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Unformatted text preview: ”’4 l
Problems and Apphcations
1. a. In the example that follows, we assume that during the school year you look for a parttime job, and that on average it takes 2 weeks to ﬁnd one. We also assume
that the typical job lasts 1 semester, or 12 weeks. If it takes 2 weeks to ﬁnd a job, then the rate of job ﬁnding in weeks is: f = (1 job/2 weeks) = 0.5 jobs/weeks.
b. If the job lasts for 12 weeks, then the rate of job separation in weeks is: s = (1 job/12 weeks) = 0.083 jobs/week.
c. From the text, we know that the formula for the natural rate of unemployment is (U/L) = (s/(s+ D), where U is the number of people unemployed and L is the number of people in the
labor force. Plugging in the values for f and s that were calculated in part (b), we ﬁnd: (U/ L) = (0.083/(0.083 + 0.5)) = 0.14. Thus, if on average it takes 2 weeks to ﬁnd a job that lasts 12 weeks, the natural rate of unemployment for this population of college students seeking parttime .
employment is 14 percent. 3. Call the number of residents of the dorm who are involved I, the number who are unin
volved U, and the total number of students T = I + U. In steady state the total number
of involved students is constant. For this to happen we need the number of newly unin volved students, (0.10)I, to be equal to the number of students who just became
involved, (0.05)U. Following a few substitutions: (0.05)U = (0.10)I
= (0.10)(T — U),
so _ 0.10
‘ 0.10 + 0.05 2 3 .
We ﬁnd that twothirds of the students are uninvolved. 5. a. The demand for labor is determined by the amount of labor that a proﬁtmaximiz
ing ﬁrm wants to hire at a given real wage. The proﬁtmaximizing condition is
that the ﬁrm hire labor until the marginal product of labor equals the real wage, W MP =—P. Chapler 6 Unemployment 51 The marginal product of labor is found by differentiating the production function
with respect to labor (see the appendix to Chapter 3 for more discussion), ”dz
MPL_ dL'
_ d(K1/3L2/3)
_ dL __2_1/3—1/3
3KL . In order to solve for labor demand, we set the MPL equal to the real wage and
solve for L: 8 W)‘3 Notice that this expression has the intuitively desirable feature that increases in
the real wage reduce the demand for labor. We asume that the 1,000 units of capital and the 1,000 units of labor are supplied
inelastically (i.e., they will work at any price). In this case we know that all 1,000 units of each will be used in equilibrium, so we can substitute them into the above
labor demand function and solve for E . P
8 —3
1,000 = 277. 1,000(?W)
E _ .2.
P 3 ' In equilibrium, employment will be 1,000, and multiplying this by 2/3 we ﬁnd that the workers earn 667 units of output. The total output is given by the production
function: Y= K1/3L2/3
= 1,0001/31,0002’3
= 1,000. Notice that workers get twothirds of output, which is consistent with what we
know about the Cobb—Douglas production function from the appendix to Chapter
3 . The congressionally mandated wage of 1 unit of output is above the equilibrium
wage of 2/3 units of output.
Firms will use their labor demand function to decide how many workers to hire at
the given real wage of 1 and capital stock of 1,000: _ _8_ «a L _ 27 1,000(1) = 296,
so 296 workers will be hired for a total compensation of 296 units of output.
The policy redistributes output from the 704 workers who become involuntarily
unemployed to the 296 workers who get paid more than before. The lucky workers beneﬁt less than the losers lose as the total compensation to the working class
falls from 667 to 296 units of output. This problem does focus the analysis of minimumwage laws on the two effects of these laws: they raise the wage for some workers while downwardsloping labor demand reduces the total number of jobs. Note, however, that if labor demand is
lea} ehd‘k W Th 171415 GMMPIQ W the 1:»; CF simply/Encod WM) be smaller) (Ami .th CW6 Tn worm“ mwwxé mng‘ be POS?+ELB_ A M 7. The vacant ofﬁce space problem is similar to the unemployment problem; we can apply
the same concepts we used in analyzing unemployed labor to analyze why vacant ofﬁce
space exists. There is a rate of ofﬁce separation: ﬁrms that occupy ofﬁces leave, either
to move to different ofﬁces or because they go out of business. There is a rate of ofﬁce
ﬁnding: ﬁrms that need ofﬁce space (either to start up or expand) ﬁnd empty ofﬁces. It
takes time to match ﬁrms with available space. Different types of ﬁrms require spaces
with different attributes depending on what their speciﬁc needs are. Also, because
demand for different goods ﬂuctuates, there are “sectoral shifts”—changes in the com position of demand among industries and regions—that affect the proﬁtability and
office needs of different ﬁrms. A production function has constant returns to scale if increasing all factors of pro
duction by an equal percentage causes output to increase by the same percentage.
Mathematically, a production function has constant returns to scale if zY = F(zK,
zL) for any positive number 2. That is, if we multiply both the amount of capital
and the amount of labor by some amount 2, then the amount of output is multi
plied by 2. For example, if we double the amounts of capital and labor we use (set
ting z: 2), then output also doubles. To see if the production function Y= F(K, L): K"2 L”2 has constant returns to
scale, we write: F(zK, 2L) = (zK)""(zL)"2 = szL"2 = zY. Therefore, the production function Y = KWLU2 has constant returns to scale.
To ﬁnd the perworker production function, divide the production function
Y = KWL”2 by L: If we deﬁne y = Y/ L, we can rewrite the above expression as: y = Km/L‘”. Deﬁning k = K / L, we can rewrite the above expression as: y=km. 56 Answers to Textbook Questions and Problems c. We know the following facts about countries A and B: 8 = depreciation rate = 0.05,
38 = saving rate of country A = 0.1,
sb = saving rate of country B = 0.2, and y = k”2 is the perworker production function derived
in part (b) for countries A and B. The growth of the capital stock Ak equals the amount of investment sﬂk), less
the amount of depreciation 8k. That is, Ak = sﬂk) — 3k. In steady state, the capital
stock does not grow, so we can write this as sﬂk) = 8k. To ﬁnd the steadystate level of capital per worker, plug the perworker pro
duction function into the steadystate investment condition, and solve for k*: ww=&.
Rewriting this:
km = s/a
k = (s/8)2.
To ﬁnd the steadystate level of capital per worker k*, plug the saving rate for
each country into the above formula:
Country A: k: = (s45? = (oi/0.05)2 = 4. Country B: k: = (sh/es)2 = (oz/0.05)2 = 16.
Now that we have found k* for each country, we can calculate the steadystate lev
els of income per worker for countries A and B because we know that y = km:
* 112
y a = (4) = 2 * 1/2 1 yb = (16) = 4.
‘ We know that out of each dollar of income, workers save a fraction 3 and con
‘ sume a fraction (1 — 8). That is, the consumption function is c = (1 — s)y. Since we
i know the steadystate levels of income in the two countries, we ﬁnd
1 Country A: c: = (1 — 3,)y: = (1 — 0.1)(2)
1 = 1.8.
Country B: c: = (1 — say: = (1 — 0.2)(4) = 3.2. d. Using the following facts and equations, we calculate income per worker y, con
sumption per worker c, and capital per worker k: 33, = 0.1. Sb = 0.2. 5 = 0.05. k, = 2 for both countries.
112 y = k . c =(1—s)y. Chapler1 Economic Growth  57 Country A
W
Year k y=km c=(1—s,)y i=s,y 8k Ak=i—8k
EM 1 2 1.414 1.273 0.141 0.100 0.041 2 2.041 1.429 1.286 0.143 0.102 0.041 3 2.082 1.443 1.299 0.144 0.104 0.040 4 2.122 1.457 1.311 0.146 0.106 0.040 5 2.102 1.470 1.323 0.147 0.108 0.039 Country B
M
Year k y=km c=(1—sa)y i=sg 5k Ak=i—8k
W 1 2 1.414 1.131 0.283 0.100 0.183 2 2.183 1.477 1.182 0.295 0.109 0.186 3 2.369 1.539 1.231 0.308 0.118 0.190 4 2.559 1.600 1.280 0.320 0.128 0.192 5 2.751 1.659 1.327 0.332 0.138 0.194 W Note that it Will take ﬁve years before consumption in country B is higher than consumption in country A. We follow Section 7 1, “Approaching the Steady State: A Numerical Erample.”
The production function is Y = K0'3L0‘7. To derive the perworker production func
tion ﬂk), divide both sides of the production function by the labor force L: Y K0.3L0.7
I: L ' Y K 0.3 E = (Z) .
Because y = Y/L and k = K / L, this becomes:
y = koa' Rearrange to obtain: Recall that Ak = sﬂk) — 6k.
The steadystate value of capital k‘ is deﬁned as the value of k at which capital
stock is constant, so Ak = 0. It follows that in steady state 0 = sf(k) — 8k,
or, equivalently,
k* _ i
f (k*) 5'
For'the production function in this problem, it follows that:
k * _ _s_
(k*)0.3 8 '
Rearranging:
k* 0.7 = i,
( ) 5
or s l/0.7
k*= —
(a) ‘ Substituting this equation for steadystate capital per worker into the perworker
production function from part (a) gives: 3 0.3/0.7
*= __
y (8] Consumption is the amount of output that is not invested. Since investment in the
steady state equals 6k‘, it follows that c* = f(k*) — 8k* = (5)0.8/03 — 5(.s_)l/0.'l 5 5 Chapter 1 Economic Growthl 59 (Note: An alternative approach to the problem is to note that consumption also
equals the amount of output that is not saved: 0.3/0.7
c* = (1 — s)f(k*) = (1 — s)(k°“)°‘3 = (1 — s)(§) Some algebraic manipulation shows that this equation is equal to the equation
above.) The table below shows k‘, y', and c" for the saving rate in the left column, using
the equations from part (b). We assume a depreciation rate of 10 percent (i.e., 0.1).
(The last column shows the marginal product of capital, derived in part (d) below). k’ y‘ c‘ MPK 0 0.00 0.00 0.00 0.1 1.00 1.00 0.90 0.30
0.2 2.69 1.35 1.08 0.15
0.3 4.80 1.60 1.12 0.10
0.4 7.25 1.81 1.09 0.08
0.5 9.97 1.99 1.00 0.06
0.6 12.93 2.16 0.86 0.05
0.7 16.12 2.30 0.69 0.04
0.8 19.50 2.44 0.49 0.04
0.9 23.08 2.56 0.26 0.03
1 26.83 2.68 0.00 0.03 Note that a saving rate of 100 percent (3 = 1.0) maximizes output per worker.
In that case, of course, nothing is ever consumed, so c‘ :0. Consumption per work
er is maximized at a rate of saving of 0.3 percent—that is, where 3 equals capital’s \ share in output. This is the Golden Rule level of s. We can differentiate the production function Y = Ko'sLM with respect to K to ﬁnd
the marginal product of capital. This gives: K0.3L0.7 Y y
= 0.3— = 0.3—.
K k MPK = 0.3 The table in part (c) shows the marginal product of capital for each value of the saving rate. (Note that the appendix to Chapter 3 derived the MPK for the general Cobb—Douglas production function. The equation above corresponds to the special
case where (1 equals 0.3.) WT 9 60 Answers to Textbook Questions and Problems 5. As in the text, let k = K/L stand for capital per unit of labor. The equation for the evolu tion of k is
Ak = Saving — (8 + n)k. e is saved and if capital earns its marginal product, then saving If all capital incom
the above equation to ﬁnd equals MPK x k. We can substitute this into Ak=MPka—(8+n)k. In the steady state, capital per efﬁciency unit of capital does not change, so Ak = 0. From the above equation, this tells us that
MPKx k = (8 + n)k, or
MPK = (8 + n). Equivalently,
MPK — 5 = n. In this economy‘ s steady state, the net marginal product of capital, MPK  8, equals the
his condition describes the Golden Rule steady state. rate of growth of output, n. But t
Hence, we conclude that this economy reaches the Golden Rule level of capital accumu lation. 7. If there are decreasing returns to labor and capital, then increasing both capital and
labor by the same proportion increases output by less than this proportion. For exam
ple, if we double the amounts of capital and labor, then output less than doubles. This
may happen if there is a ﬁxed factor such as land in the production function, and it
becomes scarce as the economy grows larger. Then population growth will increase
total output but decrease output per worker, since each worker has less of the ﬁxed fac tor to work with.
If there are increasing returns to scale, than doubling inputs of capital and labor
happen if specialization of labor becomes greater more than doubles output. This may
as population grows. Then population growth increases total output and also increases
dvantage of the scale economy output per worker, since the economy is able to take a more quickly. 9. There is no unique way to ﬁnd the data to answer this question. For example, from the
World Bank web site, I followed links to "Data and Statistics." I then followed a link to
"Quick Reference Tables" (http://www.worldbank.org/data/databytopic/GNPPC.pdf) to
ﬁnd a summary table of income per capita across countries. (Note that there are some subtle issues in converting currency values across countries that are beyond the scope
of this book. The data in Table 7—1 use what are called “purchasing power parity”) As an example, I chose to compare the United States (income per person of
$31,900 in 1999) and Pakistan ($1,860), with a 17fold difference in income per person.
How can we decide what factors are most important? As the text notes, differences in
income must come from differences in capital, labor, and/or technology. The Solow
growth model gives us a framework for thinking about the importance of these factors. One clear difference across countries is in educational attainment. One can think
abOut differences in educational attainment as reﬂecting differences in broad “human
capital” (analogous to physical capital) or as differences in the level of technology (e.g.,
if your work force is more educated, then you can implement better technologies). For
our purposes, we will think of education as reﬂecting “technology,” in that it allows
more output per worker for any given level of physical capital per worker. From the World Bank web site (country tables) I found the following data (down loaded February 2002):
Labor Force Investment/GDP Illiteracy
Growth (1990) (percent of
(1994—2000) (percent) population 15+)
United States 1.5 18 0
Pakistan 3.0 19 54 How can we decide which factor explains the most? It seems unlikely that the .5 small difference in investment/GDP explains the large difference in per capital income, leaving laborforce growth and illiteracy (or, more generally, technology) as the likely
culprits. But we can be more formal about this using the Solow model. We follow Section 71, “Approaching the Steady State: A Numerical Example.”
For the moment, we assume the two countries have the same production technology:
Y=K°'5L°5. (This will allow us to decide whether differences in saving and population
growth can explain the differences in income per capita; if not, then differences in tech nology will remain as the likely explanation.) As in the text, we can express this equa
tion in terms of the perworker production function ﬂk); y = kos. i
a
1
i
i
E 64 Answers to Textbook Questions and Problems W In steadystate, we know that Ak = sf(k) — (n + 5)k. The steadystate value of capital k* is deﬁned as the value of k at which capital
stock is constant, so Ak = 0. It follows that in steady state 0 = sf(k) — (n + 8)k,
or, equivalently, k * s
f(k*) = n + 5 ‘
For the production function in this problem, it follows that:
k * s
(k*)°'5 = n+5,
Rearranging:
0.5 S
(k *) = n + 8 ’
or “(if
— n+8 ' Substituting this equation for steady—state capital per worker into the perworker
production function gives:
a: = s
y (n + 5) , If we assume that the United States and Pakistan are in steady state and have
the same rates of depreciation—say, 5 percent—then the ratio of income per capita in the two countries is:
yus ___l: sus ][nPakistan + 005]
yParkistan sPakistnn ”US + 0’05 This equation tells us that if, say, the US. saving rate had been twice Pakistan's saving
rate, then US. income per worker would be twice Pakistan's level (other things equal).
Clearly, given that the US. has 17times higher income per worker but very similar
levels of investment relative to GDP, this variable is not a major factor in the compari son. Even population growth can only explain a factor of 1.2 (ODS/0.065) difference in
levels of output per worker. The remaining culprit is technology, and the high level of illiteracy in Pakistan is
consistent with this conclusion. l . H N
66 Answers to Textbook Questions and Problems Problems and Applications
1. a. To solve for the steadystate value of y as a function of s, n, g, and 8, we begin with
the equation for the change in the capital stock in the steady state: Ak=sﬂk)—(8+n+g)k=0. The production function y = \[k_ can also be rewritten as y2 = k. Plugging this pro duction function into the equation for the change in the capital stock, we ﬁnd that
in the steady state: sy—(8+n +g)y2= 0.
Solving this, we ﬁnd the steadystate value of y: y* = s/(S + n +g).
b. The question provides us with the following information about each country:
Developed country: s = 0.28 Lessdeveloped country: s = 0.10 n = 0.01 n = 0.04
g = 0.02 g = 0.02
a = 0.04 5 = 0.04 Using the equation for y* that we derived in part (a), we can calculate the steady
state values of y for each country. Developed country: y* = 0.28/(0.04 + 0.01 + 0.02) = 4.
Lessdeveloped country: y* = 0.10/(0.04 + 0.04 + 0.02) = 1. c. The equation for y* that we derived in part (a) shows that the lessdeveloped coun
try could raise its level of income by reducing its population growth rate n or by
increasing its saving rate s. Policies that reduce population growth include intro
ducing methods of birth control and implementing disincentives for having chil
dren. Policies that increase the saving rate include increasing public saving by
reducing the budget deﬁcit and introducing private saving incentives such as
I.R.A.’s and other tax concessions that increase the return to saving. 3. a. In the steady state, we know that sy = (5 + n + g)k. This implies that My =s/(8+n +g). Since 3, 8, n, and g are constant, this means that the ratio k/ y is also constant.
Since k / y = [K/(L x E)]/[Y/(L x E)] = K / Y, we can conclude that in the steady state,
the capitaloutput ratio is constant. b. We know that capital’s share of income = MPK x (K / Y). In the steady state, we
know from part (a) that the capitaloutput ratio K [Y is constant. We also know
from the hint that the MPK is a function of k, which is constant in the steady
state; therefore the MPK itself must be constant. Thus, capital’s share of income is
constant. Labor’s share of income is 1 — [capital’s share]. Hence, if capital’s share
is constant, we see that labor’s share of income is also constant. c. We know that in the steady state, total income grows at n + g—the rate of popula—
tion growth plus the rate of technological change. In part (b) we showed that
labor’s and capital’s share of income is constant. If the shares are constant, and total income grows at the rate n + g, then labor income and capital income must
also grow at the rate n + g. (1. Deﬁne the real rental price of capital R as: R = Total Capital Income/Capital Stock
= MPK. We know that in the steady state, the MPK is constant because capital per effec tive worker k is constant. Therefore, we can conclude that the real rental price of
capital is constant in the steady state. 68 Answers to Textbook Questions and Problems @ To show that the real wage w grows at the rate of technological progress g,
deﬁne: TLI = Total Labor Income.
L = Labor Force. Using the hint that the real wage equals total labor income divided by the labor
force: w = TLI / L.
Equivalently, wL = TLI.
In terms of percentage changes, we can write this as Aw/w + AL/L = ATLI/TLI. This equation says that the growth rate of the real wage plus the growth rate of
the labor force equals the growth rate of total labor income. We know that the
labor force grows at rate n, and from part (c) we know that total labor income
grows at rate n + g. We therefore conclude that the real wage grows at rate g. 5. a. In the twosector endogenous growth model in the text, the produc...
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