202: Dynamic Macroeconomics

Transcription

1 202: Dynamic Macroeconomics Solow Model Mausumi Das Delhi School of Economics January 14-15, 2015 Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

2 Economic Growth In this course we seek to answer the following three questions: What explains the per capita GDP growth (or lack of it) of any particular country? What explains the vast divergence in growth patterns across the world? Is there any scope for government policy in influencing the growth path of a country? We shall answer these questions in terms of three major paradigms: Neoclassical Growth Theory Endogenous Growth Theory Institution-based Models of Growth Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

3 Neoclassical Theory of Economic Growth: The first Neoclassical model of growth was developed by Solow (QJE, 1956). It is based on a classical system of aggregative macro equations which are not micro-founded. Later we are going to extend the Solow model to allow for optimizing behaviour by households. Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

4 Solow Model: The Economic Environment A closed economy producing a single final commodity which is used for consumption as well as for investment purposes (i.e, as capital.) At the beginning of any time period t, the economy starts with a historically given total endowment of labour stock (N t ) and a historically given aggregate capital stock (K t ). There are H identical households in the economy and the labour and capital ownership is equally distributed across all these households. At the beginning of any time period t, the households offer their labour and capital (inelastically) to the firms. The competitive firms then carry out the production and total output produced is distributed as factor incomes to the households at the end of the period Households consume a constant fraction of their total income and save the rest. The savings propensity is exogenously fixed (by norm, convention etc.), denoted by s (0, 1). Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

5 Solow Growth Model (Contd.) Assumption: All savings are automatically invested, which augments the capital stock in the next period. A crucial implication of the above assumption: It is the households who make the investment decisions; not firms. That is, households are the owners of capital stocks, not firms. Firms simply rent in the capital from the households for production and distributes the output as wage and rental income to the households at the end of the period. Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

6 Solow Growth Model: Production Side Story The economy is characterized by S idenical firms. Since all firms are identical, we can talk in terms of a representative firm. The representative firm i is endowed with a standard Neoclassical production technology: Y it = F (N it, K it ), which satisfies all the Neoclassical properties, namely 1 Diminishing marginal product of each factor (or law of diminishing returns): F N, F K > 0; F NN, F KK < 0; 2 Constant Returns to Scale (CRS): F (λn it, λk it ) = λf (N it, K it ); 3 The Inada Conditions: lim F N (N it, K it ) = lim F K (N it, K it ) = ; N it 0 K it 0 lim F N (N it, K it ) = lim F K (N it, K it ) = 0 N it K it In addition, F (0, K it ) = F (N it, 0) = 0, i.e., both inputs are essential in the production process. Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

7 Production Side Story (Contd.) The firms operate in a competitive market structure and take the market wage rate (w) and rental rate for capital (r) in real terms as given. The firm maximises its current profit: Π it = F (N it, K it ) wn it rk it. Static (period by period) optimization by the firm yields the following FONCs: (i) F N (N it, K it ) = w. (ii) F K (N it, K it ) = r. Identical firms and CRS technology imply that firm-specific marginal products and economy-wide (social) marginal products (derived from the corresponding aggregate production function) of both labour and capital would be the same (Why?). Thus F N (N it, K it ) = F N (N t, K t ); F K (N it, K it ) = F K (N t, K t ). Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

8 Production Side Story (Contd.) Thus we get the familiar demand for labour schedule for the aggreagte economy at time t, and a similarly defined demand for capital schedule at time t as: N D : F N (N t, K t ) = w t ; K D : F K (N t, K t ) = r t. Recall that the supply of labour and that of capital at any point of time t is historically given at N t and K t respectively. Assumption: The market wage rate and the rental rate for capital, w t and r t, adjust so that the labour market and the capital market clear in every time period. Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

9 Determination of Market Wage Rate & Rental Rate of Capital at time t: Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

10 Distribution of Aggregate Output: Recall that the firm-specific production function is CRS; hence so is the aggregate production function. We know that for any constant returns to scale (i.e., linearly homogeneous) function, by Euler s theorem: F (N t, K t ) = F N (N t, K t )N t + F K (N t, K t )K t = w t N t + r t K t. This implies that after paying all the factors their respective marginal products, the entire output gets exhausted, confirming that firms indeed earn zero profit. (One important side-result: Perfect Competition and CRS go hand in hand. With any other assumption about returns to scale (DRS or IRS), either the factors cannot be paid their marginal products, or firms will not earn zero profit.) Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

11 Dynamics of Capital and Labour: Recall that the capital stock over time gets augmented by the savings/investment made by the households. Also recall that households are identical and they invest a fixed proportion (s) of their income. Hence aggegate savings (& investment) in the economy is given by : S t = I t = sy t ; 0 < s < 1. Let the existing capital depreciate at a constant rate δ : 0 δ 1. Thus the capital accumulation equation in this economy is given by: dk dt = I t δk t = sy t δk t i.e., dk dt = sf (N t, K t ) δk t, (1) While labour stock increases due to population growth at a constant exogenous rate n: 1 dn = n. N t dt (2) Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

12 Dynamics of Capital and Labour (Contd): Equations (1) and (2) represent a 2X2 system of differential equations, which we can directly analyse to determine the time paths of N t and K t, and therefore the corresponding time path of Y t. However, given the conditions on the production function, we can transform the 2X2 system into a single-variable differential equation - which is easier to analyse. We shall follow this latter method here. Define a new variable: Then 1 dk k t dt i.e, dk dt k t K t N t. = 1 K t dk dt 1 N t dn dt = sf (N t, K t ) δk t = k t [ sf (Nt, K t ) δk t K t K t ] n. n Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

13 Dynamics of Capital and Labour (Contd): But instead of simplifying things, it seems to have compounded the problem because now we have three time-dependent variables: N t, K t and k t. We have to somehow transform the RHS into a function of a single varaible:k t. Here the neoclassical properties of the production function come in handy. Recall that the Neoclassical properties are: 1 Diminishing marginal product of each factor: F N, F K > 0; F NN, F KK < 0; 2 Constant Returns to Scale (CRS): F (λn, λk ) = λf (N, K ); 3 The Inada Conditions: lim N (N, K ) N 0 = lim K (N, K ) = ; K 0 lim N (N, K ) N = lim K (N, K ) = 0 K In addition, F (0, K ) = F (N, 0) = 0. Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

14 Capital-Labour Ratio & Per Capita Production Function: Using the CRS property, we can write: y t Y t = F (N t, K t ) = F N t N t ( 1, K ) t f (k t ), N t where y t represents per capita output, and k t represents the capital-labour ratio (or the per capita capital stock) in the economy at time t. The function f (k t ) is often referred to as the per capita production function. Notice that using the relationship that F (N t, K t ) = N t f (k t ), we can easily show that: F N (N t, K t ) = f (k t ) k t f (k t ); F K (N t, K t ) = f (k t ). [Derive these two expressions yourselves]. Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

15 Properties of Per Capita Production Function: Given the properties of the aggregate production function, one can derive the following properties of the per capita production function: (i) f (0) = 0; (ii) f (k) > 0; f (k) < 0; (iii) Lim k 0 f (k) = ; Lim k f (k) = 0. Condition (i) indicates that capital is an essential input of production; Condition (ii) indicates diminishing marginal product of capital; Condition (iii) indicates the Inada conditions with respect to capital. Finally, using the definition that k t K t N t, we can write 1 dk k dt = sf (N t, K t ) δk t n = sf (k t) (δ + n) K t k t dk dt = sf (k t) (δ + n)k t g(k t ). (3) Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

16 Dynamics of Capital-Labour Ratio: Equation (3) represents the basic dynamic equation in the Solow model. (Interpretation?) Notice that Equation (3) represents a single non-linear differential equation in k t. Once again we use the phase diagram technique to analyse the dynamic behaviour of k t. Recall that to draw the phase diagram of a single differential equation, we first plot the g(k t ) function with respect to k t. Then we identify its possible points of intersection with the horizontal axis - which denote the steady states of the system. (What is a steady state?) In plotting the g(k t ) function, note: g(0) = sf (0) 0 = 0; g (k) = sf (k t ) (δ + n) 0 according as sf (k t ) (δ + n) i.e., k t ˆk such that f ( ˆk ) = g (k) = sf (k) < 0. (δ + n) s Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

17 Dynamics of Capital-Labour Ratio (Contd.): Moreover, due to Inada Conditions, Lim (k) k 0 = slimf (k) (δ + n) = ; k 0 Lim (k) k = s Lim (k) (δ + n) = (δ + n) < 0. k We can now draw the phase diagram for k t : (How? What should it look like?) Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

18 Dynamics of Capital-Labour Ratio (Contd.): From the phase diagram we can identify two possible steady states: (i) k = 0 (Trivial Steady State); (ii) k = k > 0 (Non-trivial Steady State). Since an economy is always assumed to start with some positive capital-labour ratio (however small), we shall ignore the non-trivial steady state. The economy has a unique non-trivial steady state, given by k. Moreover, k is globally asymptotically stable: starting from any initial capital-labour ratio k 0 > 0, the economy would always move to k in the long run. (Law of diminishing returns and Inada conditions at work. But which one generates what?) Implications: In the long run, per capita output: y t f (k t ) will be constant at f (k ) and aggregate output Y t N t f (k t ) will be growing at the same rate as N t (namely at the exogenously given rate n). Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

19 Some Long Run Implictions of Solow Growth Model: Notice that the non-trivial steady state k is defined as: k : sf (k ) = (δ + n)k f (k ) k = n + δ. s Total differentiating and using the properties of the f (k) function, it is easy to show that, dk ds dk > 0; dn < 0; dk dδ < 0. A higher savings ratio generates a higher level of per capita output in the long run; A higher rate of grwoth of population generates a lower level of per capita output in the long run; A higher rate of depreciation generates a lower level of per capita output in the long run.[verify these.] Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

20 Some Long Run Implictions of Solow Growth Model (Contd.): But these are all long run level effects. What would be the impact on the long run growth? Notice that among all these parameters, only a change in the population growth rate (n) would have an impact upon long run growth. In particular, a higher rate of growth of population generates a higher rate of growth of aggregate income in the long run (although the long run rate of growth of per capita income is zero in all the three cases.) A change in the other two parameters (s and δ) has only level effect and no growth effect in the long run. This also implies that if the government tries to influence the savings rate with an objective of increasing the growth rate (by imposing a tax on consumption and then investing the tax proceeds to augment the capital stock which is redistributed to the households in the next period) such a policy would have no long run impact. Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

21 Solow Model: Long Run Growth Implications - Summary To summarise, in the benchmark Solow growth model: The per capita income remains constatnt at f (k ) - the exact level being determined by various parameters (s, n, δ). The aggregate income grows at a constant rate - given by the exogenous rate of growth of population (n). So to go back to the three question that we seek to answer: What explains the per capita GDP growth (or lack of it) of any particular country? Solow s answer: There is no growth of per capita income!!! What explains the vast divergence in growth patterns across the world? Solow s answer: Population growth!!! Is there any scope for government policy in influencing the growth path of a country? Solow s answer: No!!! Obviously these answers do not sit very well with the empirical facts that we have discussed earlier. Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

22 Transitional Dynamics in Solow Growth Model: However these are all long run (steady state) conclusions; they hold only as t. Starting from a given initial capital-labour ratio k 0 ( = k ), it will obviously take the economy some time before it reaches k. What happens during these transitional periods? In particular, what would be the rate of growth of per capita income and that of aggregate income in the short run - when the economy is yet to reach its steady state? Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

23 Transitional Dynamics in Solow Growth Model (Contd.): When the economy is out of steady state, the rate of growth of capital-labour ratio is given by: γ k 1 k t dk dt = sf (k t) (n + δ)k t k t 0 according as k t k. Moreover, dγ k dk = s [f (k) kf (k)] [k] 2 < 0. (Why?) In other words, during transition, the higher is the capital-labour ratio of the economy, the lower is its (short run) growth rate. Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

24 Implications of Solow Growth Model: Absolute Convergence The above result has important implications for cross country growth comparisons. It implies that the during transition, poorer countries (with low k 0 ) will grow at a faster rate than the rich countries (with high k 0 ); and eventually they will converge to the same level of per capita income (Absolute Convergence). This proposition (not surprisingly) has been strongly rejected by data. In fact we know (from Danny Quah s twin peak analysis) that the reality is actually opposite: richers countries have remained rich and poorer countries have remained poor and there is no significant tendency towards convergence - even when one looks at long run time series data. Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

25 Further Implications of the Solow Growth Model: Conditional Convergence The proposition of absolute convergence of course pre-supposes that the underlying parameters for all economies (rich and poor alike) are the same. If we allow rich and poor countries to have different values of s, δ, n etc. (which is plausible), then the Solow model generates a much weaker prediction of Conditional Convergence. Conditional Convergence states that a country will grows faster the further away it is from its own steady state. An alternative (and more meaningful) statement of Conditional Convergence: Among a group of countries which are similar (similar values of s, δ, n etc.), the relatively poorer ones will grow faster and eventually the per capita income of all such countries will converge. This weaker hypothesis in generally supported by data, but is not very useful in explaining the persistent differences in per capita income amongst the rich and the poor countries. Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

26 The Benchmark Solow Model: Some Thought Experiments Heterogenous Households: Suppose households differ in terms of their asset holdings. To fix ideas, suppose there are two classes of people - the capitalists (who own the entire capital stock in the economy and earn only capital income), and the workers (who provide the entire labourforce in the economy and earn only wage income). Let the savings propensity of the capitalists be s π while that of the workers is s ω, while s π > s ω. For convenience let us assume that s ω = 0. Rest of the model remains the same. Derive the dynamic equation for capital-labour ratio (k) for this economy and analyse the corresponding growth conclusions - in the long run as well as short run. If the government taxes part of the rental income in every period (say at a fixed rate τ) and redistributes the tax proceeds to the workers in the same period, how does it affect the growth rate? Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

27 The Benchmark Solow Model: Some Thought Experiments (Contd.) Endogenous Population Growth: Suppose the rate of growth of population (n), instead of being exogenous, depended on the per capita income (y) in the following way: When y = 0, n(y) = 0. As y increases n(y) initally increases (as mortality rate goes down at a faster rate than the fertility rate); then after a threshold level of per capita income, say ŷ, it begins to fall (as mortality rate becomes negligible, but now fertility rate starts declining), and eventually reaches zero again. Rest of the model remains the same. Characterise the dynamic equation for per capita income (y) for this economy and analyse the corresponding growth conclusions. Would all eceonomies now attain the same long run per capita income in the long run - no matter what their initial position is? Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28

28 Solow Model: Reference Robert Barro & Xavier Sala-i-Martin: Economic Growth, 2004 (2nd Edition), The MIT Press, Chapter 1. Das (Delhi School of Economics) Dynamic Macro January 14-15, / 28