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Boğaziçi University Department of Economics EC 205 Summer 2015 SOLUTIONS TO PROBLEM SET 1 1) Calculating percentage growth rates, and log approximations to percentage growth rates,
we obtain:
Year
Percentage Growth Rate
Log Approximation
2003
1.597484
1.584858
2004
2.466649
2.436718
2005
2.073992
2.052778
2006
1.667763
1.654008
2007
1.162146
1.155445
2008
-1.13738
-1.14389
2009
-4.41241
-4.51272
2010
2.257293
2.232193
2011
0.997377
0.992436
In this case, calculating the change in the natural logarithm from one year to the next
gives a good approximation to the percentage growth rate, as the growth rates are small. But
if we do the same thing for growth rates over ten-year periods, as below, the approximation is
poor, as the growth rates are relatively large.
Ten-year percentage growth rate Log Approximation
1960
19.09544
17.4755
1970
33.10087
28.59371
1980
22.66362
20.42756
1990
25.33007
22.57806
2000
23.14175
20.81659
1 2) Price and quantity data are given as the following:
Year 1
Good
Quantity
Price
Computers
20
$1,000
Bread
10,000
$1.00
Good
Quantity
Price
Computers
25
$1,500
Bread
10,000
$1.00
Year 2
a) Year 1 nominal GDP = 20 × $1,000 + 10,000 × $1.00 = $30,000 .
Year 2 nominal GDP = 25 × $1,500 + 12,000 × $1.10 = $50,700 .
With year 1 as the base year, we need to value both years’ production at year 1 prices. In the
base
year, year 1, real GDP equals nominal GDP equals $30,000. In year 2, we need to
value year 2’s output at year 1 prices. Year 2 real GDP
= 25 × $1,000 + 12,000 × $1.00 = $37,000 .
The percentage change in real GDP equals ($37,000 − $30,000)/$30,000 = 23.33%.
We next calculate chain-weighted real GDP. At year 1 prices, the ratio of year 2 real GDP to
year 1 real GDP equals g1 = ($37,000/$30,000) = 1.2333. We must next compute real GDP
using year 2 prices. Year 2 GDP valued at year 2 prices equals year 2 nominal GDP =
$50,700. Year 1 GDP valued at year 2 prices equals (20 × $1,500 + 10,000 × $1.10) =
$41,000. The ratio of year 2 GDP at year 2 prices to year 1 GDP at year 2 prices equals g2 =
($50,700/$41,000) = 1.2367. The chain-weighted ratio of real GDP in the two years therefore
g = g1 g2 = 1.23496
is equal to c
. The percentage change chain-weighted real GDP from year
1 to year 2 is therefore approximately 23.5%.
If we (arbitrarily) designate year 1 as the base year, then year 1 chain-weighted GDP equals
nominal GDP equals $30,000. Year 2 chain-weighted real GDP is equal to (1.23496
×$30,000) = $37,048.75.
b) To calculate the implicit GDP deflator, we divide nominal GDP by real GDP, and then
multiply by 100 to express as an index number. With year 1 as the base year, base year
nominal GDP equals base year real GDP, so the base year implicit GDP deflator is 100. For
the year 2, the implicit GDP deflator is ($50,700/$37,000) × 100 = 137.0. The percentage
change in the deflator is equal to 37.0%.
2 With chain weighting, and the base year set at year 1, the year 1 GDP deflator equals
($30,000/$30,000) × 100 = 100. The chain-weighted deflator for year 2 is now equal to
($50,700/$37,048.75) × 100 = 136.85. The percentage change in the chain-weighted deflator
equals 36.85%.
c) We next consider the possibility that year 2 computers are twice as productive as year 1
computers. As one possibility, let us define a “computer” as a year 1 computer. In this case,
the 25 computers produced in year 2 are the equivalent of 50 year 1 computers. Each year 1
computer now sells for $750 in year 2. We now revise the original data as:
Year 1
Good
Quantity
Price
Year 1 Computers
20
$1,000
Bread
10,000
$1.00
Good
Quantity
Price
Year 1 Computers
50
$750
Bread
12,000
$1.10
Year 2
First, note that the change in the definition of a “computer” does not affect the calculations of
nominal GDP. We next compute real GDP with year 1 as the base year. Year 2 real GDP in
year 1 prices is now 50 × $1,000 + 12,000 × $1.00 = $62,000. The percentage change in real
GDP is equal to ($62,000 − $30,000)/$30,000= 106.7%.
We next revise the calculation of chain-weighted real GDP. From above, g1 equals
($62,000/$30,000) = 206.67. The value of year 1 GDP at year 2 prices equals $26,000.
Therefore, g2 equals ($50,700/$26,000) = 1.95. 200.75. The percentage change chainweighted real GDP from year 1 to year 2 is therefore 100.75%.
If we (arbitrarily) designate year 1 as the base year, then year 1 chain-weighted GDP equals
nominal GDP equals $30,000. Year 2 chain-weighted real GDP is equal to (2.0075 ×
$30,000) = $60,225. The chain-weighted deflator for year 1 is automatically 100. The chainweighted deflator for year 2 equals ($50,700/$60,225) × 100 = 84.18. The percentage rate of
change of the chain-weighted deflator equals −15.8%.
When there is no quality change, the difference between using year 1 as the base year and
using chain weighting is relatively small. Factoring in the increased performance of year 2
computers, the production of computers rises dramatically while its relative price falls.
Compared with earlier practices, chain weighting provides a smaller estimate of the increase
3 in production and a smaller estimate of the reduction in prices. This difference is due to the
fact that the relative price of the good that increases most in quantity (computers) is much
higher in year 1. Therefore, the use of historical prices puts more weight on the increase in
quality-adjusted computer output.
3) Price and quantity data are given as the following:
Year 1
Good
Quantity
(million lbs.)
Price
(per lb.)
Broccoli
1,500
$0.50
300
$0.80
Cauliflower
Year 2
Good
Quantity
(million lbs.)
Price
(per lb.)
Broccoli
2,400
$0.60
350
$0.85
Cauliflower
a) Year 1 nominal GDP = Year 1 real GDP
= 1,500 million × $0.50 + 300 million × $0.80 =
$990 million.
Year 2 nominal GDP = 2,400 million × $0.60 + 350 million × $0.85 = $1,730.5 million
Year 2 real GDP = 2,400 million × $0.50 + 350 million × $0.80 = $1,450 million.
Year 1 GDP deflator equals 100. Year 2 GDP deflator equals ($1,730.5/$1,450) × 100 =
119.3.The percentage change in the deflator equals 19.3%.
b) Year 1 production (market basket) at year 1 prices equals year 1 nominal GDP = $990
million. The value of the market basket at year 2 prices is equal to
1,500 million × $0.60 + 300 million × $0.85 =$1,050 million.
Year 1 CPI equals 100. Year 2 CPI equals ($1,050/$990) × 100 = 106.1. The percentage
change in the CPI equals 6.1%.
The relative price of broccoli has gone up. The relative quantity of broccoli has also gone up.
The CPI attaches a smaller weight to the price of broccoli, and so the CPI shows less
inflation.
4 4) The answers to parts (a) and (b) are in the table.
Year
Capital when initial capital = 80
Capital when initial capital = 100
0
80
100
1
82.0
100
2
83.8
100
3
85.4
100
86.9
100
5
88.2
100
6
89.4
100
7
90.4
100
8
91.4
100
9
92.3
100
10
93.0
100
4
In the first case, where the initial quantity of capital was 80, with a constant quantity of
investment each period, the quantity of capital increases over time, but at a decreasing rate
(note the increment to the capital stock gets smaller each period). This happens because, as
the capital stock grows, the total amount of capital that depreciates each period increases. The
quantity of capital appears to be converging to some quantity, but what is this quantity? When
the quantity of capital is initially 100, then the capital stock stays at 100 indefinitely, as long
as investment is 10 each period. This is because, when the capital stock is 100, the total
quantity of depreciation each period when the depreciation rate is 10% is 10, so new
investment just replaces the capital that depreciates each period. Here 100 is what we would
call the “steady state” quantity of capital. Steady states are useful when we study economic
growth in Chapters 7 and 8.
5) As the unemployment rate is 5% and there are 2.5 million unemployed, it must be that the
labor force is 50 million (2.5/0.05). Thus, the participation rate is 50% (50/100), the labor
force 50 million, the number of employed workers 47.5 million (50-2.5), and the
employment/population ratio is 47.5% (47.5/100).
6)
a) If the Fed reduces the money supply, then the aggregate demand curve shifts down, as in
the figure below. This result is based on the quantity equation MV = PY, which tells us that a
decrease in money M leads to a proportionate decrease in nominal output PY (assuming that
velocity V is fixed). For any given price level P, the level of output Y is lower, and for any
given Y, P is lower.
5 b) In the short run, we assume that the price level is fixed and that the aggregate supply curve
is flat. As the figure below shows, in the short run, the leftward shift in the aggregate demand
curve leads to a movement from point A to point B—output falls but the price level doesn’t
change. In the long run, prices are flexible. As prices fall, the economy returns to full
employment with a lower price level at point C.
If we assume that velocity is constant, we can quantify the effect of the 5-percent reduction in
the money supply. From the quantity equation, we have that Y=MV/P. Assuming constant
velocity, we have that Y is equal to M multiplied by a constant in the short run (since prices
are also constant in the short run). Thus, a 5-percent reduction in the money supply leads to a
5-percent reduction in output in the short run.
In the long run we know that prices are flexible and the economy returns to its natural rate of
output. Thus, output is constant in the long run. From the quantity equation, we have that
P=MV/Y. Assuming constant velocity, we have that P is equal to M multiplied by a constant
in the long run (since output is also constant in the short run). Thus, a 5-percent reduction in
the money supply leads to a 5-percent reduction in the price level in the long run.
6 c) Okun’s law refers to the negative relationship that exists between unemployment and real
GDP. Okun’s law can be summarized by the equation:
Percentage Change in Real GDP = 3% – 2 [Change in Unemployment Rate].
That is, output moves in the opposite direction from unemployment, with a ratio of 2 to 1. In
the short run, when output falls, unemployment rises. Quantitatively, if velocity is constant,
we found that output falls 5 percentage points relative to full employment in the short run.
Okun’s law states that output growth equals the full employment growth rate of 3 percent
minus two times the change in the unemployment rate. Therefore, if output falls 5 percentage
points relative to full employment growth, then actual output growth is –2 percent. Using
Okun’s law, we find that the change in the unemployment rate equals 2.5 percentage points:
–2 = 3 – 2[Change in Unemployment Rate]
[–2 – 3]/[–2] = [Change in Unemployment Rate]
2.5 = [Change in Unemployment Rate]
In the long run, both output and unemployment return to their natural rate levels. Thus, there
is no long-run change in unemployment.
7)
a) An exogenous decrease in the velocity of money causes the aggregate demand curve to
shift downward. In the short run, prices are fixed, so output falls.
If the Fed wants to keep output and employment at their natural-rate levels, it must increase
aggregate demand to offset the decrease in velocity. By increasing the money supply, the Fed
can shift the aggregate demand curve upward, restoring the economy to its original
equilibrium. Both the price level and output remain constant.
If the Fed wants to keep prices stable, then it wants to avoid the long-run adjustment to a
lower price level. Therefore, it should increase the money supply and shift the aggregate
demand curve upward, again restoring the original equilibrium. Thus, both Feds make the
same choice of policy in response to this demand shock.
b) An exogenous increase in the price of oil is an adverse supply shock that causes the shortrun aggregate supply curve to shift upward. If the Fed cares about keeping output and
employment at their natural-rate levels, then it should increase aggregate demand by
increasing the money supply.
This policy response shifts the aggregate demand curve upwards. In this case, the economy
immediately reaches a new equilibrium at a higher price level. Price level becomes
permanently higher, but there is no loss in output associated with the adverse supply shock.
If the Fed cares about keeping prices stable, then there is no policy response it can implement.
In the short run, the price level stays at the higher level. If the Fed increases aggregate
demand, then the economy ends up with a permanently higher price level. Hence, the Fed
must simply wait, holding aggregate demand constant. Eventually, prices fall to restore full
employment at the old price level. But the cost of this process is a prolonged recession.
7 Thus, the two Feds make a different policy choice in response to a supply shock.
8)
a) Total planned expenditure is PE = C(Y – T) + I + G. Plugging in the consumption function
and the values for investment I, government purchases G, and taxes T given in the question,
total planned expenditure PE is
PE = 200 + 0.75(Y – 100) + 100 + 100 = 0.75Y + 325
This equation is graphed below.
b) To find the equilibrium level of income, combine the planned-expenditure equation
derived in part (a) with the equilibrium condition Y = PE:
Y = 0.75Y + 325, so Y = 1,300. The equilibrium level of income is 1,300, as indicated in the
figure above.
c) If government purchases increase to 125, then planned expenditure changes to PE = 0.75Y
+ 350. Equilibrium income increases to Y = 1,400. Therefore, an increase in government
purchases of 25 (i.e., 125 – 100 = 25) increases income by 100. This is what we expect to
find, because the formula for the government purchases multiplier is 1/(1 – MPC), the MPC is
0.75, and the government-purchases multiplier therefore has a numerical value of 4.
d) An income level of 1,600 represents an increase of 300 over the original level of income.
The government-purchases multiplier is 1/(1 – MPC): the MPC in this example equals 0.75,
so the government-purchases multiplier is 4. This means that government purchases must
increase by 75 (to a level of 175) for income to increase by 300.
9)
a) When taxes do not depend on income, a one-dollar increase in income means that
disposable income increases by one dollar. Consumption increases by the marginal propensity
to consume MPC. When taxes do depend on income, a one-dollar increase in income means
8 that disposable income increases by only (1 – t) dollars. Consumption increases by the
product of the MPC and the change in disposable income, or (1 – t)MPC. This is less than the
MPC. The key point is that disposable income changes by less than total income, so the effect
on consumption is smaller.
b) When taxes are fixed, we know that 𝛥Y/𝛥G = 1/(1 – MPC). We found this by considering
an increase in government purchases of 𝛥G; the initial effect of this change is to increase
income by 𝛥G. This in turn increases consumption by an amount equal to the marginal
propensity to consume times the change in income, MPC 𝛥G. This increase in consumption
raises expenditure and income even further. The process continues indefinitely, and we derive
the multiplier above.
When taxes depend on income, we know that the increase of 𝛥G increases total income by
𝛥G; disposable income, however, increases by only (1 – t) 𝛥G—less than dollar for dollar.
Consumption then increases by an amount (1 – t) MPC 𝛥G. Expenditure and income increase
by this amount, which in turn causes consumption to increase even more. The process
continues, and the total change in output is:
𝛥Y = 𝛥G {1 + (1 – t)MPC + [(1 – t)MPC]2 + [(1 – t)MPC]3 + ....} = 𝛥G [1/(1 – (1 –
t)MPC)].
Thus, the government-purchases multiplier becomes 1/(1 – (1 – t)MPC) rather than 1/(1 –
MPC). This means a much smaller multiplier. For example, if the marginal propensity to
consume MPC is 3/4 and the tax rate t is 1/3, then the multiplier falls from 1/(1 – 3/4), or 4, to
1/(1 – (1 – 1/3)(3/4)), or 2.
c) Introduction of this tax system makes IS curve steeper compared to the IS curve when
taxes do not depend on income. A decrease in the interest rate increases investment, which
raises income, which raises consumption, which again raises income, which again rises
consumption, and so on. When taxes depend on income, a given reduction in the interest rate
creates less of an increase in consumption in response to the same initial increase in income.
Thus, output rises by less for a given reduction in the interest rate. Hence, 𝛥Y/𝛥r falls in
absolute value. Since the slope of the IS curve is 𝛥r/𝛥Y, the slope of the IS curve rises in
absolute value, which means IS curve gets steeper if taxes depend on income.
10)
a) The variable Y represents real output or real income. The variable C represents the
consumption of goods and services. The variable I represents investment by the firms. The
variable G represents the government’s spending on newly produced goods and services. The
variable T represents lump sum taxes, and Y – T represents disposable income. The variable
M represents the nominal money supply, P is the price level, and M/P is the real money
supply. The variable r is the real interest rate. The variable (M/P) ! represents real money
demand. Consumption depends positively on disposable income, investment depends
negatively on the real interest rate, and real money demand depends positively on real income
and negatively on the real interest rate.
b) The IS curve represents all combinations of the real interest rate r and real output Y such
that the goods market is in equilibrium. The equation for the IS curve can be derived as
follows:
9 Y=C+I+G
Y = (120 + 0.5(Y – T)) + (100 – 10r) + 50
Y = (120 + 0.5(Y –40)) + (100 –10r) + 50
Y = 250 + 0.5Y – 10r
0.5Y = 250 – 10r
Y = 500 – 20r
The IS curve is illustrated below.
c) The LM curve represents all combinations of the real interest rate r and real output Y such
that the money market is in equilibrium. The equation for the LM curve can be derived as
follows:
Y-20r=M/P
Y-20r=600/2
Y=300+20r
The LM curve is illustrated in the figure above.
d) To find the equilibrium levels of the interest rate and output (or income), set the equation
for the IS curve equal to the equation for the LM curve and solve for the interest rate r to get
5. Now substitute the interest rate of 5 back into either equation to solve for Y equal to 400.
10