Dynamic Optimization Using Lagrange Multipliers

Transcription

1 Dynamic Optimization Using Lagrange Multipliers Barbara Annicchiarico Università degli Studi di Roma "Tor Vergata" Presentation #2

2 Deterministic Infinite-Horizon Ramsey Model Consider the dynamic Ramsey model. The representative household s total utility: W 0 = β t u(c t ), 0 < β < 1, (1) t=0 Preferences are additively separable. Households can hold physical capital which is used to produce output according to a standard neoclassical production function: k t+1 = (1 δ) k t + i t, 0 < δ < 1, (2) c t + i t f (k t ) (3) c t, k t+1 0 (4) The task is to find the optimal paths for consumption and capital to max (1) s.t. (2), (3) and (4). This is a typical consumption-saving decision problem, where k 0 > 0 is given.

3 Deterministic Infinite-Horizon Ramsey Model Assumptions on functional forms (standard!): u( ) : twice continuously differentiable, strictly increasing, strictly concave f ( ) : twice continuously differentiable, strictly increasing, strictly concave, f (0) = 0. Notation (again very standard!) β discount factor δ rate of depreciation of capital k t : capital (it s given at time t, it s a "state" variable) c t : consumption ("control" variable) i t : investments

4 Deterministic Infinite-Horizon Ramsey Model Combine (2) and (3), solve for c t : c t (1 δ) k t + f (k t ) k t+1 The problem: max {c t,k t+1 } t=0 t=0 β t u(c t ), s.t. c t (1 δ) k t + f (k t ) k t+1 c t 0 k t+1 0 given k 0 for t = 0, 1, 2,...

5 Deterministic Infinite-Horizon Ramsey Model The associated { Lagrangian to the above problem is } L 0 = β t u(ct ) + λ t [(1 δ) k t + f (k t ) c t k t+1 ] + t=0 +µ t c t + ω t+1 k t+1 where λ t, µ t, ω t+1 Lagrange multipliers (period t values) The FOCs can be obtained by max L t wrt {c t } t=0 and {k t+1} t=0 FOC wrt c t FOC wrt k t+1 u (c t ) = λ t µ t (5) λ t+1 β [ (1 δ) + f (k t+1 ) ] λ t + ω t+1 = 0 (6)

6 Deterministic Infinite-Horizon Ramsey Model The other conditions for a maximum λ t [(1 δ) k t + f (k t ) c t k t+1 ] = 0 µ t c t = 0 ω t+1 k t+1 = 0 Introduce two additional assumptions to rule out corner solutions 1. lim ct 0 u (c t ) = : Implies that agents hate to starve to death in any period c t > 0 µ t = 0 since u (c t ) > 0 (1 δ) k t + f (k t ) c t k t+1 = 0 2. lim kt 0 f (k t ) = k t+1 > 0 ω t+1 = 0

7 Deterministic Infinite-Horizon Ramsey Model The FOCs can be reduced to u (c t+1 )β [ (1 δ) + f (k t+1 ) ] u (c t ) = 0 Euler equation: one of the key building blocks of the DSGE methodology, describes the evolution of consumption along an optimal path (the marginal utility of current consumption must be equal to the discounted marginal utility of next period consumption adjusted for borrowing or saving between the two periods) The intertemporal marginal rate of substitution is equal to the return u (c from investing in physical capital: t ) βu (c t+1 ) = (1 δ) + f (k t+1 )

8 Deterministic Infinite-Horizon Ramsey Model Transversality Condition A further condition dictates that at the optimum: lim t βt u (c t ) }{{} k t+1 = 0 (7) λ t where β t u (c t )k t+1 denotes the present discounted utility that would derive from consuming the capital stock k t+1. If the time horizon were t, then it would be not be optimal to have any capital left at time t (it should have to be consumed). Of course the same must be true for t. From this point of view (7) provides an extra optimality condition for intertemporal infinite -horizon problems.

9 Stochastic Infinite-Horizon Ramsey Model Introduce uncertainty: a t f (k t ) where a t exogenous variable subject to shocks (assume lim ct 0 u (c t ) = lim f (k t ) = ) kt 0 The problem: {c 0, k 1 } t=0 max E 0 β t u(c t ) s.t. c t = (1 δ) k t + a t f (k t ) k t+1 } for t = 0, 1, 2,... given k 0, a 0 Remark: agents choose only current consumption because of uncertainty

10 Stochastic Infinite-Horizon Ramsey Model The associated stochastic Lagrangian to the above problem is L 0 = E 0 β t {u(c t ) + λ t [(1 δ) k t + a t f (k t ) c t k t+1 ]} t=0 The FOCs can be obtained by max L 0 wrt c 0 and k 1 FOC wrt c 0 u (c 0 ) = λ 0 (8) FOC wrt k 1 E 0 λ 1 β [ (1 δ) + a 1 f (k 1 ) ] λ 0 = 0 (9)

11 Stochastic Infinite-Horizon Ramsey Model Combining the FOCs: E 0 u (c 1 )β [ (1 δ) + a 1 f (k 1 ) ] u (c 0 ) = 0 (10) At time t=1 the agents will solve a similar problem... and so E 1 u (c 2 )β [ (1 δ) + a 2 f (k 2 ) ] u (c 1 ) = 0 (11) Generalizing.. at time t E t u (c t+1 )β [ (1 δ) + a t+1 f (k t+1 ) ] u (c t ) = 0 (12) which the stochastic Euler equation.

12 Stochastic Infinite-Horizon Ramsey Model Stochastic Transversality Condition A further condition dictates that at the optimum: the stochastic analog of (7). lim t βt E t u (c t ) }{{} k t+1 = 0 (13) λ t

13 So what? We have a stochastic Euler eq. E t u (c t+1 )β [(1 δ) + a t+1 f (k t+1 )] u (c t ) = 0 a resource constraint (capital accumulation eq.) c t = (1 δ) k t + a t f (k t ) k t+1 a stochastic process shaping the time path of a t two initial conditions k 0, a 0 we need to find a closed-form solution for consumption, that is write c t as a function of k t and a t.

14 References Heer, B. & Maussner. A. (2008), Dynamic General Equilibrium Modelling, Computational Methods and Applications, Springer, chapter 1. McCandless, G. (2008), The ABCs of RBCs, Harvard University Press, chapter 4.