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ECON 352D1 Lecture Notes - Lecture 5: Intertemporal Choice, Real Interest Rate, Consumption SmoothingExam

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ECON 352D1
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Econ 352 2018
5. A Two-Period Model: The Consumption-Saving
Decision (Ch. 8)
In this chapter we will present a model to characterize the behavior of a consumer
facing a dynamic consumption-saving decision. This consumption-saving decision
involves an inter-temporal choice, as this is fundamentally a decision involving a
tradeoff between current and future consumption.
We will present a two-period model, which is the simplest framework for
understanding dynamic issues. We will treat the first period of the model as the present
and the second period as the future. In our static model the key variable determining the
optimal choice was the real wage, the relative price (opportunity cost) of leisure in terms
of consumption. A similar role is played in the two-period model by the real interest
rate, which is the relative price (opportunity cost) of current consumption in terms
of future consumption, i.e. if you give up a amount of consumption goods today and
you save it, then you will have goods available for consumption tomorrow.
We will find that an important principle that drives household choices is
consumption smoothing. As a result of the shape of preferences (the taste for diversity)
consumers like to smooth their consumption through life independent of the path of their
income.1 They use their savings for this purpose. In periods of abnormally high income
they will save and they will decrease their saving when income is abnormally low. It is
worth noticing that the only reason to save is to consume in the future, i.e. people do not
save for the sake of it but only because saving increases their opportunities to consume
in the future.2
1 As in our one-period model households split their income between consumption and leisure, in this two-
period model, households will split their income between current and future consumption. As you will see,
the principles at work in the two-period model are very similar to the one-period model we have already
2 An additional saving motive is for bequest purposes. We could model this as saving for consumption of
the next generation.
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Econ 352 2018
To maintain simplicity and to retain focus on the important ideas of this chapter,
our two-period model will leave out production and investment. In the next chapter we
will reintroduce the consumption-leisure choice and production when we present a
complete model for the aggregate determination of output, employment, consumption,
investment, and relative prices (the wage and the interest rate).
As in the previous model we still impose our basic assumptions; full information
(in a certain world, two periods, everyone knows everything relevant, for instance
everyone knows his future income or the future path of government expenditure),
rationality/optimality (consumers know what is best for them and act accordingly),
competitive behavior (no one influences the behavior of others except through prices, no
single agent influences prices since each of them is small relative to the total), and
flexible prices (only relative prices matter, money is irrelevant or neutral).
5.1 The consumer problem
Although we will focus on a simple two-period model the results we will obtain
are easy to generalize (with the appropriate mathematical apparatus) to models with
many periods or an infinite number of periods. The reason for exploring the
consumption-saving decision with only two periods is that the resulting model is easy to
analyze while capturing the essentials of dynamic decision making by consumers.
Let’s assume the economy is populated by a large number, , of consumers
that in principle could differ in terms of the income they receive. Each consumer lives for
two periods, the current period and the future period. We will assume that consumers do
not make a work-leisure decision but simply receive an exogenous real income
stream (goods) in each period of their life. These goods are non-storable.34 Denote with
lower-case letters individual variables (we reserve upper-case for aggregate variables)
and use a prime for future-period variables. Let’s define,
, as current (real) consumption
, as future (real) consumption
, as (real) saving (in a two-period model it takes place in the first period)
, as current (real) income
, as future (real) income
, as current (real) taxes (lump-sum)
, as future (real) taxes (lump-sum)
, as the (net) real interest rate between the present and the future
3 The non-storability of goods ensures trade in financial markets without placing restrictions on the sign of
the real interest rate. If goods are storable, the real interest rate needs to be positive to ensure some lending.
4 Goods differ along several dimensions. An apple is different from and orange (goods market), but an
apple today is also different from an apple tomorrow (financial markets), finally an apple when you
are hungry is also different from an apple when you are full (insurance markets). In macroeconomics
we tend to abstract from the first distinction, by considering a single homogeneous good which is really a
bundle of goods, and we focus more on the two other cases.

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Econ 352 2018
With this information we can construct the consumer’s budget constraints for the
current and the future period as follows,
Current period: (sources of income = uses of income) (1)
If saving is positive, , the consumer is a lender on the credit market and if
he is a borrower. These borrowing-lending practices are formalized using a
financial asset, a bond, which is traded in the credit market.5 So when the consumer lends
he buys bonds (i.e. buys an I Owe yoU) and when he borrows he sells bonds (he issues a
A bond issued (sold) in the current period is a promise to pay units of
consumption (income) in the future period in exchange for one unit of consumption
(income) today.
We make two assumptions about these bonds;
1. Bonds are identical (same credit default risk that is zero, same interest rate, same
2. Bonds are traded directly in the credit market without the need for financial
intermediaries, i.e. banks. As a result the real interest rate at which a consumer
lends is the same as the real interest rate at which he can borrow.
5 This bond is a way to shift income from today to tomorrow (or from tomorrow to today) or a way to shift
income from one consumer, a lender, to another, a borrower. The lender lends income/consumption today
to the borrower. In exchange the borrower pays back the principal (plus some interest) to the lender
tomorrow. In general there are two ways of transferring income into the future: real wealth (capital) and
financial wealth. Real wealth will produce output in the future. Education (human capital), equipment,
land, machines, inventories, and plant are different forms of real wealth since these assets will be used for
production in future periods. Financial wealth are claims over future output. A bond is a claim on the
income/output of someone else, money is a claim on the output of someone that is willing to accept it as a
mean of payment, a share of a company represent the ownership of the means of production, real wealth, of
that company and therefore gives access to a fraction of the output it produces, in the form of dividends or
capital gains. Notice that future output will be determined solely by real wealth (and future conditions). In a
model without production, like the one in this chapter, there is no real wealth. Finally notice that in a closed
economy, the credit of any agent should have as a counterpart a debt for some other agent and as a result
financial wealth is in zero net supply.
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