OLG Flashcards
What is the budget constraint in this setting?
c_t^y +s_t ≤ w_t
c_{t+1}^o ≤ s_t(1+r_{t+1})
where we can write 1+r as R.
Combining those two, we create the intertemporal budget constraint.
c_{t+1}^o = w_tR_{t+1}-c_t^yR_{t+1}
This is also the budget constraint one can use to directly substitute in the utility.
What is the expression for the savings rate in OLG-setting?
s_t = w_t - c_t^y
What are the investments in the OLG model?
It = Kt+1 − (1 − δ) Kt
What condition makes our SS-capital stock to be locally stable?
dk_{t+1}/dk_t < 1
What can be said about the savings function in the OLG framework?
▶ 0 < sw < 1. Saving is an increasing function of wage-income. Separability and concavity of U (·) ensure that both goods (c Y and c o ) are normal.
▶ sr ≶ 0. The effects of interest rates are ambiguous. An increase in Rt+1 increases the relative price of ct , leading individuals to shift consumption from the first period to the second one, st ↑, this is the substitution effect.
Higher Rt +1 also increases the feasible consumption set, making it possible to increase consumption in both periods (st ↓), this is the income effect.
▶ The net result of these two effects is ambiguous, but in this two-period model is possible to show that is related to the elasticity of intertemporal substitution (see example below).
▶ Notice that with income in the second period there would be also a wealth effect.
With the investment rate It = Kt+1 − (1 − δ) Kt
How do we find an expression for the effective capital accumulation equation?
If δ =1,
It = Kt+1 = Ltst
Then multiply with 1/L_{t+1} on both sides to get
(1+n)kt_{t+1} = s_t
What is the competative equlibrium in the OLG?
A competitive equilibrium can be represented as a sequence of aggregate capital stock, household consumption, and factor prices {Kt , cY t , cO t , Rt , wt}∞_t =0, such that:
- The factor price sequence {Rt , wt }∞ t=0 is given the marginal product of capital and labor
- Individual consumption decisions {cY, cO} are given by the Euler equation and the IBC.
The aggregate capital stock evolves according to (1+n)kt_{t+1} = s_t
How do one determine the stability of kapital in the SS of the OLG?
From (1+n)kt_{t+1} = s_t we solve for k_t+1.
Since we are in SS, k_t+1 = k.
Solve for k*, which is the ss capital.
Then plug in k* in kt+1 and take the derivative w.t.r kt,
This shoud then be <1 to be stable
In the OLG framwork with CIES or CRRA utility. What happens to savings when the interest rate increases?
It depends on the elasticity of substitution. θ
With log utility, IE and SE cancels out.
How do you set up the Social planer problem in the OLG setting?
Like the original problem, but now a “weight” φ_t, is multiplied the utilities that weights between generations (this is addition to the β already included). The SP then uses the economies resource constraint when maximising.
What insigt do we get with the SP setup in the OLG model?
The social planner allocate consumption within an individual’s lifetime in the same way as the individual himself would allocate it. There are no market failures in the
allocation of consumption over time at given market prices.
However, the social planner allocation of resources across generations differs
from the competitive equilibrium. The competitive equilibrium is thus not Pareto Optimal.
That is, the steady state capital will be different from the golden rule capital. To big as a recon, then all would do better if they consumed less.
Which features in the OLG model leads to the dynamic inefficiency?
If the steady state interest rate is less than the rate of population growth, the
economy is dynamically inefficient.
In the infinite-horizon neoclassical economy the transversality condition rules out this possibility. The specific form of heterogeneity in the OLG model removes the neoclassical transversality condition. Even though time goes on forever, no individual has infinite lifetime. Debt can grow at a faster rate than the interest rate…
The inefficiency of the CE is not related to market incompleteness. The
proof of the first welfare theorem relies on a finite market value of
endowments, an assumption that can be violated in an OLG model where
there is a double infinity of households and commodities. As esemplified
below.
Explain i words what the dynamic inefficiency (kss > kgold) means,
▶ Intuitively, dynamic inefficiency is due to overaccumulation of capital
stemming from the need of the young to save for when they will be old. The
more they save, the lower is the return on savings and this might encourage
them to save even more!
▶ The effect of the saving of the young on the future return of capital is a ”pecuniary externality” which has first order effects on welfare because the first welfare theorem does not apply.
The idea behind dynamic inefficiency is that capital accumulation is not a good way to transfer resources across generations!
What does the data say about dynamic inefficiency?
There is a challenge regarding what interest rate one should use when studying this.
Abel et al (1989) suggest we should we should look at the ”average interest
rate” in the economy.
In their study, they find that the United States (1929-1985) and 6 other advanced economies (1960-1985) are dynamically efficient, i.e. they invest less than they get out from capital income.
More recently, by using a novel data set Geerolf (2018) finds that we cannot
exclude dynamic inefficiency for none of the advance economies in his study.
What is OLG with growth and how does it affect our analysis?
This is a setup where we at technological growth. We thus have a production function Yt = AtLtKt (to the power of CD preferences). So technology grows at the exonenous rate At = At-1 (1+g).
Both the growth of capital and the golden rule will depend on the growth of technology. That is, when we talk about dynamic inefficiency for an economy with growth,
we should take into account the rate of the growth of the economy, i.e. (1 + n) (1 + g ) .
