Dynamic Consumption Model Flashcards
What is the a_{T+1} ≥ condition?
This is the terminal condition, saying that there can be no lending or borrowing when the time is up, in T+1.
Households can’t accumulate debt when time is up. That is, it will hold with equality when the time is up.
When this is in the model and we solve, it is instead called the “transversality” condition. We might even be in a setting where it is possible to impose this condition directly in the exam.
What does the Euler Equation say
The marginal utility of consumption today has to be equal to the marginal utility of consumption tomorrow, discounted so they are comparable. They are discounted with the inpatient factor-beta and the interest rate. If beta = 1, then we value the future just as much as today, i.e., we are very patient.
The Euler equation specifies the slope of the optimal consumption path.
It shows that we consumption smooths. I.e., we keep the consumption path stable, with decreasing marginal utility in both periods it cannot be optimal to consume an income increase all in one period.
How do we derive the consumption function?
There are different ways, but in this course and this setting we mainly do it by constructing the intertemporal budget constraint. Which we do by iterating and substituting the household’s budget constraint in itself. Then e.g assuming B(1+r) = 1 so Uct = Uct+1, we will eventually get an expression with C by itself on the LHS. Then consumption is likely a function of the interest rate and our permanent income.
Then, the Euler equation AND the intertemporal budget constraint define the optimal consumption path.
When we have deriven the optimal path of consumption, what effects do we see from this regarding the interest rate?
An increase in the interest rate has three effects on
consumption allocations:
▶ Substitution effect. It makes consumption in period t + 1 relatively less expensive compared to ct . Thus, it increases savings, i.e. future consumption relative to consumption today.
▶ Income effect. It acts like an increase in income (assuming hh has positive assets), i.e. lowers the cost of the bundle (ct , ct +1 ) . Thus, it increases consumption in all periods.
▶ Wealth effect. It reduces the present value of wealth, hence reduces current consumption.
What is the essence of the permanent income hypotheses?
We work and save when we are young so we can continue to consume when we are old.
people will spend money at a level consistent with their expected long-term average income.
Consumption in period t is proportional to the annuity value of wealth (≈ permanent income), not simply current income.
Consumption is a forward-looking variable ⇒ anticipated changes in income in the future will have an immediate effect on current consumption.
Temporary increases in wealth should be saved, temporary falls should be offset by borrowing (= save for a rainy day).
Does t -> inf change the solution of the model where we instead have t -> T?
No
What is the No-Ponzi-Game” condition?
In infinite horizon, the terminal condition translates into a
”No-Ponzi-Game” condition:
lim qT aT +1 ≥ 0.
This condition does not prevent households from holding strictly negative assets in the long-run, it rules out a debt path growing at a higher rate than the interest rate.
Since we go to infinity, we are not thinking of a terminal point, since it is non, it is instead a very very distant future.
The value of what we are having in the very distant future is not gonna be zero, but it will be very heavily discounted.
That is, the price of the asset in the very verry distant future will be zero since it is so heavy discounted. That is, it is zero in the present value.
What is prudance?
Being worried more about the low consumption state than being pleased by the high consumption state.
This regards the case with uncertainty, when we have Et.
The third derivative of utility w.r.t consumption is a measure of individuals’ prudence. This is how the shape of the MU looks.
How MU utility is decreasing.
If the marginal utility function is curved (not constant), then marginal utility might be decreasing at a lower and lower rate.
If MU is convex: Uccc > 0, then ct < ct+1 since EtUct+1 > UEtct+1
If MU is linear: Uccc = 0, then ct = ct+1
If MU is concave: Uccc < 0, then ct > ct+1, since EtUct+1<UEtct+1
What is Precautionary saving?
This the same as prudance.
Precautionary saving is saving (non-expenditure of a portion of income) that occurs in response to uncertainty regarding future income. The precautionary motive to delay consumption and save in the current period rises due to the lack of completeness of insurance markets. Accordingly, individuals will not be able to insure against some bad state of the economy in the future. They anticipate that if this bad state is realized, they will earn lower income. To avoid adverse effects of future income fluctuations and retain a smooth path of consumption, they set aside a precautionary reserve, called precautionary savings, by consuming less in the current period, and resort to it in case the bad state is realized in the future.
What do we usually assume regarding individuals prudence?
We typically assume Uccc > 0. Households save more than in the certainty case for precautionary reasons or prudence. The downside risk affects marginal utility more than the upside risk.
Linear marginal utility Uccc = 0, is associated with a quadratic utility function and has the very special certainty equivalence property, i.e. uncertainty has no effect on expected marginal utility and thus no effect on optimal consumption.
What is meant by Intertemporal Elasticity of Substitution (IES)?
The Intertemporal elasticity of substitution shows how willing the consumer is to relocate consumption between time periods.
-dlog(ct+1/ct)/dlog(pt+1/pt)
Where (pt+1/pt) = 1/(1+rt+1)
How do you find IES?
Write Euler
Ct+1/Ct = B(1/(1+rt+1))
Take logs
Remember that
(pt+1/pt) = 1/(1+rt+1)
The, take derivative w.r.t (pt+1/pt)
With CIES utility we will have
IES = 1/p
With CIES or CRRA, what does the parameter ρ measure?
The investors attitude towards risk, i.e., the degree of risk aversion.
What happens if we add Borrowing constraints to the Dynamic model?
Constraints gives additional motives for saving since it adds more uncertainty.
What about complete markets?
Don’t understand this part
