Lecture 4: The Solow Growth Model

1. Lecture 4: The Solow Growth Model L11200 Introduction to Macroeconomics 2009/10 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 Period 1 start at a level produce , save depreciates increases by net effect on new level falling , falls produce , save depreciates increases by net effect on 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.