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In Economics, Euler equations are first order conditions to describe optimal behaviour over time. Since they form a system of nonlinear equations they often solved with Newton-Raphson or related algorithms introduced in Part 2.
- Two Period Model Taylor wants to choose how much to consume in two periods, $t=0$ and $t=1$. We won't be concerned with which bundles of goods Taylor consumes within each period. What matters in this model is total consumption by period. So the good is a composite of all goods purchased and consumed in a single period. We can, at this point, assume that this good has a price of 1 in both periods. $x = (x_0,x_1)$ is the sequence of consumptions consumed by the person over time. We call $x$ a consumption profile.
- Lifecycle Model A complete lifecycle model simply extends the two period model to $T$ periods: $$U(x) = \sum_{t=0}^{T-1} \delta^t u(x_t).$$ Taylor earns income each period, $m = (m_0, m_1, \dots, m_{T-1})$. She can borrow and save as long as she satisfies a more general lifecycle budget constraint: $$\sum_{t=0}^{T-1} { x_t \over (1+r)^t} = \sum_{t=0}^{T-1} {m_t \over (1+r)^t }.$$ That is, her consumption plan has to have the same present value as her stream of income. She can consume more than she earns ($x_t > m_t$) by borrowing from the future. Notice that this problem has a single budget equation and thus a single Lagrange multiplier, $\lambda$: $$ \delta_t u^{\,\prime} (x_t) = {\lambda\over (1+r)^t} .$$ $$ \delta_s u^{\,\prime} (x_s) = {\lambda\over (1+r)^s} .$$ $${u^{\,\prime}(x_t) \over u^{\,\prime}(x_s)} = {(1+r)\over \delta} ^{t-s}.$$ If $\delta = 1+r$ then the right side equals one, which means that the marginal utility of consumption is equal between any two periods. With utility increasing and concave that only happens if consumption is equal each period: $x_s = x_t$. In other words, Taylor wants to smooth consumption over time by saving and borrowing. The pattern of earnings do not matter directly. The level of constant consumption is determined by lifetime wealth: $$x_t = {1\over T} \sum_{t=0}^{T-1} \left[{1 \over 1+r}\right]^t m_t.$$
Very often we focus on additive utility over time: $$U(x) = u(x_0) + \delta u(x_1).$$ Here $u(x_t)$ is the utility of consuming $x_t$ within a time period. Taylor wants to allocate consumption over time so overall utility is $U(x)$. The parameter $\delta$ is Taylor's discount factor which was discussed earlier along with IRRs. Typically (but not always) the discount factor is assumed to satisfy $0 \le \delta \le 1$. The standard case is $0\lt\delta\lt 1$.
Taylor makes some income each period, $m = (m_0 m_1)$. She can borrow and save as long as she satisfies a lifecycle budget constraint. $$x_1 = A_1 = m_1 + (1+r)(m_0-x_0).\tag{LCBC}\label{LCBC}$$ This says that consumption tomorrow equals the amount of income available tomorrow, which in turn is income tomorrow plus the interest earned or paid on the difference between income and consumption today. The interest rate $r$ is, like the preference parameter $\delta$, typically between 0 and 1. $A_1$ equals the assets available in period 1.
We can substitute the budget constraint into the objective: $$U^\star(x_0) = u(x_0) + \delta u\bigl(m_1 + (1+r)(m_0-x_0)\bigr).$$ Now the choice is reduced to just determining how much to consume today (period 0), which then determines whether Taylor wants to borrow ($x_0\gt m_0$) or save ($x_0\le m_0$) in period 0.
To maximize utility take the derivative of $U^\star$ with respect to $x_0$ and set it equal to 0: $$u^{\,\prime}(x_0) - \delta (1+r) u^{\,\prime}\bigl(m_1 + (1+r)(m_0-x_0)\bigr) = 0.$$ We can then reorganize this as $$u^{\,\prime}(x_0) = \delta (1+r) u^{\,\prime}\bigl(m_1 + (1+r)(m_0-x_0)\bigr).\tag{Euler1}\label{Euler1}$$ This says that an optimal consumption profile equates marginal utility of today's consumption to the discounted marginal utility of consumption tomorrow. This is an Euler equation. If $\delta={1\over 1+r}$ then the internal discount factor $\delta$ is the same as the external or market discount factor, ${1\over 1+r}.$ This means Taylor should simply equate marginal utilities. Otherwise, Taylor is either more or less patient than the market and will adjust marginal utilities to account for that. If more patient, $\delta\gt{1\over 1+r}$, then marginal utility tomorrow will be less than today. With concave utility $u^{\prime\prime}\lt 0$, this means consumption tomorrow is greater than today: $$u^{\,\prime}(x_0) > u^{\,\prime}(x_1) \Rightarrow x_0 \lt x_1.$$