Dynamic Consumption Theory October 2007 () Consumption October - - PowerPoint PPT Presentation

Dynamic Consumption Theory

October 2007

() Consumption October 2007 1 / 20

Two-Period Planning Horizon

(discrete time, no uncertainty)

Household utility: U(c1, c2) = u(c1) + βu(c2) , ! u0(c) > 0, and u00(c) < 0 , ! subjective discount factor: β =

1 1+ρ < 1

, ! ρ = rate of time preference.

() Consumption October 2007 2 / 20

Dynamic budget constraints: A1 = (1 + r1) (A0 + w1 c1) A2 = (1 + r2) (A1 + w2 c2) Boundary condition A2 0 ) intertemporal budget constraint: c1 + c2 1 + r1 A0 + w1 + w2 1 + r1

() Consumption October 2007 3 / 20

The Euler Equation

How should household allocate consumption across the two periods? Consider saving an additional $1 at the optimum: MC of extra $ saved = u0(c1) MB of extra dollar saved = β(1 + r1)u0 (c2) . , ! optimal allocation: u0(c1) = β(1 + r1)u0 (c2) Also intertemporal budget constraint must hold with strict equality.

() Consumption October 2007 4 / 20

Diagrammatic Representation

Slope of budget constraint: dc2 dc1 = (1 + r1), Marginal rate of intertemporal substitution: dc2 dc1

• ¯

U

= u0(c1) βu0(c2). , ! optimal allocation u0(c1) βu0(c2) = 1 + r1 which is the Euler equation.

() Consumption October 2007 5 / 20

c2 c1 c1

*

c2

*

A E w1 w2 w2+(1+r2)w1 slope = - (1+r

2)

Figure: Optimal Consumption Choice

() Consumption October 2007 6 / 20

Example — CES utility function

c1θ

1

1 θ + β c1θ

2

1 θ . Euler equation cθ

1

= β(1 + r1)cθ

2 .

, ! re–arranges to c2 c1 = [β(1 + r1)]

1 θ .

Taking natural logs of both sides we get ln c2 c1 = 1 θ ln β + 1 θ ln(1 + r1). , ! It follows that d ln

• c2

c1

• d ln(1 + r1) =

% change in c2

c1

% change in 1 + r1 = 1 θ .

() Consumption October 2007 7 / 20

Three–Period Planning Horizon

Discounted utility u(c1) + βu(c2) + β2u(c3), Dynamic budget constraints: A1 = (1 + r1) (A0 + w1 c1) A2 = (1 + r2) (A1 + w2 c2) A2 = (1 + r3) (A2 + w3 c3) Boundary condition: A3 0. , ! intertemporal budget constraint c1 + c2 1 + r1 + c3 (1 + r1)(1 + r2) w1 + w2 1 + r1 + w3 (1 + r1)(1 + r2).

() Consumption October 2007 8 / 20

Optimal consumption allocation satis…es A3 = 0 and u0(c1) = β(1 + r1)u0 (c2) = β2(1 + r1)(1 + r2)u0 (c3) .

• r

u0(c1) = β(1 + r1)u0 (c2) u0(c2) = β(1 + r2)u0 (c3) CES example: c2 = [β(1 + r1)]

1 θ c1

c3 = [β(1 + r2)]

1 θ c2

c1 + c2 1 + r1 + c3 (1 + r1)(1 + r2) = w1 + w2 1 + r1 + w3 (1 + r1)(1 + r2)

() Consumption October 2007 9 / 20

T–Period Planning Horizon

Discounted utility:

T

∑

t=1

βt1u(ct) T dynamic budget constraints At = (1 + rt) (At1 + wt ct) 8 t 2 f1, ..., Tg boundary condition AT 0.

() Consumption October 2007 10 / 20

Solution

T 1 Euler equations: u0(ct) = β(1 + rt)u0 (ct+1) 8 t 2 f1, ..., T 1g and an intertemporal budget constraint

T

∑

t=1

Dtct = A0 +

T

∑

t=1

Dtwt, where Dt =

t1

∏

τ=1

• 1

1 + rτ

• =
• 1

1 + r1

• .
• 1

1 + r2

• ....
• 1

1 + rt1

• and

D

() Consumption October 2007 11 / 20

Inde…nite Planning Horizon

Boundary condition replaced by transversality condition , ! the present market value of household assets cannot be negative in the long run: lim

T !∞ DT AT 0.

) intertemporal budget constraint (at the optimum):

∞

∑

t=1

Dtct = A0 +

∞

∑

t=1

Dtwt.

() Consumption October 2007 12 / 20

Continuous Time (by analogy)

Household utility: max

Z T

eρtu(c(t))dt subject to ˙ A(t) = w(t) + r(t)A(t) c(t) 8 t and D(T)A(T) 0, where D(t) = e

R t

0 r(s)ds

intertemporal budget constraint:

Z T

D(t)c(t)dt A(0) +

Z T

D(t)w(t)dt.

() Consumption October 2007 13 / 20

Euler equation: u0(c(s)) = e

R t

s r(τ)dτeρ(ts)u0(c(t)).

Totally di¤erentiating w.r.t. to t we get ˙ c(t) = u0(c(t)) u

00(c(t)) [r(t) ρ]

CES example: u0(c) u

00(c) =

cθ θcθ1 = c θ , ) Euler equation in di¤erential form: ˙ c(t) c (t) = r(t) ρ θ .

() Consumption October 2007 14 / 20

Optimal Control Approach

General problem: max

c(t) V (0)

=

Z T

v(k(t), c(t), t)dt subject to ˙ k(t) = g(k(t), c(t), t) k(T)

• k(0)

= k0 Set up Lagrangian: L =

Z T

v(k(t), c(t), t)dt +

Z T

λ(t)

• g(k(t), c(t), t) ˙

k(t)

• dt + µk(T)

() Consumption October 2007 15 / 20

Note that using integration by parts

Z T

λ(t) ˙ k(t)dt =

Z T

λ(t)dk(t) = [λ(t)k(t)]T

Z T

k(t)dλ(t) = λ(T)k(T) λ(0)k(0)

Z T

k(t) ˙ λ(t)dt

() Consumption October 2007 16 / 20

It follows that L =

Z T

[v(k(t), c(t), t) + λ(t)g(k(t), c(t), t)] dt +

Z T

k(t) ˙ λ(t)dt λ(T)k(T) + λ(0)k0 + µk(T) The …rst–order conditions for a maximum are ∂L ∂c(t) = ∂v ∂c(t) + λ(t) ∂g ∂c(t) = 0 8 t 2 (0, T) ∂L ∂k(t) = ∂v ∂k(t) + λ(t) ∂g ∂k(t) + ˙ λ(t) = 0 8 t 2 (0, T) How do I remember all this junk?

() Consumption October 2007 17 / 20

Hamiltonians

Construct a "Hamiltonian function": H(k, c, t, λ) v(k, c, t) + λ.g(k, c, t) Derive Hamiltonian conditions ∂H ∂c = ∂H ∂k = ˙ λ ∂H ∂λ = ˙ k plus the boundary condition k(T) = 0.

() Consumption October 2007 18 / 20

Application to consumer’s problem

H(A, c, t, λ) = eρtu(c) + λ [w + rA c] Hamiltonian conditions ∂H ∂c = eρtu0(c) λ = 0 ∂H ∂A = λr = ˙ λ ∂H ∂λ = w(t) + r(t)A(t) c(t) = ˙ A

() Consumption October 2007 19 / 20

Di¤erentiate the …rst one w.r.t. time ρeρtu0(c) + eρtu(c)00 ˙ c = ˙ λ ρeρtu0(c) + eρtu(c)00 ˙ c = eρtu0(c)r ρu0(c) + u(c)00 ˙ c = u0(c)r ˙ c(t) = u0(c(t)) u

00(c(t)) [r(t) ρ] () Consumption October 2007 20 / 20