Lecture 5Dynamics of the Growth ModelNoah WilliamsUniversity of Wisconsin - MadisonEconomics 312Spring 2014Williams Economics 312An ExampleNow work out a parametric example, using standardfunctional forms. Cobb-Douglas production:y = AkαFor preferences, set:U (c) =c1−σ− 11 − σFor σ > 0. Interpret σ = 1 as U (c) = log c.These imply the Euler equation:c−σt= βc−σt+1[1 + αAkα−1t+1− δ] = βc−σt+1Rt+1For these preferences σ gives the curvature and so governshow the household trades off consumption over time.Williams Economics 312Steady State in the ExampleRecall the Euler equation:c−σt= βc−σt+1[1 + αAkα−1t+1− δ]Steady state:F0(k∗) = Aα(k∗)α−1= δ + θ⇒ k∗=αAδ + θ11−αThen we get consumption:c∗= A(k∗)α− δk∗= AαAδ + θα1−α− δαAδ + θ11−αWilliams Economics 312Comparative StaticsSteady state capital stock determined by:F0(k∗) = δ + θConsumption c∗increasing in k∗(since below golden rulelevel).If δ ↑, then k∗↓ so c∗↓ .If total factor productivity ↑ then k∗↑, so c∗↑.Williams Economics 312F(k) δk y k An increase in the depreciation rate c*+δk* k’ k* δ’k δ+ θ δ’+ θ c’+δk’ Williams Economics 312AF(k) δk y k An increase in total factor productivity k’ c’+δk’ c*+δk* k* A’F(k) δ+θ δ+θ Williams Economics 312Dynamics of the ModelLast time we analyzed the basic model of optimal growthand studied the behavior of the model in the steady state.Now will analyze the dynamics of the model.The model is nonlinear and in general does not allow for acomplete, explicit solution in general.However we can analyze the qualitative dynamics byanalyzing a phase diagram summarizing the keyequations of the model.This will allow us to analyze both the short-run andlong-run effects of changes in exogenous variables. Forexample, to see how the economy responds to a change intechnology.Williams Economics 312Qualitative DynamicsWe will analyze the joint dynamics of {ct, kt}. In anyperiod t, ktis given and ctis chosen optimally (as is kt+1).The key equations of the model are:U0(ct) = βU0(ct+1)[F0(kt+1) + 1 − δ]kt+1= (1 − δ)kt+ F(kt) − ctWe’ll use the first to determine the dynamics of c, thesecond the dynamics of k.In steady state, ∆ct+1= ct+1− ct= 0, andF0(k∗) = δ + θIf k < k∗, then F0(k) > F0(k∗), so to satisfy Euler equationwe need U0(ct+1) < U0(ct) and so ct+1> ct. Similarly ifk > k∗, ∆c < 0.Williams Economics 312∆c=0: F’(k*)=δ+θ c k Dynamics of consumption ∆c>0 ∆c<0 k* Williams Economics 312Dynamics of Capital and Phase DiagramA key equation of the model is:kt+1= (1 − δ)kt+ F(kt) − ctIn steady state, ∆kt+1= kt+1− kt= 0, andc = F (k) − δkIf ct< F (kt) − δktthen it> dkt, so ∆kt+1> 0. Similarly ifct> F (kt) − δktthen ∆kt+1< 0.Putting together the dynamics of c with the dynamics of kgives the phase diagramWilliams Economics 312∆k=0: F(k)-δk c k Dynamics of capital ∆k>0 ∆k<0 Williams Economics 312∆k=0: F(k)-δk c k Phase diagram: dynamics of consumption and capital ∆c=0: F’(k*)=δ+θ k* c* Williams Economics 312The Saddle PathFrom the phase diagram we can see the dynamics of{kt, ct} from any initial (k0, c0).But given k0the dynamics from an arbitrary c0will notconverge to the steady state.In general either ctor ktwill go to zero. These are not partof an optimal solution.However given k0there is a unique value of c0such that theeconomy converges to the steady state. This is the saddlepath.The optimal solution will be on the saddle path, as c0is afunction of k0and will be chosen so that the economy isstable and converges to the steady state.Williams Economics 312∆k=0: F(k)-δk c k Phase diagram: saddle path and dynamics from different initial consumption levels ∆c=0: F’(k*)=δ+θ k* c* k0 Williams Economics 312Comparative DynamicsWith the phase diagram, we can determine how anexogenous change affects ctand ktboth over time and inthe long run.Suppose, for example, that households became less patient,so θ increases and β falls. What would happenimmediately and in the long run?Recall the dynamics:∆c = 0 : F0(k∗) = δ + θthis curve shifts to the left, so steady state k∗would fallAnd for capital,∆k = 0 : c = F(k) − δkthis curve is unaffected.In the long run, ktand ctwill fall. When the change in θhappens ctwill jump up to the new saddle path.Williams Economics 312k Phase diagram: An increase in the discount rate (θ). F’(k*)=δ+θ’ k* c* F’(k*)=δ+θ F(k)-δk k’ c’ c0 When households become more impatient, they increaseconsumption, and save less. In the short run this leads to moreconsumption. But in the long run, the lower investment willlead to an reduction in capital and hence consumption.Williams Economics 312Improvement in TechnologyIf total factor productivity increases, we already have seenthat in steady state k∗↑, so c∗↑. But what happens alongthe transition?Recall the dynamics:∆c = 0 : AF0(k∗) = δ + θthis curve shifts to the rightAnd for capital,∆k = 0 : c = AF(k) − δkthis curve shifts upIn the long run (k∗, c∗) increase. But the effect in the shortrun depends on the slope of the saddle path, which in turndepends on how willing the household is to substitute overtime.Williams Economics 312Slope of the Saddle PathIf the household is relatively impatient (low β, high θ),and is unwilling to substitute over time (low IES), then itwill want to smooth consumption over time and value earlyperiods highly. Then ctwill increase and the economy willslowly move to the steady state. c0will increase.If the household is relatively patient (high β, low θ), andis willing to substitute over time (high IES), then it willforgo current consumption, invest more, and get to thesteady state more quickly. c0may fall.In both cases, ktincreases each period until it reaches thesteady state. In the long run ctincreases, but in theshort-run it may increase or decrease.Williams Economics 312AF(k)-δk c k Phase diagram: An increase in total factor productivity (A), with a flatter saddle path (low IES, high discount rate) AF’(k*)=δ+θ k* c* A’F’(k*)=δ+θ A’F(k)-δk k’ c’ c0 Williams Economics 312AF(k)-δk c k Phase diagram: An increase in total factor productivity (A), with a steep saddle path (high IES, low discount rate) AF’(k*)=δ+θ k* c* A’F’(k*)=δ+θ A’F(k)-δk k’ c’ c0 Williams Economics 312Solving the Model in a Special CaseThere