Unifying Time-to-Build Theory M. Bambi, and F. Gori speaker: M. Bambi (University of York, U.K.) OFCE-SKEMA, Nice, 2010
Time to Build Time to Build (TtB) means that capital takes time to becomes productive. Jevons: A vineyard is unproductive for at least three years before it is thoroughly fit for use. In gold mining there is often a long delay, sometimes even of five or six years, before gold is reached.
Business cycle literature Kydland and Prescott (Econometrica 1982), Rouwenhorst (JME 1991) - RBC models with time to build - Increase in the persistence of the business cycle more recently, Gomme et al. (JPE 2001), Casares (JME 2006), and Edge (JME 2007) Time-to-Build in a discrete time framework
Growth literature Kalecki (Econometrica 1935), Asea and Zak (JEDC 1999), and Bambi (JEDC 2008) - Capital accumulation equation K(t) = F (K(t J)) C(t) - Optimal path of capital is oscillatory: infinite number of conjugate complex eigenvalues; example with C(t) = (1 s)y (t), linear technology, and no depreciation: K(t) = sak(t J) h(λ) = λ sae Jλ = 0 Time-to-Build in a continuous time framework BUT delay in production at discrete time intervals
This Paper Attempt to unify these 2 streams of the Time-to-Build literature, continuous Vs discrete time How to do that? Step 1: Construct the continuous counterpart of the (deterministic) Kydland and Prescott discrete time model. Step 2: Compare the short run and long run dynamics induced by the 2 formulations.
Why concentrate on TtB models? 3 main reasons: TtB theories are important in replicating several stylized facts in the business cycle literature. The comparison between discrete and continuous time systems is much more challenging (and quite interesting) in the TtB context because the former are finite-dimensioned systems and the latter infinite-dimensioned. Computational interest related to a quantitative evaluation of the differences.
Playing with Time Let s consider the deterministic version of the Kydland and Prescott model with pure investment lag in production, J; time period length equals to h = 1 m with m N ; time measure equals to sh with s N {0}; beginning of time sets exogenously at t R + {0}. A sequences of relevant dates have the form t + sh. Standard KP model: t = 0, h = 1, and s time measure.
Social planner problem becomes: subject to max ( ) 1 1 + ρh } {{ } β h s=1 s 1 [u(c t+sh ) + ν(l t+sh )]h K t+j+sh+h = A hf (K t+sh, L t+sh ) + (1 δh)k t+j+sh hc t+sh and initial conditions K i, with i = t, t + h, t + 2h,..., t + nh = t + J Investment, consumption, and output as well as the depreciation rate of capital and the preference discount rate are re- scaled uniformly by h.
Optimal Equilibrium Path An optimal equilibrium path of the economy is any sequence {C t+sh } s=1,{l t+sh} s=1, and {K t+sh+j } s=1, which satisfies K t+j+sh+h = A hf (K t+sh, L t+sh ) + (1 δh)k t+j+sh hc t+sh ν L (L t+sh ) u C (C t+sh ) = AF L (K t+sh, L t+sh ) u C (C t+sh ) u C (C t+sh+h ) ] = β h [β J u h C (C t+j+sh+h ) h u C (C t+sh+h ) AhF K (K t+sh+j, L t+sh+j ) + 1 δh plus the initial history of capital and the transversality condition lim sh µ t+sh k t+sh = 0 for any choice of s N, and any t R + {0}.
Continuous time h 0 K(t + J) = A F (K(t), L(t)) δk(t + J) C(t) t R + {0} ν L (L(t)) u C (C(t)) = AF L (K(t), L(t)) t R + {0} while the continuous version of the Euler equation is Ċ(t) C(t) = u [ ] C (C(t)) uc (C(t + J)) AF K (K(t + J), L(t + J))e ρj δ ρ u CC (C(t))C(t) u C (C(t)) again t R + {0}.
Remark: The Euler equation so obtained, is exactly the same equation which can be deduced by solving the optimal growth problem: subject to max 0 [u(c(t)) + ν(l(t))]e ρt dt K(t + J) = AF (K(t), L(t)) δk(t + J) C(t) given the initial condition K (t) = K 0 (t) for t [0, J].
Result after the Game When h shrinks to zero, the stock of capital becomes K(t) = t+j I (s)e δ(s t J) ds which is exactly the same relation used in Asea and Zak, and Bambi. Gestation lags as defined by Kalecki, can be easily exploited by calling I (t) = L(t + J) + U = K(t + J) + δk(t + J). The Kydland and Prescott model with period length h, contains all the developed specifications of time to build.
However the passage from discrete time to continuous time in a TtB framework is NOT innocuous! Different dimensionality of the two systems describing the economy. What are the economic consequences of this difference? Long run: negligible differences from a qualitative/quantitative point of view... Short run: comparison is less straightforward.
Discrete time Assumption: indivisible labor supply c t = c 0 z1 t k t = c 0 Ψz1 t + 2 2 r2j t [ x 2j cos tw 2j ỹ 2j sin tw 2j ] J 1 j=1 (J is odd) J 2 unique monotonic convergence of capital and consumption. 2 J J unique oscillatory convergence of capital and unique monotonic convergence of consumption. J > J local instability.
Continuous time Assumption: indivisible labor supply c(t) = âe zt k(t) = 1 (â, Φ 1,v )e z ṽ t + 2 [ 2 (â, Φ 1,v ) cos y v t 3 (â, Φ 2,v ) sin y v t] e xv t v ṽ 0 < J < J unique oscillatory convergence of capital and unique monotonic convergence of consumption. J = J cycle through a Hopf bifurcation. J > J local instability.
Quantitative Comparison after Calibration The parameters in these simulations are chosen in line with Kydland and Prescott quarterly calibration with J = 4. (continuous case) Impossible to compute all the roots: Lambert function computes the 2m + 1 closest roots to the imaginary axis. We have checked our results for m = 100, and its robustness by choosing m = 10, 150, 300, 1000.
2 x 10-4 1 0-1 -2-3 -4-5 -6 0 2J 20 40 60 80 100 120 140 160 180 200 Figure: Continuous vs Discrete: differences in capital level
6.2% 6% 5.8% m=10 m=100 m=150 m=300 m=1000 5.6% 5.4% 5.2% 5% J J+1 J+2 J+3 2J Figure: Continuous vs Discrete: capital percentage differences for different choices of m
Results: 1. Main difference in the first 40 periods. 2. Volatility around the trend may be 6% higher in continuous time. 3. Change in stability: J = 35 = J.