Question

19. The consumer’s utility function is as follows; ?(?1, ?2 ) = ?1?2 where x1 stands for the consumption level of good 1 in period 1 and x2 refers the consumption level of good 1 in period 2. The budget constraint for the consumer is; ? = ?1 + ?2 1+? where r is the interest rate. Find ?1 ∗??? ?2 ∗ (as functions of B and r) that maximize the utility. Check whether the second order sufficient condition is satisfied with the Hessian.

Answer 1

Q UCX1, X2) = X, o X2, B.C. B=X1 + X 2 (48) By life time Income (Present value) then Max UCX, ₃X2) St B. C. ( + ) at Eqm MRS 1,2 = (1+r) MRS= aulax 12 = # x2 So at &qm x2 = (1+0), [x2 = (1+0)X/ from B.C. → B = Xit (14) X1 Exp = 8/2 % 1 *a" = (ito) B Ang NOW @ Soco- H = ƏX2² 224 = 1. aXƏX2 NOW IH)= 0-1 = -1 0 thus Soc satisfied at X* & X2* Ang - 224 /2x12 224 Xi X2 T o 17 = 22u Hessian aXX Matrix