Lecture 2: Balanced Growth

Transcription

1 Lecture 2: Balanced Growth Fatih Guvenen September 21, 2015 Fatih Guvenen Balanced Growth September 21, / 12

2 Kaldor s Facts 1 Labor productivity has grown at a sustained rate. 2 Capital per worker has also grown at a sustained rate. 3 The real interest rate or return on capital has been stable. 4 The ratio of capital to output has also been stable. 5 Capital and labor have captured stable shares of national income. 6 Among the fast growing countries of the world, there is an appreciable variation in the rate of growth of the order of 2 5 percent. Fatih Guvenen Balanced Growth September 21, / 12

3 Functional Forms for Balanced Growth max 1X t=0 t [u (C t, 1 N t )] (1) s.t. C t + K t+1 (1 K ) K t apple F t (K t, N t ). (2) u is strictly increasing, concave, twice continuously differentiable and satisfies the Inada conditions; F t is concave, twice continuously differentiable, satisfies the Inada conditions; and both capital and labor are essential inputs: F t (0, ) =F t (, 0) =0. + F t is constant returns-to-scale so that it admits natural aggregation. Definition Balanced growth is a situation in which C t, Y t, I t, K t and the wage rate grow at constant, but possibly different rates. Fatih Guvenen Balanced Growth September 21, / 12

4 Optimal Allocations Optimal allocations satisfy the following equations: t D 1 u (C t, 1 N t )= t (3) t D 2 u (C t, 1 N t )= t D 2 F t (K t, N t ) (4) t+1 [D 1 F t+1 (K t+1, N t+1 )+(1 K )] = t (5) F t (K t, N t )+(1 K ) K t K t+1 C t = 0 (6) lim tk t+1 = 0. (7) t!1 Fatih Guvenen Balanced Growth September 21, / 12

5 Theorem Three conditions jointly deliver a balanced growth path: 1 F is constant returns to scale. 2 Technology is labor augmenting. 3 Preferences satisfy one of the following two conditions: 1 Class 1: u (C, 1 N) = C1 c v (1 N) if 0 < c and c 6= 1 (8) 1 c where v(l) is increasing and concave if convex if c > 1. c < 1 and decreasing and 2 Class 2: u (C, 1 N) =log C + v (1 N) if c = 1 (9) with v (1 N) is increasing and concave. Fatih Guvenen Balanced Growth September 21, / 12

6 King-Plosser-Rebelo (2002) KPR goes through the proof step by step. Understanding these mechanics is important because in all likelihood you are going to study a model where the utility function contains more variables and the environment is more complex than the one specified above. In such cases, you can check whether your model satisfies balanced growth path. Fatih Guvenen Balanced Growth September 21, / 12

7 BGP with More General Preferences One trick that economists often use is now described and it can be extended to other applications. To see how this works, once again, consider the social planner s problem in (1)-(2), for the special case when preferences are given by u (C t, 1 N t )= C1 t 1 + t v (1 N t ). (10) Let Y t = A (K t ) (Z t N t ) 1 feasible: so along the BGP the following is Y = K = C = I = Z N = 1. where for any variable a, a = a t+1 a t. Fatih Guvenen Balanced Growth September 21, / 12

8 The Transformed Economy Express preferences and technology in terms of variables that will be constant in the steady state. To this end, we normalize all growing variables by Z t : c = C Z, k = K Z, g = G Z, y = Y Z. Start with (2), and divide both sides by Z t and multiply by Z t+1 /Z t+1 : K t+1 Z t+1 Z t+1 Z t = (1 K ) K t + A t F (K t, Z t N t ) C t Z t ) Z k t+1 =(1 K ) k t + A t F (k t, N t ) c t. (11) Fatih Guvenen Balanced Growth September 21, / 12

9 The Transformed Economy First, the assumptions we made on u and F t above ensures an interior solution. Combining (3) and (4), and substituting in u from (10) yields tv 0 (1 N t ) C t = D 2 F t (K t, N t ) (12) ) t = D 2 F t (K t, N t ) C t v 0 (1 N t ). The first term on RHS is the MPL and will grow along the BGP at rate Z, whereas N t is constant and so is v 0 (1 N t ). Hence, the RHS is growing at rate Z Z = 1 Z, so t must also grow at this same rate. Thus, letting t = t 1 1 Z, allows the model to deliver BGP. Fatih Guvenen Balanced Growth September 21, / 12

10 Balanced Growth Turning to lifetime utility, we substitute into (10) the following formulas: C t = Z t c t = Z t 1 (1 )t 0 Z c t and t = t 1 Z = 0 Z. Collecting terms we get U = Z 0 1 X t=0 where? 1 c Z. (? ) t " (Z 0 c t ) 1 c 1 c + 0 v (1 N) Then we can set Z 0 = 1 with no change the preference orderings. Notice that, with growth, we need to impose? < 1 so that lifetime utility remains finite. #, Fatih Guvenen Balanced Growth September 21, / 12

11 Balanced Growth: A Recipe The previous trick serves as a recipe and can be applied to problems with much more general utility functions. Harald Uhlig for example has shown how to do the scaling when preferences are of the Epstein-Zin form defined over consumption and leisure, as well as when they display a very rich habit formation structure. I have applied a similar scaling in my paper with Michelle Rendall, when the utility function is of a household, and features home production, leisure, and love. This method must become a part of your toolbox! Fatih Guvenen Balanced Growth September 21, / 12

12 BGP with Heterogeneity In models with heterogeneity, we must be extra careful about BGP. In those models, we will often worry about another type of balanced growth: whether there is a stationary wealth distribution. Even if aggregates grow at a stationary rate, the underlying economy could converge to one with a degenerate wealth distribution. This is the case for example if agents differ in their time discount factor, or in the curvature of their utility function. This is why models with heterogenous agents and growth rarely feature heterogeneity in preferences. You can deliver stationary wealth distribution if you assume an OLG structure. Alternatively, you can use non-homothetic preferences. For an example, see Chen and Kogan (JPE, 2002). Fatih Guvenen Balanced Growth September 21, / 12