The Stochastic Euler Equation

I introduce uncertainty into the intertemporal choice model and so derive the stochastic Euler equation. This uses a simple Lagrangian derivation used in the Euler equation video.
In real life, individuals tend to be uncertain about future income and thus future consumption. The model thus adds uncertainty into future income levels and then has the consumer maximising utility using expectations of future income. we can then later make assumptions around how uncertain this income is.
Check out the playlist for other videos on this topic, and to get up to speed on where we begin in this video. In previous videos, I derive the intertemporal budget constraint for a two-period model of intertemporal choice. I shall derive this for more periods. Check out the playlist for intertemporal macroeconomics as a whole, linked at the end of this video.
We also previously discuss the assumptions of the two-period model of intertemporal choice. This involves consumers living for 2 periods. They can consume, save or borrow in these time periods, allowing for consumption that differs from their income in that period. In order to defer income to other periods, they can buy or sell one-period bonds with interest r.
We can then mathematically write the budget constraint for each of these periods. With a bit of substitution and rearranging, this gives us the intertemporal budget constraint. This says that the present value of consumption is equal to the present value of income. The consumer can thus not spend more than she earns, but will spend all of her income since more consumption is always assumed to increase her utility.
In future videos we shall look at Hall's Random Walk Hypothesis, and begin expanding our model into choices about labour and leisure.
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