Consider a two-period constrained optimization problem. A consumer chooses a consumption plan (c,c) to maximize his utility u(c,c)=lnc+lnc subject to c+s=a+yc=(1+r)s+y where s is the saving today, y,y denote today and tomorrow's income respectively, and a is the initial endowment, which is taken as given. The market interest rate r is taken as given as well. (a) In this part, suppose there is no borrowing or liquidity constraint, which means that s can take any value (and in particular, s can be negative-the consumer can borrow as much as he wishes). 1. Combine two budget constraints into a single intertemporal budget constraint. Your answer should not include saving s and only contains c,c,y,y,r. 2. Using your answer above, set up the Lagrangian for consumer optimization and list out the first order conditions. Solve the optimal consumption today c, optimal consumption tomorrow (c) and optimal saving s. 3. Discuss how s changes with a, and r. (b) Continue with the setup in Question 1, part (a). Assume a=0 and (1+r)=1. 1. Suppose y>y. Suppose there is a rise in interest rate r. Use a (c,c) diagram to analyze the substitution effect, income effect and total effect on c and (c). 2. Suppose yy. Characterize the optimal consumption today c, optimal consumption tomorrow c and optimal saving s. Is the borrowing constraint binding in this case? 2. Suppose yy. Suppose there is a rise in interest rate r. Use a (c,c) diagram to analyze the substitution effect, income effect and total effect on c and (c). 2. Suppose yy. Characterize the optimal consumption today c, optimal consumption tomorrow c and optimal saving s. Is the borrowing constraint binding in this case? 2. Suppose y