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, BJ. 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 Price-level determination comes later, I will continue to set P = 1.0 and will also set P* =
1.0. So E – the nominal exchange rate – can be used rather than – the real exchange rate – since,
with P and P* both = 1.0,  = E. The textbook, BJ, also does this in places.]
-------------------------------------------------------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. You solved the IS, LM and BP curves
simultaneously to get Y, i, and BP. With floating exchange rates, BP always equals zero, and you
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 in 1999 for that
analysis.
----------------------------------------------------------Exchange Rates:
BJ, following US convention, give the exchange rate as the price of the US dollar expressed in
Canadian dollars: $CA/$US. This is the reverse, or technically the inverse, of the way we, in Canada,
are used to observing the exchange rate. We always observe the price of $CA expressed in $US:
$US/$CA. So be aware of this throughout as you read the textbook. This does NOT change the
meaning of depreciation and appreciation of the Canadian dollar, but it makes them somewhat counterintuitive. Depreciation of the Canadian dollar means a lower $US price of the $CA (which is intuitive),
but this is an increase in E. And appreciation of the Canadian dollar means a higher $US price of the
$CA (which is intuitive), but this is a decrease in E.
BJ, 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. His 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:
(E e  E)
i = i* +
,
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.)
Econ 3012: Notes on Symbols and Algebra
page 2
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:
E = Ēe/(1 + i – i*).
nominal exchange rate
This means that E is no longer a function of Y, and we no longer have a BP curve. This simplifies the
algebra considerably.
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*).
PART A: BALANCE OF TRADE:
1. Exchange Rate effects on Balance of Trade:
We have the balance of trade, or
net exports:
NX= (X – Q).
[NOTE: BJ 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 and what you will do in all of my
Problems.]
Graphically this is our model of the Balance of Trade:
Figure IV-1
E =
$CA/$US
S=X
(= Q-US)
D=Q
(= X-US
Quantity
of $US
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 foreigners buy
more of our goods.) And vice-versa when E goes down. So X is positively related to E : when E goes
up so does X and when E goes down so does X. To buy Canadian goods foreigners must first buy
Econ 3012: Notes on Symbols and Algebra
page 3
$CA, which is identical to selling $US. The Supply for $US, therefore, is the same thing as Canadian
Exports, and has a positive slope, as shown on Figure IV-1.
AND If E goes down and the value of the Canadian dollar goes up – the Canadian dollar appreciates –
the price of our imports from foreigners goes down. (Remember, think two transactions: first
Canadians buy foreign currency, then they buy foreign goods. If E goes down, the price of foreign
currency, E = $CA/$US, goes down. 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 down, and we buy more
foreign goods.) And vice-versa when E goes up. So Q is negatively related to E: when E goes down,
Q goes up, and when E goes up, Q goes down. To buy foreign goods Canadians must first buy $US, so
the Demand for $US has a negative slope, as shown on Figure VI-1.
X is positively related to E, so that part of NX that is due to X is postively related to E. Q is negatively
relative 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. Therefore NX is positively related to E two ways, both through exports and imports.
Exchange rates and trade balance:
NX = x0 + x2E
2. Output, or Expenditure, effects on Balance of Trade:
A constant 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*
Thus, the S and D curves on Figure IV-1 will shift when output in either Canada or the US shifts. The
Demand curve shifts to the right when Canadian output, Y increases, and vice-versa. The Supply
curve shifts to the right when US output, Y* increases, and vice-versa.
3. Summary: Balance of Trade and equilibrium Exchange Rate:
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.]
Remember that the equilibrium value of 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. So any value of NX is possible; it is NOT necessarily, or even likely, equal to zero.
PART B: 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 selling of
Canadian Bonds to foreigners, and the buying of Canadian bonds by Canadians from foreigners.
[Note: this assumes net Canadian debt, so that foreigners always hold Canadian bonds that Canadians
can buy. Since Canada has net foreign debt, this assumption is ok.]
The balance of payments must always balance. 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, D, will rise. This is denoted is D. [NOTE: This symbol is NOT in
textbook!] So if NX is negative, D is positive.
Econ 3012: Notes on Symbols and Algebra
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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.
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 negative 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, i* determines the
exchange rate, E.
2) The exchange rate, E, along with Y, determines 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.
This is shown for two situations in Figures IV-2 and IV-3 below.
Figure IV-2
E
Figure IV-3
E
S=X
X
S = X + D
E1
E2
Q
D = Q + (-D)
$US
D=Q
$US
Econ 3012: Notes on Symbols and Algebra
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E is determined by the two interest rates. On Figure IV-2, E = E1 , which is relatively high. A high E
means a low $CA, which means NX is positive. A positive NX means a negative D, so the Demand
curve is to the right of imports alone, Q. But remember, it is E that determines D, not, as you might
think just looking at the graph, that it is the intersection of Supply and Demand that determines E!
AND On Figure IV-3, E = E2 , which is relatively low. A low E means a high $CA, which means NX is
negative. A negative NX means a positive D, so the Supply curve is to the right of exports alone, X.
But remember, it is E that determines D, not, as you might think just looking at the graph, that it is the
intersection of Supply and Demand that determines E!
PART C: NEW IS and LM curves
1. 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.
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.
2. 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
3. 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.]
Econ 3012: Notes on Symbols and Algebra
and
change in debt held by foreigners:
page 6
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.