Econ 122a: Fall 2012
Wednesday, September 19th
Problem Set 1. Answer Key
Mr. Nordhaus and Staff
Economics 122a. Fall 2012. Problem Set 1. Answer Key
1)
1.a)
Real Gross Domestic
Product, Chained Dollars
Year
2007
2007
2007
2007
2008
2008
2008
2008
2009
2009
2009
2009
2010
2010
2010
2010
2011
2011
2011
2011
2012
2012
Quarter
I
II
III
IV
I
II
III
IV
I
II
III
IV
I
II
III
IV
I
II
III
IV
I
II
[Billions of chained (2005) dollars]
Seasonally adjusted at annual
rates
13,056.10
13,173.60
13,269.80
13,326.00
13,266.80
13,310.50
13,186.90
12,883.50
12,711.00
12,701.00
12,746.70
12,873.10
12,947.60
13,019.60
13,103.50
13,181.20
13,183.80
13,264.70
13,306.90
13,441.00
13,506.40
13,564.50
Annual
Growth
Rate
3.60%
2.92%
1.69%
-1.78%
1.32%
-3.71%
-9.20%
-5.36%
-0.31%
1.44%
3.97%
2.31%
2.22%
2.58%
2.37%
0.08%
2.45%
1.27%
4.03%
1.95%
1.72%
The Real Gross Domestic Product was taken from the BEA website and the annual growth rates were calculated
as described in the problem set. For example, the annual growth rate of the Real Gross Domestic Product in the
second quarter of 2007 is 4𝑥
13,173.60−13,173.60
13,173.60
= 3.60%.
In the table, we can see highlighted in bold letters the five quarters during which the US experienced a decline in
real GDP: 2008-I, 2008-III, 2008-IV, 2009-I and 2009-II. The greatest decline happened in the last quarter of 2008
and it was a 9.20% annual decline in real GDP.
Econ 122a: Fall 2012
Wednesday, September 19th
Problem Set 1. Answer Key
Mr. Nordhaus and Staff
1.b)
Price Indexes for Gross
Domestic Product
Year
Quarter
2007 I
2007 II
2007 III
2007 IV
2008 I
2008 II
2008 III
2008 IV
2009 I
2009 II
2009 III
2009 IV
2010 I
2010 II
2010 III
2010 IV
2011 I
2011 II
2011 III
2011 IV
2012 I
2012 II
[Index numbers, 2005=100]
Seasonally adjusted
105.396
106.116
106.457
106.956
107.623
108.282
109.107
109.247
109.526
109.318
109.463
109.820
110.234
110.686
111.248
111.838
112.389
113.109
113.937
114.041
114.608
115.063
Inflation Rate
2.73%
1.29%
1.87%
2.49%
2.45%
3.05%
0.51%
1.02%
-0.76%
0.53%
1.30%
1.51%
1.64%
2.03%
2.12%
1.97%
2.56%
2.93%
0.37%
1.99%
1.59%
The Price Index for Gross Domestic Product was also taken from the BEA website and the inflation rate is the
annualized growth rate of it.
In the table, we can see highlighted in bold letters the only quarter during which the US experienced a deflation:
2009-II. During the rest of the quarters, the general level of prices, measured with the Price Index for Gross
Domestic Product, increased.
2)
𝑃𝑂,𝑡 is the price of oranges in year 𝑡 and 𝑄𝑂,𝑡 is the quantity of oranges in year 𝑡. 𝑃𝐵,𝑡 and 𝑄𝐵,𝑡 are the price and
quantity of boomerangs in year 𝑡, respectively.
The Laspeyres real GDP is calculated fixing prices to 2016 prices:
𝑅𝑒𝑎𝑙𝐺𝐷𝑃2016,𝐿 = 𝑁𝑜𝑚𝑖𝑛𝑎𝑙𝐺𝐷𝑃2016 = 𝑃𝑂,2016 𝑥𝑄𝑂,2016 + 𝑃𝐵,2016 𝑥𝑄𝐵,2016 = 1𝑥100 + 3𝑥20 = 160
Econ 122a: Fall 2012
Wednesday, September 19th
Problem Set 1. Answer Key
Mr. Nordhaus and Staff
𝑅𝑒𝑎𝑙𝐺𝐷𝑃2017,𝐿 = 𝑃𝑂,2016 𝑥𝑄𝑂,2017 + 𝑃𝐵,2016 𝑥𝑄𝐵,2017 = 1𝑥105 + 3𝑥22 = 171
The Paasche real GDP is calculated fixing prices to 2017 prices:
𝑅𝑒𝑎𝑙𝐺𝐷𝑃2016,𝑃 = 𝑃𝑂,2017 𝑥𝑄𝑂,2016 + 𝑃𝐵,2017 𝑥𝑄𝐵,2016 = 1.1𝑥100 + 3.1𝑥20 = 172
𝑅𝑒𝑎𝑙𝐺𝐷𝑃2017,𝑃 = 𝑁𝑜𝑚𝑖𝑛𝑎𝑙𝐺𝐷𝑃2017 = 𝑃𝑂,2017 𝑥𝑄𝑂,2017 + 𝑃𝐵,2017 𝑥𝑄𝐵,2017 = 1.1𝑥105 + 3.1𝑥22 = 183.7
Finally, the Fisher index is the geometric mean of the Laspeyres and Paasche indexes:
𝑅𝑒𝑎𝑙𝐺𝐷𝑃2016,𝐹 = √𝑅𝑒𝑎𝑙𝐺𝐷𝑃2016,𝐿 𝑥𝑅𝑒𝑎𝑙𝐺𝐷𝑃2016,𝑃 = √160𝑥172 = 165.89
𝑅𝑒𝑎𝑙𝐺𝐷𝑃2017,𝐹 = √𝑅𝑒𝑎𝑙𝐺𝐷𝑃2017,𝐿 𝑥𝑅𝑒𝑎𝑙𝐺𝐷𝑃2017,𝑃 = √171𝑥183.7 = 177.24
Now, we can calculate the GDP deflators for each of our measures of real GDP and its growth rate to have an
inflation rate associated with each of them. In the case of the Laspeyres real GDP, we have the following GPD
𝑁𝑜𝑚𝑖𝑛𝑎𝑙𝐺𝐷𝑃2016
𝑁𝑜𝑚𝑖𝑛𝑎𝑙𝐺𝐷𝑃
= 1 and 𝑃2017,𝐿 = 𝑅𝑒𝑎𝑙𝐺𝐷𝑃 2017
𝑅𝑒𝑎𝑙𝐺𝐷𝑃2016,𝐿
2017,𝐿
𝑃2017,𝐿 −𝑃2016,𝐿
𝜋2017,𝐿 =
= 0.074 = 7.4%.
𝑃
deflators: 𝑃2016,𝐿 =
rate would be:
=
183.7
171
= 1.074. This way, the inflation
2016,𝐿
𝑁𝑜𝑚𝑖𝑛𝑎𝑙𝐺𝐷𝑃2016
160
=
𝑅𝑒𝑎𝑙𝐺𝐷𝑃2016,𝑃
172
𝑃2017,𝑃 −𝑃2016,𝑃
𝜋2017,𝑃 =
𝑃
In the case of the Paasche real GDP, we have the following GPD deflators: 𝑃2016,𝑃 =
0.930 and 𝑃2017,𝑃 =
1−0.930
0.930
𝑁𝑜𝑚𝑖𝑛𝑎𝑙𝐺𝐷𝑃2017
𝑅𝑒𝑎𝑙𝐺𝐷𝑃2017,𝑃
= 1. This way, the inflation rate would be:
2016,𝑃
=
=
= 0.0753 = 7.53%.
Finally, in the case of the Fisher real GDP, we have the following GPD deflators: 𝑃2016,𝐹 =
160
= 0.964
165.89
𝑃2017,𝐹 −𝑃2016,𝐹
𝑃2016,𝐹
and 𝑃2017,𝐹 =
=
1.036−0.964
0.964
𝑁𝑜𝑚𝑖𝑛𝑎𝑙𝐺𝐷𝑃2017
𝑅𝑒𝑎𝑙𝐺𝐷𝑃2017,𝐹
=
183.7
177.24
𝑁𝑜𝑚𝑖𝑛𝑎𝑙𝐺𝐷𝑃2016
𝑅𝑒𝑎𝑙𝐺𝐷𝑃2016,𝐹
=
= 1.036. This way, the inflation rate would be: 𝜋2017,𝐹 =
= 0.0753 = 7.45%.
In class, we learnt that Laspeyres real GDP overstated growth while Paasche real GDP understated it relative to
true and the Fisher index was intermediate. In this exercise we could see that the opposite results happen with
inflation: inflation calculated with the Laspeyres GDP deflator (𝜋2017,𝐿 = 7.4%) is smaller than the inflation rate
calculated using the Fisher GDP deflator (𝜋2017,𝐹 = 7.45%) and the largest inflation rate corresponds to the
Paasche GDP deflator (𝜋2017,𝐿 = 7.53%). We could have predicted this result without doing all these
calculations because the price deflator is calculated as the part of the change in nominal GDP which could not be
explained by real GDP and therefore is attributed to changes in the general level of prices. This way, if an index
overstates real activity, it will mechanically understate inflation.
Econ 122a: Fall 2012
Wednesday, September 19th
Problem Set 1. Answer Key
Mr. Nordhaus and Staff
3)
3a)
The endogenous variables, those which are explained by the model, are still the same: 𝑌, 𝐾, 𝐿, 𝑟 and 𝑤. The
̅ and 𝐿̅. Now the production
exogenous variables, those which are not explained by the model, are still: 𝐴̅, 𝐾
3 1
𝑌 = 𝐴̅𝐾 4 𝐿4 . Therefore, the rule of hiring capital has also changed: 𝑟 = 𝑀𝑃𝐾 =
function has changed:
3 1
𝜕[𝐴̅𝐾4 𝐿 4 ]
𝜕𝐾
3
1
1
1
̅ −4+1 𝐿 4
1 1
3
3
𝐾
3 𝐴𝐾
= 𝐴̅ 4 𝐾 4−1 𝐿4 = 4 𝐴̅𝐾 −4 𝐿4 𝐾 = 4
3 1
=
𝐾
3 1
𝜕[𝐴̅𝐾4 𝐿4 ]
3
1
3
̅
3
3 𝐴̅𝐾4 𝐿 4
4 𝐾
3
3
− +1
1
1
𝐿
1 𝐴𝐾 4 𝐿 4
𝑀𝑃𝐿 = 𝜕𝐿 = 𝐴̅𝐾 4 4 𝐿4−1 = 4 𝐴̅𝐾 4 𝐿−4 𝐿 = 4
𝐿
̅
the factors markets did not change: 𝐾 = 𝐾 and 𝐿 = 𝐿̅.
3𝑌
= 4 𝐾. And the rule of hiring labor has changed: 𝑤 =
3 1
=
1 𝐴̅𝐾4 𝐿 4
4 𝐿
1𝑌
= 4 𝐿 . Finally, the equilibrium conditions in
3b)
̅ and 𝐿∗ = 𝐿̅ because of the market equilibrium conditions in the factors markets. Then, 𝑌 ∗ =
This way, 𝐾 ∗ = 𝐾
3
1
∗
3 1
3 1
̅ ∗ ∗
̅ 4 𝐿̅4 ; 𝑟 ∗ = 3 𝑌 ∗ = 3 𝐴𝐾 4∗𝐿 4 = 3 𝐴̅
𝐴̅𝐾 ∗4 𝐿∗4 = 𝐴̅𝐾
4𝐾
4
𝐾
4
𝐾
3
∗1−4
1
1
1
∗
𝐿 4
=
∗
3
̅ ( 𝐿 ∗ )4
𝐴
4
𝐾
=
3
𝐿̅ 4
𝐴̅ (𝐾̅) ;
4
∗
𝑤 =
1 𝑌∗
4 𝐿∗
3
=
1
∗ ∗
1 𝐴̅𝐾 4 𝐿 4
4
𝐿∗
3
∗
=
1
𝐾 4
𝐴̅
1
4
∗1−
𝐿 4
=
3
̅ 4
1
𝐾
𝐴̅ ( 𝐿̅ )
4
.
3c)
∗
𝑦 =
𝑌∗
𝐿∗
3 1
=
̅ 4 𝐿̅ 4
𝐴̅𝐾
𝐿̅
3
= 𝐴̅
̅4
𝐾
1
1−
𝐿̅ 4
3
̅ 4
𝐾
= 𝐴̅ ( 𝐿̅ ) . Output per capita is a function of capital per capita.
4)
4a-c)
Capital per
Implied TFP to
Capital Per
person
Per Capita
Predicted y*
match data
Person In
Per Capita relative to the GDP relative
relative to relative to the
2007
GDP in 2007
US
to the US
Predicted y*
the US
US
(1)
(2)
(3)
(4)
(5)'
(5)
(6)
US
135,877
42,887
1.000
1.000
51.410
1.000
1.000
Canada
116,188
36,168
0.855
0.843
48.796
0.949
0.889
France
109,023
29,633
0.802
0.691
47.772
0.929
0.744
Hong Kong
123,268
43,121
0.907
1.005
49.768
0.968
1.039
South Kerea
104,864
23,850
0.772
0.556
47.157
0.917
0.606
Indonesia
9,957
5,186
0.073
0.121
21.513
0.418
0.289
Argentina
35,182
15,275
0.259
0.356
32.767
0.637
0.559
Mexico
33,168
11,204
0.244
0.261
32.130
0.625
0.418
Kenya
2,379
2,025
0.018
0.047
13.349
0.260
0.182
Ethiopia
584
1,110
0.004
0.026
8.359
0.163
0.159
Econ 122a: Fall 2012
Wednesday, September 19th
Problem Set 1. Answer Key
Mr. Nordhaus and Staff
Column (3) for each country is its column (1) divided by 135,877, which is the value of column (1) for the US.
Analogously, column (4) for each country is its column (2) divided by 42,887, which is the value of column (2) for
1
the US. Column (5)’ is the value of column (1) elevated to the 3 power, using the same formula as in problem 3c)
̅ = 1. Column (5) is the same as column (3) and (4) but using
but with a different exponent for capital and with A
the calculations in column (5)’. The relative to the US GDP per capita in column (4) includes both differences in
capital per capita and differences in TFP, while the relative to the US predicted y ∗ in column (5) includes only
differences in capital per capita, then, the ratio of the values in columns (4) and (5) are the implied TFP
differences relative to the US to match the data.
4d)
Column (4) shows us that there are big differences in real GDP per capita between countries: the average
resident in a rich country such as the US or Canada receives almost four times more income than the average
resident in a middle income country such as Argentina or Mexico and fifty or more times income than a resident
of a poor country such as Kenya or Ethiopia. These differences are explained only in part by differences in the
capital stock per worker: the figures in column (5) show differences between rich, middle income and poor
countries but these differences are not as extreme as the figures in column (4). Column (6) shows us that more
than half of the differences between countries are explained by differences in TFP because the figures in column
(6) are closer to the figures in column (4) than figures in column (5). In class, we learnt that 55% of the growth of
GDP per capita in the US was explained by growth of TFP while the remaining 45% was explained by capital per
worker accumulation. This exercise shows us that similar results can be found when we compare the differences
of GDP per capita between countries.