Deriving The Euler Equation
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- เผยแพร่เมื่อ 3 พ.ค. 2020
- I algebraically derive the Euler equation in our intertemporal choice model and give some intuition as to what this means. We show graphically how this relates to the marginal rate of substitution and the interest rate.
We derive the Euler equation by assuming additively separable utility functions with a discount factor beta. The utility functions are strictly concave.
View the playlist to see how we build to this 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 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 borrowers and savers in more detail, examining what happens with a change in the interest rate, changes in income and other exogenous changes. In this one, we focus on how to find the Euler equation.
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Best video I have seen about the Euler equation
everyone's favorite topic
This video is really helpful!! Thanks
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Really helpful for my revision, thanks very much
life-saver!!!!!
Really great video! One question - could you explain the process of "taking out the (1+r) term" at 6:00? Thank you!
Really helpful videos! Just one question - why is the utility function concave in the formal derivation but convex in the graphical one? What's the reasoning behind observing a concave utility function here? Thanks, hope your videos get good traction (I know I'll be recommending them).
Thank you! Please correct me if I'm interpreting your question incorrectly, but I believe you are not making a distinction between preferences and utility functions. If we have convex preferences over goods, we have concave (or quasi-concave) utility functions. We thus draw convex indifference curves, but concave utility functions. These are different things.
Ohh..Euler would be pleased ❤
Hi! Could you please find some questions related your derivation. I have tried to do some but I don’t get it. Hope u will save many by doing so. Thank you
This is the most satisfying explanation video about the euler equation I could find on the internet. However, I'm just still confused on some details:
1. at 8:52, when you said that the marginal cost of LHS will be equal to the marginal benefit of RHS. I keep on imagining the equation in my head that if LHS goes down, the RHS should goes down too, right?
2. Is my understanding true that: if u'(C1) goes down because a consumer is saving, the u'(C2) will get bigger because we consume more in period 2? But however, since the theory says that LHS = RHS, then β(1+r) will "deflate" that so it would be still equal to u'(C1)?
Hi - I'm glad that you found the video useful (at least to some degree)!
Regarding your question, I think this is a common query that is based on thinking about the Euler equation in the wrong way.
Hopefully this explanation makes some sense:
The Euler equation does NOT hold with equality when we are not at our optimum consumption pattern. Thus, if we start at optimum and we save more, u'(C1) will INCREASE (note this increases as we consume less in period 1 so the marginal value of an extra unit of period 1 consumption rises) and beta(1+r)u'(C2) will DECREASE (due to us consuming more in period 2, the marginal value of consuming slightly more in period 2 falls).
The Euler equation thus does NOT hold, as we are no longer at optimum. The Euler equation is a condition for optimal consumption. It does not always have to hold, and indeed it does not always hold.
Note also that here beta and r are constants from the consumer's perspective - they do not "deflate".
Let me know if I've misinterpreted your question or if I've got something wrong!
One more clarification.
The consumer chooses C1 and C2 so that the Euler equation holds.
If they then increase savings (as you put), they reduce C1 and increase C2. Thus the Euler equation will no longer hold, and they are no longer choosing the optimal C1 and C2.
Ok, I'm done now.
@@EverythingEcon Whoaa very quick replies, really really appreciate this. All has been very clear but just one more thing (hope you don't mind 😅)
You said: "u'(C1) will INCREASE (note this increases as we consume less in period 1 so the marginal value of an extra unit of period 1 consumption rises) and beta(1+r)u'(C2) will DECREASE "
It is very understandable because we reallocate some parts of C1 to C2. But then, isn't it only u'(C2) that is decreasing? Why did you say that the whole "β(1+r)u'(C2)" is decreasing? Aren't β and (1+r) exogenous/given?
Promise you this is the last missing piece of my understanding. But really, thank you so much for your previous answers!
@@LebihTenang yes, you are correct that u'(C2) is decreasing. I just said this because the whole RHS is decreasing. Beta(1+r) is just a constant, so if U'(C2) decreases, then beta(1+r)U'(C2) decreases too.
Are these equations fixed (the same) for every question?
Questions may give you specific functional forms. This video aims to give an understanding of how to solve these problems in a general sense.