International Business Cycle Benchmark Models

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1 International Business Cycle Benchmark Models All these benchmark models can be solved using the method of undetermined coefficients as in Christiano, L. (2002), Solving Dynamic Equilibrium Models by a Method of Undetermined Coefficients", Computational Economics, Vol. 20, p Can be downloaded at http: //faculty.econ.nwu.edu/faculty /christiano/research/solve/papernew.pdf 1 One Good, Small Open Economy This section is based on: Schmitt-Grohe, S. and M. Uribe (2003) Closing Small Open Economy Models, Journal of International Economics, Vol. 61, pp The seminal paper in this area is: Mendoza, E. (1991) Real Business Cycles in a Small Open Economy, American Economic Review, Vol. 81, p An endowment economy example Environment preferences: X E 0 t=0 β t log(c t ) technology: y t =(1 ρ)y + ρy t 1 + ε t where ε t is drawn from a normal distribution. markets: goods and one-period bonds at rate r w > 0 where there are adjustment costs ψ 2 bt+1 b 2 with b defined as the steady state level of bonds Equilibrium The programming problem is: max {c t,b t+1} E 0 X β t log(c t ) t=0 s.t.c t + b t+1 = y t +(1+r w )b t ψ 2 bt+1 b 2 1

2 Thinkbacktohowwesolvedthestochasticgrowthmodel: Take foc. Find a steady state. Linearize around the steady state. Solve for decision rules (using method of undetermined coefficients). The foc are: 1+ψ b t+1 b c t = β (1 + r w ) E t 1 c t+1 (1) c t = y t +(1+r w )b t ψ bt+1 b 2 bt+1 (2) 2 Note that if we substitute (2) for c t into (1), we get a second order nonlinear expectational difference equation for b t 1+ψ b t+1 b y t +(1+r w )b t ψ 2 bt+1 b 2 bt+1 " # = β (1 + r w 1 ) E t y t+1 +(1+r w )b t+1 ψ 2 bt+2 b 2 bt+2 which is why we need 2 boundary conditions b 0 and the transversality condition lim E 0β T b T +1 =0 T c T A steady state must satisfy: 1 = β (1 + r w ) (3) c = y + r w b (4) Note that the first condition doesn t pin down the steady state bond holding like the case in the growth model we studied before (i.e. 1= β f 0 (k)+(1 δ) ). Thus there is a continuum of steady states conditional on b. One solution is to let b = b 0. Linearization Consider linearizing g(x t ) around the steady state. In that case, g(x t ) g(x)+g 0 (X)Xbx t where bx t = (Xt X) X. The resource constraint (2) can be written c(1 + bc t ) y(1 + by t )+(1+r w )b(1 + b b t ) b(1 + b b t+1 ) ψ 2 (b b)2 ψ(b b)b b b t+1 2

3 or since constant drop out due to (4) bc t = y c by t +(1+r w ) b c b b t b c b b t+1. (5) The Euler equation (1) can be written 1 1 c + ψ c bb b t+1 1 c bc t β (1 + r w ) E t 1 c 1 c bc t+1 or since constant terms drop out and β (1 + r w )=1from (3) E t [bc t+1 ] bc t = ψb b b t+1 (6) Substituting (5) into (6) yields a linear second order expectational difference equation y E t c by t+1 +(1+r w ) b c b t+1 b c b t+2 (7) µ y c by t +(1+r w ) b c b t b c b t+1 = ψb b b t+1 Method of undetermined coefficients: Conjecture a solution of the form b bt+1 = κ b b bt + κ y by t. Into (7) yields y c ρby t +(1+r w ) b ³κ b b bt + κ y by t b i hκ b ³κ b b bt + κ y by t + κ y ρby t c c µ y c by t +(1+r w ) b c b t b ³κ b b bt + κ y by t c = ψb ³κ b b bt + κ y by t Grouping terms yields by t y b ρ +(1+rw )κ y [κ b κ y + κ y ρ] y b + κ y + cψκ y + b t (1 + r w )κ b κb 2 (1 + r w )+κ b + cψκ b = 0 1 Let g(b t+1,c t )= 1+ψ(b t+1 b), then c t g(b t+1,c t ) = g(b, c)+g b (b, c)b b t + g c (b, c)cbc t = 1 c + ψ c bb b t 1 c 2 cbct. 3

4 In order for this equation to hold for all by t and b t, we must have y b ρ +(1+rw )κ y [κ b κ y + κ y ρ] y b + κ y + ψcκ y = 0 (1 + r w )κ b κb 2 (1 + r w )+κ b + ψcκ b = 0 which are two nonlinear equations in κ b and κ y. The stability of the system depends critically on κ b. Notice, the second equation is a quadratic only in κ b. Furthermore, if ψ =0, the solution to this equation is given by ((1 + r w ) κ b )(κ b 1) = 0 in which case the equation for bonds is explosive (κ b =(1+r w ) > 1) or a unit root (κ b =1). That means there cannot be a solution consistent with a well-defined steady state. This is why we need adjustment costs. More generally, if we let f(κ b )=κb 2 (2+rw +ψc)κ b +(1+r w ), then f(0) = (1 + r w ) > 0 and f(1) = ψc <0, we know by continuity of the quadratic that there is at least one solution κ b (0, 1). Specifically, if we let y =1, b = 1, r w =0.01, ψ=0.01, and ρ =0.9, then κ b =0.91 (the explosive root is given by ). κ b κ y Don t be confused with κ y < 0; when starting from a negative steady state bond holding, some of the increase in income is still saved (i.e. less is borrowed). I ve plotted the levels from the Impulse Response Function in the first figure. In particular, since bx t = (Xt X), we plot X X t =(1+bx t )X. The first few terms in the sequence are: t ε t y t b t+1 c t Simulate model and calculate 2nd moments like corr(y t,nx t = y t c t ). Start from a steady state where by t =0= b b t. WLOG, normalize t =0. From a random number generator with distribution N(0,σ 2 ),draw{ε t } T t=1 (note, matlab has N(0, 1) so just multiply draws by σ 2.Once you have {ε t } T t=1, recursively calculate {b t+1,y t } T t=1. Calculate sample standard deviations and correlations. Do this N times and average the sample statistics (or just do it long enough). 4

5 An important thing to notice from the IRF is that in a small open endowment economy, if the country experiences a positive output shock, the trade balance improves since some of the income rise is consumed but some is saved so NX = S. 1.2 A stochastic production economy An important piece of data is the following (from Table 1 in B-K-K FRB- MQR): If we think of Austria as being a small open economy (i.e. not affecting prices), then the fact that its net exports are negatively correlated (-0.46) with its output is counter to the predictions of the previous model. In particular, in the preceding endowment economy corr(y t,nx t )=corr(y t,y t C t ) corr(y t,y t ) corr(y t,c t ) 1 corr(y t,c t ) > 0 where the positivity is due to consumption smoothing which says that it should be <1 (note that for Austria, corr(y t,c t )=0.65 in Table 1). So what explains the negative correlation in the trade balance? Need to study a production economy (with investment). In that case, corr(y t,nx t )=corr(y t,y t C t I t ) 1 corr(y t,c t ) corr(y t,i t ) which needn t be positive. In fact, for Austria, corr(y t,i t )=0.75, so 1 corr(y t,c t ) corr(y t,i t )= < 0. All this means is that to understand real world current account data, we need to study production economies. A positive technology shock raises the marginal product of capital which makes the country want to borrow in order to finance productive capital. Key point: NX = S I and even if S goes up, if I goes up more, capital inflow implies a trade deficit, not surplus Environment preferences: X E 0 t=0 β t u(c t,h t ) where u(c t,h t )= (c h ω ω ) 1 γ 1 γ. technology: y t = z t kt α ht 1 α with z t =(1 ρ) +ρz t 1 + ε t where ε t is drawn from a normal distribution and i t = k t+1 (1 δ)k t. The paper also includes capital adjustment costs Φ(k t+1 k t )= φ 2 (k t+1 k t ) 2, but from here we will set φ =0. markets: goods and one-period bonds at rate r w > 0 where there are adjustment costs ψ(b t+1 b)

6 1.2.2 Equilibrium The planning problem is: max {c t,h t,b t+1,k t+1 } E 0 X β t u(c t,h t ) (8) s.t.c t + k t+1 + b t+1 = z t kt α h 1 α t +(1+r w )b t +(1 δ)k t ψ(b t+1 b) 2 2 A competitive equilibrium is a set of processes {b t,c t,h t,y t,i t,k t+1 } satisfying household optimization (8). Market clearing is assured as a consequence of Walras Law (i.e. aggregating household budget constraints satisfies market clearing). In particular, notice that rearranging the budget constraint yields CA t b t+1 b t = z t kt α h 1 α t +r w b t ψ(b t+1 b) 2 c t (k t+1 (1 δ)k t ) Calibration From SG-U: γ ω α r w δ ρ σ ε ψ b where β =1/(1 + r) while ψ and b are chosen to ensure that the model matches the average current account to GDP ratio and its volatility. t= Results See Fig.1 of SG-U (Impulse Responses). 2 One Good, Two Country Model This section is based on: Backus, D., P. Kehoe, and F. Kydland (1992) International Real Business Cycles", Journal of Political Economy, Vol. 100, pp See also: Baxter, M. and M. Crucini (1993) Explaining Savings-Investment Correlations, American Economic Review, Vol. 83, pp A simple one good model sufficient to study the Quantity Anomaly (see Table 2 of BKK92): Data: Cross country consumption correlations are lower than output correlations. Puzzle: Model predicts that cross country consumption correlations are higher than output correlations. 6

7 2.1 Environment Countries i =1, 2. Capital is mobile, labor is not. Preferences E 0 P t=0 βt u(c it, 1 n it ) where Technology: c μ it u(c it, 1 n it )= (1 n it) 1 μ 1 γ. 1 γ production y it = z it kit θ n1 θ it time (J) to build k it+1 =(1 δ)k it + s 1 it where sj it+1 = sj+1 it for j =1,..., J 1 and s j it is the number of investment projects in country i at date t that are j periods from completion. Net investment is given by x it = P J s j it j=1 J. In the typical model, J =1so that x it = s 1 it. We will follow this approach but since 4 period time to build actually acts like a capital adjustment cost, in the computer assignment add a tiny capital adjustment cost. TFP (2x1 vector) z t+1 = Az t +ε z t+1 where ε z t+1 N(0,V z ) is iid over time. BKK include Government (2x1 vector) g t+1 = Bg t + ε g t+1 where εg t+1 N(0,V g ) is iid over time and independent of productivity shocks (consistent with acyclicality). We won t do so (i.e. g t =0, t) Markets: Assume complete contingent claims markets. 2.2 Equilibrium The planner s problem with equal weights is: " 2X # X max E 0 β t u(c it, 1 n it ) {c it,x it,n it } i,t 0 = 2X i=1 z it k θ itn 1 θ it i=1 t=0 c it x it g it Solve this problem by method of underdetermined coefficients for decision rules of the form k it+1 = g(k it,k it, z it,z it ) where the independence of g from i reflects the symmetry assumption. 7

8 2.3 Calibration Quarterly A = γ μ β θ δ J g and (symmetric) benchmark uses ,σ 2 1 = σ 2 2 = ,corr(ε z 1,ε z 2). Notice there is cross country spillover of technology shocks Results See IRF figure 2 of BKK. Chart 1 and 2 (impulse responses - might be easier to see in Fig 2 of JPE). Key difference, big jump up in investment (capital inflow) in country where technology rises (this generates a deficit in net exports (i.e. countercyclical net exports)) and jump down in investment (capital outflow) in the other country. This negative contemporaneous correlation of output is at odds with the positive correlation in the data given in Table 2. Eventually (since productivity shocks are positively correlated), outputs grow but there is much lower correlation than in the data and much lower than consumption correlation. THIS IS THE QUANTITY PUZZLE. For the portfolio autarky case (which implies in a one-good model that there can be no trade at all), they find (as expected) that savings and investment are highly correlated (so explain the Feldstein-Horioka puzzle). But this shouldn t be confusing since if NX =0, then S = I. The paper by Baxter and Crucini uses the above framework but allows country size to vary. The reason is that larger countries have larger effects on the world interest rate, and their model predicts that saving-investment correlations are higher for larger countries. 3 Two Good, Two Country Models This section is based on: Backus, D., P. Kehoe, and F. Kydland (1994) Dynamics of the Trade Balance and the Terms of Trade: The J-Curve", American Economic Review, Vol. 84, pp This can be consistent with z i =1even though each row of A does not add up to 1 (i.e. 1 6= a 11 1+a 12 1= =.994) provided the constant in the bivariate autoregression is 1 a11 a 12 z1. 1 a 21 a 22 z 2 8

9 In the one good model, there are not relative prices of imports (in terms of exports) so there is no way to address the Price Anomaly. Data: (Table 1 of AER) terms of trade (price of imports/price of exports) volatility is 2 to 3 times higher than output volatility (as high as 5.86 in Japan)and very persistent. Data: (Figure 1 of AER) Negative contemporaneous correlation (except for US,nearly zero for Canada and Germany) with net exports but positively correlated with future net exports. Like a J curve (they call it a horizontal S). To understand this, let p = qim q ex be the terms of trade and ex p im be net exports. If imports don t react immediately, then the impact valuation effect is for p = ex p im. Eventually if p = im (i.e. substitution effects) then p = ex p im. Theseeffects on imports and exports depend on the elasticity of substitution between foreign and domestic goods. In particular, the correlation is negative for small elasticities and positive for larger elasticities. i.e. we need a lot of imperfect substitutibility to generate negative correlation. To keep things simple, there s one final good, but two intermediate inputs which are imperfect substitutes (can think of the previous JPE paper as two intermediate inputs which are perfect substitutes - case where ρ =1 in PS#2). Again, capital is critical to getting the correlations correct (if eliminate capital, trade balance simply a reflection of output dynamics and consumption smoothing). 3.1 Environment Major difference from BKK (1992, JPE) is in the technology (i.e. 2 country, same preferences over final goods). Technology 1 (Ricardian): Each country specializes in production of an intermediate good: a for country i =1and b for country i =2. Intermediate goods are produced using technology y it = z it kit θ n1 θ it with P 2 i=1 a it = y 1t and P 2 i=1 b it = y 2t where y it denotes GDP in country i measured in units of the local good, and a it ( b it ) denotes uses of the two intermediate goods in country i. Thus, a 2t denotes exports from country 1 (to country 2) and b 1t represents imports into country 1 (from country 2). TFP (2x1 vector) z t+1 = Az t +ε z t+1 where ε z t+1 N(0,V z ) is iid over time. 9

10 Technology 2: Final goods are produced using the (Armington Aggregator) technology G it = ω 1 a ρ it + ω 2 b ρ 1/ρ it where ρ 1. Note that Git (a it,b it ) is homogeneous of degree 1. Perfect substitutes if ρ = 1. For this technology, the elasticity of substitution between foreign and domestic goods is ϕ =1/(1 + ρ). 3 Investment technology is similar (i.e. time (J=1) to build so that x it is net investment. Also consider a government but this is not important for our purposes. Government (2x1 vector) g t+1 = Bg t + ε g t+1 where εg t+1 N(0,V g) is iid over time and independent of productivity shocks (consistent with acyclicality). Markets: Assume complete contingent claims markets. 3.2 Equilibrium Final goods clearing c it + x it + g it = G it (a it,b it ). (9) Since G is homogeneous of degree 1, Euler s theorem implies c it + x it + g it = q 1t a it + q 2t b it (10) where q 1t and q 2t are the relative prices of the two intermediate goods in terms of the final good. For country 1, since a 1t = y 1t a 2t, we can rewrite (10) as c 1t + x 1t + g 1t = q 1t (y 1t a 2t )+q 2t b 1t (11) c 1t + x 1t + g 1t + q 1t (a 2t p t b 1t )=q 1t y 1t where p = q2t q 1t is the terms of trade and (a 2t p t b 1t ) are net exports. 3 Simply: foc from max a,b G(a, b) q aa q b b are: 1 1 n [] ρ 1 ρω 1 a (1+ρ)o = q a ρ µ 1 ω1 y 1+ρ a =. Then da dq a qa a = = 1 1+ρ 1 1+ρ µ ω1 y q a 1 1+ρ 1 µ ω 1 y q 2 a q a q a ³ 1 ω1 y 1+ρ q a 10

11 We can then compute the terms of trade in country 1 from the marginal rate of transformation between the two intermediate goods in country 1 evaluated at the equilibrium quantities: 4 µ p t = q 2t = q 1t à G1t(a 1t,b 1t) b 1t = ω 2 G 1t(a 1t,b 1t) ω 1 a 1t µ a1t b 1t! 1/ϕ. (12) Notice here that the way we will get variation in p t is due to the the fact that a 1t = z 1tk1tn θ 1 θ 1t a 2t b 1t z 2t k2t θ n1 θ b 2t so technology shocks will make this ratio vary and will be amplified by (actually, since ϕ =3/2, it gets dampened). 1 ϕ The planner s problem with equal weights is: " 2X # X max E 0 β t u(c it, 1 n it ) {c it,x it,n it,a it,b it } i,t 2X i=1 0 = 2t i=1 t=0 2X G it (a it,b it ) c it x it g it i=1 a it = z 1t k θ 1tn 1 θ 1t, 2X i=1 b it = z 2t k θ 2tn 1 θ 2t After we solve the simpler planner s problem for an allocation which includes decision rules a it = h i (k 1,k 2,z 1,z 2 ) and b it = e h i (k 1,k 2,z 1,z 2 ) we ³ 1/ϕ can then back out the relative price (i.e. terms of trade) from ω2 a1t ω 1 b 1t. It will not always be the case that we can solve the planner s problem and then infer prices from marginal conditions (in this case from the MRT=price ratio). 5 4 This follows since G 1t (a 1t,b 1t ) b 1t = G 1t (a 1t,b 1t ) a 1t = h ω 1 a ρ 1t +ω 2b ρ i 1/ρ 1t b 1t h ω 1 a ρ 1t +ω 2b ρ i 1/ρ 1t a 1t h ( 1/ρ) h ( 1/ρ) ω 1 a ρ 1t ω 1 a ρ 1t = ω 2b (1+ρ) 1t ω 1 a (1+ρ) = ω 2 ω 1 1t i + ω 2b ρ (1+ρ)/ρ ³ 1t + ω 2b ρ 1t µ 1/ϕ a1t i (1+ρ)/ρ ³ ρω 2 b (1+ρ) 1t ρω 1 a (1+ρ) 1t 5 Same idea as backing out interest rates (an intertemporal price) from the decision rule for k t+1 = g(k t,z t ) into r t+j = θz t+j (g(k t+j 1,z t+j 1 )) θ 1. b 1t 11

12 3.3 Calibration Quarterly (similar to above) γ μ β θ δ J ϕ =1/(1 + ρ) g and the (symmetric) benchmark uses A = ,σ 2 1 = σ 2 2 = ,corr(ε z 1,ε z 2). new part: ω 1 and ω 2 calibrated to match the average import share of To see this, note that from (12) we have in a symmetric steady state y 1 = y 2,b 1 = a 2,p=1and where b 1 y 1 a 1 = 1 b1 y 1 b b 1 1 denotes the share of imports. These imply p = ω 2 ω 1 y 1 µ 1/ϕ a1 b 1 1= ω 2 µ ω2 ω 1 ω 1 Ã 1 b 1 y 1 b 1 y 1! 1/ϕ ϕ = Ã 1 b 1 y 1 b 1 y 1 can be used to pin down ω2 ω 1 given b1 y 1 =0.15 and ϕ =1.5. For the benchmark ϕ, sd of p is 0.48 (see Table 3). When ϕ rises to 2.5 (large elasticity), sd of p falls even more to But when ϕ falls to 0.5 (small elasticity), sd of p rises to From Table 1, for the U.S. data sd of p is 2.92 (still 4x larger than the small elasticity finding). That is, the elasticity would have to be absurdly low to get even close to the data - this is the basis of the price puzzle. 3.4 Results Positives: Model replicates the high persistence of net exports and terms of trade in the data. is close to the contemporaneous countercyclicality of net exports in the data, as well as negative contemporaneous correlation in net exports and terms of trade.! 12

13 generates the cross correlation inherent in the J curve. See BKKFig4-6. Negative: See Table 3 of BKK. Model does not match the volatility of the terms of trade in the data (7 times smaller). THIS IS THE PRICE PUZZLE. The table also shows all the robustness checks they used. 4 Traded and Nontraded Goods Models This section is based on: Stockman, A. and L. Tesar (1995) Tastes and Technology in a Two- Country Model of the Business Cycle: Explaining International Comovements, American Economic Review, Vol. 85, pp (p.173) Perhaps the most striking feature of the data of the seven industrialized countries is the large share (roughly half) of a country s output consists of nontraded goods. They decompose the 10 sectors of OECD Intersectoral data base into traded (agriculture, manufacturing, mining, retail, transportation) and nontraded (water, construction, finance, insurance, real estate, private and govt services). One of the ideas of this paper was probably to help solve the consumption correlation puzzle some. If some goods cannot be traded, then there is incomplete insurance. 4.1 Environment Two Countries, each specializes in the production of a tradable and nontradable good (Ricardian). Technology: production Yt i = A i t K i 1 α i t N i α i t where i {T,NT,T,NT }, K i and N i are capital and labor in sector i. CanthinkofA i t = Ztγ i αi and then need to make correction to capital accumulation (so that everything is consistent in the resource feasibility constraint. Capital is immobile between traded and nontraded goods sectors It i = γkt+1 i (1 δ)kt i wherewewillsetγ =1here. Labor is mobile between traded and nontraded goods sectors 1= Nt T + Nt NT + L t and 1=Nt T + Nt NT + L t. The demeaned TFP (4x1 vector) bivariate process is A t+1 = ΩA t + ε t+1 where ε t+1 N(0,V z ) is iid over time and A t = 13 A T t A NT t A T t A NT t

14 Preferences E 0 P t=0 βt u(c 1t,c 2t,d t,l t ) where u(c 1t,c 2t,d t,l t )= ½ (c θ 1tc 1 θ 2t ) μ + d μ t 1 σ ¾ 1 σ 1 μ where c 1 is the traded good produced by domestic firms, c 2 is the traded good produced by foreign firms, d is the nontraded good, and L is leisure. Markets: Assume complete contingent claims markets. 4.2 Equilibrium Goods clearing Yt T = c 1t + c 1t + It T (13) Yt T = c 2t + c 2t + It T Yt NT = d t + It NT Yt NT = d t + It NT The planner s problem of choosing {c 1t,c 2t,d t,nt T,Nt NT with equal weights is given by L a t,i T t,i NT t,c 1t,c 2t,d t,n T t,n NT t,i T t,it NT } t " X max E 0 β t u(c 1t,c 2t,d t, 1 Nt T t=0 Nt NT )+u(c 1t,c 2t,d t, 1 Nt T Nt NT ) ª# subject to (13). Since S-T are also interested in price puzzles and the planner s problem only provides allocations, how do we get prices? Use the necessary conditions (like foc and budget constraint implied by the decentralized version of this environment). For a detailed treatment of the equivalence see So in that case, the household s budget constraint is given by: P 1t c1t + I T t + P2t c 2t + Pt N (d t + It NT )+ X Q t (A t+1 )B t+1 (A(14) t+1 A t+1 = w t (Nt T + Nt NT )+r 1t Kt T + rt NT K NT t + B t where B t+1 (A t+1 ) denote state contingent claims. Note that the rhs just comes from CRS and zero profit for firms π 1t = max Kt T,N t T π NT P N Kt NT,Nt NT t t = max P 1t Y T t w t N T t r 1t K T t Y NT t w t N NT t rt NT Kt NT 14

15 From this, can see that the two important relative prices for trade (termsof-trade (relative price of imports/exports) P 2t /P 1t and Pt N /P 1t can be found from the static foc after associating λ t with the hhbc (14): c 1t : u(c 1t,c 2t,d t, 1 Nt T c 1t c 2t : u(c 1t,c 2t,d t, 1 Nt T c 2t d t : u(c 1t,c 2t,d t, 1 Nt T d t so that ratios imply: N NT t ) N NT t ) N NT t ) = λ t P 1t = λ t P 2t = λ t P N t u(c 1t,c 2t,d t,1 Nt T c 2t u(c 1t,c 2t,d t,1 Nt T N NT t ) N NT = P 2t t ) c 1t u(c 1t,c 2t,d t,1 Nt T NT Nt ) d t u(c 1t,c 2t,d t,1 Nt T N NT = P N t t ) P 1t c 1t P 1t (15) Since the planner s problem gives us c 1t,c 2t,d t, 1 Nt T Nt NT, we simply need to plug them into the linearized ratios (15) to get the prices. 4.3 Calibration Annual (symmetry) α T α NT δ β 1/σ 1/(1 + μ) θ 1/a where 1/σ is the intertemporal elasticity of substitution, 1/(1 + μ) is the elasticity of substitution between traded and nontraded goods, θ is the share of domestically produced goods in the consumer s bundle of traded goods (i.e. equal consumption shares of both traded goods), 1/a is the intertemporal elasticity of substitution in leisure. The labor shares suggest labor receives more income from traded (e.g. manufactured) goods than nontraded (e.g. services) goods. The (symmetric) benchmark shock processes are calculated by a 4 variable autoregression on Solow residuals yielding Ω = Notice symmetry and that the degree of autocorrelation is low, especially in the traded goods sector. 15

16 V = Note that the variance of productivity shocks is nearly twice as high in the traded good industry as in the nontraded goods industry. The cross-country correlation of TFP shocks between the traded goods sectors is 0.33 and between the nontraded goods is within country correlation of TFP shocks between traded and nontraded goods sectors is Results Table 6 shows that the model driven by technology shocks replicates the standard deviations of oputput, labor, investment and consumption fairly closely. Two exceptions are that the model sd of investment in the traded goods sector is 30% higher than in the data and the model sd of output in the nontraded sector is 40% higher than in the data. The correlation between home and foreign output is exactly that in the data (64%). The model greatly overstates cross country correlation of traded consumption. This is not surprising given complete markets except for two nonseparabilities: MUtraded goods not independent of leisure nor nontraded goods (traded and nontraded goods are complements since their elasticity of substitution is less than 1). The model greatly overpredicts the negative correlation (-1) of the relative price of nontraded goods and the ratio of nontraded to traded goods (which is only in the data). The model greatly underpredicts the sd of the trade balance (15x smaller) and the terms of trade (2.5x). S-T suggest adding taste shocks. Do this by specifying preferences as ½ h τ 1t(c θ 1t c1 θ 2t ) μ +[τ2t d t ] μi 1 μ u(c 1t,c 2t,d t,l t )= 1 σ where τ i is a positive random variable with mean 1. ¾ 1 σ L a t. 16

17 5 Two Country Models with Money This section is based on: Finn, M. (1999) An Equilibrium Theory of Nominal and Real Exchange Rate Comovement, Journal of Monetary Economics, Vol. 44, pp The theoretical foundation of all this is from: Lucas, R. (1982) Interest Rates and Currency Prices in a Two-Country", Journal of Monetary Economics, Vol. 10, pp Environment Two countries {star, nostar}, two goods{d,f} Preferences: Technologies: U(c dt,c ft, t ) = ψ log c dt +(1 ψ)logc ft + η log(1 t ) U(c dt,c ft, t ) = ψ log c dt +(1 ψ)logc ft + η log(1 t ) Y dt = z t k (1 α) t α t Y ft = zt k (1 α) t α t z t+1 = (1 ρ)z + ρz t + ε t+1 z t+1 = (1 ρ)z + ρz t + ε t+1 Govt Reaction Function Markets: Goods: k t+1 = (1 δ)k t + i t k t+1 = (1 δ)k t + i t M t+1 µ ω Yt+1 = γ M t Y t Mt+1 µ Y Mt = γ t+1 Y t ω c dt + c dt + i t = Y dt c ft + c ft + i t = Y ft Let P t denote the domestic currency price of domestic goods and P t denote the foreign currency price of foreign goods. 17

18 Complete contingent claims on s t =(z t,zt ). Let b(s t ) denote claims paying out in domestic currency in state s t and b (s t ) denote claims paying out in foreign currency in state s t.let q(s t ) denote the domestic currency price of b(s t ) claims and q (s t ) denote the foreign currency price of b (s t ) claims. Money: m dt + m dt = M t m ft + m ft = M t Timing: Realization s t occurs, then agents choose portfolios and goods. 5.2 Equilibrium We will formulate the problem recursively (like Lucas and unlike Finn s sequence problem formulation). After the realization of s =(z t,zt ), the agent chooses m d,m f,b(s 0 ) in the asset market given current domestic good denominated wealth θ subject to budget constraint Z m d + em f + q(s 0 )b(s 0 )ds 0 Pθ (16) Future wealth evolves according to P 0 θ 0 = m d P (c d +k 0 (1 δ)k)+e 0 [m f P c f ]+PY +b(s 0 )+P 0 τ 0 (17) The c-i-a constraints are m d (c d + k 0 (1 δ)k) (18) P m f P c f Optimization: v(s, k, θ) = max U(c d,c f, )+β {c d,c f,,k 0,b(s 0 ),m d,m f } subject to (16), (17), and (18). Z f(s 0,s)v(s 0,k 0,θ 0 )ds 0 Suppose we go down the Lucas route and assume the cia constraint binds (verify later). Then (17) becomes and θ 0 = P P 0 zk(1 α) α + b(s0 ) P 0 + τ 0 (19) c d + k 0 (1 δ)k + ep P c f + X q(s 0 )b(s 0 ) P s 0 θ (20) 18

19 FOC : ψ c d : = λ(s) c d 1 ψ c f : = λ(s) ep c f P Z η : 1 = βα f(s 0,s)v θ s 0,k 0,θ 0 P P 0 (s 0 ) Z k 0 : λ(s) =β f(s 0,s)v k s 0,k 0,θ 0 ds 0 Y ds 0 b(s 0 ) : q(s 0 )λ(s) P = βf(s 0,s)v θ s 0,θ 0,b(s 0 ),k 0 1 P 0 (s 0 ) In order to derive the envelope conditions, note that the value function can be written: v(s, k, θ) = U(c d(s, k, θ),c f(s, k, θ), (s, k, θ)) Z +β f(s 0,s)v s 0,k 0 (s, k, θ),θ 0 (s, k, θ) ds 0 " θ c +λ(s) d (s, k, θ) k 0 (s, k, θ)+(1 δ)k ep P c f (s, k, θ) R q(s 0 )b (s 0,s,k,θ) P ds 0 # where θ 0 (s, k, θ) = P P 0 zk(1 α) (s, k, θ) α + b (s 0,s,k,θ) P 0 + τ 0. The envelope condition for θ is as before: v θ (s, k, θ) =λ(s) = ψ c d The envelope condition for k can be derived from dc f dc d v k (s, k, θ) = U 1 dk + U 2 dk + U 3 Z ( +β f(s 0,s) where d dk v k s 0,k 0 (k, θ),θ 0 (k, θ) dk 0 dk +v θ s 0,k 0 (k, θ),θ 0 (k, θ) dθ 0 +λ(s) dc d dk dk0 ep dc f +(1 δ) dk P dk dk ) ds 0 Z q(s 0 ) dθ 0 dk = P P 0 (1 α)zk α (k, θ) α + P P 0 αzk(1 α) α 1 d (k, θ) dk + 1 db (s 0 ) P 0. dk P db (s 0 ) 0 ds dk 19

20 After grouping terms we have Z v k (s, k, θ) = λ(s)(1 δ)+β f(s 0,s)v θ s 0,k 0 (k, θ),θ 0 (k, θ) P P 0 (s 0 ) (1 α)y k ds0 Z ψ(1 δ) = + β f(s 0 ψ P,s) c d c 0 d (s0 ) P 0 (s 0 ) (1 α)y k ds0 This implies that the following equations are necessary: r c f = : k 0 : b(s 0 ) : (1 ψ) c d (21) ψ Z η 1 = βα f(s 0 ψ P Y,s) c d (s 0 ) P 0 (s 0 ds 0 (22) ) Z 1 = β f(s 0 (1 δ),s) c d c 0 d (s0 ) ds0 (23) Z +β 2 f(s 00,s 0 )f(s 0 (1 α) P 0 (s 0 ) Y 0,s) c 00 d (s00 ) P 00 (s 00 ) k 0 ds00 ds 0 q(s 0 ) = βf(s 0 1,s) Pc d P 0 (s 0 )c d (s 0 (24) ) where µ ep r. (25) P As before, a sufficient condition for the c-i-a constraint to be binding is that the net interest rate on a nominal bond is positive. This amounts to establishing conditions from (24) under which 1 R q(s0 )ds 0 > 1 (26) Z 1 >β f(s 0 Pc d,s) P 0 (s 0 )c d (s 0 ) ds0 But these are the same kind of conditions as before: (1)β low; (2) high expected inflation; (3) high expected growth rate of consumption. 6 So, if these conditions are satisfied, then goods and money market clearing pin down the price level: M t = m dt + m dt = P t (c dt + i t )+P t c dt = P t Y t (27) P t = M t /Y t. The first condition is just the quantity equation of money where velocity is unity. 6 Actually we can go further with (26) to yield: µ 1 >β ½E s E s + cov s 1 1+π(s 0 ) 1 1+g cd (s 0 ) 1 1+π(s 0 ), 1 1+g cd (s 0 ) ¾ 20

21 So given money (exogenous, nothing to solve for) and output (from the labor and capital decision rules), we can get nominal price levels via (27). But then from the domestic and foreign consumption decision rules and (21) we can get the real exchange rate r t. Butthenfrom(25)wecanget the nominal exchange rate e t. Specifically: e = P r P = M Y (1 ψ) c d ψc f Y M. 5.3 Calibration Quarterly model. Parameters in Table Results α δ β ψ η γ ω {0, 0.25, 0.5, 0.75} Data: Table 1 (empirical regularities of nominal e and real r exchange rates. High correlation between (e, r) in excess of 0.9 High persistence in all measures in excess of 0.9. High ω matches the data well but need investment shocks too (not part of the environment I wrote up above). 21

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