportion (s) of GDP (Y)
St = sYt ----------------------------------------------------------------4.5
 Since K = vY and It = St, it follows that we can rewrite equation (4.4) as
equation (4.6):
vYt+1 = (1− δ)vYt + sYt --------------------------------------------------4.6
 Dividing by v, and subtracting Yt from both sides of equation (4.6) yields
equation (4.7):
Yt+1 − Yt = [s/v − δ]Yt -----------------------------------------------------4.7
 Dividing through by Y t gives us equation (4.8):
[Yt+1 − Yt ]/Yt = (s/v) − δ -------------------------------------------------4.8
 Here [Yt + 1 – Yt]/Yt is the growth rate of GDP. Letting G = [Yt + 1 –
Yt]/Yt,
 We can write the Harrod–Domar growth equation as (4.9):
G = s/v − δ ------------------------------------------------------------4.9

G = s/v − δ --------------------------------------4.9
NB: This simply states that the growth rate (G) of GDP is
jointly determined by
the savings ratio (s) divided by the capital–output
ratio (v).
The higher the savings ratio and the lower the capital–
output ratio and depreciation rate, the faster will an
economy grow.
By ignoring depreciation rate Harrod–Domar model as
G = s/v ---------------------------------------------4.10

 For example, if a developing country desired to achieve a growth
rate of per capita income of 2 per cent per annum and
 population is estimated to be growing at 2 per cent, then economic
planners would need to set a target rate of GDP growth (G*) equal
to 4 per cent. G/ pop%= Percapita income
4%/2% = 2%
 If v = 4, this implies that G* can only be achieved with a desired
savings ratio (s*) of 0.16, or 16 per cent of GDP.
 If s* > s, there is a ‘savings gap’, and planners needed to devise
policies for plugging this gap.

Since the rate of growth in the Harrod–Domar model
is positively related to the savings ratio,
 If domestic sources of finance were inadequate to achieve the
desired growth target,
then foreign aid could fill the ‘savings gap’
Aid requirements (Ar) would simply be calculated as
s* – s = Ar

However, a major weakness of the Harrod–Domar
 It’s assumption of a fixed capital– output ratio. K/Y
 The model assumed that aid inflows would go into investment one to one.
 But it soon became apparent that inflows of foreign aid, with the objective of
closing the savings gap, did not necessarily boost total savings.
 The assumption of zero substitutability between capital and labour (that is, a fixed
factor proportions production function).
 This is a ‘crucial’ but inappropriate assumption for a model concerned with
long-run growth.

 In the Harrod–Domar model the capital–output ratio (K/Y) and the
capital–labour ratio (K/L) are assumed constant.
 In a growth setting this means that K and Y must always grow at the same rate
to maintain equilibrium.
 However, because the model also assumes a constant capital–labour ratio
(K/L), K and L must also grow at the same rate.
 Therefore, if we assume that the labour force (L) grows at the same rate
as the rate of growth of population (n), then we can conclude that the only
way that equilibrium can be maintained in the model is
 for n = G = s/v.
 It would only be by pure coincidence that n = G.
 If n > G, the result will be continually rising unemployment.
 If G > n, the capital stock will become increasingly idle and the growth
rate of output will slow down to G = n.
 Thus, whenever K and L do not grow at the same rate, the economy falls
off its equilibrium ‘knife-edge’ growth path.

 Increasing the saving ratio in lower-income countries is not
easy.
 Many developing countries has lower marginal propensity to
save
 Many developing countries lack reliable and strong financial
system and institutions
 More challenging to achieve efficiency(K/Y) in developing
countries
 Research and development(R/D) is under funded
 Borrowing money abroad to close the saving gap can cause
external debt in the long run

4.2. The Solow Neoclassical Growth Model
 In Solow model
 capital,
 labor, and key determinants of production of goods &services
 technology
 we developed the Solow model to show
 how changes in capital (through S and I)
& affect economy’s output.
 changes in the labor force (pon growth)
 We are now ready to add the third source of growth
 changes in technology—to the mix.
 The Solow model does not explain technological progress but,
 instead, takes it as exogenously given and
 shows how it interacts with other variables in the process of economic
growth.

 The key assumptions of the Solow model are:
 The economy consists of one sector producing one type of
commodity that can be used for either
 investment or consumption purposes;
The economy is closed and the government sector is ignored;
all output that is saved is invested;
The economy is always producing its potential (natural) level
of total output;
Solow abandons the H-D assumptions of a fixed capital–output
ratio (K/Y) and fixed capital–labour ratio (K/L);
The rate of technological progress, population growth and the
depreciation rate of the capital stock are all determined
exogenously.

 The Solow growth model is designed to show how growth
in the capital stock,
growth in the labor force, and
advances in technology
affect a nation’s total output of goods and services.
 We will build this model in a series of steps.
 Our first step is to examine how the supply and demand for goods
determine the accumulation of capital.
 In this first step, we assume that
 the labor force and
 technology are fixed.
 We then relax these assumptions by introducing changes in the
labor force later in this chapter and by introducing changes in
technology in the next.

 The supply of goods in the Solow model is based on the production function,
 which states that output depends on
 capital stock and
 labor force:
Y = F(K, L).
 The Solow growth model assumes that the production function
 has constant returns to scale. zY = F(zK, zL)
 That is, if both capital and labor are multiplied by z, the amount of output is also multiplied by z.
 Production functions with constant returns to scale allow us to analyze all quantities in the economy
relative to the size of the labor force.
 To see that this is true, set z = 1/L in the preceding equation to obtain
Y/L = F(K/L, 1).

 This equation shows that the amount of output per worker Y/L
is a function of the amount of capital per worker K/L.
 (The number 1 is constant and thus can be ignored.)
 The assumption of constant returns to scale implies that
the size of the economy as measured by the number of
workers does not affect the relationship between
output per worker and capital per worker.
 y = Y/L is output per worker, and k = K/L is capital per worker.
 We can then write the production function as
y = f (k),

 Note that in Figure 4.1, as the amount of capital increases,
 the production function becomes flatter, indicating that the production
function exhibits diminishing marginal product of capital.
 When k is low, the average worker has only a little capital to work
with, so an extra unit of capital is very useful and produces a lot of
additional output.
 When k is high, the average worker has a lot of capital already, so
an extra unit increases production only slightly.
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Macro Economics-II CH-4 WSU By Zegeye
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figure 4.1 Production function
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Macro Economics-II CH-4 WSU By
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The Demand for Goods and the Consumption Function
 The demand for goods in the Solow model comes from
 consumption and
 investment.
 In other words, output per worker y is divided between
 consumption per worker c and
 investment per worker i:
y = c + i.
 This equation is the per-worker version of the national income accounts identity for an economy.
 The Solow model assumes that
 each year people save a fraction s of their income and
 consume a fraction (1 – s)
 We can express this idea with the following consumption function:
c = (1 − s)y,
 where s, the saving rate, is a number between zero and one.
 For now, however, we just take the saving rate s as given.
 To see what this consumption function implies for investment,
 substitute (1 – s)y for c in the national income accounts identity:
y = (1 − s)y + i.
 Rearrange the terms to obtain i = sy.18

 For any given capital stock k, the production function y = f(k)
 determines how much output the economy produces, and
 The saving rate s determines the allocation of that output between
 consumption and investment.
Growth in the Capital Stock and the Steady State
 At any moment, the capital stock is a key determinant of the economy’s output,
 but the capital stock can change over time, and
 those changes can lead to economic growth.
 In particular, two forces influence the capital stock:
 investment and
 depreciation.
 Investment is expenditure on new plant and equipment, and it causes the capital stock
to rise.
 Depreciation is the wearing out of old capital, and it causes the capital stock to fall.
 Let’s consider each of these forces in turn. As we have already noted, investment per
worker i equals sy.
 By substituting the production function for y, we can express investment per worker as
a function of the capital stock per worker:
i = sf(k).19

 This equation relates the existing stock of capital k to the
accumulation of new capital i.
 The production function f(k), and the allocation of that output between
 consumption
and is determined by the saving rate s.
 saving
 To incorporate depreciation into the model, we assume that a certain
fraction  of the capital stock wears out each year.
 For example, if capital lasts an average of 25 years, then the
depreciation rate is 4 percent per year ( = 0.04).
 The amount of capital that depreciates each year is k .
 Change in Capital Stock = Investment − Depreciation
 k = i −k
 where k is the change in the capital stock between one year and the
next.
 Because investment i equals sf(k), we can write this as
k = sf (k) − k .
20

 Regardless of the level of capital with which the economy begins, it ends up with the
steady-state level of capital.
 To see why an economy always ends up at the steady state,
 suppose that the economy starts with less than the steady-state level of capital, such as
level k1 in Figure 4.2.
 In this case, the level of investment exceeds the amount of depreciation.
 Over time, the capital stock will rise and will continue to rise- along with output
f(k)—until it approaches the steady state k*.
 Similarly, suppose that the economy starts with more than the steady-state level of
capital, such as level k2.
 In this case, investment is less than depreciation: capital is wearing out faster than
it is being replaced.
 The capital stock will fall, again approaching the steady-state level.
 Once the capital stock reaches the steady state, investment equals depreciation, and
there is no pressure for the capital stock to either increase or decrease.

 Figure 4-2 graphs the terms of this equation—investment and
depreciation—for different levels of the capital stock k.
 The higher the capital stock, the greater the amounts of output and
investment.
 Yet the higher the capital stock, the greater also the amount of
depreciation.
 As Figure 4.2 shows, there is a single capital stock k* at which the
amount of investment equals the amount of depreciation.
 If the economy finds itself at this level of the capital stock, the capital
stock will not change because the two forces acting on it
 investment and
 depreciation—just balance.
 That is, at k*, k = 0, so the capital stock k and output f(k) are steady
over time (rather than growing or shrinking).
 We therefore call k* the steady-state level of capital.
 The steady state is significant for two reasons.
 As we have just seen, an economy at the steady state will stay there.
 In addition, and just as important, an economy not at the steady state
will go there.22

 Figure 4.2

Approaching the Steady State: A Numerical Example
 Let’s use a numerical example to see how the Solow
model works and how the economy approaches the
steady state.
 we assume that the production function is
 The Cobb–Douglas production function with the
capital-share parameter α equal to 1/2.
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Macro Economics-II CH-4 WSU CBE, Economics By Zegeye
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 To derive the per-worker production function f(k),
divide both sides of the production function by the
labor force L:
 Rearrange to obtain
 Because y = Y/L and k = K/L, this equation becomes
 which can also be written as
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Macro Economics-II CH-4 WSU CBE, Economics, By Zegeye
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 This form of the production function states that
 output per worker equals the square root of the amount of capital per
worker.
 To complete the example, let’s assume that 30 percent of
output is saved (s = 0.3), that 10 percent of the capital stock
depreciates every year ( = 0.1), and that the economy starts
off with 4 units of capital per worker (k = 4). Given these
numbers, we can now examine what happens to this
economy over time.
 We begin by looking at the production and allocation of
output in the first year,
 when the economy has 4 units of capital per worker. Here are
the steps we follow.
 According to the production function y = , the 4 units of capital
per worker (k) produce 2 units of output per worker (y).
 Because 30 percent of output is saved and invested and 70
percent is consumed, i = 0.6 and c = 1.4.
 Because 10 percent of the capital stock depreciates, dk = 0.4.
 With investment of 0.6 and depreciation of 0.4, the change in the
capital stock is k = 0.2.26
Macro Economics-II CH-4 WSU CBE, Economics By
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 Thus, the economy begins its second year with 4.2 units of capital
per worker.
 We can do the same calculations for each subsequent year.
 Table 4.3 shows how the economy progresses. Every year, because
investment exceeds depreciation, new capital is added and output
grows.
 Over many years, the economy approaches a steady state with 9
units of capital per worker.
 In this steady state, investment of 0.9 exactly offsets depreciation of
0.9, so the capital stock and output are no longer growing.
 Following the progress of the economy for many years is one way
to find the steady-state capital stock, but there is another way that
requires fewer calculations.
 Recall that
k = sf(k) − k.
 This equation shows how k evolves over time. Because the steady
state is (by definition) the value of k at which k = 0, we know that
0 = sf (k*) − k*,
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Table 4.1
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How Saving Affects Growth
 To understand more fully the international differences in economic performance, we must consider the effects of
different saving rates.
 Consider what happens to an economy when its saving rate increases.
 Figure 4-3shows such a change. The economy is assumed to begin in a steady state with saving rate s1 and capital
stock k*1.
 When the saving rate increases from s1 to s2, the sf(k) curve shifts upward.
 At the initial saving rate s1 and the initial capital stock k*1, the amount of investment just offsets the amount of
depreciation.
 capital stock will gradually rise until the economy reaches the new steady state k*2, which has a higher capital
stock and a higher level of output than the old steady state.
 The Solow model shows that the saving rate is a key determinant of the steady-state capital stock.
 If the saving rate is high, the economy will have a large capital stock and a high level of output in the steady state.
 If the saving rate is low, the economy will have a small capital stock and a low level of output in the steady state.
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30
 Figure 4.3
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The Golden Rule Level of Capital
So far, we have used the Solow model to examine how an
economy’s rate of saving and investment determines
 its steady-state levels of capital and income.
 This analysis is might lead you to think that higher saving is
always a good thing
because it always leads to greater income.
Yet suppose a nation had a saving rate of 100 percent. That
would lead to the
largest possible capital stock and the largest possible
income.
But if all of this income is saved and none is ever consumed,
what good is it?
This section uses the Solow model to discuss the optimal
amount of capital accumulation
from the standpoint of economic well-being.
31 Macro Economics-II CH-4 WSU CBE, Economics By Zegeye
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Figure 4.4
32

 Different values of s lead to different steady states.
 How do we know which is the “best” steady state?
 Economic well-being depends on consumption, so the “best”
steady state has
the highest possible value of consumption per person:
c* = (1–s) f(k*)
An increase in s
o leads to higher k* and y*, which may raise c*
o reduces consumption’s share of income (1–s),
which may lower c*
So, how do we find the s and k* that maximize c* ?
33 Macro Economics-II CH-4 WSU CBE Economics By Zegeye Paulos

K*
golden= the Golden Rule level of capital
the steady state value of k that maximizes consumption.
To find it, first express c* in terms of k*:
c* = y*  i* Consumption is output minus
investment.
= f (k*)  i*
= f (k*)  k*
In general:
i = k + k
In the steady state:
i* = k* because k = 0.
At the golden rule level the slope of production
function (MPK) equals to the slope of  k* line that
is .34
Zegeye Paulos

35
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The transition to the Golden Rule Steady State
The economy does NOT have a tendency to move toward
the Golden Rule steady state.
Achieving the Golden Rule requires that policymakers
adjust s.
This adjustment leads to a new steady state with higher
consumption.
But what happens to consumption during the transition to
the Golden Rule?
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Macro Economics-II CH-4 WSU , CBE, Economics By Zegeye
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38 Macro Economics-II CH-4 WSU By Zegeye
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Population Growth
Assume that the population--and labor force-- grow at rate
n. (n is exogenous)
 EX: Suppose L = 1000 in year 1 and the population is growing at
2%/year (n=0.02).
 Then L = n L = 0.02  1000 = 20, so L = 1020 in year 2.
 Break-even investment
 ( + n)k = break-even investment, the amount of investment necessary to keep
k constant.
 With population growth, the equation of motion for k is
k = s f(k)  ( + n) k
 Actual invesment break even investment
L
n
L


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,
40 Macro Economics-II CH-4 WSU By Zegeye
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 The Effects of Population Growth
 An increase in n = break-even investment, leading to a lower steady-state level of k.
 Higher n  lower k*. And since y = f(k) , lower k*  lower y* .
 Thus, Solow model predicts that countries with higher popn growth will
 have lower levels of capital and income per worker in the long run.
41

The Golden Rule with Population Growth
To find the Golden Rule capital stock,
we again express c* in terms of k*:
c* = y*  i*
= f (k* )  ( + n) k*
c* is maximized when
MPK =  + n
 or equivalently,
 MPK   = n
In the Golden Rule Steady
State,
The marginal product of
capital net of depreciation
equals
The population growth rate.
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Technological Progress in Solow Model
Previously, in the Solow model
 the production technology was held constant
 income per capita was constant in the steady state.
Neither point is true in the real world
Tech. progress in the Solow model
 A new variable: E = labor efficiency
 Assume:
 Technological progress is labor-augmenting:
o it increases labor efficiency at the exogenous rate g:
(assumption about techn progress is that it causes the efficiency
of labor E to grow at some constant rate g.)
 We have to incorporate technological progress to production
function
E
g
E


43
( , )Y F K L E 
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We now write the production function as
 where L  E = the number of effective workers.
– Hence, increases in labor efficiency have the same effect on output as increases
in the labor force.
 Notation:
y = Y/LE = output per effective worker
k = K/LE = capital per effective worker
 Production function per effective worker:
y = f(k)
 Saving and investment per effective worker:
s y = s f(k)
( + n + g)k = break-even investment:
The amount of investment necessary to keep k constant.
Consists of:
  k to replace depreciating capital
 n k to provide capital for new workers
 g k to provide capital for the new “effective” workers
created by technological progress
( , )Y F K L E 
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 In effective workers due to Tech - Tend to decrease
in k
 In steady state , investment sf(k) exactly offset the
reduction in k attributable to – depreciation, popn
growth and tech progress
45

Policies to promote growth
Four policy questions:
1. Are we saving enough? Too much?
2. What policies might change the saving rate?
3. How should we allocate our investment between privately
owned physical capital, public infrastructure, and “human
capital”?
4. What policies might encourage faster technological
progress?
1. Evaluating the Rate of Saving
 Use the Golden Rule to determine whether
our saving rate and capital stock are too high, too low, or about
right.
 To do this, we need to compare
(MPK   ) to (n + g ).46

To estimate (MPK   ), we use three facts about an
economy;
1. k = 2.5 y
The capital stock is about 2.5 times one year’s GDP.
2.  k = 0.1 y
About 10% of GDP is used to replace depreciating
capital.
3. MPK  k = 0.3 y
Capital income is about 30% of GDP
1. k = 2.5 y
2.  k = 0.1 y
3. MPK  k = 0.3 y
To determine  , divided 2 by 1:
0 1
2 5
.
.
k y
k y



0 1
0 04
2 5
.
.
.
  
47

1. k = 2.5 y
2.  k = 0.1 y
3. MPK  k = 0.3 y
 To determine MPK, divided 3 by 1:
 Hence, MPK   = 0.12  0.04 = 0.08
 Real GDP grows an average of 3%/year,
so n + g = 0.03
 Thus, in this economy,
MPK   = 0.08 > 0.03 = n + g
Conclusion
 The economy is below the Golden Rule steady state:
if we increase saving rate of this economy, the economy will
have faster growth until it reaches to a new steady state with
higher consumption per capita
MPK 0 3
2 5
.
.
k y
k y


 0 3
MPK 0 12
2 5
.
.
.
 
48

2. Policies to increase the saving rate
49
 Reduce the government budget deficit (or increase the budget
surplus)
 Increase incentives for private saving:
reduce capital gains tax, corporate income tax,
estate tax as they discourage saving
replace federal income tax with a consumption
tax
expand tax incentives for individual retirement
accounts and other retirement savings accounts
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3. Allocating the economy’s investment
50
 In the Solow model, there’s one type of capital.
 In the real world, there are many types, which we can divide into three
categories:
 private capital stock
 public infrastructure
 human capital: the knowledge and skills that workers acquire
through education
How should we allocate investment among these types?
 Allocating the economy’s investment: two viewpoints
1. Equalize tax treatment of all types of capital in all industries, then let
the market allocate investment to the type with the highest marginal
product.
2. Industrial policy: Govt. should actively encourage investment in
capital of certain types or in certain industries
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4. Encouraging technological progress
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Patent laws: encourage innovation by granting
temporary monopolies to inventors of new products
Tax incentives for R&D
Grants to fund basic research at universities
Industrial policy:
encourage specific industries that are key for rapid
tech. progress

Convergence
52
 Solow model predicts that, other things equal, “poor” countries
(with lower Y/L and K/L ) should grow faster than “rich” ones.
 If true, then the income gap between rich & poor countries would
shrink over time, and living standards “converge.”
In real world, many poor countries do NOT grow faster than rich
ones.
Does this mean the Solow model fails?
 No, because “other things” aren’t equal.
What the Solow model really predicts is conditional
convergence –
countries converge to their own steady states, which are
determined by saving, population growth, and education.
And this prediction comes true in the real world.

Endogenous Growth Theory
53
 Solow model:
 sustained growth in living standards is due to tech progress
 the rate of tech progress is exogenous
 Endogenous growth theory:
 a set of models in which the growth rate of productivity and living
standards is endogenous
 A basic model
 Production function: Y = A K
 where , Y is Output , K is the capital stock, A measures the amount
of output produced for each unit of capital (A is exogenous & constant)
 Key difference between this model & Solow:
MPK is constant here, diminishes in Solow
 Investment: sY
 Depreciation:  K
 Equation of motion for total capital:
Δ K = s Y   K
Divide through by K and use Y = A K , get:
Y K
sA
Y K

 
  

54
If sA > , then income will grow forever, and
investment is the “engine of growth.”
Here, the permanent growth rate depends on s. In
Solow model, it does not.
Does capital have diminishing returns or not?
Yes, if “capital” is narrowly defined (plant &
equipment).
Perhaps not, with a broad definition of “capital”
(physical & human capital, knowledge).
Some economists believe that knowledge exhibits
increasing returns.
In the endogenous growth model, the assumption
of constant returns to capital is more plausible.

A two-sector model
55
Two sectors:
1. manufacturing firms produce goods
2. research universities produce knowledge that
increases labor efficiency in manufacturing
u = fraction of labor in research (u is exogenous)
Mfg prod func: Y = F [K, (1-u )EL]
Res prod func: Δ E = g (u )E
Cap accumulation: Δ K = s Y   K

A two-sector model
56
In the steady state, mfg output per worker and the
standard of living grow at rate
E/E = g (u ).
Key variables:
s: affects the level of income, but not its growth rate
(same as in Solow model)
u: affects level and growth rate of income
Paulos

Three facts about R&D in the real world
57
1.Much research is done by firms seeking profits.
2.Firms profit from research because
new inventions can be patented, creating a stream
of monopoly profits until the patent expires
there is an advantage to being the first firm on the
market with a new product
3.Innovation produces externalities that reduce the
cost of subsequent innovation.
 Much of the new endogenous growth theory
attempts to incorporate these facts into models to
better understand tech progress.

58
END
Macro Economics-II CH-4 WSU CBE , Economics By Zegeye
Paulos

Neoclassical Growth Models

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