Consumption Flows and Asset Costs in Continuous-Time Pricing

Summary

The document clarifies the units behind a consumption constraint in a continuous-time asset-pricing problem. In discrete time, an investor’s consumption can be written as an endowment adjusted by the cost of an asset position, with later consumption reflecting asset payoffs. In the continuous formulation, consumption and endowment are treated as rates or flows, while the asset price is an amount quoted at a point in time. Consequently, comparing a price amount with a consumption flow requires accounting for the time interval represented by dt.

The answer’s explanation is that net consumption over an interval is the difference between the endowment and consumption rates multiplied by the interval length, whereas the quoted price remains a price per unit of the asset. This dimensional interpretation explains why dt appears when translating between a flow and an asset cost. The exchange is brief and assumes the reader’s intended equation uses division by dt; it does not develop the full continuous-time budget constraint or resolve broader modeling conventions around portfolio holdings and consumption units.

Key ideas

• Continuous-time consumption and endowment are modeled as flows rather than discrete amounts.
• An asset price at time t is an amount, so it has different units from a consumption rate.
• Multiplying a consumption flow by dt gives the amount consumed over the interval.
• The dt term reconciles the units when relating consumption rates to an asset price.
• The explanation depends on interpreting the constraint with price divided by dt.

Tags

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


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












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 by ForumWhiner (score 1, accepted)

https://quant.stackexchange.com/a/52929

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 $p_t$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.