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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 ...

0

hint: This is constrained optimization problem. The constraints your objective functions are subject to are $x_1≥0,x_2≥0,x_3∈\{0,1\}$ and $1 \cdot x_1 + 1 \cdot x_2+1000 \cdot x_3≤w$. Hence consumption here of any of the good here can't reach infinity, $x_3$ is explicitly constrained to be between 0 and 1 and consumption of $x_1$, $x_2$ and $x_3$ multiplied ...

3

Here's my guess. Let use the notation $$\tilde x_t \approx \ln(x_t) - \ln(x) \approx \dfrac{x_t - x}{x}.$$ If we take logs on both sides we get: $$\ln(G_t) = \frac{1}{1 -\rho} \ln(p_t) + \ln(y_t) - \ln(1 + p_t^{\frac{\rho}{\rho-1}})$$ Subtracting the steady state gives:  \tilde G_t = \frac{1}{1 - \rho} \tilde p_t + \tilde y_t - \left[\ln(1 + p^{\frac{\...

1

The following may work. Let there be two consumers. Consumer 1 has standard (convex, continuous, strictly increasing) and let consumer 2 have constant utility (i.e. indifferent between any two consumption bundles). This still satisfies your condition. Then given some endowments, and arbitrary prices, let consumer 1 maximize her utility. Then consumer 2 will (...

3

Couldn't it be that there's some allocation that is Pareto optimal that isn't the one that maximizes the first investor's expected utility? My understanding was that Pareto optimal just meant that there was no way to make one person better off without making another person worse off. Let me consider the case of a simple resource economy such that we don't ...

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