Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 3-4 EXTENSIONS: Endogenous Investment and Population Growth

Extensions to the basic Solow model We will pursue two of them here Endogenous investment and saving Population growth We will do more (human capital, technology, efficiency) in the next classes 3-2

Extensions to the basic Solow model: endogenous investment Did we get an answer to our question of what determines income or growth gaps across countries from the basic Solow model? Yes, but only a partial one We do not know – in turn - why investment rates differ 1.Is it because of differences in saving rates? 2.Or is it because foreign capital inflows are so substantial that the extent of domestic saving ends up being irrelevant for domestic investment? –We will answer question 2 later on (speaking of globalization); for now we tackle question 1 3-3

3-4 The saving rate is not constant across income groups by decile of income per capita This suggests that the true causation between saving and per-capita Gdp may go the other way around: from Gdp to saving, and not from saving (and I) to Gdp

Implications of Solow with endogenous invesment The convergence property may break down Not necessarily true anymore that poorer countries will grow faster than richer countries –Why? See it graphically yourself 3-5

3-6 Solow Model with Saving Dependent on Income Level: result, multiple steady states & poverty traps When we allow for saving rates to differ across countries (with rich people saving more than poor people), the Solow model becomes a model with two steady state equilibria

Extensions to the basic Solow model: population growth So far: population constant What if population is not constant? What is the relation between population growth and income? What is the relation between population growth and (transitional) growth? Fortunately, we can adapt the Solow model and answer these questions in a relatively straightforward way 3-7

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4-8 Starting point: a negative relation between population growth and income. How do we make sense of this?

Idea: amend the Solow model Main point: with population growth rate, per-capita capital accumulation gets “diluted” Suppose that population growth rate = n (say 1% per year) To keep capital per worker constant, we need to supply the new workers around with the same amount of capital as the old workers 3-9

How the maths change (in a Cobb-Douglas world) The equation of k accumulation changes as follows: Δk = γAk α – δk – nk = γAk α - (δ+n) k The steady state equilibrium equation becomes: γAk α = (δ+n) k And the equation for transitional growth becomes: Δk/k = γAk α-1 - (δ+n) 3-10

Implications Once we allow for population growth, the following happens: for given investment and savings, capital accumulation per worker will be lower as well Gdp per worker will be lower as well Transitional growth of Gdp will be accordingly lower In other words, population growth is associated with: Lower steady state level of Gdp and capital per worker Lower growth along the transition Here is why we say that population growth in the Solow model leads to “dilution” 3-11

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4-12 Graphically -- The Solow Model Incorporating Population Growth

Quantitative analysis with the Solow model and population growth k ss =[γA/(n+δ)] 1/(1-α) y ss =A(k ss ) α = A 1/(1-α) [γA/(n+δ)] α/(1-α) Which is the numerical effect of “n” on y ss and k ss in practice? To make things simpler, we compare y ss for two countries (say “Oecd” and “Asia”) with same A, γ and δ but different “n” y ss Oecd /y ss Asia = [(n Asia +δ)/(n Oecd +δ)] α/(1-α) –Take δ=5% and α=1/3. Take n Oecd =0% and n Asia =4% y ss Oecd /y ss Asia = [( )/( )] ½ =1.34 An Oecd country, given that its population growth is much lower than in an Asian country, will enjoy higher Gdp per worker by some 34% This compares with usually much larger actual differences (US/India=1/0.13, which is about 8:1) This means that differences in population growth only partially account for steady state income gaps 3-13

The Solow model with different population growth and investment rates We can improve the predictive ability of the Solow model by simultaneously allowing for different investment and different population growth rates AT THE SAME TIME Things become algebraically more complicated (but conceptually NOT!). For each country: y i ss =A(k i ss ) α = A 1/(1-α) [γ i A/(n i +δ)] α/(1-α) Then take a value for A, δ and α and redo the same exercise as before (you do it yourself) 3-14