Tutorial 5: RCK Model I PDF

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Dr. Hertweck

Tutorial 5: RCK Model I

1. Show that Xt =

Z t

enτ +

0

Rt

τ rs ds

(wτ − cτ ) L0 dτ + e

Macroeconomics II

Rt

0 rs ds

(1)

is the solution to the differential equation (law of motion of household’s wealth) X˙ t = rt Xt + (wt − ct ) L t

(2)

subject to the initial condition X0 = 1. Also note that L t = L0 ent .

2. A household maximizes the intertemporal utility function Z∞

ln(ct ) exp ((n − ρ)t)dt

(3)

0

subject to the budget constraint x˙t = wt + rt xt − ct − nxt .

(4)

Assume ρ > n > 0. (a) Derive the Euler equation by using the Hamiltonian (present-value) approach. (b) How does capital per capita evolve? (c) Explain the role of the Transversality condition intuitively. (d) Draw the phase diagram and find the steady state of c and k using (a) and (b). Also sketch the path of c and k to the steady state graphically. (e) Assume an economy where c and k are lower than in the steady state, but on the stable arm. How do the following events affect thec ˙= 0 and k˙ = 0 loci and the stable arm, and thus how do they affect k and c in the short and the long run? (i) A permanent decline in the population growth rate from n to n′ (n′ < n). (ii) A rise in θ (assume the Euler equation is given by cct˙ = (rt − ρ)/θ). t

1...