6
Intuition
Well, when $r$ is zero, you are just getting value $A$ per time unit, so total present value will be $A \cdot t$.
The approximation
You are asking about the approximation
$$
\frac A {-r}(e^{-rt}-1) \approx A \cdot t.
$$
This is equivalent to
$$
\frac 1{rt}(1-e^{-rt}) \approx 1,
$$
which is true for small $rt$ values. You can see this by taking ...
4
One interpretation I can offer. The demand function can be expressed as:
$$Q_1 = Q_1(p_1,p_2)$$
Let us take the total differential:
$$dQ_1 = \frac{\partial Q_1(p_1,p_2)}{\partial p_1}dp_1+\frac{\partial Q_1(p_1,p_2)}{\partial p_2}dp_2$$
Assume that $Q_1$ remains unchanged with respect to a change in prices. This implies that $dQ_1=0$. Solving the ...
3
FE logit requires the idiosyncratic errors to be IID across $i$ and $t$, quite a strong assumption. Also the regressors should be strictly exogenous, but it's the same for linear FE models. In your application, the fact that FE logit wouldn't converge will make a good argument against FE logit, and will satisfy some referees but not all.
An important ...
3
Rationality requires the following:
Completeness
For all $x, y \in X$, either $x \succsim y$ or $y \succsim x$ or both.
Transitivity
For all $x, y, z \in X$, if $x \succsim y$ and $y \succsim z$, then $x \succsim z$.
Also note that if $x=y$ then completeness implies that $x\succsim x$. So reflexive preferences follow from completeness. So I would say ...
3
Reduced form is a regression of dependent variable on instrument directly without using some two stage approach.
Consider the following example of endogenous system
Second Stage: $$Y = \alpha + \beta X + \epsilon$$
First Stage: $$X = \mu + \gamma Z+ \eta $$
Where $Y$ is dependent variable $X$ endogenous regressor and $Z$ is your instrument.
One ...
2
I am not an econometrician so this will be a very informal explanation how to get the Instrumental Variable (IV) estimates. Since $S_i$ is endogenous,
$$\text{cov}(S_i,\epsilon_i)\neq0$$
We can think of it as having two components:
$$S_i = v_i + u_i$$
Suppose, $\text{cov}(u_i,\epsilon_i) =0 $ and $\text{cov}(v_i,\epsilon_i)\neq0$. $v_i$ is the ...
2
Remember that consistency describes how the estimator behaves in the limit as N asymptotically approaches infinity. Assuming no errors in your math up to this point, you need to consider how your error terms $U_i$ behave asymptotically as well.
2
Assumptions $1$ to $3$ are sufficient to obtain a linear representation when $X$ is open and convex. We proceed in two steps.
Step $1$:
We will repeatedely use the following consequence of continuity and $A1$:
If $x \sim x^{\prime}$, then $x \sim x + \lambda (x^{\prime} - x)$ for every $\lambda \in \mathbb{R}$ such that $x + \lambda (x^{\prime} - x) \in X$....
2
Your value function is as follows:
$$
V_t[w] = \max_{c_t \in[0,w]} \left\{u(c_t) + \frac{1}{2}V_{t+1}[\alpha(w_t - c_t)] + \frac{1}{2}V_{t+1}[\beta(w_t-c_t)] \right\}
$$
with the terminal condition
$$
V_{T}[w_T] = \max_{c_T \in [0,w_T]} u(c_T)
$$
So, we can solve this via backward induction. Clearly, at the final period $T$, since $u$ is monotonic, we ...
1
Captive insurance is an insurance for purpose of insuring the owners of the said insurance. It’s a kind of self-insurance where multiple agents pool their capital together to insure themselves. So the term in that sentence refers to the pool of resources for insurance.
It’s called captive because the modern concept of this kind of insurance was first ...
1
Well resource depletion could create recession as it could reduce the productive capability of economy. For example, if oil would suddenly run out tomorrow it would definitely cause supply side shock that would result in recession.
Rather, the fallacy behind resource depletion scare is the resource depletion itself. For example, more oil is available now ...
1
If GDP is measured (as it usually is) for a period such as a year, then number of hours worked ($h$) should be measured for the same period. Thus $h$ is dimensionless and the dimension of the right hand side is also value / time.
1
This problem is quite specific to economics. The correct statement is:
Proposition
If $u(\cdot)$ is quasiconcave, strictly increasing, and continuous, then $\forall x$, there exists $p \gg 0$ and $w \geq 0$ such that $x \in x^*(p, w)$, where $x^*(p, w)$ is the Marshallian demand correspondence.
Proof
Quasiconcavity of $u$ means the upper-contour set $\...
1
What one needs are two facts:
Scalarization/Sufficiency for Pareto Optimality
Suppose the economy has finitely many agents $u_i$, $i = 1, \cdots, I$, with feasible allocations given by some $X \subset \mathbb{R}^I$. If
$x = (x_1, \cdots, x_I) \in X$ solves the problem
$$
\max_{x \in X} \sum_{i=1}^I \lambda_i u_i(x_i),
$$
where $\lambda_i > 0$ for all $i$,...
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