How is the Euler Equation for Consumption derived from from intertemporal budget constraint and lifetime utility function in basic macroeconomics

Question

I suspect that what I'm actually asking here is just a basic calculus question, which I have overwrought, but I wanted to ask it here to make sure before taking it to SEMaths.

In the Jones Introductory Macroeconomics textbook, he says that we don't really need to know the calculus for HOW Euler's Equation is derived, which is true, but I'm having a hard time moving on without it.

We start with the lifetime utility function:

$U= u(c_{today}) + \beta u(c_{future})$

$c$ is a quantity consumed, $u()$ is a utility function, $\beta$ is a ratio parameter to account for impatience.

Then we use the intertemporal budget constraint to express $c_{future}$ in terms of $c_{today}$.

$U= u(c_{today}) + \beta u[(1 + R)(\bar{X} - c_{today})]$

$R$ is real interest rate, $\bar{X}$ is total lifetime wealth

The next step is the one I don't understand, and probably just comes down to my own poor calculus. We are told to take the derivative of the previous equation with respect to $c_{today}$ in order to arrive at

$u'(c_{today}) + \beta u(c_{future})(1+R)(-1)=0$ OR $u'(c_{today}) = \beta (1+R)u'(c_{future})$

I understand that we've applied the sum rule of differentiation, meaning we take the derivative of $u(c_{today})$ and $\beta u[(1 + R)(\bar{X} - c_{today})]$ separately. And since in the context of this problem the utiltiy function is a black box, we just express it as $u'(c_{today})$. No problem.

However, when I look at the second part, $\beta u[(1 + R)(\bar{X} - c_{today})]$, I do not understand what rule has been applied. It appears to be the chain rule, but if I'm reading it correctly, then $(1 + R)(\bar{X} - c_{today})$ is the argument of the utility function. The chain rule, as I understand it, would only apply to the value of the function. After performing the differentiation, some values have somehow moved from being arguments of a function to being part of the equation. Considering that, for this exercise, the utility funciton is a black box with an unknown relationship, how are we allowed to apply any of our differentiation rules to the arguments?

Answer 1

It is simply the chain rule, or you could see it from the total differential.

Let $C_1$ be consumption today and $C_2$ be future consumption.

In the second part you have $U(C_2 (C_1) )$, i.e. a utility function dependending on $C_2$, which is itself (through the budget constraint you used) a function of $C_1$. You take the derivative of U with respect to $C_1$, which equals:

$d U/d C_1 = \partial U(C_2)/ \partial C_2 * d C_2 / d C_1$.

The first part is just $ \partial U(C_2)/ \partial C_2 = U'(C_2)$.

From what you have written, it seems that you understand that $C_2(C_1)=(1 + R)(\bar{X} - C_1)$.

Taking the derivative of that for the second factor in the multiplication above yields the result.

Answer 2

With thanks to @bbk and their probably more straightforward answer above, here is how I managed to understand it without partial differentiation. The key was understanding lifetime consumption as a function of a function.

Start with maximum lifetime utility as the sum of two utility functions

$U_M = u(C_T) + \beta u(C_F)$

Then use the intertemporal budget constraint to show that, when we're maximising consumption, future consumption can be expressed as a function of today's consumption

$C_F(C_T) = (1 + R)(\bar{X} - C_T)$

Go back to our original equation and replace $C_F$ with $C_F(C_T)$

$U_M = u(C_T) + \beta u(C_F(C_T))$

And when I look at it this way, the application of the chain rule is intuitive.