Sectoral Price Changes and Output Growth - Supply and Demand in General (Appendix)

Appendix 1 Linear Approximation and State Space Solution of the Multisector Flexible Price Model In this appendix we show how we approximate the optimizing conditions (19) (32) linearly around a steady state and solve the resulting linear dynamic system. Note that the percentage deviations of the variables from steady state levels are denoted by a circumflex (^). Combining equations (19), (22), (24) and (32) and taking linear approximation we obtain, ĉ i, t + p̂ i, t = 0 (A.1) We substitute for yi,t from the production function into (25) and solve for wt. Then substituting for wt into (20) and linearizing, we obtain J J i =1 j=1 (1 − b i ) ĥ i,t + ∑ sh i ĥ i,t − ∑ a i , j n̂ i , j,t − p̂ i,t = Â i,t + α i k̂ i,t − E ν̂ t +1 t (A.2) where shi is the ratio of labor input in sector i to leisure. Substitution for yi,t from the production function into (28) and linear approximation yields J b i ĥ i,t + ∑ a i, j n̂ i, j,t − n̂ i, j,t + p̂ i,t − p̂ j,t = −α i k̂ i,t − Â i,t j =1 (A.3) Substituting for ξt into (29) from (28) and then linearizing we obtain, J x̂ i, t − ∑ d j x̂ j, t + p̂ i, t = q̂ i, t j =1 Linear approximation of the market clearing condition (31) yields (A.4) J J j =1 j =1 − sc i ĉ i,t + b i ĥ i,t + ∑ a i, j n̂ i, j,t − ∑ sn j, i n̂ j, i,t − sx i x̂ i,t = −α i k̂ i,t − Â i,t (A.5) g g + (1 + ~ gi ) ĝ t + ĝ i,t yi yi where sci is the fraction of sector i’s output allocated to consumption. Similarly, snj,i is the fraction of good i output used as material inputs in sector j, and sxi is its fraction used for investment. Note that in order to linearize the market clearing condition (31), we first re-write the equation as follows: ⎛ g i, t − g t y i, t = c i, t + x i, t + g t × ⎜⎜ gt ⎝ J ⎞ ⎟ + g t + ∑ n j, i, t ⎟ j =1 ⎠ (A.6) g i, t − g t , where gi,t is the government (autonomous) expenditure on We define ~ g i, t = gt good i in period t and gt is the aggregate government expenditures. Therefore, ~ g i, t can be interpreted as percentage deviation of sectoral government expenditures from the mean. We substitute for yi,t from the production function into (26) and solve for ri,t. Then substituting for ri,t into (21) and linearizing, we obtain ⎡⎛ ⎛ ⎛ β ⎞ β ⎞ β ⎢⎜⎜1 − (1 − δ i ) ⎟⎟(α i − 1) k̂ i , t +1 + ⎜⎜ (1 − δ i ) ⎟⎟ q̂ t + ⎜⎜1 − (1 − δ i ) ηk ⎠ ηk ⎠ ηk ⎝ ⎝ ⎢⎝ E⎢ t J ⎫⎪ ⎢× ⎧⎪⎨b ĥ + ∑ a n̂ i i,t +1 i , j i , j,t +1 + p̂ i,t +1 + Â i,t +1 ⎬ − q̂ t − ν̂ t + 2 + ν̂ t +1 ⎢ ⎪ j=1 ⎪⎭ ⎣ ⎩ ⎞⎤ ⎟⎟⎥ ⎠⎥ ⎥=0 ⎥ ⎥ ⎦ (A.7) We substitute for zi,t in (30) from the capital accumulation technology and linearize around steady state to obtain J ∑ sz i i =1 J J ηk sz i (1 − δ i ) k̂ i,t +1 = ∑ k̂ i,t + ∑ d i x̂ i,t {η k − (1 − δ i )} i =1{η k − (1 − δ i )} i =1 (A.8) where szi is the share of investment in sector i to aggregate investments in the economy. Using notation in King, Plosser, and Rebelo (1987), equations (A.1) - (A.5) can be written as: Mcc clt = Mcs st + Mce et (A.9) where Mcc is a (J2+4J) × (J2+4J) matrix of the coefficients; clt is a (J2+4J) × 1 vector of control variables, namely ci, hi, xi, ni,j and pi , and i,j = 1,2,3,….J; Mcs is a (J2+4J) × (J+1) matrix of coefficients; st is a (J+1)×1 vector of J predetermined variables, ki, i= 1,2,….J, and one non-predetermined variable, q; Mce is a (J2+4J) × (2J+1) matrix of coefficients and et is a (2J+1) ×1 vector of exogenous variables. Given this system of equations, (A.9) can be solved for the control variables as functions of predetermined, non-predetermined and exogenous variables. Given these optimal solutions, equations: (A.7) and (A.8) imply a first-order dynamic system in st, i.e. in kis and q, E s t +1 = Ws t + R E e t +1 + Qe t t t (A.10) where W is a (J+1) × (J+1) matrix and R and Q are (J+1) × (2J+1) matrices. To obtain the solution to this system of difference equations we decompose W as PμP-1, where P is the matrix of eigenvectors of W and μ is a diagonal matrix with the eigenvalues on the diagonal. For our system since there is only one non-predetermined variable (qt), in order for this system to have a unique solution there should be just one eigenvalue of W outside the unit circle. After algebraic manipulation, the optimal time path for capital accumulation may be written as follows: k̂ t +1 = π kk k̂ t + π ke ê t (A.12) where πkk and πke are respectively J × J and J × (2J+1) matrices of coefficients. These coefficients are complicated functions of the underlying parameters of preferences and technology. The dynamics of capital accumulation depend on the previous period’s capital stock, current exogenous factors and on the entire future time path of these factors. Appendix 2 Construction of Data on Sectoral Autonomous Expenditures For the simulation of the model, we need to calibrate shocks to the sectoral government expenditures, which are assumed to be exogenous. Data on government expenditures by sectors are, however, not readily available. Therefore, we decompose sectoral real GDP into three final expenditure components, namely, consumption, investment and government expenditures. This decomposition scheme exactly matches the market clearing conditions of our model. The method we use for decomposition involves the following steps: 1) We obtain data on ‘personal consumption expenditures’ and ‘gross fixed investments’ by major type of products from unpublished tables of the National Income and Product Accounts (NIPA) for a period from 1947 to 1997. Note that consumption data are available for 83 product categories and investment data are available for 26 product categories. 2) These consumption and investments by products are then mapped into consumption and investments by two-digit input-output (I-O) industries using a mapping scheme outlined in Survey of Current Business (April, 1994) for 1987 Benchmark InputOutput Tables. Note that there are ninety-five two-digit I-O industries. 3) These consumption and investments data are then consolidated to match thirty-five sectors of the KLEM data set. 4) We obtain data on GDP by 2-digit SIC industries from the National Income and Product Accounts (NIPA) “Income, Employment and Product by Industry” unpublished detail tables and consolidate, wherever necessary, for thirty-five sectors. 5) Data on government expenditures by sectors are then obtained as residuals by subtracting consumption and investments from sectoral GDP. Note that all these values are in current dollars. In order to convert them into 1987 constant dollar we deflate the government expenditures by the price indices obtained from the KLEM data set. Note that sectoral GDP also includes a ‘net exports’ component and it is impossible to separate out this component from the residuals we obtain in the final step. Therefore, we prefer the nomenclature ‘sectoral autonomous expenditures’ to ‘sectoral government expenditures’. Sectors that have relatively higher shares of net exports in their respective total GDP will be subject to volatility that arises from changes in international market conditions. Table 1. Summary Statistics of Correlations across Sectors Mean Median Standard Deviation Minimum Maximum (1) (2) (3) (4) (5) Corr(Δpi ,Δyi ) -0.27 (0.040) -0.31 0.24 -0.60 0.33 Corr(Δpi ,ΔAi ) -0.51 (0.033) -0.53 0.19 -0.89 0.01 Corr(Δyi ,ΔAi ) 0.45 (0.045) 0.49 0.27 -0.18 0.91 Note: Sample period 1949-1989. Estimated standard errors are in parentheses. Table 2. Intermediate and Final Uses of Sectoral Output in 1987 Fractions of output in different uses Sector Intermed Consum (1) (2) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Agriculture Metallic ores mining Coal mining Crude oil and natural gas Nonmetallic minerals mining Construction Food and kindred products Tobacco products Textile Apparel Lumber and wood products Furniture and fixtures Paper Printing Chemicals Petroleum refining and related prod Rubber Footwear, leather, and leather Stone Primary metal Fabricated metal Machinery Electrical machinery Motor vehicle Transportation equipment Instruments Miscellaneous manufacturing Transport Communications Electric utility service Gas utility service Wholesale and retail trade FIRE Services Government enterprises 0.87 1.07 0.84 1.43 1.01 0.20 0.38 0.14 0.85 0.29 0.94 0.11 0.89 0.63 0.66 0.51 0.90 0.41 0.98 1.08 0.93 0.43 0.61 0.31 0.19 0.24 0.30 0.59 0.51 0.51 0.68 0.28 0.38 0.44 0.56 0.12 0.00 0.01 0.00 0.00 0.00 0.61 0.79 0.12 0.98 0.02 0.53 0.11 0.27 0.28 0.44 0.09 1.55 0.07 0.00 0.04 0.03 0.23 0.52 0.09 0.09 0.87 0.25 0.39 0.40 0.31 0.62 0.57 0.54 0.39 Invest (3) 0.00 0.07 0.00 0.00 0.00 0.61 0.00 0.00 0.05 0.00 0.05 0.42 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.51 0.20 0.34 0.09 0.38 0.12 0.01 0.03 0.00 0.00 0.05 0.02 0.01 0.00 Govt.pur Net exp (4) (5) 0.01 -0.02 0.00 0.00 0.00 0.19 0.02 0.00 0.01 0.02 0.00 0.05 0.03 0.08 0.05 0.08 0.02 0.02 0.01 0.01 0.02 0.08 0.09 0.04 0.48 0.31 0.05 0.06 0.06 0.09 0.03 0.01 0.02 0.01 0.04 0.04 -0.12 0.10 -0.40 -0.01 0.00 -0.02 0.06 -0.05 -0.31 -0.04 -0.13 -0.04 0.00 -0.01 -0.05 -0.02 -1.03 -0.08 -0.11 -0.03 -0.05 -0.15 -0.27 0.12 -0.03 -0.42 0.09 0.02 -0.01 -0.02 0.04 0.02 0.00 0.00 Note: These ratios have been calculated from BEA’s 1987 Benchmark I-O Use Table Table 3. Simulation Results: Average Sectoral Correlations Data Baseline Model No I-O structure and independent technology shocks (1) (2) (3) (4) (5) -0.27 -0.27 (-0.41, -0.14) -0.84 (-0.87, -0.81) -0.61 (-0.67, -0.55) -0.39 (-0.49, -0.29) Corr(Δpi ,ΔAi ) -0.51 -0.56 (-0.67, -0.45) -0.92 (-0.95, -0.89) -0.79 (-0.84, -0.74) -0.68 (-0.75, -0.61) Corr(Δyi ,ΔAi ) 0.45 0.71 (0.67, 0.75) 0.84 (0.81, 0.87) 0.83 (0.81, 0.84) 0.75 (0.71, 0.78) Corr(Δpi ,Δyi ) I-O structure and correlated Add sectoral technology demand shocks shocks Note: The five and ninety-five percentiles for the simulations are in parentheses. Table 4. Simulation Results: Aggregate Volatility and Aggregate Correlations No I-O I-O structure structure and and Add sectoral independent correlated demand technology technology shocks shocks shocks Data Baseline Model (1) (2) (3) (4) (5) SD (Y) 0.05 0.06 0.03 0.06 0.06 SD(C) / SD(Y) 0.56 0.61 0.65 0.44 0.55 SD(X) / SD(Y) 1.48 5.83 5.70 5.16 5.70 SD(H) / SD(Y) 0.52 0.80 0.82 0.70 0.74 Corr (Y, C) 0.79 0.15 0.15 0.30 0.21 Corr (Y, X) 0.85 0.86 0.88 0.94 0.86 Corr (Y, H) 0.76 0.80 0.80 0.89 0.85 Note: The statistics for the U.S. economy in column 1 are calculated by aggregating the data for 35 sectors from the KLEM data set. Sample period 1947 - 1989. Table 5. Simulation Results: Sectoral Price-Output Correlation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Agriculture Metallic ores mining Coal mining Crude oil and natural gas Nonmetallic minerals Construction Food and kindred produ Tobacco products Textile Apparel Lumber and wood produ Furniture and fixtures Paper Printing Chemicals Petroleum refining and Rubber Footwear, leather and Stone Primary metal Fabricated metal Machinery Electrical machinery Motor vehicle Transportation equip Instruments Misc. manufacturing Transportation Communications Electric utility service Gas utility service Wholesale and retail FIRE Services Govt. enterprises Mean Data Baseline Model (1) -0.24 0.28 0.07 -0.17 -0.52 -0.39 -0.56 -0.20 0.10 -0.44 0.33 -0.38 -0.18 -0.45 -0.42 -0.18 -0.26 -0.23 -0.34 0.12 -0.27 -0.18 -0.32 -0.31 -0.05 -0.17 -0.51 -0.32 -0.56 -0.60 -0.19 -0.48 -0.46 -0.57 -0.47 -0.27 (2) -0.86 -0.17 -0.85 -0.87 -0.37 0.48 0.09 -0.90 -0.36 -0.39 0.17 0.50 -0.57 -0.60 -0.75 -0.87 -0.24 -0.89 0.20 0.21 0.32 0.52 0.49 -0.37 -0.10 0.51 -0.44 -0.40 -0.65 -0.30 -0.77 0.09 -0.20 -0.40 -0.75 -0.27 I-O No I-O Structure & structure & independent correlated technology technology shocks shocks (3) (4) -1.00 -0.99 -0.99 -0.54 -1.00 -0.96 -1.00 -0.97 -1.00 -0.65 -0.29 0.53 -1.00 -0.98 -1.00 -1.00 -0.96 -0.71 -1.00 -0.94 -0.96 0.04 -0.18 0.25 -1.00 -0.90 -1.00 -0.97 -1.00 -0.95 -1.00 -0.98 -1.00 -0.63 -1.00 -1.00 -1.00 -0.10 -1.00 -0.29 -0.96 -0.09 -0.27 0.05 -0.29 0.19 -0.64 -0.87 -0.02 -0.56 -0.10 0.15 -0.88 -0.81 -1.00 -0.81 -0.99 -0.90 -1.00 -0.67 -1.00 -0.92 -0.95 -0.71 -0.99 -0.86 -1.00 -0.77 -1.00 -0.99 -0.84 -0.61 Add sectoral demand shocks (5) -0.91 -0.26 -0.92 -0.91 -0.58 0.54 0.08 -1.00 -0.51 -0.59 0.06 0.51 -0.80 -0.82 -0.89 -0.93 -0.45 -0.98 0.09 0.11 0.22 0.44 0.46 -0.55 -0.30 0.43 -0.65 -0.52 -0.84 -0.55 -0.87 0.09 -0.15 -0.68 -0.89 -0.39 Figure 1(a). Effects of a Positive Supply Shock Figure 1 (b). Effects of a Positive Monetary Shock Figure 2. Sectoral Distribution of Impulse Responses to a Monetary Shock Output Responses 1 Price Responses Period 0 Period 0 1 0 -1 2 0 5 10 15 20 25 1 30 35 Period 1 0 0 5 10 15 20 25 2 30 35 Period 1 1 0 -1 0 0 5 10 15 20 25 1 30 35 Period 5 0 5 10 15 20 25 2 30 35 Period 5 1 0 -1 0 0 5 10 15 20 25 1 30 35 Period 10 0 5 10 15 20 25 2 30 35 Period 10 1 0 -1 0 0 5 10 15 20 Sectors 25 30 35 0 5 10 15 20 25 30 Sectors Note: The height of each bar represents the percentage deviation of output (price) of the corresponding sector from its steady state level of output (price). Refer to Table 2 for a list of the 35 sectors represented along the horizontal axis. 35 Figure 3. Sectoral Distribution of Impulse Responses to a Technology Shock to the Crude Oil and Natural Gas Sector Output Responses Price Responses Period 0 1 0.5 Period 0 0 0.5 -0.5 -1 0 0 5 10 15 20 25 30 35 Period 1 1 0 5 10 15 20 25 0.5 30 35 Period 1 0 0.5 -0.5 -1 0 0 5 10 15 20 25 30 35 Period 5 1 0 5 10 15 20 25 0.5 30 35 Period 5 0 0.5 -0.5 -1 0 0 5 10 15 20 25 30 35 Period 10 1 0 5 10 15 20 25 0.5 30 35 Period 10 0 0.5 -0.5 -1 0 0 5 10 15 20 Sectors 25 30 35 0 5 10 15 20 Sectors 25 30 35 Figure 4. Sectoral Distribution of Impulse Responses to a Technology Shock to the Rubber Sector Output Responses Price Responses Period 0 1 Period 0 0 -0.5 0.5 -1 0 0 5 10 15 20 25 30 35 Period 1 1 0 5 10 15 20 25 30 35 Period 1 0 -0.5 0.5 -1 0 0 5 10 15 20 25 30 35 Period 5 1 0 5 10 15 20 25 30 35 Period 5 0 -0.5 0.5 -1 0 0 5 10 15 20 25 30 35 Period 10 0.4 0 5 10 15 20 25 30 35 Period 10 0.2 0 0.2 -0.2 0 -0.4 0 5 10 15 20 Sectors 25 30 35 0 5 10 15 20 Sectors 25 30 35 Figure 5. Sectoral Distribution of Impulse Responses to a Technology Shock to the Motor Vehicle Sector Output Responses 1.5 Price Responses Period 0 1 0.5 Period 0 0 0.5 -0.5 0 -1 -0.5 0 5 10 15 20 25 1.5 30 35 Period 1 1 0 5 10 15 20 25 0.5 30 35 Period 1 0 0.5 -0.5 0 -1 -0.5 0 5 10 15 20 25 30 35 0 5 10 15 20 25 1 Period 5 30 35 Period 5 0.2 0.5 0 0 -0.2 -0.5 0 5 10 15 20 25 30 35 Period 10 0.4 0 0 0 -0.2 5 10 15 20 Sectors 25 30 35 10 15 20 25 30 35 Period 10 0.2 0.2 0 5 0 5 10 15 20 Sectors 25 30 35 Figure 6. Sectoral Distribution of Impulse Responses to a Technology Shock to the Wholesale and Retail Sector Output Responses Price Responses Period 0 1 0.5 Period 0 0 0.5 -0.5 0 -1 0 5 10 15 20 25 30 35 Period 1 1 0 5 10 15 20 25 0.5 30 35 Period 1 0 0.5 -0.5 0 -1 0 5 10 15 20 25 30 35 Period 5 1 0 5 10 15 20 25 0.5 30 35 Period 5 0 0.5 -0.5 0 -1 0 5 10 15 20 25 30 35 Period 10 0.4 0 5 10 15 20 25 0.5 30 35 Period 10 0 0.2 -0.5 0 -0.2 -1 0 5 10 15 20 Sectors 25 30 35 0 5 10 15 20 Sectors 25 30 35 Figure 7. Sectoral Distribution of Impulse Responses to a Technology Shock to the Construction Sector Output Responses 3 Price Responses Period 0 1 Period 0 2 0 1 0 -1 0 5 10 15 20 25 3 30 35 Period 1 0 5 10 15 20 25 1 30 35 Period 1 2 0 1 0 -1 0 5 10 15 20 25 3 30 35 Period 5 0 5 10 15 20 25 1 30 35 Period 5 2 0 1 0 -1 0 5 10 15 20 25 3 30 35 Period 10 0 5 10 15 20 25 1 30 35 Period 10 2 0 1 0 -1 0 5 10 15 20 Sectors 25 30 35 0 5 10 15 20 Sectors 25 30 35 Figure 8. Sectoral Distribution of Impulse Responses to a Technology Shock to the Machinery Sector Output Responses 2 Price Responses Period 0 0.5 Period 0 0 1 -0.5 0 -1 0 5 10 15 20 25 2 30 35 Period 1 0 5 10 15 20 25 0.5 30 35 Period 1 0 1 -0.5 0 -1 0 5 10 15 20 25 2 30 35 Period 5 0 5 10 15 20 25 0.5 30 35 Period 5 0 1 -0.5 0 -1 0 5 10 15 20 25 2 30 35 Period 10 0 5 10 15 20 25 0.5 30 35 Period 10 0 1 -0.5 0 -1 0 5 10 15 20 Sectors 25 30 35 0 5 10 15 20 Sectors 25 30 35 Figure 9. Response of Aggregate Value Added and Value Added Deflator to Technology Shocks in the Rubber and Machinery Sectors Response of value added to rubber technology shock 0.06 Response of value added deflator to rubber technology shock 0.0025 0.0000 0.05 -0.0025 0.04 -0.0050 -0.0075 0.03 -0.0100 0.02 -0.0125 -0.0150 0.01 -0.0175 0.00 -0.0200 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 horizon horizon Response of value added to machinery technology shock Response of value added deflator to machinery technology shock 0.144 0.10 0.08 0.136 0.06 0.128 0.04 0.120 0.02 0.00 0.112 -0.02 0.104 -0.04 0.096 -0.06 0.088 -0.08 0 1 2 3 4 5 horizon 6 7 8 9 10 0 1 2 3 4 5 horizon 6 7 8 9 10