A Textbook on Macroeconomics in Business

 To finance the projects borrowers need a unit of capital from a bank. The bank in turn announces a repayment amount in the event that the borrower’s project does not fail. If the project fails, borrowers owe nothing (they declare bankruptcy). If the project succeeds, borrowers consume their output minus , if the project fails, borrowers consume zero. Assume that borrowers are risk neutral so that their utility function is just their expected consumption. x x r There is a risk-free interest rate of that banks must pay to their depositors (thus they have to realize at least 1 + in expected value on their loan to meet their deposit liability). r xrp p 1. Write down a bank’s balance sheet (in terms of , , S , and R ) assuming that, with the borrower is risky. probability the borrower is safe and with probability 1 x 2. Assume that banks compete by offering the lowest value of that gives them nonnegative profits in expectation. Determine the equilibrium interest rate ? ( ) as a function of the interest rate and the proportion of safe agents . r x r; V r 3. Find the expected utility of a safe agent who borrows, S ( ), as a function of the interest rate when is given by ? ( ). Repeat for a risky agent. r x x r; 4. Agents stop borrowing if the expected utility of being a borrower falls below zero. Show that if a safe agent decides to borrow, a risky agents will too. Find the critical interest rate ? at which safe agents stop borrowing. At interest rates greater than or equal to this critical value,  ? all safe agents leave the pool, so = 0. What happens to the equilibrium payment ? r r r x Exercise 17.3 (Moderate) Consider the model of costly audits again. Now suppose that intermediaries gain access to a technology which allows them to extract more from each borrower (that is, for each value ) shifts up). What happens to of announced repayment and audit cost , suppose ( x  x; Financial Intermediation 208 the demand schedule of capital? What happens to the supply schedule of capital? What happens to the equilibrium interest rate? What happens to equilibrium economy-wide output? Are agents made better off or worse off? Exercise 17.4 (Moderate) Yale University costs 1 dollars to attend. After graduation, Yalies (that is, graduates of Yale) either land good jobs paying or no job at all, paying nothing. The probability of landing the good job is where is hidden effort exerted by the Yalie. Yalies are born with wealth  0, and those Yalies born with wealth 1 must have a loan of 1 to attend. Yale University will act as a lender to those students. Yale must borrow at the risk-free gross 1 to finance the loans. Student borrowers who get the good job must interest rate repay Yale University some amount out of their wages . Student borrowers who do not land the good job pay nothing. All students have preferences over lifetime expected consumption ( ) and private labor effort of:  a  w a< r> x w Ec Assume 0 a  2 V (E (c); ) = E (c) w 2 : < < 1. a > 1. Show that her optimal effort ? is . 2. Now consider poor Yalies, with a < 1, who must borrow to finance their education. Calculate a borrower’s optimal effort  (x) as a function of x. 3. Write down Yale University’s expected profit on a loan to a student with wealth a < 1 as a function of x, assuming that Yale University knows  (x) from Exercise (2). 1. Start with a rich Yalie, with 4. Assume that Yale University operates a “fair lending policy” in which borrowers of ) if they get the good job. What is wealth must repay an amount ( ) = (1 “fair” about this lending policy? Given this policy and your answer to Exercise (2) above, calculate a borrower’s optimal effort as a function of their wealth. That is, write down [ ( )], and call it ( ). a xa xa r a= a 5. Show that, given Yale University’s “fair lending policy”, all Yalie borrowers exert less ? 1, show that ( ) effort than rich Yalies, that is, for Yalies with wealth 0  ? ? and that ( = 1) = , where is from Exercise (1) above. a   a< a < 6. Finally, show that given its fair lending policy, that Yale loses money on student loans, and that the loss is increasing in loan size. Why does the fair lending policy cost Yale money? Variable Definition Proportion of population who are type-1 workers. The audit cost of agent . Consumption in the first period of life (type-1 agents only). Labor effort in the first period of life (type-1 agents only). Consumption in the second and last period of life (both types of agent). Preferences of type-1 agents. Preferences of type-2 agents (risk neutral). Capital input to type-2 agent’s project, can take on only two values, = 0 or = 1. Output of agent ’s project, or just output. Shock to output, distributed uniformly on [0 1]. Maximum labor effort by a type-1 agent, 1. Savings of a type-1 agent, or of the representative type-1 agent (aggregate supply of capital). Economy-wide equilibrium interest rate on capital. Repayment amount, = 1 + . Revenue of bank on a loan (gross of the borrowing cost ) to an agent with audit cost , when the repayment amount is and the production shock is . Expected revenue (gross of the borrowing cost ) on a loan to an agent with audit cost , when the repayment amount is . Expectation taken over the production shock . The highest possible expected revenue (gross of the borrowing cost ) on a loan to an agent with an audit cost of . The repayment amount that results in the highest revenue to the bank on a loan to an agent of audit cost . Largest value of the audit cost at which the bank can make enough revenues to cover the cost of borrowing, . Aggregate demand for capital. i i c0 `0 c1 U 1(c0; `0; c1) U 2(c1) k yi; y " h s; S (r) r X; x (x; ; ") i k ; h< X r ) x x " (x; k r x " ? ( ) x? ( ) r ?( ) K d (r ) r r Table 17.2: Notation for the model of audit costs in Section 17.2 Variable k a ` c q r X a?(r) Definition Capital input to productive technology, can take only two variables, = 0 or = 1. Wealth of agent. Private labor effort of agent. Consumption of agent. High output of technology (the low output is zero). Disutility of effort in agent’s preferences. Economy-wide risk-free rate on capital. Repayment amount. Threshold credit rationing wealth. k k Table 17.3: Notation for model with moral hazard in Section 17.3 Variable F U (c1; c2 ; ) ;  Q r1 ; r2 cij ? c11 ; c22? Definition Technology parameter: growth of asset between periods = 1 and = 2. Utility function over consumption in period = 1, = 2 and shock term . Shock term: = 1 means that the agent is urgent to consume (probability ). Preference parameter: marginal utility of type-2 agents, 1. Interest rate promised by the bank on deposits held until period = 1 ( 1 ) or period = 2 ( 2 ). Consumption by agent to type in period . Optimal consumption in period = 1 by type1 agents and the optimal consumption in period = 2 by type-2 agents. t t t t    Q< t r i t t r j t Table 17.4: Notation for model of bank runs in Section 17.4 Chapter 18 Fiscal and Monetary Policy In Chapter 14 we described how the government changes its outstanding debt over time so as to match its revenues and expenditures. In that framework, there was nothing intrinsically harmful about government debt. Now we turn our attention to the inflationary effects of persistent government budget deficits. This will give us a theory about the interplay between fiscal and monetary policies. Imagine a government comprised of two competing authorities: a fiscal authority (in the U.S., the Congress and the President) and a monetary authority (in the U.S., the Federal Reserve System). The fiscal authority wants to finance government spending, while the monetary authority wants to keep inflation low. But inflation produces revenue for the government through a process known as seigniorage. If the monetary authority is dominant, it simply announces a sequence of inflation rates, which in turn implies a sequence of seigniorage revenues, and the fiscal authority takes this as given when making spending decisions. Completely dominant central banks are, however, extremely rare. Even the U.S. Federal Reserve System is statutorily a creature of the Congress and the Executive, and must, by law, balance the twin goals of fighting inflation and targeting full employment. The case of a central bank (i.e., a monetary authority) that is not fully dominant is much more interesting. Note that this does not mean that the fiscal authority controls the money supply absolutely, merely that the fiscal authority does not have to credibly commit to a sequence of taxes sufficient to finance, in present value, its spending. In particular, we are going to assume that there is some limit on the debt-to-GDP ratio. That is, investors will only accept government debt up to some ceiling, defined in proportion to output. The monetary authority will control the money supply until this ceiling is reached, and thereafter it will fully accommodate government spending with seignorage revenue. This is the fashion in which government deficits are inflationary. After presenting the theory, we will discuss the evidence. In a study of post-WWI hyperin- Fiscal and Monetary Policy 212 flations in Germany, Austria, Hungary and Poland titled “The Ends of Four Big Inflations”, Thomas Sargent illustrates this effect clearly. More recent monetary disturbances in some the successor states of the old Soviet empire can also be traced to persistent government budget deficits. This will provide us with a practical guide on how to end hyperinflations. 18.1 Are Government Budget Deficits Inflationary? The model for this section is taken from a paper by Thomas Sargent and Neil Wallace, “Some Unpleasant Monetarist Arithmetic”. The interested reader is advised to read the original paper, since it doesn’t require very much math and is, despite the title, actually quite pleasant. Government Budget Constraint We will consider the problem of a government which must cover a sequence of real core deficits f t g1 t=0 : D Dt = Gt T t; for all t = 0; 1; : : : ; 1; G where t is the real value of government expenditures and T t is the real value of government revenues in period . Notice that interest payments on the debt are not included in t (see Chapter 14 for more on the government budget constraint). t D B The government has some amount tg 1 of real debt outstanding at the beginning of each period . The government must pay its creditors a real amount (1 + ) tg 1 in period . Hence the total real excess spending of the government on goods and services and debt service, net of tax revenue, is: t rB borrowing demand = t Dt + (1 + r)Btg 1: The government will finance this in two ways: (1) By issuing more bonds, dated endof-period (call these bonds tg ) and (2) By printing money and realizing the seigniorage revenue (more on what that is in a second). Hence government borrowing is: t B borrowing supply = M t 1 Btg + Mt PM : t Here t is the end-of-period quantity of pieces of paper with the words “Federal Reserve Note” and “In God We Trust” printed on them, also known as fiat currency. Take t to be strictly high-powered money, or the monetary base, which is under the control of the government. M 18.1 Are Government Budget Deficits Inflationary? 213 For the government’s books to balance it must borrow as much as it needs to, so: Dt + (1 + r)Btg 1 = Btg + Mt PMt 1 ; for all t = 0; 1; : : : ; 1: t (18.1) Another way to write this is: t 1 ; for all t = 0; 1; : : : ; 1: Dt + rBtg 1 = (Btg Btg 1) + Mt PM t This form says that the government’s real deficit plus the interest on the debt may be paid g for by net new bonds ( tg t 1 ) or seigniorage. B B Seigniorage The government has a monopoly on issuing pieces of paper with the words “Federal Reserve Note” written them. People want this stuff for transactions purposes, so they hold it even though it pays zero interest. As a result, the government can print more of the stuff and trade it for goods and services. We will not model the precise way in which the government does this. The effectiveness of this practice depends on how the general price level t responds to an increase in t . P M Although seigniorage revenue in developed countries like the United States is currently very low, developing countries or countries in turmoil use it heavily. Internal bond markets and tax collection systems are often the first instruments of state power to vanish in turbulent times. Governments also often find direct taxation to be unpalatable for domestic political reasons, but are unable to sell bonds on international markets. Consider the case of Zaire, an African country which is now called the Democratic Republic of the Congo. This government practiced a very bald form of seigniorage in which it would introduce a new denomination of the currency (the zaire), print up a bunch of notes and pack some of the print run into suitcases which were then distributed among government ministers. These ministers would then use the notes to purchase foreign currency on the black market as well as domestic goods and services. In the waning days of the rule of former president Mobuto Sese Seko, the government introduced the 500 zaire note and the 1000 zaire note. These were used, in part, to finance the president’s cancer treatments in France. The population derisively termed the notes “prostates” and refused to accept them as payment in any transaction. The government’s seigniorage revenue fell to zero and it succumbed to the rebels shortly thereafter. More formally, the value of the seigniorage revenue in our model is the real value of net new notes: Mt Mt 1 : Pt Notice that we will have to take a stand on how Pt varies with Mt to fully determine the seigniorage = seigniorage revenue. Fiscal and Monetary Policy 214 Model Assumptions To make this model work, we will have to specify a rule for output, population growth, how the price level is determined and what limits there are on borrowing. I list all of the model’s assumptions here for convenience: y nN y N > N 1. Output per capita t is constant and t = 1, but population t grows at the constant rate , so t = (1 + ) t 1, where 0 0 is given. So total GDP each period t = t t is just equal to population. n N r Y yN r 2. The real interest rate on government debt is constant at t = , and the government never defaults on its debt. This includes default by unexpected inflation when bonds are denominated in dollars. Thus we are dealing with inflation-indexed bonds. We . Without this assumption, most also make the very important assumption that of the “arithmetic” is not so “unpleasant”. r>n 3. A stark monetarist Quantity Theory of Money relation with a constant velocity, v = 1: PtYt = vMt: Combine this with the definition of Yt in Assumption (1) above to find the price level in period t, Pt , is: t Pt = M Nt : 4. There is an upper bound on per-capita bond holdings by the public of b̄. That is, Btg =Nt  b̄. (18.2) In addition, to make life easier, we will specify that the government’s fiscal policy, which is a sequence of deficits f t g1 t=0 , is simply a constant per-capita deficit of . Thus: D d Dt = d; for all t = 0; 1; : : : ; 1: Nt g Define bt to be the level of per-capita bond-holdings bgt  Btg =Nt . Assumption 4 states that bgt  b̄ for some b̄. Notice that with the assumption that the constant per-capita output level is yt = 1, bgt is also the debt-to-GDP ratio. Also, Dt =Nt becomes the deficit-to-GDP ratio. Monetary Policy M The monetary authority (in the U.S., the Fed) produces a sequence of money stocks f t g1 t=0 . These then feed through the quantity theory of money relation (18.2) to produce a sequence of inflation rates. A monetary policy will be a choice for the growth rate of money. If the stock 18.1 Are Government Budget Deficits Inflationary? 215 of debt is growing, eventually the bond ceiling will be reached and the Fed will no longer be able to pick an inflation rate, it will be forced to provide enough seigniorage revenue to cover the government’s reported deficit. We call this the catastrophe. The catastrophe happens at date . The catastrophe date is itself as a function of choice made by the government. T T t Given money supply growth, the gross inflation rate in period is: Pt Mt Nt Pt 1 = Nt Mt The net inflation rate is defined as Pt =Pt 1 1 1 = 1 1+ Mt n Mt 1 : 1. For simplicity, assume (with Sargent and Wallace) that the Fed picks a constant growth rate for money, , in the periods before the catastrophe. Thus:  Mt Mt 1 = 1 + ; for all t = 0; 1; : : : ; T: This implies that inflation is: Pt = 1 +  ; for all t = 0; 1; : : : ; T: Pt 1 1 + n >n For , the net inflation rate will be strictly positive. If the Fed dislikes inflation, it will seek to minimize the growth rate of money t t 1 by picking a low . Such a policy will decrease seigniorage revenue in the short run (until period ), forcing the fiscal authority to rely more on bond finance of deficits, bringing closer the catastrophe date at which g = and no more bonds may be sold. From period on, the money supply expands to T produce enough revenue to satisfy the government budget constraint. b M =M b̄  T T T Analysis b Our goal is to determine the time path of per-capita bond holdings gt and to determine when (if ever) the limit of is reached. Table (18.1) lists all of the variables and their meanings. In addition, let’s list again all of the equations we know about this model: b̄ (Gov. Budget Constraint) (Fiscal Policy Rule) (Monetary Policy Rule) (Population Growth Rate) (Quantity Theory of Money) (1 + ) tg   t D B r B 1 + Mt PM t Dt=Nt = d; for all t = 0; 1; : : :; 1: Mt = (1 + )Mt 1; t = 0; 1; : : : ; T: Nt = (1 + n)Nt 1: Pt = Mt=Nt: g t= t 1 : Fiscal and Monetary Policy 216 Begin by dividing the government budget constraint (18.1) by Nt on both sides to produce: t 1 ; for all t = 0; 1; : : : ; 1: r BNt 1 + N1t Mt PM t Now, we use the fact that 1=Pt = Nt =Mt to write this as: g Nt d = bgt (1 + r) NBt 1 NNt 1 + N1 (Mt Mt 1) M t 1 t t t 1 + r M M = bgt bg + t M t 1 1+n t 1 t   M 1+r g t 1 g = bt b + 1 Mt ; for all t = 0; 1; : : : ; 1: 1+n t 1 Solving for bgt yields:   M 1+r g t 1 g (18.3) bt = 1 + n bt 1 + d 1 Mt ; for all t = 0; 1; : : : ; 1: Notice that the evolution of per-capita borrowing bgt determined in equation (18.3) holds in all periods, including those after the catastrophe period T . Before period T the monetary policy specifies a growth rate of money, Mt =Mt 1 = 1 + , so seigniorage is constant and potentially low. The remaining borrowing is done by issuing bonds. After the catastrophe date T , monetary policy must produce enough seigniorage revenue to completely meet the government’s borrowing needs, and per-capita bonds are constant at bgT = bgT +1 =


= b̄. Dt = Btg Nt Nt g (1 + ) t After the catastrophe the evolution of the money supply is determined by the post catastrophe government budget constraint, so we replace gt with : b̄ = 11 ++ nr b̄ + d b  b̄ Mt 1  ; for all t  T + 1: Mt 1 We manipulate this equation to solve for the growth rate of money: Mt = 1 Mt 1 1 d r1+nn
b̄ ; for all t  T + 1: Notice that after period T , money supply growth is increasing in the terms d and b̄. Not only does the Fed have to pay for the deficit d entirely out of seigniorage, it also has to pay the carrying costs on the public debt b̄. (18.4) Thus the money stock must evolve as: (18.5) Mt Mt 1 = ( 1+  1 ; d b̄ 1+r 1+n 1
1 t = 1; : : : ; T ; t = T + 1; T + 2; : : : ; 1: 18.1 Are Government Budget Deficits Inflationary? 217 Equation (18.5) gives us the evolution of the money supply in all periods, including those after the catastrophe. Notice that the money supply growth rate after is not affected by the value of . In other words, after the catastrophe hits, the inflation rate will be the same, no matter when it hit. T T  How much seigniorage revenue does the government raise, given , each period prior to the catastrophe? That is, what happens when we substitute in the Fed’s monetary policy into equation (18.3)? From equation (18.3): bgt = 11 ++ nr bgt 1+ d  Mt 1  ; for all t = 1; 2; : : : ; 1: Mt 1  But in the periods before the catastrophe, money growth is simply , so: (18.6) b g = 1+ t 1+ r bg nt 1 + d  1 1+ 1   = 1+ 1+ T r bg nt 1 + d  ; for all: t = 1; 2; : : : ; T: 1+ Notice this interesting result: Before period , the government takes as seigniorage a fraction (1 + ) of GDP. Any remaining portion of the per-capita deficit must be raised by net new bonds. =  b d b Finally, let’s calculate gt without reference to gt 1 . We can do this with recursive substitution from equation (18.6), using the assumption that g0 = 0: b bg1 = 11 ++ nr bg0 + d 1 +  = d 1 +  :      1+r g 1+r   g b2 = 1 + n b 1 + d 1 +  = 1 + 1 + n d 1 +  :  " 2 #    1+r  1+r g 1+r g b3 = 1 + n b2 + d 1 +  = 1 + 1 + n + 1 + n d  1+   : And so on. The pattern should be clear from these first terms. In general:  (18.7) Recall that bgt = d  1+  X t  i=1 1+ 1+ r i 1 ; for all t = 0; : : : ; T: n r > n, hence the summation term is explosive. Equation (18.7) neatly captures the Fed’s dilemma in this model. By setting a low value for , the Fed trades low inflation today for an earlier onset of the hyperinflationary catastrophe. On the other hand, by choosing a relatively high value for the Fed suffers high inflation today but staves off the catastrophe point. Indeed, if:     d;  1+ then there will be no catastrophe. Fiscal and Monetary Policy 218 Determining The Catastrophe Date T Given the time path for debt in equation (18.7), we can determine roughly in which period the catastrophe hits. I say “roughly” because to keep the algebra neat we are going to assume that, at the monetary policy , end-of-period debt gT is perfectly equal to . You can see that it is easy to imagine cases in which gT is slightly less than , in which case in period + 1 a residual amount of borrowing is allowed. However if is large, this effect is unimportant. Thus at the end of period : T  b T T b T T  d For notational convenience, let  1+  T  i X 1+  i=1 r 1+n 1 = b̄:  (1 + r)=(1 + n). Thus: TX1 j =0 j= b̄ : d 1+ Recall that the sum on the left hand side of this equation is equal to (1 1 1 where I have introduced b̄ b̄ T = d b̄  1+ = T ) (1  J; J to keep the notation down. Manipulation produces: T = 1 (1 )J: Taking logarithms of both sides produces: (18.8) T ln( ) = ln (1 (1 )J ) ; so: T (; b̄) = ln (1 ln((1 ) )J ) ; where: 1+r = ; and: 1+n J = d b̄  : 1+ Notice that T is increasing in  and b̄. Indeed, for T to be finite, we must have:  < d; 1+ so that the government must resort to bond financing. ). Thus: 18.1 Are Government Budget Deficits Inflationary? 219 Some Examples b  In Figure (18.1) we present the time path of debt, gt , under two different values of , 1 = 0 03 and 2 = 0 10. In this model = 0 02 = 0 05 = 0 10 and = 1 5. That is, the government is trying to finance a persistent core deficit of 10% of GDP and the maximum value of total debt is 150% of GDP. The government does not have to pay a very high real interest rate on its debt, but output is growing at the relatively low rate of 2% a year. With the tight monetary policy ( 1 = 0 03), the government hits the catastrophe 16 years into the policy, while with the loose monetary policy ( 2 = 0 10), the catastrophe occurs 61 years in the future. :  : n  : ;r : : ;d  : b̄ : : Time paths of bonds: bt when θ1=0.03, θ2=0.10 Time paths of inflation: π when θ =0.03, θ =0.10 1 1.5 2 0.16 0.14 0.12 1 t bt 0.1 π 0.08 0.06 0.5 0.04 0.02 0 0 10 20 30 40 50 60 70 0 0 80 t Figure 18.1: Evolution of the stock of percapita debt holdings t under two monetary policies: the solid line under the tight money ( = 0 03) policy and the dotted line under the loose money ( = 0 10) policy. b  :  : 10 20 30 40 t 50 60 70 80 Figure 18.2: Evolution of the inflation rate t under two monetary policies: the solid line under the tight money ( = 0 03) policy and the dotted line under the loose money ( = 0 10) policy.    : : In Figure (18.2) we plot the inflation rates over time associated with the two monetary policies. Notice that the inflation rate t does not quite equal the growth rate of money since:    PPt t 1 = 1 +1 n MMt t 1 : Before the catastrophe date T , inflation is constant at  where: 1+ 1 +  = ; for all t = 0; 1; : : : ; T:; so: 1+n  = 11 ++ n 1 = 1 + nn : 1+ t Fiscal and Monetary Policy 220  d b̄ Note that  will not vary with the deficit or the maximum debt load . On the other hand, the catastrophe date and the post-catastrophe inflation rate will vary with and . After the catastrophe, inflation T ( ) will not vary with the pre-catastrophe monetary policy . We can calculate T ( ) from the evolution of the money supply, equation (18.5). Thus: b̄ T   d; b̄ d  d; b̄  d; b̄) = 1 +1 n 1 d b̄1 1+r 1+n 1 1 T (d; b̄) = 1 + n 1 d b̄ 1+r 1+n 1 + T( 1 ;
so: rb̄ + d n(b̄ + 1 d) : d)(1 + n) b̄(r n) The tight monetary policy is associated with very low inflation initially,  = 0:0098 but, as noted above, the catastrophe happens relatively early. The loose monetary policy is associated with a relatively high inflation rate initially,  = 0:0784 but the catastrophe is staved off for over 60 years. After the catastrophe the inflation rate is T = 0:1455, or about (18.9) 1
1= (1 1 2 twice the rate with the loose monetary policy. Application: Optimal Inflationary Policies In this section we consider the trade-off between two monetary policies: (1) A policy of high inflation in which the catastrophe never occurs and (2) A low-inflation policy which brings forward the catastrophe date. Notice from equation (18.7) that if the government sets ?  1 d d ;  = ? , where: then each period’s seigniorage revenue is: ? = d: 1 + ?  That is, with the money supply growth rule set to ? as defined above, the government raises enough seigniorage revenue to completely finance the real deficit each period. As a 1 and the result the government never resorts to bond finance, so gt = 0 all = 0 1 2 catastrophe never happens. When = ? inflation satisfies: b   d n nd ? ? = 1 + nn = 1 d n= n t ; ; ;::: ; ;  where = + . Notice that if = (1 + ) then ? = 0. That is, the government can pay for the real deficit entirely with seigniorage revenue and have zero inflation. < ? , the government must resort to perOn the other hand, for any monetary policy sistent debt financing and eventually face the catastrophe. We know from equation (18.9) 18.2 The Ends of Four Big Inflations 221  d; b̄). By examination, we see that: T (d; b̄) > ? : above that after the catastrophe, inflation is T ( T Intuitively, by waiting until period to begin financing excess government spending by printing money the monetary authority has allowed the fiscal authority to borrow up to its limit. The Fed then has to repay creditors out of seigniorage as well.   If the Fed dislikes inflation, it has an unpleasant choice: suffer inflation of ? now or T at some future date . As you can see, the Fed’s choice of which policy to pursue depends in large part on how varies with . T T Variable Gt Tt Dt Yt Nt n r Mt Pt Btg bgt b̄ Bg 1 T  d  Definition t Real government spending at Real government tax revenues at Real government core deficit at GDP at , t = t Population at Constant population growth rate Constant real net return on debt, end-of-period stock of money at exchange rate of money for goods at real par value of outstanding end-of-period debt per-capita debt, gt = tg t maximum possible value of gt initial stock of debt g 1 = 0 “catastrophe date”–when gT = Money supply growth rate before catastrophe constant per-capita deficit (fiscal policy) t tY N t t r>n t t b B =N B b b b̄ Y Nt, variables Table 18.1: Notation for Chapter 18. Note that, with the assumption that t = denoted as per-capita are also expressed as fractions of GDP. 18.2 The Ends of Four Big Inflations The most dramatic evidence of the validity of the Sargent-Wallace argument comes from the post-WWI hyperinflations in Germany and the successor states to the AustroHungarian Empire in a paper by Sargent, “The Ends of Four Big Inflations”. What makes that case so special is that, not only was there a deficit-driven hyperinflation, once the fiscal 222 Fiscal and Monetary Policy authorities had made credible commitments to back all government debt by tax revenues, the inflation stopped (even though the printing presses were still running). These histories are valuable also because the problems facing the four nations in question bear much in common with those facing some of the successor states of the old Soviet empire. The post-war central European inflations of 1919-1924 were a new, and deeply unpleasant, experience for its citizens. It is a commonplace to ascribe modern Germany’s strong commitment to low inflation to a national horror of repeating those days. Yet it was not the abstract experience of seeing prices (and wages) climb to 1012 times their pre-war level that was so traumatic, nor was it the mild “shoeleather cost” studied in Chapters 4 and 8. As a result of the inflation, there were tremendous social dislocations as creditors were impoverished, as enterprises failed, as speculation flourished and as households hoarded illiquid assets rather than trading them for a currency whose value was essentially unknown. These were new phenomena at the time, but unfortunately since then they have been consistent hallmarks of monetary crises to the present day. Sargent identifies four characteristics common to the hyperinflationary experiences in Poland, Hungary, Austria and Germany: 1. All four countries ran very large budget deficits. 2. All four countries took very similar, very dramatic, monetary and fiscal steps to end the hyperinflations. 3. In all cases, the inflation stopped very quickly. 4. After the inflationary episodes, there was a large and persistent rise in the level of “high-powered” money. Governments ran deficits because, in the aftermath of the war, they made payments to the large numbers of unemployed workers, because state monopolies (such as railroads) kept prices artificially low and lost money, because governments subsidized basic necessities such as food and housing, and, in certain cases, because they had been ordered to pay war reparations of unknown amounts. Sargent draws a clear distinction between government actions and government regimes. An action takes the form of a one-period decision of the government (cutting the subsidy on heating oil for one month, for example), with no credible assurance that the action will be repeated. In contrast a regime is a credible commitment to a sequence of actions, for example selling off the state railroad or making the central bank independent. The solution to the hyperinflations, in all cases, was a switch in regime: governments abandoned deficits and seigniorage financing in favor of balanced budgets and independent central banks. In many cases, at least part of the credibility of the new regimes derived from international obligations. For example, in August of 1922 Austria signed agreements with the League of Nations binding her to fiscal balance and monetary stability. Exercises 223 In Germany, where the inflation was most dramatic, the largest single fiscal liability was the bill for war reparations. In the original treaty negotiations at Versailles, the Great Powers had been unable to fix a firm value on Germany’s war reparations. In theory all of Germany was mortgaged for reparations, and indeed, in 1923 France occupied the Ruhr to drive home this point. In October of that year Germany issued a new currency, the rentenmark, whose initial value was 1012 reichsmarks. Yet in 1924 the catastrophic German inflation stopped. Sargent reports several deliberate, permanent actions that constituted a regime shift. Among these, the government fired 25% of its workforce and cut employment in the state railroad system by about 180,000. Germany also negotiated a fixed, reasonable, value for its reparations bill with the treaty powers. In all of these inflations, at some point the central banks were called upon to purchase almost all of the net new national debt issues. A common reform was to prohibit the central bank from purchasing government debt. This was a statutory commitment to fiscal discipline, and a good example of the difference between regimes and actions. Once households were assured that the hyperinflationary regime was over, their holdings of currency rebounded remarkably. Thus even after the inflations had ended, governments continued to issue large quantities of new base money. This money was absorbed by households which had economized dramatically on their currency holdings during the hyperinflation. The parallel to present-day countries such as Ukraine and Russia is clear. These too are new states without a history to guide investors, with bloated public sectors and inefficient systems of tax collection. In contrast with the earlier examples, they are committed, as much as possible, to fiscal discipline, although in some cases this required defaulting on some of the government’s obligations (for example, Russian government employees must often wait months for paychecks). International organizations such as the IMF and foreign governments, just as in the 1920s, have acted as commitment devices to prevent the Russian government from using the printing press to meet its obligations. However, until either government spending obligations diminish or tax collections increase, there will be a persistent possibility of hyperinflation, with its attendant social dislocations. Exercises Exercise 18.1 (Easy) The answers to these exercises can be found in Friedman and Schwartz (1963) A Monetary History of the United States, Chapter 7, entitled, “The Great Contraction”; the Barro textbook Chapters 4 and 18 but especially Chapters 7, 8, and 17; the article by Richard D. Porter, “The Location of U.S. Currency: How Much is Abroad?”; and Sargent Rational Expectations and Inflation (either the 1st or 2nd edition), Chapter 3, entitled “The Ends of Four Big Inflations”, Fiscal and Monetary Policy 224 and Chapter 5, entitled, “Some Unpleasant Monetarist Arithmetic”. 1. What was the path of the money stock in the U.S. from January 1929 to March 1933? How did household’s holdings of currency change over the same period? 2. What was the path of real income in the U.S. from January 1929 to March 1933? How did prices change over the same period? 3. In the period 1948-1991 have American real interest rates ever been negative? In the same period, has the U.S. inflation rate ever been negative? If so, when? 4. What is the evidence that inflation, in Milton Friedman’s words, “is always and everywhere a monetary phenomenon”? In the long run? In the short run? 5. If you take the population of the U.S. to be 260 millions, roughly how many dollars of currency were in circulation for every U.S. citizen at the end of 1995? How much currency are you carrying right now? How do you account for this discrepancy? 6. Why do people hold currency and keep part of their wealth in low interest bearing accounts (like the Hyde Park Bank’s zero interest checking account)? 7. Explain how a rational expectations view of agent’s behavior (as defined by Sargent) can explain why inflation seems to have momentum, while in fact it does not. 8. What is seignorage? How much money did the U.S. raise via seignorage in 1991? 9. What is the Quantity Theory of Money? Explain the sense in which it is “just” an accounting identity. 10. What is a gold standard? True or false: Under a gold standard the quantity of money is fixed. Exercise 18.2 (Easy) Evaluate this statement: Government austerity programs cause civil unrest. Exercise 18.3 (Moderate) In each period the government raises real tax revenue of t and spends (in real terms) t . Let t  t t be the real deficit at time . At the suggestion of a revered elder whose initials are M.F., the government is allowed to finance this deficit only by issuing fiat currency and obtaining the seignorage revenue. The government’s budget constraint is thus: G D t G T t T Dt = Mt PMt 1 ; t where Dt is the real government deficit at t, Mt is the stock of money at t and Pt is the price level at t. Prices are related to the money supply by the Quantity Theory of Money relation with a constant velocity v = 1: Pt Yt = Mt: Exercises 225 Y Y Nt, where Nt is population at t, and evolves according to: Nt = (1 + n)Nt 1; with N0 = 1. The government runs a constant per capita real deficit of d, so Dt = dNt for all t. Answer the following questions: Output t satisfies t = Mt evolve given Mt 1 and d? 2. For what value of d is the inflation rate zero? That is, for what value of d will Pt = Pt 1? 3. A reasonable estimate for n is about 0.03. At this value, how large a deficit, expressed 1. How must as a fraction of GDP, can the government cover by printing money and still not cause inflation? 4. Assume d = 0. What happens to prices? Exercise 18.4 (Fun) Through a map-making error in 1992 the Absolutely Autonomous People’s Republic of Kolyastan (hereafter known as Kolyastan) was created out of the more rubbishy bits of neighboring successor states to the Soviet Union. The Kolyastani central bank is run parttime by a popular local weatherman on the state-run television station. The market for Kolyastan’s chief export, really really big statues of Lenin, seems to have collapsed. Most of its citizens continue to work in the enormous state-run Lenin Memorial Lenin Memorial factory, which is currently producing no revenue at all. The government subsidizes consumption of bread and kirghiz light (the local liquor) by paying merchants to keep their prices artificially low. The Kolyastani currency, the neoruble, is made up of old Soviet rubles with the top left corner cut off. Inflation is currently running at 400% per month. Although the Kolyastani government claims to be financing most of its big budget deficits through bond sales, most of these bond sales, it turns out, are to the central bank. In desperation the Kolyastani government have turned to you, a University of Chicago undergraduate, for economic advice. Briefly outline your plan for Kolyastan’s recovery. Be specific. How can the Kolyastani people be certain that the reforms proposed by the government will be maintained after you graduate? Chapter 19 Optimal Monetary Policy As we have discussed, expansionary monetary policies include decreases in the Fed funds rate and unexpected growth in the money supply. In the U.S., such expansionary monetary policies have tended to produce real expansions in output and increases in inflation. Conversely, contractionary monetary policies have tended to produce real contractions in output and decreases in inflation. In Chapter 18 Barro claims that these effects have been quite moderate, but recent empirical work lends support to the opposite view, that monetary shocks can have large effects on real variables in the short run. Everyone agrees that expansionary monetary policies tend to lead to increases in inflation, while contractionary policies produce decreases in inflation. At this broad level, monetary policy would appear to be a matter of trading off inflation and output. Since unemployment tends to decrease as output increases, this is often cast a choice between inflation and unemployment. The empirical relationship between the two is called the Phillips curve. In the U.S., as in most countries, monetary policy is under the control of the government. This immediately raises the question of how best to conduct monetary policy. As we shall see, this not so much a question of when and how to time expansions and contractions of the money supply, as economists used to think, as it is a question of what the private sector predicts the government will do and how the government can influence those predictions. Before we can think fruitfully about monetary policy, we will have to have a reasonable model of how monetary shocks can influence the real economy. Our model will be a simplification of the seminal paper by Robert E. Lucas, Jr, “Expectations and the Neutrality of Money”. In that model, the private sector is divided into different industries (called “islands”) which observe only the price for their own product. This price is made up of a general price level (unobserved) and an industry-specific shock (also unobserved). The private sector has some forecast about inflation (never mind for the moment its origin) and uses this to derive an estimate of the industry specific shock it faces. If the estimated shock 228 Optimal Monetary Policy is high, the private sector increases production. If it is low, the private sector decreases production. The government chooses an inflation rate. An unexpected monetary expansion will produce a temporary increase in output. Thus Lucas’s model highlights the role of expectations in the conduct of monetary policy. In Lucas’s model, only unanticipated changes in the price level have real effects. If a monetary expansion is completely expected, it has no real effects. This points to something quite important in the real conduct of monetary policy: only surprises matter. Moreover, the private sector does not enjoy being surprised, even if the monetary surprise produced a temporary boom. An older tradition in macroeconomics holds that governments should try to manipulate the money supply to cushion supply and demand disruptions. The central lesson from Lucas’s research is that governments should instead strive to minimize the uncertainty surrounding monetary policy. We then move away from the specific form of the Phillips curve derived from Lucas’s model and start using a simple generalization in which inflation, inflationary expectations and unemployment are all related by a very simple formula. The government will have some preferences (and thus indifference curves) over unemployment and inflation (both will be bad), and monetary policy, if we ignore how expectations are formed, can be seen as a simple choice of unemployment and inflation. Once we begin modeling the formation of expectations, we will see that the ability of the government to commit credibly to a particular inflationary path is critical. We will model explicitly a two-person game between the private sector and the government. With a socalled commitment device, the government will be able to play the Ramsey strategy and realize the Ramsey outcome. Recall the Ramsey optimal tax problem from Chapter 14. In that chapter we assumed that the government could commit to a particular tax sequence, hence the term “Ramsey”. We did not consider what would happen if the government could not commit to a particular tax sequence. In this chapter we will see that without a commitment device, the government and the private sector will play Nash strategies and achieve the Nash outcome. The fundamental result of this chapter is that Ramsey is better than Nash. Both the government and the private sector are better off in the Ramsey outcome than in the Nash outcome. Indeed, under certain circumstances, the Nash outcome involves (temporarily) high inflation and high unemployment, the so-called “stagflationary” episode of the 1970s. At the time, stagflation was blamed on an oil price shock. We have to reconsider, and say that possibly it was the result of a lack of credible commitment by the government. The theory in this chapter will give us an explanation for the “pain” associated with fighting inflation. There is a powerful maintained assumption in the media that policies that are anti-inflationary require some sacrifice of real output. As we shall see, when the private sector has formed strong expectations about continued high inflation, confounding those expectations with sudden, unexpected, low inflation can have a severe cost in terms of real output. This is not a reason to oppose anti-inflationary policies, it is a reason to campaign for a credible commitment to low inflation. Finally, it is worth noting that, in this chapter, we will ignore the government budget con- 19.1 The Model of Lucas (1972) 229 straint. In Chapter 18 we were very concerned about the relationship between persistent government budget deficits and inflation. In this chapter we will assume the government budget is more or less in balance, and that the government does not particularly need the seignorage revenue generated by high inflation. This is a safe assumption when thinking about inflation in the U.S. In thinking about inflation across different countries, though, the analysis of Chapter 18 is probably more appropriate in countries, like Brazil, that experience persistent inflation and large budget deficits. The “pain” of fighting inflation in those countries is the pain of raising direct taxes and decreasing government spending. Both this chapter and Chapter 18 highlight the importance of credible government policies. In Chapter 18, to stop hyperinflations the government had to credibly commit to balancing its fiscal books. In this chapter, to prevent milder inflations, the government will have to credibly commit to keep its hands off of the monetary spigot. In both cases there is a role for international institutions as commitment devices. 19.1 The Model of Lucas (1972) In this section we consider a simplified version of the important model of Lucas. We are going to get a relationship between the anticipated price level, the actual price level and something that looks like unemployment. We will use this relationship to argue for a particular functional form for the Phillips curve. We will not derive precisely a Phillips curve since our model is going to be static, to keep the exposition simple. The dynamic generalization is very elegant, and the interested reader is referred directly to the Lucas paper. This model turns on the decisions made by many separated industries in the private sector. These industries cannot communicate with one another about prices. They will hire labor according to their estimate of the true state of demand for their product. Q i L Let i be output in industry . Assume that all industries use only one input, labor. Let i be the number of workers hired in industry . Assume that all industries have the common production function: i Qi = Li ; where the technology parameter satisfies 0 < < 1. Assume that all workers are paid the common wage of unity for their unit of labor supplied. To produce an output Qi therefore 1= requires labor input (and total costs) of Li . Thus the cost function in industry i is: Total Cost(Qi ) = Li : In industry i there will be a price Pi for that industry’s output. It is known that this price is made up of two parts: a general price level P , common to all industries, and a shock term Zi specific to industry i. These terms are related by the price equation: (19.1) Pi = PZi: 1 Optimal Monetary Policy 230 Z i P The shock term i gives the real price of output in industry . The general price level will not be revealed until the end of the period, since the industries are on islands and cannot communicate during production. All private-sector industries begin the period with a common forecast of note by e . Thus an industry ’s best estimate of its real price i is: P i Zie = PPie : (19.2) i Z P P , which we de- Recall that industry only observes i . i Equilibrium in industry , assuming that it is competitive, requires that marginal cost equal estimated real price ie . Since we know the total cost curve, marginal cost must just be its derivative with respect to output i . That is, equilibrium requires: Z Q 1 Qi 1 1 Z: = ie We can solve this to produce the equilibrium demand for labor conditional on the estimated shock ie : Z Li = ( Zie ) (19.3) 1 : 1 As expected, industries will demand more labor if they estimate that demand for their product is unusually strong (if ie is large). Z Z P The estimated shock ie is comprised of two parts: the known estimate of the price level e and industry-specific price level i , related by equation (19.2). Thus we can substitute from that equation into equation (19.3) to find the industry-specific demand for labor conditional on e and i : P P P Li =  Pi  Pe 1 1 : Now we take logarithms of both sides. From now on, let lower-case variables denote logarithms. Thus i = ln( i ) is given by: ` L `i = 1 z p 1  ln Pi  = 1 Pe 1 ln( ) + p 1 1 substitute i + for i from equation (19.1) above, and let (19.4) p pe ) ; ( i A = [1=(1 )] ln( ) to produce: `i = A + 1 1 (zi + p pe ): Equation (19.4) captures the log of labor demand as a function of the (log of the) shock, the common price level and the common price forecast e . p p 19.2 Monetary Policy and the Phillips Curve u 231 Define to be the “not employed rate” (not quite the unemployment rate, but something close).1 If is the total workforce, and = ln( ), then define as: N n N X u=n i u `i: Assume for a moment that there are only two industries. Now: u=n A 2 + p p) 2 ( e 1 2 1 ( z1 + z2): Define further: u? = n 2 A; u where ? is something like the natural rate of not-employment, "= 2 1 2 = 1 ( z1 + z2); and: : Now we can write the aggregate not-employment rate as: u = u? + (pe p) + ": (19.5) We will use some version of this equation throughout this chapter. p From the point of view of the government, the common price level is a control variable. The government picks a level for with monetary policy. Notice what equation (19.5) says about the relationship of unemployment (or not-employment), the price level and the forecast price level: unemployment is decreasing in the price level but increasing in the forecast price level e . From the point of view of private industry, if the actual price level e , the industry has produced too much and suffers exceeds the forecast price level, e , it can stimulate a losses as a result. From the point of view of the government, if one-period boom in which unemployment is below its natural level. p p p p>p p>p 19.2 Monetary Policy and the Phillips Curve For the rest of this chapter we will be using a modified version of equation (19.5). Assume that: u = u? + (e ): (19.6) 1 The unemployment rate is 1 (1=N ) P L which doesn’t translate well into logarithms. i i Optimal Monetary Policy 232 u u  Here is the unemployment rate, ? is the “natural rate” of unemployment, e is the expected inflation rate and is the actual inflation rate. The natural rate of unemployment is the level of unemployment when inflation is perfectly anticipated, so no industries are fooled into thinking that relative demand is unusually high or low. The slope of this Phillips curve where we assume 0 (monetary expansions reduce unemployment). If we think is that there is uncertainty about the state of the real economy, we can add a mean zero shock term, , to produce:  > " u = u? + (e ) + ": For the most part we will assume that the monetary authority knows the state of the real economy with certainty.  In Figure (19.1) we plot Phillips curves with two different values of expected inflation e , a low value in which the expected inflation rate is zero, and a high value, in which the expected inflation rate is 8.3%. The dotted line gives the natural rate of unemployment (here ? = 5%), and = 0 3. Notice that when inflationary expectations are high, to achieve any given unemployment rate requires a higher inflation rate, and to achieve zero inflation requires an unemployment rate well above the natural rate. : Phillips curves 0.25 0.2 Inflation rate π u 0.15 0.1 0.05 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Unemployment rate u Figure 19.1: Phillips curves under two different expectations about inflation. The bottom curve assumes e = 0 and the top curve assumes e = 0 0833. The dotted line gives the natural rate of unemployment.   : 19.2 Monetary Policy and the Phillips Curve 233 Monetary Policy with Fixed Expectations Assume that the government (or which ever arm of the government controls monetary policy) has a utility function over unemployment and inflation of g ( ) given by: 2 2 g( )= V u;  u : V u;  That is, the government dislikes unemployment and inflation equally. We will assume this form for for the rest of the chapter, so it’s worth mentioning that the Federal Reserve Board is, by law, supposed to balance the twin goals of full employment and price stability. Thus this utility function seems to be written in law. V  If we assume that e is given exogenously and fixed, we can substitute the Phillips curve in equation (19.6) into the government’s utility function above to produce a maximization problem. Thus if the private sector has fixed expectations about the inflation rate given by e , then the government’s optimal choice of inflation is given by:  2 [ ?+ ( e )]2 max   u     : The first-order condition with respect to inflation  is: 2 [u? + ( e  )] 2 = 0: We can solve this for  to get the optimal inflation choice when expected inflation is fixed at  e and the natural rate is u? (call it  ? ( e )): ?(e ) = 1 + 2 (u? + e): We can plug  ? ( e ) into the Phillips curve in equation (19.6) to produce the associated unemployment rate, u0 ( e ): u0(e ) = 1 +1 2 u? + 1 + 2 e : Notice that if  e is “small” that u0 ( e ) will lie below u? . The government trades off some inflation for a lower unemployment rate.    We plot ? ( e ) in Figure (19.2) below. Notice that for low values of expected inflation, e , the government chooses inflation rates above expectations and for high values of e , the government chooses inflation rates below expectations. At one unique expected inflation rate, the government’s best response is to choose an actual inflation rate exactly equal to the expected inflation rate. This will play a special role, as we shall see.  Two Stories About Inflationary Expectations We are not yet ready to discuss the strategic interactions between the private sector and the government that determine inflationary expectations. However, we can study the outcomes under two different stories about inflationary expectations. These will help us to Optimal Monetary Policy 234 * e Government’s best inflation choice: π (π ) 0.5 0.45 Inflationary choice: π 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 e 0.4 0.45 0.5 Inflationary expectations: π Figure 19.2: Government’s optimal choice of inflation ? as a function of different expectations of the private sector (solid) and the 45-degree line (dotted). Where the two lines cross is the inflationary expectation at which the government’s best response is to meet expected inflation. In other words: the Nash inflation level.  think about the government’s problem. First, we will assume that expectations are fixed, but that the private sector knows the government’s maximization problem. If this is the case, then the private sector will set expectations to a unique value such that the government chooses to set inflation at exactly the same value the private sector anticipated. Second, we will assume that expected inflation exactly equals actual inflation in all cases. The private sector has a crystal ball (or a spy) which informs it precisely of the government’s inflationary plan, no matter what the government picks. Imagine for a moment that the private sector understands the government’s maximization problem and correctly anticipates inflation. That is, assume that inflationary expectations satisfy:  e =  ? ( e ): From Figure (19.2) below, we see that there is exactly one such expected inflation rate. Expanding produces: e = 1 + 2 (u? + e ):  We can solve for this special value of e , call it (19.7) 1 = 1, to get: u?; 19.2 Monetary Policy and the Phillips Curve  235  where 1 is unique inflation rate such that when expectations satisfy e = ment’s inflation target is also 1 . The associated unemployment rate is: = ?  u1 u ;    1 , the govern- u since e = . Thus the unemployment rate is at the natural rate ? and inflation is relatively high at 1 . This will be the Nash equilibrium in inflation (as we shall see below). Now imagine that the government is forced by law to correctly announce its inflation target each period. The private sector anticipates this and sets e = . Thus the Phillips curve in equation (19.6) becomes: = ?+ ( )= ?  u u    u: In other words, inflation does not affect output. If this is the case, the government chooses an inflation rate of zero (since inflation is costly and now provides no benefit), and the unemployment rate again goes to the natural rate. This will turn out to be the Ramsey equilibrium as we shall see below. Contrast the Ramsey and the Nash equilibria. Both produced the natural rate of unemployment, but the Nash equilibrium also had a high inflation rate. Thus the government and the population are better off if the government is able to announce the inflation rate and be believed. As we shall see below, unfortunately, when the private sector expects inflation to be low, there is a temptation for the government to inflate. Ramsey Monetary Policy This last example was the Ramsey problem. If the government can credibly commit to a particular inflation rate, the private sector responds by setting inflationary expectations to the announced inflation target. As a result, the government announces an inflation target of zero, and the result is the natural rate of unemployment. What are some commitment devices? By making the monetary authority completely independent of the fiscal authority it can be insulated from political pressure. Further, if the central banker has a reputation for being an unpleasant misanthrope who cares only about defeating inflation, the private sector can become convinced over time that in fact the central bank will set = 0 for all time.  Indeed, one reading of the deeply unpleasant recession in the early 1980s is that the private sector had to be convinced of the new central banker’s commitment to low inflation. Paul Volcker arrived as Chairman of Federal Reserve Board at a time of high inflation and high unemployment. He announced that there would be low inflation in the future. The private sector did not adjust its expectations, but Volcker followed through on his promise. The result was the unusual case in which inflationary expectations exceeded actual inflation, . As a result, unemployment shot above its natural rate in one of the deeper that is e recessions of the century. After two years of this treatment, the private sector adjusted its expectations, convinced that Mr. Volcker was committed to low inflation.  > Optimal Monetary Policy 236 Other countries, without the benefit of the tradition of anti-inflationary policies of the Fed to reassure the private sector, will completely let go of the reins of monetary policy. In Hong Kong, for example, the local currency is pegged to the U.S. dollar in an arrangement known as a currency board. For every 7.8 Hong Kong dollars issued, one U.S. dollar must be placed on deposit, so the currency is fully backed. The Hong Kong government cannot print money. Thus the exchange rate is immutably fixed, and there can be no depreciation of the local currency against the U.S. dollar. Countries will go to great lengths to convince the private sector that they are really committed to low inflation. They have to work so hard at it, we shall see, precisely because, if expectations are low, there is always a temptation to inflate. 19.3 Optimal Monetary Policy without Commitment: The Nash Problem In this section we will explicitly model the strategic interaction between the private sector and the government when forming inflationary expectations. We will force the government to choose from only two possible inflation levels, and the private sector to pick from only two possible inflationary expectations. The results we derive here generalize to the case in which both choose from continuous distributions.  ; Inflation can only take on one of two values: f0 1 g. That is, inflation can be zero or the high level we derived in equation (19.7). The private sector expects e which can also only take on the values f0 1 g, since it wouldn’t make sense for the private sector to anticipate inflation rates that the government can’t pick. ;   ; There are four possible combinations of expected and actual inflation, f e g. At each one of these four combinations we will specify the payoff to the private sector and to the government. These payoffs will be known by both players. We will look for a Nash equilibrium, which is simply a pair of choices (one for the private sector, one for the government) such that, given the other player’s choice, no player can do better. V  ; We now consider each of the four possible combinations. Let g ( e ) be the payoff to the government and p ( e ) be the payoff to the private sector at each possible f e g combination. We will assume that the private sector suffers a penalty of 1 if it does not correctly forecast the inflation rate and gets a payoff of zero otherwise (this is just a normalization). We assume that the baseline government payoff (at zero inflation and the natural rate of unemployment) is 0, and that otherwise the government dislikes inflation and unemployment. At each of the four possible outcomes, the payoffs of the two players are: V  ;  ; 19.4 Optimal Nominal Interest Rate Targets Private Sector ee = 0  = 1 237 Government = 1 g=1 p= 1 g= 05 p=0 1 =0 V gg = 0; V p =p 0 V = 1; V =   V ;V V : ;V   ; Notice that the government really dislikes f e = 1 = 0g; this corresponds to the Volcker play of low inflation when expectations are high. The result is unemployment above the = 1 g; here natural rate. Also, the government dislikes (but not as much) f e = 1 inflation is high, but unemployment is at the natural rate. The government would prefer to be at f e = 0 = 1 g; here inflation is unexpectedly high, so unemployment is below the natural rate.    ;  ;  Now let us work through these payoffs to find the Nash equilibrium. If the household plays e = 0, the best response of the government is to set = 1 . If the household plays e = 1 , the best response of the government is to play = 1 . If the government plays = 0 the best response of the household is e = 0, but this is not a Nash equilibrium since, if the household does play e = 0, we saw that the government will want to deviate to = 1 . If the government plays = 1 then the household’s best response is to play e = . Since = 1 is the government’s best response to a household play of e = 1 , this is the only Nash equilibrium in this example.                     The Nash equilibrium then is unemployment at the natural rate combined with high inflation. Compare this to the Ramsey outcome of unemployment at the natural rate and inflation of zero. 19.4 Optimal Nominal Interest Rate Targets In this section we will consider the government’s optimal choice of nominal interest rates. We will consider the real cost of inflation, whereas previously we had simply taken it as given that the government disliked inflation. We will use the simple inventory model of cash holdings from Chapter 4 to show that households are best off when the nominal interest rate is zero. This is a form of what is known as the Friedman rule. It appears frequently in monetary economics. R r Recall that the nominal interest rate , the real interest rate and the expected inflation rate e are related by the Fisher formula: = + e . For this discussion we will take the real interest rate as fixed and beyond the control of the government. Furthermore, we will assume that the government cannot directly manipulate inflationary expectations, and that the private sector correctly forecasts inflation. That is: e = . Thus the government influences the nominal interest rate only through its choice of the actual inflation rate, . The  R r     Optimal Monetary Policy 238 intuition behind the Fisher formula is quite compelling: households demand a premium of for holding assets denominated in money, which is losing value at the rate of inflation.  In our model there will be no production. Households own a stock of interest bearing assets, which earn a nominal rate of return of , and a stock of zero interest money. Money must be used for transactions. There is a fixed cost of of converting the interest bearing assets into money, which must be paid every time the household goes to the bank to replenish its cash inventory. The household has real consumption at a rate per period which it does not vary. R  c x =x The household goes to the bank times in one year, so it goes 1 of a year between trips to the bank. To have enough cash on hand to meet its consumption requirement per period at every trip. over those 1 periods, the household has to withdraw a real amount Thus average real cash holdings over the entire year are (2 ). Those cash balances could have been invested in interest bearing assets earning an amount over the year. Thus the foregone interest cost is: ( ) (2 ). Each time the household goes to the bank to replenish its cash inventory, it incurs a real cost of . Thus, transactions costs are: . Total costs for a particular policy are: =x c= x Rc = x c=x R  x c x x + R 2cx : The household minimizes total costs. The minimization problem has a first order condition of: cR 1 +  = 0: 2 x2 Solving for x produces Baumol and Tobin’s famous square-root rule for trips to the bank: s x = cR : 2 x ! ; c; R p !(; c; R) = 2cR: We can plug the household’s decision back into its cost function to determine the house): hold’s annual cash management costs, (  It is increasing in the fixed charge of going to the bank, , the rate of consumption, the nominal interest rate . R c and R A benevolent government that wishes to minimize the household’s costs by choice of would clearly choose to set = 0. At this interest rate, the household goes to the bank only once in its lifetime and incurs no interest penalty for holding money. This is because money also earns in a real interest rate of . This can only be the case if inflation is negative. From the Fisher formula = + , we see that = 0 implies that = . So if = 0, money is a perfect substitute for bonds. Holding a dollar isn’t so bad, because next year the household will be able to purchase more with that dollar than it can now. R R r  r R  r R Exercises 239 Although this model is quite limited, it points to one of the important real costs of inflation. Inflation causes households to engage in privately useful but socially useless activities. In times of high inflation, households find it in their interest to spend time and real resources economizing on cash balances. Notice that this is the first indication we have had this chapter that perfectly anticipated inflation is harmful. If the idea of negative inflation rates seems outlandish, think of the Friedman rule instead as advocating paying interest on money. It is difficult (though not impossible) to pay interest on cash holdings (C. A. E. Goodhart, an English central banker, suggested having a lottery based on cash serial numbers), it is quite easy to pay interest on demand deposits. Exercises Exercise 19.1 (Easy) True, False, or Uncertain (and explain): 1. The Consumer Price Index overstates increases in the “true” cost-of-living index. 2. Inflation is bad because to fight it the Fed increases interest rates, which hurts Americans. Exercise 19.2 (Easy) Do governments prefer Phillips curves that are relatively flat (low value of ) or relatively steep (high values of )? Exercise 19.3 (Moderate) Assume that the government has a payoff over inflation  and unemployment u of: V g (u; ) = u2 2: >  Here 0. The larger is, the nicer the central banker (that is, the more the central banker cares about the unemployed. Assume that there is a Phillips curve of the form in equation (19.6). Answer the following questions:  1. Assume that inflationary expectations are fixed at e . Find the optimal inflation rate choice of the government, 0 ( ).   2. For fixed inflationary expectations, find the corresponding choice of unemployment rate, 0 ( ). u  3. Now assume that the private sector is aware of the government’s maximization problem and knows perfectly. Find the inflation rate 1 at which expectations are met. What is the associated unemployment rate, 1 ?  u  Optimal Monetary Policy 240 Variable Qi; qi Li; `i P; p Pi ; pi Zi ; zi P ;p Z ;z N; n A u e e e e i i u?  e ?  ( e ) V u; ) g( Vp 1 u1 Definition i Output (and its log) in industry . Employment (and its log) in industry . Production parameter common across industries. Common but unobserved general price level (and its log). Observed price (and its log) in industry . Shock (and its log) specific to industry , also industry ’s relative price. Common price forecast (and its log). Estimated industry-specific shock and its log. Population and its log. Parameters (used to make notation neat). In Lucas model: the “not-employment” rate, elsewhere, the unemployment rate. The natural rate of unemployment, that is, the rate of unemployment when all industries correctly estimate their specific shocks. Slope of Phillips curve. Actual inflation (chosen by government). Expected inflation (chosen by private sector). Government’s optimal choice of inflation when inflationary expectations are e . Government’s preferences over unemployment and inflation (it dislikes both equally). Private sector payoff: industries dislike making errors in estimating the inflation rate. Inflation rate at which expected and chosen inflation coincide, the Nash equilibrium. Unemployment rate at Nash, just equal to ? . i i i i  u Table 19.1: Notation for Chapter 19 4. Would you prefer to live in a country whose government has a high value of low value of ?   or a Exercise 19.4 (Moderate) For this exercise, we will consider what happens when the government and the private sector repeatedly interact. Unemployment in period t , inflation t and inflationary ex- tu  Exercises 241  pectations te are related by the simple Phillips curve: ? e t= t+ ( t t ) for all = 0 1 u u   ; t ; ; : : : ; 1: The parameter is fixed over time. The government knows about the Phillips curve, but the private sector does not. The government has preferences over unemployment and inflation in period of: g 2 2 1 t t for all = 0 1 t ( t t) = t V  ;u u  ; t ; ;::: ; : The private sector sets inflationary expectations based on last period’s inflation. This is known as adaptive expectations. As a result, te is given by: e 1 t = t 1 for all = 1 2 Assume that e = 0, that is, the private sector begins by believing that inflation will be zero.    ; 0 t ; ;::: ; : Answer the following questions:  1. Assume that the government takes as given expectations in a period te and picks the inflation rate t which gives it the highest payoff in period . Find the government’s choice rule t? ( te ). 2. If the government sets inflation t = ? ( te ), how do expectations evolve over time? Thus right down a law of motion for inflation, t ( t 1 ).    t      3. What do the trajectories of inflation and unemployment look like over time? Are they rising or falling? Do they settle down? If so, where? 4. How would your answer have been different if, instead of the initial expected inflation being zero, it had been some very large number instead? " 5. Now assume that the Phillips curve is augmented with a mean zero shock term, , so: ? e t= + ( t t) + t u u   " ": Assume that the government knows the value of t and reacts appropriately. Now what happens? Exercise 19.5 (Easy) To answer this exercise, you need to answer Exercise 19.4 above. Imagine that the private sector has adaptive expectations about the government’s inflationary policy over time, but that part of expected inflation is the government’s announced inflation target. This announced inflation target is merely an announcement and has nothing to do with reality. If a t is the announced target for period- inflation, expectations satisfy: e ) ta for all = 1 2 1 t = t 1 + (1   Æ <Æ< t Æ ; t ; ;::: ; : Here 0 1 is a parameter indexing how much weight the private sector puts on the government’s announced inflation rate target. Assume that the government lies constantly, and announces ta = 0 always. Assume that the government, as in Exercise 19.4, always chooses the inflation rate that maximizes its one-period payoff. Find the steady-state levels of inflation and unemployment.  Solutions to Exercises Exercise 1.1 You may have noticed that this question glosses over the compounding issue. You were intended to assume that the APR was quoted as a simple interest rate. Accordingly, the daily interest rate is just: :8%  0:046027%: R = 16365 Exercise 1.2 This exercise glossed over the compounding issue again. Assuming no compounding over the week, the interest rate is:  R = 1;25000  : (52) = 1 3 = 130% : Exercise 1.3 The key to this question is that the units you use to measure time in the exponent are the same units of time for the resulting interest rate. For example, if you measure in years, then solving for gives you an annual interest rate. If you measure in “quarters”, then will be a quarterly interest rate. R R n Since this question asks you to annualize the answer, you want to measure time interval is 3 months, which is 1/4 of a year. Accordingly: : h n n in years. The i e(R)(1=4) (156:7); so: ln(156:7)]  0:02798104 = 2:798104%: 157 8 = R = 4[ln(157:8) Exercise 1.4 You do not want to annualize these interest rates, so you measure R1 = ln(156:7) R2 = ln(157:8) R3 = ln(158:6) R4 = ln(160:0) 1st quarter: 2nd quarter: 3rd quarter: 4th quarter: : : : : : : : : : : : : n in quarters, i.e., n = 1: ln(155 7)  0 00640207 = 0 640207%. ln(156 7)  0 00699526 = 0 699526%. ln(157 8)  0 00505690 = 0 505690%. ln(158 6)  0 00878851 = 0 878851%. Solutions to Exercises 244 You can see that by adding these four lines together, all but two terms cancel, leaving: = ln(160 0) ln(155 7)  0 02724274 = 2 724274%. And of course, this is precisely the formula for the annual growth using a continuous interest rate. R : : : : Exercise 1.5 h i e(0:02)(n) (GDP); so: = 34:66 years: n = ln(2) 0:02 (2)(GDP) = The Rule of 72 says that it should take about 72/2 = 36 years, which is pretty close (it is off by about 3.9%). Of course, you are smart enough to look at: n = ln(2) 0:02 and notice that a better rule would be the “Rule of 69”, but nobody is very good at dividing into 69 in their head. Exercise 1.6 The first thing you need to do is calculate the number of whole (i.e., undivided) years this investment will require. There is some number of years such that: n : ; < ; ; (1 + 0 065)n($10 000) $15 000 but where: (1 + 0 065)n+1($10 000) $15 000 This implies that : ; > ; : n = 6. After the 6th year, the investment has grown to: (1 + 0:065)6($10; 000) = $14; 591:42: That becomes the principal of the investment in the 7th year, since interest was able to compound at the end of the 6th year. Now you need to figure out the number of days of simple interest the investment will need in the 7th year. It is short of $15 000 by $408 58, and each day the investment earns: ;  : 0 065 365  ; : : : : ($14 591 42) = $2 60 You use these facts to calculate the required number of days: : : : $408 58 = 157 23 $2 60 : Since interest only accrues after a full day, the investment would not earn the interest from the last 0 23 days until 158 days had passed. All in all then, the investment would require 6 years and 158 days. : Exercises from Chapter 2 245 Exercise 1.7 Let t be the acreage of forest in year . Then: F t Fn = F0 (1 0:046)n: You are looking for the n such that Fn = (0:5)F0 . Plugging this into the equation and taking logs of both sides yields: n  14:72, so half will be cleared in 15 years. Exercise 1.8 1. The relevant formula is: 679 e100R = 954: R  3:4  10 3. 2. Letting x be the number of years, the relevant formula is: 2exR = 679; 000; 000: You use R from the previous part. Solving yields x  5777, so they would have left so the Garden of Eden in about 4075 BCE. (Population growth was probably slower in the past, so this is likely not early enough.) Exercise 1.9 1. You are solving for n in the following equation: $10; 600(1:047)n = $15; 400(1:017)n: This implies that n  12:85 years, so the incomes would be the same sometime in 1996. (Nb: They were not. Japan’s income was still less at that date.) n 2. Just plug in the value of : ; : ; $15 400(1 017)12:85  $19 124 Exercise 2.1 c l ; l n w: max fln( ) + ln( )g such that: c;l;ns = 4 0s:5 + (24 s) c n Exercise 2.2 First, write out the maximization problem: max c;l  c l)1 ; such that: c = y = Al : (1 : Solutions to Exercises 246 Plug the constraint into the objective to get an unconstrained maximization problem:  max ( l Al : l )1 ) (1 There is only one first-order condition: l (FOC ) Al [ ( ?) ] 1 [ A (l ? ) 1 l )1 ](1 Al = [ ( ? ) ] (1 )(1 l? ) : After a bunch of canceling and rearranging, this reduces to: l? = 1 + : c y = f (l) constraint yields: Plugging this back into the = c? = A   1 + : Exercise 3.1 1. The marginal period utility is: u0(ct) = 12 ct : 1 2 Plugging this into equation (3.7) yields: c c 1 ? 2( 1) 1 ? 2( 2) 1 2 R; = (1 + ) or: 1 2  ?  12 c2 c?1 c c R: = (1 + ) b 2. We have three unknowns: ?1 , ?2 , and 1? . The three equations relating them are: the Euler equation above and the two budget equations. Solving these is an unpleasant exercise in algebra. Solve the Euler equation for ?2 : c?2 = c c 2 (1 + R)2 c?1 : Use this to remove 2? from the second-period budget: Py2 + b1?(1 + R) = P  2  (1 + )2 ?1 R c : c Solve this for 1? , and plug the result into the first-period budget:   ? Py1 = P PyP2 +2(1b1 +(1R+)2R) + b1?: Exercises from Chapter 3 247 This looks awful, but it reduces to: P [y2 + y1(1 + R)] ; 2 (1 + R)](1 + R) which is the answer for the household’s choice of b1 . Plugging this back into the first-period budget gives the optimal c1 : + R) c?1 = [1 +y22+(1y+1(1R)](1 : + R) Finally, we plug the answer for c?1 into the second-period budget equation to get: 2 c?2 = (1 +1R+)[y22(1++y1R(1) + R)] : 3. In equilibrium, b?1 = 0, so: + R? )] Py1 = [1 +P [y22(1++y1R(1?)](1 ; + R? ) Solving for R? yields:   y 2 ? R = 2 y1 1: 4. From the above equation, we see that an equal percentage increase in y1 and y2 will have no effect on the equilibrium interest rate R? , just like under logarithmic preferb1? = Py1 [1 + 1 2 ences. Exercise 3.2 1. @R? = @ y2 < 0: 2y 1 Greater impatience means decreases (say, from 0.95 to 0.9), and opposite direction, so the equilibrium interest rate increases. 2. R? moves in the @R? = y2 < 0; @y1 y12 so smaller first-period income causes the equilibrium interest rate to increase. Exercise 3.3 1. c c ; c s e; c e Æ s: max fln( 1 ) + ln( 2 )g subject to: c1 ;c2 ;s and: 1+ = 1 ) 2 = 2 + (1 Solutions to Exercises 248 2. The Lagrangean for this problem is: L = ln(c1 ) + c ln( 2 ) + 1 [e1 c1 s] + 2 [e2 + (1 Æ)s c2 ]: The first-order conditions are: (FOC c1) 1 c1 1 = 0; c2 2 = 0; and: 1 + 2 (1 Æ) = 0: c2) (FOC s) (FOC We also have two first-order conditions for the Lagrange multipliers, but we leave those off, since they just reproduce the constraints. We can quickly solve the above equations to remove the Lagrange multipliers, giving us: c2 = c1 Æ ): (1 We combine this with our two constraints to get: s= c1 = e1 (1 (1 (1 (1 c2 = e2 + Æ)e1 e2 ; Æ)(1 + ) Æ)e1 e2 ; and: Æ)(1 + ) (1 Æ )e1 e2 : 1+ Æ 3. We just take the derivatives of the above answers with respect to : e2(1 + ) @c1 = @Æ (1 Æ)2(1 + @c1 = e1 < 0; @Æ 1 + @s = e2(1 + ) @Æ (1 Æ)2(1 + )2 < 0; and: )2 < 0: Æ Here, Maxine has learned how to defend against rats, so we are interested in going down. The negative derivatives above imply that all the choices change in the opposite direction, so consumption in both periods and first-period saving all increase. Exercise 3.4 ( max s0 ;::: ;s4 4 X t=0 ) c ; such that: t ln( ) t ct = (1 st)xt ; for t = 0; : : : ; 4; xt+1 = (1 + )sxt; for t = 0; : : : ; 4; and: x0 is some given constant. Exercises from Chapter 4 249 Exercise 4.1 1. Take the derivative of real money demand with respect to the interest rate @(R; c; =P ) =  1  c  2  @R 2 PRc 1 2  R:  PR2c < 0; 2 so the interest rate and real money holdings move in opposite directions. An increase in the interest rate causes the consumer to hold less real money. c 2. Differentiate with respect to : @(R; c; =P ) =  1  c  1   2  @c 2 2 PRc   = = PRc 1 4 PRc > 0; 2  PRc 2 c PRc  + 2 1 4  12  2     12 1 2   2 1 2 2 PRc  12    12   1 2 1 c so consumption and real money holdings move in the same direction. If the consumer consumes more, then the consumer will hold more real money. 3. First, we replace =P with , giving us:    (R; c; )= Taking the derivative with respect to @(R; c; =P ) = @ 1 2 gives us:     1 2  12 c Rc : 2 c 1 2 2 Rc  1 2   Rc > 0; 2 so real money holdings and real transactions costs move in the same direction. If the consumer faces higher real transactions costs, the consumer will hold more real money. Exercise 5.1 The budget constraint for the first period was given by: (3.2) Py1 = Pc1 + b1: The condition for clearing the goods market in the first period was: (3.10) This implies Ny1 = Nc2: y1 = c1. Plugging this into (3.2) gives: Pc1 = Pc1 + b1; or: b1 = 0; which is the market-clearing constraint for bonds. Solutions to Exercises 250 Exercise 5.2 The price for a good can only be zero if all consumers are satiated with that good, that is, if they cannot increase their utility by consuming more of it. In our model this is ruled out because all utility functions are strictly increasing in all arguments. This implies that the consumers always prefer to consume more of each good. If the price for a good were zero, they would demand infinite amounts, which would violate market-clearing. Therefore, with strictly increasing utility functions, all prices are positive. If utility is not strictly increasing, zero prices are possible. In that case, Walras’ Law might not hold, because total demand by consumers can be less than the total endowment. The proof of Walras’ Law fails once we use the fact that the price of each good is positive. On the other hand, the First Welfare Theorem still goes through, since it does not rest on the assumption of positive prices. Exercise 6.1 1. The first-order condition with respect to da is: l l : l w = 0: (0 5)( da?) 0:5 (FOC da ) l l Solving for da? yields: da? = 4w1 2 . The farm’s profit is: l ( da? )0:5  wlda? = 4w1 2 0:5 w = 1 = a? : w 2 4w 4 l 2. The first-order condition with respect to db is: l : l (0 5)(2)( db?) 0:5 (FOC db ) l w = 0: l Solving for db? yields: bd? = w12 . The farm’s profit is: l 2( bd?)0:5 wl d? b =2  1 w2 0:5 w 1 b? w2 = w =  :  j a 3. For economy, we will work this out for an unspecified j? , where is either or . We’ll plug those in later. Substitute the constraint into to objective in order to eliminate j . This gives us: b c  max ln( ljs wlsj + j? ) + ln(24 lsj ) : We carry out the maximization: l (FOC sj ) w wl  lj? = 0; so: l w wlsj? + j? : + j s + j? 24 j? (24 s ) = 1 Exercises from Chapter 6 251 l Solving for sj? yields: lsj? = 24w2w  : j? (S.8)  For this exercise, we are using a? , so we plug that in to get: lsa? = 24w2w 4w 1 w2 : 1 = 12 8  4. We can just re-use equation (S.8), but this time we plug in b? , yielding: lsb? = 24w2w 1 w2 : 1 w = 12 2 5. 700 800 ld? = 400lda? + 700ldb? = 4400 w2 + w2 = w2 : 6. ls? = 400lsa?+ 700lsb? 1 = (400) 12 8 ; = (1 100)(12)  w2  + (700) 12 1 2 w2 :  w2 400 l l 7. We want to set d? = s? , or: 400 w2 = (1; 100)(12) w2 ; which reduces to w2 = 1=11, or w  0:3015. 800 Exercise 6.2 1. We take the first-order condition of equation (6.9) with respect to d : l @ =  7  Ak (l?) d @ld 10 l 3 10 (FOC d ) 3 10 l When we solve that for d? , we get: l ? d=  A k w 7 10 10 3 w=0 Solutions to Exercises 252 2. The result is as follows: ? = Ak (ld?) 3 10 wld? 7 10 " # 107    A 7A = Ak w 10w k w k     7A 7A = Ak k w k 10w 10w      7A 7A A w =k 10w 10w    3A 7A = k: 10 10w 3 10 10 3 7 10 7 3 3 10 10 3 10 3 7 10 7 3 7 3 3. c ls) g; subject to: c = ? + wls : maxf 2 (1 c;ls 1 1 2  4. To begin, we can leave the ? term in the Lagrangean: L=c 1 2 ls ) + [? + wls c]: 1 2 (1 Our first-order conditions are:   c (FOC ) l c 1 2 ( ?) c 1 2 (1 ls?) + ? [ (1 ls?)   1 ( ?) 2 (FOC s ) 1 2 1 2 1 2 ; 1] = 0 and:  w ( 1) + ? [ ] = 0 :   We leave off the FOC for . Combining the above FOCs to get rid of ? yields: c? = w(1 ls?): We plug this result and our expression for  ? into the budget equation  + wls = c, (S.9) yielding:  l A   7A  k = w(1 l?) wl?: s s 10w 7 3 3 10 Solving this for s? gives us: l " ? 1 s=2 1 3 7  # A k : w 7 10 10 3 Exercises from Chapter 7 253 l When we plug this value of s? back into equation (S.9), we get the optimal consumption ? : c c l l ?= w "  3 1+ 2 7 w "   7A 1 k=2 1 10w? # A k : w 10 3 7 10 w 5. We just set s? = d? , solve for , and call the result ? : 10 3 3 7  A w 7 10 ?  103 # k : After a bunch of algebra, we get: w? =  A 7 10 k 17 7 3 10 ! : w k @w? =  7A   3   17k   17  =  51A   17k  : @k 10 10 7 7 100 7 Since this derivative is positive, w? increases as k does. 6. We are interested in the derivative if ? with respect to : 7 10 7 10 7. The U.S. has a much larger stock of capital (per capita) than Mexico does. According to this model, that difference alone causes wages to be higher in the U.S. From the equation for the equilibrium wage ? , we see that increasing the per-capita capital stock by a factor of two causes the wage to increase, but by less than a factor of two. Hence, wages between the two countries differ by less (in percentage terms) than their per-capita capital stocks. k w Of course, owners of capital try to export it to wherever labor is cheapest. In this case, the households in the U.S. try to send some of their capital to Mexico in order to take advantage of lower wages there. If this movement of capital is restricted, then the wage difference will persist, and there will be an incentive for workers to move to the country with more capital. In this case, Mexican workers will see higher wages across the border and will immigrate to the U.S. where they will earn more. Exercise 8.1 According to the quantity theory, the inflation rate is approximately equal to the difference between the growth rate of money supply and the growth rate of output. Since the question assumes that velocity is constant, the quantity theory applies. The annual rate of inflation is therefore two percent. Exercise 8.2 In Chapter 8 we determined that velocity is inversely related to the time spent between two trips to the bank. In Chapter 4 we saw that the time between two trips to the bank Solutions to Exercises 254 decreases when the nominal interest rate increases. Therefore velocity and the nominal interest rate are positively related. In Section 8.3 we found out that inflation and nominal interest rates are positively related. Therefore, a high inflation rate results in high nominal interest rates and high velocity. This is also true in the real world: velocity is much higher in countries with high inflation than in countries with moderate inflation. Intuitively, high inflation means that money quickly loses value. It is therefore not attractive to hold a lot of money, so money circulates quickly. In countries with hyperinflation, wages are often paid daily, and workers usually spend wages the same day they receive them. Exercise 9.1 Of course, the solution depends on the country you pick. As an example, Figure S.3 displays GDP and its trend for Germany. You can see that the trend does not look that much smoother than the actual series. This shows that our method of computing the trend is not especially good. 16000 14000 GDP per person 12000 10000 8000 GDP Trend 6000 4000 2000 0 1950 1955 1960 1965 1970 1975 1980 1985 1990 Years Figure S.3: GDP and Trend Exercise 9.2 Figure S.4 shows the cyclical component for Germany. Your business cycle should look similar, unless your country is a former member of the communist block. Those countries either had radically different business cycles, or, more likely, they adjusted their statistics in order to get nice, smooth figures. Exercise 9.3 For Germany, there are ten peaks in the cyclical component. The duration of a full cycle is between three and six years, with the average slightly above four years. The overall Exercises from Chapter 9 255 0.06 0.04 0.02 0 1950 1955 1960 1965 1970 1975 1980 1985 1990 -0.02 -0.04 -0.06 Figure S.4: The Cyclical Component amplitude of the cycles is relatively stable. Although there are some general similarities, the cycles are of quite different shape. The process generating the cycles seems not to have changed much, however. The cycles in the fifties and sixties are not much different from those in the eighties and nineties. Exercise 9.4 By using the resource constraints, we can write the problem as: max ln( p Bkt + t it) + A ln((1 Æ)kt + it): The first-order condition is: 0= i Solving for t , we get: pBk +1  i + (1 ÆA)k + i : t t t t t p it = A[ Bkt +1+t]A (1 Æ)kt : c Using the resource constraint for the first period, we can solve for t : pBk +  + (1 Æ)k t t t ct = : 1+A Solutions to Exercises 256 Exercise 9.5 The derivatives are: @it A @t = 1 + A = 0:8; and: @ct = 1 = 0:2: @t 1 + A The numbers correspond to the value A = 4 that is used for the simulations. Investment reacts much stronger to shocks than consumption does, just as we observe in real-world data. Exercise 9.6 Figure S.5 shows consumption and investment, and Figure S.6 is GDP. Investment is much more volatile than consumption. The relative volatility of consumption and investment is comparable to what we find in real data. We simulated the economy over 43 periods, because there were also 43 years of data for German GDP. In the simulation there are nine peaks, which is close to the ten peaks we found in the data. The length of the cycle varies from four to seven years. The average length is a little less than five periods, while the German cycles lasted a little more than four years on average. 1.4 1.2 1 0.8 Consumption 0.6 Investment 0.4 0.2 0 1 6 11 16 21 26 31 36 41 -0.2 -0.4 Figure S.5: Simulated Consumption and Investment Exercise 9.7 The aim of real business cycle research is to gain a better understanding of business cycles. The theory differs from other approaches mainly by the methods that are applied. Real business cycle models are fully specified stochastic equilibrium models. That means that the microfoundations are laid out in detail. There are consumers with preferences, firms Exercises from Chapter 10 257 1.8 1.6 1.4 GDP 1.2 1 0.8 0.6 0.4 0.2 0 1 6 11 16 21 26 31 36 41 Periods Figure S.6: Simulated GDP with technologies, and a market system that holds everything together. Real business cycle theory takes the simplest models of this sort as a point of departure to explain business cycles. Model testing is most often done with the “calibration” method. This means that first the model parameters are determined by making them consistent with empirical facts other than the business cycle facts that are supposed to be explained. The parameterized model is then simulated, and the outcomes are compared with real world data. Exercise 9.8 Plosser’s model does not contain a government, and even if there were one, there would be no need to stabilize the economy. There are no market frictions in the model; the outcomes are competitive equilibria. By the First Welfare Theorem we know that equilibria are efficient, so there is nothing a government could do to improve economic outcomes. It is possible to extend the model to allow for a government, and we could add frictions to the model to make intervention beneficial, without changing the general framework very much. Also, any government is certainly able to produce additional shocks in the economy. Still, real business cycle theory works fine without a government, both as a source of disturbance and as a possible stabilizer. Exercise 10.1 1. This is the myth of small business job creation again. The SBA has every reason to tout the influence of small small businesses, but, as DHS point out, the dominant job market role if played by large, old firms and plants. 2. This rather entertaining quote has several immediate and glaring errors, but it does contain an argument quite in vogue at the moment. There is a common idea that Solutions to Exercises 258 jobs are a scarce resource, and that the pool of jobs is shrinking under pressure from greedy company owners, slave labor factories abroad and so on. In reality, as we’ve seen in this chapter, the pool of jobs is churning all the time. Ten percent of all jobs are typically destroyed in a year, and ten percent are created. In the face of this turmoil, one or two high profile plant closings is simply not important. Exercise 10.2 The term est is defined as: g For a new plant Xes;t 1 gest = 0:5(XXes;tes;t +XXes;tes;t 1 1) : Xes;t = 0. Thus for a new plant: = 0 and for a dying plant gest = 0:5(XXes;t 0 =2 + es;t 0) : And for a dying plant: 0 Xes;t 1 = 2: gest = 0:5(0 + Xes;t 1 ) Exercise 10.3 The only thing tricky about this problem is remembering how to deal with absolute values. if is negative. For shrinking plants, If = , then j j = if is positive and j j = es;t is negative, so for shrinking plants: a b a b a a b a
X j
Xes;t j : es;t gest =
X = Zest Zest Now we work through the algebra required to get the answer to the first identity. We begin with the definition of st : c cst = CZstst = Z1st X e2S +
Xes;t = Z1st X e2S + Zest
Xes;t = 1 X Zest gest: Zest Zst e2S + Turning to the next identity, we begin with the definition of netst : Cst Dst ; Zst 1 X 1 X =
X es;t Zst e2S Zst e2S j
Xes;t j; 1 X 1 X = Z Zest( gest); est gest Z Z netst = + st e2S + 1 X = est est st e2S Z Z g : st e2S + Exercises from Chapter 10 259 Exercise 10.4 This question really just boils down to plugging the definitions of t and NETst into the definition of covariance. However, the algebra shouldn’t detract from an interesting statistical regularity. Begin with the definition of covariance (supplied in the question): R R ; NETt) < 0: N 1 X cov( t N N 1 X ( i+ i=1 N i=1 R R)(NETi NET ) < 0: ( i C Di C D)(Ci Di C + D) < 0: N N 1 X i=1 C C ) + (Di D) (Ci C ) (Di D) < 0: ( i N 1 X N i=1 N X C C )2 N1 ( i i=1 D D )2 < 0 : ( i Using the definition definition of variance supplied in the question, this last inequality can be written var( ) var( ) 0, so var( ) var( ). That was a lot of algebra, but it was all straightforward. Thus if periods of large net job loss coincide with periods of larger than normal job reallocation, it must be the case that job destruction has a higher variance than job creation. C D< C < D Exercise 10.5 Here is the original chart, now augmented with the answers. X X Year X 1990 1000 0 500 1991 800 100 800 1992 1200 200 1993 1000 1994 c d t netst UB LB 0.250 0.125 0.125 600 200 700 0.263 0.053 0.210 600 400 400 600 0.098 0.146 -0.048 500 100 800 800 500 0.195 0.146 0.049 700 100 1995 400 1200 600 0.233 0.186 0.047 900 100 1996 200 1400 600 0.091 0.091 0 400 0 1997 0 2000 500 0.255 0.128 0.127 900 300 1;t 2;t 3;t t 260 Solutions to Exercises Exercise 10.6 All of these statements referred to specific charts or graphs in DHS. This question was on the Spring 1997 midterm exam in Econ 203. 1. Most students were at least able to say that this hypothesis wasn’t exactly true, even if they couldn’t identify specifically why. Any two of the following facts were acceptable: (a) Even the industries in the highest important ratio quintile had an import penetration rate of about 13.1%, which is pretty low. (b) The relationship between import penetration quintile and net job growth and job destruction is not monotone. (c) For the highest import penetration quintile, net job growth averaged nually. : 2 8% an- 2. Robots replacing workers is another favorite canard (thankfully less common recently) of the chattering classes. The reality is reflected in DHS Table 3.6 showing gross job flows by capital intensity decile. The most fascinating part of this table is the final entry, showing an average annual net employment growth rate of 0.7% for plants in the highest capital intensity decile. Plants in the lowest capital intensity decile shed about 10% of their jobs, net, each year. That is, over the 15-year sample period, they must have become nearly extinct. Thus high capital plants (plants with lots and lots of robots, one presumes) have been steadily adding excellent jobs of the past 20 years. 3. What we were looking for here was some version of Figure 2.2 in DHS, giving the distributions of plant-level job creation and destruction by employment growth . They have a distinctive “double hump” shape with the first peak at about = 0 10. However, we accepted more general statements about how most destruction occurs at plants which are shutting down and so on. g : g 4. This question is drawn directly from Table 3.6, showing that highly specialized plants have high job creation and destruction rates, and a net growth rate of -2%. Because of their high job destruction rate, and the tendency of plants to close in recessions (the cyclical behavior of job destruction), highly specialized plants are indeed at risk of closing in recessions. 5. For this question we wanted students to tell us about job creation and destruction rates by wage quintile (Table 3.4 in DHS). Any two of the following facts were acceptable: (a) Job creation and destruction are falling by plant wage quintile. (b) Of all jobs destroyed each year, only about 18% are accounted for by the highest wage quintile, while about 26% are accounted for by the lowest wage quintile. (c) High wage jobs tend to be more durable (longer creation persistence). Exercises from Chapter 12 261 Exercise 11.1 The aggregate production technology is = 3 :7 :3 , and we have = 0 2. The law of motion for capital is given by: s Y : LK L = 150, Æ = 0:1, and Kt = (1 Æ)Kt 1 + sYt : Therefore the steady state level of capital K̄ has to satisfy: K̄ = (1 Æ)K̄ + s(3L0:7 K̄ 0:3): Plugging in the values for labor, depreciation, and the saving rate yields: K̄ = 0:9K̄ + (0:6)(150)0:7 K̄ 0:3; or: 0:1K̄ = (0:6)(150)0:7 K̄ 0:3 ; or: K̄ = (10)(0:6)(150)0:71=0:7 : Evaluating this expression results in K̄  1940. Steady state output Ȳ is given by: Ȳ = 3L0:7 K̄ 0:3; which gives us the solution Ȳ  970. Exercise 11.2 In terms of the Solow model, the war temporarily reduced the capital stock in Kuwait. Given the lower capital stock, per capita incomes will be lower in the next years. In the long run, the economy reaches the steady state again, so the war does not affect per capita income any more. Similarly, the effect on the growth rate of per capita income is also temporary. In the short run, the growth rate will be higher, because the growth rate of per capita income is inversely related to the capital stock. In the long run, the growth rate of per capita income is determined by the rate of technological progress, so the war does not have an effect on the long-run growth rate. Recovery will be faster if foreigners are allowed to invest, because more investment implies that the economy returns faster to the steady state level of capital. The gains and losses of workers and capitalists depend on the reaction of wages and the return on capital to a higher capital stock due to foreign investment. Our formulas for wage and interest, equations (11.3) and (11.4), indicate that the wage is positively related to the capital stock, while the return to capital is negatively related to the capital stock. Since a prohibition of foreign investment lowers the capital stock, workers would lose, and capitalists would gain by a prohibition. Exercise 12.1 1. True. Under an unfunded pension system payments to the old are made by taxing the young, not by investing in the bond market. Hence the volume of physical savings between periods of life is higher under a funded than an unfunded pension system. n 2. Check the Economic Report of the President to get a good sense of , and the back of the Economist magazine to get the latest value for . Unless something very odd is happening, is probably considerably lower than . n r r Solutions to Exercises 262 3. From the Economic Report of the President we see that (among others) the U.S government spends more than 20% of its total outlays on interest payments on the Federal debt, social security and defense. We shall have quite a bit more to say about the Federal debt in Chapter 14 and Chapter 18. Exercise 12.2 The household’s budget constraint is: C + I + G = Y P + Y G: We are given that private output Y P is fixed at Y and that government output Y G is G. Thus government spending G must satisfy: G = Y + G C I; or: (1 )G = Y C I: Obviously, as G grows, C and I are going to have to shrink (although not one-for-one with G). The maximum allowed level for government spending occurs when consumption and investment are each zero, so C = I = 0. In that case: G = 1 Y : The government can spend more than total private output since its spending is productive. As is closer to zero, the closer must be to . As is closer to unity, the larger may be relative to .  G Y Y  G Exercise 12.3 To calculate the market-clearing interest rate, we have to find the interest rate that makes the household want to consume precisely its endowment stream net of government taxes. Since in this question consumption in each period must just be t = t t , we find that: t 1 U 0 (Y G) 1 + r0 = U 0(Y G) ; so: r0 = ; and: 1 U 0 (Y G0 ) 1 + r0 = U 0 (Y G ) : C Y G 1 We cannot characterize r further without more information about Y; G and U , but we can G >G 0 say that, since 0 1 , the marginal utility in the first period must be greater than the 0( marginal utility in the second period, that is, 0 ( 0) 1 ). Thus: r >r U Y G >U Y G U 0(Y G0) > 1: U 0(Y G1) As a result, 0 0 . This fits well with the results of this chapter, which hold that temporary increases in government spending increase the real interest rate. Exercises from Chapter 12 263 Exercise 12.4 The household’s maximization problem becomes: n p  )y St + 2 max 2 (1 St p S The first-order condition with respect to t is: 1 p  )y St (1 rS p +p o n y : (1 + ) t + (1 + ) rS 1+ r n y (1 + ) t + (1 + ) =0 : S Solving for t produces: St = 1 + (12 (1+ +r)r) y r n (1 + ) + 1+ 1+r 1 + 2 (1 + ) 2 2  r y: n Notice that private savings is (as usual) decreasing in . Also notice that the larger is relative to , the greater this effect. When , contributions to the social security system supplant private savings at a greater rate than in a funded system. The reason is because, , the social security system is more attractive than private savings. when r n>r n>r Exercise 12.5 1. Grace trades consumption today for consumption tomorrow via schooling there is no bond market). Her maximization problem is: S) + max fln(1 S ab a ln( S (since AS )g : b Recall that ln( ) = ln( ) + ln( ). Hence the first-order condition is: S + S = 0: Solving for Grace’s optimal schooling provides S = =(1 + 1 1 ). In this setup, 2. Now Grace’s problem becomes: max fln(1 S S G) + A S G)]g : ln[ ( + The first-order condition for maximization is: S G + S + G = 0: 1 1 Thus Grace’s optimal schooling choice becomes: S = 1+ + 1+  G: K1 = S . Solutions to Exercises 264 G Grace’s schooling is certainly decreasing in (thus investment is, to a certain extent, being crowded out). Grace’s human capital is 1 = + , so substituting in provides: K1 = 1 + +(  K 1) S G G1 + : < Notice that if 1, the government is less efficient at providing schooling than the private sector, and Grace’s human capital decreases in . G 3. Now Grace’s maximization problem becomes: S) + max fln(1 S A S G)]g ; ln[ ( + since Grace does not have to pay a lump-sum tax in the first period. The first order condition is now: 1 1 S + S + G = 0: Grace’s optimal schooling choice is: S = 1+ G 1 +1 ; and her human capital becomes: K1 = 1 + (1 + G): Notice that Grace’s schooling is still being crowded out, but that her human capital is increasing in no matter what the value of , as long as 0. G  > Exercise 13.1 1. If the agent works, i ( i = 1) = 1 c`  , while if the agent does not work, ci (`i = 0) = 0. i while if the agent does not work, ui (`i = 2. If the agent works, ui (`i = 1) = 1  0) = 0. An agent will work if the utility of working is greater than the utility of not i  0. working, or if 1  3. From our previous answer, it is easy to see that  ( ) = 1  . 4. We know that the fraction of agents with less than or equal to some number, say  , is just  if 0    1. Thus aggregate labor supply as a function of the tax rate is . On each agent who works, the government collects revenue just ( ) =  ( ) = 1 . Thus T ( ) = (1 ). This is sketched in Figure (13.1).  `      5. There is a Laffer curve in the tax system. Exercises from Chapter 13 265 Exercise 13.2 Briefly, although such a result might be evidence for a Laffer curve, the regression does not control for changes in real income over time. There may not truly be a Laffer curve, but it would look like there was one if real incomes were high when taxes were low and low when taxes were high. Exercise 13.3 The point of this simple problem was to clear up the difference between the tax system H( ; ) and the government’s revenue function T ( ). This problem should also give you some practice in thinking about exemptions. a  1. The parameters of the tax system are the choices of the flat tax rate and the lumpsum tax . The household chooses an effort level in response. Thus = [ ] and = here. a L S L a 2. The tax system H( ; ) maps household actions an amount of tax: ; S a and tax system parameters into H(L; [; S ]) = S +  (L S ): S Recall that income directed towards the lump-sum tax is exempt from the flat tax. We do not consider (yet) that is itself a function of and . L S  3. A household’s tax bill is always the same as the tax system. In this case, if the house). hold works an amount it owes + ( L S L S 4. The household’s income as a function of L is just L. Hence the household consumes L H(L; [; S ]) or: C = L [S +  (L S )] = (1  )(L S ): 5. Substituting in to the household’s utility function gives: U (C; L) = 2 p (1  )(L S ) L: The first-order condition for maximization with respect to L is: 1  L S = 1: L produces: L(; S ) = S + 1 : (Where did the 2 go?) Solving for This gives the household’s optimal response to the tax system H. In the chapter we called this max ( ). a Solutions to Exercises 266 6. The government revenue function is the tax system with the household’s action optimized out. That is: T (psi) = H [amax ( ); ] a : In this case, this produces: T ([; S ]) =  (1  ) + S:  Notice that there is a Laffer curve (as expected) in the tax parameter . For further practice: Assume that income spent on the lump-sum tax is no longer exempt from the flat tax. How do your answers change? You should be able to show that the Laffer curve in vanishes.  Exercise 13.4 1. If the household works , it raises gross income of and must pay a tax bill of consumes the residual, (1 ) . ` c `;  ` ` `. It 2. Substitute ( ) into the household’s utility function to find utility purely as a function of labor effort. The household’s maximization problem becomes: o n p max 4 ` Taking the derivative with respect to tion: r 2 (1  )` ` : ` gives the first order condition for maximiza- 1 `  1=0 : We can solve this to find the household’s optimal choice of labor effort given taxes, ( ): ` `( ) = 4(1  ): 3. That government’s tax revenue is: T ( ) = `( ) = 4 (1  ): = 4. The government wishes to raise revenue of 3 4. We are looking for the tax rate satisfies:  4 (1  that T ( ) = 3=4; or:  ) = 3=4: =; = Inspection reveals that there are two such tax rates: f1 4 3 4g. Since the government is nice, it will choose the lower tax rate, at which the household consumes more. Exercises from Chapter 13 267 Exercise 13.5 In the question, you were allowed to assume that = 0 and that Tammy had an implicit discount factor of = 1. These solutions are a little more general. To check your solutions, substitute = 1 and = 0. r r y w`  b 1. In the first period of life, Tammy earns an income of = of which she must pay in taxes. Thus her income net of taxes is (1 ). She splits this between 1 1 consumption in the first period of life, 1 and savings, . Thus: w` w` c b c= c1 + b  (1 1 )w`; and: c2  (1 + r)b: r Notice that = 2 (1 + ) so we can collapse the two one-period budget constraints into a single present-value budget constraint. Thus: c1 + 1 +1 r c2  (1 1 )w`: 2. Tammy’s Lagrangian is: L(c ; c ; `) = pc 1 2 1 + pc 2 `+  This has first-order conditions with respect to 1 1 p1 2 p1 2 1 +1 2 c c 1+ w(1 1 )w` c1 (1 c1; c2, and ` of: 1 1+  r c2 :  = 0; r  = 0; and: 1) = 0: Manipulating each of these equations produces the system:  2 c1 = 14 1 ;  1 (1 + r) c2 = 4  2 ; and: 1  = (1 1)w: We can further manipulate these three equations, by substituting out the multiplier to find the optimal choices of consumption: c1 = 41 (1 1)2w2 ; and: c2 = 41 ((1 + r) )2 (1 1 )2w2:  Solutions to Exercises 268 ` We can find labor effort by substituting the optimal consumption decisions (calculated above) into the budget constraint. This will tell us how many hours Tammy must work in order to earn enough (after taxes) to afford to consume 1 2 . The budget constraint is: c ;c w(1 1 )` = c1 + 1 +1 r c2 w2 (1 1)2 1 + (1 + r)2 2  = 4 (1 + r) 2 2 w (1 1) (1 + (1 + r) 2 ); so: = 4 w (1 1 ) ` = 4 (1 + (1 + r) 2 ): Notice that Tammy’s effort is strictly decreasing in 1 and that at 1 = 1, ` = 0. In other words, if the government taxes Tammy to the limit, we expect her not to work at all. This will induce a Laffer curve. Once we’ve figured out how much Tammy works, it’s an easy matter to deduce how much revenue the government raises by taxing her. The government revenue function here is: 2 T 1 ( 1 ) = 1 = 1 (14 1 ) [1 + (1 + ) 2 ] In terms of 1 , this is just the equation for a parabola:  Hence   w` w   r : H1 (1 ) = 1 (1 1 )(constant term): 1 = 1=2, and:    2   1  H1 (1 ) = 4 (constant term) = w16 1 + (1 + r) 2 : Thus there is a strict limit on the amount of revenue that the government can squeeze out of Tammy. As the tax rate 1 increases, Tammy works less, although if 1 1 2, the government collects more revenue.   < = 3. There is indeed a Laffer curve in this problem. We should have expected it the instant we saw how Tammy’s hours worked, , responded to the tax rate. ` Exercise 13.6 Now Tammy is allowed to deduct savings held over for retirement. This is also known as being able to save in “pre-tax dollars.” Almost all employers feature some kind of taxsheltered savings plan. 1. Tammy’s tax bill at the end of period 1 is budget constraints: 2(wn b). c1 = (1 2 )(w` b); and: c2 = (1 + r)b: Tammy faces a sequence of Exercises from Chapter 13 269 Once again, we use the trick of constraint becomes: b = c2=(1 + r), so that Tammy’s present-value budget   c1 = (1 2 ) w` cR2 ; so: 2 ) 1 +1 r c2 = (1 2 )w` c1 + (1  Note that as 2 ! 1, Tammy’s ability to consume in the first period of life goes to zero, but her ability to consume in the second period of life is unchanged. 2. Tammy’s Lagrangian is: L(c ; c ; `) = pc 1 2 1 + pc `+ 2 The first-order conditions with respect to  (1 2)w` c1 c1 ; c2 and ` are: 1 1 p1 2 p1 11 + 2 2 c  c2 : r (1 2) 1+  = 0; 2  = 0; and: r 1 + (1 2 )w = 0: We can write c1 and c2 easily as a function of : c c1 = 412 ; and: c2 = 42 2  So it’s an easy matter to substitute out for Thus: 1+ 1 r 2 : 2  and calculate optimal consumption c1; c2. c1 = 14 w2 (1 2 )2; and: 2 c2 = ( (1 + r)) w2 : c  4 Notice that 2 does not depend on 2 . We can substitute in the optimal consumptions above into the budget constraint to determine how many hours Tammy has to work to be able to afford her optimal consumption plan: (1 2)w` = c1 + (1 2 ) 1 +1 r c2 w2 (1 2)2 + (1 2 ) 2(1 + r)w2 ; so: = 4 4 w 2 ` = 4 [(1 2 ) + (1 + r) ]: Solutions to Exercises 270 Notice from the first line above that: w` = 1 c1 2 + b: The trick here is to substitute back into the right budget constraint. The first couple of times I did this I substituted back into the budget constraint from Exercise (13.5) and got all sorts of strange answers. Notice that Tammy always consumes a certain amount of 2 , no matter what 2 is, so she always works a certain amount. However this may not overturn the Laffer curve since she is paying for 2 with pre-tax dollars. c  c 3. Once again, this is a bit tricky. Remember that Tammy’s tax bill is = 2 (1 + ). Thus: b c= r T 2 (2 ) = 2 (w` b)  = 2 w`  = 2 w` c2 r  1+ 2 (1 + ) 4 r w2    2 w w4 (1 2 ) + w4  2 w : = 2 (1 2 ) = 2 (w` b) and that 2 r (1 + )  2 r w4 2  (1 + ) 4 This was a matter of remembering to substitute into the right revenue equation. Although Tammy always works at least enough to finance a certain amount of consumption while old, this amount of income is tax-deductible, so the government can’t get at it.  As 2 ! 1, government revenue goes to zero, as before. The maximizing tax rate, is 2 = 0 5 and the maximum amount of revenue that government can raise is:  : 2 , T 2 (2 ) = w16 : 2  < T 1(1). Notice that T 2 ( 2 ) 4. There is still a Laffer curve present. Unfortunately for the government, tax revenue is now lower. 5. Our answers are indeed different. Because Tammy is able to shelter some of her income from the government, total tax revenue will be lower. Exercise 14.1 1. About $5.2 trillion/$7 trillion. 2. About 1.07 in 1945 (Barro p.362). Exercises from Chapter 14 271 3. About 0.38 in 1981 (Barro p.341). 4. 3%. Exercise 14.2 The consumer’s problem is thus: x p t x max fln( 1 ) + x1 ;x2 ( 1 + 1) 1 + ( x2 g ; subject to: p2 + t2)x2 = M: The two first-order conditions for this problem are: x1 = (p1 + t1); and: 1 = (p2 + t2 ): 1 These first order conditions plus the budget constraint can be used to solve for the three unknowns , 1 , and 2 in terms of the givens in the problem, 1 , 2 , 1 , 2 , and . Solving: x x p p t t x1 = pp2 ++ tt2 : 1 1 M x2 = p + t 1: 2 2 M The government’s revenue function can be calculated accordingly: T (t1 ; t2 ; p1 ; p2 ; M ) = t1 x1 + t2 x2 = t1p(p2++tt2 ) + pt2+Mt 1 Substitute the above demand functions for the household’s indirect utility: 1 2 2 t2 : x1 and x2 into the objective function to obtain   V (p1 + t1; p2 + t2 ; M ) = ln pp21 ++ tt21 p2M+ t2 + 1: So at this point, we have found the government’s revenue function which tells us how much the government can raise from taxes given that consumers respond optimally to the given tax rates. By deriving the consumer’s indirect utility function, we know how consumers compare different tax rates and income levels in utility terms. G Potentially the government is faced with the need to raise a certain level of revenue, . It can raise this revenue a number of different ways by taxing the two goods in different amounts with the constraint that in the end, it must have raised in revenues. G Solutions to Exercises 272 t ;t A benevolent government could decide to choose the combination of taxes ( 1 2 ) such that consumer utility is maximized, subject to the constraint that it raises the necessary revenue . The government’s optimal-tax problem would then be: G max t1 ;t2 V (p1 + t1; p2 + t2; M ); subject to: T (t1 ; t2 ; p1 ; p2 ; M ) = G; M and some given . Make sure you understand the intuition of this problem. Both the indirect utility function and the government revenue function account for the fact that households respond optimally to the given tax policy. Before you ever write down the optimal tax problem, you must know how consumers will respond to any possible tax policy given by ( 1 2 ). Implicit in the indirect utility function and government revenue function is the fact that consumers are responding optimally to their environment. t ;t Exercise 14.3 The key to this problem is realizing that the household’s budget set will be kinked at the point f 1 T 1 2 T 2 g. For points to the left of this kink, the household is saving, and the budget set is relatively flat. For points to the right of this kink, the household is borrowing, and the budget set is relatively steep. The government’s optimal plan will be to levy very low taxes initially and then high taxes later, in essence borrowing on behalf of the household. y ;y 1. If the household neither borrows nor lends, it consumes: c1 = y1 T 1; and: c2 = y2 T 2: This is the location of the kink in the budget constraint: to consume more in period = 1 than 1 T 1 , it will have to borrow at the relatively high rate 0 and the budget set will have a slop of (1+ 0 ). The government will be able to move the kink around, increasing or decreasing the number of points in the household’s budget set. t y r r 2. For convenience, all of the answers to the next three questions are placed on the same set of axes (below). The solid line gives the answer to the first question. 3. The dotted line gives the answer to this question. Notice that the household has more points to choose from. 4. The dashed line gives the answer to this question. Notice that the household has fewer points to choose from. 5. The government chooses behalf of the household. T 1 = 0, in essence borrowing at the low interest rate r on Exercises from Chapter 15 273 Budget Sets 1.4 1.2 1 c2 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 c1 0.6 0.7 0.8 0.9 1 Exercise 14.4 In this question the government runs a deficit of unity in the first period (period = 0), because expenditures exceed revenues by exactly unity. In all subsequent periods, government revenues just match direct expenditures in each period, but are not enough to repay the interest cost of the initial debt. As a result, the government will have to continually roll over its debt each period. Under the proposed plan, the government has not backed the initial borrowing with any future revenues, so it does not ever intend to repay its debt. From the government’s flow budget constraint, assuming that g 1 = 0: t B B0g = G0 T 0 = 1: B1g = 1 + r: B2g = (1 + r)2 : .. .. . . g = (1 + )t t B r : So the government debt level is exploding. Substituting in to the transversality condition, we get: r B r r lim (1 + ) t tg = lim (1 + ) t (1 + )t = 1 t!1 t!1 : Since the limit does not equal zero, we see that the government’s debt plan does not meet the transversality condition. Exercise 15.1 The Lagrangean is: L = (cPw ) (cPb )1 m + [ P cPw pw + cPb pb]: Solutions to Exercises 274 The first-order conditions are: c c (FOC P w) (FOC P b) c c  p ; c  p : ? 1 P ? 1 + ?[ ( P w ] = 0 and: w ) ( b ) ? ? ? ( P )( P w ) (1 b ) + [ b] = 0 c Combining these to get rid of ? yields: ? 1 P? 1 ? ? (cP (cP )(cP w ) (cb ) w ) (1 b ) ; or: = pw pb P ? pw = cb : pb (1 )cPw? ? Solving this for cP b and plugging back into the budget equation gives us:  ? (1 )pw cP w P ? cw pw + pb = mP : pb After some algebra, we get the first result: P cPw? = pmw : b When we plug this back into the budget equation and solve for P ? , we get the other result: cPb ? = (1 pw)m : P Exercise 15.2 1. See Figure S.7. 2. See Figure S.7. p =p 3. From the graph, we know that w b = 3. Suppose the relative price is less than 3. Then only Pat will make wine, and supply will be 4 jugs. Plugging 4 jugs into the demand function gives a relative price of w b = 7 2, but 7 2 3, which is a contradiction, so the equilibrium relative price can’t be less than 3. p =p = = > By a similar argument, you can show that the equilibrium relative price can’t be more than 3. 4. Pat makes (i) 4 jugs of wine and (ii) 0 jugs of beer. Chris makes (iii) 1 jug of wine and (iv) 1 jug of beer. < 6. 6. Pat has a comparative advantage in wine production, since 2=1 < 6=2. Chris makes 5. Pat has an absolute advantage in wine production, since 2 wine anyway, since the equilibrium price is so high. Exercises from Chapter 17 pw pb 275 6 A At A A 3 QSw A A A A 2 A A QDw - Q = # of jugs 5 4 A A Figure S.7: The Supply of Wine by Pat and Chris Together Exercise 17.1 We know that: 1+ r2 = F 1 1(1 + r1 ) : We can manipulate this to produce: r2 = F 1 r  Fr1 : 1  This is an interesting result. Essentially, 2 is the net return on turnips held until period = 2 (that is the 1 term) minus a risk premium term that is increasing in 1 . t F r Exercise 17.2 1. The bank’s assets are the value of the loans outstanding net of loss reserve, in other words, the expected return on its loans. The bank’s liabilities are the amount it owes its depositors. Considering that the bank must raise a unit amount of deposits to make a single loan, this means that the bank must pay 1 + to make a loan. Thus the bank’s expected profits are: r pS x + (1 px ) R r: (1 + ) x only if the borrower does not default. 2. Now we solve for the lowest value of x which generates non-negative expected profThat is, the bank gets Solutions to Exercises 276 its. Setting expected profits to zero produces: pS x + (1 px x r; r; ) R = (1 + ) so: 1+ ?( ) = + ) R S (1 p r p : Since safe borrowers repay more frequently than risky ones because: x pS > pR ; the amount repaid, , is decreasing as the mix of agents becomes safer, that is, as increases. As expected, is increasing in the interest rate . x r 3. We assumed that agents are risk neutral. Thus if the project succeeds (with probabil? ( ) and a risky agent (with probability R ) ity S ), a safe agent consumes S ? ( ). If the project fails, agents consume nothing. Their expected consumes R utilities therefore are: ? ( )] and: S( ) = S[ S ? ( )] R( ) = R[ R Where ? ( ) is the equilibrium value of . Notice that since S S = R R and S R that we can write S and R as: ? R( ) = S ( ) + ( S R) ( ) p  x r;  x r; V r p  V r p  x r; ; x r; : x p p V V V r V r p p x r; : Thus at any given interest rate r > 0, the expected utility of risky borrowers is greater than the expected utility of safe borrowers, VR (r) > VS (r). 4. Since VR (r) > VS (r), it is easy to see that if VS (r) > 0 then VR (r) must also be greater than zero. Next we find r? such that VS (r? ) = 0. Substituting: pS 0 = VS (r? ) = pS [s x? (r? ; )] = pS S (1 + r? ) pS + (1 )pR ; so: 1 + r? = S + S (pS pR ): At interest rates above r? all safe agents stop borrowing to finance their projects. Realizing this, banks adjust their equilibrium payments to: x? (r; = 0), so (1 + r)=pR . p >p x r; p Exercise 17.3 ) all shift up by some amount, then, for any given interest If the revenue functions ( rate , intermediaries can make loans to agents with higher audit costs. That is, ? ( ) also shifts up as a result. This shifts the demand for capital up and out, but leaves the supply schedule untouched. As a result the equilibrium interest rate increases, as does the equilibrium quantity of capital saved by type-1 (worker) agents. As a result, type-1 agents work harder, accumulate more capital and more type-2 (entrepreneurial) agents’ projects are funded so aggregate output goes up. Type-1 agents are made better off by the increase in the interest rate because their consumption goes up (although they are working harder too). Type-2 agents who had been credit rationed are made better off, but type-2 agents who previously had not been credit rationed are made worse off because the interest rate paid on their loans goes up. r  x; r Exercises from Chapter 17 277 Exercise 17.4 This question uses slightly different notation from that used in this chapter. Most bothersome is probably the fact that here denotes the gross interest rate, which elsewhere is denoted 1 + . This question is a reworking of the model of moral hazard from this chapter. This question was taken directly from the Spring 1998 Econ 203 final exam. r r a> 1. A rich Yalie can finance the tuition cost of Yale from her own wealth (that is, 1). If she gets the good job, she consumes + ( 1), if she does not, she consumes ( 1). Hence her maximization problem is: w ra ra  max   [w + r (a 1)] + (1 )[r(a 1)] w 2  : 2  is: w w  = 0: We can easily solve this to find that  = w. 2. Poor Yalies are required to repay an amount x only if they land the good job. Hence if they land the good job, they consume w x, while if they go unemployed, they The first-order condition with respect to consume 0. Thus their optimization problem may be written as:  max  (w x) + (1 )
0 w 2  : 2  is now: w x w  = 0: The first-order condition with respect to effort We can solve this to find the optimal effort as a function of repayment amount:   (x) = 1 wx : Notice that effort is decreasing in x. 3. Yale University must also pay r to raise the funds to loan to its students. If it is making these loans out of its endowment, then it is paying an opportunity cost of r. A student of wealth a < 1 needs a loan of size 1 a, which costs Yale an amount r(a 1). Thus Yale’s profit on this loan is: x(x) + 0
[1 (x)] r(a 1): But we know  (x) from the previous question, so:   x 1 wx r(a 1): This is the usual quadratic in x. Solutions to Exercises 278 4. Yale’s “fair lending policy” guarantees that all borrowers pay the same interest rate, regardless of wealth. Since we know ( ) from above, and we are given ( ), it is an easy matter to calculate ( ): x a xa r w (1 a):  ( a) = r a Notice that effort is decreasing in and increasing in wealth . a   a< 1 5. Here we are supposed to show that ( )  ? from above, where ? = . If 0, and 0 and 0 by assumption. It’s easy to see that this must be then 1 true. a> r> w> 6. Now we are supposed to show that Yale’s profits are negative on loans and that poor borrowers cost it more than richer borrowers. The fair lending policy charge all borrowers the same interest rate. Further, this interest rate guarantees Yale zero profits assuming that they exert effort. Poor borrowers will exert less than effort, and so Yale will lose money. Return to Yale’s profit function: (a)x(a) r(1 a): a xa r (1 a)i  r(1 a)  r(1 a): w Substituting in for ( ) and ( ) we get: h We can manipulate this to produce: r [ (1 a)]2 : w a All other terms canceled out. This is certainly negative, and increasing in . Thus Yale loses no money on “borrowers” of wealth = 1, and loses the most money on borrowers of wealth = 0. a a Exercise 18.1 This question has been given on previous problem sets. In particular, we have amassed a few years’ data on students currency holding habits. 1. According to Friedman and Schwartz, the stock of money fell 33% from 1929 to 1933. Household holdings of currency increased over the period. 2. Real income fell by 36% over the same period and prices decreased. 3. From the Barro textbook: Real interest rates have been negative in the years 1950-51, 1956-57 and 1973-79. Inflation was negative in 1949 and 1954. Exercises from Chapter 18 279 4. From the Barro textbook: There is evidence in looking cross-sectionally at different countries that changes in money stocks are positively correlated with changes in prices, or inflation. Long run time-series evidence demonstrates a positive correlation between money growth and inflation as well. 5. From the Porter article on the location of U.S. currency: The stock of Federal Reserve notes outside of banks (vault cash) at the end of 1995 was about $375 billion, or about $1440 per American. Nobody had quite this much cash on them, although some students were carrying over $100. I assume these students were well trained in selfdefense. According to Porter, between $200 and $250 billion, that is, more than half, was abroad, primarily in the former Soviet Union and South America. 6. Generally people keep their money in low interest assets because they are liquid and provide transactions services. It’s tough to buy lunch with shares of GM stock rather than Hyde Park bank checks. 7. Sargent states that inflation can seem to have momentum if people have persistent expectations that the government will continue to pursue inflationary fiscal and monetary policies. 8. Since currency is a debt of the government, whenever the government prints money, it is devaluing the value of its debt. This is a form of taxation and the value by which its debt is reduced is called seignorage. The government obtained $23 billion in seignorage in 1991. 9. The quantity theory is the theory that the stock of money is directly related to the nominal value of output in the economy. It is usually written as the identity: M = PY=V M P Y where is the money stock, is the price level, and is the real amount of output. It is an accounting identity in that the velocity of money, , is defined residually as whatever it takes to make the above identity true. V 10. A gold standard is a monetary system where the government promises to exchange dollars for a given amount of gold. If the world quantity of gold changes (for example, gold is discovered in the Illinois high country) then the quantity of money also changes. Our current monetary system is a fiat system, where money isn’t backed by any other real asset. It is simply money by “fiat”. Exercise 18.2 Government austerity programs involve reducing government expenditures and increasing tax revenue. Both cause immediate and obvious dislocations. Governments typically reduce spending by firing lots of government workers, closing or privatizing loss-making government-owned industries and reducing subsidies on staples like food and shelter. Governments increase revenue by charging for previously-free services and pushing up the tax rates. From the point of view of a typical household, expenses are likely to go up Solutions to Exercises 280 while income is likely to fall. Thus austerity programs can indeed cause immediate civil unrest. On the other hand, we know that subsidies are a bad way to help the poor (since most of the benefit goes to middle-class and rich households), that state-owned businesses tend to be poorly run, depressing the marginal product of workers and tying up valuable capital and that bloated government bureaucracies are rarely beneficial. Leave all this to one side: the fact is that no government willingly embarks on an austerity program. They only consider austerity when they are forced to choose between austerity and hyperinflation. Like Germany in 1921, an austerity program has to be seen as better than the alternative, hyperinflation. The central European countries in the early 1920s tried both hyperinflation and austerity, and found austerity to be the lesser of the two evils. That early experience has since been confirmed by a host of different countries. Austerity may indeed be painful, but it is necessary in the long run and better than hyperinflation. Exercise 18.3 1. We know that the money supply must evolve to completely cover the constant percapita deficit of . So we know that: d Mt Mt Pt (S.10) 1 = Dt = dNt: We know from the Quantity Theory of Money given in the problem that: t Pt = MYtt = M Nt : (S.11) Thus we can put equations (S.10) and (S.11) together to produce: 1 (S.12) M = d M Mt Mt 1 ; so: t 1 dNt = Mt MM = Nt t Mt Nt d = Mt MMt t 1 ; so: Mt 1 = d; and: Mt Mt 1 Mt 1 = 1 d : )] t 1 . This gives us an expression by how much the total stock Thus t = [1 (1 of money must evolve to raise enough seignorage revenue to allow the government to run a constant per-capita deficit of each period. d 2. To answer this question we will use the quantity-theoretic relation, equation (S.11) above and the effect of on the evolution of money in equation (S.12) above to a find a value for at which prices are stable, that is, at which t = t 1 . Notice that: d (S.13) d P P Pt = Mt=Nt = Nt 1 Mt = 1 Mt = 1 1 : Pt 1 Mt 1=Nt 1 Nt Mt 1 1 + n Mt 1 1 + n 1 d Exercises from Chapter 19 P =Pt If t 1 281 = 1 then, continuing from (S.13): 1 1+ (S.14) n 1 d = 1; so: d = 1 +n n : 1 n= n By (S.14) we see that the government can run a constant per capita deficit of (1 + ) by printing money and not cause any inflation, where is the growth rate of the economy/population (they are the same thing in this example). n n n 3. As ! 0, the non-inflationary deficit also goes to zero. At = 1 (the economy doubles in size every period) the non-inflationary per-capita deficit goes to 1 2. That is, the government can run a deficit of 50% of GDP by printing money and not cause inflation. At the supplied estimate of = 0 03, the critical value of is 0 03 1 03 or about 0.029 or 2.9% of GDP. n 4. From equation (S.13) above, if : d = 0 then: 1 Pt Pt 1 = 1 + n < 1; so: Pt = 1 +1 n Pt 1; and: Pt < Pt 1 : d = : =: n So there will be deflation over time—prices will fall at the rate . Exercise 18.4 Although we will accept a variety of answers, I will outline briefly what we were looking for. As with the central European countries in 1921-23, Kolyastan is politically unstable and in economic turmoil. Many of the same policies that worked in those countries should also work in Kolyastan. The government should move quickly to improve its tax collection system and radically decrease spending. This will probably mean closing down state-run factories and ending subsidies. The argument, often advanced, that such direct measures will hurt the citizens ignores the fact that the people are already paying for them through the inefficient means of the inflation tax. With its fiscal house in order, the government should reform the monetary sector by liberating the central bank, appointing a dour old man to be its head and undertaking a currency reform. For these changes to be credible, Kolyastan must somehow commit not to return to its bad old ways. It could do so by signing treaty agreements with the IMF, World Bank or some other dispassionate outside entity. Further, it should write the law creating the central bank in such a way that it is more or less independent from transitory political pressures. The bank ought to be prohibited from buying Kolyastani Treasury notes. Exercise 19.1 1. True: The CPI calculates the change in the price of a market-basket of goods over fairly short time periods. If one element of that basket were to increase in price dra- Solutions to Exercises 282 matically, even if they were compensated enough to buy the new market basket, consumers would choose one with less of the newly-expensive good (substituting away from it). 2. Inflation is bad because it leads consumers to undertake a privately useful but socially wasteful activity (economizing on cash balances). The Fed cannot effectively fight inflation with short-term actions, it must maintain a long-term low-inflation regime. Exercise 19.2 The slope of the Phillips curve gives the relative price (technological tradeoff) between inflation and unemployment. If inflationary expectations are fixed, the government can achieve a higher utility if it does not have to accept more inflation for lower unemployment. In other words, if the Phillips curve is flatter. It is interesting to note that a perfectly flat Phillips curve would mean that unemployment was purely a choice of the government and did not affect inflation at all. If the government and the private sector engage in a Nash game, the Nash outcomes inflation rate is directly proportional to , so low values of mean lower Nash inflation. Exercise 19.3 The point of this question was bested summed up by Goethe in Faust. His Mephistopheles at one point describes himself: “That Power I serve / Which wills forever Evil / And does forever good.” Or as Nick Lowe put: “You’ve got to be cruel to be kind.” The higher is the higher the inflation rate, but unemployment is only marginally lower (depending on expectations).  1. The government’s maximization problem is: max  We can solve this to find:  (u + e  )2  2 : 2 0 () = 1 + 2 u + 1 +  2 e:  Thus the optimal inflation choice is increasing in . 2. The corresponding unemployment rate is: u0() = 1 +1 2 u + 1 +  2 e : 3. Now we assume that government continues to take expectations as fixed, but that the private sector adjusts its expectations so that they are perfectly met. Recall that, given expectations e , the government’s optimal inflationary response is:  2  = 1 + 2 u + 1 +  2 e : Exercises from Chapter 19 Now define 1 as: We can solve this for 283 2 1 = 1 + 2 u + 1 +  2 1: 1 to find: 1 =  u: The associated unemployment rate is u1 = u , since  e =  in this case. 4. Given that agents form expectations rationally, eventually  e will converge to  . If the government is playing Ramsey (because it has a commitment device), then  = 0 and u = u no matter what  is. If the government is playing Nash, then unemployment is still at the natural rate, u = u , but inflation is  =  u . Thus the lower the value of , the lower the Nash inflation rate. The point of this question is that if  = 0, the Nash and Ramsey inflation rates coincide. Having  = 0 is an effective device with which to commit to low inflation. Exercise 19.4 Think of the dynamics in this question as sliding along the government’s best response curve, as depicted in Figure 19.2. Expectations will creep up, always lagging behind actual inflation, until the gap between the two vanishes and the private sector expects the Nash inflation, and the government (of course) delivers it. t  1. In period , given inflationary expectations te , the government solves: max t  fu? + (te t )g2 t2 : The government’s optimal choice is: t? (te) = 1 + 2 u? + 1 + 2 e : 2 2. Since expectations are just last period’s inflation rate, and since we know that the government inflation policy rule is given by t? above, the dynamics of the system are given by the pair of equations:  t = A(u? + te); for all t = 0; 1; : : : ; 1; and: te = t 1; for all t = 1; 2; : : : ; 1: Recall that initial inflationary expectations are 0e = 0. We can substitute out the expectations term to produce a single law of motion in inflation: t = Au? + At 1; for all t = 1; 2; : : : ; 1: For notational convenience we have defined A = =(1 + 2 ). Solutions to Exercises 284 3. Since expectations start at zero, the first period’s inflation rate is: 0 = Au?: Where is as defined above. Thus in the first few periods inflation evolves as: 0 = Au?: 1 = Au? + A0 = Au? + A2u? = Au?(1 + A): 2 = Au? + A1 = Au? + (1 + A) A2u? = Au?(1 + A + ( A)2): 3 = Au? + A2 = Au? + Au(1 + A + ( A)2) = Au?(1 + A + ( A)2 + ( A)3): The pattern ought to be pretty clear. In general, inflation in period t will be: t = Au? t X i=0 So as time moves forward, we have: lim t!1 t = Au? ( A)i: 1 X i=0 ( A) i : We can solve the summation using the geometric series to get:  Recall that we defined A to be: lim = t!1 t 1 A u?: A A = 1 + 2: So we can further simplify to get:  = lim t!1 t u?: This is just the Nash inflation rate. Expectations are also converging to this level, so at the limit, unemployment will also be at the Nash level of the natural rate ? . u Given that inflationary expectations were initially low, the government was able to surprise the private sector and push unemployment below its natural level. Over time the private adapted its expectations and as expected inflation rose, so did unemployment. Thus the time paths of inflation and unemployment are both rising over time, until they achieve the Nash level. 4. The steady-state levels of inflation and unemployment are not sensitive to the initial expected inflation. If the private sector were instead anticipating very high inflation levels at the beginning of the trajectory, the government would consistently produce surprisingly low inflation levels (but still above the Nash level) and the unemployment rate would be above its natural rate. Over time both inflation and unemployment would fall to their Nash levels. Exercises from Chapter 19 285 t ; ;::: ; : 5. The government’s optimal choice of inflation in period , t , now becomes: ? 1 t = ( + t 1 + t ) for all = 1 2  Au  " ; t Since the shock term is mean zero, over time we would expected the inflation rate to settle down in expectation to the same level as before, although each period the shock will push the inflation rate above or below the Nash level. In Figure (c19:fa3) we plot the mean and actual trajectories for inflation and unemployment. Inflation and Unemployment 0.4 inflation πt 0.3 0.2 0.1 0 0 20 40 60 80 100 120 140 160 180 200 unemployment ut 0.1 0.05 0 0 50 100 time t 150 200 Figure S.8: The dotted line gives the actual time paths for inflation and unemployment with adaptive expectations when there is a mean-zero i.i.d. Normal shock to the Phillips curve, while the solid lines give the same thing with the shock turned off. Exercise 19.5 As in the previous question, the dynamics of expectations and inflation are given by the system: ? e 1 and: t = ( + t ) for all = 0 1 e= 1 t 1 for all = 1 2 t  Au  ;  Æ ; t t ; ;::: ; ; ; ;::: ; : Recall that initial inflationary expectations are defined to be 0e = 0. Again, the term A is defined to be: A = 1 + 2: Solutions to Exercises 286 We can substitute out the expectations term above to determine the law of motion for inflation: t = Au? + ÆA t 1 ; for all t = 1; 2; : : : ; 1:   Eventually this will converge to a steady-state level of inflation, at which t+1 = t = Substituting in: 1. 1 = Au? + Æ A1 : Solving for 1 produces: The associated inflation rate, Æ u1, is: 1 = 1 AÆ A u?: u1 = 11 Æ AA u?: Æ Notice that if = 1 this is just the normal Nash outcome. As moves closer to zero, so that the private sector puts more and more weight on the government’s (utterly mendacious) announcement, inflation and unemployment both fall.