macro3 solow growth

8/9/2019 Macro3 Solow Growth

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The Solow Growth

Model (Part One)

The steady state level of capital and

how savings affects output andeconomic growth.


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Model Background

• Previous models such as the closed economy andsmall open economy models provide a static viewof the economy at a given point in time. The Solow

growth model allows us a dynamic view of howsavings affects the economy over time.


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Building the Model: goods market supply

• We egin with a production function and assume constantreturns.

!"#$%&'( so) z !"#$z %&z '(

• By setting z "*+' we create a per worker function.

!+'"#$%+'&*(

• So& output per worker is a function of capital per worker. Wewrite this as&

y"f$k(


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Building the Model: goods market supply

• The slope of this functionis the marginal product ofcapital per worker.MP% " f$k,*(-f$k(

k

y

Change in y

Change in k

y=f(k)k inchange

yinchange MPK =

• t tells us the change inoutput per worker thatresults when we increasethe capital per worker yone.


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Building the Model:goods market demand

• We egin with per worker consumption and investment.$Government purchases and net e/ports are not included in the

Solow model(. This gives us the following per workernational income accounting identity.

y " c,• Given a savings rate $s( and a consumption rate

$*-s( we can generate a consumption function.c " $*-s(y )which makes our identity&y " $*-s(y , )rearranging&

i " s0y )so investment per workere1uals savings per worker.


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Steady State 21uilirium

• The Solow model long run e1uilirium occurs at thepoint where oth $y( and $k( are constant. These arethe endogenous variales in the model.

• The e/ogenous variale is $s(.


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• By sustituting f$k( for $y(& the investment per worker function$i " s0y( ecomes a function of capital per worker $i" s0f$k((.

• To augment the model we define a depreciation rate $3(.

• To see the impact of investment and depreciation on capitalwe develop the following $change in capital( formula&4k " i - 3k )sustituting for $i( gives us&4k " s0f$k( - 3k


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• 5t the point where oth $k( and $y(are constant it must e the casethat& 4k " s0f$k( - 3k " 6 )or& s0f$k(

" 3k)this occurs at our e1uiliriumpoint k0.

khighklow

• f our initial allocation of $k(were too high& $k( would

decrease ecause depreciation

e/ceeds investment.

• 5t k0 depreciation e1uals

investment.

k

s0f$k(&3k

k0

s0f$k0("3k0 s0f$k(

3k• f our initial allocation were too

low& k would increase ecauseinvestment e/ceeds

depreciation.


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5 ;umerical 2/ample

• Starting with the ouglas productionfunction we can arrive at our per workerproduction as follows&

!"%*+7

'*+7

)dividing y '& !+'"$%+'(*+7 )or&y"k*+7

• recall that $k( changes until& 4k"s0f$k(-3k"6 ...i.e. until& s0f$k("3k


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5 ;umerical 2/ample

• Given s& 3& and initial k& we can computetime paths for our variales as we approach

the steady state.

• 'et?s assume s".9& 3".6@& and k"9.

• To solve for e1uilirium set s0f$k("3k. This

gives us .90k*+7

".6@0k. Simplifying gives usk"*@.A8*& so k0"*@.A8*.


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5 ;umerical 2/ample

• But what it the time path toward k0 To get this usethe following algorithm for each period.

• k"9& and y"k*+7 & so y"7.• c"$*-s(y& and s".9& so c".Cy"*.7

• i"s0y& so i".D

• 3k ".6@09".8C

• 4k"s0y-3k so 4k".D-.8C".99

• so k"9,.99"9.99 for the ne/t period.


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5 ;umerical 2/ample

• Eepeating the process gives)

period k y c i 3k 4k

* 9 7 *.7 .D .8C .99

7 9.99 7.*6A... *.7C9... .D97) .8@@) .998)

. . . . . . .

*6 D.898... 7.DDD... *.CD@... *.*7C... .A*8) .9*7)

. . . . . . .

F *@.A... 9.99) 7.CCA... *.AAA... *.AAA... 6.666...


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5 ;umerical 2/ample

• Graphing our results in Mathematica gives us&


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macro3 solow growth

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