Instructor : Tony Smith November 30, 2007T.A. : Evrim Aydin SaherCourse : Econ 510a (Macroeconomics)Term : Fall 2007 (second half)Problem Set 3 : Suggested Solutions1. .(a) A competitive equilibrium for the economy is a set of objects (p0; p1) and (ci0; ci1) such thati. Given (p0; p1); (ci0; ci1) solves consumer i’s problem for i = A; B:max(ci0;ci1)u(ci0) + u(ci1)subject top01 + ici0+ p11 + ici1= p0!i0+ p1!i1+ p0S0+ p1S1St= AcAt+ (1  )BcBtii. Markets clear in each period:cAt+ (1  )cBt= ct= !t= !At+ (1  )!Btfor t = 0; 1Solving:max(ci0;ci1)u(ci0) + u(ci1) + ip0!i0+ p1!i1+ p0S0+ p1S1p01 + ici0+ p11 + ici1First order conditions:u0(ci0) = ip01 + ifor i = A; Bu0(ci1) = ip11 + ifor i = A; BThese four …rst-order conditions, along with the two budget constraints and market clearingconditions determine the competitive equilibrium.Pareto Optimality:The …rst step in checking for Pareto optimality is to check the MRS and relative price ratio.MRSA=u0(cA0)u0(cA1)=p0p1=u0(cB0)u0(cB1)= MRSB:1Since the MRS of both consumers are equal to each other the allocation is Pareto optimal.Next step is to compare our …rst order conditions with the FOC we would obtain from thesocial planning problem where the planner solves:max(cA0;cA1;cB0;cB1)u(cA0) + u(cA1) +(1)u(cB0) + u(cB1) (since cit= citin equilibrium),subject top01 + AcA0+ p11 + AcA1+(1  )p01 + BcB0+ p11 + BcB1=p0!A0+ p1!A1+ p0S0+ p1S1+(1  )p0!B0+ p1!B1+ p0S0+ p1S1which simpli…es top0cA0+ (1  )cB0+ p1cA1+ (1  )cB1= p0!A0+ (1  )!B0+ p1!A1+ (1  )!B1when we substitute inS0andS1:So the planner’s FOC are:u0(ci0) = ip01 + ifor i = A; Bu0(ci1) = ip11 + ifor i = A; BwithMRSA=u0(cA0)u0(cA1)=p0p1=u0(cB0)u0(cB1)= MRSB:Since in equilibrium we have cit= cit; the FOC is equivalent to the one we found for thecompetitive equilibrium and hence the competitive equilibrium allocation is Pareto optimal.(b) A competitive equilibrium for the economy is a set of objects (p0; p1) and (ci0; ci1) such thati. Given (p0; p1); (ci0; ci1) solves consumer i’s problem for i = A; B:max(ci0;ci1)u(ci0) + u(ci1)subject top01 + i0ci0+ p11 + i1ci1= p0!i0+ p1!i1+ p0S0+ p1S1St= AtcAt+ (1  )BtcBtfor t = 0; 1:ii. Markets clear in each period:cAt+ (1  )cBt= ct= !t= !At+ (1  )!Btfor t = 0; 12Solving:max(ci0;ci1)u(ci0) + u(ci1) + ip0!i0+ p1!i1+ p0S0+ p1S1p01 + i0ci0+ p11 + i1ci1First order conditions:u0(ci0) = ip01 + i0for i = A; Bu0(ci1) = ip11 + i1for i = A; BThese four …rst-order conditions, along with the two budget constraints and market clearingconditions determine the competitive equilibrium.Pareto Optimality:MRSA=u0(cA0)u0(cA1)=p01 + A0p11 + A16=p01 + B0p11 + B1=u0(cB0)u0(cB1)= MRSB:Since the MRSA6= MRSB; this allocation is not Pareto optimal.(c) A competitive equilibrium for the economy is a set of objects (p0; p1) and (ci0; ci1) such thati. Given (p0; p1); (ci0; ci1) solves consumer i’s problem for i = A; B:max(ci0;ci1)u(ci0) + u(ci1)subject top0ci0+ p1ci1= p01  i0!i0+ p11  i1!i1+ p0S0+ p1S1St= At!At+ (1  )Bt!Btii. Markets clear in each period:cAt+ (1  )cBt= ct= !t= !At+ (1  )!Btfor t = 0; 1Solving:max(ci0;ci1)u(ci0) + u(ci1) + ip01  i0!i0+ p11  i1!i1+ p0S0+ p1S1p0ci0+ p1ci1First order conditions:u0(ci0) = ip0for i = A; Bu0(ci1) = ip1for i = A; BThese four …rst-order conditions, along with the two budget constraints and market clearingconditions de termine the competitive equilibrium.Note that the tax does not enter into the3…rst order condition.Pareto Optimality:MRSA=u0(cA0)u0(cA1)=p0p1=u0(cB0)u0(cB1)= MRSB:We can check for Pareto optimality, exactly as we did in part (a).Note: Since we have lump-sum taxation, there is no distortion in the relative prices soPareto optimality does not break down.(d) A competitive equilibrium for the economy is a set of objects (p0; p1) and (ci0; ci1) such thati. Given (p0; p1); (ci0; ci1) solves consumer i’s problem for i = A; B:max(ci0;ci1)u(ci0;cA0+ (1  )cB0) + u(ci1;cA1+ (1  )cB1)subject top0ci0+ p1ci1= p0!i0+ p1!i1ii. Markets clear in each period:cAt+ (1  )cBt= ct= !t= !At+ (1  )!Btfor t = 0; 1;Solving:max(ci0;ci1)u(ci0;cA0+ (1  )cB0)+u(ci1;cA1+ (1  )cB1)+ip0!i0+ p1!i1p0ci0+ p1ci1First order conditions:u1(ci0;cA0+ (1  )cB0) = ip0for i = A; Bu1(ci1;cA1+ (1  )cB1) = ip1for i = A; BThese four …rst-order conditions, along with the two budget constraints and marketclearing conditions determine the c ompetitive equilibrium.Note that the tax does notenter into the …rst order condition.Pareto Optimality:In this case, the social planner solves:max(cA0;cA1;cB0;cB1)u(cA0;cA0+ (1  )cB0) + u(cA1;cA1+ (1  )cB1) +(1  )u(cB0;cA0+ (1  )cB0) + u(cB1;cA1+ (1  )cB1) ;subject top0cA0+ p1cA1+(1  )p0cB0+ p1cB1=p0!A0+ p1!A1+(1  )p0!B0+ p1!B1:4So the planner’s FOC are:with respect to cA0uA1+ uA2 + (1  )uB2= p0uA1+ uA2+ (1  )uB2= p0:with respect to cA1uA1+ uA2 + (1  )uB2= p1uA1+ uA2+ (1  )uB2 = p1with respect to cB0(1  )uA2+ (1  )uB1+ (1  )uB2 = (1  )p0uA2+ uB1+ (1  )uB2= p0:with respect to cB1(1  )uA2+ (1  )uB1+ (1  )uB2 = (1  )p1uA2+ uB1+ uB2 = p1:As we can see, the FOC are very di¤erent from what we found above, and so thisallocation is not Pareto optimal.Note that