# Tag Info

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). ... 1Var[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[...
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 $\lambda \in [0,1]$,...