Dynamics of the OLG model

The dynamics of the capital per unit of effective labor ˆkt obeys

ˆkt+1=s(f′(ˆkt+1))[f(ˆkt)−ˆktf′(ˆkt)](1+g)(1+n),

This is an implicit system (the left- and right-hand sides both contain ˆkt+1).

kt+1 may not be a function of kt in which case, there is multiple possibilities for the time path.

Cobb-Douglas Production and Log Utility

To obtain a clear result, we assume that

f(k)=kα,0<α<1

and the log instantaneous utility function: i.e., θ=1.

Prove that the dynamics is characterized by

ˆkt+1=(1−α)(1+g)(1+n)(2+ρ)ˆkαt

Looks like the Solow model.

Cobb-Douglas Production and Log Utility (Stability)

It can be shown that for any ˆk0>0

ˆkt→ˆk∗=[1−α(1+n)(1+g)(2+ρ)]11−α

as t→∞.

The prediction of the Solow model is preserved. For instance, per-capita capital

kt=Atˆkt

grows at the rate of g in the long run.

Dynamic Inefficiency

OLG models may have dynamic inefficiency in that at least one generation can increase utility without nobody else reducing utility.

Let’s continue to assume that the production function is C-D and utility is log. The steady state for capital per unit of effective labor satisfies

ˆk∗=[1−α(1+n)(1+g)(2+ρ)]11−α

f′(ˆk∗)=α([1−α(1+n)(1+g)(2+ρ)]11−α)α−1=α(1+n)(1+g)(2+ρ)1−α.

Dynamic Inefficiency (cont’d)

The golden rule capital stock ˆkG satisfies (under δ=0):

f′(ˆkG)=n+g

Thus,

α(1+n)(1+g)(2+ρ)1−α<n+g⟺ˆk∗>ˆkG

If α is sufficiently small, the above inequalities hold and it gives rise to dynamic inefficiency.

In OLG models, the fact that there is no coordination between generations may result in too much saving.

Reallocation

Let’s suppose ˆk∗>ˆkG and that the economy is on the BGP.

If some social planner forces

• young people in period t0 to consume more and save less by some amount Δ>0 by which the economy moves to ˆkG,
• people in subsequent periods consume the golden rule level

Consumption per effective worker in period t0 satisfies:

f(ˆk∗)+(ˆk∗−ˆkG)−(g+n)ˆkG>f(ˆkG)−(g+n)ˆkG>f(ˆk∗)−(g+n)ˆk∗.

Reallocation (cont’d)

f(ˆk∗)+(ˆk∗−ˆkG)−(g+n)ˆkG

-> consumption in period t0

f(ˆkG)−(g+n)ˆkG

-> consumption in period t>t0

f(ˆk∗)−(g+n)ˆk∗ -> consumption without the reallocation.

All generations potentially become better off by this reallocation. The original saving rate was too high.

Dynamic efficiency of the Ramsey model

In the Ramsey model, it always holds that

ˆk∗<ˆkG

So, there is no dynamic inefficiency in the Ramsey model.

Multiple equilibrium

Let’s go back to the implicit equation. By rearranging terms, we have

f(ˆkt)−ˆktf′(ˆkt)=(1+g)(1+n)ˆkt+1s(f′(ˆkt+1))

Define w(k)=f(k)−kf′(k). Since

• w(k) is increasing in k
• w(0)=0, and
• the right-hand side is positive

Given ˆkt+1, we can solve for a unique ˆkt that satisfies the above equation. We have the inverse dynamics (ˆkt+1↦ˆkt)

Code

OLG = function(theta, rho, g, n, alpha, gamma, A) {
  f = function(k) A * (alpha * k ^ gamma + 1 - alpha) ^ (1 / gamma)
  df = function(k) A * alpha * (alpha + (1 - alpha) * k ^ (-gamma)) ^
                                              ((1 - gamma) / gamma)
  w = function(k) f(k) - k * df(k)
  w_inv = function(y) {
    sol = nleqslv::nleqslv(100, function(x) w(x) - y)
    if (sol$termcd == 1) return(sol$x) else return(NA)
  }
  s = function(r){
    sigma = (1 - theta) / theta
    return((1 + r) ^ sigma) / ((1 + rho) ^ (1 / theta) + (1 + r) ^ sigma)
  }
  w2 = function(k) (1 + g) * (1 + n) * k / s(df(k))
  return(list(f=f, df=df, w=w, w_inv=w_inv, w2=w2))
}

Typical Case

model = OLG(theta=0.8, rho=0.4, g=0.2, n=0.2, 
            alpha=0.33, gamma=0.1, A=7)

k1 = seq(0.0, 10, length.out=100)
k0 = lapply(model$w2(k1), model$w_inv)
k0 = unlist(k0)
qplot(k0, k1, geom='line') + geom_line(data=tibble(x=k0, y=k0), aes(x,y)) +
  xlim(0.0, 10) + ylim(0.0, 10)

Multiple Equilibrium (Poverty Trap)

model = OLG(theta=8, rho=0.4, g=0.3, n=0.2, 
            alpha=0.33, gamma=-0.7, A=7)

k1 = seq(0.0, 7, length.out=100)
k0 = lapply(model$w2(k1), model$w_inv)
k0 = unlist(k0)
qplot(k0, k1, geom='line') + geom_line(data=tibble(x=k0, y=k0), aes(x,y)) +
  xlim(0.0, 5) + ylim(0.0, 5)

Multiple Steady States (Self-fulfilling prophecy)

model = OLG(theta=8, rho=0.4, g=0.3, n=0.2, 
            alpha=0.33, gamma=-2.9, A=7)

k1 = seq(0.00, 6, length.out=100)
k0 = lapply(model$w2(k1), model$w_inv)
k0 = unlist(k0)
ggplot(data=tibble(k0=k0, k1=k1)) + geom_path(aes(x=k0, y=k1)) + 
  geom_line(aes(k0,k0)) + xlim(0.0, 6) + ylim(0.0, 6)

Multiple Steady States (Self-fulfilling prophecy)

Comments on the previous figure.

A path that starts from somewhere very close to the middle steady state (which is unstable) is not entirely determined by the initial condition. (Indeterminacy)

Because the agents have three possible ˆkt+1. The path they choose is affected by non-fundamental factors (called sunspots).

If they believe the economy experiences boom, they choose the higer ˆkt+1 and the economy converges to the higher steady state. Their belief has the power to make it true.