6
The utility function under consideration is $v(c,q)$ and then
$$MRS(c,q) = \frac{\partial v/\partial q}{\partial v/\partial c} = v_2/v_1$$
make the functional denpendency of on $u$ explicit then you have
$$\frac{\partial}{\partial u}MRS(c(u),q(u)) = \frac{\partial MRS(c(u),q(u))}{\partial c} \frac{\partial c(u)}{\partial u} + \frac{\partial MRS(c(u),q(u))}{\...
5
I think you have a typo in your $q_0$: the exponent of $N$ should be $-\frac{b}{a+b}$. I did the whole calculus with this corrected type of $q_0$ and I was able to replicate your results (this is why assume $q_0$ has only a typo and you actually did the algebra with the correct $q_0$).
I suggest that the discrepancies to the paper are indeed connected to ...
5
The assumption $dm =0$ says that we examine the behavior of the consumer under a fixed nominal income, and this is something interesting to study, because it aligns to a large degree with the observed reality of many people that have approximately constant income.
The assumption $dp_2=0$ assumes away general equilibrium effects, since we are looking at ...
4
\begin{align}y_t &= \alpha + \theta_1y_{t-1}+u_t \\
&= \alpha+\theta_1(\alpha + \theta_1y_{t-2}+u_{t-1}) + u_{t} \\
&= (1+\theta_1) \alpha + \theta_1^2y_{t-2} + \theta_1u_{t-1}+u_{t} \\
&= (1+\theta_1) \alpha + \theta_1^2(\alpha + \theta_1y_{t-3}+u_{t-3}) + \theta_1u_{t-1}+u_{t} \\
&= (1+\theta_1 + \theta_1^2) \alpha + \theta_1^3y_{t-3} ...
1
So graphically this is pretty straight forward, as you may have already understood: Since the LM curve has a positive slope the entire shift of IS curve ($\frac{\partial Y}{\partial G})$ is not fully translated into final output. It would happen when LM curve is flat (which is usually the case at very low interest rates in liquidity trap situation).
...
1
$$Var[y_t] = E[(u_t+\theta_1u_{t-1}+\dots+\theta_1^{t-1}u_1)^2] = E[u_t^2]+\theta_1^2E[u_{t-1}^2]+\theta_1^4 E[u_{t-2}^2] + \dots +\theta_1^{2t-2}E[u_1^2]$$
the latter equality follows from the assumption that $u_t$ is not serially correlated (i.e. $E[u_{i}u_{j}] = 0\ \forall\ i \neq j$). Then, since $E[u_{t}^2] = \sigma^2\ \forall\ t$ it follows that
$$Var[...
1
Assuming the following:
$ S > 0 $, $T > 0 $,
$ c(S) = \left\{c\in\Bbb{R}^T_{+} : \sum_{t=1}^T{c_t} = S \right\} $, and
$ U(\cdot) $ continuous.
Let's start with convexity. To demonstrate convexity, we need to show that for any two points $c^1$ and $c^2$ in $c(S)$, any linear combination of the two is also an element of $c(S)$.
Let $\lambda \in [0,1]$,...
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