Graph the resource constraint with 0 : kt+1= F(kt) -kt - ct = 0 kt+1= so: ct = F(kt) -kt is the 0 locus (see graph) kt+1= (locus: slope of locus is F'(k) - =0 at Golden rule k (kGR); >0 below Golden rule k <0 above Golden rule k.) - How does k evolve? on the 0 locus investment just offsets kt+1= depreciation. - if above the k=0 curve: ↓k (c so high can’t replace depreciating capital) - if below the k=0 curve: (c low enough that k rises) ↑k - kmax: is the maximum sustainable level of k (here c=0 and F(kmax)=kmax). - How does c evolve at different starting points? - Take the Euler equation: - What does it say about the locus? ct+1=0 if ct+1=ct the Euler eqn then says: + (1- ct+1=0 [F'(kt+1) or: F'(kt+1)=  - so at the level of k where F'(kt+1)= this k*) ct+1=0 (call i.e. locus is a vertical line at k*. ct+1=0
Graph the resource constraint with kt+1= 0 : kt+1= F(kt) -kt - ct = 0 so: ct = F(kt) -kt is the kt+1= 0 locus (see graph) (locus: slope of locus is F'(k) - =0 at Golden rule k (kGR); >0 below Golden rule k <0 above Golden rule k.) - How does k evolve? on the kt+1= 0 locus investment just offsets depreciation. - if above the k=0 curve: ↓k (c so high can’t replace depreciating capital) - if below the k=0 curve: ↑k (c low enough that k rises) - kmax: is the maximum sustainable level of k (here c=0 and F(kmax)=kmax). - How does c evolve at different starting points? - Take the Euler equation: - What does it say about the ct+1=0 locus? ct+1=0 if ct+1=ct the Euler eqn then says: [F'(kt+1) + (1- or: F'(kt+1)=  - so ct+1=0 at the level of k where F'(kt+1)= (call this k*) i.e. ct+1=0 locus is a vertical line at k*.