1. Lecture 4: The Solow Growth Model
L11200 Introduction to Macroeconomics 2009/10
Lecture 4: The Solow Growth Model
Reading: Barro Ch.3 : p52-67
2 February 2010

2. Introduction Last time: first lecture on economic growth
Considered data on cross-country growth rates
Began model of economic growth
This time: expand the model of economic growth
Develop the ‘Solow Growth Model’
Aim: to understand what determines economic growth and explain cross-county growth rates

3. Where did we get to last week?
Setup a production function with attributes of diminishing marginal product, constant returns to scale:
Showed that with this production function and fixed A, growth in per capita output only possibly by increasing capital per worker:

4. Next steps
Growth in per capita output depends on growth in capital per worker, given by:
What determines the growth of the capital stock?
What determines the growth rate of labour?
Can then calculate growth in capital per worker

5. 1. Growth of Capital Stock
Growth of capital stock depends on:
How much new capital is added ‘investment’,
How much existing capital depreciates (wears out)
Assume a fraction of the capital stock δ, depreciates each period and has to be replaced.
So household income (after depreciation) is given by:
Households save some fraction, s, which they invest in new capital.

6. 1. Growth of Capital Stock
So have equation for change in capital:
Can convert this into the growth rate of capital stock by dividing both sides by K
This is the equation for growth of capital stock

7. 2. Growth rate of labour
Capital investment depends on how much people decide to save
Labour force growth depends on how much people decide to reproduce.
Assume this is constant growth rate, ‘n’ So

8. Growth of capital per worker
So now have:
Can express in per worker terms by dividing through by , so
From earlier, can now substitute:

9. So growth rate of capital stock per worker depends on:
Labour force growth, n: negatively
Depreciation, δ: negatively
Saving rate, s: positive
All of the above are fixed
‘Average Product of Capital’ : what determines this?

10. Average Product of Capital
Marginal Product of capital is given by:
e.g. if a 1 unit increase in K causes a 10 unit increase in Y, then MPk=10
Average product of capital is simply
e.g. 10 units of K produce 50 units of Y, so average product per unit is 5

11. Intuition
The more capital you add to production, the less each additional unit adds to output
So as capital increases, average product decreases.
This explains the final part of:

12. There are two ‘forces’ on the growth rate of capital stock per worker:
Saving raises capital stock per worker. But as the capital stock grows, the average product of capital falls. So a fixed s (e.g. 5%) translates to a lower growth rate of capital at higher levels of capital
Depreciation and population growth lower capital per worker
So there is a level of capital per worker at which these two forces are equal: an equilibrium

13. Production and Investment Depreciation and labour force growth
start at a level produce , save depreciates increases by net effect on new level falling , falls produce , save depreciates increases by
Period 1
Period 2
This is lower than before
So net positive effect is smaller

14. Intuition Can increase y by increasing K to a point:
Depreciation and population growth lowers k
At high levels of K, the saved part of the marginal product of additional capital is only just enough to offset depreciation and population growth
So diminishing marginal product limits the impact of increasing K upon k, and hence upon y

15. Explaining k*
k* is the level of capital per worker at which the positive effect of new investment is exactly matched by the negative effect of δ and n
When k reaches k* it stops at the equilibrium level of capital per worker.
We call this the steady state level of k*

16. Implications for y* From earlier:
We now know what factors determine and so what determines
So starting from 1 unit of capital, per capita output will grow until and then stop growing at the steady state

17. Summary Developed a growth model
Capital and labour produce output
They exhibit diminishing marginal returns: so adding labour cannot increase per capita GDP and capital investment can only increase it to a point.
Next lecture: more on what the model predicts for growth rates, and for the impact of changing s, δ and n.