Homework 3
Economics 215
Intermediate Macroeconomics
Assigned: Monday, April 23, 2002
Due: Friday, May 3, 2002
1. Consider an economy, in which there is a proportional tax on labor income, .
W
When a worker is paid a pre-tax real wage,
, the government collects a
P
W
W
fraction,  , and the government keeps an after tax real wage wAT = (1-) .
P
P
Since the after tax real wage is the reward that workers keep for their work,
the labor supply is an increasing function of the after tax real wage,


2
LS = 4 w AT .
Since the pre-tax real wage is the cost of labor for employers, employers
choose labor supply so that the pre-tax real wage is equal to the marginal
product of labor. In the economy, the production function is Cobb-Douglas
with capital, K=1, and technology, T=1:
1
.
Q  L MPL  12
L
a. Calculate the equilibrium labor, output, and the pre-tax real wage when
 = 0, .15, .5, and .9
The firm will set the marginal product of labor equal to the pre-tax
1
real wage. The after-tax real wage is w AT  (1   ) 12
. The labor
L
2
1 

supply can then be written as L  4 (1   ) 12
  (1   ) . Output is
L

W 1 1
equal to Q  (1  
. Government revenue is

P 2 1
S
Q
1
2
(1   
W
L. Calculate government revenue
P
when the tax rate is  = 0, .15, .5, and .9.
b. Government revenue is equal to 
Tax Rate Labor
0
1
0.15
0.85
0.5
0.5
0.9
0.1
Output
Wage Rate Revenue
1
0.5
0
0.921954 0.542326 0.069147
0.707107 0.707107 0.176777
0.316228 1.581139 0.142302
2. Assume that the demand for M1 in Hong Kong was given by the equation
PQ
. Hong Kong’s central bank, the Hong Kong monetary
M 1D  2
100i
authority operates a fixed exchange rate. This implies that the interest rate in
Hong Kong will be determined by the US interest rate denoted i*. Thus, a
credible fixed exchange rate implies i = i*.
a. In 2001, the average interest rate in the US was about i* = .04. In 2002,
the interest rate is about i* = .01. Calculate the velocity of money in
HK in 2001 and 2002. (Reminder: The velocity of money is the ratio of
nominal GDP to M1).
M 1D
M 1D
1
Velocity is
; in 2001 it is
2
 1 ; in 2002 it is
PQ
PQ
4
M 1D
1
2
 2.
PQ
1
b. For each dollar of demand deposits acquired by banks in HK, the
banks want to keep rd = $.10 on reserve. For each dollar in demand
deposits/checking accounts HK, households want to keep $.20 in cash.
Calculate the money multiplier in Hong Kong (Reminder: The money
multiplier is the ratio of M1 to Mh).
Money Multiplier is
M 1 Currency  DemandDepo sits
cd  1 1.2



4
Mh
Currency Re serves
cd  rd
.3
c. The nominal GDP (measured in HK dollars) in Hong Kong was
HK$1.3 Trillion in 2001 and is projected at $1.22 Trillion in 2002.
Calculate M1 and Mh in 2001 and 2002. The exchange rate with the
US dollar is E = 7.8. Calculate the change in the HKMA’s holding of
foreign currency assets (measured in US dollars) between 2001 and
2002.
M1 demand in 2001 is equal to nominal GDP is 1.3 Trillion since
velocity is 1. and Mh = 1.3 Trillion/Money Multiplier is 325 Billion.
Money demand in 2002 is equal to half of GDP (since velocity is 2) or
601 Billion. Mh is 150.25 Billion. The change in high powered money
is 174.75 Billion. Since with a fixed exchange rate
Mh2002  Mh2001  E( BC* , 2002  BC* , 2001) . This implies that changes in
reserves
Mh2002  Mh2001
174.75
=22.40384615.
 ( BC* , 2002  BC* , 2001) 
E
7.8
3. The Central Government is in the process of constructing a 5-year plan.
Assume that the government has zero debt and plans to finish 5 years from
now with zero debt. This implies that the present value of government outlays
must equal the present value of government revenues
G  I G1
G  I G4
TX 1
TX  4
G  I G  1
 .....  4
 TX 
 ... 
4
1 r
1 r
(1  r )
(1  r ) 4
In each year, the government plans to spend 250 billion RMB (in constant
price terms) on government consumption and investment G+IG = 250 in every
period of the 5 year plan. Beginning next year, the government plan implies
constant tax revenue, TX  TX 1  TX 2  TX 3  TX 4 If the government
runs a balanced budget today, TX= 250, then the present value condition
implies that it will run a balanced budget in the future TX  250 . Calculate
the level of taxes TX that the government must collect in the future if the
government runs a deficit this year of G+IG-TX =100 when r = .1. (Hint:
1 
 1


 1
1  r (1  r ) 5 
1 

 ).

...



4 
1
1

r
(
1

r
)




1


1 r
1
 1


 1
1  r (1  r ) 5
1 


...




1
(1  r ) 4  
1  r
1

1 r
G  IG 
G  I
G
1
  1
  1.1  (1.1) 5

.1
 
 
1 .1


1 
  10 1 
  3.169865

 1.4641

250
250
TX
TX
 .....
 TX 
 ... 
4
1 r
1 r
(1  r )
(1  r ) 4
 1
1
1
1 
 TX  250[
 .....
]  TX 
 ... 

4
1 r
(1  r )
(1  r ) 4 
1  r



 1
1 
100  TX  250 
 ... 

(1  r ) 4 
1  r
100
TX  250 
 281.55
3.169865
4. The attached schedule shows a long term forecast for real government
revenues and real government outlays over the next 10 years. Assume that the
government begins with an outstanding set of assets of 200. This implies an
initial negative government debt equal to DG-1 = -200. Use the attached
spreadsheet to project the government’s debt over the next 10 years under the
assumption that real interest rates are constant and equal to r = .05.
II G
G
500
525
551
579
608
638
670
704
739
776
TX
200
210
221
232
243
255
268
281
295
310
650
676
703
731
760
791
822
855
890
925
DG1
-200
-160
-109
-45.74
31.1489
123.1527
231.883
359.0867
506.6557
676.6374
G
D^G
D
-160
-109
-45.74
31.1489
123.1527
231.883
359.0867
506.6557
676.6374
871.2464