Question

An agent lives for five periods and has an edible tree. The agent comes into the world at time t=0, at which time the tree is of size Xo. Letc, be the agent's consumption at time t. If the agent eats the whole tree at time t, then ce=X, and there will be nothing left to eat in subsequent periods. If the agent does not eat the whole tree, then the remainder grows at the simple growth rate a between periods. If at time t the agent saves 100 se percent of the tree for the future, then Xt+1 = (1 + asext. All the agent cares about is consumption during the five periods. Specifically, the agent's preferences are U = 24 B'In(c). The tree is the only resource available to the agent. a. Write out each period's budget constraint for the agent. b. Write out the agent's optimization problem.

Answer 1

a) Red Budget constraint to B Cocosxo. t=1 BC, : c < (142) Stolee Casa of n o t because x = (1+2) Soto 0 ay is the amount of tree left after consumption and savings en persod o. t=2 BG: C2 = (1+2)-5,24. i.es C = x2 [az = (1+255124] - t=3 : BC3: C S (1+2) Sq2 ie, Cz C 203 (23 = (1 +d)9223) -> t=4 By: Cy E (HQ) Sz X3 de 4 = 14 LEX- [x4 = (1+2) Sz X₂ - 4 for 2, N, Xz and as are the amounts of Consumption in periods 1, 2, 3 and 4 tree left respetively. 09 = (10) Solo x = (1+2) sisu = (1+2) Sosing. Bol Rox x = (1+2) S N = (1+2) So. S1.52.Xo xy = (1+2) S3xz = (1+2)" so. Sa . S2. 53.80 X4 = (1+2) "So. S. 82.53.00

Lou= - U = 7 to 8 ln (CG). Incot Bln (4) + 8? ln (c2) + B²ln (C₂) + B incons. Ophonization problem: the constrain mounts on in the part a. maxinize a lo, u, Cs, ez u and subject to 4 mentioned Problem: man. I & ln (4) subject to t=0 Cos do GS (1H2) Sono (22 (1225o.Si.to : (25 (1to3 so. S1.S2.Xo ; y = (1+0)" so. S,.52.5₂.20.