0y1y2c1c21 2 1 2 1 2 U c c c c c c ln ln 0 0 = + 2 2 1 1 1 1 c y c y r r + = + + + +ab c1 c2c d ry1y2e r = 0= 10%

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Suppose a consumer who can borrow or lend at an annual rate of interest r0 anticipates receiving positive income y1 this year and y2 next year. (The consumer ignores the future more than one year ahead.) The same consumer chooses the levels of consumption c1 this year and c2 next year in order to maximize the utility function U(c1,c2)=lnc1+lnc2(c1>0,c2>0) subject to the budget constraint c1+1+rc2=y1+1+ry2 (a) Write out the Lagrangian for the constrained maximization problem. (b) Show that the Lagrangian is concave as a function of (c1,c2). (c) Write out the first-order conditions for a constrained maximum. (d) Find the utility maximizing expenditures in both periods, as well as the Lagrange multiplier , all as functions of the parameter triple (r,y1,y2). (e) Find the utility maximizing expenditures in both periods for the special case when the interest rate r=0, and also when r=10%