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In the first chapter of John Cochrane's Asset Pricing textbook, in order to calculate the price in discrete time, we solve the maximization problem of $Max\space E(\Sigma\beta^j U(c_{t+j}))$ when our $c_t = e_t - \xi p_t $ and $c_{t+j} = e_{t+j} + \xi D_{t+j} $ (e as endownment, c as consumption and $\xi$ as quantity of the asset purchased.)

So, when we move to continuous time, the problem becomes to maximize $E\int e^{-\delta t} U(c_{t})\space dt$. In this form the constraint is written as $c_t = e_t - \xi p_t/d_t$.

What I don't understand is why does $dt$ show up in the constraint. Isn't it still true that the price of $\xi$ units of asset is $p_t$ so that we should write it similar to the discrete time i.e ($e_t - \xi p_t$)?

Any advice would be appreciated.

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I guess you meant to write $c_t = e_t - \xi p_t/dt$. Think of $c_t$ and $e_t$ as the 'flow' of consumption and endowment, whereas $p_t$ is the price of good at time $t$. Within the span of time $dt$, the net consumption is $(e_t - c_t)*dt$ whereas price per good is still $p_t$.

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    $\begingroup$ Thinking of c and e as 'flow' is very helpful, thank you. But why to call (e-x)*dt the "net consumption" though? To me it seems more like a "net saving" or something like that, because it's the accumulation of all endowments which do not get consumed in the span of dt. $\endgroup$ – MarcusAerlius Mar 30 '20 at 16:44
  • $\begingroup$ Yes "savings" is a good term here I think. $\endgroup$ – noob2 Mar 30 '20 at 16:54
  • $\begingroup$ I agree that 'savings' is a good term. It's just a matter of terminology though. $\endgroup$ – ForumWhiner Mar 30 '20 at 17:13

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