Chapter 9
A Dynamic Consumption-Saving Model
Modern macroeconomics is dynamic. One of the cornerstone dynamic models is the simple
two period consumption-saving model which we study in this chapter. Two periods (the
present, period t, and the future, period t + 1) is sufficient to think about dynamics, but
considerably simplifies the analysis. Through the remainder of the book, we will focus on two
period models. The key insights from two period models carry over to models with multiple
future periods.
In the model, there is a representative household. There is no money in the model and
everything is real (i.e. denominated in units of goods). The household earns income in the
present and the future (for simplicity, we assume that future income is known with certainty,
but can modify things so that there is uncertainty over the future). The household can save or
borrow at some (real) interest rate rt, which it takes as given. In period t, the household must
choose how much to consume and how much to save. We will analyze the household’s problem
both algebraically using calculus and using an indifference curve - budget line diagram. The
key insights from the model are as follows. First, how much the household wants to consume
depends on both its current and its future income – i.e. the household is forward-looking.
Second, if the household anticipates extra income in either the present or future, it will want
to increase consumption in both periods – i.e. it desires to smooth its consumption relative
to its income. The household smooths its consumption relative to its income by adjusting its
saving behavior. This has the implication that the marginal propensity to consume (MPC) is
positive but less than one – if the household gets extra income in the present, it will increase
its consumption by a fraction of that, saving the rest. Third, there is an ambiguous effect of
the interest rate on consumption – the substitution effect always makes the household want
to consume less (save more) when the interest rate increases, but the income effect may go
the other way. This being said, unless otherwise noted we shall assume that the substitution
effect dominates, so that consumption is decreasing in the real interest rate. The ultimate
outcome of these exercises is a consumption function, which is an optimal decision rule which
relates optimal consumption to things the household takes as given – current income, future
income, and the real interest rate. We will make use of the consumption function derived in
this chapter throughout the rest of the book.
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We conclude the chapter by consider several extensions to the two period framework.
These include uncertainty about the future, the role of wealth, and borrowing constraints.
9.1 Model Setup
There is a single, representative household. This household lives for two periods, t (the
present) and t+1 (the future). The consumption-saving problem is dynamic, so it is important
that there be some future period, but it does not cost us much to restrict there to only be one
future period. The household gets an exogenous stream of income in both the present and
the future, which we denote by Yt and Yt+1. For simplicity, assume that the household enters
period t with no wealth. In period t, it can either consume, Ct, or save, St, its income, with
St = Yt - Ct. Saving could be positive, zero, or negative (i.e. borrowing). If the household
takes a stock of St into period t + 1, it gets (1 + rt)St units of additional income (or, in the
case of borrowing, has to give up (1 + rt)St units of income). rt is the real interest rate.
Everything here is “real” and is denominated in units of goods.
The household faces a sequence of flow budget constraints – one constraint for each period.
The budget constraints say that expenditure cannot exceed income in each period. Since the
household lives for two periods, it faces two flow budget constraints. These are:
Ct + St ≤ Yt (9.1)
Ct+1 + St+1 ≤ Yt+1 + (1 + rt)St. (9.2)
The period t constraint, (9.1), says that consumption plus saving cannot exceed income.
The period t + 1 constraint can be re-arranged to give:
Ct+1 + St+1 - St ≤ Yt+1 + rtSt. (9.3)
St is the stock of savings (with an “s” at the end) which the household takes from period
t to t +1. The flow of saving (without an “s” at the end) is the change in the stock of savings.
Since we have assumed that the household begins life with no wealth, in period t there is no
distinction between saving and savings. This is not true in period t + 1. St+1 is the stock of
savings the household takes from t + 1 to t + 2. St+1 - St is its saving in period t + 1 – the
change in the stock. So (9.3) says that consumption plus saving (Ct+1 + St+1 - St) cannot
exceed total income. Total income in period t + 1 has two components –Yt+1, exogenous flow
income, and interest income on the stock of savings brought into period t, rtSt (which could
be negative if the household borrowed in period t).
We can simplify these constraints in two dimensions. First, the weak inequality constraints
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