What is the consumption constraint in writing the continuous version of Asset Pricing Model?

Question

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.

Answer 1

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 $p_t$.