Robinson Crusoe is stranded on an island with 1 unit of rice. He can count upon being rescued after two periods. Any fraction of the endowment of 1 unit of rice can either be consumed or used as seed. Crusoe also has a unit of leisure in each period.

If s; theamount of rice used as seed, is combined with labor l; the rice output y that can beconsumed in the second period is 2pls: Crusoeís utility corresponding to x; y; s is xyz where x = 1l and z = 1s:

(a) What is the optimal allocation in the above economy?

(b) In the following problems, show directly that the optimal allocation in a: is part ofa competitive equilibrium.

i. Compute the competitive equilibrium prices p of rice and q of seed, letting theprice of leisure be 1.
ii. Show that the Crusoe is maximizing utility at the allocation in part a, giventhe prices and ëincome.
íiii. Show that at the allocation and prices above, the producer (Robinson Crusoewith a di§erent hat) is maximizing proÖt.