Foundations of International Macroeconomics1 Workbook2 Maurice Obstfeld, Kenneth Rogoff , and Gita Gopinath Chapter 2 Solutions 1. (a) The current account identity can be written as Bs+1= (1+r)Bs+TBs. Now just plug in the assumed trade balance rule. (b) Using the answer to part a, for any 0, X s=t 1 1 + r st rBs= X s=t 1 1 + r st r1 + (1 )rstBt = rBt 1 1 + (1 )r 1 + r = (1 + r)Bt. (c) Under the rule above, debt grows without bound if 0, logC, = 1. Using the intertemporal Euler equation, we thus obtain, (1 + r) = 1 = 1 Et ( Ct+1 Ct 1/). (1) Since consumption has a conditional lognormal distribution, the natural log of the gross consumption growth rate is conditionally normally distributed: log Ct+1 Ct N Et log Ct+1 Ct ,Vart log Ct+1 Ct . Thus Et C t+1 Ct 1 =Et exp 1 log C t+1 Ct =exp 1 Et log Ct+1 Ct + 1 22 Vart log Ct+1 Ct .(2) 15 Consult footnote 41 on p. 313 of the book. Equation (2) follows from com- puting the mean and variance of the random variable (1/)log(Ct+1/Ct), which is normally distributed when log(Ct+1/Ct) is. Combining eqs. (1) and (2) above and taking natural logs of the result, we arrive at Et log Ct+1 Ct = 1 2 Vart log Ct+1 Ct or logCt+1 logCt= 1 2 Vartt+1 + t+1, where t+1 logCt+1 EtlogCt+1. Since t+1is a normal random vari- able that is uncorrelated with past information (because it is a pure fore- cast error), it is also statistically independent of that information on the assumption that the past information itself is generated by a jointly normal (i.e., Gaussian) stochastic process. In that case the conditional variance in the preceding equation actually is a time-invariant constant, so the natural log of consumption follows a random walk with a constant drift equal to 1 2Vart+1. 4. (a) Using eq. (32) in Chapter 2, we can write Ct+1 Ct=r(Bt+1 Bt) + r 1 + r X s=t+1 1 1 + r s(t+1) Et+1Ys X s=t 1 1 + r st EtYs . The current account identity gives Bt+1 Bt= Yt+ rBt Ct= Yt r 1 + r X s=t 1 1 + r st EtYs, which can be substituted into the previous equation for consumption to give the result that the change in consumption equals the present value of changes in expected future output levels. 16 (b) If the process for output follows Yt+1 Yt= (Yt Yt1) + t+1, then (Et+1Et)Yt+1= t+1, (Et+1Et)Yt+2= (1+)t+1, (Et+1Et)Yt+3= (1 + + 2)t+1, and so on. Therefore, for s t, (Et+1 Et)Ys= 1 st 1 t+1. (c) Substituting the last expression into the equation for the change in con- sumption derived in part a, we get the following Ct+1 Ct= r 1 + r t+1 “ 1 1 ! + 1 1 ! 1 1 + r + . 1 2 (1 + r)(1 ) . # = r 1 + r t+1 “ (1 + r) (1 )r 1 1 + r 1 + r !# = 1 + r 1 + r t+1. (3) As a result, provided that 0 1, the desire to smooth consumption makes consumption innovations more variable than output innovations. (d) The current account identity for date t + 1 is CAt+1= Bt+2 Bt+1= Yt+1+ rBt+1 Ct+1. Because Yt+1 EtYt+1= t+1and, by eq. (3) from part c above, Ct+1 EtCt+1= Ct+1 Ct= 1 + r 1 + r t+1, the preceding current account identity gives a current account innovation of t+1 1 + r 1 + r t+1 = 1 + r t+1 0. 17 Thus, a positive output innovation leads to a current account decit, as claimed at the end of section 2.3.3 in the book. 5. Work backward from the equation CAt+1 Zt+1 (1 + r)CAt= t+1, where t+1is uncorrelated with date t or earlier information. Taking expec- tations with respect to date t information yields EtCAt+1 EtZt+1 (1 + r)CAt= 0. The previous equation can be rearranged to express CAtas CAt= 1 1 + r EtCAt+1 1 1 + r EtZt+1. Through forward recursive substitution (and using the law of iterated condi- tional expectations ) we obtain CAt= X s=t+1 1 1 + r st EtZs because as j , 1 1+r j EtCAt+j 0. This is Campbells (1987) “saving for a rainy day” equation, eq. (43) in Chapter 2. The equation can alterna- tively be derived using the lag and lead operator methodology described in supplement C to Chapter 2. Start again with CAt+1 Zt+1 (1 + r)CAt= t+1 and take expectations with respect to date t information to get EtCAt+1 EtZt+1 (1 + r)CAt= 0. Using the lead operator we write this as L1CAt L1Zt (1 + r)CAt= 0, 18 or, dividing by 1 + r and rearranging, as 1 1 1 + r L1 CAt= 1 1 + r L1Zt. Inversion of the lag polynomial on the left-hand side above gives CAt= 1 1 + r 1 1 1 + r L1 1 EtZt+1 = 1 1 + r X s=t 1 1 + r st EtZs+1 = X s=t+1 1 1 + r st EtZs. To derive the converse, that the last equation implies CAt+1Zt+1(1+ r)CAt= t+1, one can simply reverse the steps above. 6. Write the expression for the current account as follows CAt=Zt Et e Zt= Zt r 1 + r Et X s=t 1 1 + r st Zs =Zt r 1 + r 1 1 1 + r L1 1 EtZt, where the last equality is suggested in the hint. Then, multiplying both side