2. Intertemporal Trade
To understand international finance and open-economy macroeconomics, we must first
explain why individuals might want to use international financial markets. One such
reason is intertemporal trade.
Our objective is to understand the gains from intertemporal trade. We will analyze these
gains using a 2 period barter, endowment economy. We will consider the equilibrium for
2 versions: an economy in autarky (closed economy) and an economy with trade (open
economy). We will introduce investment and production once we are familiar with the
endowment economy.
2.1 A Simple Economy in Autarky
Our simple barter economy has the following characteristics:
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The temporal universe only has 2 periods.
Its population consists of a large number of identical individuals. Accordingly, the
population behaves as if there was only one (aggregate) individual. We call this
individual the representative consumer.
This representative consumer has preferences over two goods: consumption today and
consumption tomorrow. The goods are perishable.
The representative consumer receives an endowment of consumption good in every
period.
In equilibrium, the representative consumer will consume all his endowment in each
period, because consumption goods are perishable.
The Budget Constraint:
Definition:
The budget constraint represents all the possible combinations of two
commodities that a consumer can purchase, given the prices of the
commodities and some fixed amount of money at her disposal.
In general, the consumer's budget is written as:
P1C1 + P2C 2 = Y
This budget constraint has a graphical representation. The slope of the budget constraint
is given by the price ratio. It describes the amount of one commodity that the market
requires the consumer to give up in order to obtain an additional unit of another
commodity, without any change in the amount of money spent.
How does this relate to our problem? Consider the choices that our representative
consumer faces. That is, what can our representative consumer do with his income
(endowment) in the first period? He can consume and save:
C1 + S1 = Y1
Question:
Why do people save?
Answer:
1. To transfer wealth through time, and
2. To protect themselves against an uncertain future (precautionary
savings).
In the second period, the consumer can consume her endowment and the gross return on
its savings:
C2 = Y2 + (1 + r a ) S1
Should the consumer save part of her income in the second period? Obviously, the
consumer will not save for a third period that will never come.
These two constraints can be merged to form the intertemporal budget constraint:
 1 
 1 
C1 + 
C = Y1 + 
Y
a  2
a  2
1+ r 
1+ r 
This intertemporal constraint states that the consumer will equate the present value of its
lifetime consumption to the present value of its lifetime income. It has a graphical
representation, just as the original budget constraint.
A comparison of the general budget and our intertemporal budget suggests that
Y / P2 = (1 + r a )Y1 + Y2
P1 / P2 = (1 + r a )
Consumer Preferences:
A consumer's preferences are generally described by her utility function.
Definition:
Total utility is the total amount of satisfaction obtained from consumption of a
good. Thus, total utility comes from all units consumed.
Definition:
The marginal utility is the additional satisfaction gained by the consumption of
one more unit of something. Thus, the marginal utility only comes from the last
unit consumed.
Total utility has two important characteristics:
• It is increasing in the amount consumed: more is better.
• It is subject to the law of diminishing marginal utility.
For our purpose, we will be concerned with a graphical representation involving the
indifference curve.
Definition:
An indifference curve connects all combinations of commodities that are
equally desirable to the consumer.
The indifference curve has a graphical representation.
As long as the consumer desires more of each commodity, every point on a higher
indifference curve is preferred to any point on a lower indifference curve. This can be
shown graphically.
The slope of an indifference curve is called the marginal rate of substitution.
Definition:
The marginal rate of substitution is the maximum amount of one
commodity that the consumer is willing to give up in exchange for one
more unit of another commodity.
How does this relate to our problem? Our consumer has preferences over consumption
today and consumption tomorrow. For our purpose, we will describe the utility function
as:
U (C1 , C 2 ) = u(C1 ) + βu(C 2 )
The parameter β is called the subjective discount factor, with ρ being the rate of time
preference: β = 1/(1+ρ).
Using this notation the marginal rate of substitution is:
MRS =
u ' (C1 )
βu ' ( C 2 )
The notation is:
u ' (C ) =
∂u (C )
∂C
The Consumer's Choice:
The representative consumer will select the most desired combination of goods
obtainable for her money. The choice will be that point on the budget line at which the
budget line is tangent to an indifference curve.
The consumer will get the most benefit from their money by choosing a combination of
commodities whose marginal rate of substitution is equal to the ratio of their prices:
MRS = P1 / P2 = (1 + r a )
The Autarkic Equilibrium:
The last piece of our puzzle is market clearing. Market clearing refers to the conditions
that stipulate that supply must equal demand.
In our problem, the market clearing condition is:
C1 = Y1
C 2 = Y2
This is so because goods are perishable.
In macroeconomics, the market clearing conditions have another signification. From
national income accounting, recall that the national income account identity is:
Y = C + I + G + EX - IM
How does this relate to our problem? We are analyzing a closed economy with no
government and no production abilities (I = G = EX = IM = 0). Accordingly,
Y=C
A market clearing condition is simply the national income account identity for our model
economy.
The Equilibrium:
The equilibrium for this endowment economy is an allocation and prices such that the
consumer chooses her preferred consumption bundle for her income. Formally, the
equilibrium is characterized by:
MRS = (1 + r a )
C1 = Y1
C 2 = Y2
This equilibrium can be shown graphically.
2.2 A Simple Small Open Economy
Our simple small open economy has the following characteristics:
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The temporal universe has 2 periods.
The population is aggregated in a representative consumer.
The representative consumer has preferences over two goods: consumption today and
consumption tomorrow. The goods are perishable.
The representative consumer receives an endowment of consumption good in every
period.
The consumer can borrow and lend abroad at the world real interest rate.
The Budget Constraint:
The consumer can consume or save his endowment at the world rate of interest:
C1 + B1 = Y1
C 2 = Y2 + (1 + r w ) B1
As before, these constraints can be merged to form the intertemporal budget constraint:
 1 
 1 
C1 + 
C = Y1 + 
Y
w  2
w  2
1+ r 
1+ r 
The Consumer's Choices:
As before, the representative consumer will select the most desired combination of goods
obtainable for their money. The choice will be that point on the budget line at which the
budget line is tangent to an indifference curve.
The consumer will get the most benefit from their money by choosing a combination of
commodities whose marginal rate of substitution is equal to the ratio of their prices:
MRS = (1 + r w )
The Equilibrium:
The equilibrium for this small open economy is the allocation and prices such that the
consumer maximizes its utility and market clears:
Formally, the equilibrium is characterized by:
 1 
 1 
C1 + 
C = Y1 + 
Y
w  2
w  2
1+ r 
1+ r 
MRS = (1 + r w )
The equilibrium can be shown graphically.
The Current Account:
A country's current account balance is the change in the value of its net claims on the rest
of the world.
The national income account identity for our problem with no production and no
government (I =G = 0) is:
Y = C + EX - IM
CA = EX - IM
Note that in the National Income Accounts, the current account includes the balance on
goods and services and the balance on income.
For our problem, the country's current account in the first period is:
CA1 = B1 − B0 = Y1 − C1
Note that it also corresponds to the balance on goods and services:
TB1 = Y1 − C1
The current account in the second period is:
CA2 = B2 − B1 = Y2 − C 2 + r w B1
The balance on goods and services is:
TB2 = Y2 − C 2
Question:
Answer:
Is it better for a country to run a trade surplus or a trade deficit for ever?
It is not possible to run either a surplus or a deficit for ever. It can be
shown that the present value of all trade balance surpluses and deficits is 0.
 1 
TB1 + 
TB2 = CA1 + CA2 = 0
w 
1 + r 
The Gains from Trade:
A comparison of the autarkic and open economy equilibrium suggests that there are gains
from trade.
These gains from trade can be illustrated graphically.
Two points must be highlighted from this graphical analysis:
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The representative consumer must be able to still consume its endowments.
The representative consumer's utility goes up, since every point on a higher
indifference curve is preferred to any point on a lower indifference curve.
Comparative Advantage and Trade Patterns:
As can be seen from the graphical analysis, a country's net exports depend on the relation
between the autarky prices for its goods and the world prices.
For example, if the autarkic interest rate is larger than the world interest rate, the country
will run a current account deficit in the first period and a surplus in the second period.
That is, the country will be a net importer of the consumption good in period 1 and a net
exporter of the consumption good in the second period.
How are the autarkic and world interest rates related to goods prices?
P1 / P2 = (1 + r )
So that
 P1a   P1w 
r ⟩ r ⇔  a ⟩  w 
 P2   P2 
a
w
Then the period 1 consumption good is produced more cheaply in the world than in the
small country.
Capital Mobility and Interest Rate Parity:
In our simple model, we have assumed that the consumer can freely borrow and lend
abroad at the world interest rate. That is, we have assumed perfect capital mobility. A
result of this assumption is that the real interest rate prevailing at home and abroad are
equalized in the open economy equilibrium.
2.3 A Small Open Economy with Production
Why consider production in our simple model? A discussion of investment opportunities
requires production.
We use the assumptions of our small open economy, Section 2.2 above, and add the
following assumption: in addition to the endowment, it is possible to produce second
period consumption using a production technology, that we call a production function.
To simplify, we assume that there is only one factor of production, namely capital. That
is, we abstract from labor market issues, to concentrate on investment.
The Production Function:
Definition:
The production function shows the maximum level of output attainable for
every level of input or factors of production.
For our case, the production function can be written as:
Y = F (K )
It has the following properties:
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The production process yields no output if there is no input.
The production function is increasing in the amount of capital.
The production function exhibits decreasing returns.
This production function can be shown graphically.
Investment:
Generally, economic investment is described as fixed capital formation. Also, investment
today only produces capital available for production tomorrow:
K1 = I + (1 − δ ) K 0
The parameter δ is the depreciation rate. For the remaining, we will assume full
depreciation: δ = 0.
The Firm's Choices:
The representative firm will select the most desired investment level to maximize its
value. The firm's value is the present value of its profits. These profits will be distributed
as dividends to the firm's shareholders.
In the first period, the firm's profits are:
Π 1 = Y1 − I = Y1 − K1
In the second period, the firm's profits are:
Π 2 = Y2 + F ( K 1 )
The present value of these profits is:
 1 
 1 
Π1 + 
Π 2 = Y1 − K1 + 
[Y + F ( K1 )]
w 
w  2
1+ r 
1+ r 
The optimal choice of investment is the point where the firm is indifferent between
investing at home in fixed capital or abroad in foreign assets. The return from investing at
home is given by the marginal product of capital, and it is the slope of the production
function. The return from investing abroad is one plus the world interest rate. This can be
shown graphically.
The Production Possibility Frontier:
The production possibility frontier illustrates the principle of opportunity cost. The
opportunity cost of producing more second period consumption is that fewer first period
consumption goods will be consumed.
Definition:
The production possibility frontier shows all the efficient combinations of
commodities that an economy can produce.
The production possibility frontier can be derived from the economy's choices.
In the first period, the representative consumer can consume, invest in machinery, or
invest abroad her income. This income only comes from her endowment:
C1 + K1 + B1 = Y1
In the second period, the representative consumer can consume all of her income. It
comes from the endowment, production, and returns on foreign investment:
C2 = Y2 + F ( K1 ) + (1 + r w ) B1
We can rewrite these equations as:
 1 
 1 
[Y + F ( K1 )]
C1 + 
C = Y1 − K1 + 
w  2
w  2
1 + r 
1+ r 
The production possibility frontier can be shown graphically.
The slope of the production possibility frontier is called the marginal rate of
transformation.
Definition:
The marginal rate of transformation is the amount of one commodity that
the economy must give up to obtain one more unit of another commodity.
For our problem, the marginal rate of transformation is:
MRT = F ' ( K1 ) =
∂F ( K1 )
K1
The Equilibrium:
The equilibrium for this small open economy is the allocation and prices such that the
representative consumer chooses her preferred bundle of consumption, and that the firm
maximizes its value.
The equilibrium is characterized by:
 1 
 1 
C1 + 
C = Y1 − K1 + 
[Y + F ( K1 )]
w  2
w  2
1 + r 
1+ r 
MRS = (1 + r w ) = MRT
This equilibrium can be shown graphically.
The Current Account:
The national income account identity for our problem with no government (G = 0) is:
Y = C + I + EX - IM
CA = EX - IM
Note that in the National Income Accounts, the current account includes the balance on
goods and services and the balance on income.
For our problem, the country's current account in the first period is:
CA1 = TB1 = Y1 − C1 − K1
The current account in the second period is:
CA2 = B2 − B1 = Y2 + F ( K1 ) − C 2 + r w B1
The trade balance is:
TB2 = Y2 + F ( K1 ) − C 2
The Gains from Trade and Trade Patterns:
As in our previous examples, we can graphically compare the autarkic equilibrium to the
small open economy equilibrium, and obtain the gains from trade and the trade patterns.
Capital Mobility and the Savings and Investment Puzzle:
The above analysis suggests that investment is a domestic event and saving is an
international event. That is, investment should respond to domestic opportunities
affecting the return on fixed capital. Savings should respond to international opportunities
that affect the return on home and foreign assets. Accordingly, savings and investment
should not be related.
There is a lot of empirical research that estimates the correlation between savings and
investments. Empirical evidences suggest that the savings rate and the investment rate are
highly correlated. Does this contradict the theory? The answer is no.
2.4 Summary
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International financial markets permit the exploitation of the gains from intertemporal
trade.
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International financial markets ensure that it is not possible to run trade surpluses or
trade deficits for ever.
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Intertemporal trade patterns respond to the forces of comparative advantages:
country's export goods that they produce relatively more cheaply and import those
that are produced relatively more cheaply abroad.
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Perfect capital mobility ensures that interest rates at home and abroad are equated.
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Investment responds to domestic opportunities affecting the return on fixed capital.
Saving responds to international opportunities affecting the return on home and
foreign assets.