004: Macroeconomic Theory

Transcription

1 004: Macroeconomic Theory Lecture 16 Mausumi Das Lecture Notes, DSE October 28, 2014 Das (Lecture Notes, DSE) Macro October 28, / 24

2 Solow Model: Golden Rule & Dynamic Ineffi ciency In the last class we have defined the golden rule savings ratio s g and the corresponding steady state capital-labour ratio k g in the context of Solow model. The golden rule k g represents one particular steady state (among all the possible steady states - corresponding to various values of the savings ratio s [0, 1]) which maximises the steady state level of per capita consumption, given by: c (s) = f (k (s)) sf (k (s)) = f (k (s)) (n + δ) k (s). [Using the definition of k ] Accordingly, the golden rule capital-labour ratio that maximises the above expression is defined by the following equation: k g : f (k ) = (n + δ). Das (Lecture Notes, DSE) Macro October 28, / 24

3 Digrammatic Representation of the Golden Rule Steady State: The point (k g, c g ) in some sense represents the best or the most desirable steady state point (although in the absence of an explicit utility function, such qualifications remain somewhat vague). Das (Lecture Notes, DSE) Macro October 28, / 24

4 Alternative Digrammatic Representation of the Golden Rule Steady State: There are many possible steady states to the left and to the right of k g - associated with various other savings ratios. Das (Lecture Notes, DSE) Macro October 28, / 24

5 Golden Rule & the Concept of Dynamic Ineffi ciency Importantly, all the steady states to the right of k g are called dynamically ineffi cient steady states. From any such point one can costlessly move to the left - to a lower steady state point - and in the process enjoy a higher level of current consumption as well as higher levels of future consumption at all subsequent dates. (How?) Notice however that the steady states to the left of k g are not dynamically ineffi cient. (Why not?) Das (Lecture Notes, DSE) Macro October 28, / 24

6 Cause of Dynamic Ineffi ciency in Solow Model Notice that dynamic ineffi ciency occurs because people oversave. This possibility arises in the Solow model because the savings ratio is exogenously given - it is not chosen through households optimization process. If the steady state of an economy indeed happens to be dynamically ineffi cient, then it justifies an active, interventionist role of the government in the Solow model - even though government cannot affect the long run rate of growth of the economy. Das (Lecture Notes, DSE) Macro October 28, / 24

7 Limitations of the Solow Growth Model: Even though the Solow model is supposed to be a growth model - it cannot really explain long run growth: The per capita income does not grow at all in the long run; The aggregate income grows at an exogenously given rate n, which the model does not attempt to explain. The steady state in the Solow model might be dynamically ineffi cient. It is not clear why households will not correct this ineffi cinency by choosing their savings ratio optimally. But this latter possibility is simply not allowed in the Solow model. Das (Lecture Notes, DSE) Macro October 28, / 24

8 Extensions of Solow Growth Model: The basic Solow growth model has subsequently been extended to counter some of these critisisms. The primary challenge is to retain the basic result of the Solow model (namely, existence of a unique and globally stable steady state) while relaxing various restrictive assumptions. We now look at two such extensions: 1 Solow Model with Technological Progress: This extension allows the per capita income to grow in the long run; developed by Solow himself (Solow (1957)). 2 Neoclassical Growth Model with Optimizing Households: This extension allows the households to choose their consumption/savings behaviour optimally over infinite horizon; developed independently by Cass (1965) & Koopmans (1965). Das (Lecture Notes, DSE) Macro October 28, / 24

9 Solow Model with Exogenous Technological Progress: Let us now introduce a productivity-specific parameter into our Solovian firm-specific production function: Y it = F (K it, N it, A t ); F A > 0, where F satisfies all the standard Neoclassical properties specified earlier. The term A t captures the state of the technology at time t. Since all firms have access to identical technology, this technology-specific parameter is the same for all firms (hence no i-subscript here). The assumption of identical firms and CRS implies that the aggregate production function will also take similar form: Y t = F (K t, N t, A t ); F A > 0. Technological progress implies that the productivity-specific term, A t, increases in value over time. Thus with the same amount of labour and capital, the economy can now produce greater amount of output. Das (Lecture Notes, DSE) Macro October 28, / 24

10 Different Types of Technological Progress: Technological progress can be of three types: 1 Labour Augmenting or Harrod-Neutral: Technical improvement enhances the productivity of labour alone: Y t = F (K t, A t N t ). 2 Capital Augmenting or Solow-Neutral: Technical improvement enhances the productivity of capital alone: Y t = F (A t K t, N t ). 3 Capital & Labour Augmenting or Hicks-Neutral: Technical improvement enhances the productivity of both capital and labour in equal proportion and thus augments total factor productivity: Y t = F (A t K t, A t N t ) = A t F (K t, N t ). Notice that with Cobb-Douglas Production Function, all the three notions of technical progress are equivalent. (Prove this.) Das (Lecture Notes, DSE) Macro October 28, / 24

11 Characterization of Technological Progress in terms of Isoquants: Technological progress implies that the isoquant for some given level of output will shrink inward (same output can now be produced with less inputs). The exact nature of the shift depends on the type of technological progress. With Harrod-neutral technological progress, along any ray through the origin (say along the 45 O line) the curve becomes steeper. (Prove this.) Das (Lecture Notes, DSE) Macro October 28, / 24

12 Characterization of Technological Progress in terms of Isoquants (Contd.): With Solow-neutral technological progress, along any ray through the origin (say along the 45 O line), the curve becomes flatter. (Prove this.) Das (Lecture Notes, DSE) Macro October 28, / 24

13 Characterization of Technological Progress in terms of Isoquants (Contd.): With Hicks-neutral technological progress, along any ray through the origin (say along the 45 O line), the slope of the curve remains unchanged. (Prove this.) Das (Lecture Notes, DSE) Macro October 28, / 24

14 Technological Progress and Balanced Growth: Modern Growth Theory often focuses on Balanced Growth Path: a scenario when every variable in the economy grows at some constant rate (not necessarily equal for all variables). Recall that steady state is also a special case of balanced growth (when the growth rate is constant at 0). It can be shown that only Harrod-neutral technological progress can generate balanced growth path. (Proof is available in Acemoglu, pages Will not be discussed in class; is not part of the syllabus). Henceforth, we shall therefore restrict our analysis only to Harrod-neutral technological progress. (Question: Does the steady state of the original Solow model (without technological progress) satisfy the requirements of a balanced growth path?) Das (Lecture Notes, DSE) Macro October 28, / 24

15 Solow Model with Harrod-Neutral Technological Progress: Suppose all assumptions of the original Solow model remain unchanged, except that we now have a firm-specific production technology, given by: Y it = F (K it, A t N it ). The corresponding aggregate production technology is given by Y t = F (K t, A t N t ) F (K t, ˆN t ), where we denote the productivity-augmented labour term: A t N t ˆN t (effective labour). Labour productivity increases automatically over time (like manna from heaven ) at an exogenous rate m: A t+1 = (1 + m)a t. Each firm now equates the marginal product of effective labour with the wage rate per unit of effective labour (ŵ) and the marginal product of capital with the rental rate of capital (r). Das (Lecture Notes, DSE) Macro October 28, / 24

16 Solow Model with Harrod-Neutral Technological Progress (Contd.): CRS and identical firms implies that the firm-specific marginal product and the (social) marginal product for the aggregate economy are the same. Thus we get the aggregate demand functions for effective labour and capital as: F ˆN (K it, ˆN it ) = F ˆN (K t, ˆN t ) = ŵ t ; F K (K it, ˆN it ) = F K (K t, ˆN t ) = r t. At the beginning of any time period t, the economy starts with a given stock of population (N t ), a given stock of capital (K t ) and a given level of labour productivity (A t ). The wage rate and rental rate adjust so that that there is full employment of all the factors at every point of time t. Das (Lecture Notes, DSE) Macro October 28, / 24

17 Capital- Effective Labour Ratio & Output per unit of Effective Labour: Using the CRS property, we can write: ŷ t Y t = F (K t, A t N t ) = F ˆN t A t N t ( ) Kt, 1 f ( ˆk t ), A t N t where ŷ t represents output per unit of effective labour, and ˆk t represents the capital-effective labour ratio in the economy at time t. Using the relationship that F (K t, ˆN t ) = ˆN t f ( ˆk t ), we can easily show that: F ˆN ( ˆN t, K t ) = f ( ˆk t ) ˆk t f ( ˆk t ) = ŵ t ; F K ( ˆN t, K t ) = f ( ˆk t ) = r t. [Derive these two expressions yourselves]. Das (Lecture Notes, DSE) Macro October 28, / 24

18 Properties of the Reduced Form Production Function: Once again, given the properties of the aggregate production function, one can derive the following properties of the reduced form production function (in terms of effective labour) f ( ˆk): (i) f (0) = 0; (ii) f ( ˆk) > 0; f ( ˆk) < 0; (iii) Limf ( ˆk) = ; Lim f ( ˆk) = 0. ˆk 0 ˆk Finally, using the definition that ˆk t K t A t N t, we can write ˆk t+1 K t+1 A t+1 N t+1 = sf (K t, A t N t ) + (1 δ)k t (1 + m)a t (1 + n)n t ˆk t+1 = sf ( ˆk t ) + (1 δ) ˆk t (1 + m)(1 + n) g(ˆk t ). (1) Das (Lecture Notes, DSE) Macro October 28, / 24

19 Dynamics of Capital-Effective Labour Ratio: Equation (1) represents the basic dynamic equation in the discrete time Solow model with technological progress. Once again we use the phase diagram technique to analyse the dynamic behaviour of ˆk t. In plotting the g(ˆk t ) function, note: g(0) = g (k) = g (k) = sf (0) + (1 δ).0 = 0; (1 + n) 1 [ sf ( ˆk t ) + (1 δ) ] > 0; (1 + m)(1 + n) 1 (1 + m)(1 + n) sf ( ˆk t ) < 0. Das (Lecture Notes, DSE) Macro October 28, / 24

20 Dynamics of Capital-Effective Labour Ratio (Contd.): Moreover, as before, Lim g ( ˆk t ) = ; ˆk t 0 Lim g ( ˆk t ) = ˆk t (1 δ) (1 + m) (1 + n) < 1. We can now draw the phase diagram for ˆk t : Das (Lecture Notes, DSE) Macro October 28, / 24

21 Dynamics of Capital-Effective Labour Ratio (Contd.): From the phase diagram we can identify two possible steady states: (i) ˆk = 0 (Trivial Steady State); (ii) ˆk = ˆk > 0 (Non-trivial Steady State). As before, ignoring the non-trivial steady state, we get a unique non-trivial steady state, ˆk, which is globally asymptotically stable: starting from any initial capital-effective labour ratio ˆk 0 > 0, the economy would always move to ˆk in the long run. Implication: In the long run, output per unit of effective labour: ŷ t f ( ˆk t ) will be constant at f ( ˆk ). Das (Lecture Notes, DSE) Macro October 28, / 24

22 Long Run Growth Implications of Solow Model with Technological Progress: How about the long run growth rate of per capita income and aggregate income? Notice that now per capita income grows at a constant rate m, While aggregate income grows at a constant rate (m + n). Thus incorporating technical progress in the Solow model indeed allows us to counter the critisism that per capita income does not grow in the long run. Notice however that both the growth rates are still exogenous; we still do not know what determines m and n. Thus Solow model still does not tell us what determines long run economic growth!! Das (Lecture Notes, DSE) Macro October 28, / 24

23 Long Run Growth of Per Capita Income without Technological Progress? Exogenous technological progress is not a very satisfactory way to generate long run growth of per capita income in the Solow model. Without a proper theory to explain this phenomenon, it remains a mere technical exercise. Can we have long run growth of per capita income in the Solow model - even without exogenous technical progress? The answer is: yes, but only if you allow some of the Neoclassical properties of the production function to be relaxed. Recall that the long run constancy of the per capita income in the Solow model arises due to the strong uniqueness and stability property of the steady state - which in turn depends on two key assumptions: the property of diminishing returns and the Inada Conditions. The simplest way to generate long run growth of per capita income in the Solow model is to allow for relaxation of one of the Inada conditions. Das (Lecture Notes, DSE) Macro October 28, / 24

24 An Exercise: Find an example of a well-known production function which satisfies all the standard Neoclassical properties except the Inada conditions and explore the possibility of perpetual (long run) growth in per capita income in the context of this production function. Das (Lecture Notes, DSE) Macro October 28, / 24