macro chapter 3 capital accumulation and growth
the previous chapter raised a basic economic question: how can a nation escape
from poverty and ultimately become rich? Or more precisely: how can a country
initiate a growth process that lead it to a higher level of GDP and consumption
per person?
This chapter presents a fundamental economic model that delivers some first
answers. The model show how the long-run evolutions of income and consumption
per worker in a country are allected by structural parameters such as the country's rate of
saving and investment and the growth rate of its population. The model is known as the
Solow model of economic growth.
In 19 56 the economist and Nobel Prize Laureate Robert M. Solow published a
seminal article called 'A Contribution to the Theory of Economic Growth'. The article
presented a coherent dynamic model with an explicit description of the process of capital
accumulation by which saving and investment become new capital.
In the Solow model. competitive clearing of factor markets implies that output in each
period is determined by the available supplies of capital and labour. Furthermore. total
saving and investment is assumed to be an exogenous fraction of total income, and the
labour force is assumed to grow at a given rate.
The essential additional feature of the Solow model is that it incorporates the dynamic
link between the !lows of savings and investment and the stock of capital. Solow's model
accounts for the fact that between any two successive periods, the stock of capital will
increase by an amount equal to gross investment minus depreciation on the initial capital
stock. 2
The model describes how capital evolves as a result of capital accumulation. how the
labour force evolves as a result of population growth, and how total production and
income evolves as a consequence of the evolutions of the total inputs of capital and labour.
The model therefore involves a certain evolution of income per worker as well. It thereby
contributes to answering the limdamental question of what determines 'the wealth of
nations'.
This chapter presents the Solow model in its most basic version. Considering the
simple framework, the model will take us remarkably far in understanding the process of
economic growth and the sources oflong-run prosperity. There will be much more to say
after this chapter, but the basic Solow model makes a great contribution and is indeed an
important workhorse model in economics.
3.1 Basic Solow model
The model presented in this chapter describes a closed economy (the next chapter will
present a Solow model for an open economy). Initially we will not explicitly include the
public sector in the model, but we will show that it can easily be interpreted as including
government expenditure and taxation .
Agents, commodities and markets
The economic agents in the model are households, also referred to as consumers; firms,
also called producers; and possibly a government. Time runs in a discrete sequence of
periods indexed by t. A period should be thought of as one year. There are three commodities
in each period, and for each commodity there is a market. The commodities are
output, capital services and labour services.
In the market for output the supply consists of the total output of firms . Y1• The
demand is the sum of total consumption, C,. and total investment, I, . Hence output can be
used either for consumption, or it can be transformed into capital via investment (at no
cost. we assume). The model is thus a one-sector model and does not distinguish between
production of consumption goods and production of capital goods. The price in the output
market is normalized to one. so other prices are measured in units of output.
We may think of the accumulated stock of physical capital as being directly owned by
the households w h o lease it to th e firms. Hence. in th e market for capital services the
supply comes from the consumers. By a convenient definition of the unit of measurement
for capital services (machine years) a capital stock of K, units of capital can give rise to a
supply of K, units of capital services during period t. The demand for capital services. K;1
,
comes from the firms. The real price, r 1, in this market is the amount of output that a firm
must pay to a consumer for leasing one unit of capital during period t. Thus r1 is a real
rental rate.
This rental rate has a close association with, but is not equal to, the model's real
interest rate. The reason is that capital depreciates. We assume that the use of one unit of
physical capital for one period implies that an amount<~. where 0 <c)< 1, must be set aside
to compensate for depreciation. that is, for the capital th at is worn out by one period's use.
The name for c) is the rate of depreciation. The model's real interest rate, p 1, is the return to
capital net of depreciation, that is, p 1 = r, - <~. This is the rate of return on capital comparable
to an interest rate earned from a flnancial asset like a bond.
Alternatively, one can think of physical capital as being owned by firms who finance
their acquisition of capital by issuing debt to consumers. In the latter case the real price of
the use of a unit of capital for one period. also called the 'user cost', is the real interest rate
on debt plus the depreciation on one unit of capital. In this interpretation one can trunk of
the frrm's 'rental rate' as r 1 = p, + o. For our purposes it does not matter whether the user
cost of capital. r 1, is a direct leasing rate or the sum of an interest rate and a depreciation
rate.
In the labour market the supply of labour services, L1
• comes from the households,
while the demand. L~1• comes from the firms . We measure labour flows in man years, so a
labour force of L1 can give rise to a labour supply of L1• The real wage rate in period tis
denoted by w ~'
All three markets are assumed to be perfectly competitive, so economic agents take
the prices as given. and in each market the appropriate price adjusts so that price taking
supply becomes equal to price taking demand. This implies that available resources are
fully utilized in all periods or, in an alternative interpretation. that they are utilized up to
the 'natural rate' defined in Chapter 1.
from poverty and ultimately become rich? Or more precisely: how can a country
initiate a growth process that lead it to a higher level of GDP and consumption
per person?
This chapter presents a fundamental economic model that delivers some first
answers. The model show how the long-run evolutions of income and consumption
per worker in a country are allected by structural parameters such as the country's rate of
saving and investment and the growth rate of its population. The model is known as the
Solow model of economic growth.
In 19 56 the economist and Nobel Prize Laureate Robert M. Solow published a
seminal article called 'A Contribution to the Theory of Economic Growth'. The article
presented a coherent dynamic model with an explicit description of the process of capital
accumulation by which saving and investment become new capital.
In the Solow model. competitive clearing of factor markets implies that output in each
period is determined by the available supplies of capital and labour. Furthermore. total
saving and investment is assumed to be an exogenous fraction of total income, and the
labour force is assumed to grow at a given rate.
The essential additional feature of the Solow model is that it incorporates the dynamic
link between the !lows of savings and investment and the stock of capital. Solow's model
accounts for the fact that between any two successive periods, the stock of capital will
increase by an amount equal to gross investment minus depreciation on the initial capital
stock. 2
The model describes how capital evolves as a result of capital accumulation. how the
labour force evolves as a result of population growth, and how total production and
income evolves as a consequence of the evolutions of the total inputs of capital and labour.
The model therefore involves a certain evolution of income per worker as well. It thereby
contributes to answering the limdamental question of what determines 'the wealth of
nations'.
This chapter presents the Solow model in its most basic version. Considering the
simple framework, the model will take us remarkably far in understanding the process of
economic growth and the sources oflong-run prosperity. There will be much more to say
after this chapter, but the basic Solow model makes a great contribution and is indeed an
important workhorse model in economics.
3.1 Basic Solow model
The model presented in this chapter describes a closed economy (the next chapter will
present a Solow model for an open economy). Initially we will not explicitly include the
public sector in the model, but we will show that it can easily be interpreted as including
government expenditure and taxation .
Agents, commodities and markets
The economic agents in the model are households, also referred to as consumers; firms,
also called producers; and possibly a government. Time runs in a discrete sequence of
periods indexed by t. A period should be thought of as one year. There are three commodities
in each period, and for each commodity there is a market. The commodities are
output, capital services and labour services.
In the market for output the supply consists of the total output of firms . Y1• The
demand is the sum of total consumption, C,. and total investment, I, . Hence output can be
used either for consumption, or it can be transformed into capital via investment (at no
cost. we assume). The model is thus a one-sector model and does not distinguish between
production of consumption goods and production of capital goods. The price in the output
market is normalized to one. so other prices are measured in units of output.
We may think of the accumulated stock of physical capital as being directly owned by
the households w h o lease it to th e firms. Hence. in th e market for capital services the
supply comes from the consumers. By a convenient definition of the unit of measurement
for capital services (machine years) a capital stock of K, units of capital can give rise to a
supply of K, units of capital services during period t. The demand for capital services. K;1
,
comes from the firms. The real price, r 1, in this market is the amount of output that a firm
must pay to a consumer for leasing one unit of capital during period t. Thus r1 is a real
rental rate.
This rental rate has a close association with, but is not equal to, the model's real
interest rate. The reason is that capital depreciates. We assume that the use of one unit of
physical capital for one period implies that an amount<~. where 0 <c)< 1, must be set aside
to compensate for depreciation. that is, for the capital th at is worn out by one period's use.
The name for c) is the rate of depreciation. The model's real interest rate, p 1, is the return to
capital net of depreciation, that is, p 1 = r, - <~. This is the rate of return on capital comparable
to an interest rate earned from a flnancial asset like a bond.
Alternatively, one can think of physical capital as being owned by firms who finance
their acquisition of capital by issuing debt to consumers. In the latter case the real price of
the use of a unit of capital for one period. also called the 'user cost', is the real interest rate
on debt plus the depreciation on one unit of capital. In this interpretation one can trunk of
the frrm's 'rental rate' as r 1 = p, + o. For our purposes it does not matter whether the user
cost of capital. r 1, is a direct leasing rate or the sum of an interest rate and a depreciation
rate.
In the labour market the supply of labour services, L1
• comes from the households,
while the demand. L~1• comes from the firms . We measure labour flows in man years, so a
labour force of L1 can give rise to a labour supply of L1• The real wage rate in period tis
denoted by w ~'
All three markets are assumed to be perfectly competitive, so economic agents take
the prices as given. and in each market the appropriate price adjusts so that price taking
supply becomes equal to price taking demand. This implies that available resources are
fully utilized in all periods or, in an alternative interpretation. that they are utilized up to
the 'natural rate' defined in Chapter 1.
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