International Macroeconomics and Finance: Theory and Empirical Methods Phần 5 docx

5.1. CALIBRATING THE ONE-SECTOR GROWTH MODEL 145 where a 0 = U c g 1 + βU c g 2 =0, a 1 = βU cc g 1 g 2 , a 2 = U cc g 2 1 + βU cc g 2 2 + βU c g 22 , a 3 = U cc g 1 g 2 , a 4 = βU c g 32 + βU cc g 2 g 3 , a 5 = U cc g 1 g 3 . The derivatives are evaluated at steady state values. A second but equivalent option is to take a second—order Taylor approximation to the objective function around the steady state and to solve the resulting quadratic optimi zation problem. The second o ption is equivalent to the Þrst because it yields linear Þrst—order conditions around the steady state. To pursue the second option, recall that λ t = (k t+1 ,k t ,A t ) 0 . Write the period utility function in the unconstrained optimization problem as R(λ t )=U[g(λ t )]. (5.20) Let R j = ∂R(λ t )/∂λ jt be the partial derivative of R(λ t ) with respect to the j−th element of λ t and R ij = ∂ 2 R(λ t )/(∂λ it ∂λ jt ) be the second cross-partial derivative. Since R ij = R ji the relevant derivatives are, R 1 = U c g 1 , R 2 = U c g 2 , R 3 = U c g 3 , R 11 = U cc g 2 1 , R 22 = U cc g 2 2 , +U c g 22 R 33 = U cc g 2 3 , R 12 = U cc g 1 g 2 , R 13 = U cc g 1 g 3 , R 23 = U cc g 2 g 3 + U c g 23 . The second-order Taylor expansion of the period utility function is R(λ t )=R(λ)+R 1 (k t+1 − k)+R 2 (k t − k)+R 3 (A t − A)+ 1 2 R 11 (k t+1 − k) 2 146 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLES + 1 2 R 22 (k t − k) 2 + 1 2 R 33 (A t − A) 2 + R 12 (k t+1 − k)(k t − k) +R 13 (k t+1 − k)(A t − A)+R 23 (k t − k)(A t − A). Suppose we let q =(R 1 ,R 2 ,R 3 ) 0 be the 3 × 1 row vector of partial derivatives (the gradient) of R,andQ be the 3 × 3matrixofsecond partial derivatives (the Hessian) multiplied by 1/2whereQ ij = R ij /2. Then the approximate period utility function can be compactly written in matrix form as R(λ t )=R(λ)+[q +(λ t − λ) 0 Q](λ t − λ). (5.21) The problem is now to maximize E t ∞ X j=0 β j R(λ t+j ). (5.22) The Þrst order conditions are for all t 0=(βR 2 + R 1 )+βR 12 (k t+2 − k)+(R 11 + βR 22 )(k t+1 − k)+R 12 (k t − k) +βR 23 (A t+1 − 1) + R 13 (A t − 1). (5.23) If you compare (5.23) to (5.19), you’ll see that a 0 = βR 2 + R 1 , a 1 = βR 12 ,a 2 = R 11 + βR 22 ,a 3 = R 12 ,a 4 = βR 23 ,a 5 = R 13 .This veriÞes that the two approaches are indeed equivalent. Now to solve the linearized Þrst-order conditions, work with (5.19). Since the data that we want to explain are in logarithms, you can con- vert the Þrst-order cond itions into near logarithmic form. Let ˜a i = ka i for i =1, 2, 3, and let a “hat” denote the approximate log difference from the steady state so that ˆ k t =(k t − k)/k ' ln(k t /k) and ˆ A t = A t − 1 (since the steady state value of A =1). Nowlet b 1 = −˜a 2 /˜a 1 , b 2 = −˜a 3 /˜a 1 , b 3 = −a 4 /˜a 1 , and b 4 = −a 4 /˜a 1 . The second—order stochastic difference equation (5.19) can be writ- ten as (1 − b 1 L − b 2 L 2 ) ˆ k t+1 = W t , (5.24) where W t = b 3 ˆ A t+1 + b 4 ˆ A t . 5.1. CALIBRATING THE ONE-SECTOR GROWTH MODEL 147 The roots of the polynomial (1 − b 1 z − b 2 z 2 )=(1− ω 1 L)(1 − ω 2 L) satisf y b 1 = ω 1 + ω 2 and b 2 = −ω 1 ω 2 . Using the quadratic formula and evaluating at the parameter values that we used to calibrate the model, the roots are, z 1 =(1/ω 1 )=[−b 1 − q b 2 1 +4b 2 ]/(2b 2 ) ' 1.23, and z 2 =(1/ω 2 )=[−b 1 + q b 2 1 +4b 2 ]/(2b 2 ) ' 0.81. There is a stable root, |z 1 | > 1 which lies outside the unit circle, and an unstable root, |z 2 | < 1 which lies inside the unit circle. The presence of an unstable root means that the solution is a saddle path. If you try to simulate (5.24) directly, the capital stock will diverge. To solve the difference equation, exploit the certainty equivalence property of quadratic optimization problems. That is, you Þrst get the perfect foresight solution to the problem by solving the stable root backwards and the unstable root forwards. Then, replace future ran- dom variables with their expected values conditional upon the time-t information set. Begin by rewriting (5.24) as W t =(1− ω 1 L)(1 − ω 2 L) ˆ k t+1 =(−ω 2 L)(−ω −1 2 L −1 )(1 − ω 2 L)(1 − ω 1 L) ˆ k t+1 =(−ω 2 L)(1 − ω −1 2 L −1 )(1 − ω 1 L) ˆ k t+1 , and rearrange to get (1 − ω 1 L) ˆ k t+1 = −ω −1 2 L −1 1 − ω −1 2 L −1 W t = − µ 1 ω 2 L −1 ¶ ∞ X j=0 µ 1 ω 2 ¶ j W t+j = − ∞ X j=1 µ 1 ω 2 ¶ j W t+j . (5.25) The autoregressive speciÞcation (5.18) implies the prediction formulae E t W t+j = b 3 E t ˆ A t+j+1 + b 4 E t ˆ A t+j =[b 3 ρ + b 4 ]ρ j ˆ A t . Use this forecasting rule in (5.25) to get ∞ X j=1 µ 1 ω 2 ¶ j E t W t+j =[b 3 ρ + b 4 ] ˆ A t ∞ X j=1 µ ρ ω 2 ¶ j = " ρ ω 2 − ρ # (b 3 ρ + b 4 ) ˆ A t . 148 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLES It follows that the solution for the capital stock is ˆ k t+1 = ω 1 ˆ k t − " ρ ω 2 − ρ # [b 3 ρ + b 4 ] ˆ A t . (5.26) To recover ˆy t , note that the Þrst-order expansion of the produc- tion function gives y t = f(A, k)+f A ˆ A t + f k k ˆ k t ,wheref A =1, and f k =(αy)/k. Rearrangement gives ˆy t = ˆ A t + ˆ k t .Torecover ˆ i t , subtract the steady state value γk = i +(1−δ)k from (5.8) and rearrange to get ˆ i t =(k/i)[γ ˆ k t+1 −(1−δ) ˆ k t ]. Finally, get ˆc t =ˆy t − ˆ i t from the adding-up constraint (5.9). The log levels of the variables can be recovered by ln(Y t )=ˆy t +ln(X t )+ln(y), ln(C t )=ˆc t +ln(X t )+ln(c), ln(I t )= ˆ i t +ln(X t )+ln(i), ln(X t )=ln(X 0 )+t ln(γ). Sim u lating the Model We’ll use the calibrated model to generate 96 time-series observations corresponding to the n umber of observations in the data. From these pseudo-observations, recover the implied log-levels and pass them through the Hodrick-Prescott Þlter. The steady state values are y =1.717,k=5.147,c=1.201,i/k=0.10. Plots of the Þltered log income, consumption, and investmen t o bserva- tions are given in Figure 5.3 and the associated descriptive statistics are given in Table 5.2. The autoregressive coefficien t and the error variance of the technology shock were selected to match the volatility of output exactly. From the Þgure, you can see that both consumption and in- vestment exhibit high co-movemen ts w ith output, and all three series display persistence. However from Table 5.2 the implied inv estment series is seen to be more volatile than output but is less v olatile than that found in the data. Consumption implied by the model is more volatile than output, which is counterfactual. 5.2. CALIBRATING A TWO-COUNTRY MODEL 149 -0.2 -0.15 -0.1 -0.05 0 0.05 73 75 77 79 81 83 85 87 89 91 93 95 Investment GDP (broken) Consumption Figure 5.3: Hodrick-Presco tt Þltered cyclical observations from the model. Investment has been shifted down b y 0.10 for visual clarity. This coarse overview of the one sector real business cycle model shows that there are some aspects of the data that the model does not explain. This is not surprising. Perhaps it is more surprising is how well it actually does in generating ‘realistic’ time series dynamics of the data. In any event, this perfect markets—no nominal rigidities Arrow- Debreu model serves as a useful benchmark against which reÞnements can be judged. 5.2 Calibrating a Two-Country Model We now add a second country. This two-country model is a special case of Backus et. al. [5]. Each county produces the same good so we will not be able to study terms of trade or real exchange rate issues. The presence of country-speciÞc idiosyncratic shocks give an incentive to individuals in the two countries to trade as a means to insure each 150 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLES Table 5.2: Calibrated Closed-Economy Model Std. Autocorrelations Dev. 1 2 3 4 6 y t 0.022 0.90 0.79 0.67 0.53 0.23 c t 0.023 0.97 0.89 0.77 0.63 0.31 i t 0.034 0.70 0.50 0.36 0.19 -0.04 Cross correlation with y t−k at k 6 4 1 0 -1 -4 -6 c t 0.49 0.77 0.96 0.90 0.79 0.33 0.04 i t 0.29 0.11 0.41 0.74 0.73 0.61 0.44 other against a bad relative technology shock so we can examine the behavior of the current account. Measurement We will call the Þrst c ountry the ‘US,’ and second country ‘Europe.’ The data for European output, government spending, investment, and consumption are the aggregate of observations for the U K, France, Ger- many, and Italy. The aggregate of their current account balances suf- fer from double counting and does not make sense because of intra- European trade. Therefore, we examine only the US current account, which is measured as a fraction of real GDP. Table 5.3 displays the features of the data that we will attempt to explain–their volatility, persistence (characterized by their autocorre- lations) and their co-movements (characterized by cross correlations). Notice that US and European consumption correlation is lower than the their output correlation. The Two-Country Model Both countries experience identical rates of depreciation of phy sical capital, long-run technological growth X t+1 /X t = X ∗ t+1 /X ∗ t = γ,have 5.2. CALIBRATING A TWO-COUNTRY MODEL 151 Table 5.3: Open-Economy Measurements Std. Autocorrelations Dev. 1 2 3 4 6 ex t 0.01 0.61 0.50 0.40 0.40 0.12 y ∗ t 0.014 0.84 0.62 0.36 0.15 -0.15 c ∗ t 0.010 0.68 0.47 0.30 0.04 -0.15 i ∗ t 0.030 0.89 0.75 0.57 0.40 0.07 Cross correlations at lag k 6 4 1 0 -1 -4 6 y t ex t−k 0.43 0.42 0.41 0.41 0.37 0.03 0.32 y t y ∗ t−k 0.28 0.22 0.21 0.36 0.43 0.36 0.22 c t c ∗ t−k 0.26 0.39 0.28 0.25 0.05 0.15 0.26 Notes: ex t is US net exports divided by GDP. Foreign country aggregates data from France, Germany, Italy, and the UK. All variables are real per capita from 1973.1 to 1996.4 and have been passed through the Hodrick—Prescott Þlter with λ = 1600. the same capital shares and Cobb-Douglas form of the production func- tion, and identical utility. Let the social planner attach a weight of ω to the domestic agent and a weight of 1−ω to the foreign agent. In terms of efficiency units, the social planner’s problem is now to maximize E t ∞ X j=0 β j [ωU(c t+j )+(1− ω)U(c ∗ t+j )], (5.27) subject to, y t = f(A t ,k t )=A t k α t , (5.28) y ∗ t = f(A ∗ t ,k ∗ t )=A ∗ t k ∗α t , (5.29) γk t+1 = i t +(1− δ)k t , (5.30) γk ∗ t+1 = i ∗ t +(1− δ)k ∗ t , (5.31) y t + y ∗ t = c t + c ∗ t +(i t + i ∗ t ). (5.32) In the market economy interpretation, we can view ω to indicate the size of the home country in the world economy. (5.28) and (5.29) are the 152 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLES Cobb—Douglas production functions for the home and foreign counties, with normalized labor input N = N ∗ = 1. (5.30) and (5.31) are the domestic and foreign capital accumulation equations, and (5.31) is the new form of the resource constraint. Both countries have the same technology but are subject to heterogeneous transient shocks to total productivity according to " A t A ∗ t # = " 1 − ρ − δ 1 − ρ − δ # + " ρδ δρ #" A t−1 A ∗ t−1 # + " ² t ² ∗ t # , (5.33) where (² t ,² ∗ t ) 0 iid ∼ N(0, Σ). We set ρ =0.906, δ =0.088, Σ 11 = Σ 22 = 2.40e−4, and Σ 12 = Σ 21 =6.17e−5. The contemporaneous correlation of the innovations is 0.26. Apart from the objective function, the main difference between the two-county and one-country models is the resource constraint (5.32). World output can either be consumed or sav ed but a country’s net sav- ing, which is t he current account balance, can be non—zero (y t − c t − i t = −(y ∗ t − c ∗ t − i ∗ t ) 6=0). Let λ t =(k t+1 ,k ∗ t+1 ,k t ,k ∗ t ,A t ,A ∗ t ,c ∗ t ) be the state vector, and i ndi- cate the dependence of consumption on the state by c t = g(λ t ), and c ∗ t = h(λ t )(whichequalsc ∗ t trivially). Substitute (5.28)—(5.31) into (5.32) and re-arrange to get c t = g(λ t )=f(A t ,k t )+f(A ∗ t ,k ∗ t ) − γ(k t+1 + k ∗ t+1 ), +(1 − δ)(k t + k ∗ t ) − c ∗ t (5.34) c ∗ t = h(λ t )=c ∗ t . (5.35) For future referen ce, the derivatives of g and h are, g 1 = g 2 = −γ, g 3 = f k (A, k)+(1− δ), g 4 = f k (A ∗ ,k ∗ )+(1− δ), g 5 = f(A, k)/A, g 6 = f(A ∗ ,k ∗ )/A ∗ , g 7 = −1, h 1 = h 2 = ···= h 6 =0, h 7 =1. 5.2. CALIBRATING A TWO-COUNTRY MODEL 153 Next, transform the constrained optimization problem into an un- constrained problem by substituting (5.34) and (5.35) into (5.27). The problem is now to maximize ωE t ³ u[g(λ t )] + βU[g(λ t+1 )] + β 2 U[g (λ t+2 )] + ··· ´ (5.36) +(1 −ω)E t ³ u[h(λ t )] + βU[h(λ t+1 )] + β 2 U[h(λ t+2 )] + ··· ´ . At date t, the choice variables available to the planner are k t+1 ,k ∗ t+1 , and c ∗ t .Differentiating (5.36) with respect to these variables and re- arranging results in the Euler equations γU c (c t )=βE t U c (c t+1 )[g 3 (λ t+1 )], (5.37) γU c (c t )=βE t U c (c t+1 )[g 4 (λ t+1 )], (5.38) U c (c t )=[(1− ω)/ω]U c (c ∗ t ). (5.39) (5.39) is the Pareto—Optimal risk sharing rule which sets home marginal utility proportional to foreign marginal utility. Under log utility, home and foreign per capita consumption are perfectly correlated, c t =[ω/(1 − ω)]c ∗ t . The Two-Country Steady State From (5.37) and (5.38) we obtain y/k = y ∗ /k ∗ =(γ/β+δ−1)/α.We’ve already determined that c =[ω/(1 − ω)]c ∗ = ωc w where c w = c + c ∗ is world consumption. From the production functions (5.28)—(5.29) we get k =(y/k) 1/(α−1) and k ∗ =(y ∗ /k ∗ ) 1/(α−1) . From (5.30)—(5.31) we get i = i ∗ =(γ + δ −1)k. It follows that c = ωc w = ω[y + y ∗ −(i + i ∗ )] =2ω [y − i]. Thus y − c − i =(1− 2ω)(y − i) and unless ω =1/2, the current account will not be balanced in the steady state. If ω > 1/2thehome country spends in excess of GDP and runs a current account deÞcit. How can this be? In the market (competitive equilibrium) interpreta- tion, the excess absorption is Þnanced by interest income earned on past lending to the foreign country. Foreigners need to produce in excess of their consumption and investment to service the debt. In a sense, they ha ve ‘o ver-invested’ in ph ysical capital. In the planning problem, the social planner simply takes away some of the foreign output and gives it to domestic agents. Due to the 154 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLES concavity of the production function, optimality requires that the world capital stock be split up between the two countries so as to equate the marginal product of capital at home and abroad. Since technology is identical in the 2 countries, this implies e qualization of national capital stocks, k = k ∗ , and income levels y = y ∗ , even if consumption differs, c 6= c ∗ . Quadratic Approximation You can solve the model by taking the quadratic approximation of the unconstrained objective function about the steady state. Let R be the period weighted average of home and foreign utility R(λ t )=ωU[g(λ t )] + (1 − ω)U[h(λ t )]. Let R j = ωU c (c)g j +(1− ω)U c (c ∗ )h j , j =1, ,7betheÞrst partial derivative of R with respect to the j−the e lement of λ t .Denotethe second partial derivative of R by R jk = ∂R(λ) ∂λ j ∂λ k = ω[U c (c)g jk +U cc g j g k ]+(1−ω)[U c (c ∗ )h jk +U cc (c ∗ )h j h k ]. (5.40) Let q =(R 1 , ,R 7 ) 0 be the gradient vector, Q be the Hessian matrix of second partial derivatives whose j, k−th element is Q jk =(1/2)R j,k . Then the second-order Taylor approximation to the period utility func- tion is R(λ t )=[q +(λ t − λ) 0 Q](λ t − λ), and you can rewrite (5.36) as max E t ∞ X j=0 β j [q +(λ t+j − λ) 0 Q](λ t+j − λ). (5.41) Let Q j• be the j−th row of the matrix Q.TheÞrst-order conditions are (k t+1 ): 0=R 1 + βR 3 + Q 1• (λ t − λ)+βQ 3• (λ t+1 − λ), (5.42) (k ∗ t+1 ): 0=R 2 + βR 4 + Q 2• (λ t − λ)+βQ 4• (λ t+1 λ), (5.43) (c ∗ t ): 0=R 7 + Q 7• (λ t − λ). (5.44) [...]... forwards and the stable root backwards The solution for the world capital stock is ˜w ˜w kt+1 = ω1 kt − ´ (ρ + δ) ³ ˜ [m3 (ρ + δ) + m4 ]Aw + m5 t ω2 − (ρ + δ) (5. 53) Now you need to recover the domestic and foreign components of the world capital stock Subtract (5. 49) from (5. 48) to get ˜ kt+1 − ˜∗ kt+1 = Ã ! Ã ! ˜7 − a7 ˜8 − a8 b ˜ b ˜ ˜ ˜ At+1 + A∗ t+1 a3 − a4 ˜ ˜ a3 − a4 ˜ ˜ (5. 54) Add (5. 53) to (5. 54)... + fk kt = y At + α ∗ kt , ˜ (5. 57) k 158 CHAPTER 5 INTERNATIONAL REAL BUSINESS CYCLES and investment rates are ˜ ˜ ˜t = γ kt+1 − (1 − δ)kt , i ˜∗ ˜∗ ˜∗ = γ kt+1 − (1 − δ)kt it (5. 58) (5. 59) Let world consumption be cw = ct + c∗ = yt + yt − (˜t + ˜∗ ) By the ˜t ˜ ˜t ˜ ˜∗ i it ∗ optimal risk-sharing rule (5. 39) ct = [(1 − ω)/ω]˜t , which can be used ˜ c to determine ct (5. 60) ct = ω˜w ˜ It follows... Q74 0 0 Q 75 Q76 0 Q77 R7 Mimicking the algorithm developed for the one-country model and using (5. 47) to substitute out c∗ and c∗ in (5. 45) and (5. 46) gives t t+1 0 = a1 kt+2 + a2 kt+2 + a3 kt+1 + a4 kt+1 + a5 kt + a6 kt + a7 At+1 ˜ ˜ ˜ ˜∗ ˜ ˜ ˜ ˜∗ ˜ ˜ ˜ ˜∗ ˜ ˜ +˜8 A∗ + a9 At + a10 A∗ + a11 , a ˜t+1 ˜ ˜ ˜ ˜t ˜ (5. 48) b ˜∗ b ˜ b ˜∗ b ˜ b ˜∗ b ˜ 0 = ˜1 kt+2 + ˜2 kt+2 + ˜3 kt+1 + ˜4 kt+1 + 5 kt + ˜6... Autocorrelations 1 2 3 4 0.66 0.40 0. 15 0.07 0.63 0.42 0.18 0.12 0. 05 -0.13 -0.09 -0.10 0.09 -0.09 -0.09 -0.10 0. 65 0.32 0.07 -0. 15 0.63 0.42 0.18 0.12 0.03 -0. 15 -0.07 -0.08 6 0.04 -0.04 0.03 -0.00 -0.27 -0.04 0.00 Cross correlations at k 6 4 1 0 -1 -4 0.00 0.18 0.41 0.44 0.21 0. 15 0.10 0.06 0.27 0.18 0.06 0.28 -6 0. 15 0. 05 97 160 CHAPTER 5 INTERNATIONAL REAL BUSINESS CYCLES International Real Business Cycles... DEVIATIONS FROM UIP 169 Table 6.3: Estimates of Regression Equations (6.3) and (6.4) US-BP ˆ2 β -3.481 t(β2 = 0) (-2.413) t(β2 = 1) (-3.107) ˆ 4.481 β1 US-JY US-DM DM-BP DM-JY -4.246 -0.796 -1.6 45 -2.731 (-3.6 35) (-0 .54 2) (-1.326) (-1.797) (-4.491) (-1.222) (-2.132) (-2. 455 ) 5. 246 1.796 2.6 45 3.731 BP-JY -4.2 95 (-2.626) (-3.237) 5. 2 95 Notes: Nonoverlapping quarterly observations from 1976.1 to 1999.4 t(β2... b ˜ b (5. 49) t+1 t 156 CHAPTER 5 INTERNATIONAL REAL BUSINESS CYCLES At this point, the marginal beneÞt from looking at analytic expressions for the coefficients is probably negative For the speciÞc calibration of the model the numerical values of the coefficients are, ˜1 = 0.1 05, a1 = 0.1 05, ˜ b ˜2 = 0.1 05, a2 = 0.1 05, ˜ b ˜3 = −0.212, b a3 = −0.218, ˜ ˜4 = −0.218, b a4 = −0.212, ˜ 5 = 0.107, a5 = 0.107,... Slope Coefficients and T-ratios using Overlapping and Nonoverlapping Observations Overlapping percentiles Relative T Observations 2 .5 50 97 .5 Range 50 yes slope 0.778 0.999 1.207 0.471 tNW (-2.738) (-0.010) (2.716) 1.207 tHH [-2.998] [-0.010] [3.248] 1.383 16 no slope 0 .54 3 0.998 1. 453 tOLS ((-2.228)) ((-0.008)) ((2.290)) 100 yes slope 0.866 0.998 1.126 0.474 tNW (-2.286) (-0.0 25) (2. 251 ) 1.098 tHH [-2.486]... ˜7 ˜w ˜ b a11 + ˜11 ˜ b a1 kt+2 + ˜ ˜w kt+1 + 5 kt + a ˜w At+1 +˜9 Aw + a ˜t = 0 (5. 50) 2 2 2 ˜w ˜ ˜∗ (5. 50) is a second—order stochastic difference equation in kt = kt + kt , 4 which can be rewritten compactly as ˜ ˜ ˜ k w − m1 k w − m2 k w = W w , (5. 51) t+2 t+1 t t+1 w ˜t+1 ˜t where Wt+1 = m3 Aw + m4 Aw , and m1 m2 m3 m4 m5 4 −(˜3 + a4 )/(2˜1 ), a ˜ a − 5 /˜1 , a a −(˜7 + ˜7 )/(2˜1 ), a b a −˜9 /˜1... = −0.128, ˜ ˜8 = −0.130, a8 = −0. 159 , ˜ b ˜9 = 0. 158 , a9 = 0. 158 , ˜ b ˜10 = 0. 158 , b a10 = 0. 158 , ˜ ˜11 = 0.007 a11 = 0.007, ˜ b You can see that a3 + a4 = ˜3 + ˜4 and a7 + ˜7 = a8 + ˜8 which means ˜ ˜ b b ˜ b ˜ b that there is a singularity in this system To deal with this singularity, ˜t ˜ ˜t let Aw = At + A∗ denote the ‘world’ technology shock and add (5. 48) to (5. 49) to get a3 + a4 ˜w ˜ ˜ a7 + ˜7... b5 kt + b6 kt + b7 At+1 ˜ ˜ ˜ +b8 A∗ + b9 At + b10 A∗ + b11 c∗ + b12 c∗ + b13 , ˜ ˜ (5. 46) t+1 t t+1 t ˜ ˜∗ ˜ ˜∗ ˜ ˜ 0 = d3 kt+1 + d4 kt+1 + d5 kt + d6 kt + d9 At + d10 A∗ t +d12 c∗ + d13 , ˜t (5. 47) where the coefficients are given by j aj 1 βQ31 2 βQ32 3 βQ33 + Q11 4 βQ34 + Q12 5 Q13 6 Q14 7 βQ 35 8 βQ36 9 Q 15 10 Q16 11 Q37 12 Q17 13 R1 + βR3 bj βQ41 βQ42 βQ43 + Q21 βQ44 + Q22 Q23 Q24 βQ 45 βQ46 Q25 . ˜a 4 ! ˜ A t+1 + Ã ˜ b 8 − ˜a 8 ˜a 3 − ˜a 4 ! ˜ A ∗ t+1 . (5. 54) Add (5. 53) to (5. 54) to get ˜ k t+1 = 1 2 [ ˜ k w t+1 +( ˜ k t+1 − ˜ k ∗ t+1 )]. (5. 55) The d ate t +1 world capital stock is predetermined. economy. (5. 28) and (5. 29) are the 152 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLES Cobb—Douglas production functions for the home and foreign counties, with normalized labor input N = N ∗ = 1. (5. 30). for the one-country model and using (5. 47) to substitute out c ∗ t and c ∗ t+1 in (5. 45) and (5. 46) gives 0=˜a 1 ˜ k t+2 +˜a 2 ˜ k ∗ t+2 +˜a 3 ˜ k t+1 +˜a 4 ˜ k ∗ t+1 +˜a 5 ˜ k t +˜a 6 ˜ k ∗ t +˜a 7 ˜ A t+1 +˜a 8 ˜ A ∗ t+1 +˜a 9 ˜ A t +˜a 10 ˜ A ∗ t +˜a 11 ,