Homework Assignment 1
Economics 514
Macroeconomic Analysis
Due: Mid-term Exam, 2014
1. Interest Rates
a. Use intertemporal theory to construct a model of interest rates for world financial
markets. In this spreadsheet, you will find measures of the quantity of GDP (Yt),
government consumption (Gt), and gross capital formation/investment (It) for the
years 1980-2012 for Europe, the US, and Japan. The data for each series is
converted into constant US$ so it is comparable across time and countries. For
each country a, construct net output, Yt a  Yt a  I ta  Gta , for each year. Estimate
world net output, YtW  Yt EU  Yt JPN  YtUS . By definition, YtW  CtW . Assume that
1
felicity is u (Ct ) 
Ct
1

1
1
and the subjective discount, β, are the same for all
1

countries. Use the Consumption Euler Equation to construct a model of the world
real interest rate. For each period t for 1980-2011, use the future growth path of
YW
net output t 1 W to construct an estimate of the World real interest rate under the
Yt
assumption of β = .98 and   1. For each period between 1984-2011, construct 5
year moving averages, rt W  15  rtW  rtW1  rtW 2  rtW3  rtW 4  . Describe the general
tendency of interest rates.
Under the Euler equation for country I,

1

Ct ,i

1
 u (Ct ,i )   (1  r )u (Ct 1,i )   (1  r )Ct 1,i 
Ct ,i    (1  r )  Ct 1,i
Adding up acros the countries of the world,

World
C
i 1
t ,i
C
World
t
   (1  r )

World
C
i 1
t 1,i
   (1  r ) CtWorld
1

1
(1  r ) 

1  CtWorld
1
 World 
  Ct

b. Get data on US real interest rates from World Development Indicators. Link .
Observe the series, Real Interest Rate (%) for the period 1984-2011. Compare the
dynamics of actual real interest rates with the prediction from the model.
Over time, global growth rates in consumption have been slowing down. This is
consistent with falling global interest rates. Indeed, examining global real interest rates,
they also seem to be on a downward path over the last 30 years.
c. Examine the Autarky interest rate for China. Get data on GDP (Yt), general
government final consumption expenditure (Gt), and gross capital formation (It)
for China, People’s Republic of from United Nations Main Aggregates Database
using GDP by Expenditure, at constant 2005 prices - US Dollars for .
2. Growth Accounting
Researchers at the University of Pennsylvania have assembled a database on
comparable levels of output and investment for a large number of countries. Use the
data from Penn-World Tables. Data can be obtained here: Link, Get data:
Under National accounts-based variables
Real GDP at constant 2005 national prices (in mil. 2005US$) (Yt)
Capital stock at constant 2005 national prices (in mil. 2005US$) (Kt)
Under
Real GDP, employment and population levels
Number of persons engaged (in millions) (Lt)
Under Regions select China and India. Under Periods select 1960, 1970,
1980, 1990, 2000, and 2010.
Yt  K t ( At Lt )1  TFPt K t ( Lt )1
ln Yt  ln TFPt   ln K t  (1   ) ln Lt
ln Yt T  ln Yt  ln TFPt T  ln TFPt   ln K t T  ln K t   (1   ) ln Lt T  ln Lt 
ln Kt T  ln Kt   (1   ) ln Lt T  ln Lt 
ln Yt T  ln Yt ln TFPt T  ln TFPt


T
T
T
T
Y
TFP
K
L
 t:T   t:T   t:T  (1   ) t:T
a. Calculate the average compound growth rate γ for GDP, Capital, and Number of
Employees for each country for each decade.
Capital stock at
constant 2005
national prices (in
mil. 2005US$)
China
India
Number of persons
engaged (in millions)
China
India
Real GDP at constant
2005 national prices
(in mil. 2005US$)
China
India
1950
1960
1970 0.0413437 0.0415364 0.0269221 0.0082999 0.0324192 0.0425407
1980 0.0736858 0.0382248
1990 0.0682065
0.033125 0.0310804 0.0603408 0.0294052
0.045351 0.0284942 0.0262899 0.0887216
2000 0.0949762 0.0544588
0.056888
0.011874 0.0190415 0.0992371 0.0532701
2010 0.1105556 0.0857832 0.0085431 0.0213727 0.0995473
0.072738
b. Assume a Cobb-Douglas utility function with α = ⅓. Calculate the growth of TFP
in each country in each decade.
TFP
China
India
0.0006899 0.0231619
0.0136955
-0.004057
0.0469899 0.0242444
0.0596624 0.0224229
0.057 0.0298951
c. Calculate the share of growth due to each factor and TFP for each country.
China
India
Capital
0.013781
0.024562
0.022736
0.031659
0.036852
Labor
TFP
0.017948 0.0006899
0.022083 0.0136955
0.018996 0.0469899
0.007916 0.0596624
0.057
0.005695
Capital
0.013845
0.012742
0.015117
0.018153
0.028594
Labor
0.017948
0.022083
0.018996
0.007916
0.005695
TFP
0.023162
-0.00406
0.024244
0.022423
0.029895
3. Development Accounting
Compare the level of technology in Hong Kong with that in the USA. Assume a
production function in each country of the form:
Yt  Kt ( At H t Lt )1

H t  e1
yt  kt ( At )1 e  yearst
yearst
where Ht is human capital which is a function of the number of years of schooling.
A. Data on output, labor and capital can be obtained here: Link
Under Current price GDP, capital and TFP
Output-side real GDP at current PPPs (in mil. 2005US$) , (Yt)
Capital stock at current PPPs (in mil. 2005US$) (Kt)
Labor can be calculated as the product of
Under
Real GDP, employment and population levels
Number of persons engaged (in millions)
Average annual hours worked by persons engaged
Under Regions select Hong Kong and United States.
Toggle
display
Hong
Kong
United
States
Under Periods select
Average annual
hours worked by
persons engaged
Capital stock at
current PPPs (in
mil. 2005US$)
Number of
persons
engaged (in
millions)
2010
2010
2010
Output-side
real GDP at
current PPPs
(in mil.
2005US$)
2010
2343.53
1005043.938
3.4521
271894
1695.03
40731044
140.8549
12993576
B. Data on Human Capital can be obtained here Link
2010.
Select China, Hong Kong Special Administrative Region and USA for
Gender select Total; under Age group select 25+. Press
Average years of schooling for both economies.
2010; under
. Report
Average years of schooling
Hong Kong
11.02
United
States
13.42
Under the assumption that α = ⅓ and  = .08, calculate the level of technology for
both economies in 2010.

 yt 1
  yt
k
At   t 
TFPt  At1
Ht
H
Toggle
display
y
A
TFP
Y/K
Hong
Kong
3.752416 33.60824 0.270529 4.658456 2.789296
United
States
5.004813 54.42261 0.319009 6.141763 3.353735
Computational Problems
Macroeconomics often uses computer simulations to study theoretical models. Do a
couple of simulation exercises for an economy with a Cobb-Douglas production
function
1
2
1
2
Yt  Kt 3 ( At Lt ) 3
yt  kt 3 ( At ) 3
a depreciation rate of 8% (δ=.08), a population growth rate of 1% (n = .01),and an
annual technology growth rate of 2% (gA = .02).
1. Golden Rule
Assume that the economy was on its balanced growth path. Normalize technology at
L
time t to At = 1. Assume a constant labor hours per population of t
= 500.
POPt
Output is used for consumption and investment, Ct + It = Yt and investment is a
constant share of output. Calculate consumption per capita at different levels of invest
rates s = 1100 , 1 4 , 1 3 , and 1 2 . Which investment rate generates the highest steady
state labor productivity.

 y 1
Ct
Y
Lt
Lt
 (1  s) t  (1  s) yt
 (1  s)  t  At
POPt
POPt
POPt
POPt
 kt 


1
Lt
s
 (1  s) 
 At
POPt
 n    
s
0.01
0.25
0.333333
0.5
consumption per capita
149.2481156
565.3337711
580.2588532
533.0017909
ss labor productivity
0.301511
1.507557
1.740777
2.132007
s= 13 offers the highest level.
2. Convergence
Assume that the investment rate for the economy is at the level which generates
the highest level of consumption per capita. Examine the dynamics of the neoclassical model. Start in period 0. In that period, technology is A0 = 1 and k0 = 1.5.
Calculate labor productivity in period 0 and the labor productivity if the economy
were on its balanced growth path. What is the percentage gap between the actual
level of the economy and the balanced growth path (y0 and yBGP). Calculate the capital
productivity in period 0. Use this to calculate the growth rate of the capital labor ratio,
k  g k  kt 1  kt between period 0 and period 1. Use the growth rate of the capital
t 1
k
kt
A  At
labor ratio and the growth rate of technology A A  gtA1  t 1
to calculate k1 and
At
A1. Use these numbers to calculate actual and balanced growth path labor productivity
and capital productivity in period 1. Use capital productivity to calculate the growth
rate of the capital-labor ratio in period 1. Calculate technology and the capital labor
ratio in period. 2. Repeat. Calculate the path of actual labor productivity as well as the
balanced growth path for periods 0 through 30. How large is the percentage gap at
period 30.
Labor productivity along the balanced growth path is


 y 1

1
s
BGP
ytBGP   t  At  
 At In this case it is yt 
k
n






 t
k  g k  kt 1  kt  s yt  (n   )
t 1
k
kt
kt
time
k
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
growth rateAof k
1.5
1.746571
1.996143
2.247422
2.499417
2.751368
3.002701
3.252993
3.501942
3.749345
3.995078
4.239083
4.481354
4.721929
4.960882
5.198314
5.434351
5.669133
5.90282
6.135579
6.367587
6.599029
6.830092
7.060968
7.29185
7.522931
7.754407
7.986471
8.219316
8.453133
8.688111
0.164381
0.142892
0.125883
0.112126
0.100804
0.091348
0.083356
0.076529
0.070647
0.06554
0.061076
0.057152
0.053684
0.050605
0.047861
0.045406
0.043203
0.041221
0.039432
0.037814
0.036347
0.035015
0.033803
0.032698
0.03169
0.030769
0.029927
0.029155
0.028447
0.027798
1
1.02
1.0404
1.061208
1.082432
1.104081
1.126162
1.148686
1.171659
1.195093
1.218994
1.243374
1.268242
1.293607
1.319479
1.345868
1.372786
1.400241
1.428246
1.456811
1.485947
1.515666
1.54598
1.576899
1.608437
1.640606
1.673418
1.706886
1.741024
1.775845
1.811362
growth rateyof A
0.02 1.144714
0.02 1.220288
0.02 1.292798
0.02 1.362789
0.02 1.430694
0.02 1.496868
0.02 1.561606
0.02 1.625156
0.02 1.687734
0.02 1.749522
0.02 1.810685
0.02 1.871365
0.02 1.931692
0.02 1.99178
0.02 2.051735
0.02 2.111653
0.02 2.171623
0.02 2.231726
0.02 2.292038
0.02 2.352632
0.02 2.413574
0.02 2.474927
0.02 2.536752
0.02 2.599106
0.02 2.662045
0.02 2.72562
0.02 2.789882
0.02 2.854881
0.02 2.920665
0.02 2.98728
0.02 3.05477
1
3
APK
APK^BGP
0.763143
0.33
0.698676
0.33
0.647648
0.33
0.606379
0.33
0.572411
0.33
0.544045
0.33
0.520067
0.33
0.499588
0.33
0.481942
0.33
0.466621
0.33
0.453229
0.33
0.441455
0.33
0.431051
0.33
0.421815
0.33
0.413583
0.33
0.406219
0.33
0.39961
0.33
0.393663
0.33
0.388296
0.33
0.383441
0.33
0.379041
0.33
0.375044
0.33
0.371408
0.33
0.368095
0.33
0.365071
0.33
0.362308
0.33
0.35978
0.33
0.357465
0.33
0.355342
0.33
0.353393
0.33
0.351604
0.33
.11
At . Calculate
yBGP
1.740777
1.775592
1.811104
1.847326
1.884273
1.921958
1.960397
1.999605
2.039597
2.080389
2.121997
2.164437
2.207726
2.25188
2.296918
2.342856
2.389713
2.437507
2.486258
2.535983
2.586702
2.638436
2.691205
2.745029
2.79993
2.855928
2.913047
2.971308
3.030734
3.091349
3.153176
percentage gap%
0.342412
0.312743
0.286182
0.262291
0.240718
0.221176
0.203424
0.187261
0.172516
0.159041
0.146707
0.135403
0.125031
0.115504
0.106744
0.098684
0.091262
0.084423
0.078117
0.0723
0.06693
0.061972
0.057392
0.053159
0.049246
0.045628
0.04228
0.039184
0.036318
0.033665
0.031208