ECONOMICS 3012
Notes IV: Symbols and basic Macroeconomic Algebra
Foreign Sector
As I add the foreign sector to the model, rather than make up an additional complex parameter and a
new variable replacing A, I will just use the subscript “f”, so Af is autonomous-like stuff with the foreign
sector included and f is the spending multiplier with the foreign sector included. The economics of
what is going on is well described in the textbook, BJM. I won’t add to that. I’ll just give you the
symbols and a bit of the reasoning behind them.
FOREIGN SECTOR:
[Note: because I ‘m just developing the algebra you will use solving problems, and because we are
through with the Price-level determination, I will set P = 1.0 and P* = 1.0. So E can be used rather than
 since, with P and P* both = 1.0,  = E. The textbook, BJM, do this also in Chapter 8.]
-------------------------------------------------------Some history:
Earlier models, “imperfect-capital” models, have a BP curve, a balance of payments curve, with as E
as a function of Y and i. This is solved for reduced form in Y and is upward-sloping in the i, Y space.
Changes in E shift the curve. Adding that to the IS/LM system, there were three equations and three
endogenous variables – three unknowns. These needed to be solved simultaneously to get full
equilibrium – that is simultaneous equilibrium in the Real, Financial and Foreign sectors. With fixed
exchange rates, E was an exogenous, fixed, variable, and one solved the IS, LM and BP curves
simultaneously to get Y, i, and BP. With floating exchange rates, BP always equals zero, and one
solved the three curves simultaneously to get Y, i, and E. Robert Mundell, a Canadian economist,
worked out this analysis in the 1960s. He won the Nobel Prize in Economic Science for that analysis
about five years ago.
----------------------------------------------------------Now, Exchange Rates:
BJM, and I, use the most current theory of the determination of the nominal exchange rate E. (Our own
late Trevor Dick was one of the economists who developed this theory, and whose work was
instrumental in its acceptance by economists.) In this theory the exchange rate – the price of the US
dollar expressed in $Canadian – is determined exclusively in speculative markets by the requirement
that the expected rate of return must be equated in financial markets across all financial instruments.
Here the only financial instruments we “model” are Bonds, as before. So the condition, which is known
as “interest parity” is that the expected rate of return is equated for both Canadian and US bonds.
Simplifying somewhat, this requirement means that:
i = i* +
(E e  E)
,
E
where Ēe is the expected value of the nominal exchange rate, and i* is the US interest rate. (In a
perfect long-run equilibrium, Ēe would be the purchasing power parity value of the nominal exchange
rate.)
It is the nominal exchange rate that moves. The interest rates in both countries are determined by the
respective Central Banks. The nominal exchange rate adjusts to changes in the interest rates, not viceversa. So, writing with the endogenous variable on the LHS:
nominal exchange rate
E = Ēe/(1 + i – i*).
This means that E is no longer a function of Y, and we no longer have a BP curve. This simplifies the
algebra considerably.
Econ 3012: Notes on Symbols and Algebra
page 2
However, the equation for the nominal exchange rate is non-linear. To make the algebra workable, I
give here, and use below:
linear approximation, nominal exchange rate
E = Ēe – e1(i – i*).
BALANCE OF TRADE:
Exchange Rate effects on Balance of Trade:
We have the balance of trade, or
net exports:
NX= (X – Q).
[NOTE: BJM write this as (X – EQ), to convert the imports into Canadian dollars. This is unnecessarily
fussy. We can just measure Q in Canadian dollars, as the Canadian national income accountants do
anyway, and avoid the extra use of the variable, E. This is what I do.]
If E goes up, then the value of the Canadian dollar goes down – the Canadian dollar depreciates – and
the price of our exports to foreigners goes down. (Remember, think two transactions: first the
foreigners buy Canadian currency, then they buy Canadian goods. If E goes up, the price of Canadian
currency, 1/E = $US/$CA, goes down. Since we are holding both Price levels constant at 1.0, the price
of Canadian goods to foreigners, relative to the prices of foreign goods, goes down.) And vice-versa
when E goes down. So X is positively related to E and that part of NX that is due to X is positively
related to E.
AND If E goes up and the value of the Canadian dollar goes down – the Canadian dollar depreciates –
the price of our imports from foreigners goes up. (Remember, think two transactions: first Canadians
buy foreign currency, then they buy foreign goods. If E goes up, the price of foreign currency, E =
$CA/$US, goes up. Since we are holding both Price levels constant at 1.0, the price of foreign goods to
Canadians, relative to the price of Canadian goods, goes up.) And vice-versa when E goes down. So
Q is negatively related to E. But Q is in NX as a negative, so that part of NX that is due to Q is also
positively related to E. NX is positively related to E.
Exchange rates and trade balance:
NX = x0 + x2E
Output, or Expenditure, effects on Balance of Trade:
A fraction of Canadian purchases of final goods and services are purchases from foreigners. So as
Aggregate Expenditure = Aggregate Demand goes up, so do our imports. And vice-versa when AE
goes down. Q is positively related to Y, and
Q = q1Y
From the other side of the border, a fraction of US purchases of final goods and services are purchases
from Canadians. So as US Aggregate Expenditure = US Aggregate Demand goes up, so do their
imports, which are Canadian exports. And vice-versa when US AE goes down. X is positively related
to Y*, where Y* denotes US Income, and
X = x1Y*
Summary: Balance of Trade:
Adding the three effects above, we get
Balance of trade or net exports
NX = x0 + x1Y* – q1Y + x2E
[NOTE: x0 must be negative. The minus sign will be given in problems.]
Econ 3012: Notes on Symbols and Algebra
page 3
BALANCE OF PAYMENTS
The balance of payments, sometimes called the balance of international payments, is the sum of two
accounts: the current account, which is just the balance of trade, NX, and the so-called capital account,
which is financial flows across borders. Here we model the capital account as just the buying of
Canadian Bonds by foreigners, and the selling of Canadian bonds by foreigners to Canadians.
The balance of payments must always balance. So, if the balance of trade is negative, Canadians
must borrow to pay for that part of imports, Q, not matched by exports, X. That borrowing is measured
in the capital account. To borrow from foreigners, Canadians must sell Canadian Bonds to foreigners.
As they do, Canadian foreign-held debt will rise. This is denoted is D. [NOTE: NOT in textbook!] So if
NX is negative, D is positive.
AND, if the balance of trade is positive, foreigners must borrow to pay for that part of their imports
(which are Canadian exports, X) not matched by their exports (which are Canadian imports, Q).
Canada is a net debtor to foreigners. So this “borrowing” by foreigners is best modeled as having
Canadians buy bonds back from foreigners. As they do, Canadian foreign-held debt will fall. So if NX
is positive, D is negative.
In more detail, if Canadians borrow, we sell our bonds to foreigners. To buy Canadian bonds,
foreigners must first buy Canadian dollars. Buying Canadian dollars is the same as selling US dollars.
So when D is positive it increases the supply of US dollars. This increase in supply must be adequate
to cause the quantity supplied to equal the quantity demanded of Canadian dollars at the equilibrium
exchange rate, E.
REMEMBER: The equilibrium exchange rate, E, is exclusively determined by the difference between
the Canadian interest rate and the US interest rate. That is, it is exclusively determined by actions by
the two Central Banks.
If Canadian foreign-held debt is being paid down, Canadians buy Canadian bonds back from
foreigners. To do this, Canadians must first buy US dollars. So when D is positive it increases the
demand for US dollars. This increase in demand must be adequate to cause the quantity demanded
to equal the quantity supplied of Canadian dollars at the equilibrium exchange rate E.
The balance of payments must balance; ie it must equal zero, when exchange rates float. The balance
of payments is the sum of the current account and the capital account. With floating rates we have BP
= 0,
so
BP = NX +D) = 0 ,
and we complete the foreign sector:
D = -NX.
Summarizing, the way the foreign sector works is:
1) The difference between the Canadian interest rate, i and the US interest rate, Ii* determines the
exchange rate, E.
2) The exchange rate, E, along with Y, determine Net exports, NX, or the balance of trade.
And 3) the balance of trade, net exports, NX, determines the change in Canadian foreign-held debt,
D.
REAL SECTOR, or the GOODS MARKET: ISf curve
We have an IS curve developed in Notes II, Part C: Y =  A – b2 i , where A is autonomous-like stuff:
A = c0 – c1T + bo + G, and = 1/(1 – c1). We need to modify that now because now:
Aggregate Expenditure:
AE = C + I + G + NX
and Equilibrium is:
Y = AE = C + I + G + NX.
Econ 3012: Notes on Symbols and Algebra
page 4
The Net Export Function is:
NX = x0 + x1Y* – q1Y + x2E
The solution is the ISf curve:
Y = f Af – f b2i – f x2E.
Note that the ISf curve has three endogenous variables: Y , i , and E. Below, we get rid of one.
Financial Markets, or the Financial Sector: LM curve
This doesn’t change.
LM curve:
i = (d1/d2)Y – (1/d2)(M/P)
P is the aggregate price level, and will always be equal to 1.0
WORKING IS curve:
E is a function of only i and an exogenous variable, i* and we no longer have a BP curve to deal with.
So substitute for E in the ISf curve to give a working ISf curve. This has Y as a function only of i and
exogenous variables:
ISf curve:
Y = f Af – f b2 i + f x2E
Substitute for E:
Y = f Af – f b2 i + f x2Ēe –x2 e1 i +x2 e1 i*
And the working ISf curve is
Y = f Af – i
where
one complex parameter is:
and the other complex parameter is:
Af = (c0 – c1 T + b0 + G + x0 + x1 Y* + x2 Ēe+ x2 e1 i*)
f= 1/[1 – c1 + q1]
 = f (b1 + x2 e1)
The system now consists of:
working ISf curve:
Y = f Af – i
LM curve:
i = (d1/d2)Y – (1/d2)(M/P)
linear approximation, nominal exchange rate:
E = Ēe – e1(i – i*)
Balance of trade or net exports:
NX = x0 + x1Y* – q1Y + x2E
[NOTE: x0 must be negative. The minus sign will be given in problems.]
and
change in debt held by foreigners:
D = -NX.
The model now has been reduced to IS and LM, but the IS curve is now a bit more elaborate. There
are the same two exogenous policy variables: G for fiscal policy, and M for monetary policy. Since G is
part of A, A is the vehicle for fiscal policy. There are two new exogenous variables, both US values: US
Income, Y*, and the US interest rate, i*.
There is one more new exogenous variable, Ēe. This is an expected value. A more elaborate model
would try to specify how these expectations are formed. The default value is purchasing power parity,
but the way the world has been working strongly suggests that ppp is inadequate.
The system is now five equations with five endogenous variables: Y , i, E, NX , and D. To solve the
system, the first two steps are the same as with the second unit of the course. 1) Substitute LM into IS
and solve for equilibrium Y. 2) Substitute the equilibrium value of Y back into LM and solve for
equilibrium i. 3) With equilbrium i, you can find equilbrium E (and equilbrium I). 4) With equilbrium Y
and E, you can find equilbrium NX. And 5) with equilbrium NX, you can find equilibrium D.