文档介绍：The Ramsey model
z How can we determine the saving rate?
1. Assumptions
Firms Y = F(K, AL) They are owned by households.
Households: ( H ). The size of each household grows at rate n .
Labor: Each member supplies one unit of labor.
Capital: K(t) / H
e: from the labor and capital it supplies, and potentially from profits
Markets: competitive factor markets and petitive output market.
How does the household divide its e at each point in time between
consumption and saving so as to maximize lifetime utility?
∞ L(t)
Lifetime Utility function: U = e−ρtu(C(t)) dt
∫t=0 H
C(t)1−θ
Instantaneous utility function: u(C(t)) = , θ> 0, ρ− n −(1−θ)g > 0
1−θ
What is θ?
Cu"(C)
z The measure of relative risk aversion: −
u"(C)
1
z The elasticity of substitution between consumption in different periods:
θ
z θ→1 u(C) = ln(C)
1
2. The behavior of households and firms
z Firms
The real interest rate: r(t) = f '(k(t ))
The real wage per unit of effective labor: w(t) = f (k(t)) − k(t) f '(k(t ))
The labor e of a worker: A(t)w(t )
z Households’ maximization problem
∞ L(t) K(0) ∞ L(t)
The budget constraint ∫ e−R(t)C(t) dt ≤+ ∫ e−R(t) A(t)w(t) dt
t=0 H H t=0 H
t
where R(t) = r(τ)dτ
∫τ=0
K(0) ∞ L(t)
+ ∫ e−R(t)[A(t)w(t) − C(t)] dt ≥ 0
H t=0 H
Normalization by the quantity of effective lab