Suppose that the consumer's utility function is logarithmic, u(ct)=ln(ct), find the optimal consumption in the first and in the second periods. In particular, suppose that the consumer solves the following maximization problem:

maxc1,c2ln(c1)+ln(c2)

subjec to:

c1+1+rc2=y1+1+ry2

Set up the Lagrangian and take FOCs to find the Euler equation. Then, use the lifetime budget constraint to leave the Euler equation only in terms of eitherc1 orC2 (whatever you find easier). This process will give the consumption level in say period 1, just in terms of the lifetime wealth, , and the interest rate. Finally, compute the consumption level in the other period, which will also be in terms of the lifetime wealth, , and the interest rate. For simplicity, just call m the lifetime budget constraint, so that m=y1+1+ry2

a)C1=1+m;C2=1+(1+r)m

b)C1=1+m;C2=1+m

C)C1=1+(1+r)m;C2=1+(1+r)m

d)C1=1+(1+r)m;C2=1+(1+r)m