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It is the first one, $TC(0) = FC$. This is the definition. Also consider that it is not clear what is "transformed by $q$ in some way". In case of $$\frac{5q}{q+1} + \frac{5}{q+1}$$ are the two fractions transformed by $q$, or should I just sum them up to 5? With your function, one can rearrange it to $$TC(q) = \frac{5}{q+1} + 5 + 5q + q^2 = -\frac{5q}{... 4 Let me take an example based on the estimation of returns to education, which has been a well-studied problem. The usual result is that researchers find the 2SLS estimate to be larger than the OLS estimate by approximately 25%-50%, e.g. Card (1999, 2001). Three reasons : An omitted variable that could be negatively correlated with the amount of education. ... 4 This is only partial answer since your question is quite broad but in my work I had similar problems. When ti comes to optimal values and derivatives I choose to use Leibniz's notation as it does the job quite nicely: instead of writing this \pi^{'*}, I would write this \frac{\partial \pi^*}{\partial x} or if you like it more \frac{\partial }{\... 3 I’ve worked in a few areas of applied mathematics, and each field has its own conventions. You just need to be internally consistent. Just some comments. As noted in another answer, using ‘ to denote derivatives is awkward. (I’ve not seen it done since high school physics.) If you are indexing things, subscripts are pretty much a necessity. Superscripts ... 2 Actually, neither demand for Veblen good nor for Giffen good is strictly increasing in price. In case of Giffen good the demand actually looks as shown below in picture 1. The reason for this is that you can only increase demand for the Giffen good up until you consume your entire budget. Once the price gets higher then that you still get normal downward ... 2 So we have S_n \thicksim^{iid} \ ? \bar{s} = \frac{1}{n}\sum{s_n} Exponential Suppose S_n follows the exponential distribution.$$f(s|\beta) = \frac{1}{\beta} e^{-\frac{s}{\beta}} \quad , \quad 0 \leq s < \infty \quad , \quad \beta > 0$$Take the simple bivariate case. Say Z = \frac{S_1 + S_2}{2} and W = S_1. So S_2 = 2Z - W and S_1 ... 2 The above answer by Regio covers most of the answer Just to emphasize further : Consider a 2 goods case. We define MRS as the opportunity cost of consuming one more unit of good 1. Opportunity cost means "What Am I Giving Up ?". Since you are consuming only two goods, you are giving up some amount of good 2 in order to consume this one more unit of good 1. ... 2 It is implicit in the interpretation: Mas-Collel: the amount that must be given (+) to compensate for a reduction (-). Reny: The rate at which good j can be exchanged (+ & -) for good i. The derivation from total differentiation only requires the utility to be constant, so the derivative must be negative to express that if the quantity of good i... 2 This is not true. Let n=1 and define u(x)=\min{\{x,0\}}. Let \succsim be the preference relation represented by u. This preference relation is continuous and convex. We also have x\sim y implies x+a\sim y+a for any a\geq0 and x,y\in\mathbb R. But let x=0, y=1, and a=-1. Then x\sim y, but y+a=0\succ -1=x+a, thus x+a\nsim y+a and \... 1 From your formula for x when y=0, you should be able to find U in terms of M and p_x when y=0. Similarly, U in terms of M and p_y when x=0. The key then is to find the critical price ratio at which, to maximise U, the switch needs to occur from y=0 to x=0. Can you take it from there? 1 TC(q)=10+3q+0.5q^2 is a quadratic cost function and has a shutdown point at P=3. 1 It is not true. Let us consider \mathbb{R}^2 so bundles are x = (x_1,x_2). Consider the preference: (i) If x_1 \leq 0, preferences are lexicographic, i.e.$$ x \succ y \Leftrightarrow \begin{cases} x_1 > y_1 \\ \text{ or } \\ x_1 = y_1 \text{ and } x_2 > y_2 \end{cases}  (ii) If $x_1 \geq 0$, $u(x_1,x_2)=x_1+x_2$. Notice that no ...