A TWO PERIOD MODEL
Up to now all the “action” in our model economy has taken place during one period. It is
as if the economy exists for one period in time and then disappears.
Obviously this is not realistic and we need to remedy the situation.
When economists deal with changes over time we move from “static” models to
“dynamic” models. This can involve some level of mathematical rigor, but we can
summarize the gist of a fully dynamic model by simply limiting the number of periods to
This is the tact we will take. Our two periods will be the current period (period 1) and the
future (period 2).
To begin we need to define the agents in our dynamic world.
First we will have many IDENTICAL consumers who will exist for TWO periods.
Second we will not consider explicitly the consumption leisure choice made by these
consumers, but rather their consumption and saving choice. Notice we have allowed the
consumer to save some of REAL income. The reason we can now do this is that we have
a second period in which the consumer could consume some of the income produced in
the first period. Therefore it makes sense to introduce saving.
Our consumers are price takers in a competitive economy. NOTE – WE are still dealing
with a REAL economy. All prices are still denominated in terms of the consumption
good. So far, the only price in our model has been the wage rate. We will now add a
second price, namely the real interest rate.
One important concept we must elucidate is “date contingent goods”. Now this is a rather
a fancy way of simply saying that goods consumed in the current period are treated as
distinct from goods consumed in the second period. Simply put: current goods and future
goods are treated as different goods. This means that there will be two markets- one for
current goods and one for future goods.
Another assumption of the model is that consumers prefer to have a smooth consumption
pattern over time. That is to say that our economy will operate such that goods are
demanded in both periods.
We have set up the description of the dynamic economy now we need to price the goods
in the future period.
This is easily done if you recall that in economics, generally speaking, we are concerned
with relative prices. For example, we can say that diamonds are relatively more
expensive that carrots (pun intended). However it wouldn’t matter if the price of a
diamond was 100 and the price of a carrot was 1 or if the price of diamond was 1000 and
a carrot were 10. What matters is that a diamond is RELATIVELY 100 times more
expensive than a carrot. Another way of stating this is to set up a ratio of diamonds/
carrots. In this case that ratio would be 100. Thus when we want to engage in exchange
we need 100 times more consumption units to trade for 1 diamond. Now just map this idea onto our problem at hand.
In order to “price” future goods all we need to do is relate their value to current goods.
Number of current goods/ number of future goods
Suppose our consumer was willing to give up 5 units of consumption today (i.e. save 5
units) for 10 units of consumption in the future. Now our “price ratio” would be:
5 units of current consumption / 10 units of future consumption
This equals ½ or .5
Thus we would say that the “the price of future goods is .5 current goods ”.
Now let’s make this a little more realistic. (Notice the model adds more realism as we
continue to construct it).
We can’t really intertemporaly trade something. That is we can not transport goods
produced today into the future. However, we can promise to give someone goods we
produce in the future in exchange for goods today.
This promise can be thought of as a bond. An old expression is “My word is my bond”.
This means that your promise (your word) is a good and binding contract.
A bond is therefore just a debt instrument that is a promise to “pay” in the future.
Now this means that the market for future consumption will be in fact a credit market. In
this credit market agents will exchange “promises” or bonds.
Therefore in the current period, if someone wants to exchange present goods for future
goods they will enter the credit market and buy bonds. Alternatively if someone wants to
exchange future consumption for consumption in the current period, they will enter the
credit market and sell bonds.
We have just modeled a bond market – demanders and suppliers – buyers and sellers of
As in all markets, this exchange will take place at a price and this price is called the real
interest rate (r).
Thus the real interest rate will be the rate at which current goods trade for future goods.
If you don’t want to consume all your production in the present period, you can now save
some of it and lend it in the credit market at a rate of interest.
That is you lend 10 units today at a rate of 10% -- you will get back 11 units in the future.
The 10% is the price that lenders and borrowers take from the credit market.
If you were the saver (lender) then you would buy a bond that promises to pay you 11
units of consumption in the future if you lend 10 units today. You are compensated 10%
for the foregone consumption.
Now that we have a bond market we could turn around and sell the bond we have just
bought. But what price would we get for the bond. – This is one of the fundamental
concepts in finance and financial economics- the process of bond pricing. A bond promises 1+r units of future consumption for 1 unit of current consumption
So its relative price is the ratio:
1 current consumption/ (1+r) future consumption.
Bond price = 1 / 1+r
In our example 10 units of current goods traded for 11 units of future goods Æ the real
interest rate (r) is thus 10% or .10.
1+ r = 1.10
Thus we could sell the bond for 1/1.10 or .909090909090 of its face value.
Since we bought a bond that costs us 10 consumption goods we would get about 9
consumption goods for that bond in the current period.
This may seem strange; after all we just lent 10 units why can’t we get our ten units
The answer is that we are dealing with a BOND MARKET. When we initially entered the
market there were a certain number of borrowers (sellers) and lenders (buyers). An
equilibrium price emerged –10%. Now there is one more seller in the market –us. Thus
there is one more seller and if the number of buyers stays the same the price must fall.
More sellers –supply—than buyers---demand—means the price falls.
Therefore if we sell our bond, the market will “discount “it and we will get about 9 units
of consumption goods.
In the real world, there are millions of buyers and sellers of bonds and the market trades
all day long almost every day of the year. However, the principle is same as the example
above. The price depends on the real interest rate.
Now notice that if the real interest rate changes so does the price of the bond. In fact the
price is inversely related to the interest rate.
Let’s do another stylized example.
Suppose we buy a bond in one period with an interest rate of 10% attached to it.
Suppose we lend 1000 – buy a 1000 bond.
For the moment we are ignoring the duration of the bond—which IS important!!
However for now just suppose the bond will pay 10% a year for a very long time.
Now suppose one year goes by. --- You receive a 100 unit payment from the bond.
Suppose during the second year you need to sell your bond because you need the
consumption due to unforeseen circumstances. How much will you get for your bond? It depends on the interest rate. Suppose the interest rate in the market during the second
year has gone up to 20% (this is a huge increase, but it is just an example).
How much will you get for the bond? – Only 500!!!
Why? Because someone with 500 units to lend would get a 100 unit payment in today’s
500 @ 20% = 100
Your bond will also pay 100 , so all it is worth is 500.
Notice what this demonstrates—the price of a bond varies inversely with the market rate
In the real world bonds are of different duration and risk. Both of these factors have to
taken into account when pricing bonds. This is the subject matter of finance and financial
economics and we will we not cover it here. Nonetheless the basic principle is sound:
Interest up-Æ bond prices down
Interest downÆ bond prices up
III. Consumers Budget Constraints
The next step in building our model is to incorporate the consumer’s budget constraint
into the two period model.
Remember that in a two period model the consumer can save some income in period one
and then consume that that income in period two. Therefore the consumer has to make
choices that cover consumption over an entire lifetime.
But this poses a problem since the future income has to be priced today. In other words,
when the consumer makes their consumption decision for the entire lifetime the value of
the future income has to be taken into account.
We therefore have to ‘discount” anything that occurs in the future to the present.
Why do we say ‘discount’? The reason has to do with opportunity cost.
We have a credit market where we can lend our savings and receive an interest payment
For example, suppose we have 100 units of income and that the real interest rate is 10%.
This means that if we consume the 100 units we give up 10 extra units we could have
consumed in the future. ( 100 * 1.10 = 110). Thus the foregone FUTURE consumption
represents the opportunity cost of PRESENT consumption.
Thus 110 units of consumption in the future are worth 100 units today at an interest rate
Likewise 100 units of future consumption are worth 90.90909090 units of current
This is called the present discounted value or just the present value. It is calculated as
PV = Future amount / (1+r) Because of the opportunity cost the “PRICE” of ONE unit current consumption is 1(1+r),
whereas the price of ONE unit future consumption is just 1. Remember that that
consumption good is the numeraire in this model so the price of one unit is just one.
The ratio of these two prices is Current consumption/ Future Consumption = (1+r)/ 1
We will return to this point shortly.
In order to model lifetime consumption in the economy we will need to model the
lifetime consumption of the representative consumer. Aggregate consumption will then
simply be the consumption decisions of the representative multiplied by m (the total
number of consumers in the economy). We can do this since we have assumed that all
consumers are identical.
To distinguish between the consumer and the entire economy, the variables associated
with the representative consumer will be depicted in lower case and the variables
associated with the economy as a whole will be in upper case.
The consumer’s current period budget constraint will be:
c + s = y- t
c: current period consumption
y: current period income
t: current period
The left hand side of this equation represents current period consumption and saving.
The right hand side represents current period disposable income.
The consumer’s future period budget constraint will be:
c′ = y′ - t′ + (1 + r)s
c′: future period consumption
y′: future period income
t′: future period taxes.
s: savings Æ that come from the first period
The left hand side of this equation represents future period consumption.
The right hand side represents future period disposable income
Notice that savings occur ONLY in the current period. We can see this since there is no ′
on the savings variable. Savings must be attained in the first period and consumed in the
second period. Our economy lasts for TWO periods - there is no third period where
savings could be “spent”.
Notice also that (1+r)s represents the income that first period savings have generated.
This amount depend on the level of savings AND the interest rate.
Therefore the interest rate is going to play a crucial role in amount of consumption
available in the future period. Obviously this will affect the consumption/saving decision
the consumer will make in the first period.
Our next step will be to model this decision process. Lifetime budget constraint
Notice that the two budget constraints have the s term in common. We can use this fact
to get an equation that represents the consumer’s budget constraint over a lifetime.
First remember that a decision to consume less in the current period is automatically a
decision to consume more in the future period.
This means that the consu