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Given no arbitrage, the price of the machine of vintage $\tau$ should be $q_{k t}(\tau)=p_{k t}(\tau)-\frac{p_{k t+1}(\tau)}{1+r_{t+1}}$... "No arbitrage" is a bit of a misnomer here. In this sentence, it really means "zero profit". The rental price must be equal to the price of owning a machine for one period. For example, if $$ q_{k t}(...

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