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15

For a shorter proof, here are a few things we need to know before we start: $X_1, X_2 , ..., X_n$ are independent observations from a population with mean $\mu$ and variance $\sigma^{2}$ $\mathbb E(X_i) = \mu$ , $\mathbb{Var}(X_i)= \sigma^{2}$ $\mathbb E(X^2) = \sigma^{2} + \mu^{2}$ $\mathbb{Var}(X)=\mathbb E(X^2)-\mathbb [E(X)]^2$ $\mathbb E(\bar{X}^2) ... 9 I think what you need is that if$U(x,y)$is homothetic then $$\forall \alpha \in \mathbb{R}_{++}, \forall (x,y) : \hskip 6pt \frac{\frac{\partial U(x,y)}{\partial x}}{\frac{\partial U(x,y)}{\partial y}} = \frac{\frac{\partial U(\alpha \cdot x,\alpha \cdot y)}{\partial x}}{\frac{\partial U(\alpha \cdot x,\alpha \cdot y)}{\partial y}}$$ and love. 9 I know that during my university time I had similar problems to find a complete proof, which shows exactly step by step why the estimator of the sample variance is unbiased. The proof I used can be found under http://economictheoryblog.wordpress.com/2012/06/28/latexlatexs2/ The proof itself is not very complicated but rather long. That also the reason why ... 8 While taking Industrial Organization I remember working with: Strategies and games: theory and practice by Dutta Introduction to industrial organization by Cabral Industrial organization: theory and applications by Shy Industrial Organization: Markets and Strategies by Belleflamme and Peitz The first two are rather introductory while third and forth are ... 7 From Varian (7th edition): Consumer behavior: at least chapters 1–6; preferably also chapters 7, 8, and 12. Perfect competition and Theory of production: chapters 15, 16, and 18-23. Monopoly price discrimination: chapter 25 (you might want to look at chapter 24; it seems a bit odd to study price discrimination without first looking at a non-discriminating ... 7 Let$z_1$and$z_2$be$\geq 0$and solution to $$\min_z \{w^\top z\lvert f(z)\geq q\}$$ then clearly$f(z_1)\geq q$and$f(z_2)\geq q$and since$\{z\geq 0\lvert f(z)\geq z \}$is convex it then follows that$z_3 := \lambda z_1 + (1-\lambda)z_2$must satisfy the constraint$f(z_3)\geq q$. Since$z_1$and$z_2$are both minimizers it cannot be the case that$...

6

My prof is always telling us, if we want to pursue PhD level Econ in the future, we should master the full content of the following book: Microeconomic Theory. Andreu Mas—Colell Michael D. Whinston and. Jerry R. Green. New York Oxford OXFORD UNIVERSITY PRESS 1995. He also mentioned that there's main difference in viewpoint between graduate-level and top-...

6

Don't commit the cardinal mistake of equating preferences with choices. In the context of Expected Utility Theory, the fact that a risk-averse agent ($RA$) would choose $N$ over $M$ implies that $$E[u_{RA}(N)] > E[u_{RA}(M)]$$ The fact that a risk-neutral agent ($RN$) could choose $M$ over $N$ implies that $$E[u_{RN}(N)] < E[u_{RN}(M)] \implies ... 6 Hidden information concerns characteristics that are unobservable by one side of the market. For example, a consumer's willingness to pay, a worker's productivity, the quality of a used car all fall under this category. The characteristics in question are typically assumed to be fixed or very costly to modify. Moral hazard concerns actions that are ... 6 It's great that you're developing an interest in economics. I would suggest Mankiw's Principles of Economics to start with. I believe it meets both your requirements and covers the two major areas of economics, microeconomics and macroeconomics, so you would get a decent overview of this field of study. Good luck! 6 A couple hints. Regarding the lower bound on \epsilon: What happens if deviation occurs at stage T? In other words, there is no opportunity for your so-called "punishment stages". Regarding the upper bound on \epsilon: Suppose player 2 deviates at stage T-1 but player 1 does not. What must be true about \epsilon in order for player 1 to ... 5 In an intertemporal maximization problem, we seek to find the optimal sequence of the control and the state variables. It is the recursive nature of the problem that permits us to consider a "typical" point in time and just one condition per variable. For each such problem, we need to find out (carefully) in how many distinct periods a specific ... 5 In the preface, Takayama writes that the book was written with the intention to keep the prerequisites to a minimum: elementary calculus and matrix algebra. Perhaps he was exaggerating a little, but I suspect, after skimming the table of contents, that knowledge of the aforementioned subjects and experience working with (i.e. reading/understanding and ... 5 I've just solved this problem. First of all, your solution does not make too much sense, as in a simple interest rate rule it must hold that the sum of all coefficients must be greater than one. In your case this means that \phi>1. Therefore, the series would converge not to zero. Second, an interest rate rule should try to offset fluctuations. This ... 5 The way you show that v(p,m) is homogeneous of degree one in m is correct, but the reason why this implies that, e(p,u) is homogeneous of degree one in u, is not very precise in your argument. For example, duality tells us$$v(p,e(p,u))=u,where u is just a target utility level, but should not be u(x) as in your proof. Here is one possible way ... 5 (i) A & B If player 1 play A with probability p and B with probability (1-p), where 0<p<1, then player 2's expected payoff from playing D is 4p+4(1-p) = 4 E is 6p + 2(1-p) = 4p + 2 F is 6p + 4(1-p) = 2p + 4 Since payoff from playing F is more than the payoff from playing any other strategy for player 2, he will ... 5 The following argument assumes that we are dealing only with interior solutions. Let (x_1,x_2,x_3) be an optimal demand bundle at prices (p_1,p_2,p_3) and income m and assume that (x_1,x_2,x_3) is an optimal demand bundle at prices (p_1',p_2,p_3) and income m. We want to show that one of the following three cases holds: x_2=x_2' and x_3=x_3'... 5 All four of those books are widely used at the undergraduate level. They are written with the intention to be text-books of a class, so they should be good for self-study. I recommend you to read 1 and 2 before reading 3 and 4. Will give you a basic overview of the way we think in economics, clarify some basic concepts and explain work-horse models in the ... 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 Let's improve the "answers per question" metric of the site, by providing a variant of @FiveSigma 's answer that uses visibly the i.i.d. assumption (showing also its necessity). We want to prove the unbiasedness of the sample-variance estimator,s^2 \equiv \frac{1}{n-1}\sum\limits_{i=1}^n(x_i-\bar x)^2 using an i.i.d. sample of size $n$, from a ...

4

As far as I remember, Varian's book is aimed at second year undergraduates who are normally studying a micro 2 module (or something similar). As a result, he does tend to assume some prior knowledge so it would be beneficial for you to plug the gaps in your knowledge (assuming you haven't) before you proceed with this book. Even though you are studying ...

4

Note that I am not 100% sure. But in my understanding, we have Year 1 Price for a product in the US : $p_{US}=v$ \$Exchange rate:$x$pesos for$1$\$ Price of the product in the Philipines: $p_{Ph})=v.x$ pesos Year 2 Price for the same product in the US : $p^\prime_{US} = (1+\alpha_v)v$ \$. The price increased due to the inflation$\alpha_v$. Nominal ... 4 I assume what you're asking based on your comments is: "How can I visualize indifference curves for 3 goods?" I can think of three options: 1) Use a tool like Matlab, or its open-source equivalent, Octave, to plot 3 dimensional indifference curves. Here is a tutorial on how to do that. 2) Make a series of 2-dimensional indifference curves for two of the ... 4 While textbooks are the best way to learn the material (MC=MR etc), here are some suggestions for improving your intuitive understanding of economics. Books The Undercover Economist by Tim Harford The Armchair Economist by Steven E. Landesburg Blogs Marginal Revolution Noahpinion Conversable Economist The Enlightened Economist (Great for book ... 4 Firm 1's profit maximization problem is : \begin{eqnarray*} \max_{q_1} & \ p(q_1 + q_2 + q_3)q_1 - c(q_1) \end{eqnarray*} Firm 1's response to firm 2's and 3's quantity choices will satisfy the following: \begin{eqnarray*} p(q_1 + q_2 + q_3) + p'(q_1 + q_2 + q_3)q_1 - c'(q_1) = 0 \end{eqnarray*} Likewise, we'll get the implicit best response functions ... 4 In part a, if$B = D_1 + D_2$, then the SGPE should be$\left\lbrace D_1 = \frac{W}{3},\ D_2 = \frac{W}{3}, \left\lbrace P = \alpha (D_1 + D_2), \ S = (1 - \alpha) (D_1 + D_2) \right\rbrace \right\rbrace$Don't say$P = \alpha \frac{2W}{3}$. That's an action, and the second stage best respond should a strategy (function) to make the equilibrium subgame ... 4 We'll first find manager's strategy. Manager of the charity chooses$S$and$P$by solving the following problem : \begin{eqnarray*} \max_{S, P} & \ \frac{P^a S^{1-a}}{B^a} \\ \text{s.t.} & \ P+S = B \end{eqnarray*} where$B = D_1+ D_2$. Solving it we get the manager's strategy as a function of donations$D_1, D_2$as : \begin{eqnarray*} P &=&... 4 "Endogenous Growth" is actually the short version of saying "Endogenous Technology Growth" Exogenous (Technology) Growth Models The rate technological progress$g$is Exogenously given. In both Solow and RCK, we can find$A_t = (1 + g)^t A_0 \ \ $(or$A(t) = A(0) e^{gt}$if in continuous time).$Y$increases over time because$A$increases over time. This ... 4 Prove that$\Pi_j(x_j)$is strictly concave in$x_j$.$\Pi_j(x_j) = G(x_j) + F\left(\frac{x_j}{y}\right) =G(x_j) + F\left(R_j(x_j)\right) $Differentiating it we get$\Pi_j'(x_j) =G'(x_j) + F'\left(R_j(x_j)\right)R_j'(x_j) $Differentating$\Pi_j'(x_j)$, we get$\Pi_j''(x_j) =G''(x_j) + F''\left(R_j(x_j)\right)(R_j'(x_j))^2 + F'\left(R_j(x_j)\right)R_j''...

4

Firm $i$'s profits $(\pi_i)$ as a function of its own price $(p_i)$ and the other firm's price $(p_j)$ are as follows : \begin{eqnarray*} \pi_i(p_i, p_j) = \begin{cases} (p_i-200)\min(1000- p_i, 300) & \text{if } p_i < p_j \\ (p_i-200)\min\left(\frac{1000- p_i}{2}, 300\right) & \text{if } p_i = p_j \\ 0 & \text{if } p_i > p_j\end{cases} \...

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