Exercise 1 Flashcards
What is the elasticity of f (x ) wrt. x. What are the two commonly used approaches
The elasticity of f (x ) wrt. x is the percentage change in f (x ) when x
increases with one percent.
See the two main approaches in exercise 1.
What is the elasticity of marginal utility. How do you calculate the elasticity of marginal utility.
Shows how fast the utility changes with consumption. The second order derivative (u´´(c)), shows how fast the marginal utility changes (u´(c)).
How to calculate see exercise 1.
What is the product rule
d/dx(fg)=f´g+f*g´
Why do we use L´Hospitals rule
L’Hôpital’s rule is very useful for evaluating limits involving the indeterminate forms 00 and ∞/∞.
However, we can also use L’Hôpital’s rule to help evaluate limits involving other indeterminate forms that arise when evaluating limits. The expressions 0⋅∞,∞−∞,1∞,∞0, and 00 are all considered indeterminate forms.
What is the chain rule
Take the derivative to the outer function and times that with the derivative of the inner function.
d/dx(f(g(x)))=f´(g(x))*g´(x)
How do you use L´Hospitals rule
When we have indeterminate forms in a fraction we can use L´Hospitals rule.
Se exercise 1 how to write down the rule.
When we want to evaluate the limits involving indeterminate forms when x approaches zero. First we take the derivative of both functions. If we still get an indeterminate form, when x goes towards zero, then we have to take the derivative again. Until we have an expression, where there are no indeterminate forms.
Is it logical that σ = 1/ε?
Vi har to forbrugsbundter. I hvert forbrugsbundt er der 2 varer med forskellige priser. ε = elasticity of the marginal utility and σ = substitution effect.
Hvis ϵ er høj, så vil agenten vælge at beholde sit fobrugsbundt, fordi dens marginale nytte fra vare 1 til 2 ændrer sig meget, hvis vi ændrer forbrugsbundtet. Derfor er substitutions effekten mellem de to forbrugsbundter lav, hvis der kommer en ændring i den relative pris på varene i det ene bundt.
Omvendt hvis substitutionseffekter høj, hvilket betyder at agenten er mere villig til at skifte mellem de to forbrugsbundter, så vil en relativ prisændring i det ene bundt ikke betyde så meget for agentens marginale nytte, da han bare vil skifte (substituerer) over til det andet forbrugsbundt. Altså ændrer agentens marginale nytte fra vare 1 og 2 ved en ændring af forbrugsbundt sig ikke særlig meget, hvis den vælger at skifte til det andet forbrugsbundt.
What does σ measure in the Ramsey Model
σ measures the preference for consumption
smoothing across periods, which is why CRRA utility functions are
useful, as they imply a constant σ. They are easy to use.
What is the Euler equation, u′(c_t)/u′ (c_t+1 ) = βR_t+1, used for.
Which 3 factors is the slope determined by?
What is the intuition of the Euler equation, when we think about the effects of changes in the interest rate. R_t.
The Euler equation is used to characterize the slope of the optimal consumption path. The slope of the consumption path is determined by 3 factors.
When patience rises, β ↑, more value is placed on future consumption, c_t+1/c_t ↑.
A higher interest rate, R_t+1 ↑, makes future consumption cheaper, c_t+1/c_t ↑, because current consumption is relatively more expensive
Lastly, a larger curvature of the marginal utility
function, σ ↑, yields a lesser degree of changes in the consumption path of changes in β and Rt+1. If σ <1, then the substitution effect dominates the income effect. If σ =1, then the substitution and income effect cancels out (when we assume log-utility, the income and substitution effects of interest rate changes cancel each other out). The income effect dominates when σ > 1.
It gives us som intuition about the effects of changes in the interest rate, R_t.
The negative first income effect (also known as the wealth effect), as R_t+1 ↑ lowers the discounted value of future income and thus total consumption, the positive second income effect, as Rt+1 ↑ lowers the price of a given consumption bundle, and the substitution effect towards the cheaper good such that Rt+1 ↑ means c_t+1/c_t ↑.
What is the No-Ponzi-Game condition
In this case, the no Ponzi game condition is such that growth rate of debt has to be lower than the real interest rate. Public debt is then exactly equal to the present value of all the discounted future primary (positive or negative) surpluses.
Se the condition in exercise 1
What does the No-Ponzi-Game condition (NPGC) and Transversality condition (TVC) combined prevent?
The NPGC constraint combined with the TVC prevents households from
rolling over debt infinitely without serving it. I.e. agents cannot finance
interest payments by taking up new loans. The activities at the end of the
period must be 0.
Se the conditions in exercise 1
What is the law of large numbers
It states that if you repeat an experiment independently a large number of times and average the result, what you obtain should be close to the expected value.
What is the Intertemporal Budget Constraints (IBC)
Since consumption decisions are taken over a period of time, consumers face intertemporal budget constraint, which shows how much income is available for consumption now and in the future. This constraint reflects a consumer’s decision on how much to consume today and how much to save for the future.
c_0+c_1/R1 =w_0+w_1/R1
In the second period of consumption c_1, the price of one unit of consumption depends on the the size of R1. The same for the salary. Therefore when R1 rise –> c_1 and w_1 falls. Lower consumption gives lower wages.
Explain the dynamic budget constraint
Assuming an inelastic labour supply and ruling out risk and leisure, the DBC is given as
a_t+1 =a_tR_t +w_t −c_t ⇔∆a_t+1 =atrt +w_t −c_t,
where Rt = 1 + r_t − δ is the gross interest rate, at is assets from which the household receives interest
incomes, wt is salary, and ct is consumption.
What is CRRA-preferences.
Our utility function has CRRA-preferences.
We usually assume CRRA-preferences, where σ is the degree of constant relative risk aversion.
If σ <1, then the substitution effect dominates the income effect. If σ =1, then the substitution and income effect cancels out (when we assume log-utility, the income and substitution effects of interest rate changes cancel each other out). The income effect dominates when σ > 1.
Our utility function, u(c) has either preferences like
u(c) = (c^(1−σ)−1)/c-σ if σ ≥ 0, σ ikke er 1
or
u(c) = ln(c) if σ = 1