Euler Equation and Marginal Rate of Substitution

Question

I was wondering if someone could help me clarify a result from my lecture notes. I have put them as a picture. It concerns the result on the last slide (the other three slides are included as well because it may be useful).

The problem is, I do not see how it follows from the Euler equation that $MRS=1+r$.

So from the lecture notes, the Euler equation is

\begin{equation*} u'(C_1) = \beta(1+r)u'(C_2) \end{equation*} This is the first order condition for an optimisation problem with respect to $C_1$ so really we have \begin{equation*} u_{C_1}(C_1) = \beta(1+r)u_{C_1}(C_2) \end{equation*} To get the marginal rate of substitution, we need $u_{C_2}(C_2)$. So somehow we must go from $u_{C_1}(C_2)$ to $u_{C_2}(C_2)$.

From the budget constraint \begin{equation*} C_1 + \frac{C_2}{1+r} = Y_1 + \frac{Y_2}{1+r} \implies C_1 = \frac{-C_2}{1+r} + Y_1 + \frac{Y_2}{1+r} \end{equation*} We expressed $C_2$ in terms of $C_1$, i.e. as a function of $C_1$, i.e. $C_2=C_2(C_1)$. Then by the chainrule \begin{equation*} \frac{\partial u}{\partial C_2} =\frac{\partial u}{\partial C_1}\frac{\partial C_1}{\partial C_2} = \frac{\partial u}{\partial C_1}\frac{-1}{1+r} \end{equation*} This is where I am stuck. I cannot get the result from the Euler equation as the lecture notes do.

Answer 1

The OP's confusion comes from the fact that he or she does not apply the chain rule correctly and interprets the Euler condition

$u'(c_1) = \beta (1+r) u'(c_2)$

as

$ u_{C_1}(C_1) = \beta(1+r) u_{\boldsymbol{C_1}}(C_2) $

This is not the correct way to read the Euler condition and one should really read

$ u_{C_1}(C_1) = \beta(1+r) u_{\boldsymbol{C_2}}(C_2) $

Any doubt can be cleared by rederiving the FOC. The maximisation problem is

$\max_{C_1,C_2} u(C_1) + \beta u(C_2) \qquad s.t. \qquad $

\begin{equation*} C_1 + \frac{C_2}{1+r} = Y_1 + \frac{Y_2}{1+r} \implies C_2 = {-C_1}({1+r}) + Y_1(1+r) + {Y_2} \end{equation*}

So substituting the constraint, the problem can be written as

\begin{align}\max_{C_1} u\Big(C_1\Big) + \beta u\Big(\underbrace{{-C_1}({1+r}) + Y_1(1+r) + {Y_2}}_{=C_2}\Big)\end{align}

To find the FOC, one needs to apply the chain rule to the second period's utility function. Applying the chain rule carefully we have

\begin{align} \frac{\partial \beta u(C_2)}{\partial C_1} = \beta \frac{\partial u(C_2)}{\partial C_2} \frac{\partial C_2}{\partial C_1}\end{align}

In our case that is

\begin{align} \frac{\partial \beta u(C_2)}{\partial C_1} = \beta \frac{\partial u(C_2)}{\partial C_2} (-1)(1+r)\end{align}

Or to put it in yet another way

\begin{align}\frac{\partial \beta u(C_2)}{\partial C_1} & = \frac{\partial \big[\beta u\big({-C_1}({1+r}) + Y_1(1+r) + {Y_2}\big)\big]}{\partial C_1}\\ & = \beta \frac{\partial \big[\ u\big({-C_1}({1+r}) + Y_1(1+r) + {Y_2}\big)\big]}{\partial \big({-C_1}({1+r}) + Y_1(1+r) + {Y_2}\big)} \frac{\partial \big({-C_1}({1+r}) + Y_1(1+r) + {Y_2}\big)}{\partial C_1} \\ & = \beta \frac{\partial \big[\ u\big({-C_1}({1+r}) + Y_1(1+r) + {Y_2}\big)\big]}{\partial \big({-C_1}({1+r}) + Y_1(1+r) + {Y_2}\big)} (-1) (1+r) \\ &= \beta \frac{\partial u(C_2)}{\partial C_2} (-1)(1+r) \end{align}

Then, by definition, given a utility function $U(c_1,c_2)$, the MRS of good one with respect to good two is

$MRS := \frac{{\partial U}/{\partial c_1}}{{\partial U}/{\partial c_2}}$

In your case, as stated in your notes, you get

$MRS = \frac{{\partial u}/{\partial c_1}}{\beta{\partial u}/{\partial c_2}} = \frac{u'(c_1)}{\beta{u'(c_2)}}$

At an equilibrium, when consumers maximize utility, the Euler condition must be satisfied, that is

$u'(c_1) = \beta (1+r) u'(c_2)$

So replace this specific value for $u'(c_1)$ in the formula for the MRS and you get

$MRS = \frac{u'(c_1)}{\beta{u'(c_2)}} = \frac{\beta (1+r) u'(c_2)}{\beta{u'(c_2)}} = (1+r)$.

As a good exercise, try to convince yourself that if the MRS takes another value, then the consumer can benefit from reallocating resources between the first and the second period.