Deriving the Intertemporal Budget Constraint (Detailed)

I derive the intertemporal budget constraint for a two-period model of intertemporal choice. In a later video, 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 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.
This is a general solution so that this can be solved algebraically. This is intended to give an intuitive explanation of how to find the intertemporal budget constraint for two periods. This equation also should give some understanding with regards to the concepts of time preference and present value. Let me know if you have any questions, and I can look to offer some examples in the future.
That is all for deriving the intertemporal budget constraint. Hopefully, this was a sufficiently detailed explanation!
In future videos we shall look at borrowers and savers in more detail, examining what happens with a change in the interest rate, changes in income and other exogenous changes.
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[@EverythingEcon]

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[@econecone]

I was looking for an explanation just like this. Thanks

[@krish8873]

Thank you for explaining it in simple terms

[@cosmicgaming2588]

Very nice explanation

[@daudbakianadia9490]

Why did you have to divide the substituted expression through by the (1+r)?

[@Grandfinal43]

thanks so much!

[@iremiposiajayi122]

Why isn’t the second budget constraint equation: Y_2 = C_2 + B_2(1+r)

[@suryaj_]

how would you answer a question that says, what is the total income in terms of period 1 consumption?

[@calidere7153]

what would you do when theres a transaction cost involved so say there is a fixed transaction fee of 50 with U=C1C2^0.85 and the interest rate being 0.08 as well as Y1 = 2000 and Y2 = 2200

[@cristianenriquez6591]

what if utility consumption is T periods with w_t ≥ 0 for all t?

[@monsan6327]

Why is the subscript of B 2 at BC1? Isnt it supposed to be 1 since you're borrowing at time 1? Thanks for your answer!

[@EverythingEcon]

It doesn't matter too much (you could also write this term as 'S' for saving). However, I think of this term as "bonds maturing in period 2", so it is a B subscripted with the number 2. As long as your notation is consistent and you explain what your terms mean, you can write this however you like (and however makes the most sense to you).

[@monsan6327]

@@EverythingEcon Thank you so much!

[@athreyakb1]

Hey if we were borrowing instead of saving in period 1, would the intertemporal budget constraint then be Y2/1-r instead of 1+r?

[@EverythingEcon]

Hi. This is not the case. The intertemporal budget constraint looks as shown in the video whether we save or borrow. I tried to give some intuition as to why this is the case when explaining the period 1 and period 2 constraints near the start of the video - it doesn't matter if our "B2" term is positive or negative, we still get the same constraint

[@athreyakb1]

@@EverythingEcon thank you so much for the quick reply. Liked and subscribed :)

[@athreyakb1]

@@EverythingEcon mate I'm struggling with a question related to this. I could use some help! Is there any way I can contact you? My mail is 'athreyakb' @ Gmail