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You interview one employer per week, and your discount rate is $\delta = 1/(1+.07)$. Suppose $x$ is the lowest quality job you would accept: anything worse than $x$, and you reject and keep searching. Let $J$ be the value of searching. Then we have the functional equation: $$J = \max_{x} \int_{x}^{1}z dF(z) + \int_{0}^x \delta J dF(z)$$ The first term is ...

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You want to prove that $U(\alpha) = \int_z u(\alpha\cdot z) dF(z)$ is continuous, given that $u(w)$ is continuous? By definition $|U(\alpha)-U(\alpha')| = |\int_z u(\alpha\cdot z) - u(\alpha' \cdot z) dF(z) | <\int_z |u(\alpha\cdot z) - u(\alpha' \cdot z)| dF(z)$. Take the supremum over $z$, to get  \int_z |u(\alpha\cdot z) - u(\alpha' \cdot z)| dF(z) &...

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At the end of this answer there is an explication of the fractional reserve system. It is taken from another question that was closed as duplicate: The problem I have with this concept is the following: A bank give a Merry loan of 100,000\$. Merry buys a house with this money. For a period of 1 year, Merry has paid back 10,000 to the bank. Afterward ...

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