Answer for Homework 6: Modern Macroeconomics I

1. Assume that population is equal to 1. Consider the following IS-LM curve.
y = c (y(2))+ ( )+g

ms= k( )y(3)P
where price level P , tax revenue , government expenditure g and money supply msare given. Two equations determine GDP, y and the interest rate . Suppose thatc (y) = c0 + c1kk( ) =( ) =(y)awhere c0 , c1 , k, and a are parameters.
(a) Assume that the interest rate is constant. Derive GDP y, as a function ofgovernment expenditure and tax revenue. Show that dy> 1 and explain itsdg
reason.
Answer Equation (2) can be rewritten as
y = c (y)+ ( )+g= c0 + c1 (y) + a + g:
By solving this equation with respect to y, we have
y=g + c0
1c1
c1a

GDP is a function of government expenditure and tax revenue. By di¤erentiating this equation with respect to g,1dy
=>1dg
1 c1

I thank to Wataru Tamura who made these sample answers.

1Figure 1: GDP and interest rate in equilibrium.
Figure 2: E¤ect of an increase in tax revenue.
for all c1 2 (0; 1). This implies that an increase in the government expenditure
generates the multiplier e¤ect. The increase in government expenditure rises
not only the nation’s income directly but also the consumption indirectly.
(b) Derive the GDP, y, and the interest rate, , as the function of P , g, ms , .
Answer Given P , g, ms , , there are two equations (2) and (3) and two unknown
variables, y and . From (2),ay=

1

c1+g + c0c1;
1 c1

(IS)

and from (3), we havekPy:(LM)
ms
as an intersection of these two lines (see=Therefore, we can derive y andFigure 1).

ms (c0 c1 + g)y(g; ; m ; P ) =akP + ms (1 c1 )P k(c0 c1 + g):
(g; ; ms ; P ) =akP + ms (1 c1 )s(c) What is the e¤ect of an increase in tax revenue, , on GDP and the interest rate?
Explain economic logic behind your answer.
Answer We analyze the e¤ect of an increase in tax revenue, , on GDP and theinterest rate. Note that IS curve depends on but LM curve does not dependon . When tax revenue increases, IS curve shifts down (See Figure 2). BothGDP and the interest rate decrease. The economic logic for this result is
as follows. First, IS curve implies that an increase in directly decreases
demand and GDP (see (IS)):") y #
Second, the decrease in GDP reduces the demand for money (md ). When theprice level and money supply are constant, this actually decreases the interest
rate in the money market (see (LM)):") y #) md #)#Since investment i = ( ) = a , a decrease in leads to an increase in theinvestment i :") y #) md #) #) i "2

Figure 3: E¤ect of an increase in government expenditure.
Figure 4: E¤ect of an increase in money supply.
This indirectly increases y") y #) md #)#) i ") y "In this model, the direct e¤ect dominates the indirect e¤ect. Therefore, theincrease in decreases both GDP and the interest rate.
(d) What is the e¤ect of an increase in government expenditure, g, on GDP and theinterest rate? Explain economic logic behind your answer.
Answer Note that IS curve depends on g but LM curve does not depend on g.
When government expenditure, g, increases, IS curve shifts up (see Figure4). Both GDP and the interest rate increase. The economic logic for thisresult is as follows. First, the IS curve implies that an increase in g directly
increases demand and GDP (see (IS)):g ") y "Second, the increase in GDP increases the demand for money. When theprice level and money supply are constant, this actually increases the interest
rate in the money market (see (LM)):g ") y ") md ")Since i = ( ) =i:"a , an increase in leads to the decreases in the investmentg ") y ") md ")") i #

This indirectly decreases yg ") y ") md ")") i #) y #In this model, the direct e¤ect dominates the indirect e¤ect. Therefore, theincrease in g increases both GDP and the interest rate.
(e) What is the e¤ect of increase in money supply, ms , on GDP and the interest rate?
Explain economic logic behind your answer.
Answer Note that IS curve does not depend on ms but LM curve depends on
ms . When money supply, ms , increases, LM curve shifts down (see Figure4). GDP increases but the interest rate decreases.
The economic logic for this result is as follows. From the LM curve (LM), theincrease in the money supply lowers the market value of money, and therefore
directly lowers the interest ratems ")3#However, the IS curve (IS) implies that a decrease in the interest rate directly
increases the investment and GDPms ")#) i ") y "The increase in GDP increases the demand for money. When the price leveland money supply are constant, this actually increases the interest rate inthe money market:ms ")#) i ") y ") md ")"This indirectly decreases investment i and GDP y:ms ")#) i ") y ") md ")") i #) y #In this model, the direct e¤ect dominates the indirect e¤ect. Therefore, theincrease in ms increases GDP but decreases the interest rate.
(f) What is the limitation of IS-LM model?
Answer IS-LM model is static. In order to analyze long-term changes or e¤ectsby policies, we need a dynamic model. A static model cannot answer what isthe temporal e¤ect and what is the permanent e¤ect for example.
2. Answer the following questions.
(a) A sticky nominal wage is considered as one of the source of an increasing aggregate
supply curve. Explain its logic.
Answer Assume that the nominal wage is …xed in the short run: W = W . Notethat in the short run equilibrium,w=0(le ) T;
00

where (le ) F (1; le ) and 0 (le ) > 0, (le ) < 0. Since labor demand curve
is a decreasing in the real wage, if the nominal wage is rigid, labor demand
must be increasing in P :P ") w =W#)P

0(le ) T #) le " :Therefore, the sticky nominal wage is one of the source of an increasing aggregate supply curve.
(b) An imperfect information can be considered as another reason for an increasing
aggregate supply curve. Explain its logic.
Answer Assume that …rms observe output price P , but workers cannot. Hence,workers must make information about price. Workers’expected price is de-4

noted as P e . Then workers respond to PWe :heWPe= he= heW PP PePw e :PSuppose that the overall price level P goes up. However, workers do not knowa change in aggregate price. Hence, P e stays the same. Hence, PPe goes up
and supply curve shifts to right.
(c) How do two explanations change policy implications?
Answer Because there is unemployed workers under sticky nominal wage model,
an increase in demand can increase GDP by employing more workers. However, if the imperfect information model is correct, there are no unemployedworkers. If active stabilization policy itself brings the uncertain movement ofprice, it may increase workers’further misperception. It reduces the welfareof an economy.

5