10.2 Saving and Borrowing
In this section we analyze the problem of a consumer who lives and receives incomes in two periods (period 0 and period 1), and must decide how much to consume in each period. We will denote by $C_0$ the period-0 consumption and by $C_1$ the period-1 consumption. The incomes in the two periods are denoted by $M_0$ and $M_1$, while the price of consumption is $P_0$ in period 0 and $P_1$ in period 1.
Intertemporal Budget Constraint
Which consumption bundles can the consumer afford? Let $R$ denote the interest rate at which the consumer can lend or borrow money. Let $S$ denote the consumer’s savings, with $S>0$ meaning lending money, and $S<0$ meaning borrowing money. Then it should be clear that a bundle $(C_0,C_1)$ is affordable if and only if both of the following two conditions are satisfied:
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Expenditure for consumption in period 0 equals period-0 income less money saved (or plus money borrowed): $$ P_0 C_0 = M_0 - S $$
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Expenditure for consumption in period 1 equals period-1 income plus money saved (or less money borrowed) and corresponding interest: $$ P_1 C_1 = M_1 +(1+R) S $$
Putting together the two conditions, (e.g. computing $S=M_0-P_0C_0$ from the first equation, and plugging into the second equation) we obtain the answer to our question. The bundles $(C_0,C_1)$ that the consumer can afford are those lying on the intertemporal budget constraint
\(\begin{gathered} C_1 = \dfrac{(1+R)M_0+M_1}{P_1} - \dfrac{(1+R)P_0}{P_1} C_0 \end{gathered}\)
Before proceeding, it is worth making a few observations:
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The vertical intercept of the intertemporal budget constraint (i.e. the value of $C_1$ corresponding to $C_0=0$) equals $[(1+R)M_0+M_1]/P_1$. The numerator of this ratio is the future value of the consumer's flow of incomes: this is the amount of money the consumer has available in period 1, if he decides save the whole period-0 income. Dividing that value by the future price of consumption, gives the maximum possible quantity of future consumption that the individual can achieve.
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The horizontal intercept of the intertemporal budget constraint (i.e. the value of $C_0$ corresponding to $C_1=0$) equals $[M_0+M_1/(1+R)]/P_0$. The numerator of this ratio is the present value of the consumer's flow of incomes: this is the amount of money the consumer has available in period 0, if he decides to borrow as much as possible. Dividing that value by the present price of consumption, gives the maximum possible quantity of present consumption that the individual can achieve.
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The slope of the intertemporal budget constraint, $(1+R)P_0/P_1$, measures the market value of present consumption in terms of future consumption. Indeed, giving up one unit of present consumption means saving $P_0$ euros more. This in turn means having $(1+R)P_0$ euros more, and hence consuming $(1+R)P_0/P_1$ units more, in the future.
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Whatever the interest rate $R$, the consumer can always choose to consume $C_0=M_0/P_0$ in period 0 and $C_1=M_1/P_1$ in period 1. This consumption bundle (named "initial situation" in the figure below), in fact, does not require saving or borrowing: in each period the consumer consumes exactly as much as the income of that period allows.
The figure below illustrates the intertemporal budget constraint and how the constraint depend on its parameters (incomes, prices, interest rate).
Optimal Intertemporal Choice
Assuming that the consumer’s preferences are of the Cobb-Douglas type
\(\begin{gathered} U(C_0,C_1) = C^a_0 C^b_1 \end{gathered}\)
the following figure illustrates the consumer’s optimal intertemporal choice, and how that choice depends on the preference parameters ($a$ e $b$) and on incomes, prices, and interest rate.