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The answer to 2. is no. One way to see this is from MWG's Property 6.D.2: $F$ SOSD $G$ if and only if $$\int_0^xF(t)\mathrm dt \le \int_0^xG(t)\mathrm dt \quad\text{for all }x.$$ Dixit calls the two integrals super-cumulative functions of $F$ and $G$, respectively. Hence, a characterization of SOSD is that the super-cumulative of ...

7

The answer to 1. Your conjecture is correct. Consider lotteries $A,B$ where $A$ guarantues a payoff of 1 while $B$ yields 0 or 4, each with 50% probability. $B$ does not SOSD $A$, as you can easily find an agent risk averse enough that they will prefer $A$, e.g. an agent whose preferences are described by $u(x) = \ln(x)$. $A$ does not SOSD $B$ either, as $E(... 3 I don't think it even makes sense to talk about risk without specifying the payoffs. Take your two examplary gambles and suppose that$a=b=c=d=e=0$. In that case, there is no risk involved at all. It is like flipping a coin and if it lands Heads, nothing happens, and if it lands Tails, nothing happens. This would not even be considered a gamble. ... 3 Let$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)}.$...

3

$p\circ x\oplus(1-p)\circ y$ is a lottery that gives you the prize $x$ with probability $p$ and the price $y$ with probability $(1-p)$. Unless $x,y$ can be identified with numbers, such as amounts of money, it makes no sense to take the expectation of this lottery. What is the expected value of a lottery that gives you a cow with a probability of $0.5$ and a ...

2

I’d start by briefly noting that “the Fed” didn’t say anything; that’s a research paper (by some smart people), but it’s disclaimed and doesn’t necessarily reflect the views of anyone other than the authors. Having said that, I would then rephrase your question as follows: “what happens if everyone decides to cash in their retirement funds at once?” The ...

2

Let $\pi$ be the fine that needs to be paid besides the parking fee (say, $\phi$) if he gets caught. Further, he may or may not get caught for not paying. So let $p$ be the probability perceived by John about whether he will be caught. Let $u(x)$ be utility function of paying for amount $x$, with $u(0)=0$, $u'(x)<0$. If he pays the parking fee $\phi$ he ...

2

I am not going to give a solution to you question (as I am not able too). However, let me try to indicate the difficulty of obtaining one if you cannot find a closed form solution for $q_1$ and $q_2$ in the last stage of the game. Let $\Pi(q_1, q_2, p_1, p_2, B)$ be the objective function that we want to maximize. Assuming we can totally differentiate this, ...

1

They differ in the type of Derivative Contract that is chosen. A Swap would be an agreement with a second counterparty, in which in your example, the bank would swap or trade their interest rate asset, with a second counterparty, and which the bank would receive another asset from the second counterparty, and they agree to hold these swapped assets for a ...

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