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4

The literature is full of examples in which either individual rationality leads to aggregate rationality individual rationality does not yield aggregate rationality (when public goods or externalities are considered; Arrow's impossibility theorem also falls into this case) lack of individual rationality yields lack of aggregate rationality is obvious and ...

2

Satiation means there is a consumption bundle that is at least as good as any other consumption bundle. Monotonicity means that larger bundles are always better. Since there is always a larger bundle, no bundle can be at least as good as any other consumption bundle when preferences are monotone. So monotone preferences are never satiated.

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This paper describes the distinction between implicit, tacit, and explicit knowledge: http://www.mkdavies.net/Martin_Davies/Recent_files/KnowledgeExpImpTacit.pdf If utility functions are tacit and implicit there is no secondary explicit description of whether or not behavior motivated by such utility functions is "rational" or "irrational"...

5

Can you explain the rationale behind the above statement in further details? This is an example of an emergent property or sometimes called just emergence. Emergence in layman's terms denotes situations where the whole is more than just a sum of its parts. For example, individual molecules of $\text{H}_2\text{O}$ (water) are not wet themselves, but if you ...

3

As you can tell from the comments, there are multiple versions of monotonicity in preference theory, but let us assume the (in my opinion) most intuitive one for now, and say A preference $\succeq$ represented by utility function $u(\cdot)$ is strictly monotonic if whenever $x > y$, $u(x) > u(y)$ Let us consider the most degenerate satiated preference ...

2

Here's one way to think about it: Assume your preferences are differentiable, U = f(x). If your preferences are monotonically increasing, meaning: $U_1 = f(x)$ $U_2 = f(x+e)$ $U_2 > U_1; e > 0$ Then the first derivative is always positive, since a derivative is: $\lim\limits_{e \to 0} \frac{f(x+e)-f(x)}{e}$ If a first derivative is positive, the ...

1

Let $c(p, u)$ be the expenditure function. The Hicksian demand for good $j$ is the derivative of $c$ with respect to $p_j$. $$\frac{\partial c(p,u)}{\partial p_j} = h_j(p,u).$$ From this, it follows (by Young's theorem) that: $$\frac{\partial h_j(p,u)}{\partial p_i} = \frac{\partial^2 c(p,u)}{\partial p_j \partial p_i} = \frac{\partial^2 c(p,u)}{\partial ... 2 I wouldn't say that Varian's diagram is wrong: it's more that his explanation of the MC_F function (see 34.3 on p 652) is incomplete (and to be fair textbooks do sometimes need to simplify matters in order to focus on key points). This is how I would analyse the effect on the fishery of changes in the quantity of pollution. The fishery's profit function is:... 0 This can be easily done by taking price index and by dividing each index value by the index value in the new base year you selected and then multiplying by 100. For example, if you have two indexes with different base year (2001 and 2002 respectively): Index 2000 2001 2002 I_1 75 100 125 I_2 50 75 100 You can create new ... 1 You are both right. Varian doesn't include the f-argument in his MC_f function, so his graph is correct if that function is your red one. Note that he nowhere claims that his MC_f function shows the one of the fishery before integration. 0 Suppose the voters are A,B, and C, and the players are x,y, and z, and the preferences are A: x,y,z B: z,x,y C: y,z,x All of their totals are 6, so it's a 3-way tie. However, two-thirds of the voters prefer x to y, so if z were removed, x would win. So this violates the irrelevant alternatives condition. It is a weak violation, though, as y goes from a tie ... 4 You have to show that something does not hold universally true. To show this, you just have to show that there is at least one exception- a counterexample. For this counterexample, you can make any assumption that does not contradict the assumptions of your problem. 5 (1) Satisfying completeness, independence, and continuity but not transitivity: Take two outcomes, \{0,1\}, and the associated lottery space [0,1]. Consider the preference relation \succsim where x\sim y if and only if x=y, x=0 and y=1 or vice versa, x\succ y if and only if x>y, except for x=0 and y=1 or vice versa. Transitivity is ... 2 Usually, there won't be equal-treatment. This follows from the results in Green, Jerry R. "On the inequitable nature of core allocations." Journal of Economic Theory 4.2 (1972): 132-143. Green shows, as a simple corollary tp the classical equal treatment result of Debreu and Scarf, that if preferences are continuous, strictly monotone, and strictly ... 3 Assume that, towards a contradiction, that both \succeq and \succeq^\ast rationalise the choice function and that they are different. The fact that \succeq and \succeq^\ast are different means that there should exist options x, y such that \succeq and \succeq^\ast disagree on the preference over x and y. So, for example, x \succeq y and ... 2 When solving this problem, it is important to notice that whenever you have a tax, there will be a difference between the price that the buyers pay and the price that the sellers will receive. Let p_D be the price that the buyers pay and let p_S be the price that the sellers receive. The price p_D is the one that is relevant for the buyers, so it ... 3 Let c be the cost per unit of insurance, so the premium is equal to cn. Then the agent maximises:$$ p(u(w - L - cn + n) + (1-p)u(w - cn). $$The first order condition with respect to n is given by:$$ (1 - c) p u'(w - L - cn + n) - c (1-p) u'(w - cn) = 0 $$Rearranging gives:$$ \frac{p}{1 - p} = \frac{c}{1 - c} \frac{u'(w - cn)}{u'(w - L - cn + n)}. $... 0 I have a different take on the question: Suppose a representative shareholder owns$n$firms. The shareholder owns equal shares of the$n$firms, and the firms are identical and independent. For simplicity, assume the shareholder has starting starting wealth one, and their preferences are an increasing function only of ending wealth. Firm$j$has rate of ... 2 Let$p$be price vector, let$m$be income and let$u$be utility. Let$e(p,u)$be the expenditure function which gives the minimal expenditure necessary to get utility level$u$and let$v(p,m)$be the indirect utility function, which gives the maximal utility that can be obtained ad prices$p$given income$m$. The Hicksian demand for good$i$,$h_i(p,u)$... 4 Suppose you are an analyst studying a Bayesian game. You know the players, the possible states of nature, the common prior, the action spaces, the payoff functions, and you know about some information channels available to the players, the latter given via some information structure. However, you can't rule out that the players have additional information ... 2 Touch wood that I did not make any mistakes. Consider the production function$X = (K + \alpha)(L + \beta)$. The elasticity of substitution is given by: $$\frac{\partial \ln(K/L)}{\partial\ln(MP_L/MP_K)} = \frac{\partial(\ln(K) - \ln(L)}{\partial(\ln(K + \alpha)- \ln(L + \beta))}$$ Let's take the derivative of both numerator and denominator with respect to ... 3 The initial definition of a correlated equilibrium in [Aumann, Robert J. "Subjectivity and correlation in randomized strategies." Journal of Mathematical Economics 1.1 (1974): 67-96.] was with respect to some underlying probability space (more actually, to take account of the "subjectivity part"), and private information represented by$\...

0

First, $l(\tau)(y)$ is a function of both $\tau$ and $y$, and the dependence on $\tau$ is essential; this is how communication happens. Since $T$ and $Y$ are finite, there is no point in introducing integrals. $\mathbb{E}_{l(\tau)}g(F(y))$ is the expectation over the function when the random value $y$ is distributed according to $l(\tau)$. That is, \mathbb{...

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