# 9.4: Derivation of the AA Curve

- Page ID
- 20266

learning goals

- Learn how to derive the AA curve from the money-Forex model.

The AA curve is derived by transferring information described in
the money market and foreign exchange market models onto a new
diagram to show the relationship between the exchange rate and
equilibrium GNP. (At this point we will substitute GNP for its
virtually equivalent measure, GDP, as a determinant of real money
demand.) Since both models describe supply and demand for money,
which is an asset, I’ll refer to the two markets together as the
asset market. The foreign exchange market, depicted in the top part
of Figure 9.4.1 , plots the rates of return on domestic U.S.
assets (*\(R_{0}R_{$}\)*) and foreign British assets
(*\(R_{0}R_{£}\)*). (See __Chapter 5 "Interest Rate
Parity"__, __Section 5.3 "Forex Equilibrium with the Rate
of Return Diagram"__ for a complete description.) The
domestic U.S. money market, in the lower quadrant, plots the real
U.S. money supply (*\(M_{$}^{S}/P_{$}\)*) and real money
demand (*\(L(i_{$}, Y_{$})\)*). The asset market
equilibriums have several exogenous variables that determine the
positions of the curves and the outcome of the model. These
exogenous variables are the foreign British interest rate
(*\(i_{£}\)*) and the expected future exchange rate
(*\(E_{$/£}^{e}\)*), which influence the foreign British
rate of return (*\(R_{0}R_{£}\)*); the U.S. money supply
(*\(M_{$}^{S}\)*) and domestic U.S. price level
(*\(P_{$}\)*), which influence real money supply; and U.S.
GNP (*\(Y_{$}\)*), which influences real money demand. The
endogenous variables in the asset model are the domestic interest
rates (*\(i_{$}\)*) and the exchange rate
(*\(E_{$/£}\)*). See __Table
9.2 "Asset Market (Money + Forex)"__ for easy
reference.

Figure
\(\PageIndex{1}\):* Derivation of the AA
Curve*

Exogenous Variables | \(i_{£}, E_{$/£}^{e}, M_{$}^{S},
P_{$}, Y_{$}\) |

Endogenous Variables | \(i_{$}, E_{$/£}\) |

Figure \(\PageIndex{2}\): Table 9.2 Asset Market (Money + Forex)

Initially, let’s assume GNP is at a value in the market given
by *\(Y_{$}^{1}\)*. We need to remember that all the
other exogenous variables that affect the asset market are also at
some initial level such
as* \(i_{£}^{1}\)*, *\(E_{$/£}^{e1}\)*, *\(M_{$}^{S1}\)*,
and *\(P_{$}^{1}\)*. The real money demand function
with GNP level *\(Y_{$}^{1}\)* intersects with
real money supply at point *\(G_{1}\)* in the
money market diagram determining the interest
rate *\(i_{$}^{1}\)*. The interest rate in turn
determines *\(R_{0}R_{$}^{1}\)*, which intersects
with *\(R_{0}R_{£}\)* at
point *\\(G_{2}\)*, determining the equilibrium
exchange rate* \(E_{$/£}^{1}\).* These two values
are transferred to the lowest diagram at point *\(G\)*,
establishing one point on the AA curve (*\(Y_{$}^{1},
E_{$/£}^{1}\)*).

Next, suppose GNP rises, for some unstated reason,
from *\(Y_{$}^{1}\)* to *\(Y_{$}\)*,
ceteris paribus. The ceteris paribus assumption means that all
exogenous variables in the model remain fixed. Since the increase
in GNP raises real money demand, *\(L(i_{$}, Y_{$})\)*,
it shifts out to *\(L(i_{$}, Y_{$}^{2})\)*. The
equilibrium shifts to point *H*_{1}, raising
the equilibrium interest rate to *\(i_{$}^{2}\)*.
The *\(R_{0}R_{$}\)* line shifts right with the
interest rate, determining a new equilibrium in the Forex at
point *\(H_{2}\)* with equilibrium exchange
rate *\(E_{$/£}^{2}\)*. These two values are then
transferred to the diagram below at point *\(H\)*,
establishing a second point on the AA curve (*\(Y_{$}^{2},
E_{$/£}^{2}\)*).

The line drawn through
points *G* and *H* on the lower
diagram in Figure 9.4 is called the AA curve. The AA
curve plots an equilibrium exchange rate for every possible GNP
level that may prevail, ceteris paribus. Stated differently, the AA
curve is the combination of exchange rates and GNP levels that
maintain equilibrium in the asset market, ceteris paribus. We can
think of it as the set of aggregate asset equilibriums.

## A Note about Equilibriums

If the economy were at a point off the AA curve, like
at *I* in the lower diagram, the GNP level is
at *\(Y_{$}^{1}\)* and the exchange rate
is *\(E_{$/£}^{2}\).* This corresponds to
point *I* in the upper diagram
where *\(R_{0}R_{£} \ > R_{0}R_{$}\)*. In the Forex
model, when foreign assets have a higher rate of return than
domestic assets, investors respond by buying pounds in exchange for
dollars in the foreign exchange market. This leads to a
depreciation of the dollar and an increase in *\(E_{$/£}\)*.
This continues until *\(R_{0}R_{£} \ =
R_{0}R_{$}\)* at point *G*. For all points
below the AA curve, *\(R_{0}R_{£} \ > R_{0}R_{$}\)*;
therefore, the behavior of investors would cause an upward
adjustment toward the AA curve from any point
like *I* to a point like *G*.

Similarly, at a point such as *J*, above the DD
curve, the GNP level is at *\(Y_{2}\)* and the
exchange rate is *\(E_{$/£}^{1}\)*. This corresponds to
point *J* in the upper diagram
where *\(R_{0}R_{$} \ > R_{0}R_{£}\)* and the
rate of return on dollar assets is greater than the rate of return
abroad. In the Forex model, when U.S. assets have a higher rate of
return than foreign assets, investors respond by buying dollars in
exchange for pounds in the foreign exchange market. This leads to
an appreciation of the dollar and a decrease
in *\(E_{$/£}\)*. This continues
until *\(R_{0}R_{£} \ = R_{0}R_{$}\)* at
point *H*. For all points above the AA
curve, *\(R_{0}R_{$} \ > R_{0}R_{£}\)*; therefore,
the behavior of investors would cause a downward adjustment to the
AA curve from a point like *J* to a point
like *H*.

Figure \(\PageIndex{3}\): A 3-D AA Curve

As with the DD curve, it is useful to think of the AA curve as a river flowing through a valley. (See the 3-D diagram in Figure 9.4.3 .) The hills rise up both above and below. Just as gravity will move a drop of water down the hill to the river valley, in much the same way, investor behavior will move the exchange rate up or down to the lowest point lying on the AA curve.

key takeaways

- The AA curve plots an equilibrium exchange rate level for every possible GNP value that may prevail, ceteris paribus.
- Every point on an AA curve represents an equilibrium value in the money-Forex market.
- The AA curve is negatively sloped because an increase in the real GNP lowers the equilibrium exchange rate in the money-Forex model.

exercise

**Jeopardy Questions**. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”- This is what has happened to its GNP if an economy’s exchange rate and GNP combination move downward along a downward-sloping AA curve.
- Of
*greater than*,*less than*, or*equal to*, this is how the rate of return on domestic assets compares to the rate of return on foreign assets when the economy has an exchange rate and GNP combination that places it above the AA curve. - Of
*greater than*,*less than*, or*equal to*, this is how the rate of return on domestic assets compares to the rate of return on foreign assets when the economy has an exchange rate and GNP combination that places it on the AA curve. - The equilibriums along an AA curve satisfy this condition.