The Solow-Swan growth model

The Solow‐Swan growth model 1. Description of the model The economy evolves in discrete time. Time is measured in periods, denoted by , and indexed by integers: ∈ 1, 2, 3, … . There is only one good in each period. The good is produced. The amount of good in is . Its price is normalized to 1 in every . There are two types of agents: consumers (individuals or families) and firms. Consumers are globally characterized by a constant saving rate : in total, each , consumers save a fixed proportion of . Consumers supply labour inelastically in a competitive market in exchange for a wage rate . Labour is identified with employment. The total supply of labour (total employment) in period is and is assumed to grow at a constant rate . Firms are represented by a profit‐maximizing aggregate firm that produces the good using an aggregate production function ,
,
exhibiting constant returns to scale [alternative interpretation: all firms have access to the same (free) production technology]. is the capital stock: good used to produce more good. The good can be consumed by consumers or used in production by firms. represents the state of technology in period . It is defined by a number but it has no natural units. It is rather an index for a family of production functions that depend on just and . captures everything affecting the way of using and efficiently. The production function is assumed to have the following properties. (i)
It takes non‐negative values: 0. (ii)
It is twice differentiable with respect to and . (iii)
Positive marginal productivity of and : (iv)
Decreasing marginal productivity of and : (v)
It exhibits constant returns to scale (is homogeneous of degree 1) in and [ is , , ]. homogeneous of degree in and if, for all 0 and , , ,
≔
0 and ≔
The above assumptions make concave. They also imply that Solow-Swan growth model ǀ 16 December 2014 ǀ 1 ≔
0 and , ,0
0. ≔
, 0,
0. 0. Euler’s homogeneous function theorem Euler’s adding‐up theorem . If , , is twice differentiable with respect to and , and homogeneous of degree in and , then: (i)
for all , , , , ,
; (ii)
and are homogeneous of degree 1 in and . If 1, then , ,
means that output is exhausted if factors of production are paid according to their marginal productivities. All markets (for output , capital , labour ) are competitive. The price of and in period are denoted, respectively, by and . The capital stock depreciates at a constant rate 0: 1 unit of in period , becomes 1
units in period 1. The (real) interest rate is = . If one unit of the good is used as capital at the start of time , then, at the end of , its marginal productivity (equal to , see below) is obtained, but a fraction of the capital is lost. Firms maximize profits and can be represented by a unique firm with production function . Hence, the firms’ decisions in period are determined by the solution to the problem , ,
, where ,
, and are given and the price of output in is normalized to 1. As is concave, by the first order condition for maximization, and . By Euler’s theorem, , so profits are zero. The Inada conditions (after Ken‐Ichi Inada) state that (a) lim →
∞ and lim →
0 (b) lim →
∞ and lim →
0. Interpretation: the initial units of an input are highly productive but, for a sufficiently high amount of input, additional units are unproductive. Inada’s conditions are helpful to ensure interior solutions (
0 and 0). They are not satisfied by production functions that are linear ). in or ( like, for instance, Fig. 1 next depicts what may be termed the economy’s life cycle. Solow-Swan growth model ǀ 16 December 2014 ǀ 2 Fig. 1. B
Basic descrription of th
the econom
my represen
nted by thee model Thee macroeco
onomic eq
quilibrium condition
n is or , wh
here . Thee capital sto
ock accum
mulation condition is 1
1
. To ssum up, th
he model iss given by the follow
wing equatiions (when
n the first ffour are insserted into
o the lastt one, the laast equatio
on summarrizes the w
whole modeel).  Aggreg
gate produ
uction funcction ,
,  Uses off aggregatee output  Aggreg
gate saving
gs function
n , 0
1 ( aand, therefo
ore, 1
)  Macroeeconomic eequilibrium
m  Capitall stock accu
umulation
n ∆
ven populaation , an initial capital stock 1 , and ttechnology
y representted by , a
an equilibrrium Giv
path
h for the ecconomy is a sequencce ,
,
,
,
such thatt: (i)
1
,
,
1
(ii)
,
, (iii)
1
(iv)
(v)
. Kno
owing the sequence 1 ,
2 ,
3 ,…,
… of capital sstocks, the rest of varriables can
n be deteermined. H
Hence, solv
ving the m
model is red
duced to deetermining
g that sequ
uence. Solow-Swan grow
wth model ǀ 16 December 2014 ǀ 3 2. Basic analysis of the model It is more convenient to express the model in per capita terms. Define capital per capita (or capital‐
labour ratio) as and output (or income) per capita as
. By constant returns, 1
,
,
,
,
,
,1
This says that output per capita is a function of capital per capita. Therefore, . Given that , ′
1
′
′
. Consequently, the marginal product of coincides with the price of capital. And since follows that By assumption, 0, it 0: the output per capita function is increasing. ≔
0. In view of this, as ′
0
≔
, 1
′′
which implies 0. Accordingly, is a concave (increasing) function. By Euler’s theorem, . By the profit maximizing assumption, and . As just shown, ′
. Summarizing ,
,
′
. If both sides are divided by , ′
. Solving for yields ′
The dynamics of capital accumulation is represented by the equation 1
,
,
1
. If both sides are divided by , 1
1
1 Solow-Swan growth model ǀ 16 December 2014 ǀ 4 whiich represeents the dy
ynamics of capital p
per capita accumulat
a
ion. Equattion 1 su
ummarizess the mod
del when tthere is neiither popu
ulation grow
wth nor technologica
al progresss. A steady‐state equilib
brium (witthout pop
pulation growth g
no
or technollogical pro
ogress) iss an equ
uilibrium p
path wheree, for some and all , (capital p
per capita rremains constant). Bein
ng constaant, having
g constan
nt implies that is co
onstant (he
ence, and
d are also
o constant). Stea
ady‐state eequilibria aare obtaineed from 1 by setting
g 1
. Thus, 0 satisfiees .. 0 such that 0
Pro
oposition 11. Given thee assumptioons on , thhere is a un
nique capiita and conssumption peer capita ̅
1
Fig.. 2 represen
nts 1 and
d the only steady‐staate equilibrrium , w
with outputt per . 0. Fiig. 2. Graphical repreesentation of the mod
del and its steady‐staate equilibrria On the one hand, as sshown below, any other Thee value ssynthesizess the soluttion of thee model. O
possitive valuee for is un
nstable. An
nd, on the other, the rest of varriables can n be determ
mined once is kno
own. Speccifically, , ̅
, , , ̅
̅
, ̅
̅
̅
… Fig.. 3 next prrovides an
n alternativ
ve graphiccal represeentation off the mod
del, based on display
ying separately thee (per capiita) producction functtion , the savings fu
unction fun
nction .. Solow-Swan grow
wth model ǀ 16 December 2014 ǀ 5 , and thee depreciaation Fig. 3. A
Alternativee graphicall representtation of th
he model an
nd its stead
dy‐state eq
quilibria In a a steady‐sttate equilib
brium, thee curves reepresenting
g the new capital (
) and th
he lost cap
pital (
) intersecct. This inteersection is point in
n Fig. 3. Fig.. 4 illustrattes the posssibility of non‐existeence and no
on‐uniqueness of thee non‐trivial steady‐sstate equ
uilibrium. IIn case , fails to
o be concaave (margiinal produ
uctivities n
need not be b decreasiing). Sev
veral interior steady states may
y exist. Caase sho
ows that the failure of the Ina
ada conditiions may
y be lead to
o the non‐eexistence o
of interior ssteady stattes. Fig. 4. Failure off uniqueneess and existence Pro
oposition 22. Given thee assumption
ns on : , which correspondds to (i)
the steady‐statte equilibriu
um of 1
1
, is globaally asymptootically stabble, which m
means that conve
verges to ; aand (ii)
for aall 1
0, coonverges moonotonically
y to . Hen
nce, the dy
ynamics of capital acccumulation
n is similarr to the dyn
namics in tthe OLG m
model. Solow-Swan grow
wth model ǀ 16 December 2014 ǀ 6 Fig.. 5 shows h
how capitaal per capitta convergees to when the initiial value 1 of capiital per cap
pita is sm
maller than
n . A sim
milar reason
ning appliees when th
he initial va
alue is largger than . Fig. 5. Converrgence to th
he interior steady‐sta
ate equilibrrium It iss worth no
oticing thatt, if 1
, then th
he sequence of wage
es is increa
asing, whereas is a decreasing
g sequencee (↑ ⇒↓ ⇒↓ ). W
When 1
, conv
vergence to
o means that cap
pital accum
mulates ( iincreases). Since ↑ implies ↑ and ↓ ′, it follows s from ′ that fa
alls. Intuitiively, moree makes labour mo
ore productive, so rrises. If 1
, ccapital decumulatess, which im
mplies tha
at is decrreasing an
nd is incrreasing. Thee following
g comparattive staticss analysis rrelies on th
he fact thatt is a d
decreasing function o
of , sincce 0.  Efffect of a ch
hange in th
he deprecia
ation rate: ↑
Giveen  Efffect of a ch
hange in th
he savings rate: ↑
,↑
⇒↓ .
⇒↑ ⇒↑
⇒↓
⇒
. ⇒
⇒↑ . Giveen ,↑
⇒↓ ⇒↓
⇒↑
⇒
. hift upwarrds of to yields↑ .  Efffect of a teechnological improveement: a sh
Giveen , so↑ is n
needed to g
get ↓ folllows from
m . Therefore
.
e, Solow-Swan grow
wth model ǀ 16 December 2014 ǀ 7 , . Lettting ∆
≔
1
, equation 1 can be eq
quivalently
y expresse d as ∆
Theen the grow
wth rate of .
is ∆
. 0 ( aaccumulatees) if and o
only if Theerefore, imp
plies that . Wiith 1
gence to , converg
falls an
nd goes to 0. Fig. illu
ustrates this. Fig. 6. A
Alternative illustration
n of the con
nvergencee to the inte
erior steadyy‐state equ
uilibrium
The golden
n rule of caapital accu
umulation
3. T
Stea
ady‐state p
per capita cconsumptiion ̅, defin
ned as a fun
nction of th
he savingss rate , is ̅
≔ 1
. Thee golden ru
ule savingss rate ∗ ma
aximizes stteady‐statee per capita consump
ption ̅. ̅
0
′
′
. ∗
Theerefore, ∗ ssatisfies ′
and
d the steady
y‐state con
ndition . The valu
ue ∗
∗
∗
∗
ca
an be comp
puted from
m ′
∗
∗
, then satisfies ′
. Given , whiich is the sshare of iin (in the steady‐staate equilibrrium). Fig.. 7 illustrattes this. Solow-Swan grow
wth model ǀ 16 December 2014 ǀ 8 ∗
(i)
(ii)) Fig. 7. Thee golden ru
ule solution
n Thee golden ru
ule can be interpreted as deriviing from a
an equilibrium in retu
urns. Conssider capital and
d labour aas two asseets producced in the eeconomy.
Thee rate of rreturn of capital can
n be defin
ned as : the productivit
p
ty of capital minus the dep
preciation o
of capital. Thee rate of retturn of lab
bour can bee taken to b
be zero: po
opulation d
does not grrow. If both returns are to be the same, then 0; thatt is, . An economy iis dynamiccally inefficient if perr capita con
nsumption
n can be inccreased by
y reducing
g the cap
pital stock ((someone iis better off, and no o
one worse off, with le
ess capital)). ∗
Pro
oposition 33. If , then (at thee correspondding steady
y state) the eeconomy is dynamicallly inefficient. Cap
pital overaaccumulatio
on (point in Fig. 7((i)) generattes dynamic inefficieency ( doees not). If is ∗
reduced to , per capiita consum
mption is h
higher du
uring the transition tto the new
w steady state s
(immediate efffect) and aat the new steady staate (perman
nent effect). Dynamics o
of capital aaccumulattion with p
population
n growth
4. D
Let 1
1
. The ccondition ffor capital stock accumulation iis, as beforre, 1
1
. 1 , viding both
h sides by Div
1
1
Tha
at is, 1
or 1
1
1
1
1
Solow-Swan grow
wth model ǀ 16 December 2014 ǀ 9 . ∆
In a
a steady staate, ∆
1
. 1
0. Therefore, capitaal per capiita in a steady statee satisfies . Desspite the prresence of the denom
minator 1
;; see Fig. 8.. The grow
wth rate , can b
be calculate
ed by equaating of is now; see Fig. 9. 1
1
. Fig
g. 8. Steady
y‐state equ
uilibrium w
with population grow
wth Fig. 9. Growth raates with population growth Solow-Swan grow
wth model ǀ 16 Deecember 2014 ǀ 10
0 with In a
a steady staate, no mattter wheth
her 0 orr 0, peer capita va
ariables rem
main consttant: , , … Butt, when 0, aggreg
gate variab
bles , , … also rremain con
nstant, becaause is co
onstant: and co
onstant imply consstant, and
d constan
nt imply constant, etc. Wh
hen 0, / co
onstant and
d growin
ng at rate imply g
growing att rate . By
y constant retu
urns, if aand grow
w at rate , then also
o grows att rate . To d
determine the golden
n rule solu
ution with p
population
n growth, iin a steadyy state, consumption per cap
pita is ̅
≔ 1
. Thee golden ru
ule savingss rate ∗ ma
aximizes stteady‐statee per capita consump
ption ̅. ̅
0
′
′
Theerefore, ∗ ssatisfies ∗
′
. ∗
can be ccomputed ffrom . Usin
ng ∗
an
nd the stea
ady‐state coondition ∗
∗
∗
it fo
ollows thatt ∗ satisfiees ∗
Thee situation is analogo
ous to the o
one arising
g when ′
. 0; see Fig
g. 10. Fig
g. 10. The g
golden rulee solution w
with population grow
wth Solow-Swan grow
wth model ǀ 16 Deecember 2014 ǀ 111 5. Dynamics of capital accumulation with population growth and technological progress For a production function of the sort , , , there are three basic types of technological progress.  Hicks‐neutral technological progress means that , , can be expressed, for some , as , . A technological change just relabels isoquants (no change in shape).  Solow‐neutral (or capital‐augmenting) technological progress means that , , can be written as , , for some . Higher is equivalent to having more .  Harrod‐neutral (or labor‐augmenting) technological progress makes , , equal to ,
, for some (higher is like more ) Suppose population grows at rate and technology accumulates at a constant rate : 1
1
and 1
1
. Technological progress is assumed to be Harrod‐neutral. Define per capita variables in terms of efficiency units of labour (also called “effective labour units”): . The capital stock accumulates following the usual rule 1
1
After dividing both sides by 1
1 , 1
1
. 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
or, equivalently, ∆
1
1
. 0. Accordingly, capital per capita in a steady state satisfies 1
1
In sum, In a steady state, ∆
1
. Solow-Swan growth model ǀ 16 December 2014 ǀ 12 Desspite the p
presence of o 1
witth 1
1
1
iin the den
nominator,, can bee obtained
d by equaating ; seee Fig. 11. Th
he growth rate oof is n
now 1
1
1
1
. 2 Fig. 111. Steady‐sttate equilib
brium with
h populatio
on growth and techn
nological prrogress Tecchnologicall progress produces the follow
wing effectt: as con
nverges to , output per efficieency unit of labourr also con
nverges (to
o some ), so output per efficien
ncy unit dooes not gro
ow in the llong run
n. Formally
y, is ev
ventually constant ( ). Denotiing growth rates byy , that is consstant imp
plies that 0. That is, . Summing up, ⁄
. Outtput per p
person gro
ows at thee same raate as tech
hnology: technologiccal progreess offsets the dim
minishing rreturns to ccapital. Sum
mmarizing
g, the Solow
w‐Swan mo
odel exhib its the follo
owing characteristicss.  There is a unique (no
on‐trivial) steady sta te.  The steady
y state is assymptotica
ally stable and comparative sta
atics is simp
ple. no sustaineed growth:: growth o
occurs only
y when the
e economyy shifts fro
om one steeady  There is n
state to an
nother stead
dy state wiith a higheer . hnological change.  Sustained growth requires sustained tech
of output (llong term g
growth) iss not affected by : it only affectts the levells of  The rate of growth o
, , and (more saavings only
y depress th
he margin
nal producttivity of cap
pital). Solow-Swan grow
wth model ǀ 16 Deecember 2014 ǀ 13
3 6. C
Convergen
nce Con
nvergence (or catch‐‐up effect) is the hy
ypothesis according a
to which there is a a tendency
y for output per cap
pita in poo
orer econom
mies to gro
ow faster tthan outpu
ut per capitta in richerr economiees. Abssolute (inconditionall) convergeence holdss that the ccatch up occcurs regarrdless of th
he characteeris‐
tics of the economies. Th
here are tw
wo basic re asons to ju
ustify the existence off absolute convergen
nce. (i)
Technological trransfer. Th
he “advanttage of bacckwardnesss”: poorerr economy
y can repliicate techno
ology, prod
duction methods, m
an
nd instituttions develloped in th
her first place p
by riccher econom
mies witho
out having
g to bear th
he cost of crreating or developingg them. (ii)
Capitaal acumulaation. Incrreasing acccumulatio
on from a a low leveel has a bigger b
imp
pact becausse diminish
hing return
ns in pooreer economies are nott as strong as in richeer economiies. Thee empiricall evidence does not y
yet offer ev
vidence for absolute convergen
nce (at leasst Africa iss not catcching up w
with the reest of the world); seee Fig. 12(a
a) and 12(b). For con
nvergencee to occur in a giveen time in
nterval, it is to be exp
pected a n
negative reelationship between output per capita att the starrt of the intterval and the averag
ge growth rate of outtput per ca
apita in thaat interval. Fig. 12. Co
onditional convergen
nce (MP To
odaro, SC S
Smith (2011): Econom
mic developm
ment, p. 80)) Problem of sselection biias: converrgence is teested for rich countrie
es so, for th
hose who w
were initiaally relatively
y poor, it iss plain thatt they had to grow fa
aster than tthe rich cou
untries inittially rich.
Solow-Swan grow
wth model ǀ 16 Deecember 2014 ǀ 14
4 Thee Lucas paaradox (Ro
obert E. Lu
ucas, Jr, 19990) referss to the cla
aim that ccapital is apparently
a
not flow
wing from richer to p
poorer desspite the faact that po
oorer econo
omies hav
ve lower leevels of cap
pital per capita, wh
hich should correspo
ond to high
her returnss on capita
al and, therrefore, ma
ake investm
ment morre attractiv
ve in poorrer countriies. Recentt research suggests that, t
for th
he period 1970‐2000, the lead
ding explan
nation for the lack off capital flo
ows to poo
orer countrries is low iinstitution
nal quality. Fig. 13. C
Convergencce as a funcction of cou
untry size P Todaro, SC Smith ((2011): Economic development, p.. 82) (MP
of conditional converrgence meaans that po
oorer econ
nomies grow
w faster if they havee the Thee concept o
sam
me technolo
ogy and fu
undamenta
al variabless for capita
al accumulation , ,, as rich
her econom
mies. Con
nditional cconvergen
nce occurs in the SSS model: economiess with thee same technology and fun
ndamental variables cconverge tto the sam
me steady sstate (simillar countrie
ies converg
ge). In factt, by 2 , , the SS m
model predicts that poorer p
cou ntries wou
uld grow faster f
than
n richer ecconomies. In a poo
or economy
y, is com
mparatively
y small. Th
herefore, / is h
high. Given
n the rest of parameeters ( , , , and ), high / impliees, by 2 , h
high . Th
his, in turn
n, implies h
high . Butt since marrginal prod
ductivities are decreaasing, grow
wth cannott last. In th
he SS modeel, (per cap
pita) growth occurrs only wh
hile the ecconomy co
onverges to t the stea
ady state. Once theere, per caapita variables rem
main stagnaant. The em
mpirically observed convergen
nce has beeen lower tthan prediccted by tthe SS mod
del. Conveergence: OE
ECD, Asian
n tigers. Divergence: Africa. Solow-Swan grow
wth model ǀ 16 Deecember 2014 ǀ 15
5 According to the hypothesis of conditional convergence, output per capita in an economy converges to a certain level dictated by the structural characteristics of the economy. Given that what guides convergence is not the initial value of output per capita, there is no need to transfer income from richer to poorer nations: foreign aid should concentrate of establishing the necessary structural characteristics (related to, for instance, the ability to absorb new technology, attract capital and participate in global markets). The concept of strong conditional convergence expresses convergence conditional on having the same determinants of capital accumulation: , , and . It relies on the presumption that any technology is eventually available to all economies. What separates economies is capital accumulation not technology. The term sigma‐convergence refers to a reduction in the dispersion of output per capita levels across economies. Beta‐convergence is said to occur when poorer economies grow faster than richer ones. Conditional beta‐convergence means beta‐convergence conditional on certain variables held constant. Inconditional (or absolute) beta‐convergenceʺ takes place when the growth rate of an economy declines as it approaches its steady state; see http://en.wikipedia.org/wiki/Convergence_%28economics%29. Convergence in terms of rates contends that differences in growth rates (mainly, ) tend to vanish. It does not imply convergence in the level of output per capita. There is convergence in rates in the SS model between economies with the same technology: the same rate of technological change yields the same growth rate of output per capita (convergence club = those who innovate). The expression club convergence captures the observation that one can identify groups of economies with similar growth trajectories. In particular, there are poor (low income) economies that simultaneously experience low growth rates. Certain structural barriers (limited access to education, lack of resources, poor infrastructure) may prevent a poor economy from migrating to a richer convergence club. Consequently, it may prove extremely difficult to move from one to another convergence club. The SS model displays a negative correlation between and the speed of convergence. The lower , the more decreasing returns to capital are. This means that, to converge, capital should accumulate faster. With more capital, rates of returns are lower. Capital migrates to poorer economies, where rates of return are higher. Through diffusion of knowledge, poorer economies can adopt technological advances developed in richer economies. Solow-Swan growth model ǀ 16 December 2014 ǀ 16 Fig
g. 14. Conv
vergence cclub, Europ
pe, 1990‐20
005 http:///www.ub.edu/sea20009.com/Pa
apers/129.p
pdf 7. S
Stylized faccts of grow
wth Thee following
g are “Rem
markable h
historical co
onstanciess revealed by recent eempirical investigatiion” acco
ording to N
Nicholas K
Kaldor (195
57, 1961). 1. / grrows (at a rroughly co
onstant ratee) 2. / grrows contiinuously (ffollows fro
om 1 and 4)) of return on capital stable oveer the long run 3. Rate o
4. / has been rou
ughly stab
ble for long
g periods 5. The sh
hares of aand in ou
utput havee remained
d approxim
mately consstant wage grows over timee (follows from 2, 4, a
and 5) 6. Real w
7. Outpu
ut and prod
ductivity g
growth difffer substan
ntially acro
oss countriees Alteernatively::  “Laborr productiv
vity has gro
own at a su
ustained ra
ate.  Capitall per work
ker has also
o grown at t a sustaineed rate.  The reaal interest rrate or retu
urn on cap
pital has beeen stable.
 The rattio of capittal to outpu
ut has also
o been stable.  Capitall and laborr have capttured stablle shares o
of national income.  Among
g the fastt growing
g countriees of the world, th
here is an
n apprecia
able variaation in the rrate of grow
wth “of thee order of 2–5 percen
nt.” Solow-Swan grow
wth model ǀ 16 Deecember 2014 ǀ 17
7 Pau
ul Romer (11989) sugg
gested more facts. 8. In cross‐seection, the mean grow
wth rate sh
hows no va
ariation with income per capita
a levels. 9. The rate off growth o
of inputs is insufficien
nt to expla
ain the grow
wth of outp
put. 10. Growth in
n the volum
me of tradee is positiveely correla
ated with growth in ooutput. 11. Population
n growth rrates are neegatively ccorrelated w
with the level of incoome. 12. Skilled and
d unskilled
d workers tend to miigrate towa
ards high‐iincome cou
untries New
w facts prroposed by
y Charles I. Jones aand Paul Romer R
(20
009); see h
http://www
w.stanford..edu /~ch
hadj/Kaldo
or200.pdf. 8. E
Exercises Ejerrcicio 1. T
Teorema de d Euler. Demuestraa el teorema de Eu
uler. [Sugeerencia: pa
artiendo de d la defiinición dee homogen
neidad de grado , deriva am
mbos lados de la eccuación co
on respecto al parámetro y considera el valor = 1; p
para la seg
gunda parte del teorrema, deriiva ahora con resp
pecto a y
y divide po
or .] Solow-Swan grow
wth model ǀ 16 Deecember 2014 ǀ 18
8 Ejercicio 2. Cobb‐Douglas. Considera la función Cobb‐Douglas , ,
0 y 0
1. (i)
Comprueba si se satisfacen todas las condiciones impuestas sobre la función de producción en modelo (productividades, rendimientos, Inada). (ii)
Definiendo y , obtén la expresión correspondiente (esto es, (iii)
, con determina ). Asumiendo mercados competitivos de y , indica la fórmula que expresa en términos de y la que expresa en términos de . Ejercicio 3. Función demuestra que / . En el caso sin crecimiento de la población ni progreso tecnológico, es una función decreciente de . Ejercicio 4. Precios de los factores. (i) Suponiendo que no hay crecimiento de la población ni progreso tecnológico, explica porqué, si 1
, la secuencia de salarios es una secuencia creciente y la secuencia de precios del factor capital es decreciente (
0 es el valor de en el estado estacionario). (ii) Muestra que es decreciente y creciente si 1
. Ejercicio 5. Modelo de Solow y Swan. Considera el modelo de Solow y Swan sin crecimiento de /
la población ni progreso tecnológico con , tasa de depreciación 0’1 y tasa de ahorro 0’4. (i)
Calcula los valores de la producción per cápita, el consumo per cápita, la inversión per cápita y la depreciación per cápita en el estado estacionario. Señala estos valores en una representación gráfica del modelo. (ii)
Indica en la representación gráfica cómo variarían esos valores si (a) la tasa de depreciación aumentara; (b) la tasa de ahorro aumentara; (c) se produjeran simultánemante (a) y (b). (iii) Obtén la tasa de ahorro y el consumo per cápita correspondientes a la regla de oro. ¿Es la economía dinámicamente ineficiente si inicialment es 20? (iv) Representa gráficamente la tasa de crecimiento de en función de . (v)
Respón a las preguntas anteriores si la tasa de crecimiento de la población (en tanto por uno) es 0’1. Ejercicio 6. Análisis gráfico de estática comparativa. Muestra gráficamente el efecto sobre el capital per cápita de estado estacionario (así como de la producción per cápita y el consumo per cápita de estado estacionario) si: (i) aumenta ; (ii) aumenta ; (iii) tiene lugar progeso tecnológico; (iv) ocurren al mismo tiempo (ii) y (iii); (v) ocurren al mismo tiempo (i) y (ii); (vi) disminuye y aumenta ; y (vii) se produce un regreso tecnológico al mismo tiempo que disminuye . Ejercicio 7. Progreso tecnológico. Verifica que los tres tipos de progreso tecnológico neutral (en el sentido de Harrod, de Hicks y de Solow) son equivalentes en la función de producción Cobb‐
Douglas , ,
, con 0 y 0
1. Solow-Swan growth model ǀ 16 December 2014 ǀ 19 Ejercicio 8. Progreso tecnológico y crecimiento de la población. Considera el modelo de Solow con crecimiento de la población y progreso tecnológico con función de producción /
/
. La población crece a la tasa constante
0 y la tecnología se acumula a la tasa constante 0, de manera que 1
1
y 1
1
. Define el capital per cápita como como
y la producción per cápita como . (i)
Determina la ecuación en diferencias que establece la dinámica del capital per cápita, la fórmula del capital per cápita en el estado estacionario y la fórmula de la tasa de ahorro que cumple con la regla de oro. (ii)
Respón a las mismas preguntas si . Ejercicio 9. Función de producción. Verifica que, para una función de producción Cobb‐Douglas , 0. Prueba que, para la expresión per cápita de , ′ y ′. Exercici 10. Pago a los factores. Demuestra que, en la siguiente gráfica, cuando el stock de capital per cápita es ǩ (la distancia ), entonces la distancia representa el salario y la distancia representa el beneficio por trabajador ǩ. …
ǩ Exercici 11. Modelo de Solow y Swan. Considera el modelo de Solow y Swan sin progreso tecnológico en el que la población crece a la tasa (neta) pero con la siguiente variante: la tasa de ahorro es si el stock de capital per cápita es superior a 3 y es si el stock de capital per cápita es igual o inferior a 3. La función de producción per cápita es depreciación es . /
. La tasa de (i)
Calcula los valores de todas las variables endógenas en todos los estados estacionarios en los que el capital per cápita sea positivo. (ii)
Indica los valores obtenidos en (i) en una representación gráfica del modelo y explica si los estados estacionarios son estables. Solow-Swan growth model ǀ 16 December 2014 ǀ 20