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Macroeconomics
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Ch VII. Investment and Growth Theories
Chapter VII. Investment and Growth Theories
1.Investment
The first and foremost thing about investment is that we have to make a sharp distinction
between real investment and financial investment. Real investment is on capital stock or
inventory. Financial investment does not necessarily lead to a change in capital stock. The
tenuous or ‘weak’ relationship between financial investment and capital stock is presented
by Tobin’s Q-theory. Even it is not really strong, theoretically or empirically. Thus when
we talk about investment, we refer to real investment or replenishment or addition of
capital stock or inventories.
Investment is important for two reasons: First, it is an important component of aggregate
expenditures, and it is the most volatile component. Thus changes in investment are the
major causes of ups and downs of national income over time. In other words, investment is
the major cause of business cycles over the medium term. This is best explained by J. M.
Keynes’ theory of investment.
Second, investment leads to an accumulation of capital. And capital is one of the three
most important components of aggregate production function: Y = ff(K, L: T). Thus
investment determines the rate of growth of national income in the long-run. This side of
investment can be best explained in terms of theories of economic development.
1) Definition of Investment
Investment is a flow concept or a concept of flows for a given period of time, say, one year.
Specifically, in the national income accounting system, the government statistics author
calculate Investment by aggregating



Fixed Capital Investment: Machinery, Equipment;
Non-residential Building, and Residential Construction; and
Addition to Inventory: finished goods and materials on the pipeline, and also buffer
stock of finished goods.
The total Investment is usually a Gross Investment. The gross investment consists of two
parts: The Net Investment which leads to an increase in the stock of capital; and
depreciation allowances for wears and tears of capital.
Gross Investment – Depreciation = Net Investment = K
There are some suggestions to include Consumer Expenditures for Durables in Investment.
In this chapter, we will go over investment theories of various macroeconomic schools as
follow:
i)Keynesian Theory of Investment: The Accelerator Model of Investment
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ii)Neo-classical Theory of Investment: Rental Cost of Capital Theory
iii)Present Value Method of Investment Evaluation
iv)Tobin’s Q Theory
And we will have a brief discussion about investment tax cuts, and their difference from
consumption tax cuts.
2) Keynesian Investment Theory: The Accelerator Model of Investment
Biggest Issue: Volatility of Investment
Investment is much more volatile than income or consumption; Inventory Investment is
still more volatile.
(1)Keynesian Accelerator Model of Investment
Inverting the following Aggregate Production Function with labor input being held
constant
Y t = F ( K t , N t ).
we can rewrite the above equation into;
K t =  Y t , > 1.
We can also apply this to the last period;
K t -1 = Y t -1 .
Investment is the increase in capital stock which is proportional to the increase in (the
production of aggregate output, which is equal to) national income;
I t = K t - K t -1 =  ( Y t - Y t -1 ).
Differentiating both sides with respect to time t, we get
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( )
dI t
=  Y t Y t -1 .
dt
dt
The rate of change in investment depends on the acceleration/deceleration of the growth
rate of income, or the change in the rate of change in income
(2)Numerical Example:
Assumption: Ct = 50 + 0.8 Yt, It = 3 (Yt - Yt-1).
Note that v =3 here.
──────────────────────────────────────────────────
Year Yt
% change
Ct
% change
Kt
It
% change
──────────────────────────────────────────────────
1
450
N.A.
410
N.A.
1350
N.A.
N.A.
2
500
11 %
450
10 %
1500
150
3
600
20 %
530
17 %
1800
300
100 %
4
660
10 %
580
9.5%
1980
180
-40 %
5
726
10 %
630
8.6%
2178
190
5%
─────────────────────────────────────────────────
Note that the % change in income and the % change in consumption go hand in hand in
a similar proportion. However, the % changes in investment are much more volatile
than those in national income or consumption.
In fact it is in a proportion to the % change of the % change in income; the % change in
the % change in Y between years 2 and 3 (from 11% to 20%) is 82%. The % change in
the % change in Y between years 3 and 4 (from 20% to 10%) is -50%. The % change
for the subsequent period is 0%. These numbers, 82%, -50%, and 0%, are in line with
the % changes in investment, 100%, -40%, and 5%.
Therefore,

Acceleration in Y (an increase in the growth rate of national income)  I

Deceleration in Y (a slow-down of the growth rate)  I

Whether investment will increase or decrease this year in comparison to the
last year's investment depends on whether the growth rate of this year is
larger or smaller than the growth rate of the last year;
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For instance, suppose that the real income grew 3% last year, and grows 1% this
year. The economy is still growing; income increases this year and so does the
consumption. But the investment will decrease compared to the last year's level
because the growth rate drops from 3 to 1 % or the growth decelerates.
Implications of the Keynesian Accelerator Model;
i) When the above investment function as a function of changes in income is
substituted in the equilibrium national income equation, the only exogenous
variables left over are autonomous consumption (C) and government
expenditure (G). So what ultimately determines the equilibrium national income
is G.
ii) Substituting the above investment function into the equilibrium income
function, we get a first-order difference equation; Yt = A (G + C) + B Yt-1.
Depending on the value of B, there could be different patterns of business cycles.
Problems
i) This is a circular argument; Y changes as I changes, which changes as Y
changes. Therefore this is rather a mechanical illustration than a explanation
which touches the fundamental causes of volatility of investment.
ii) There is some factor which attenuates the volatility of investment; the
adjustment cost makes actual fixed capital, investment or increases in fixed
capital, take place over time in a gradual fashion rather than over night. But the
adjustment cost is minimal for inventory investment.
The actual change in capital stock or dK cannot take place overnight. The time
lag involved in increasing K is fairly long (think about the construction period,
and the time lag between the order and the shipment of equipment and
machinery). Inventory Investment does not involve any significant lag.
(2) A Declining Marginal Efficiency of Capital over time
Keynes came up with the concept of ‘Marginal Efficiency of Capital(MEC hereafter)’,
which is essentially similar to the marginal productivity of capital. MEC is a decreasing
function of the amount of capital: Investors start investing on the most productive or
lucrative project, and gradually move onto increasingly less productive projects. Thus,
eventually a falling MEC may even hit zero if capital stock reaches a certain level. On
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the other hand, according to Keynes, the interest rate is determined in the money
market: The equilibrium interest rate equates real money demand and real money supply.
Thus, in the Keynesian investment theory, the MEC and interest rate are independently
determined.
Only when the MEC is higher than interest rate, investment is responsive to a fall in
interest rate. Suppose that initially the economy has the capital stock of K1. The
interest rate is being determined in the money market. It can be at any level. Suppose
that the equilibrium interest rate is determined at i* in the money market. Then, MEC
is higher than interest rate. Thus, marginal benefit from investment would be higher
than marginal cost of investment. K1 will increase to K2 level.
Marginal Efficiency of Capital
i*
ifrom money market
Capital Stock (K)
K1
K2
In the short-run, a shifting interest rate may go up or down, depending on the situations
of the money market. In the case illustrate below, interest rate may go up from i1 to i2 .
The last few projects of investment have MEC lower than the new interest rate, and thus
will be eventually pushed out.
Marginal Efficiency of Capital
i2
i1
ifrom money market
Capital Stock (K)
K2
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3) Neoclassical Model of Investment: “Optimal Stock of Capital”
Unlike the Keynesian theory of MEC, the Neo-classical theory of investment uses the
marginal product MP instead of MEC, and redefines the interest rate in the real, and
states that where MP = real interest rate, there is the equilibrium in the capital market.
For reality, deprecation of capital, tax, and inflation are introduced.
(1) Rental Cost of Capital Theory
Basic Model without Inflation(for simplicity)
Investment is the demand for new capital stock or an increase in capital stock(ΔK). The
demand for ΔK is determined by weighing the cost and the benefit (or revenue) of the
capital good at the margin.
Marginal Revenue = Marginal Cost.
Let's suppose that you are an investor or entrepreneur. You are borrowing money from a
bank at the interest rate of i for a year and buy a capital good. You produce outputs
from the use of the capital good, and sell it in a year to repay your loan from the bank.
Your revenues come from two sources: During the year, there will be product generated
from the machine. At the end of year when you sell the machine you will have gains or
loss as the price of the machine has changed.
If there is no change in the price level –and thus no inflation-, the marginal revenue is
the same as the marginal product of capital.
MPK is primarily a decreasing function of capital stock. And thus, the curve of MPK
will have a downward sloping line as shown below. It is also an increasing function of
technical innovation and a decreasing function of any event which adversely affects
productivity of capital (for instance, oil shocks). These two, technical innovations and
productivity shocks, will be the MPK curve’s shift variables.
MPK
Capital Stock (K)
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There is a government tax on the revenue. If the tax rate is t, then the after-tax revenue
is:
MPk(1-t)
MPK(1-t)
Capital Stock (K)
However, if the price level changes, then we have to consider the impact of changes in
the price level on the marginal revenue.
For the investor who uses capital, there occur two kinds of cost: one is the interest ("i")
the investor pay to the bank. The other is the maintenance cost as the machine needs
repairs over time. That is depreciation. Here the interest rate i is determined in the
money market. Thus the user cost of capital is the sum of the two costs:
MC = i +  .
This user cost of capital is also called ‘rental’ cost of capital.
The interest rate i and the depreciation rate  are all given for the investors, and the
investors have no control over the two variables.
i+
Capital Stock (K)
The investor will keep increasing investment until MR = MC, where he stops. That is
the optimal point of investment, which maximizes the investor’s accumulated profits or
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the integrated areas of MR- MC. If he stops short of that point, he is not fully taking
advantage of the profit opportunity. And if he goes beyond that point, he starts incurring
loss.
By transposing the rate of inflation, we rewrite the equilibrium condition as follows:
MP K ( 1  t) = i +  .
This can be re-written as:
MP K ( 1  t) = i +  .
Here i-is defined as a real interest rate as opposed to the nominal interest rate of i.
Conventionally we use r to denote the real interest rate.
MP K ( 1  t) = i +  .
We call the right-hand side express the user (rental) cost of capital.
MPK(1-t)
i
Capital Stock (K)
K*
Two Realistic Complications/Modifications for the Model
i)Inflation: How does inflation alter the above model and thus affect the optimal
stock?
Note that the interest rate minus the rate of inflation is the real interest rate. Therefore,
the equilibrium condition is that the marginal product of capital is equalized with the
user cost of capital, the sum of the real interest rate and the depreciation rate.
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The optimal stock of capital K* maximizes the accumulated sum of profits for the
investor.
The marginal revenue is the sum of the marginal product of capital MPK (for a year) and
the capital gains or loss due to the changes in price of the capital good (when you are
selling your company at the end of a year):

MR = MP K + P K .
PK
The percentage change in the price of the capital good may be in line with the rise of the
general price level. Suppose that it is given and constant as:
 PK
= .
PK
There will be no tax on these ‘hidden’ capital gains, and thus the after-tax marginal
revenue would be
MR = MP K (1  t ) + 
We know that the marginal cost is given by
MC = i +  .
Is this impervious to inflation? Not exactly. According to Irving Fischer, the nominal
interest rate i is the real interest rate r plus the expected rate of inflation:
i = r + e.
Thus the equilibrium condition is now modified as follows:
e
MP K ( 1  t)   = r   +  .
Note that the actual rate of inflation on the left hand side and the expected inflation rate
of inflation may have a gap in the short-run. This is particularly a case of the short-run
when the rate of inflation is increasing or when inflation accelerates. While the actual
rate of inflation on the left hand side of the equation goes up, the expected inflation rate
on the right hand side may not go up quickly due to lagging perception of inflation. In
the long-run, however, the expected inflation will be equal to the actual inflation rate.
We call the right-hand side express the user (rental) cost of capital.
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MPK(1-t) +
 r + e+ 
Capital Stock (K)
K*
The optimal stock of capital K* maximizes the accumulated sum of profits for the
investor.
So, what is the impact of inflation on investment?
Unless inflation occurs in fully anticipated manners, actual inflation rate exceeds
expected inflation rate at least in the short-run.
In the short-run, according to this theory of rental cost of investment, when the rate of
inflation goes up, then the after-tax marginal revenue curve goes up, and the horizontal
cost curve does not go up very much. Thus there will be an increase in the optimal K
stock K*, and there will be new investment.
MPK(1-t) +
 r + e+ 
Capital Stock (K)
K*
Here, inflation does affect the real variable of investment and thus inflation is nonneutral to investment. This is called ‘Non-Neutrality of Inflation” (on real variables).
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In the long-run, as the expectation gets revised in line with the reality and e rises to the
level of , the optimal level of capital will fall back the previous equilibrium level.
MPK(1-t) +
 r + e+ 
Capital Stock (K)
K*
Here, inflation does not affect the real variable of investment and thus inflation is
neutral to investment. It is called “Neutrality of Inflation” on real variables. As we will
see it later, Tobin has a different idea about this.
Last but not least, we would like to emphasize that this investment theory is not the
theory of interest rate determination: The optimal stock of capital is determined at the
point where the declining MPk is equated with the real interest rate, which is set
separately and externally. Interest rate is determined in the money market where the
demand for liquidity or real money is equal to the supply of liquidity or real money.
This is in line with the Keynesian LM curve. The preceding classical economists had
the Loanable Fund Theory for the determination of interest rates. It may be worthwhile
to carve up the differences between the Keynesian liquidity theory and the Loanable
Fund Theory: The first concerns liquidity in the money market, and the second concerns
credit available in the financial market. While liquidity and credit are closely related,
they are separate entities. Liquidity is of a shortest duration, and credit is inclusive of
medium- and long-term investment funds. Thus liquidity is often interchangeable with
‘cash’. However, in a strict sense, there is a continuum of liquidity in the scale of
monetary supply items, ranging from cash to M3.
According to Keynes, the loanable fund theory is flawed: The major supply of loanable
fund is savings, and the major demand for loanable fund is investment. Investment
affects aggregate expenditures and thus national income, which in turn affects savings.
Thus, the demand is not independent of the supply, and a shift of the loanable- fund
demand curve leads to a shift of the supply curve. In reality, the government control
some interest rates by controlling the supply of liquidity. It seems that the Keynesian
liquidity theory is more convincing that the Loanable Fund Theory when it comes to
interest rate determination.
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ii)Partial Adjustment over Time
In the short-run, the actual stock of capital K and the optimal stock of capital K* can
diverge. However, a change in investment will bridge the gap between the two over
time. If the current actual stock of capital is less than K*, then investment will take
place to increase the stock of capital to K*. If the current actual stock of capital exceeds
K*, then investment will not take place so that over time K will fall to K*. This gradual
adjustment is particularly true when there is an adjustment cost, including the cost of
time for adjustment. Thus only a portion, say , will be done in this year. In the next
period, an investment will be made for a portion of the remaining (1- portion of the
gap between K and K*. This is called a ‘Partial Adjustment Model’ of the optimal
capital stock.
It = K* - Kt)
The volatility of investment can be explained in this setting as the value of  changes
from one year to another, or from one aspect of business cycles to another. As its value
fluctuates, the investment will fluctuate as well.
iii) Does the presence of tax really matter for investment?
In general the tax system allows some investors to write off depreciation. This is
particularly the case with machine and equipment. This is called the Capital Cost
Allowance. The CCA system was first introduced in Canada with the Income Tax Act
of 1949, but substantially amended in 1972 and again in 1987. In 1949, when the CCA
system was introduced, 12 classes of assets were established; in 1972, the number of
classes reached 28, in 1987 – 37, whereas currently, 56 classes exist.Thus, in those
cases depreciation may not be an important part of user cost.
If the invetor is incorporated, the tax rate applied to the corporate profit may differ from
that applied to personal income. Tax rates on dividend income is lower than those on
personal labor income. In fact, effectively all corporate income can be retained within
the corporate, carrying zero tax.
3) Present Value Method Theory of Investment
The above theory of Rental Cost of Investment just looks at the one-period of values of
interest, etc., on a convenient yet unrealistic assumption that the values of the relevant
variables are to be constant over time.
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Investment is the demand for new capital stock (dK). The demand for dK is determined
by weighing the cost (Present Price) and the benefit (PDV) of the capital good;
so It = dKt = f (Cost versus Benefit of New Capital Stock)
The cost is equal to the present price of New Capital Stock.
The benefit comes over time in the form of the stream of revenues generated from the
capital stock over its life-time. To compare it with the present price of new capital
stock, we should get the present value equivalence of the revenue streams by
discounting the revenues with interest rate of the times and summing them up. This
leads to the Present Discounted Value of the stream of future revenues. So the Benefit
of New Capital Stock is a function of the stream of future revenues and future interest
rates. In reality where there is uncertainty, the future values are unknown. The best the
investor can do is to make a rational guess about the future variables. This means that
the PDV becomes the function of expected future variables such as expected future
revenues and expected future interest rates;
*
PDV =
Rt +
Rt+1
+ ......
(1 + r) (1 + r t )( 1 + r t+1* )
*
+
Rt+20
.
(1 + r t )(1 + r t+1* )......( 1 + r t+20* )
where * denotes expected future variables, Rt+i is the stream of net revenues from the
investment project, r interest rate. The Rt is determined by MPk, depreciation rate,
inflation rate(of outputs and inputs), tax rate of each period and so forth.
*
*
*
*
I t = f ( Rt , r t , Rt+1 , r t+1 , Rt+2 , r t+2 ..... ).
The revenue is the value of sales of output (= the price of output multiplied by the
amount of output demanded and thus produced) minus tax, and so on;
I t = f (M Pk t ,T t , r t , , Pt
*
*
*
MPk t+1 , T t+1 , r t+1 , Pt 1
*
*
*
MPk t+2 , T t+2 , r t+2 , Pt  2 ....).
Implications:
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(1) There are a lot more expected variables in the investment function than in
any other function; the expectations matter more in the investment function than
in other functions.
(2) The expectations change all the time, reacting to 'News', which may not be
necessarily correct.
(3) Among the expected variables which affect the PDV of the investment
project, in percentage terms, the interest rate is the most volatile. For instance,
at the aggregate level, revenues rarely change by 50% (due to such changes in
sales or price) while the interest rate often changes by 50 % (from the 11 % to
15 % level or the other way around). So ultimately, a substantial part of the
volatility of investment can be explained by the volatility of interest rate.
What makes interest rate volatile? It should be considered in the context of
Money Supply and Demand. Naturally this has a lot to do with the next topic of
this course.
4) Tobin's Q Theory
According to James Tobin, the `Q' index larger than one is a green-light signal for
expansion of facilities or new investment. The Q index is defined as being equal to the
market value of a firm over its replacement cost: The market price incorporates the
market’s expectations as to the prospect of future business returns to the firm, while the
replacement cost is simply the present market price of capital required to set up the firm.
The market value can be easily approximated by the number of stocks or equities and
the price of each stock or equity in the stock market.
Tobin's Q shows how or through what transmission mechanism, for instance, an
increase in money supply leads to an increase in investment. If money supply increases,
other things being equal, expenditures on all assets will rise. As the demand for stocks
rises, the stock prices will go up. The market value of a firm is the stock volume times
the stock price. As the market value of stocks rises, Tobin's Q exceeds one. The
entrepreneurs will try to leap capital gains by selling their firms and by setting up new
companies. There occurs a new physical investment.
This is the only theory that links the stock market and the physical investment. It is
rather tenuous for the following reasons: First, the Q does not take into consideration all
other invisible costs of new investment: Can we really reproduce a firm without
incurring high personal costs? Second, empirical evidence shows that in the western
countries a firm depends less and less on funds raised from the stock market for a new
investment project. The stock market boom does not much lead to financing of a new
investment project. The stock market ups and downs seem to be happening, to a large
extent, regardless of the actual economy or the real physical investment.
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5) Permanent versus Temporary Investment Tax Credits?
By nature, investment can be done in the discrete manner; investment spurts, making
the best use of an auspicious investment environment, which comes occasionally
("Make hay while the sun shines").
Implication: A temporary tax cut on investment will have a larger expansionary impact
on investment than a permanent tax cut. This contrasted with the case of tax cut on
income; a permanent income-tax cut has a larger impact on consumption and aggregate
expenditures than a temporary income-tax cut.
"....... Congress may revive the investment tax credit (ITC) in hopes of
boosting spending on factories and equipment. Bush would probably sign on.
Experts caution that ITC would be truly helpful only if the credit is
temporary...." (The Times, "Does America Need a New Deal for the Nineties?",
January 13, 1992).
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2.Long-run Growth Theories
Economic growth refers to the change in national income. To be precise, it refers to the
annual rate of percentage change in real national income. And if it is to be a measure of
economic development and economic welfare, it has to be a ‘per person’ or ‘per capita’
basis.
The aggregate output is a function of inputs called ‘production factors’ such as capital K,
labour (force) L and technology T.
Y = F(K, L; T)
Technology is a difficult and elusive concept, and in most cases is assumed to be
constant for simplicity.
Out of these three production factors or inputs, the conventional growth theories or the
neo-classical theories of economic growth focus on capita K. This is only natural as
most of poor countries have a relatively abundant population and a limited amount of
capital. In other words, they have too many people and too little a capital stock. Their
economic growth is constrained by the ‘bottleneck’ of capital, and thus the main focus
is on how to get over the limitation posed by the capital stock.
On the basis of the above production function, first we will discuss two major neoclassical theories, i.e., Harrod-Domar Theory and the Solow Model.
The above neo-classical models do not adequately explain some countries, particularly
of the East Asia, and their differences from other developing countries. How come
would some countries have a continued economic growth without slowing down over a
long period of time while other countries cannot get on a path of economic
development? Here comes the Endogenous Economic Growth Theories, which seeks
another variable related to endogenous aspects of a society. The basic tenet of these
Endogenous Economic Growth Theories is that in addition to K, L, and T something
coming from the inside of the society is necessary for economic growth. In concrete
they focus on Human Capital, economic system, financial institutions, or/and the quality
of government and policies.
1)Harrod-Domar Model
This model is older and cruder than Solow Model.
This model links economic growth with the level of capital stock K:
dY/Y = dY/dK x
dK/Y.
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dY/Y is a percentage change in Y in fraction. The above is a simple mathematical
manipulation of linking dY/Y to dK.
And the stock of capital or K increases through investment, and investment is financed
by savings:
dK = I = S
Combining the above two equations, we get
dY/Y = dY/dK x
S/Y.
This means that
Income Growth Rate = Marginal Product(ivity) of Capital x Savings Rate
= Efficiency in use of Capital x Savings Rate
Alternatively we can say that
Economic Growth Rate =
=
S/Y
dK/dY
Average Propensity to Save
Incremetal Capital Output Ratio
\
Here the Incremental Capital Output Ratio(ICOR) dK/dY is an inverse of the marginal
product of capital(MPk) dY/dK. The MPk measures the efficiency of capital, and the
ICOR measures an inefficiency of capital: The larger the MPk and the smaller the
ICOR, the higher of efficiency of use of capital.
According to this mode, a country with a high rate of economic growth should have a
high saving rate, and an efficient use of capital. This model explains some experiences
of economic growth, such as the economic growth of Japan in the 1960s to 70s. Japan’s
savings rate was high. And the ICOR was low and continued to fall, reflecting an
improved efficiency of capital use.
Why was the Japanese saving rate high or higher than most countries? In fact, the
reasons could be found in Japan’s unique economic and social institutions: First, there
was no adequate social welfare system, which forced people to save for their postretirement life or for possible misfortunes. Second, mortgage loans were not readily
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available and thus people had to save cash for the purchase of houses. Third, Japanese
companies have a unique way of setting salaries for employers: the annual salaries were
divided into the ‘regular’ pay and the ‘bonus’ part. The bonus part fluctuates depending
on financial conditions of the companies and individual performances. This bonus
system was devised to give a financial flexibility to the companies which had to go
through business cycles and fluctuating profits so that they did not have to lay off
workers even during the recession time. At the same time, it had an impact of increasing
savings on the part of employees. They would regard the bonus part as a ‘transitory
income’, which we have examined would be mostly saved in the consumption theory of
permanent income.
The Japanese government was careful of the use of scarce resources of capital. It
facilitated an efficient allocation of capital by preventing an overlapping investment.
The Japanese government would designate one or a few companies for development in
a specific industry, and investment. Each company should have the ‘main transaction
bank’, and other banks would not fund or finance investment projects which did not
carry the government’s approval.
These are not what Harrod and Domar have pointed out. However, their theory points
to the possibility that economic and social institutions make differences in savings rates
and efficiency of capital.
2)The Solow Model
‘The’ representative Neo-Classical Growth Model: focusing on savings and investment.
It explains the long-run evolution of economy quite well with all being held constant.
It is a Dynamic Model.
Focusing on capacity of Savings to meet the demand for Investment as Capital
Requirements and, beyond that, as Capital Accumulation for expansion of Production
capacity;
Assumptions of the model are as follows:
•
•
The population grows at rate n per annum
L’ = (1 + n) L ;
Population growth rate equals the rate at which new labor force enters the work
place;
•
No productivity growth or technical innovation for now;
•
Capital depreciates at rate  ;
The aggregate production function is the same as before:
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Ch VII. Investment and Growth Theories
Y = F(K, L: T)
Solow expresses all the variable for one person or per capita.
Y
L
K
k
L
y
y is per capita income, and k the capital equipment for one worker or per-capita capital
equipment ratio.
Now the per-capita production function is
y = f(k)
Per-person or per-capita income level (y) depends on each worker’s capital
equipment(k). And of course, y=f(k) shows Declining Marginal Product of Capital as
well. Thus, there shape of per-capita production curve corresponding to y =f(k) is
essentially similar to the regular aggregate production curve.
Per-capita Production Curve: f(k)
f(k)
k
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Out of all the possible levels of k and y, is there any equilibrium level which is
sustainable in the long-run? Yes, there is. And it is the point where the supply of capital
is equal to the demand for capital. There the supply of capital is equal to the demand for
capital, and thus there is no net investment or an increase in capital. Thus the level of k
is constant and the level of y is constant. It is a steady-state equilibrium.
(1)Supply of Capital
Solow assumes that the savings rate is constant regardless of income level: Unlike the
usual Keynesian model of consumption function, the average propensity to consume
does not fall as the level of income rises.
We may recall that dK = I = S: the supply of capital comes from savings.
The savings rate is defined as
s =S//Y = savings /income.
There is no difference between the aggregate level and the individual level as
s = S/Y = S/N * N/Y
On a per capital basis, for a given income of y, the actual savings S/L per capita is
savings rate times income per capita, such as:
S = s · y = s f(k)
This is the supply of capital per capia.
We can take a numerical example: When the savings rate is given as 30% or s =0 .3,
the supply function of capital per capita is
S = 0.3 f(k).
We can draw the actual savings per capita curve as follows:
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Ch VII. Investment and Growth Theories
Curve for the supply of capital: s f(k)
f(k)
Consumption
s f(k)
Savings
k
*Note: the graph is not to scale: consumption should be much larger than savings.
(2)Demand for Capital
There are two sources of demand for capital, which are necessary to keep constant the
level of capital equipment per capita: Capital wears and tears over time, and thus needs
replenishment. That is the replenishment of capital for depreciation. The second comes
from the need to equip new workers with the same level of capital as the existing
worker has.
The minimum capital requirement to just keep up for each work is proportional to
population growth rate(n) and capital depreciation rate(d):
k  (  n)k
Without this minimum capital requirement is met, the level of capital per worker will
fall over time, and the level of per-capita income will fall as well.
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Demand curve for capital
(+n)k
k
(3) Steady State Equilibrium
When the supply of capital exceeds the demand for capital, the left over capital will lead
to capital accumulation or an increase in capital stock. Per-capita income will rise.
When the supply of capital is the same as the demand for capital, there is no capital left
over for accumulation. The level of capital stock is constant, and so is the level of percapita income.
s f(k*) = (+n)k*.
Supply and Demand for K
f(k)
(+n)k
y*
k < 0
s f(k)
E
k > 0
k0
k*
k
This is a
steady
state
equilibri
um.
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Ch VII. Investment and Growth Theories
Suppose that we are located at point k0 , then the supply of capital given by s f(k)
exceeds the demand for capital given by (n)k. Thus the left over capital will lead to
an increase in capital and the economy will move to the right.
At E or the steady state equilibrium, the supply of capital is equal to the demand for
capital. And the corresponding per-capita income or y* = Y/N is determined along with
the production function.
(4) Major Implications of the Solow Model
-A higher level of income can be achieved in three ways:
i)A higher rate of savings leads to a higher level of income
A rising savings rate
(+n)k
s2f(k)
s1f(k)
k*
k*’
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Note that an increase in savings rate does increase the level of income, but not the rate
of growth of income.
Does this mean that the higher the savings rate, the better, and thus the highest possible
savings rate leads to the best result? Not really. Recall that in economics the ultimate
goal is to maximize consumption as consumption increases utility or material wellbeings. Only to the extent that a rise in income leads to an increase in consumption, the
income can be a measure of material well-being. However, a level of income with an
excessively high savings rate means too little consumption.
As we can see in the following graph, there is a unique savings rate which maximizes
the consumption. The unique savings rate is called ‘the Golden Rule’ of economic
growth. And it is not the highest possible savings rate. The point of the Golden Rule is
obtained by getting the tangent straight line to the aggregate production function, which
is parallel to the demand curve for capital and thus has the slope of delta and n.
If we assume 0for simplicity of presentation, then
In the above graph, c1* is consumption for Golden Rule where MPk = n + d , which is
lager than c2* or any other levels of consumption.
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ii)A lower rate of population growth leads to a higher level of income
A lower rate of population growth means a lower requirement or demand for capital
which is needed to keep constant the level of capital equipment per capita. A given
amount of savings is released from this requirement and can be given additionally to the
existing workers. Each existing worker’s capital equipment will rise and each worker’s
output or income would rise as well.
A falling population growth rate and growth
(+n1)k
(+n2)k
sf(k)
k1
k2
k
Population control raises the level of per capita national income.
iii)Technical Innovations lead to a higher national income
Technical innovation shifts the entire production function upwards. With the
unchanged amount of capital, more outputs will be made. This not only increases per
capita income but also raises the rate of changes or growth rates. This means that
growth rates for an economy can be sustained, as opposed to declining, in the long-run.
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- Convergence
The economy converges, over time, to its steady state.
If the economy starts BELOW the steady state, it accumulates capital until it reaches the
steady state.
If the economy starts ABOVE the steady state, it reduces capital until it reaches the
steady state.
Convergence is usual. However, there is a possibility of Technical Innovation for
Sustained Economic Growth. The best example is the U.S. economy over time.
- Uselessness of a one-time Capital Injection or Economic Aids
A one-time injection of capital from outside increases the level of income only for the
short-run. Over time, the increased level of capital cannot be sustained. A higher level
of capital requires a higher level of replenishment of capital for depreciation and more
capital to maintain the new higher level of capital equipment for each worker in the face
of the increasing number of workers. Eventually the newly injected capital will wear
out. The economy slides back to the steady state equilibrium.
This also explain why economic aids given to any country is useless unless it comes
with a fundamental change in the country’s savings rate and forth.
3) Other Growth Theories