The Solow Growth Model

Transcription

1 The Solow Growth Model Seyed Ali Madanizadeh Sharif U. of Tech. April 25, 2017 Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

2 Economic Growth Facts 1 In the data, real GDP/capita has been growing over time for most countries. 2 There are large differences in per capita income levels across countries: $34k vs $ In similar countries, economies with lower real GDP per capita had faster growth rates. But this is not true globally. 4 Some economies were converging to each other. 5 Still there are some stagnations. Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

3 Questions 1 What are the factors that lead to economic growth? 2 Why do some countries grow fast and others slow? (East Asians vs sub-saharan African countries) 3 What policies increase real GDP per capita? 4 How did rich countries sustain growth rates of 2%? Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

4 Solow Model Production function: Y t = A t.f (K t, L t ) Capital accumulates through investment: K t+1 = (1 δ) K t + I t and Y = C + I I is the investment: Purchases of capital: In the data 14% of GDP. F is diminishing marginal product of capital and labor. Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

5 Solow Model F is constant returns to scale, therefore Real GDP per capita is Y /L = AF (K /L, L/L) y = AF (k, 1) y = Af (k) Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

6 Solow Model Example: Therefore Y = AK α L 1 α y = Ak α Output Y A F(K, L) As K increases, the marginal product of K (slope) decreases. Capital K Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

7 Growth Accounting log Y t = log A t + α log K t + (1 α) log L t g Y = g A + αg K + (1 α) g L or Y Y = A A + α K K L + (1 α) L y y = A A + α k k Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

8 Solow Model Simplifying assumption national savings rate: s I t = sy t Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

9 Solow Model Population growth rate: g n Rewriting the law of motion: K t+1 K t+1 = (1 δ) K t + sy t L t+1 = (1 δ) K t + s Y t L t+1 L t L t L t k t+1 (1 + g n ) = (1 δ) k t + sy t k t+1 = sy t + 1 δ k t 1 + g n 1 + g n k t+1 = saf (k t) 1 + g n + 1 δ 1 + g n k t = G (k t ) Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

10 Capital choices and capital dynamics Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

11 Steady state level k = G (k ) = saf (k ) 1 + g n + 1 δ 1 + g n k = sak α + 1 δ k 1 + g n 1 + g n ( ) 1 k = A 1 α 1 s 1 α g n + δ ( ) i gn + δ = k 1 + g n ( y = A 1 α 1 s g n + δ ( Y t = A 1 α 1 s g n + δ ) α 1 α ) α 1 α L0 (1 + g n ) t Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

12 Average product of capital Output per worker y Average product of capital with k a A f(k) Average product of capital with k b k a k b Capital per worker k Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

13 Capital growth rate Determinants of k/k k/k > 0 since s (Y/K) > sδ + n sδ + n s (y/k) k 0 k* Capital per worker k Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

14 Growth variable Y Y k k y y = A A + α K L + (1 α) K L ( y ) = s δ n k ( ( y ) ) = α s δ n k Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

15 Predictions Convergence Data: There is conditional convergence: economies with similar characteristics converge Lower K, higher growth Capital per worker k* k(0) 2 k(0) 1 Time Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

16 Comparative statics Changes in s, g n, A, δ, α Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

17 A Change in the Saving Rate Determinants of k/k For any level of k, k/k is larger the higher the saving rate s 2 δ + n s 1 δ + n s 2 (y/k) s 1 (y/k) k(0) k 1 * k 2 * Capital per worker k Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

18 A Change in the Saving Rate Compare two countries with different savings rates. How does that affect k/k, k* and y*. A higher saving rate raises k/k, which rate remains higher during the transition to the steady state. In the long run, k/k and y/y are equal to zero for any saving rate, but a higher saving rate leads to higher steady-state k* and y*. Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

19 A Change in the Technology Level Determinants of k/k The growth rate of capital per worker, k/k, is higher at any k when the technology level is higher k(0) k 1 * k 2 * sδ + n sa 2 f(k)/k sa 1 f(k)/k Capital per worker k Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

20 A Change in the Technology Level What happens to k/k, k* and y* if there is a change in technology? In the short run, an increase in A raises k/k and y/y, which remain higher during the transition to the steady state. In the long run, k/k and y/y are equal to zero for any technology level, but a higher technology level leads to higher k* and y*. Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

21 A Change in Labor Determinants of k/k An increase in L 0 reduces k 0 and increases k/k sδ + n s (y/k) k(0) k(0) k* Capital per worker k Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

22 A Change in Labor What happens to k/k, k* and y* if there is a change in the amount of labor, L? In the short run, an increase in L raises k/k and y/y, which remain higher during the transition to the steady state. In the long run, k/k and y/y are equal to zero for any L. Also, k* and y*, are the same for any L. Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

23 A Change in Population Growth Determinants of k/k An increase in n reduces k/k sδ + n sδ + n s (y/k) k(0) (k*) k* Capital per worker k Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

24 A Change in Population Growth What happens to k/k, k* and y* if there is a change in the population growth rate, n? In the short run, an increase in n reduces k/k and y/y, which remain lower during the transition to the steady state. In the long run, k/k and y/y equal to zero for any, but a higher population growth rate leads to lower k* and y*. A change in the depreciation rate, δ, has the same effect. Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

25 Summary Increase in Effect on k* and y* Saving rate, s INCREASE Technology, A INCREASE Population growth, n DECREASE Depreciation DECREASE Labor input, L NO EFFECT Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

26 Optimum saving rate: Golden Rule Maximize consumption with respect to s subject to the steady state condition: max c s c = y i = y sy = A (1 s) f (k ) = (1 s) A ( s g n + δ ) α 1 α Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

27 Optimum saving rate: Golden Rule Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

28 Optimum saving rate: Golden Rule take logs: log c = log (1 s) + α 1 α log s + log A (g n + δ) 1 α α FOC: α 1 1 s + 1 α = 0 s α (1 s) = s 1 α s = α Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

29 Optimum saving rate: Golden Rule Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

30 Solow Model The Biggest Failure of the model: No sustained per capita growth of GDP and capital Due to DECREASING RETURNS to scale. Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

31 Convergence Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

32 Convergence Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

33 Convergence Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

34 Conditional Convergence: Different Saving Rates Determinants of k/k The poor country grows from k(0) 1 to k 1 *; the rich from k(0) 2 to k 2 *. Which country grows faster depends on the initial k/k. s 2 δ + n s 1 δ + n s 2 (y/k) s 1 (y/k) k(0) 1 k(0) 2 k 1 * k 2 * Capital per worker k Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

35 Conditional Convergence: Different Population Growth Rates Determinants of k/k The poor country grows from k(0) 1 to k 1 *; the rich from k(0) 2 to k 2 *. Which country grows faster depends on the initial k/k. sδ + n 1 sδ + n 2 s (y/k) k(0) 1 k(0) 2 k 1 * k 2 * Capital per worker k Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

36 Transition Paths For Two Economies Capital per worker k* k* 1 k(0) 2 k(0) 1 Time Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

37 Exogenous Growth Why don t we see sustained growth in the simple Solow model? How can we get it? Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

38 Exogenous Growth Exogenous productivity growth!!! How does it work? log Y t = log A t + α log K t + (1 α) log L t g Y = g A + αg K + (1 α) g L g y = g }{{} A + αg }{{} k Technical Growth Capital Deepening Intuition: How does productivity growth leads to capital deepening. Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

39 Exogenous Growth From I = sy we had: So k t+1 = sy t + 1 δ k t 1 + g n 1 + g n ( ) k t+1 s yt = + 1 δ k t 1 + g n k t 1 + g n Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

40 Exogenous Growth Balanced Growth path: All variables have a constant growth rate. Thus y t k t is constant. Therefore y and k have the same growth rate; call it g = g Y = g k g = g A + αg Amplification mechanism! Explain graphically! g = g A 1 α Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

41 Exogenous Growth Also: ( ) s yt 1 + g = + 1 δ 1 + g n k t 1 + g n y t = (1 + g) (1 + g n) (1 δ) k t s y 0 g + g n + δ k 0 s A 0 k0 α 1 g + g n + δ s ( k 0 = sa 0 g A 1 α + g n + δ ) 1 1 α Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

42 Growth Accounting y t and k t are observable. How about A t? How about α? Recall that with Cobb Douglas production function the shares of factor payments are constant: Empirically we can test this: In US: α 1 3 In Iran: α 2 3 Y t = rk t + wl t wl t Y t = α Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

43 Growth Accounting Now we know a log A t = log y t α log k t A t computed this way is called the SOLOW RESIDUAL. It comes from Solow (1957) growth accounting framework. He applied this framework to United States data. Found that changes in A t are responsible for 80% of changes in y t. Thus, changes in k t are responsible for ONLY 20%! Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

44 Growth Accounting More sophisticated studies include more inputs than capital: Male and female labor force participation. Education. Land and natural resources. For specific episodes the importance of input accumulation can be even higher: The USSR had input accumulation as the main growth strategy. Young (95), Krugman (94) and others provide evidence that the large post-war growth in Honk Kong, Singapore, South Korea and Taiwan was driven by input accumulation. Hsieh (99) contest their ndings Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

45 Growth Accounting Country log y t+1 y t α log k t+1 k t log A t+1 A t US ( ) USSR ( Singapore Note: This does not include other inputs. Source: Ofer (1987), Bosworth and Collins (2003) Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46

46 Growth Accounting What is A t? Main problem with this methodology: Recall Moses Abramovitz (1956): At is the measure of our ignorance. In our theory A t is EXOGENOUS. In the data it is the part of y t+1 y t not explained by k t. Yet, it is fundamental to understand growth. Opennig the black box of A t Endogenous Growth models (new growth theory) Seyed Ali Madanizadeh Sharif U. of Tech. () The Solow Growth Model April 25, / 46