Macroeconomics+

ECON+321+

Solow+Growth++Prac:ce+Problems+

1.+If+Y!=!AKαL1(α,!!and+we+assume+that+the+popula:on+=+the+labor+force+(L),+find+GDP+per+capita+in+terms+of+A,+α,+and+the+capital+to+labor+ra:o.+Take+the+natural+logs+and+:me+deriva:ves+to+find+the+growth+rate.++Explain+the+growth+rate+in+terms+of+its+components.+

2.+In+a+graph+with+k+on+the+xQaxis+and+y,+c,+and+i+on+the+yQaxis,+show+the+loca:on+of+the+steadyQstate+capitalQlabor+ra:o,+k,+steadyQstate+investment+i,+steadyQstate+consump:on+c,+and+steadyQstate+output+y+(all+in+per+capita+terms).++

Solow+Growth+Model+

–  We+really+care+about+per+capita,+so+lets+simplify+things+and+assume+the+popula:on+is+L.!

y!=!Akα!!where+++++++++++++!!is+the+capital'to'labor'ra+o.!!

The+beauty+of+LogQLinear+Approxima:ons+ln(y)!=!ln(A)!+!αln(k)!

gy=gA+αgk!

€

YL

=AKαL1−α

L

€

k =KL

€

d ln(y)dt

=d ln(A)dt

= αd ln(k)dt

€

Y = AKαL1−α

Capital+and+Growth:+Diminishing+Marginal+Returns+

Marginal+Product+of+Capital+(MPK)+

Aggregate+Investment+or+Change+in+Capital+Stock+

•  ΔK!=!I!–!δK!– ΔK!is+Net+Investment!

– Where+I!is+Gross+investment+– δ!is+the+deprecia:on+rate:!0<δ<1!

•  I!=!S!;!or!Investment+equals+Savings+with+market+clearing.+

•  I!=!sY!=!sAKαL1(α!– Where+s!is+the+savings+rate!

Per+Capita+

•  i+=+sAkα!

•  !Δk!=!i!–!(n+δ)k!– Where+n+is+the+popula:on+growth+rate+

•  Δk!=!sAkα!–!(n+δ)k!!– Where+(n+δ)!is+effec:ve+deprecia:on!– With!k=K/L,!as!L!!!!!!!!!k!!!!!!!

The+NeoQClassical+/+Solow+Growth+Model+

y,i,c++

y!=!Akα!+

(n+δ)k+

i!=!sAkα!+

k+Kss*+

So…growth+depends+on+Total+Factor+produc:vity+and+and+the+Capital+to+Labor+Ra:o.++The+Capital+to+Labor+Ra:o+can+be+increased+by+Investment,+which+equals+savings,+which+equals+the+savings+rate+mul:plied+by+produc:on.++Growth+is+constrained+by+effec:ve+deprecia:on.+

y+

i+

yQi=c++

Steady+State+y,i,c++

y!=!Akα!+

(n+δ)k+

i!=!sAkα!+

k+Kss*+

There+is+a+natural+tendency+to+move+to+the+steady+state.++If+there+is+too+much+capital,+we+can’t+keep+up+with+it+all+and+it+the+extra+will+depreciate+away.++If+there+is+not+enough+capital,+we+will+keep+growing.+

Klow+

ylow+

Khigh+

y*+

yhigh+

Solve+for+Steady+State+

•  NoHce:!!gy=gA+αgk!!!

So+if+A+doesn’t+grow+(+i.e.+no+produc:vity+growth),+then+we+don’t+grow.+

€

kss =sAn +δ

#

$ % &

' (

11−α

€

kss1−α =sAn +δ

€

sAkα ss = (n +δ)kss

€

Δkss = 0 = sAkα ss − (n +δ)kss

Growth+Rate+of+Capital+

!!!gk!!!!+

!!!k!!!!+

sAkα(1+

The+difference+is+the+growth+rate.+

So….+!gk!>0!when!k<kss!gk!<0!when!k>kss+

!!!kss!!!!+

€

Δkss = sAkα ss − (n +δ)kss

€

Δksskss

= sAkα−1ss − (n +δ)

Produc:on+Func:on:+NeoQClassical+Growth+Model:++Assume+L=Pop+

Convergence+y,i,c++

y!=!Akα!+

(n+δ)k+

i!=!sAkα!+

k+Kss*+

Uncondi+onal'Convergence:'The+tendency+of+real+GDP+per+capita+in+poor+countries+to+grow+faster+than+in+rich+countries,+so+that+poor+economies+will+catch+up+over+:me+to+richer+economies.'

kUK+kBrazil+

Uncondi:onal+Convergence+Doesn’t+Hold+Up+with+the+Empirics+

gy++++++++++++++++++++++++++THEORY+ +++++++++++++++++++++++++Actual+

Ini:al+GDP+per+capita+

Why+it+didn’t+doesn’t+hold+up+

Why+it+doesn’t+hold+up+

Why+it+didn’t+hold+up+

– Because+if+the+savings+rate+and+produc:vity+are+different,+then+we+have+different+steady+states.++As+well,+differing+popula:on+growth+rates+and+deprecia:on+rates+will+affect+growth.+€

kss =sAn +δ

#

$ % &

' (

11−α

Let’s+Tweek+It+

•  Condi+onal'Convergence:+The+idea+that+real+GDP+per+capita+in+poor+countries+grows+faster+than+in+rich+countries,+condi:onal+on+savings+rates,+access+to+technology,+popula:on+growth,+government+policies...ect.'

How+Does+it+Hold+Up?+

How+Does+it+Hold+Up?+

Savings+Rate+

Posi%ve(effect.'

€

kss =sAn +δ

#

$ % &

' (

11−α

€

dkssds

=1

1−αA

n +δ

%

& '

(

) *

11−αs11−α

−1> 0

Savings+Rate+

y,i,c++

y!=!Akα!+(n+δ)k+

i!=!s1Akα!+

k+Kss1*+

The effect of an increase in the savings rate with s2>s1.

What is the effect on the growth rate???

What is the effect on consumption???

i!=!s2Akα!+

Kss2*+

i1+

y1+

i2+

y2+

Savings+Rate+and+Consump:on+

€

css = A sAn +δ

#

$ % &

' (

α1−α

− As sAn +δ

#

$ % &

' (

α1−α

€

yss = css+ iss

€

css = yss − iss

€

css = Akα ss − sAkα ss

A:+TFP+Q+Technology+

y,i,c++

y!=!A1kα!+(n+δ)k+

i!=!sA1kα!+

k+Kss1*+

i!=!sA2kα!+

Kss2*+

i1+

y1+

i2+

y2+y!=!A2kα!+ The effect of an

increase in the Technologywith A2>A1.

What is the effect on the growth rate???

What is the effect on consumption???