Review Session 7 Solow model (continued)

1. Review Session 7Solow model (continued) Catalina Martinez catalina.martinez@graduateinstitute.ch Office hours: Tuesdays 6-8pm Rigot 27 Economics and Development MDev 2012-2013 THE GRADUATE INSTITUTE | GENEVA

2. Change in class dynamics • The RS will cover the topic one week after • Like this you will have more time to absorb the information • The quizzes will be solved during the RS

3. Today’s RS • Complete Solow model (with population and technological growth) • Based on RomerCh 1(sent by email and also on the website) • Focus on intuitions • Solve Solow Quiz

4. Building the Model: The production function • We begin with a production function Y=F(K,AL) K is capital L is labor A is knowledge or effectiveness of labor AL are the effective units of labor available in the economy Intuition: Inputs other than capital, labor and knowledge are relatively unimportant. In particular, the model does not take into account land and natural resources.

5. Building the Model: The production function • We assume constant returns zY=F(zK,zAL) Intuition: The economy is big enough that the gains from specialization are exhausted. Indeed, in a small economy it is likely that if the inputs are doubled the output is more than doubled because there are many opportunities for further specialization. Here we are assuming that this possibilities have all been explored.

6. Building the Model: The production function • By setting z=1/AL we create a per efficient worker function. Y/AL=F(K/AL,1) Y/AL=y is the output per efficient worker K/AL=k is the capital per efficient worker • So, output per efficient worker is a function of capital per efficient worker. We write this as, y=f(k) Intuition: If the economy is divided into AL small economies, each with one unit of effective labor AL and K/AL=k units of capital, since the production function has constant returns to scale, this small units will produce 1/AL what is produced in the overall economy. Therefore, y depends only on k and not on the overall size of the economy.

7. Building the model: the production function • The production is Cobb Douglas (it has the same shape that we have always seen), which implies that inputs can be substituted Y K Holding AL constant, we get….. Y Y K AL

8. Building the model: the production function • The production function presents positive but decreasing marginal productivity of capital Output per efficient unit of labor (y) f(k) Capital per efficient unit of labor (k)

9. Building the model: the production function • The production function presents positive but decreasing marginal productivity of capital (This is the intuition behind the Inada conditions and corresponds to the shape of the production function that we have always seen) • This is a key assumption because it guarantees the existence of the equilibrium: i.e. the existence of a point beyond which accumulating more capital is not efficient anymore (because it does not add up any productivity)

10. Building the Model: closed economy • The basic accounting national income identity of any economy says that income is equal to consumption (C), investment (I), net exports (X exports – M imports) and public expenditure (G): Y=C+I+(X-M)+G • In the Solow model, the economy is closed, therefore government purchases and net exports are not included. This gives us the following per efficient worker national income accounting identity. y = c+i Where c is C/AL, the consumption per efficient worker And i is I/AL, the investment per efficient worker

11. Building the model: closed economy • Given a savings rate (s) and a consumption rate (1–s) we can generate a consumption function. c = (1–s)y y = (1–s)y + i i = s*y • Therefore investment per worker equals savings per worker (indeed, if there is no trade or public expenditure, the only way to finance investment is through savings) • Savings are assumed to be exogenous (not determined by the model): important and criticizedbecause it is a key parameter in the model. • By substituting f(k) for (y), the investment per worker function (i = s*y) becomes a function of capital per worker i= s*f(k)

12. Building the model: depreciation, population growth and technological progress • Population grows at a constant rate n • Technology A grows at a constant rate g • The depreciation rate δis the share of capital is lost every period (due to use, etc…) • To see the impact of investment and depreciation on capital we develop the following (change in capital) formula, ΔK = I – δK ΔK = s*f(K) – δK • Which means that capital changes (over time) depending on how much we invest and how much capital depreciates.

13. Steady State • Our purpose is to understand how the economy achieves the steady state (SS) and what happens in it. • Think about the SS as the equilibrium of the overall economy: as when we studied the equilibrium in micro we want to see how resources get to their most efficient use. • We want to study the point in which the savings in the economy are invested in the exact amount of capital that can be used by the economy. • If we go beyond, we would be accumulating capital unnecessarily.

14. Steady State and LR vs SR growth • Moreover, we are interested in the SS because it represents the long run (LR) path of the economy (trend). • We are interested in understanding how structural changes in growth happen, and not only in how business cycles ocur. • Business cycles are more closely related with short run (SR) growth.

15. Steady State Equilibrium • In the SS capital does not change anymore because we invest exactly the amount of capital that is used: Δk = 0, i.e. kt=kt+1 Δy = 0 i.e. yt=yt+1 • We are interested in this point because we want to understand how the economy behaves in equilibrium between capital creation and capital destruction.

16. Derivation for incorporating n and g (population and technology growth)

17. Derivation for incorporating n and g (population and technology growth)

18. Derivation for incorporating n and g (population and technology growth)

19. BASIC EQUATION OF THE MODEL • Actual investment = break even investment. • In the SS we invest up to the point in which capital is used by the new population, by the new knowledge and by the depreciation. • If we go beyond, we would be accumulating too much capital. • If we do not achieve this level, then capital is eroding with time.

20. Steady State Equilibrium (n+g+δ)k s*f(k) s*f(k*)=(n+g+δ)k* k klow k* khigh • If our initial allocation of (k) were too high, (k) would decrease because depreciation exceeds investment. • If our initial allocation were too low, k would increase because investment exceeds depreciation.

21. Steady State Equilibrium δk s*f(k) s*f(k*)=δk* k klow k* khigh • KEY: • For convergence to happen we need positive but decreasing marginal productivity of capital (The shape of the red curve)

22. Convergence Growth is faster for low levels of capital than for higher ones. It gets slower and slower up to the point that it is zero in the SS. Conditional convergence: If we assume that two economies have the same characteristics, the one that has lower capital will grow faster than the one with higher capital. This holds in the data (Example of OECD countries discussed in lecture)

23. IMPORTANT: LR Growth in terms of what?.... • Y/AL=y is not growing because capital per efficient unit of labor is being accumulated at exactly the same way in which it is destructed. • Y/L, i.e. GDPpc or output per worker is growing at the growth rate of A knowledge, i.e. g. • Y, GDP or output of the whole economy is growing at the growth rate of population and knowledge (n+g).

24. Problem: What is A?? • The conclusion of the model is that in the long run growth per capita depends on the growth rate of A. • This is completely exogenous: not determined by the model. • It is everything that makes labor efficient: • Human capital: quality and education • Institutions and property rights • Infrastructure • Culture….

25. So, what can developing countries do? • The parameter that is most likely affected by policy is s: the savings rate • But a temporary increase in the savings rate induces only a temporary increase in the growth rate. • This is due to the fact that capital has positive but decreasing marginal productivity. Therefore investing more and more cannot be good forever… • If the marginal productivity of capital is higher than the break even investment (n+g+d)k, then it is possible to keep growing. • But if what an extra unit of capital produces is less than what it costs, then growth is not sustainable anymore. • There is a level effect but not a growth effect.

26. SS and savings rate

27. The Golden Rule level of capital C2 C1 • What we do not invest we consume • There is an optimal level of savings that leads to the maximum consumption • Note that the consumption implied by s1 is higher than that implied by s2

28. Quiz

29. Question 1 • What is the main purpose of the Solow model? Explain why is it important to understand the role that capital and investment play on long-term growth from the point of view of a developing country. • Main purpose: Explain long run growth • Role of capital and investment: Understand if and how policy can change long run growth

30. Question 2 • What is the steady state? Explain why it is important to think about the equilibrium between capital creation and capital destruction, even if in the steady state there might be growth due to other factors, such as exogenous technology growth. • SS: Break even investment, savings are allocated to increase capital stock up to the point in which capital is depleated. • It is important because it represents the long run equilibrium, the long run growth path of the economy.

31. Question 3 • What happens if a country raises its savings rate? Explain why the k* in the steady state increases. • There is an increase in short term growth but not in long term growth • There is an increase in the level of output, put not in the long term growth of output.

32. Question 4 • What is absolute convergence? Does it hold? • According to absolute convergence all countries irrespective of their characteristics converge to the same SS level. • It does not hold in practice.

33. Question 5 • What is conditional convergence? Explain how it works. Does it hold in the data? • According to conditional convergence countries with the same characteristics will converge to the same SS level. • Countries with lower capital level will grow faster than those with higher capital level, up to the point in which all reach the SS. • It does hold in practice: OECD study mentioned in lecture.

34. Question 6 • What is the main and most criticized assumption of the Solow model? • Constant and exogenous savings rate s • Exogenous technology A