REExam in Economic Growth: Spring 2011
1. Growth accounting question: 2 points
Assume Yt  A  Kt  L1t  and that the growth rate of L is 2 percent, that the growth rate of K
is 1 percent, and that the growth rate of Y is 4 percent, and that   0.5 , what is the growth
rate of A? What is the growth rate of labor productivity; that is, what is the growth rate of
production per worker (Y/L)?
2. Calculate average annual growth rate: Assume that your income is 100 kronor and that it
grows to become 200 kronor in 30 years time, what is the average annual growth rate of your
income during this period? 1 point
5 points
3.Assume that the economy is described by the 2-period life-cycle model without a
government sector and without trade or exchange of factors of production between countries.
(The economy is thus a closed economy.) To simplify assume that the long-run growth rate of
the technology is zero; that is, A / A =0. In this model:
(1   )  (1   ) A 
Yt  A  K t  L1t  ; Ut  cyt  c1ot1 ; kt 1 
 kt
1 n
where n is the growth rate of the size of the young population.
a) (i) Derive the steady-state for k, that is, solve for k as a function of the
exogenous variables/parameters  , A , n and  .
(ii) Derive the steady-state expressions for y, w, and r as functions of the
exogenous variables/parameters.
b) What is the effect on k , y , w, and r if A increase? .
c) If  increase, what happens to saving of a young working individual? What is
the effect on k , y , w, and r ?
d) What is the effect of a higher n on k , y , w, and r ?
Why do workers tend to be against labor immigration, which raises n; whereas capital-owners
(the old generation) tend to favour labor immigration?
4. Voting for redistributive taxation 5p
Suppose voters are considering adoption of a transfer scheme structured so that every person
is going to receive the same transfer, g. The transfers are to be financed by proportional
taxation. Proportional taxation means that richer people pay more in taxes. Assume that the
utility function of individual i is:
U i  Ci  g
C i is the consumption of the individual i of a private good, which is the only good in this
model. This means that g is the amount of this good that is received as a transfer. Assume that
the number of individuals=number of voters= n. The budget constraint of individual i is:
Ci  (1  t )  Yi
where t is the tax rate and Yi is the income of the individual i which is assumed to be
exogenously given; that is, Yi is determined outside the model.
Assume that the budget constraint of the government is:
ng  tnY where n is number of voters, t is proportional tax rate and Y is average income in
the economy.
a.For a voter with income above average what is the optimal tax rate he will vote for?
b. For a voter with income below average what is the optimal tax rate he will vote for?
c. In the real world, do more or less voters/people have an income below average? Or are
equally many people above average as there are people below average? Explain why!
d. Applying the answer in c. about the real world to this model, what is the voting equilibrium
regarding the tax rate if using the median voter theorem? What are the government revenues
in this case?
e. Explain verbally why we do or do not observe this equilibrium in the real world.
5. 5 points. Assume an economy, which is described by the life-cycle model for a closed
economy. To simplify assume that the growth rate of the technology is zero; that is, A / A =0,
and that the growth rate of the size of the young generation (n) is zero. Suppose that utility
and production are described by the usual Cobb-Douglas functions (see above), and that the
following parameter values hold:
Parameter
A
a.
b.
c.
d.
e.
Value
4
0


0.5
N
100
Calculate the steady-state capital-labor ratio and the level of output and level of output
per worker. Note: N is the size of a generation.
Suppose that the government decides to purchase 1 unit of output for government
consumption each period. To pay for it, the government collects taxes equal to 1 from
the old each period. What will the steady-state capital-labor ratio and output level be?
Compare these answers to those calculated in a. Explain any differences or
similarities.
Now suppose that the government changes its policy and pays for its spending by
collecting taxes equal to 1 from the young each period. How does the capital-labor
ratio and the output level compare to the answers in b). Give a qualitative answer!
That is, higher, lower, the same. Explain any differences or similarities.
If the old (due to altruism) compensate the young by a transfer of 1, what is the effect
on capital-labor and the output level. Compare these answers to the steady state values
calculated in part b. Explain any differences or similarities.
Now disregard the possibility of voluntary transfers from old to young. Suppose that
the government continues to collect taxes equal to 1 from the young each period. Now,
however, the government uses the tax revenue to purchase 0.5 units of capital and 0.5
unit of consumption instead of 1 unit of consumption. What will be the impact of the
policy change on the capital-labor ratio and output level compared to the values in c.
Higher, lower, the same..? Explain why!
How to allocate time optimally across family members: 7 points
6. A family with two adult members seeks to maximize a utility function of the form:
U  C  (lm)  (lw)
where 0  lm  1, 0  lw  1 , and C  0
where C is family consumption, lm is leisure of the man and lw is leisure of the woman.
 ,  ,  are preference parameters that are assumed to be positive; that is, > 0. Assume that the
time endowment of each individual of the household is 1 unit. (It is actually 24 hours per day.
But to simplify we assume that the time each individual has to his/her disposal is 1.) Each
individual spends his and her time working and/or enjoying leisure. If the man only works and
does not enjoy any leisure, he receives the real wage, wm. Similarly, if the woman devotes
her one unit of time to work, she receives the real wage ww. Moreover, the household
receives a non-labor income (N); e.g. transfers from the government. (The price of
consumption (C) is as usual assumed to be 1.)
a) Write up the budget constraint of the household.
b) Derive the household’s optimal demands of C, lm and lw as functions of the
exogenous variables: ww, wm, and N. Moreover, what is the optimal labor supply of
the man and what is the optimal labor supply of the woman as functions of the
exogenous variables?
c) What happens to the labor supply of the man and to the labor supply of the woman if
wm increases exogenously?
d) What happens to the labor supply of the man and to the labor supply of the woman if
ww increases exogenously?
(e) What happens to the optimal levels of lm and lw if N increases?
Instruction: Even if you have not come up with the correct answers in b). Try to answer c)-e).
Instruction how to solve this problem: In case of a Cobb-douglas utility function (like the one
above), the share of expenditures spent on each good equals is constant; that is, regardless of
of the level of income. For example: If the individual maximizes U ( x, y )  x   y 
subject to the budget constraint: p x  x  p y  y  I
where y= quantity of good y, x=quantity of good x, p x = price of good x. p y = price of good
y. I = income. The optimal demand of x and y are such that:
px  x*

 I
 I
if   1    x * 
 x* 

(   )  p x
px
I
 
p y  y*


 y* 
 I
(   )  p y
if   1  

y* 
 I
I
 
py
If you use this information you do not have to do the actual mathematical calculation (based
on the Lagrangetechnique) to come up with the right answers.
Answers to REExam in Economic Growth: Spring 2011
Answer1: 4 = growth rate of A + 0.5*1+0.5*2.
Growth rate of A = 4-1.5=2.5. The growth rate of labor productivity = 4-2=2.
Answer2:
200  100(1  x)30
21/30  1  x , 21/30  1  x
Answers3:
1/(1  )
A
 (1  n)
 (1   )(1   ) A 
A) k  
, y  Ak  , w  (1   ) y , r  1  

1 n
k
(1   )(1   )


B) If A increases k, y, and w increases and r remains unchanged.
C) Saving of an individual decreases: a=(1-alfa)*w, where a is saving.
k, y, w decreases and r increases.
D) k, y, w decreases and r increases. As immigration decreases w and increases r workers are
against immigration and capital-owners favors immigration.
Answer4:
1a. 0. 1b. 1. 1c. More. 1d. 1, nY 1e. If the tax rate =1; that is, 100 percent, people won’t
work so that the assumption of exogenously given incomes are unrealistic.
Answers5:
1/(1  )
 22  4 , y=4*2=8. Y=800.
A. k   (1   ) A
B. Same.
C. Lower because the young generation saves.
D. Same as in b.
E. They will be Higher in e. compared to c.
Answers6:
A. C  wm *(1  lm)  ww *(1  lw)  N
In other words:
C  wm * lm  ww * lw  [ wm  ww  N ]
B. C 
lm 
lw 

 [.]
   



[.] 

[ ww  N ]
(     )* wm
(     ) (     ) wm

(     )* ww
[.] 



[ wm  N ]
(     ) (     ) ww
Labor supply of man = 1-lm. Labor supply of woman=1-lw.
C. Labor supply of man increases and labor supply of woman decreases.
D. Labor supply of man decreases and labor supply of woman increases.
E. Leisure of both woman and of man increases.
Exam in Economic Growth: SPRING 2011.
2 points.
Question 1. Use 2-period consumption model from the AK-book to answer the following
question:
If the individual’s utility function is such that he prefers/values current consumption more
than future consumption, do we know for sure that his current consumption will be higher
than his future consumption? Why or why not! Explain!
3 points.
Question 2. Give 3 explanations of why income inequality may influence the growth rate of
GDP per capita. Robert Barro describes 4 different theories of why income inequality may
influence the growth rate of GDP per capita in his article. Explain briefly 3 of these 4 theories.
2 points.
Question 3. Assume an economy without production. The United Nations fly in 100
kilograms of rice to the country. There are 3 starving individuals in the country each with the
utility function: U  q0.5 , where q is the quantity of rice in kilogram.
How should the rice be distributed among the 3 individuals if the UN bases the allocation of
rice on utilitarism? That is, what’s the optimal allocation if the UN wants to maximize the
sum of the individuals’ utilities. Explain your answer.
5+2=7 points
Question 4. The optimal allocation of time between 3 activities.
Assume 3 types of activities: work (L), leisure (R) and raising kids (K).
Assume that the individual chooses how to devote her time to these 3 different activities.
Assume that amount of time available to the individual = 1.
If the individual raise no kids(K=0): the time constraint is L+R=1,
where L is amount of work and R is amount of leisure.
If the individual uses all her time (=1) to work, she receives the real wage w.
In this case, R=0, and number of kids (K)=0.
Regarding the time required for raising a kid:
One kid requires 20 percent of the individual’s time available. Two kids require 40 percent of
the individual’s time available, 3 kids 60 percent, 4 kids 80 percent and 5 kids 100 percent.
(so in this problem the individual cannot raise more than 5 kids).
Apart from the time constraint of the individual, he has a budget constraint: C = wL + N,
where C= quantity of consumption good. N real non-labor income; that is, non-labor income
in terms of the consumption good. wL= real labor income.
(The price of the consumption good = 1.)
Assume the utility function:
U  C a  R b  K 1 a b
where 0  a  1, 0  b  1 ,0< (1  a  b) <1.
A1. Derive the optimal demands of C, R, L and K as functions of the exogenous variables: w,
N, and of the parameters a, b.
A2. What happens to optimal R and K and L and C when N increases.
A3. What happens to optimal R and K and L and C when w increases.
Students unable to derive optimal demands should nevertheless be able to answer A2, and
possibly also A3.
Instruction how to solve this problem: In case of a Cobb-douglas utility function (like the one
above), the share of expenditures on each good equals that good’s preference parameter in the
utility function. For example: If the individual maximizes U ( x, y )  x   y 
subject to the budget constraint: p x  x  p y  y  I
where y= quantity of good y, x=quantity of good x, p x = price of good x. p y = price of good
y. I = income. The optimal demand of x and y are such that:
px  x*

 I
 I
 x* 
if   1    x * 

(   )  p x
px
I
 
p y  y*


 y* 
 I
(   )  p y
if   1  

y* 
 I
I
 
py
If you use this information you do not have to do the actual mathematical calculation (based
on the Lagrangetechnique) to come up with the right answers.
2+2+1+1+2+1+2= 11 points
Question 5. Suppose that the world consists of only two countries. Savum and Spendum. The
following values apply to the cobb-douglas production functions and utility functions of each
country. Use the values in the table to answer the questions below.
Parameter
A


N
Savum
4
0.5
0.5
100
Spendum
4
0.6
0.5
100
a. calculate the steady state values for capital per worker (k=K/L), production per worker
(y=Y/L), the real wage per worker (w), and the real return on capital (r) for the two
countries: Savum and Spendum under autarky; that is, when the countries are closed
with respect to each other (=autarky).
.
If you cannot answer with numbers, explain in words which country has higher or lower
values on the variables mentioned above for partial credit.
b. calculate steady state values of GDP (Y), of consumption when young ( c y ), of
consumption when old ( co ), and of utility (U) for Savum and Spendum.
c. Calculate the steady state values of y, k, w, and r for each country under free trade; that is,
when capital mobility is allowed for. Help: The world-wide value of  equals
N spendum
N savum

  savum 
  spendum
N savum  N spendum
N savum  N spendum
d. From what country will capital move? In other words, which of the countries will end up
owning assets (=capital) in the other country?
e1. calculate steady state values of Y , c y , co , U for the two countries under free trade.
e2. Compare the steady-state utility of a young person in Spendum under autarky with the
steady-state utility of a young person in Spendum under free trade.
Is it higher or lower or unchanged?
e3. Compare the steady-state utility of a young person in Savum under autarky with the
steady-state utility of a young person in SAVUM under free trade.
Is it higher or lower or unchanged?
f. During the transition to the new steady state which group (current young or current old)
gain or loose in the two countries?
g. Now suppose that the world consists of the countries. Savum and Efficient. (Note the
country Spendum does no longer exist.) Country Efficient is identical to Savum except that A
equals 9 in country efficient. Thus, Efficient have the following parameter values:
A=9,  =0.5,  =0.5, N=100.
If we allow for capital mobility between the country Savum and the country efficient, from
what country will capital move? Explain why?
Answers spring 2011:
Answer1: no it can also be lower or the same depending on the value of the interest rate.
Co/cy=(1+r)(1-a)*w/aw)=(1-a)(1+r)/a
Co/cy<1 if (1-a)(1+r)/a<1 which is not likely if r is high.
Answer2:
1. More inequality, means more riots, revolutions, and social unrest, and hence lower
saving rates in physical and human capital, and hence lower growth.
2. More inequality, means lower saving rates in physical and human capital due to
CREDIT MARKETS CONTRAINS. Credit market constraints imply that profitable
project do not take place.
3. More inequality, means higher saving rates as richer people have higher saving rates.
4. More inequality, higher demand for redistribution, hence higher tax rates, and hence lower
growth.
Answer3:
Equal distribution due to diminishing marginal utility of consumption.
Answers4:
A1.
L+R+0.2*K=1L=1-R-0.2*K.
C=w*L+N
C=w*(1-R-0.2K)+N
C+wR+0.2wK=w+N
C/I=a
C=a(w+N)
wR/I=b
R=b(w+N)/w=b+(N/w)
0.2wK/I=(1-a-b)
K=(1-a-b)(w+N)/0.2w=(1-a-b)/0.2+(1-a-b)N/0.2w.
L=1-R-0.2K.
A2. If N increases R and K will increase and L will decrease.
A3. Cup, Rdown, K down, L up
Answers5: A+B
K=((1-a)*(1-b)A)^(1/1-b)
Y=Ak^0.5
W=(1-b)*y=0.5*y
r=A*b*k^-0.5
SAVUM
1
4
2
2
SPENDUM
(0.4*0.5*4) )^2=0.64
3.2
1.6
2.5
Y=y*N
Cy=a*w
400=200+200
1
320
0.96
Co=(1+r)(1-a)w
U=cy^a*co^(1-a)
3
3^0.5=1.73
3.5*0.4*1.6=2.24
U=0.96^0.6*2.24^0.4=0.98*1.38=1.3524
answers5: C.-e.
K=((1-a)*(1-b)A)^(1/1-b)
Y=Ak^0.5
W=(1-b)*y=0.5*y
r=A*b*k^-0.5
Y=y*N
WORLD
(0.45*0.5*4) )^2=0.81
3.6
1.8
2.2222
720
Cy=a*w: spendum
Co=(1+r)(1-a)w: spendum
U=cy^a*co^(1-a): spend.
0.5*1.8=0.9
3.222*0.5*1.8=2.9
U=0.9^0.5*2.9^0.5=0.95*1.70=1.615
Cy=a*w: spendum
Co=(1+r)(1-a)w: spendum
U=cy^a*co^(1-a): spend.
0.6*1.8=1.08
3.222*0.4*1.8=2.32
U=1.08^0.6*2.32^0.4=1.05*1.40=1.47
d. Savum will end up owning capita in spendum
e. Steady state utility goes down in savum and up in spendum.
f.during the transition: in savum: workers loose and old people gain.
In spendum: workers gain and old people loose.
g. no capital mobility as r is the same in the 2 countries.
SPRING 2010.
Question 1: 5 points
1. In the life-cycle model for a closed economy without a government sector and assuming
that the long-run growth rate of the size of the young generation is n, what is the effect of a
one-time increase in A on the steady state values of r and w? Show mathematically! Thus, in
this model:
Yt  A  Kt  L1t  ; Ut  cyt  c1ot1 ; kt 1 
(1   )  (1   ) A 
 kt
1 n
If you cannot do the calculation, answer in words what the effect on w and r is of a higher A.
Such an answer will give you some points.
Answer: n up leads to k and y and w down and r up.
A up leads to k and y and w up and r unchanged.
Question 2: 5 points
Assume an economy, which is described by the life-cycle model for a closed economy
(without altruism between generations). To simplify, assume that the individual’s utility
function is such that the individual only consumes when she is old; this means that U t  co ,t 1 .
That is,  = 0 in the utility function: Ut  cyt  cot11 . When the individual is young she only
works and saves. To simplify, assume also that the long-run growth of the size of the young
generation, n, is zero.) Assume that the economy is in its steady state.
Suddenly it is decided to permanently introduce public consumption: G .
a) If public consumption is financed by taxing the current old generation and the present
young and all future generations when they are old, what is the impact on the steady
state levels of capital per worker and production per worker relative to the case when
there was no public consumption? Explain your results.
b) If the public consumption is financed by taxing the current young generation and the
future generations when they are young, what is the impact on the steady state levels
of capital per worker and production per worker relative to the case when there was no
public consumption? Explain your results.
c) If the public consumption is financed by borrowing from the current young generation
and the future generations when they are young, what is the impact on the steady state
levels of capital per worker and production per worker relative to the case when there
was no public consumption? Explain your results. The government is assumed to
repay its loans to the individuals when they are old.
d) If the tax revenues raised in b. instead of financing public consumption finance public
investment (and public capita is a perfect substitute to private capital, what is the
effect on capital per worker relative to the case when there is no government sector.
Answer:
A. No effect on k and y.
B. Lower k and y.
C. Same effect as in b.
D. No effect on k and y relative to the case when there is no government sector.
Question 3: 3 points
A. Write up the government’s intertemporal budget constraint.
B. Analyze the statement “a government budget decifit is a tax on future generations”.
Analyze the effect of lowering taxes today. According to the intertemporal budget constraint
what will happen to taxes in the future?
Question 4: 2 points
If average hourly wage in the industry in 1970 year prices (constant prices) is 150 kronor in
2010 whereas it was 56 kronor in 1970, what was the average annual growth rate of the real
hourly wage between 1970 and 2010?
Answer:
(150/56)^^(1/40)-1= 0,0249.
Question 5: 10 points
An individual’s optimal choice regarding the allocation of time.
A one-period model. Assume the following utility function:
U  C   R1
where 0    1 , 0  (1   )  1 .
where C is real consumption, R is hours of leisure.
The individual can spend time on 2 types of activities: hours working (L), hours enjoying
leisure (R). The total number of hours per day is 16.
(Thus, 24-16=8 is the number of hours sleep the individual needs.)
In other words: 16 = L + R.
The hourly real wage is w. The individuals only source of income is from working.
A.Write up the individual’s budget constraint.
B.Derive the optimal demands of C, and R as functions of the exogenous variable: w.
Derive also the optimal labor supply as a function of w.
C.Analyze the effect on C, and R, on labor supply (L) of a higher real wage?
D. Now assume that the individual, in addition to labor income, receives a non-work-related
transfer: N.
Derive the optimal demands of C, and R as functions of the exogenous variables: w and N.
Derive also the optimal labor supply as a function of the exogenous variables: w and N.
E. Analyze the effect on C, and R, on labor supply (L) of a higher real wage?
F. What happens to the optimal choice of C, R and L when N increases?
Even if you cannot do the math try to answer some of the questions in words!
Answer:
A. C=w*L, which means that C=w*(16-R), which means that
C+w*R=w*16.
B. C=alpha*w*16
R=16*(1-alfa)
L=16*alfa.
C. Higher C, but no effect on R and L.
D. C=alpha*w*16+N
R=16*(1-alfa)+(N/w)
L=16*alfa- (N/w).
E. when w increases, C increases and R decreases and L increases.
F. C increases, R increases and L decreases.
Instruction how to solve this problem: In case of a Cobb-douglas utility function (like the one
above), the share of expenditures on each good equals that good’s preference parameter in the
utility function. For example: If the individual maximizes U ( x, y )  x   y 
subject to the budget constraint: p x  x  p y  y  I
where y= quantity of good y, x=quantity of good x, p x = price of good x. p y = price of good
y. I = income. The optimal demand of x and y are such that:
px  x*
 I
 I
if   1    x * 
 x* 

px
p x  (   )
I
p y  y*

 y* 
 I
p y  (   )
if   1  

y* 
 I
py
I
If you use this information you do not have to do the actual mathematical calculation (based
on the Lagrangetechnique) to come up with the right answers. The formulas above come from
Lagrange-optimization.
SPRING 2009.
5 points
1. If the following regression is run on a sample consisting of the countries of the world:
GINI i    1  GDPpci   2  GDPpci 2  ei
Note: i denotes country i,
GDPpci is real GDP per capita in country i.
According to the hypothesis of a Kuznets Curve, what are the expected signs of  1 and  2 ?
In other words, according to the hypothesis of the Kuznets Curve what sign (positive,
negative or zero) should  1 be, and what sign (positive, negative or zero) should  2 be?
Explain how you came up with your answer!
5 points
2. Assume a closed economy, which is described by the life-cycle model (without a
government sector). Moreover, assume that the individual’s utility function is such that the
individual only consumes when she is old; this means that U t  co ,t 1 . When the individual is
young she only works and saves. (To simplify, assume that the growth rate of the technology
is zero; that is, A / A  0 , and that the long-run growth of the size of the young generation, n,
is zero.) Assume that the economy is in its steady state.
Assume that suddenly a pay-as-you-go Social Security system is introduced. This pension
system means that the current young generation pays x units of goods (corn) per young
worker to the current old generation. When the current young generation becomes old, which
occurs in the next period, they will each receive x units of goods (corn) from the generation
which is young in this next period. The pension system means that each generation will pay x
units of goods (corn) per person when being young, and that they will each receive x units of
goods (corn) when they are old. One underlying assumption here is that the sizes of the
different generations are the same.
This pensions scheme means that the current old generation is a winner as they receive x units
of goods (corn) but did not pay anything to the previous generation.
a) In present value terms is the current young generation and all future generations losers or
winners due to the introduction of this pension system?
b) What is the effect on the steady state values of k and y of the introduction of the pension
system? Give a verbal qualitative answer (that is, whether the steady state values of k and y
increase, decrease or are unchanged). Also show the effect on k in a transition diagram.
Explain why an effect occurs or does not occur!
C. Assume now that the current old generation (the generation that is old when the
pension system is introduced) care so much for the current young generation that they
transfer resources, x units of goods (corn), to the current young generation
immediately after the pension system is introduced. (The current young and all future
generations act in the same way, when they are old.) Compared to the steady state
values of k and k prior to the pension reform, what are the steady state values of k and
y after the pension reform in this case. Give a verbal qualitative answer, and explain
why! Possibly by referring to the transition equation.
7 points
3. Assume that the economy is described by the 2-period life-cycle model without a
government sector and without trade or exchange of factors of production between countries.
(The economy is thus a closed economy.) To simplify assume that the long-run growth rate of
the technology and of the population are zero; that is, A / A =0 and n=0. In this model:
Yt  A  Kt  L1t  ; Ut  cyt  c1ot1 ; kt 1  (1   )(1   ) A  kt
e) Derive the steady-state expression for k; that is, solve for k as a function of
the exogenous variables/parameters  , A , and  .
f) Derive the steady-state expressions for y, and w as functions of the exogenous
variables/parameters  , A , and  .
c)Assume now instead that the production function is: y  A  k  L1  . This is an
so-called endogenous growth model.
Rewrite the transition equation, and derive an expression for the growth rate of k;
k k
that is, derive t 1 t .
kt
Is the growth rate of k dependent on whether the economy starts out rich or poor;
That is, does the growth rate depend on the value of kt .
IN other words, does this endogenous growth model say that the lower initial k (and y) the
higher the growth rate of k (and y) when holding the steady state constant (which depends on
 ,  , A )? What is the result if we assume the standard production function: Yt  A  Kt  L1t  ,
which in per worker terms is: y  A  k 
What does the empirical evidence say about convergence in per capita income across
production among economies that are similar with respect to institutions etc; for example,
countries in Europe or regions within a country?
d) In the endogenous growth model, what happens to the growth rate of k if the number of
workers (L) increases?
8 points
4. Consumption theory: Intertemporal Choice.
Assume the following equations:
U  C1  C21 where alpha is between 0 and 1.
Restrictions: C1 + S = w (1)
In other words, consumption in the first period of life and saving equals the income in the first
period of life, w.
C2= (1+r)*S (2)
Where C2 is consumption in period 2.
The individual is assumed to live 2 periods and is assumed not to receive or leave any
inheritance.
a. Derive optimal demand for C1 and C2, and optimal saving as functions of the
exogenous variables: w, r.
b. Study how an increase of the real interest rate (r) impacts optimal saving and optimal
consumption in the first and second periods of life?
Instruction how to solve this problem: In case of a Cobb-douglas utility function (like the one
above), the share of expenditures on each good equals that good’s preference parameter in the
utility function. For example: If the individual maximizes U ( x, y )  x   y 
subject to the budget constraint: p x  x  p y  y  I
where y= quantity of good y, x=quantity of good x, p x = price of good x. p y = price of good
y. I = income. The optimal demand of x and y are such that:
 I
 I
px  x*

if   1    x * 
 x* 

px
p x  (   )
I

p y  y*


 y* 
 I
p y  (   )
if   1  

y* 
 I
I

py
If you use this information you do not have to perform any mathematical calculation to come
up with the right answers.
c. Now allow for a pension in second period of life, N 2 .
Write up the individual’s intertemporal budget constraint in terms of present value.
Derive the optimal demands of C1, C2, and optimal saving as functions of the exogenous
variables: w, r (=the real interest rate), and N 2 .
What happens to optimal C1, C2 and optimal saving if N 2 increases?
d. Now allow for 3 types of activities in the first period of life: work, leisure and raising kids.
Assume that the individual chooses the time devoted to the 3 different activities in the first
period of life. Assume that amount of time available to the individual = 1. If the individual
has no kids: the time constraint is L+R1=1, where L is amount of work and R1 is amount of
leisure.
If the individual uses all her time in period 1 (=1) to work, she receives the real wage w.
In this case, R1=0, and number of kids (K)=0.
Regarding the time required for raising a kid:
One kid requires 20 percent of the individual’s time available. Two kids require 40 percent of
the individual’s time available, 3 kids 60 percent, 4 kids 80 percent and 5 kids 100 percent.
(so in this problem the individual cannot have more than 5 kids).
Write up the time constraint with all 3 activities: work (L), leisure (R1) and raising kids (K).
Assume the following utility function:
U  C1  C2  K   R11   
where 0    1, 0    1 , 0  (1       )  1 , 0    1 .
Also allow for a pension in second period of life, N 2 .
Write up the individual’s intertemporal budget constraint in terms of present value.
Derive the optimal demands of C1, C2, and R1 and L and K as functions of the exogenous
variables: w, r (=the real interest rate), and N 2 .
Analyze the effect on optimal C1, C2, R1 and K and L if w increases?
REEXAM SPRING 2009
The open-economy question är från april-tentan resten är från januari-tentan.
3 points
1. Analyze the following statement:
” A higher level of production (GDP) inevitably leads to a deterioting environmental quality”.
Deterioting means worse.
Is there a difference between local pollution or global pollution? If so explain, why?
By local pollution I mean pollution that primarily affect the local community such as smoke
in the air and pollution in rivers and lakes. Global pollution on the other hand is pollution that
affect the environment around the world regardless where it is emitted. For example, carbon
dioxide (CO2) contributes to the green house effect regardless where the carbon dioxide is
emitted; Sunne or Cairo.
Answer: no, when countries become richer local air and water pollution tend to become
better.
7 points
2. Assume an economy, which is described by the life-cycle model for a closed economy. To
simplify assume that the growth rate of the technology is zero; that is, A / A =0, and that the
growth rate of the size of the young generation (n) is zero.
Suppose that utility and production are described by the usual Cobb-Douglas functions:
Yt  A  Kt  L1t  ; Ut  cyt  c1ot1 and that the following parameter values hold:
Parameter
A
f.
g.
h.
i.
j.
Value
4
0

0.5

N
100
Calculate the steady-state capital-labor ratio and the level of output.
Suppose that the government decides to purchase 1 unit of output for government
consumption each period. To pay for it, the government collects taxes equal to 1 from
the old each period. What will the steady-state capital-labor ratio and output level be?
Compare these answers to those calculated in a. Explain any differences or
similarities.
Now suppose that the government changes its policy and pays for its spending by
collecting taxes equal to 1 from the young each period. Calculate the capital-labor
ration and the output level for the 2 periods following the change. Compare these
answers to the steady state values calculated in part b. Explain any differences or
similarities.
If the old (due to altruism) compensate the young by a transfer of 1, what is the effect
on capital-labor and the output level. Compare these answers to the steady state values
calculated in part b. Explain any differences or similarities.
Now disregard the possibility of voluntary transfers from old to young. Suppose that
the government continues to collect taxes equal to 1 from the young each period. Now,
however, the government uses the tax revenue to purchase 0.5 units of capital and 0.5
unit of consumption instead of 1 unit of consumption. What will be the impact of the
policy change on the capital-labor ratio and output level in the 2 periods after the
change? Do these differ from the values calculated in part c. Why or why not?
Show the your calculations!
Svar:
1/(1  )
4
A. k   (1   ) A
y  Ak   4  2  8
B. same effect as in A.
C. k1   (1   ) A k0  1  2  40.5  1  4  1  3
k2   (1   ) A k1  1  2  30.5  1  3.46  1  2.46
y1  Ak1  4  3  6.93
y2  Ak2  4  2.46  6.27
D. same effect as in A and B.
E. k1   (1   ) A k0  0.5  2  40.5  0.5  4  0.5  3.5
k2   (1   ) A k1  1  2  3.50.5  .5  3.74  0.5  3.24
y1  Ak1  4  3.5  7.48
y2  Ak2  4  3.24  7.2
8 points
3. Suppose that the world consists of only two countries. Savum and Spendum. The following
values apply to the cobb-douglas production functions and utility functions of each country.
Use the values in the table to answer the questions below.
Parameter
A


N
Savum
4
0.3
0.25
100
Spendum
4
0.6
0.25
200
a. Calculate the steady state values of k, y, w, and r for each country under autarky; that is,
when the countries are closed with respect to each other (=autarky).
b. Compare the steady-state values in a. Do the differences make economic sense?
c. Calculate the world-wide value of  . Recall that it equals
N spendum
N savum

  savum 
  spendum
N savum  N spendum
N savum  N spendum
d. Calculate the steady state values of k, y, w, and r for each country under free trade; that is,
when capital mobility is allowed for.
e. Compare Savum’s steady-state values under autarky to those under free trade. Explain any
differences.
Which of the countries own assets (=capital) in the other country?
7 points
4. Labor supply and saving, the effect of a lower pension in the future.
Use the life-cycle model and assume that an individual lives 2 periods. In the first period she
works and consumes, in the second period she also consumes. In this period she is retired.
While working in the first period, she values leisure. Assume the following utility function:
where 0    1 , 0    1 , 0  (1     )  1 .
U  C1  C2  R11 
where C1 is good consumption in period 1 when she is young and works, C2 is good
consumption in period 2 when she is retired, and R1 is leisure that the individual enjoys in
period 1. In the second period the individual is not allowed to work. As a result she faces no
choice between labor and leisure in this period. The time constraint in period 1 is: L+R1=1.
If the individual uses all her time in period 1 (=1) to work, she receives the real wage w.
In this case, R1=0. The individual also receives a non-work-related income (pension), N 2 , in
the second period.  and  are preference parameters. (The price of good consumption in
the first period is assumed to be 1 as usual.)
A.Write up the individual’s intertemporal budget constraint in terms of present value.
B.Derive the optimal demands of C1, C2, and R1 as functions of the exogenous variables:
w, r (=the real interest rate), and N 2 . Derive also the optimal labor supply.
C.(i)Analyze the effect on C1, C2, R1 and L if N 2 decreases?
In other words, what happens if the individual expect to receive a lower pension in the
future?
(ii) What happens to optimal levels of C1, C2, R1 and L when r increases?
Instruction how to solve this problem: In case of a Cobb-douglas utility function (like the one
above), the share of expenditures on each good equals that good’s preference parameter in the
utility function. For example: If the individual maximizes U ( x, y )  x   y 
subject to the budget constraint: p x  x  p y  y  I
where y= quantity of good y, x=quantity of good x, p x = price of good x. p y = price of good
y. I = income. The optimal demand of x and y are such that:
 I
px  x*
(1   )  I

if   1    x* 
 x* 

px
p x  (   )
I
 
p y  y*


 y* 
 I
p y  (   )
if   1  

y* 
 I
I
 
py
If you use this information you do not have to perform any mathematical calculation to come
up with the right answers.
Svar:
A. C1 
C2
N2
 w  R1  w 
1 r
1 r
B.
SPRING 2008.
EXAM IN MACROECONOMICS INTERMEDIATE LEVEL AT KARLSTAD
UNIVERSITY, SPRING 2008.
Maximum points: 25 points. To pass 12.5 points are required.
1. 12 points Use the overlapping generation model with the following production function:
Y  AK  L1 , and the following utility function: U  cyt  cot11
Assume also that the growth rate in A is zero. Assume that the world consists of 2 economies:
Savum and Spendum.
Parameter
SAVUM SPENDUM EFFICIENT
A
4
4
16
0.5
0.6
0.6


0.5
0.5
0.5
Population growth
0.1
0.1
0.1
rate (n)
N(0)
100
200
-1a. calculate the long-run equilibrium values for capital per worker (k=K/L), consumption per
person when being young (cy), production per worker (y=Y/L), the real wage per worker (W),
and the real return on capital for the two countries: SAVUM and SPENDUM.
1b. Calculate long-run GDP (Y) for the 2 countries: SAVUM AND SPENDUM. Explain.
If you cannot answer with numbers, explain in words which country has higher or lower
values on the variables mentioned above in 1a. and 1b. for partial credit.
c. If we allow for capital mobility (K) between the countries, from what country (SAVUM
and SPENDUM) will the capital move? Why? Does capital mobility diminish initial
differences in k,y,W, and real return on capital?
2a. Assume another country: EFFICIENT. Under the assumption that this country is a closed
economy calculate capital per worker (k), production per worker (y=Y/L), consumption per
worker, real wage per worker, and the real rental price per unit of capital.
2b. If now assuming 2 countries: SPENDUM and EFFICIENT, and allowing for capital
mobility (K) between these 2 countries. (Note: The country SAVUM does not exist any
longer.) Do you expect capital to move between the countries SPENDUM and EFFICIENT?
Explain.
If you cannot answer with numbers, explain in words which country has higher or lower
values on the variables for partial credit.
answers:
K=((1-alpha)*(1-beta)A/(1+n))^2
Y=Ak^0.5
W=beta*y=0.5*y
Cy=apfa*y
r=A*beta*k^-0.5
SAVUM
0.826
3.636
1.818
1.818
2.200
SPENDUM
0.529
2.909
1.4546
1.7454
2.749
EFFICIENT
8.463
46.55
23.275
27.93
2.749
3. 4 poiints. What is the effect on the long-run equilibrium values of k, y, consumtion when
being young and consumption when being old, the real wage, and the real return on capital if
n increases? No need to calculate with numbers. Who tend to welcome immigration and
whom tend to oppose immigration of workers and employer? Explain why.
4. 4 points If the government tax the old, can it actually increase production per capita relative
to the case when there is no government? Use our overlapping generations model and explain
why.
5. 2.5 points write the government’s intertemporal budget constraint.
6. 2.5 points Mention and explain two reasons why inequality may impact the growth rate of
per capita income according to the article by Barro.
FALL 2008
4 points
1. Analyze the following statement:
” A higher level of production (GDP) inevitably leads to a deterioting environmental quality”.
Deterioting means worse.
Is there a difference between local pollution or global pollution? If so explain, why?
By local pollution I mean pollution that primarily affect the local community such as smoke
in the air and pollution in rivers and lakes. Global pollution on the other hand is pollution that
affect the environment around the world regardless where it is emitted. For example, carbon
dioxide (CO2) contributes to the green house effect regardless where the carbon dioxide is
emitted; Sunne or Cairo.
10 points
2. Assume an economy, which is described by the life-cycle model for a closed economy. To
simplify assume that the growth rate of the technology is zero; that is, A / A =0, and that the
growth rate of the size of the young generation (n) is zero.
Suppose that utility and production are described by the usual Cobb-Douglas functions:
Yt  A  Kt  L1t  ; Ut  cyt  c1ot1 and that the following parameter values hold:
Parameter
A
Value
4
0

0.5

N
100
k. Calculate the steady-state capital-labor ratio and the level of output.
l. Suppose that the government decides to purchase 1 unit of output for government
consumption each period. To pay for it, the government collects taxes equal to 1 from
the old each period. What will the steady-state capital-labor ratio and output level be?
Compare these answers to those calculated in a. Explain any differences or
similarities.
m. Now suppose that the government changes its policy and pays for its spending by
collecting taxes equal to 1 from the young each period. Calculate the capital-labor
ration and the output level for the 2 periods following the change. Compare these
answers to the steady state values calculated in part b. Explain any differences or
similarities.
n. If the old (due to altruism) compensate the young by a transfer of 1, what is the effect
on capital-labor and the output level. Compare these answers to the steady state values
calculated in part b. Explain any differences or similarities.
o. Now disregard the possibility of voluntary transfers from old to young. Suppose that
the government continues to collect taxes equal to 1 from the young each period. Now,
however, the government uses the tax revenue to purchase 0.5 units of capital and 0.5
unit of consumption instead of 1 unit of consumption. What will be the impact of the
policy change on the capital-labor ratio and output level in the 2 periods after the
change? Do these differ from the values calculated in part c. Why or why not?
Svar:
1/(1  )
4
A. k   (1   ) A
y  Ak   4  2  8
B. same effect as in A.
C. k1   (1   ) A k0  1  2  40.5  1  4  1  3
k2   (1   ) A k1  1  2  30.5  1  3.46  1  2.46
y1  Ak1  4  3  6.93
y2  Ak2  4  2.46  6.27
E. same effect as in A and B.
E. k1   (1   ) A k0  0.5  2  40.5  0.5  4  0.5  3.5
k2   (1   ) A k1  1  2  3.50.5  .5  3.74  0.5  3.24
y1  Ak1  4  3.5  7.48
y2  Ak2  4  3.24  7.2
7 points
3. Labor supply and saving, the effect of a lower pension in the future.
Use the life-cycle model and assume that an individual lives 2 periods. In the first period she
works and consumes, in the second period she also consumes. In this period she is retired.
While working in the first period, she values leisure. Assume the following utility function:
where 0    1 , 0    1 , 0  (1     )  1 .
U  C1  C2  R11 
where C1 is good consumption in period 1 when she is young and works, C2 is good
consumption in period 2 when she is retired, and R1 is leisure that the individual enjoys in
period 1. In the second period the individual is not allowed to work. As a result she faces no
choice between labor and leisure in this period. The time constraint in period 1 is: L+R1=1.
If the individual uses all her time in period 1 (=1) to work, she receives the real wage w.
In this case, R1=0. The individual also receives an income that is not job-related, N1 (which
can be negative), in the first period and the non-work-related income (pension), N 2 , in the
second period.  and  are preference parameters. (The price of good consumption in the
first period is assumed to be 1 as usual.)
A.Write up the individual’s intertemporal budget constraint in terms of present value.
B.Derive the optimal demands of C1, C2, and R1 as functions of the exogenous variables:
w, r (=the real interest rate), N1 och N 2 . Derive also the optimal labor supply.
C.(i)Analyze the effect on C1, C2, R1 and L if N 2 decreases?
In other words, what happens if the individual expect to receive a lower pension in the
future?
(ii) What happens to optimal levels of C1, C2, R1 and L when r increases?
Instruction how to solve this problem: In case of a Cobb-douglas utility function (like the one
above), the share of expenditures on each good equals that good’s preference parameter in the
utility function. For example: If the individual maximizes U ( x, y )  x   y 
subject to the budget constraint: p x  x  p y  y  I
where y= quantity of good y, x=quantity of good x, p x = price of good x. p y = price of good
y. I = income. The optimal demand of x and y are such that:
px  x*
 I
(1   )  I
if   1    x* 
 x* 

px
p x  (   )
I
p y  y*

 y* 
 I
p y  (   )
if   1  

y* 
 I
py
I
If you use this information you do not have to perform any mathematical calculation to come
up with the right answers.
Svar:
A. C1 
C2
N2
 w  R1  w  N 1 
1 r
1 r
B.
4 points
4.A Show graphically and explain how economic inequality is measured.
4B. Describe 3 theories why economic inequality may affect the growth rate of GDP per
capita, at least during the transition to steady state.
REEXAM FALL 2008.
2 points
1. Use 2-period consumption model from the AK-book to answer the following questions:
If the individual’s utility function is such that he prefers/values current consumption more
than future consumption, do we know for sure that his current consumption will be higher
than his future consumption? Why or why not! Explain!
5 points
2.Assume that the economy is described by the 2-period life-cycle model without a
government sector and without trade or exchange of factors of production between countries.
(The economy is thus a closed economy.) To simplify assume that the long-run growth rate of
the technology is zero; that is, A / A =0. In this model:
Yt  A  Kt  L1t  ; Ut  cyt  c1ot1 ; kt 1 
(1   )  (1   ) A 
 kt
1 n
where n is the growth rate of the size of the young population.
g) (i) Derive the steady-state for k, that is, solve for k as a function of the
exogenous variables/parameters  , A , n and  .
(ii) Derive the steady-state expressions for y, w, and r as functions of the
exogenous variables/parameters.
h) What is the effect on k , y , w, and r if A increase? .
i) If  increase, what happens to saving of a young working individual? What is
the effect on k , y , w, and r ?
j) What is the effect of a higher n on k , y , w, and r ?
Why do workers tend to be against labor immigration, which raises n; whereas capital-owners
(the old generation) tend to favour labor immigration?
2 points
3. Explain shortly what the Environmental Kusnets curve and the theory behind it.
6 points
4. A. If the government imposes a tax on the young to finance government consumption, what
happens to saving by the young and the steady state capital labor ratio?
B. Now suppose that the elderly care enough about their children to provide them with a gift
equal to the amount of the government tax. Relative to the initial steady state (i.e., with no
fiscal policy), what will be the effect on the steady-state capital-labor ratio? What will happen
to the composition of consumption? Use the model in the textbook (chapter 6) and the
assumptions of this model to answer the question.
C. Now assume that the government instead imposes a tax on the old to finance government
consumption, what happens to saving by the young and the steady state capital labor ratio?
D. If instead the tax revenues that are raised by taxing the old, instead is used to finance
government investment (roads railroads etc.), what happens to the steady state capital labor
ratio?
5points
5. Suppose that the world consists of only two countries. Savum and Spendum. The following
values apply to the cobb-douglas production functions and utility functions of each country.
Use the values in the table to answer the questions below.
Parameter
A


N
Savum
4
0.3
0.25
100
Spendum
4
0.6
0.25
200
a. Calculate the steady state values of y, k, w, and r for each country under autarky; that is,
when the countries are closed with respect to each other (=autarky).
b. Compare the steady-state values in a. Do the differences make economic sense?
c. Calculate the world-wide value of  . Recall that it equals
N spendum
N savum

  savum 
  spendum
N savum  N spendum
N savum  N spendum
d. Calculate the steady state values of y, k, w, and r for each country under free trade; that is,
when capital mobility is allowed for.
e. Compare Savum’s steady-state values under autarky to those under free trade. Explain any
differences.
Which of the countries own assets (=capital) in the other country?
4 points
6. Calculate the steady-state levels of consumption per person for old people and young
people in each country under autarky.
a. Calculate the steady-state utility of each young person in each country under autarky.
b. Calculate the steady-state utility of each young person in each country under free
trade. Compare this to the utility value under autarky and explain any differences.
c. During the transition to the new steady state which group (current young or current
old) gain or loose in the two countries?
Exam in Economic Growth: SPRING 2011.
2 points.
Question 1. Use 2-period consumption model from the AK-book to answer the following
question:
If the individual’s utility function is such that he prefers/values current consumption more
than future consumption, do we know for sure that his current consumption will be higher
than his future consumption? Why or why not! Explain!
Answer1: no it can also be lower or the same depending on the value of the interest rate.
Co/cy=(1+r)(1-a)*w/aw)=(1-a)(1+r)/a
Co/cy<1 if (1-a)(1+r)/a<1 which is not likely if r is high.
3 points.
Question 2. Give 3 explanations of why income inequality may influence the growth rate of
GDP per capita. Robert Barro describes 4 different theories of why income inequality may
influence the growth rate of GDP per capita in his article. Explain briefly 3 of these 4 theories.
Answer2:
4. More inequality, means more riots, revolutions, and social unrest, and hence lower
saving rates in physical and human capital, and hence lower growth.
5. More inequality, means lower saving rates in physical and human capital due to
CREDIT MARKETS CONTRAINS. Credit market constraints imply that profitable
project do not take place.
6. More inequality, means higher saving rates as richer people have higher saving rates.
4. More inequality, higher demand for redistribution, hence higher tax rates, and hence lower
growth.
2 points.
Question 3. Assume an economy without production. The United Nations fly in 100
kilograms of rice to the country. There are 3 starving individuals in the country each with the
utility function: U  q0.5 , where q is the quantity of rice in kilogram.
How should the rice be distributed among the 3 individuals if the UN bases the allocation of
rice on utilitarism? That is, what’s the optimal allocation if the UN wants to maximize the
sum of the individuals’ utilities. Explain your answer.
Answer3:
Equal distribution due to diminishing marginal utility of consumption.
5+2=7 points
Question 4. The optimal allocation of time between 3 activities.
Assume 3 types of activities: work (L), leisure (R) and raising kids (K).
Assume that the individual chooses how to devote her time to these 3 different activities.
Assume that amount of time available to the individual = 1.
If the individual raise no kids(K=0): the time constraint is L+R=1,
where L is amount of work and R is amount of leisure.
If the individual uses all her time (=1) to work, she receives the real wage w.
In this case, R=0, and number of kids (K)=0.
Regarding the time required for raising a kid:
One kid requires 20 percent of the individual’s time available. Two kids require 40 percent of
the individual’s time available, 3 kids 60 percent, 4 kids 80 percent and 5 kids 100 percent.
(so in this problem the individual cannot raise more than 5 kids).
Apart from the time constraint of the individual, he has a budget constraint: C = wL + N,
where C= quantity of consumption good. N real non-labor income; that is, non-labor income
in terms of the consumption good. wL= real labor income.
(The price of the consumption good = 1.)
Assume the utility function:
U  C a  R b  K 1 a b
where 0  a  1, 0  b  1 ,0< (1  a  b) <1.
A1. Derive the optimal demands of C, R, L and K as functions of the exogenous variables: w,
N, and of the parameters a, b.
A2. What happens to optimal R and K and L and C when N increases.
A3. What happens to optimal R and K and L and C when w increases.
Students unable to derive optimal demands should nevertheless be able to answer A2, and
possibly also A3.
Instruction how to solve this problem: In case of a Cobb-douglas utility function (like the one
above), the share of expenditures on each good equals that good’s preference parameter in the
utility function. For example: If the individual maximizes U ( x, y )  x   y 
subject to the budget constraint: p x  x  p y  y  I
where y= quantity of good y, x=quantity of good x, p x = price of good x. p y = price of good
y. I = income. The optimal demand of x and y are such that:
 I
 I
px  x*

 x* 
if   1    x * 

(   )  p x
px
I
 
p y  y*


 y* 
 I
(   )  p y
if   1  

y* 
 I
I
 
py
If you use this information you do not have to do the actual mathematical calculation (based
on the Lagrangetechnique) to come up with the right answers.
Answers4:
A1.
L+R+0.2*K=1L=1-R-0.2*K.
C=w*L+N
C=w*(1-R-0.2K)+N
C+wR+0.2wK=w+N
C/I=a
C=a(w+N)
wR/I=b
R=b(w+N)/w=b+(N/w)
0.2wK/I=(1-a-b)
K=(1-a-b)(w+N)/0.2w=(1-a-b)/0.2+(1-a-b)N/0.2w.
L=1-R-0.2K.
A2. If N increases R and K will increase and L will decrease.
A3. Cup, Rdown, K down, L up
2+2+1+1+2+1+2= 11 points
Question 5. Suppose that the world consists of only two countries. Savum and Spendum. The
following values apply to the cobb-douglas production functions and utility functions of each
country. Use the values in the table to answer the questions below.
Parameter
A


N
Savum
4
0.5
0.5
100
Spendum
4
0.6
0.5
100
b. calculate the steady state values for capital per worker (k=K/L), production per worker
(y=Y/L), the real wage per worker (w), and the real return on capital (r) for the two
countries: Savum and Spendum under autarky; that is, when the countries are closed
with respect to each other (=autarky).
.
If you cannot answer with numbers, explain in words which country has higher or lower
values on the variables mentioned above for partial credit.
b. calculate steady state values of GDP (Y), of consumption when young ( c y ), of
consumption when old ( co ), and of utility (U) for Savum and Spendum.
c. Calculate the steady state values of y, k, w, and r for each country under free trade; that is,
when capital mobility is allowed for. Help: The world-wide value of  equals
N spendum
N savum

  savum 
  spendum
N savum  N spendum
N savum  N spendum
d. From what country will capital move? In other words, which of the countries will end up
owning assets (=capital) in the other country?
e1. calculate steady state values of Y , c y , co , U for the two countries under free trade.
e2. Compare the steady-state utility of a young person in Spendum under autarky with the
steady-state utility of a young person in Spendum under free trade.
Is it higher or lower or unchanged?
e3. Compare the steady-state utility of a young person in Savum under autarky with the
steady-state utility of a young person in SAVUM under free trade.
Is it higher or lower or unchanged?
f. During the transition to the new steady state which group (current young or current old)
gain or loose in the two countries?
g. Now suppose that the world consists of the countries. Savum and Efficient. (Note the
country Spendum does no longer exist.) Country Efficient is identical to Savum except that A
equals 9 in country efficient. Thus, Efficient have the following parameter values:
A=9,  =0.5,  =0.5, N=100.
If we allow for capital mobility between the country Savum and the country efficient, from
what country will capital move? Explain why?
Answers5: A+B
K=((1-a)*(1-b)A)^(1/1-b)
Y=Ak^0.5
W=(1-b)*y=0.5*y
r=A*b*k^-0.5
SAVUM
1
4
2
2
SPENDUM
(0.4*0.5*4) )^2=0.64
3.2
1.6
2.5
Y=y*N
Cy=a*w
Co=(1+r)(1-a)w
U=cy^a*co^(1-a)
400=200+200
1
3
3^0.5=1.73
320
0.96
3.5*0.4*1.6=2.24
U=0.96^0.6*2.24^0.4=0.98*1.38=1.3524
answers5: C.-e.
K=((1-a)*(1-b)A)^(1/1-b)
Y=Ak^0.5
W=(1-b)*y=0.5*y
r=A*b*k^-0.5
Y=y*N
WORLD
(0.45*0.5*4) )^2=0.81
3.6
1.8
2.2222
720
Cy=a*w: spendum
Co=(1+r)(1-a)w: spendum
U=cy^a*co^(1-a): spend.
0.5*1.8=0.9
3.222*0.5*1.8=2.9
U=0.9^0.5*2.9^0.5=0.95*1.70=1.615
Cy=a*w: spendum
Co=(1+r)(1-a)w: spendum
U=cy^a*co^(1-a): spend.
0.6*1.8=1.08
3.222*0.4*1.8=2.32
U=1.08^0.6*2.32^0.4=1.05*1.40=1.47
d. Savum will end up owning capita in spendum
e. Steady state utility goes down in savum and up in spendum.
f.during the transition: in savum: workers loose and old people gain.
In spendum: workers gain and old people loose.
g. no capital mobility as r is the same in the 2 countries.