Intermediate Macroeconomics

Transcription

1 Intermediate Macroeconomics Lecture 3 - The Solow model Zsófia L. Bárány Sciences Po 2014 February

2 Recap of last week two distinct phases of growth: 1. pre-industrial revolution - stagnation: growth in total output is offset by population growth, constant per capita consumption 2. post-industrial revolution - sustained growth: around 1.5% to 2% growth per year sustained over 150 years different models can explain the different stages: 1. Malthusian model - main assumptions: population growth depends on consumption per capita & land is available in limited supply 2. Solow model - the topic of today s lecture

3 The Solow model key: the description of the saving - investment - capital accumulation - growth process the model has stark predictions relating to the growth facts main prediction: technological progress is necessary for sustained growth in living standards note: not a micro-founded model, i.e. the goals and constraints of the agents are not explicit the Solow model is the basis for the modern theory of economic growth

4 Assumptions on Consumers/Workers Labor supply there are N consumers who do not make a labor-leisure choice each provides his one unit of labor N is also the size of the labor force the population is growing at exogenous constant rate n N = (1 + n) N

5 Consumers/Workers Income and consumption consumers are assumed to save an exogenous, constant fraction s of their income, and consume the rest the total income of consumers is Y, which is equal to total output, as there is no government note: we do not need to distinguish between labor income and capital income or profits (since households own the firms and do not make a labor supply decision) the consumer s budget constraint is Y = C + S total saving in a period is s Y total consumption in a period is (1 s) Y

6 Production Production function Y = z F (K, N) Y total output z total factor productivity (TFP) K stock of capital N labor input Important features: output depends on the quantity of labor and capital, and TFP no other inputs, such as land or natural resources, matter Production function is assumed to be neoclassical positive, but diminishing marginal products Inada conditions constant returns to scale (CRS)

7 Characteristics of the neoclassical production function I. positive, but diminishing marginal products: z F (K, N) is monotone increasing in K and N positive marginal product of each inputs MPK = z F K = z F K > 0: if K increases by 1, z F (K, N) increases by MP K MP N = z F N = z F N > 0: if N increases by 1, z F (K, N) increases by MP N z F (K, N) is concave - decreasing returns to K and N decreasing marginal product of both inputs FKK = 2 F K = F 2 K K < 0 and F NN = 2 F N = F 2 N N < 0 as K increases more and more, F (K, N) increases by less and less as N increases more and more, F (K, N) increases by less and less

8 Characteristics of the neoclassical production function II. Inada conditions: the marginal product of an input goes to zero as that input goes to infinity (holding all other inputs constant): lim K F K (K, N) = 0 and lim N F N (K, N) = 0 the marginal product of an input goes to infinity as that input goes to zero (holding all other inputs constant): lim K 0 F K (K, N) = and lim N 0 F N (K, N) =

9 Characteristics of the neoclassical production function III. constant returns to scale: F (λk, λn) = λf (K, N) for any λ > 0 e.g. λ = 2: if K and N double, then F (K, N) and Y doubles consequence of CRS: since we can pick λ = 1/N, output per worker can be written as: y = Y N = z F ( K N, N N ) = z F (K N, 1) = z f (k) }{{} f (k) allows us to work with per capita quantities denote: k = K N, capital per capita; y = Y N output per capita f inherits the properties from F : f k (k) > 0, f kk (k) < 0

10 Shape of the production function Y z F (K, N) K For a given labor supply (N) Y as a function of capital (K) looks something like this.

11 Shape of the production function Y N z F ( K N, 1) K In per capita terms, Y /N looks something like this.

12 Shape of the production function y = Y N z f (k) k = K N The per capita production function, f (k) = F ( K N, 1) looks therefore something like this.

13 Example: the Cobb-Douglas production function Y = z K α N 1 α has constant returns to scale as the coefficients sum to 1: zf (λk, λn) = z(λk) α (λn) 1 α = λ α+1 α zk α N 1 α =λzk α N 1 α Per capita output is y = Y /N = zk α =λf (K, N) positive, but diminishing marginal products, which are given by: MP K = αzk α 1 N 1 α Inada conditions hold MP N = (1 α)zk α N α

14 The capital accumulation equation Capital stock increases - due to investment decreases - due to depreciation, d Capital evolves according to: K = (1 d)k + I How do we determine I? This is part of the equilibrium condition of the model.

15 Competitive equilibrium Two options: 1. Income - expenditure identity holds as an equilibrium condition i.e. the goods market is in equilibrium Y = C + I combine this with the consumer s budget constraint: Y = C + S gives us market clearing on the capital market: S = I 2. Capital market is in equilibrium total saving = total investment S = I combine this with the consumer s budget constraint: Y = C + S gives us the income-expenditure identity: Y = C + I

16 Equilibrium dynamics of capital per worker/per capita The equilibrium dynamics of capital is then: K = I +(1 d)k = S+(1 d)k = s Y +(1 d)k = s z F (K, N)+(1 d)k Transforming this into per capita terms: K N = s z F (K, N) N + (1 d) K N

17 Equilibrium dynamics of capital per worker/per capita The equilibrium dynamics of capital is then: K = I +(1 d)k = S+(1 d)k = s Y +(1 d)k = s z F (K, N)+(1 d)k Transforming this into per capita terms: K N = s z F (K, N) N + (1 d) K N N K }{{} N N =1 = s z F (K, N) N + (1 d) K N

18 Equilibrium dynamics of capital per worker/per capita The equilibrium dynamics of capital is then: K = I +(1 d)k = S+(1 d)k = s Y +(1 d)k = s z F (K, N)+(1 d)k Transforming this into per capita terms: K N = s z F (K, N) N + (1 d) K N N K }{{} N N =1 (1 + n)n N = s z F (K, N) N + (1 d) K N K F (K, N) =s z + (1 d) K N N N

19 Equilibrium dynamics of capital per worker/per capita The equilibrium dynamics of capital is then: K = I +(1 d)k = S+(1 d)k = s Y +(1 d)k = s z F (K, N)+(1 d)k Transforming this into per capita terms: K N = s z F (K, N) N + (1 d) K N N K }{{} N N =1 (1 + n)n N = s z F (K, N) N + (1 d) K N K F (K, N) =s z + (1 d) K N N N (1 + n)k =s z f (k)+(1 d)k

20 the key equation of the Solow model k = s z f (k) 1 + n + (1 d)k 1 + n determines capital per worker in the next period as a function of capital per worker today capital per capita in the next period is investment in the current period plus whatever is left of capital after depreciation adjusting for the fact that the population is higher in the next period

21 Finding the steady state the steady state k is where k = k if k < k k > k if k > k k < k

22 The steady state Equation determining the steady state of the economy: k = k = k in the steady state capital per worker is constant In the steady state k = s z f (k ) (1 d)k n 1 + n (1 + n)k = s z f (k ) + (1 d)k (n + d)k }{{} = s z f (k ) }{{} break-even investment actual investment the amount of actual investment is exactly such that it compensates for depreciation and the growth in population, i.e. for the break-even investment the level of capital per worker stays constant

23 A useful graphical representation: The savings curve (actual investment) and the effective depreciation line (break-even investment) break-even investment actual investment steady state: where the break-even investment = = actual investment This graph is very useful for studying the effects of n, s and z

24 Three important questions about the Solow model: 1. does the steady state exist? 2. is the steady state unique? 3. is the steady state stable, i.e. does the economy converge to k from any initial k 0?

25 Three important questions about the Solow model: 1. does the steady state exist? YES, if the two curves cross 2. is the steady state unique? 3. is the steady state stable, i.e. does the economy converge to k from any initial k 0?

26 Three important questions about the Solow model: 1. does the steady state exist? YES, if the two curves cross 2. is the steady state unique? YES, if the two curves only cross once 3. is the steady state stable, i.e. does the economy converge to k from any initial k 0?

27 Three important questions about the Solow model: 1. does the steady state exist? YES, if the two curves cross 2. is the steady state unique? YES, if the two curves only cross once 3. is the steady state stable, i.e. does the economy converge to k from any initial k 0? YES, if the actual investment curve is above the break-even investment curve for k < k, and if the actual investment curve is below the break-even investment curve when k > k

28 Three important questions about the Solow model: 1. does the steady state exist? YES, if the two curves cross 2. is the steady state unique? YES, if the two curves only cross once 3. is the steady state stable, i.e. does the economy converge to k from any initial k 0? YES, if the actual investment curve is above the break-even investment curve for k < k, and if the actual investment curve is below the break-even investment curve when k > k This is due to the neoclassical production function.

29 Example 1. i (d + n)k k

30 Which of the assumptions on a neoclassical production function is violated? Example 1. i (d + n)k szf (k) k

31 Which of the assumptions on a neoclassical production function is violated? Example 1. i szf (k) (d + n)k k

32 Which of the assumptions on a neoclassical production function is violated? Example 2. i (d + n)k szf (k) k

33 Which of the assumptions on a neoclassical production function is violated? Example 3. i szf (k) (d + n)k k

34 With a neoclassical production function A unique and stable steady state exists. break-even investment actual investment steady state: break-even investment=actual investment k = k = k

35 Implications 1. In the steady state k is constant y = f (k ) is constant as well c = (1 s)y is constant as well K = N k grows at rate n Y = N y grows at rate n C = N c grows at rate n in the long-run no growth in output per worker

36 Implications 2. if two countries have the same n, s, and d, as well as the same production function they have the same steady state capital per worker, k they converge to the same steady state conditional convergence along this convergence path poorer countries grow faster (why?)

37 Implications 3. what happens if two countries have different saving rates, i.e. s A < s B? their steady state per capita capital is different, i.e. k A k B both countries converge to their own steady state, but these are different no absolute convergence when saving rates are different poorer countries don t necessarily grow faster

38 Saving rate and the steady state per capita capital i (d + n)k k

39 Saving rate and the steady state per capita capital i (d + n)k s A zf (k) k

40 Saving rate and the steady state per capita capital i (d + n)k s A zf (k) k A k

41 Saving rate and the steady state per capita capital i (d + n)k s B zf (k) s A zf (k) k A k

42 Saving rate and the steady state per capita capital i (d + n)k s B zf (k) s A zf (k) k A k B k

43 Saving rate and the steady state per capita capital i (d + n)k s B zf (k) s A zf (k) k A k B k The steady state capital per worker is higher, when the savings rate is higher. s A < s B k A < k B

44 Data: Real income per capita and the investment rate per capita GDP and the investment rate are positively correlated

45 Income per capita and saving rate in the model What is the implication of a higher saving rate in the Solow model? intuitively, a higher saving rate implies that the saving curve, the actual investment curve is higher higher steady state k higher y What is the effect of an increase in the saving rate? long-run increase in the level of per capita capital long-run increase in the level of per capita output temporary increase in the growth rate of per capita capital and per capita output no effect on the long-run growth rate of capital per worker and output per worker, which are equal to zero A change in the savings rate has a long-run level effect, but does not have a long-run growth effect.

46 Saving rate and the steady state per capita capital so a higher saving rate, s, leads to higher per capital capital in the steady state, k, and thus to a higher steady state per capita income, y should people increase their saving rate in order to increase output? would this be good for them? put another way: Does it mean that a higher saving rate is always better? what is the best steady state?

47 Saving rate and the steady state per capita capital so a higher saving rate, s, leads to higher per capital capital in the steady state, k, and thus to a higher steady state per capita income, y should people increase their saving rate in order to increase output? would this be good for them? put another way: Does it mean that a higher saving rate is always better? what is the best steady state? define the best steady state as the one that maximizes consumption per capita, c

48 Consumption and steady state capital c = (1 s)zf (k ) = zf (k ) szf (k ) i zf (k) szf (k) k

49 Consumption and steady state capital c = (1 s)zf (k ) = zf (k ) szf (k )= zf (k ) (n + d)k i zf (k) (d + n)k szf (k) k k

50 Consumption and steady state capital c = (1 s)zf (k ) = zf (k ) szf (k )= zf (k ) (n + d)k i zf (k) (d + n)k c szf (k) k k To maximize c, need to find the biggest distance between zf (k) and (d + n)k.

51 The golden rule what are the effects of a higher s on c? higher s higher k higher y higher s smaller consumption share in output c }{{}? = (1 s) }{{} y }{{}

52 The golden rule what are the effects of a higher s on c? higher s higher k higher y higher s smaller consumption share in output c }{{}? = (1 s) }{{} y }{{} net effect in general is ambiguous

53 The golden rule what are the effects of a higher s on c? higher s higher k higher y higher s smaller consumption share in output c }{{}? = (1 s) }{{} y }{{} net effect in general is ambiguous due to the assumptions made on the production function

54 The golden rule what are the effects of a higher s on c? higher s higher k higher y higher s smaller consumption share in output c }{{}? = (1 s) }{{} y }{{} net effect in general is ambiguous due to the assumptions made on the production function c first increases, then decreases in s

55 The golden rule what are the effects of a higher s on c? higher s higher k higher y higher s smaller consumption share in output c }{{}? = (1 s) }{{} y }{{} net effect in general is ambiguous due to the assumptions made on the production function c first increases, then decreases in s golden rule of saving: s such that c is the highest how to find it? by choosing s maximize c = (1 s)zf (k ) = zf (k ) szf (k )= zf (k ) (n + d)k note: the last equality makes use of the steady state condition (szf (k ) = (n + d)k )

56 The golden rule c = (1 s)zf (k ) = zf (k ) szf (k ) = zf (k ) (n + d)k take the derivative with respect to s, and find s where c s = 0 where does s enter the right hand side?

57 The golden rule c = (1 s)zf (k ) = zf (k ) szf (k ) = zf (k ) (n + d)k take the derivative with respect to s, and find s where c s = 0 where does s enter the right hand side? only through k, as the steady state level of per capita capital increases in s c s =z f (k ) (n + d) k s s = zf (k ) k (n + d) k s s = ( zf (k ) (n + d) ) k s = 0 zf (k ) = n }{{}}{{ + d } MP k effective depreciation rate

58 The golden rule c = (1 s)zf (k ) = zf (k ) szf (k ) = zf (k ) (n + d)k take the derivative with respect to s, and find s where c s = 0 where does s enter the right hand side? only through k, as the steady state level of per capita capital increases in s c s =z f (k ) (n + d) k s s = zf (k ) k (n + d) k s s = ( zf (k ) (n + d) ) k s = 0 zf (k ) = n }{{}}{{ + d } MP k effective depreciation rate First find the k that maximizes c (where c k = 0). Then find the s that achieves this k using the steady state condition szf (k ) = (n + d)k.

59

60 The golden rule so the marginal product of capital should equal n + d in the steady state to achieve the highest per capita consumption we can find the saving rate that results in this specific steady state: s gr if the government, central planner would prescribe this saving rate, the economy would reach the steady state and the maximum possible consumption in practice: we can estimate MP K, we know n, we can calculate d we can actually calculate s gr should the government do something, try to impose this? 1. redistribution across generations 2. consumers save optimally (given their preferences) is there a market failure that prevents them from achieving the correct trade-off between current consumption and savings?

61 Data: Income per capita and the population growth rate GDP per capita and pop growth rate negatively correlated

62 Income per capita and the population growth rate in the model What is the effect of a higher population growth rate in the Solow model? a higher n increases the break-even investment, rotates the effective depreciation line up lower steady state per capita capital and output: k, y temporary decrease in the growth rate of output per worker long-run decrease in the level of output per worker no effect on the long-run growth rate of capital per worker and output per worker, which are equal to zero of course the growth rate of aggregate variables increases

63 Income per capita and the population growth rate in the model

64 Income per capita and technology in the model What is the effect of higher productivity in the Solow model? a higher productivity, z increases actual investment in the model the savings curve pivots up higher steady state per capita capital and output: k, y temporary increase in the growth rate of output per worker long-run increase in the level of output per worker no effect on the long-run growth rate of capital per worker and output per worker, which are equal to zero

65 Income per capita and technology in the model

66 Income per capita and technology in the model but what is the effect of a sustained productivity growth?