Life time utility function is ln C1+(10/11)lnC2 . Compute the marginal rate of substitution as MRS = MUC1/MUC2
MRS = (1/C1)/(10/11C2) = 1.1C2/C1. Now see the inter-temporal budget constraint
C1 + C2/(1+r) + B0*(1 + i) = Q1 + Q2/(1+r)................. r is the interest rate on saving and i is the interest rate on debt
C1 + C2/1.1 + 5*(1 + 0.2) = 10 + 10/1.1
C1 + C2/1.1 = 4 + 10/1.1
1.1C1 + C2 = 14.4
Utility maximization requires MRS = slope of the budget constraint
1.1C2/C1 = 1.1/1
C2 = C1
Placing this in the budget equation
1.1C1 + C1 = 14.4
C1* = 6.8571 and C2* = 6.8571
These are the current consumption levels. In period 1 when consumers receive 10, they consume 6.8571, save the remaining and invest the same at 10% so that in period 2 they have (10 - 6.8571)*(1 + 0.1) = 3.4572. In the second period, they consume 6.8571, and add the two period savings = 3.4572 + (10 - 6.8571) = 6.6 to pay off the liability.
