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004: Macroeconomic Theory Lecture 22 004: Macroeconomic Theory Lecture 22 Mausumi Das Lecture Notes, DSE November 11, 2014 Das (Lecture Notes, DSE) Macro November11,2014 1/12 AK Production Technology: First Step Towards Endogenous Growth? In our earlier discussion we have seen that we can have long run growth of per capita income in the Solow model without technical progress if we relax some of the properties of the neoclassical production function. In particular, steady growth along a balanced growth path is possible if the production function exhibits non-diminishing returns. A specific example of such non-diminishing returns technology is the AK technology where output is a linear function of capital: Yt = AKt . The AK model is the precursor to the latter-day ‘endogenousgrowth theory’which (unlike Solow) explains technological progress in terms endogenous factors within the economy. When technology interacts with the factor accumulation, there is no reason why the production function would exhibit diminishing returns. Das (Lecture Notes, DSE) Macro November11,2014 2/12 Justifications for AK Technology: There are many justifications as to why the ‘aggregate’production technology might be linear. Here we look at three distinct examples - each providing a different justification as to why diminishing returns might not work: 1 Fixed Coeffi cient Technogy - exhibiting complemetarity between factors of production; 2 Production Technology with Learning-by-Doing and Knowledge Spillover (Frankel-Romer); 3 Production Technology with Producible Inputs (Barro). Das (Lecture Notes, DSE) Macro November11,2014 3/12 Justification for AK Technology: Leontief Production Function The first example of an AK technology is of course the Fixed Coeffi cient (Leontief) Production Function: Yt = min [aKt , bNt ] , where a and b are the constant coeffi cients representing the productivity of capital and labour respectively. Suppose the economy starts with a (historically) given stock of capital and a given amount of labour force at time t. Then fixed coeffi cient technology implies: Yt = aKt if aKt < bNt (capital constrained economy); Yt = bNt if bNt < aKt (labour constrained economy). In the first case, the production technology would in effect be AK - although notice that labour is still used in the production process. Das (Lecture Notes, DSE) Macro November11,2014 4/12 Justification for AK Technology: Learning by Doing & Knowledge Spillover A more nuanced justification was provided by Frankel (1962), which was later exploited by Romer (1986) in developing the first model of endogenous growth. Consider an economy with S identical firms - each having access to an identical firm-specific technology: a 1 a Yi = A¯ t F (Kit , Nit ) A¯ t (Kit ) (Nit ) ; 0 < a < 1. Note that the firm-specific production function exhibits all the neoclassical properties. The term A¯ t represents the current state of the technology in the economy, which is treated as exogenous by each firm. Frankel relates the A¯ t term to the aggregate capital labour-ratio in the economy - due to ‘learning by doing’: K A¯ = g t ; g > 0; where K = SK ; N = SN . t N 0 t it it t Das (Lecture Notes, DSE) Macro November11,2014 5/12 AK Technology: Frankel-Romer Explanation Without any loss of generality, let us assume: b Kt Kt g = ; b > 0, N N t t Corresponding Aggregate Production Function: ¯ a 1 a Yt = ∑ Yit = S At (Kit ) (Nit ) a h 1 a i = A¯ t (SKit ) (SNit ) a 1 a = A¯ t (Kt ) (Nt ) . Notice that A¯ t is the total factor productivity term - which is given for each firm, but not so for the aggregate economy. Replacing the value of A¯ t in the aggregate production technology: a 1 a a+b 1 a b Yt = A¯ t (Kt ) (Nt ) = (Kt ) (Nt ) . In the special case where a + b = 1, the aggregate production technology is AK. Das (Lecture Notes, DSE) Macro November11,2014 6/12 Justification for AK Technology: Producible Inputs Another interesting justification for the AK Technology comes from Barro (2004). Barro considers a production technology which uses capital and human capital (skilled labour) as the two factors of production. The important difference between this production function and the Solovian production function is that now both the factors are ‘producible’inputs. Just as one can invest in physical capital formation to augment capital stock, similarly, one can invest in skill formation (education) to augment the stock of human capital. In fact Barro makes the extreme assumption that physical capital and human capital can be instantaneously converted into each other - one to one. However we do not need such a strong assumption. Let us assume instead that physical capital and human capital are ex ante convertible, but not ex post. Das (Lecture Notes, DSE) Macro November11,2014 7/12 AK Technology: Barro Explanation (Contd.) Suppose the economy starts with some historically given stocks of physical capital and human capital at time 0. Then these stocks cannot be converted into one another immediately. But in deciding about tommorrow’sstock, a household can invest in either form of capital. One unit of investment in terms of final commodity can be converted either into one unit of physical tomorrow or into one unit of human capital tomorrow. Let us assume that there is 100% deprciation of both types of capital so that the entire stock of tomorrow’sphysical as well as human capital must come from today’sinvestment. Let us also endow the households with perfect foresight - such that in making their portfolio decision toady they can perfectly guess the returns to the two factors that will prevail tomorrow. What would be these returns? For that we have to turn to the production side story. Das (Lecture Notes, DSE) Macro November11,2014 8/12 AK Technology: Barro Explanation (Contd.) As before, consider an economy with S identical firms - each having access to an identical firm-specific technology: a 1 a Yit = F (Kit , Hit ) (Kit ) (Hit ) ; 0 < a < 1. (One can think of Hit as At N¯ where At is the skill level of per unit of labour and N¯ is the fixed labourforce.) Corresponding aggregate production technology: a 1 a a 1 a Yt = ∑ Yit = S (Kit ) (Hit ) = (SKit ) (SHit ) a 1 a Yt = (Kt ) (Ht ) ) Perfect Competition would imply that the two factors are paid their respective marginal products: a 1 1 a (rK )t = a (Kt ) (Ht ) ; a a (rH ) = (1 a) (Kt ) (Ht ) . t Das (Lecture Notes, DSE) Macro November11,2014 9/12 AK Technology: Barro Explanation (Contd.) Since physical and human capital are convertible to each other ex ante, and since households have perfect foresight so that they can foresee these returns when they make their portfolio choice, at any point of time after the initial point, returns from the two forms of capitals must be the same. Thus, (rK )t = (rH )t for all t > 0 a 1 1 a a a i.e., a (Kt ) (Ht ) = (1 a) (Kt ) (Ht ) Ht 1 a i.e., = for all t > 0. K a t Thus for all t > 0 the reduced form aggregate technology would look as follows: 1 a 1 a a 1 a Ht 1 a Yt = (Kt ) (Ht ) = Kt == Kt . K a t Once again the aggregate production technology is de-facto AK. Das (Lecture Notes, DSE) Macro November11,2014 10/12 AK Technology: General Insight Notice that the AK explanations do not necessarily rule out role of labour in the production process. But the role of labour becomes implicit - hidden within a broader concept of capital. Even though the AK models hint at some explanation for the technology parameter (learning by doing, skill formation), they do not model technological progress explicitly. The subsequent endogenous growth models (developed in 1980s) explicitly specify a process of technological change in terms of R&D, education etc. They also move away from the CRS technology to allow for IRS - which necessitates a movement away from competitive market structure. Even more recent developments in growth theory go beyond technology and explain growth in terms of institutions, political economy, income distribution etc. These are however beyond the purview of the current course. Das (Lecture Notes, DSE) Macro November11,2014 11/12 Reference for AK Technology: 1 P. Aghion & P. Howitt (1998): Endogenous Growth Theory, pages 24-26. 2 R.Barro & X. Sala-i-Martin (2004): Economic Growth, pages 211-213. (Scanned pages avilable in the CDE server Q: Course C004) n n Das (Lecture Notes, DSE) Macro November11,2014 12/12.
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