macro chapter 6 the solow model with human capital
chapter 6 the solow model with human capital
the Solow model of Chapter 5 performed quite well by empirical standards, but we
identified two systematic discrepancies between the international evidence and the
predictions of the model: the steady state of the model seemed to substantially
underestimate how strongly savings and population growth rates influence GDP per
person, and the model seemed to substantially overestimate the speed by which economies
converge to their steady states.
Is there a single modification of the Solow model that can redress both of its two
empirical weaknesses? Indeed there is. and the idea behind it is brilliant. It is due to the
economists. N. Gregory Mankiw. David Romer and David N. Wei!. who presented it in an
article entitled 'A Contribution to the Empirics of Economic Growth'. 1 We will now
explain the idea intuitively.
Consider the problem concerning the speed of convergence. In the Solow model
convergence is slower the larger the elasticity of output with respect to capital. o.. We
explained this fact in Chapter 3 and again in Chapter 5 where we found the rate of
convergence to be A= ( 1 - a )(n + g + f~) . If o. could be chosen freely. we could simply set o.
sutllciently large to bring A in accordance with the empirics, but a is one minus labour's
share. so realL~tically o. cannot be set much higher than t. However. suppose there were
also another kind of capital '>\lith an output elasticity and 'income share' of its own, cp say,
and that this capital were accumulated in the same way as physical capital {and therefore
delayed convergence similarly). Furthermore, suppose the income share of this form of
capital were accruing to workers. Then, lor a maintained around ~. one would have a
slower convergence since o. + cp > ~.and still a labour's share, 1 - o., around i·
This suggests the idea of adding human capital to the model: each year there is a
certain stock of human capital, which can be thought of as the sum of all the education
and training the workers have had, and this stock (per worker) serves to increase the
productivity of workers. Just like physical capital, total human capital is assumed to
accumulate by a certain exogenous fraction of GDP being added to it each year. The
income share of human capital goes to workers. because human capital (embedded education) is linked to workers and its services therefore sold inseparably together with
work hours.
Since human capital accumulates like physical capital. the inclusion of it will lower
the model's speed of convergence. Because the output elasticity of physical capital is
unchanged. the owners of physical capital will earn the same fraction of GDP . The fraction
earned by human capital goes to the workers, so labour's share will also be unchanged.
Thus. adding human capital should work to overcome the second of the empirical
problems.
What about the first problem concerning the model's steady state prediction? Assume
that the rate of investment in physical capital, increases. This will create higher GDP per
person in steady state because more physical capital per person is accumulated. If a
certain fraction of GDP is invested in human capital, then the higher GDP first caused by
the higher rate of investment in physical capital will also imply that more humnn capital
per worker is accumulated. This reinforces the increase in output. Thus, the increase in
GDP per person caused by a higher investment rate in physical capital is larger in a model
including human capital than in one with only physical capital (other things, e.g. a. being
equal).
The introduction of human capital sh ould thus work to mitigate both of the empirical
failures of the Solow model. How well a job it does in this respect is the subject of th is
chapter, and we will see that it does a very fine job.
6.1 The solow model with human capital
The 'micro world' of the model to be presented is similar to that of the Solow models
considered previously. There are the same kinds of economic agents, a representative
profit maximizing Hrm and consumers (government can again be interpreted into the
model, a possibility that will be considered in an exercise) . The production function of the
firm is now different as it includes human capital, and the consumers will also have to
decide on how large a fraction of income to accumulate as human capital each period.
We assume that each unit of output can be used either as consumption. investment in
physical capital. or as investment in h uman capital.
There are also the same competitive markets as before, one lor output with real price
normalized to 1. one for the services of physical capital with real rental rate, r 1, and one for
labour with real wage rate. w ,. Hence we do not assume th at there is a separate market for
the services of human capital with a rental rate of its own. This is because human capital
cannot be separated from workers. Its services have to be sold together with labour. So the
services traded on the labour market will no longer be units of 'raw' labour, man-years,
but man-years endowed with a certain level of human capital or education. This level is
assumed to be the same for all workers. Hence the total stock of human capital, H,,
manifests itself as each worker in the labour Ioree. L,. holding a human capital of h, = Hj L,
that goes inseparably with the worker. Employing one more worker means hiring an
additional unit oflabour equipped with h 1•
The Solow model with human capital thus has two new features compared to the
original Solow model: the inclusion of human capital in the production function and the
accumulation of h uman capital by the consumers.
the Solow model of Chapter 5 performed quite well by empirical standards, but we
identified two systematic discrepancies between the international evidence and the
predictions of the model: the steady state of the model seemed to substantially
underestimate how strongly savings and population growth rates influence GDP per
person, and the model seemed to substantially overestimate the speed by which economies
converge to their steady states.
Is there a single modification of the Solow model that can redress both of its two
empirical weaknesses? Indeed there is. and the idea behind it is brilliant. It is due to the
economists. N. Gregory Mankiw. David Romer and David N. Wei!. who presented it in an
article entitled 'A Contribution to the Empirics of Economic Growth'. 1 We will now
explain the idea intuitively.
Consider the problem concerning the speed of convergence. In the Solow model
convergence is slower the larger the elasticity of output with respect to capital. o.. We
explained this fact in Chapter 3 and again in Chapter 5 where we found the rate of
convergence to be A= ( 1 - a )(n + g + f~) . If o. could be chosen freely. we could simply set o.
sutllciently large to bring A in accordance with the empirics, but a is one minus labour's
share. so realL~tically o. cannot be set much higher than t. However. suppose there were
also another kind of capital '>\lith an output elasticity and 'income share' of its own, cp say,
and that this capital were accumulated in the same way as physical capital {and therefore
delayed convergence similarly). Furthermore, suppose the income share of this form of
capital were accruing to workers. Then, lor a maintained around ~. one would have a
slower convergence since o. + cp > ~.and still a labour's share, 1 - o., around i·
This suggests the idea of adding human capital to the model: each year there is a
certain stock of human capital, which can be thought of as the sum of all the education
and training the workers have had, and this stock (per worker) serves to increase the
productivity of workers. Just like physical capital, total human capital is assumed to
accumulate by a certain exogenous fraction of GDP being added to it each year. The
income share of human capital goes to workers. because human capital (embedded education) is linked to workers and its services therefore sold inseparably together with
work hours.
Since human capital accumulates like physical capital. the inclusion of it will lower
the model's speed of convergence. Because the output elasticity of physical capital is
unchanged. the owners of physical capital will earn the same fraction of GDP . The fraction
earned by human capital goes to the workers, so labour's share will also be unchanged.
Thus. adding human capital should work to overcome the second of the empirical
problems.
What about the first problem concerning the model's steady state prediction? Assume
that the rate of investment in physical capital, increases. This will create higher GDP per
person in steady state because more physical capital per person is accumulated. If a
certain fraction of GDP is invested in human capital, then the higher GDP first caused by
the higher rate of investment in physical capital will also imply that more humnn capital
per worker is accumulated. This reinforces the increase in output. Thus, the increase in
GDP per person caused by a higher investment rate in physical capital is larger in a model
including human capital than in one with only physical capital (other things, e.g. a. being
equal).
The introduction of human capital sh ould thus work to mitigate both of the empirical
failures of the Solow model. How well a job it does in this respect is the subject of th is
chapter, and we will see that it does a very fine job.
6.1 The solow model with human capital
The 'micro world' of the model to be presented is similar to that of the Solow models
considered previously. There are the same kinds of economic agents, a representative
profit maximizing Hrm and consumers (government can again be interpreted into the
model, a possibility that will be considered in an exercise) . The production function of the
firm is now different as it includes human capital, and the consumers will also have to
decide on how large a fraction of income to accumulate as human capital each period.
We assume that each unit of output can be used either as consumption. investment in
physical capital. or as investment in h uman capital.
There are also the same competitive markets as before, one lor output with real price
normalized to 1. one for the services of physical capital with real rental rate, r 1, and one for
labour with real wage rate. w ,. Hence we do not assume th at there is a separate market for
the services of human capital with a rental rate of its own. This is because human capital
cannot be separated from workers. Its services have to be sold together with labour. So the
services traded on the labour market will no longer be units of 'raw' labour, man-years,
but man-years endowed with a certain level of human capital or education. This level is
assumed to be the same for all workers. Hence the total stock of human capital, H,,
manifests itself as each worker in the labour Ioree. L,. holding a human capital of h, = Hj L,
that goes inseparably with the worker. Employing one more worker means hiring an
additional unit oflabour equipped with h 1•
The Solow model with human capital thus has two new features compared to the
original Solow model: the inclusion of human capital in the production function and the
accumulation of h uman capital by the consumers.
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