1
Answers to Questions for Chapter 5
1. The IS relation traces out combinations of the interest rate and the level of GDP such
that total output is actually sold to households, firms, and governments (ignoring foreign
trade, for simplicity). Along the IS schedule, Y  C  I  G implies that inventories are
neither rising nor falling. [In an open economy, Y  C  I  G  Net Exports at every
point on the IS curve.]
The IS relation is inverse: a lower interest stimulates demand for durable goods,
especially capital goods (Investment). Then the multiplier process comes into play so
that GDP rises by more than the initial stimulus to Investment caused by lower interest
rates. The additional GDP is mainly Consumption. However, there will be more
Investment (beyond the initial stimulus) if the IS relation incorporates a marginal
propensity to invest.
Reversing the argument, higher interest rates reduce interest sensitive spending in the
goods market. This causes a fall in GDP, which is larger than the fall in Investment, as
the multiplier process works itself out, reducing consumption and investment further.
Output moves horizontally towards the IS relation from points off the curve. To the right
of the relation, both output and expenditure are higher. Inventories are rising, however,
because the additional spending is less than the additional output owing to the fact that
the marginal propensity to spend is a fraction: c1 (1  t1 )  b1  1 . Firms react to rising
inventories by reducing output. To the left of IS, both output and expenditure are lower.
However, inventories are falling because the reduced spending is less than the reduced
output, again because the marginal propensity to spend is a fraction. Now, firms react by
increasing output in order to stop the fall inventories.
Investment can be more than or less than private Saving at a point on the IS relation.
However, if the word “Saving” means total saving (private saving plus the government
surplus plus the trade deficit, which requires an inflow of foreign saving), then one can
say that Investment = National Saving plus the Inflow of Foreign Savings.1 Recall the
national income accounting relation: I  G  EX  S  T  IM so that:
I  [ S  (T  G )]  ( IM  EX )  National Saving  the Trade Deficit
 National Saving  Foreign Inflow of Saving
If foreign trade is ignored, Investment is equal to National Saving. The latter is defined
as Private sector saving plus Government saving (the budget surplus). Government
saving is negative when there is a budget deficit. Therefore, a budget deficit reduces
national saving.
1
The latter is an Outflow for countries with a trade surplus. Then, the addition is a negative
number.
2
2. The first equation states that Capital Spending changes in the same direction as current
GDP (the plus sign). One reason is that higher GDP, by creating the expectation of an
increase in future sales, will encourage firms to prepare for that increase by undertaking
capital spending in the current period; lower GDP, by creating the expectation of a
decrease in future sales, will discourage firms from purchasing additional capital goods in
the current period. Another reason is that higher GDP increases retained earnings, which
is the main source of financing for capital spending. Therefore higher GDP results in
higher capital spending. Lower GDP puts everything in reverse.
The first equation also states that Capital Spending varies inversely with the interest rate.
A simple explanation is that higher interest rates increase the cost of borrowing to finance
the purchase of capital goods. Lower interest rates decrease the cost of borrowing. A
more sophisticated explanation recognizes the importance of the useful life of the
physical asset being purchased. The longer the useful life of a capital asset, the more
sensitive its present value is to changes in the interest rate.
The right side of the first equation is a linear form of the investment function (there are
no reciprocal effects or squared terms). It contains a constant term, which is useful for
representing changes in Investment that are unrelated to current changes in GDP or to the
interest rate. For example, advances in technology force competitive firms to modernize
their capital equipment. This can be captured by an increase in b0 . Excess capacity, can
cause a sudden drop in the constant term. Not long ago, a large increase in investment in
fiber optic cables was followed by huge drop once the new capacity to transmit data was
installed. More recently, it has become very clear that there is excess capacity in the
automobile industry worldwide.
In words, the equation for the IS relation states that GDP is equal to the basic multiplier
times an adjusted level of exogenous demand. The adjustment takes account of the effect
of interest rates on capital spending and therefore depends on the inverse sensitivity of
Investment to changes in interest rates.
In terms of the model, Y  C  I  G can be solved, after substituting the linear
equations for consumption and investment, to yield:
Y

1
c0  c1t0  b0  G  b2 r 
1  c1 (1  t1 )  b1
1
1
c0  c1t0  b0  G  
b2 r
1  c1 (1  t1 )  b1
1  c1 (1  t1 )  b1
1
1
50  0  150  400 
2000r
1  .9(.75)  .075
1  .9(.75)  .075
1
Y
(600  2000r )
1  .75
Y  4(600  2000i )  2400  8000r
Y
3
It follows that when r  .05  5% , equilibrium GDP is 2400  400  2000 .2
To check this answer for goods market equilibrium, substitute both the given interest rate,
and the solution for GDP into the consumption function and the investment function.
Tax revenue is 25% of GDP. Therefore disposable income is 1500.
C  50  .9(2000  500)  50  1350  1400
I  150  .075(2000)  2000(.05)  150  150  100  200
Because Government spending is fixed at 400, the answer for GDP checks out:
Y  2000  1400  200  400  C  I  G
As for the Budget Surplus, this is now easy to calculate:
T  .25Y  500
G  400
BS  T  G  100
In a closed economy (no inflow or outflow of savings to finance a non-zero trade
balance), there must be an offsetting excess of Investment over private Saving. To check,
subtract consumption from disposable income to obtain private saving:
S  Y  T  C  2000  500  1400  100
I  200
I  S  200  100  100
Therefore, as expected, Investment exceeds private saving by exactly the amount of
government “saving”, namely, the budget surplus. The government does not directly
finance private investment. Rather, the government is retiring debt, using the budget
surplus. The excess of tax receipts over government spending thus flows back to the
private sector, as bonds are paid off. These funds are available to finance the excess of
capital spending over private saving. You can imagine the previous holder of a maturing
government bond using the proceeds to buy a new corporate bond.
At any other point on the IS schedule, the solution for GDP will differ. At r  .06 , the IS
equation tells us that GDP is 1920. This must be checked. Disposable income has fallen
to 1920(1 - .25) = 1440, using the tax rate, so that consumption is down to .9(1440) =
I have been a little cavalier about the symbol for the interest rate. Strictly speaking, it
should be r for “real interest rate” in the goods market and i for “nominal interest rate”
in the money market. When there is no inflation, the two coincide. When inflation is
brought into the argument, i  r will be allowed for by placing the equilibrium solution
for output to the right of the intersection of IS and LM.
2
4
1346, using the marginal propensity consume. Investment has fallen for two reasons. It
is lower by 20 (2000 times the one percentage point increase in r ). It is also lower by 6
(.075 times the fall of 80 in Y ). Therefore Investment is down by 26 to 174.
C  c1 (1  t1 )Y  .675  (80)  54
I  b1Y  b2 r  .075  (80)  2000  (.01)  6  20  26
Now, the answer can be checked using levels or changes:
Y  1920  1346  174  400  C  I  G
Y  80  54  26  C  I
Government spending is unchanged along a given IS curve; it is assumed to be
independent of current tax receipts and current interest rates in this simple model.3
Therefore the Budget surplus has fallen as tax receipts go down. The budget surplus is
now (.25  1920)  400  480  400  80 . We should therefore expect the gap between
capital spending and private saving to fall to 80 as well. As a check, disposable income
of 1440 minus consumption of 1346 leaves private saving of 94 . And we have shown
that Investment is down to 174 . As expected, the gap is down to 80 . This is a further
check on the coherence of the solution for goods market equilibrium.
In general, if government spending is fixed in a closed economy, a movement up the IS
relation lowers the budget surplus. If the surplus falls to zero at some point on the IS
relation, then there is a deficit at still higher interest rates. The reason is that a movement
up along IS reduces tax receipts as income falls.
As noted above, Investment is not equal to private saving unless the budget surplus is
zero (in a closed economy). Therefore the correct statement should have been Investment
is equal to the sum of private saving plus the budget surplus at each point on the IS curve.
If there is a budget deficit, the statement is still correct: “plus the budget surplus” means
adding a negative number.
3. The equation for the IS relation makes it clear that any change in exogenous demand,
that is, any change in c0 , t0 , b0 or G will shift the IS relation in a parallel fashion: to the
right c0 , b0 or G is higher or t 0 is lower; and to the left in the opposite case. The
horizontal shift is greater than the change c0 , t0 , b0 or in G because of the multiplier.
In fact, state and local government spending is likely to respond to both interest rates
(on state and municipal bonds issued in connection with some types of expenditure) and
GDP. At the present time, many state governments are cutting back on spending as tax
revenues fall.
3
5
If the multiplier itself increases because of a fall in the tax rate, t1 , the IS relation shifts to
the right and becomes flatter. Any change in the interest rate would then have a larger
effect on aggregate demand because the basic multiplier is larger.
4. The first three equations are identical. Note, however, that the third equation expresses
the equilibrium condition as an equation between the supply of money measured in terms
of its purchasing power over goods (the real supply of money) and the real demand for
money. Thus, the right side of the third equation depends on real income, Y, as opposed
to nominal or money income, $Y or PY.
The two forms of the “real money supply equals real money demand” equation, using
L (Y , i ) and YL (i ) are different. The first simply says that real demand depends on real
income and the interest rate without saying how. The second says something more
restrictive: real demand for money is proportional to real income at any given rate of
interest. If GDP is higher by, say 10%, this second equation says that the real demand for
money is also higher by 10% (at a given interest rate). Empirical evidence shows that
this is not true. Therefore, the more general form L(Y , i ) , by not assuming such
proportionality is superior. A plus sign below output indicates that a change in output
and therefore income and expenditure is associated with a change in the demand for
liquidity in the same direction. A minus sing below the interest rate indicates an inverse
relation between changes in the interest rate and changes in the quantity of liquid
purchasing power demanded by the private sector. The derivation of a particular example
of the function L(Y , i ) is included at the end of this answer sheet.
The LM relation traces out combinations of the interest rate and the level of GDP such
that the demand for liquid purchasing power L is equal to the supply of currency plus
demand deposits at commercial banks, also expressed in real terms: M / P . This relation
is direct because an increase in real GDP raises the volume of transactions in goods and
services. A larger stock of liquid purchasing power is demanded. As firms and
households attempt to increase their real money holdings to satisfy this demand, interest
rates rise. The increase can be explained in two ways. Firms and households sell bonds
in order to obtain money, thereby causing bond prices to fall, which increases yields and
therefore interest rates. Or, increasing demand for bank loans allows banks to demand
higher interest rates from borrowers. In either event, higher interest rates provide an
incentive for everyone to economize on money. The consequence for the economy as a
whole is that the average rate of turnover of the stock of money increases. This allows a
given stock of money to finance a larger flow of transactions in goods and services as
GDP rises. The whole process works in reverse when GDP is lower, causing interest
rates to fall, slowing down the average rate of turnover of the money stock.
Above the LM curve there is excess supply of money: at high interest rates, everyone is
being careful to economize on their holding of money, trying to get rid of it, but, as a
group, failing to do so. The counterpart to this activity is the purchase of bonds, which
drives up bond prices, lowers yields, and pushes interest rates down until firms and
households, as a group, stop trying to become less liquid. The existing stock of money is
then willingly held at a point on the LM curve.
6
Below the LM curve, there is excess demand for money: at low interest rates there is little
incentive to economize on holdings of money. Everyone is trying to be more liquid. The
counterpart is a sale of bonds (which have high prices at low interest rates). But this sale
causes bond prices to fall, raises yields, and pushes interest rates back up to a point on the
LM schedule where, once again, the existing stock of money is willingly held.
On the LM relation, households and firms, as a whole, are not trying to become either
more or less liquid. All this assumes the money supply is a definite quantity at each
interest rate. In the simplest case (in your text) that quantity never changes unless the
central bank acts by engaging in open market purchases or sales of government bonds.
Additional argument: Note that if the reserve ratio  (see chapter 4) falls as interest rates
rise, then banks must be lending out excess reserves as interest rates rise. The deposit
creation multiplier (chapter 4) therefore rises. This means that the existing stock of highpowered money (currency and bank reserves) corresponds to a larger stock of money
(currency plus demand deposits) at higher interest rates. As a result, the supply curve of
money is upward sloping and the LM curve is somewhat flatter than it would be with a
vertical supply curve corresponding to a constant M / P .
The demand for money is higher as we move up the LM curve because GDP is increasing
(and vice versa on the way down). However, the quantity of money demanded is not
higher if the supply is fixed. This is because the interest rate keeps rising to push the
economy to a point where the demand curve for money intersects a vertical supply curve.
5. The linear form of the LM relation can be written as:
d 2i  (d 0  M s / P)  d1Y
i
d 0  M s / P d1
 Y
d2
d2
The first term is the intercept of the LM relation on the interest rate axis; the second term
shows the slope as the ratio d1 / d 2 . It follows that the simplest explanation for a
downward shift in the LM relation is an increase in M s caused by a purchase of
government bonds by the central bank. This purchase ends up raising the money supply
as the high-powered money provided by the central bank finds it way into increased
holdings of cash and commercial bank reserves (the base for more lending). The
consequent drop in interest rates at each level of GDP can be thought of as the result of
the central bank bidding up the price of bonds, combined with the effect of competition
among commercial banks, eager to lend out the increased reserves created by the central
bank open market purchase of government bonds.
6. Substitute the data into the equation for the LM relation:
1400  600  .5Y  4000i
7
i
600  1400
.5
Y

Y  .20 
4000
4000
8000
When GDP is 2000, the equilibrium interest rate in the money market is i  .05  5% .
When GDP is 3000, i  .175  17.5% . Along this LM schedule a reasonable range of
variation in GDP would be between, say, 1800 and 2400; the corresponding range for the
equilibrium interest rate is from 2.5% to 10%. With no inflation, the nominal interest rate
that clears the money market can double as the real rate that affects expenditure on
durable goods (Investment in the goods market).
7. Using the answers to questions 2 and 6, it should be obvious that the solution to the
model is (Y , i  r )  (2,000, .05) . The reason is that when we assume that the interest
rate is 5%, the goods market equilibrium solution is Y  2000 ; and when we assume that
output is 2,000, the money market equilibrium solution is 5%. Thus, the two markets are
in simultaneous equilibrium. If instead we had assumed Y  2500 , equilibrium in the
money market would require i  11.25% , using LM. However, at this interest rate
equilibrium in the goods market would be only Y  1500 , using IS. Likewise, if we
assumed an output below 2000, the lower interest rate from LM would, upon substitution
in IS, have yielded an output above 2000. In neither case is there an overall equilibrium
in both the goods market and the money market.
The general solution to the IS/LM model forces LM to be satisfied in the IS relation.
Therefore substitute the above equation for the interest rate i into the IS relation where
r appears (assuming no inflation so that nominal and real interest rates are equal):
Y
 d  M s / P d1 
1
1
c0  c1t0  b0  G  
b2  0
 Y
1  c1 (1  t1 )  b1
1  c1 (1  t1 )  b1 
d2
d2 
GDP appears on both sides. It is necessary to group the terms in Y on the left side and
factor out Y. First, write:
 d  M s / P d1 
Y [1  c1 (1  t1 )  b1 ]  c0  c1t0  b0  G   b2  0
 Y
d
d2 
2

Then, move the new term in output, which is introduced into the right side when LM is
substituted into IS, to the left side. Factor out GDP and divide by the term in square
brackets on the left:
8

bd 
b Ms
Y 1  c1 (1  t1 )  b1  2 1   [c0  c1t0  b0  G ]  2 
d2 
d2 P

M s


Y
c0  c1t0  b0  G  
 d0 

bd
bd

1  c1 (1  t1 )  b1  2 1
1  c1 (1  t1 )  b1  2 1  P
d2
d2
1
b2 / d 2
The ratio that multiplies the first bracketed term is the IS/LM multiplier (smaller than the
basic multiplier because of an additional term in the denominator) for shifts in exogenous
demand. Such shifts include changes in Government spending. The ratio that multiplies
the second bracketed term is the multiplier for shifts in the real money supply and for
shifts in real money demand unrelated to changes in income, i.e., for changes in d 0 . The
last term in the denominator of both these multipliers can be written as:
b2 
d1
: the sensitivity of Investment to the interest rate TIMES the slope of LM
d2
The important difference between the IS/LM multipliers for changes in exogenous
demand and for changes in the real money supply is in the numerator. In the first
multiplier, it is simply 1; in the second, we find the ratio of parameters measuring the
responses of Investment and of Money Demand to changes in the interest rate. Using our
parameter values, we confirm the answer already found by guesswork:
50  0  150  400
.5
1  .675  .075  2000
4000
2000 / 4000
1,400  600

.5
1  .675  .075  2000
4000
Y  2[600]  1[800]  2,000
Y
1
We must now use this answer to find the interest rate. Substitute Y  2000 into the LM
equation:
i
600  1400
.5
Y
2000

Y  .20 
 .20 
 .05
4000
4000
8000
8000
This confirms what was already known from questions 2, 6, and 7. The exercise now
gives us a general formula for determining GDP, based on substituting LM into IS and
solving for Y. Then, output is put back into LM to find the interest rate. In the example,
the IS/LM multiplier for a shift in exogenous demand is 2. The IS/LM multiplier for an
increase in the real money supply (or a drop in real money demand) is 1.
9
8. If Government spending increases by 30, our general formula tells us that GDP
increases by 60, but we need to check this. First, find the change in the interest rate
required to clear the money market following an increase in the demand for money
associated with the higher level of GDP. The new rate is:
Y
2060
 .20 
 .0575  5.75%
8000
8000
i  .20 
The interest rate has increased by 75 basis points (a basis point is 1/100 of a percentage
point) in order to clear the money market at the higher level of GDP. The quantity of
money demanded has not changed because the supply is fixed. Average cash balances
have fallen (as the interest rate increases) to allow a fixed stock of money to finance a
larger flow of transactions. Second, confirm that the goods market clears by calculating
Consumption and Investment, which together with the new level of Government spending
must add up to 2060. Note how the solution is used to find consumption and investment,
which are partly endogenous:
C  50  .9(1  .25)2060  1440.5  C  40.5
I  150  (.075)2060  (2000)(.0575)  189.5  I  4.5  15  10.5
G  430  G  30
C  I  G  2060  Y
 Y  60
The fiscal stimulus has raised aggregate demand. Consumption is higher by 40.5 because
disposable income is higher. Investment has fallen by 10.5 because the effect of the rise
in the interest rate has more than offset the effect of higher GDP working through the
marginal propensity to invest. When Investment falls as a result of the higher interest
rates, indirectly caused by a fiscal stimulus, economists say that Investment has been
“crowded out”. Some capital spending is curtailed as a result of higher borrowing costs
caused by the sale of new government bonds to finance the increase in government
spending, which exceeds the increase in tax revenue caused by higher GDP.
Tax receipts are higher by an amount equal to the tax rate times the change in GDP:
.25  60  15 . Therefore, the budget surplus has declined by 15: government expenditure
increase by 30 while tax revenue increased by only 15. As a final confirmation of our
solution, the excess of Investment over Savings must also decline by this reduction in the
budget surplus. To check, find disposable income:
Y  T  2060  (.25)2060  (.75)2060  1545
Disposable income minus Consumption is private saving: 1545  1440.5  104.5 .
Finally, the excess of Investment over private Saving is: 189.5  104.5  85 , which is
lower than what is was (100 before the fiscal stimulus) by exactly 15, as expected.
9. By making the sensitivity of money demand to the interest rate larger, the slope of the
LM relation changes from
10
d1 / d2  .5 / 4000  1 / 8000 to d1 / d2  .5 / 12,000  1 / 24,000
The new IS/LM multiplier for changes in exogenous demand is therefore:
1
1  .75  2000
1
24,000

1
1

3
3 1
2
1 
1
4 12
3
This multiplier is larger because, when GDP rises, thereby raising the demand for money,
the interest rate does not have to increase as much to clear the money market (money
demand is more sensitive to changes in the interest rate). The negative feedback on
Investment is therefore lower and so the effect of the fiscal stimulus on output is larger.
As for the effect of reducing the sensitivity of Investment to changes in the interest rate,
this too will increase the IS/LM multiplier for changes in exogenous demand, such as a
change in Government spending. Using the original slope for LM, the fiscal policy
multiplier becomes:
1
1  .75  1000
1
8000

1
1

 8 / 3  2.67  2
3 1
5
1 
1
4 8
8
This shows that the IS/LM fiscal multiplier can rise for two different reasons. If the
model were reset to yield the same initial solution, one would see a steeper IS curve
shifting to the right (as government spending increases) to intersect an unchanged LM.
In the previous case, where the slope of LM was reduced, the original IS relation shifts
out to intersect the flatter LM at a higher GDP and an interest rate only slightly higher
than the initial rate. In each case the rise in GDP is greater, but in one case the rise in the
interest rate is small (flat LM) and in the other is large (steep IS).
10. In the general solution equation for GDP, it is clear the multiplier for an increase in
the real money supply differs from that for an increase in government spending because
the numerator, instead of being 1, is b2 / d 2  2000 / 4000 . This ratio could be larger or
smaller than one. The monetary policy multiplier could be larger or smaller than the
fiscal policy multiplier. In the example, the monetary policy multiplier is:
b2 / d 2
1  c1 (1  t1 )  b1 
b2 d1
d2

.5
1
1  .75  .25
An increase in the real money supply of 100 raises aggregate demand by 100. Because
interest rates move faster than GDP, the initial effect (before GDP rises) of the extra 100
in real money is to force interest rates down sufficiently to clear the money market. This
11
decrease is found by solving the LM curve at the initial equilibrium output, but with a
larger real money supply:
1400  100  1500  600  .5(2000)  4000i
4000i  100
i  1 / 40  .025  2.5%
The interest rate has fallen from 5% to 2.5%. The mechanism by which monetary policy
raises aggregate demand in the IS/LM model is through its effect on capital spending.
Given the sensitivity of Investment to a change in the interest rate, it follows that Capital
spending will rise by  b2 i  2000  (.025)  50 . The IS/LM multiplier for any
change in aggregate demand now gets to work. In our example, it is 2 and, of course, 2
times 50 is 100, as we expect—the monetary policy multiplier is 1.
Note that the monetary policy multiplier is equal to the fiscal policy multiplier times the
initial effect of a change in the real money supply on capital spending, namely:
I
b2 / d 2 
. Some of this stimulus is then eroded as the rise in GDP increases
( M s / P)
the demand for money. Therefore the interest rate rises back up from the initial drop,
cutting into some of the initial stimulus to capital spending.
To check our answer, it is necessary to first calculate the equilibrium interest rate. We
use the new money supply and the new solution for GDP:
1500  600  .5(2100)  4000i
4000i  150
i  150 / 4000  .0375  3.75%
The interest rate has risen somewhat, after its initial fall from 5% to 2.5%. Now, let us
check to be sure that the answer is correct. Use the initial value for Government
spending in order to focus on monetary policy by itself. We check as before:
C  50  .9(1  .25)2100  1467.5  C  67.5
I  150  (.075)( 2100)  (2000)(.0375)  232.5  I  32.5
G  400  G  0
Y  C  I  G  2100  Y  100
There is an important difference between a fiscal stimulus and a monetary stimulus.
Fiscal policy raises aggregate demand by increasing Government spending and
Consumption spending. Capital spending often falls although it could rise if the marginal
propensity to invest is sufficiently large relative to the sensitivity of Investment to interest
rates. Monetary policy raises aggregate demand by increasing Capital spending and
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Consumption spending, with no change in Government spending. Interest rates rise
under fiscal stimulus and fall under monetary stimulus.
11. The fiscal policy IS/LM multiplier increased when d 2 was higher or when b2 was
lower. The monetary policy multiplier moves the other way for these same parameter
changes.
Under fiscal stimulus, the rise in the interest rate is greater and the fall in investment is
greater when b2 is low and d 2 is high. But such parameter values are just what is
needed to give monetary policy an edge in terms of its effect on aggregate demand.
Interest rates fall more and investment rises more, under monetary stimulus, when b2 is
low and
d2
is high. Using
d 2  12,000  4,000 for example::
b2 / d 2
1  c1 (1  t1 )  b1 
If instead b2
b2 d1
d2

2,000 / 12,000
 .5  1
2,000
1  .75  (.5)
12,000

1,000 / 4,000
 2/3 1
1,000
1  .75  (.5)
4,000
 1000  2000 :
b2 / d 2
1  c1 (1  t1 )  b1 
b2 d1
d2
In each case, the monetary policy multiplier is smaller. In the first case, the initial drop in
the interest rate is smaller. Therefore, the rise in Investment is smaller. In the second
case, the initial drop is the same as before, but the rise in Investment is smaller because
investment is assumed to be less sensitive to changes in the interest rate.
12. Other things equal, a higher b2 / d 2 increases the effect on GDP of a monetary
stimulus while reducing the effect of a fiscal stimulus. Predictions about the effects of
fiscal and monetary policy on aggregate demand must therefore come to grips with the
problem of pinning down this ratio empirically. The other parameters matter too, but
they affect both fiscal policy and monetary policy multipliers in the same way. Thus,
both multipliers are higher when c1 and d1 are higher; both are lower when t1 and d1
are higher.
In addition to the problem of estimating the sensitivity of investment and money demand
to interest rate changes, namely, b2 and d 2 , there is the further question of how stable
the investment and money demand equations are. In terms of the notation, instability
would mean that b0 and d 0 , the intercepts of the investment and money demand
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equations, jump around a lot. For example, at the present time, it appears that investment
has shifted down and the demand for liquidity in virtually insatiable, both reflecting
uncertainty about the future course of the economy. This translates into a flat money
demand equation and therefore a flat LM curve, while the IS curve has shifted left while
becoming steeper—low interest rates just don’t seem to have much effect on capital
spending. This is consistent with a low interest rate recession.
Notes on the demand function for liquidity purchasing power.
An example of an explicit demand function for liquid purchasing power is the famous
“square root formula” first proposed by William Baumol and James Tobin (in separate
papers):
L(Y , i )  bY / 2i
In this formula, b is the “brokerage cost” per financial transaction in a model where
households, for example, choose an optimal number of financial transactions “between
pay checks” where Y is the real value of the paycheck. This formula is easy to derive,
and can be modified to take account of the price level, as shown below.
Consider the lost interest or opportunity cost of real money held as cash (or in a noninterest bearing account). On average the purchasing power held as money held is related
to real income: L  Y / 2n . This assumes income is spent at a steady rate between pay
checks (hence the “2”) and that the number of times you go to the bank to obtain money
is n . If n  1 , this just means that you go to the bank once to cash your whole check,
and then spend the money at a steady rate until the next paycheck—no interest is earned.
Two trips means you cash half your check when you receive it and invest the other half in
an interest bearing asset (one trip to the bank), and then go back at the mid-point between
pay checks to liquidate the assets (second trip to the bank), earning interest for the first
half of the interval between paychecks. For simplicity, this interest income is not added
to Y . If n  3 , cash out one third of your paycheck on the first visit and invest the
remaining 2/3; go back ten days later (monthly paycheck) and cash out half your interest
bearing assets (1/3 of your paycheck); and finally go back ten days before the next check
and cash out the remaining half of your interest bearing assets. In effect, you have
simulated a more frequent paycheck, 1/3 of your monthly income every ten days. As the
number of financial transactions rises, your simulated paycheck gets smaller and more
frequent while holding constant your actual monthly paycheck.
The average cash balance falls as the number of financial transactions increases. The
interest cost of holding L  Y / 2n is just iY / 2n . The interest rate is for the period
during which you are earning interest, which may be quite short.
The downside of all these “trips to the bank (or broker)” is that each financial transaction
entails a “brokerage fee” b . The cost of the financial transactions is just the product of
the brokerage fee times the number of financial transactions: nb .
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It can now be seen that the cost of being liquid is made up of two parts, the lost interest
plus the “brokerage” fees:
iY
 nb
2n
Choose the best value for n by ensuring that this total cost is neither falling at the chosen
value for n (if so, n is too small) nor rising (if so, n is too large). This requires setting
the derivative of total cost equal to zero:
d  iY
iY

2
2
  nb   iY / 2n  b  0  iY  2n b  n 
dn  2n
2b

This solution for the optimal number of financial transactions can now be substituted
back into the expression for money demand as a function of the number of financial
transactions:
L
Y Y 2b
bY


2n 2 iY
2i
This is a non-linear function of real income, the real brokerage fee, and the interest rate.
If the real value of liquidity, the real fee, and real income were all multiplied by the price
level, the equation would still hold:
PL 
( Pb)(PY )
bY
P
2i
2i
 L
bY
2i
This means that a higher price level increases the demand for nominal liquidity
proportionally without changing the real demand for liquid purchasing power.