Download Macroeconomics 1 Ch VII. Investment and Growth Theories Chapter

Macroeconomics
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Ch VII. Investment and Growth Theories
Chapter VII. Investment and Growth Theories
1.Investment
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
ii)Neo-classical Theory of Investment: Rental Cost of Capital Theory
iii)Present Value Method of Investment Evaluation
iv)Tobin’s Q Theory
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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
( )
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
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(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;
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.
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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.
3) Neoclassical Model of Investment: “Optimal Stock of Capital”
(1) Rental Cost of Capital Theory
Basic Model without Inflation
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.
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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)
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)
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However, if the price level changes, then we have to consider the impact of changes in the price
level on the marginal revenue.
You incur two kinds of cost: one is the interest ("i") you pay to the bank. Here the interest rate i
is determined in the money market. The other is that the machine needs repairs, that is,
depreciation. Let's suppose that with the payment for depreciation the machine is maintained in
as good a condition as a new one:
MC = i +  .
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 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.
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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*
Applications of Rental Cost of Capital Model: How does this model work in respond to
variety of changes?

For some reason of external shocks (such as an increase in real interest rate) the MC may
rise. To ensure the equality between the user cost and the MPK, the MPK should rise to
re-establish the equality. MPK will rise when K decreases. Investment should decrease.
Intuitive explanation is that when the user cost of capital rises, the least productive
project of investment should go to enhance the marginal productivity of capital of the
existing project. We will observe that the negative correlation between the interest rate
and investment: An increase in the real interest rate will lead to a decrease in investment.

For some reason (such as oil shocks, which lead to cumbersome and disruptive energy
saving measures) the MPK may decrease. The two forces will start working. In order to
re-establish the equality, the MPK of the left-hand side should rise back. The capital
stock should decrease to have an increase in MPK. The least productive project should
go to enhance the productivity of capital. This decrease in the demand for capital will
lead to a fall in the interest rate or th lending rate of the bank. In the right-hand side of
the equality, the real interest starts falling. We will observe a positive correlation
between the interest rate and investment.

A decrease in the nominal interest rate will not necessarily lead to an increase in
investment; For instance, in 1990, the nominal interest rate was about 12% and the rate
of inflation stood around 7%. The user cost of capital was then 12 minus 7 % plus
depreciation rate. Now the nominal interest rate is only 8%, and the inflation rate is 2%.
The current user cost of capital is 8 minus 2 % plus depreciation rate. The current user
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cost is higher than that of 1990. What matters to the investor is not the nominal but the
real interest rate.
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.
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:
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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.
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?
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)
Here, inflation does affect the real
K* variable of investment and thus inflation is non-neutral 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.
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.
3) Present Value Method Theory of Investment
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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.
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;
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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:
(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
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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.
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).
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 neoclassical 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 neo-classical
theories, i.e., Harrod-Domar Theory and the Solow Model.
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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.
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
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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 post-retirement life or for possible
misfortunes. Second, mortgage loans were not readily 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.
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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:
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 percapita production curve corresponding to y =f(k) is essentially similar to the regular aggregate
production curve.
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Per-capita Production Curve: f(k)
f(k)
k
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)
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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:
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
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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.
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 per-capita income.
s f(k*) = (+n)k*.
This is a steady state equilibrium.
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Supply and Demand for K
f(k)
(+n)k
y*
k < 0
s f(k)
E
k > 0
k0
k*
k
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) Three Major Implications of the Solow Model
(1) A higher level of income can be achieved in two ways:
i) A higher rate of savings leads to a higher level of income
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A rising savings rate
(+n)k
s2f(k)
s1f(k)
k*
k*’
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 well-beings. 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.
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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.
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.
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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.
(2) 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.
(3) 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
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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 a country is useless unless it comes with a
fundamental change in the country’s savings rate and forth.
3) Endogenous Growth Theories
The Solow model with technical innovations explains experiences of advanced countries quite
well.
However, it has a few short-comings:
First, it does not explain the cases of the economic growth of East Asian countries, which the
World Bank has called ‘Miracles’. A. Young and Paul Krugman have examined economic
growth of the East Asian countries and found that there was very little of technical innovations
and their contribution to economic growth.
They used the conventional Cobb-Douglas production function such as
Y = A K L1-
dY = d A + dK + ( dL
They also assumed that =1/3 (recall v=3). Then,
dY = d A + 1/3 dK + 2/3 dL.
They collected data for dK and dL for the East Asian countries, and plugged the numbers for
them. They calculated the residual of dY which cannot be explained by dK or dL:
dY – 1/3 dK – 2/3 dL = dA.
This residual is the part of economic growth to be attributed to technical innovations.
They have found that dA is negligible for the most of the East Asian economies.
Yet, economic growth continues or accelerates in the East Asia. How can we explain the
continued economic growth? With what variable or variables, other than technical innovations,
can we use for explanation?
One of the most prominent theories was by Paul Romer, Mankiew, and Weil. They adopt
Human Capital in addition to the conventional production factors.
Y = F (K, H, L; T)
In concrete mathematical equation, the conventional Cobb-Douglas production function
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Y = A K L1-
is modified into
Y = A K H L.
Here H represents human capital. It works in the same manner as technology: H works to lift
the limit on production imposed by the decreasing MP of capital and labour.
Other authors have used different explanatory variables, such as the quality of government, the
level of development of financial system, endowment of resources, and so forth. Their empirical
results are mixed due to ambiguity of measuring these qualitative variables.
Still other authors are trying to relate economic growth to the value system of a society, such as
religion and ethics. For example, it seems that the East Asian economic growth seems to be
positively correlated with the high level of work ethics which is an integral part of NeoConfucianism. The Neo-Confucianism also emphasizes the importance of education, which is
conducive for the accumulation or development of human capital as well.
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