8.4. VAR ANALYSIS OF MUNDELL—FLEMING 259 When this is multiplied out, you get 0=−θ 1 α + β/λ, (8.77) 0=πδα −[θ 1 + π µ δ + σ λ ¶ ]β. (8.78) It follows that α = β/θ 1 λ. (8.79) Because α is proportional to β,weneedtoimposeanormalization. Letthis normalization be β = p o − ¯p where p o ≡ p(0). Then α =(p o − ¯p)/θ 1 λ = −[p o − ¯p]/θλ,whereθ ≡−θ 1 .Usingthesevaluesofα and β in (8.63) and (8.64), yields p(t)=¯p +[p o − ¯p]e −θt , (8.80) s(t)=¯s +[s o − ¯s]e −θt , (8.81) where (s o − ¯s)=−[p o − ¯p]/θλ. This solution gives the time paths for the price level and the exchange rate. To characterize the system and its response to monetary shocks, we will w ant to phase diagram the system. Going back to (8.58) and (8.61), we see that ús(t)=0ifandonlyifp(t)=¯p, while úp(t)=0ifandonlyif s(t) −¯s =(1 + σ/λδ)(p(t) − ¯p). These points are plotted in Figure 8 .10. The system displays a saddle path solution. 260 CHAPTER 8. THE MUNDELL-FLEMING MODEL p=0 . p s . s=0 Figure 8.10: Phase diagram for the Dornbusch model. 8.4. VAR ANALYSIS OF MUNDELL—FLEMING 261 Problems 1. (Static Mundell-Fleming with imperfect capital mobility). Let the trade balance be given by α(s + p ∗ − p) − ψy . A real depreciation raises exports and raises the trade balance whereas an increase in income leads to higher imports which lowers the trade balance. Let the capital account be given by θ(i−i ∗ ), where 0 < θ < ∞ indexes the degree of capital mobility. We replace (8.3) with the external balance condition α(s + p ∗ − p) − ψy + θ(i − i ∗ )=0, that the balance of pa yments is 0. (We are ignoring the service ac- count.) When capital is completely immobile, θ = 0 and the balance of pa yments reduces to the trade balance. Under perfect capital mobility, θ = ∞ implies i = i ∗ which is ( 8.3). (a) Call the external balance condition the FF curv e. Draw the FF curve in r, y space along with the LM and IS curves. (b) Repeat the comparative statics experiments cov ered in this chap- ter using the modiÞed external balance condition. Are any of the results sensitive t o the degree of capital mobility? In particular, how do the results depend on the slope of the FF curve in relation to the LM curve? 2. How would the Mundell-Fleming model with perfect capital mobility explain the international co-mo vements of macroeconomic variables in Chapter 5? 3. Consider the Dornbusch model. (a) What i s the instantaneous effect on t he exchange rate of a shock to aggregate demand? Why does an aggregate demand shock not pro duce overshooting? (b) Suppose output can change in the short run by replacing the IS curve (8.7) with y = δ(s − p)+γy − σi + g, replace the price adjustment rule (8.8) with úp = π(y−¯y), where long-run output is given by ¯y = δ(¯s − ¯p)+γ ¯y −σi ∗ + g. Under w hat c ircumstances is the overshooting result (in response to a change in money) robust? 262 CHAPTER 8. THE MUNDELL-FLEMING MODEL Chapter 9 The New International Macroeconomics The new international macroeconomics are a class of theories that em- bed imperfect competition and nominal rigidities in a dynamic general equilibrium open economy setting. In these models, producers have monopoly power and charge price above marginal cost. Since it is op- timal in the short run for producers to respond to small ßuctuations by changing output, these models explain why output is demand de- termined in the short run when current prices are predetermined due to some nominal rigidity. It follows from the imperfectly competitive environment that equilibrium output lies below the socially optimal level. We will see that this feature is instrumental in producing re- sults that are very different from Mundell—Fleming models. Because Mundell—Fleming predictions can be overturned, it is perhaps inaccu- rate to characterize these models as providing the micro-foun dations for Mundell-Fleming. These models also, and not surprisingly, are sharply distinguished from the Arrow-Debreu style real business cycle models. Both classes of theories are set in dynamic general equilibrium with optimizing agents and w ell-speciÞed tastes and technology. Instead of being set in a per- fect real business cycle world, the presence of market imperfections and nominal rigidities permit international transfers of wealth in equi- librium and prevent equilibrium welfare from reaching the socially op- timal level of welfare. It therefore makes sense here to examine the 263 264CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICS welfare effects of policy interventions whereas it does not make sense in real business cycle models since all real business cycle dynamics are Pareto efficient. The genesis of this literature is the Obstfeld and Rogoff [113] Re- dux model. This model makes several surprising predictions that are contrary to Mundell—Fleming. The model is somewhat fragile, however, as we will see when we cover the pricing-to-market reÞnement by Betts and Devereux [10]. In this chapter, stars denote foreign country variables but lower case letters do not automatically mean logarithms. Unless explicitly noted, variables are in levels. There is also a good deal of notation. For ease of reference, Table 9.1 summarizes the notation for the Redux model and Table 9.2 lists the notation for the pricing-to-market model. The terms household, agent, consumer and individual are used interchangeably. The home currency unit is the ‘dollar’ and the foreign currency is the ‘euro.’ 9.1 The Redux Model We are set in a deterministic environment and agents have perfect foresigh t. There are 2 countries, each populated by a continuum of consumer—producers. There is no physical capital. Each household produces a distinct and differentiated good using only its labor and the production of each household is completely specialized. Households are arranged on the unit interval, [0, 1] with a fraction n living in the home country and a fraction 1−n living in the foreign country. We will index domestic agents by z where 0 <z<n, and foreign agents by z ∗ where n<z ∗ < 1. When we refer to both home and foreign agents, we will use the index u where 0 <u<1. Preferences. Households derive utility from consumption, leisure, and real cash balances. Higher output means more income, which is good, but it also means less leisure which is bad. Money is introduced through the utility function where agents value the real cash balances of their own country’s money. Money does not have intrinsic value but 9.1. THE REDUX MODEL 265 0 n 1 z z* Home Country Foreign Country Figure 9.1: Home and foreign households lined up on the unit interval. provides individuals with indirect utility because higher levels of real cash balances help to lower shopping (transactions) costs. We assume that households have identical utility functions and w e will work with a representative household. Representative agent (household) in Redux model.Letc t (z)bethe home representative agent’s consumption of the domestic good z,and c t (z ∗ ) be the agent’s consumption of the foreign good z ∗ .Peoplehave tastes for all varieties of goods and the household’s consumption basket is a constant elasticity of substitution (CES) index that aggregates across the available varieties of goods C t = · Z 1 0 c t (u) θ−1 θ du ¸ θ θ−1 = · Z n 0 c t (z) θ−1 θ dz + Z 1 n c t (z ∗ ) θ−1 θ dz ∗ ¸ θ θ−1 , (9.1) where θ > 1 is the elasticity of substitution between the varieties. 1 Let y t (z)bethetime-t output of individual z, M t be the domestic per capita money stock and P t be the domestic price level. Lifetime utility of the representative domestic household is given by ⇐(147) U t = ∞ X j=0 β j   ln C t+j + γ 1 − ² à M t+j P t+j ! 1−² − ρ 2 y 2 t+j (z)   , (9.2) 1 In the discrete commodity formulation with N goo ds, the index can be written as C = · P N z=1 c θ−1 θ z ∆z ¸ θ θ−1 where ∆z = 1. The representation under a continuum of goods takes the limit of the sums given by the integral formulation in (9.1). 266CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICS where 0 < β < 1 is the subjective discount factor, C t+j is the CES index given in (9.1) and M t /P t are real balances. The costs of forgone leisure associated with work are represented by the term (−ρ/2)y 2 t (z). Let p t (z) be the domestic price of good z, S t be the nominal ex- change rate, and p ∗ t (z) be the foreign currency price of good z.Akey assumption is that prices are set in the producer’s currency. It follows that the law of one price holds for every goo d 0 <u<1 p t (u)=S t p ∗ t (u). (9.3) The pricing assumption also implies that there is complete pass through of nominal exchange rate ßuctuations. That is, an x−percent depre- ciation of the dollar is fully passed through resulting in an x−percent increase in the dollar price of the imported good. Since utility of consumption is a monotone transformation of the CES index, we can begin with some standard results from consumer theory under CES utility. 2 First, the correct domestic price index is(148)⇒ P t = · Z 1 0 p t (u) 1−θ du ¸ 1 1−θ (9.4) = · Z n 0 p t (z) 1−θ dz + Z 1 n [S t p ∗ t (z ∗ )] 1−θ dz ∗ ¸ 1 1−θ . Second, household demand for the domestic good z, and for the foreign good z ∗ are c t (z)= " p t (z) P t # −θ C t , (9.5) 2 In the static problem facing a consumer who wants to maximize U =(x θ−1 θ 1 + x θ−1 θ 2 ) θ θ−1 subject to I = p 1 x 1 + p 2 x 2 , where I is a given level of nominal income, the indirect utility function is v(p 1 ,p 2 ; I)= I [p (1−θ) 1 + p (1−θ) 2 ] 1 1−θ , the appropriate price index is, P =[p (1−θ) 1 +p (1−θ) 2 ] 1 1−θ , and the individual’s demand for g ood j = 1, 2isx d j =[p j /P ] −θ (I/P ), where (I/P) is real income. 9.1. THE REDUX MODEL 267 c t (z ∗ )= " S t p ∗ t (z ∗ ) P t # −θ C t . (9.6) Analogously, foreign household lifetime utility is ⇐(150) U ∗ t = ∞ X j=0 β j   ln C ∗ t+j + γ 1 − ² à M ∗ t+j P ∗ t+j ! 1−² − ρ 2 y ∗2 t+j (z ∗ )   , (9.7) with consumption and price indices ⇐(151) C ∗ t = · Z n 0 c ∗ t (z) θ−1 θ dz + Z 1 n c ∗ t (z ∗ ) θ−1 θ dz ∗ ¸ θ θ−1 , (9.8) P ∗ t =   Z n 0 à p t (z) S t ! 1−θ dz + Z 1 n [p ∗ t (z ∗ )] 1−θ dz ∗   1 1−θ , (9.9) and individual demand for z and z ∗ goods c ∗ t (z)= " p t (z) S t P ∗ t # −θ C ∗ t , c ∗ t (z ∗ )= " p ∗ t (z ∗ ) P ∗ t # −θ C ∗ t . Every good is equally important in home and foreign households utility. It follows that the elasticity of demand 1/θ, in all go ods mar- kets whether at home or abroad, is identical. Every producer has the identical technology in production. In equilibrium, all domestic produc- ers behave identically to each other and all foreign producers behave identically to eac h other in the sense that they produce the same level of output and charge the same price. Thus it will be the case that for any two domestic producers 0 <z<z 0 <n y t (z)=y t (z 0 ), p t (z)=p t (z 0 ), and that for any two foreign producers, n<z ∗ <z ∗ 0 < 1 y ∗ t (z ∗ )=y ∗ t (z ∗ 0 ), p ∗ t (z ∗ )=p ∗ t (z∗ 0 ). 268CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICS It follows that the home and foreign price levels, (9.4) and (9.9) simplify to P t =[np t (z) 1−θ +(1− n)(S t p ∗ t (z ∗ )) 1−θ ] 1 1−θ , (9.10) P ∗ t =[n(p t (z)/S t ) 1−θ +(1− n)p ∗ t (z ∗ ) 1−θ ] 1 1−θ , (9.11) and that PPP holds for the correct CES price index P t = S t P ∗ t . (9.12) Notice that PPP will hold for GDP deßators only if n =1/2. Asset Markets. The world capital market is fully integrated. There is an internationally traded one-period real discount bond which is de- nominatedintermsofthecomposite consumption good C t . r t is the real interest rate paid by the bond between t and t + 1. The bond is available in zero net supply so that bonds held by foreigners are issued by home residents. The gross nominal interest rate is given by the Fisher equation 1+i t = P t+1 P t (1 + r t ), (9.13) and is related to the foreign nominal interest rate by uncovered interest parity 1+i t = S t+1 S t (1 + i ∗ t ). (9.14) Let B t be the stock of bonds held by the domestic agent and B ∗ t be the stock o f bonds h eld by the foreign agent. By the zero-net supply constraint 0 = nB t +(1− n)B ∗ t , it follows that B ∗ t = − n 1 −n B t . (9.15) The Government. For 0 <u<1, let g t (u)behomegovernmentcon- sumption of good u. Total home and foreign government consumption is given by a the analogous CES aggregator over government purchases of all varieties(153)⇒ [...]... (1 68) ⇒ (169)⇒ (9 .87 ) ˆ ˆ ˆ But ˆ b/(1 − n) = yt (z) − yt (z ∗ ) − St − (Ct − Ct∗ ), which follows from ˆ ˆ∗ subtracting (9 .82 ) from (9 .81 ) and noting that ˆ = ˆt In addition, b b ˆ yt (z) − yt (z ∗ ) = θSt , which you get by subtracting (9. 48) from (9.47), ˆ ˆ∗ using PPP and noting that pt (z) − p∗ (z ∗ ) = 0 Now you can rewrite ˆ ˆt (9 .87 ) as (θ 2 − 1)r ˆ ˆ ˆ St , C − C∗ = (9 .88 ) r(1 + θ) + 2θ and. .. r(1 + θ) + 2θ and solve (9 .85 ) and (9 .88 ) to get ²[r(1 + θ) + 2θ] ˆ ˆ (Mt − Mt∗ ), − 1) + ²[r(1 + θ) + 2θ] ²[r(θ2 − 1)] ˆ ˆ = (Mt − Mt∗ ) r(θ2 − 1) + ²[r(1 + θ) + 2θ] ˆ St = ˆ ˆ Ct − Ct∗ (170)⇒ (9 .89 ) (9.90) From (9 .87 ) and (9.90), the solution for the current account is ˆ= b (171)⇒ r(θ2 2θ²(1 − n)(θ − 1) ˆ ˆ (Mt − Mt∗ ) r(θ2 − 1) + ²[r(1 + θ) + 2θ] (9.91) (9 .83 ), (9.90) and (9.69) together give the... 274CHAPTER 9 THE NEW INTERNATIONAL MACROECONOMICS Together, these relations tell us that 0-steady-state output at home and abroad are equal to consumption y0 (z) = ∗ y0 (z ∗ ) " θ−1 = ρθ #1/2 ∗ w = C0 = C0 = C0 (9.37) Nominal and real interest rates in the 0-steady state are equalized with (1+i0 )/i0 = 1/(1−β) By (9. 28) and (9.29), 0-steady state money demand is " #1/² ∗ M0 M0 γy0 (z) = ∗ = (9. 38) P0 P0 1−β... unanticipated and permanent monetary shock The analysis of governments spending shocks is treated in the end-of-chapter problems Monetary Shocks Set Gt = 0 for all t in the preceding equations and subtract (9. 78) from (9.77), (9 .80 ) from (9.79), and use PPP to obtain the pair of equations ˆ ˆ ˆ ˆ C − C ∗ = Ct − Ct∗ , 1 ˆ β ˆ ˆ ˆ ˆ ˆ ˆ Mt − Mt∗ − St = (Ct − Ct∗ ) − (S − St ) ² ²(1 − β) (9 .83 ) (9 .84 ) ˆ Substitute... interest rate gives the liquidity effect à ! β ˆ rt = − ² + ˆ Mtw (1 − β) (9.94) A home monetary expansion lowers the real interest rate and raises average world consumption From the world demand functions (9.47) 282 CHAPTER 9 THE NEW INTERNATIONAL MACROECONOMICS (175)⇒ and (9. 48) it follows that domestic output unambiguously increases following a the domestic monetary expansion The monetary shock raises home... the international inequality of real interest rates implies that home and foreign consumption will be not be perfectly correlated The bond is sold at discount and has a face value of one dollar Let Bt be the dollar value of bonds held by domestic households, and ∗ Bt be the dollar value of bonds held by foreign households Bonds ∗ outstanding are in zero net supply nBt + (1 − n)Bt = 0 The dollar 288 CHAPTER... 290CHAPTER 9 THE NEW INTERNATIONAL MACROECONOMICS " qt (z ∗ ) ct (z ) = Pt ∗ #−θ Ct , (9.119) Foreign household demand for domestic z-goods and for and foreign z ∗ -goods are " #−θ ∗ qt (z) ∗ ct (z) = Ct∗ , (9.120) Pt∗ c∗ (z ∗ ) t " p∗ (z ∗ ) = t ∗ Pt #−θ Ct∗ (9.121) Firms Firms only employ labor There is no capital in the model The domestic and foreign production technologies are identical and are linear... qt (z ∗ ) θ p∗ (z ∗ ) = = W ∗ (9.125) t St θ−1 t Using (9.124) and (9.125), the formulae for the price indices (9.1 08) and (9.109) can be simpliÞed to ⇐( 186 - 187 ) Pt = Pt∗ = h npt (z)(1−θ) + (1 − n)qt (z ∗ )(1−θ) h i ∗ nqt (z)(1−θ) + (1 − n)p∗ (z ∗ )(1−θ) t 1 (1−θ) i 1 (1−θ) , (9.126) (9.127) Output is demand determined in the short run and can either be sold to the domestic market or made available... − Pt = C − β rt + ˆ ² t 1−β (164)⇒ (9.77) (9. 78) (9.79) (9 .80 ) Using the consolidated budget constraints, (9.40)—(9.41) and the price level response (9.73) and (9.74), the current account responds by 9.1 THE REDUX MODEL (165-166)⇒ 279 ˆt = yt (z) − (1 − n)St − Ct − gt , ˆ ˆ b ˆ ˆ ˆ∗ = y ∗ (z ∗ ) + nSt − C ∗ − g ∗ = −n ˆt ˆ ˆ bt ˆt ˆt b t 1−n (9 .81 ) (9 .82 ) We have not speciÞed the source of the underlying... or by lump—sum taxes Tt , and Tt∗ Negative values of Tt and Tt∗ are lump—sum transfers from the government to residents The budget constraints of the home and foreign governments are Mt − Mt−1 Gt = Tt + , (9.16) Pt M∗ − M∗ G∗ = Tt∗ + t ∗ t−1 (9.17) t Pt Aggregate Demand Let average world private and government consumption be the population weighted average of the domestic and foreign counterparts . = −[p o − ¯p]/θλ,whereθ ≡−θ 1 .Usingthesevaluesofα and β in (8. 63) and (8. 64), yields p(t)=¯p +[p o − ¯p]e −θt , (8. 80) s(t)=¯s +[s o − ¯s]e −θt , (8. 81) where (s o − ¯s)=−[p o − ¯p]/θλ. This solution. level and the exchange rate. To characterize the system and its response to monetary shocks, we will w ant to phase diagram the system. Going back to (8. 58) and (8. 61), we see that ús(t)=0ifandonlyifp(t)=¯p,. money) robust? 262 CHAPTER 8. THE MUNDELL-FLEMING MODEL Chapter 9 The New International Macroeconomics The new international macroeconomics are a class of theories that em- bed imperfect competition and nominal