A New Keynesian model for the analysis of       energy shocks in Pakistan                          Dario Debowicz         ...
Motivation: energy crisis in PakistanWe realize the suffering that load shedding causes our people. We are painfullyaware ...
Some stylized facts- The world price of crude oil and LNG increases more than 300% from 2000- Energy is perceived as a bot...
World energy prices             (US dollars per physical unit)       160       140       120       100        80        60...
Net barter terms of trade, export and        import value index (2000=100)           450           400           350      ...
Energy related subsidies in Pakistan                (millions of rupees)                                     2006-07    20...
Fiscal balance (share of GDP)          -               FY00   FY01   FY02   FY03   FY04   FY05   FY06   FY07   FY08   FY09...
Money supply (yearly growth)      7.00      6.00      5.00      4.00      3.00      2.00      1.00      0.00      Source: ...
Load shedding in Pakistan                                       National   Total                                       loa...
Real Gross Domestic Product growth        (at market prices of 1999-00)                9.0                8.0             ...
Inflation (GDP deflator % changes)        25.0        20.0        15.0        10.0         5.0         0.0               F...
Real effective exchange rate            140            120            100             80             60             40    ...
Current account balance              Source: Handbook of Statistics of Pakistan• Low competitiveness leads Pakistan to fac...
Central questions• How are the increases in world energy prices affecting  growth, employment and inflation in Pakistan? W...
Building a New Keynesian model• To answer these questions, we need a model that, capturing  the channels by which energy-r...
Main characteristics of New Keynesian       model we are buildingThe model has production, consumption, international trad...
Main characteristics of New Keynesian       model we are buildingInternational tradeExports (imports) are determined by gl...
Next steps• Calibrate the model based on macroeconomic data and information specific to the  energy sector (Handbook of St...
BibliographyAn, S and Kang, H. (2011), “Oil Shocks in a DSGE Model for the Korean Economy”,Chapter in NBER book Commodity ...
BibliographyFriends_of_Democratic_Pakistan and Energy_Sector_Task_Force (2010). IntegratedEnergy Sector Recovery Report & ...
Model equations*                                                                                           µΕ0 ∑∞ βt Ut   ...
Model equations                                                                      𝜂𝑐𝑡 𝑗 = 𝛿         𝑒𝑒 𝑡 𝑗         + 1−...
Model equations𝑝𝑡 =     𝛿𝑝𝑝𝑝 𝑡             + 1 − 𝛿 𝑝𝑝 𝑡                                                        1          ...
Model equationsΕt ∑∞ 𝛽𝜙 𝑙 𝑖 𝑙 𝑡+𝑖 𝑗    𝑖=0                                       � 𝑙 𝑡+𝑖 εl + wt+i                        ...
Model equations               =        𝑧𝑧 𝑡 𝑗  𝑒 𝑧𝑧,𝑡 (𝑗)       𝜒𝑒              𝜙𝑒  𝑘 𝑡−1    (𝑗)     𝜙𝑒                   ...
Model equations  𝐹 𝑡 𝑠 = 𝑝𝑝 𝑡 𝑠 𝑦𝑦 𝑡 𝑠 − 𝑦𝑦 𝑡 𝑠 𝑚𝑚 𝑡                        (31)                                      εh𝑦𝑦...
Model equations                            −𝜂∗𝑐𝑐∗      =    𝛾∗                    𝑐∗              𝑟𝑟𝑟 𝑡 =   𝑒𝑡 𝑝          ...
Model equations𝑝 𝑡 𝑔 𝑡 = 𝑡 𝑡 + 𝑏 𝑡 − 1 + 𝑖 𝑡 𝑏 𝑡−1 + 𝑚 𝑡 − 𝑚 𝑡−1                     (47)𝑡𝑡 𝑡 = 𝑐 𝑝 𝑡 𝑐 𝑡                (...
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A New Keynesian model for the analysis of energy shocks in Pakistan by Dario Debowicz, IFPRI

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Presentations made at the PSSP First Annual Conference - December 13, 14, 2012 - Planning Commission, Islamabad, Pakistan

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A New Keynesian model for the analysis of energy shocks in Pakistan by Dario Debowicz, IFPRI

  1. 1. A New Keynesian model for the analysis of energy shocks in Pakistan Dario Debowicz (International Food Policy Research Institute) and Alejandro Quijada (Inter-American Development Bank) December, 2012
  2. 2. Motivation: energy crisis in PakistanWe realize the suffering that load shedding causes our people. We are painfullyaware of the darkness it spreads, how children study by candlelight, and howthe wheels of industry often stop. President Asif Ali Zardaris Speech at the Joint Session of Parliament Islamabad, April 5, 2010The scale of the present crisis is formidable and requires persistent structuraland pricing reforms in the sector, increased implementation of loss-reductionprograms, and expanded investments. The government recognizes the need forsubstantial external assistance to help overcome the energy crisis. Pakistan Yearly Energy Book 2010
  3. 3. Some stylized facts- The world price of crude oil and LNG increases more than 300% from 2000- Energy is perceived as a bottleneck constraining growth and employment opportunities in Pakistan (Friends of Democratic Pakistan Energy book 2010).- The energy constrain leads to a low rate of utilization of the capital stock (State Bank of Pakistan 2012).- The energy sector of Pakistan is financially unsustainable today. Notified electricity tariffs are below the cost recovery level as per determined tariffs, so the government therefore subsidizes tariffs by providing tariff differential subsidies in the budget (Friends of Democratic Pakistan Energy book 2010).
  4. 4. World energy prices (US dollars per physical unit) 160 140 120 100 80 60 40 20 0 2000M01 2000M07 2001M01 2001M07 2002M01 2002M07 2003M01 2003M07 2004M01 2004M07 2005M01 2005M07 2006M01 2006M07 2007M01 2007M07 2008M01 2008M07 2009M01 2009M07 2010M01 2010M07 2011M01 2011M07 2012M01 Crude oil, Dubai ($/bbl) Liquefied natural gas, Japan ($/mmbtu) Source: World Bank Commodity Price Data (“Pink sheet”) BBL: oil barrel. MMBTU: Million Metric British Thermal Units• World crude and gas prices grow by more than 300% from the start of 2000.
  5. 5. Net barter terms of trade, export and import value index (2000=100) 450 400 350 300 250 200 150 100 50 0 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Export value index (2000 = 100) Import value index (2000 = 100) Net barter terms of trade index (2000 = 100) Source: World Development Indicators (World Bank)• The country’s terms of trade deteriorate during the last decade as the import prices (led by energy prices) increase significantly more than export prices.• Overall, the country’s terms of trade fall almost 40% from 2000 to 2010
  6. 6. Energy related subsidies in Pakistan (millions of rupees) 2006-07 2007-08 2008-09 2009-10 2010-11 2011-12WAPDA Budget 41,934 52,893 74,612 62,903 84,000 122,700 Revised 42,464 113,658 92,840 147,005 295,827 -KESC Budget 13,938 19,596 13,800 3,800 20,447 28,588 Revised 17,699 19,596 18,800 32,521 64,447 -Oil Refineries/OMCs Budget 10,000 15,000 140,000 15,000 10,807 7,921 Revised 25,000 175,000 70,000 11,224 10,807Fertilizer Manufacturers Budget 980 10,360 12,860 210 185 162 Revised 978 6,360 21,268 439 985 -Total Budget 66,852 97,849 241,272 81,913 115,439 159,371 Revised 86,141 314,614 202,908 191,189 372,066 -Source: PSSP elaboration based on 2011-12 ‘Federal budget in brief’• Outstanding energy-related subsidies are implemented, exceeding 372 billions of Pakistan rupees in 2010-11, and explaining more than half of the 680-billion fiscal deficit.
  7. 7. Fiscal balance (share of GDP) - FY00 FY01 FY02 FY03 FY04 FY05 FY06 FY07 FY08 FY09 (1) (2) (3) (4) (5) (6) (7) (8) Source: Handbook of Statistics of Pakistan• Recurrent large fiscal deficit, which exceeds 7% of GDP in FY 2008
  8. 8. Money supply (yearly growth) 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00 Source: Broad Money Supply, Handbook of Statistics of Pakistan• Through fiscal deficit monetization, money supply accelerates without interruption during last decade, reaching a 6% rate of growth in 2010
  9. 9. Load shedding in Pakistan National Total load National Load National shedding Demand Shedding Year sales (GWH) (GWh) (GWh) (%) 2003 52,661 52,661 0.0% 2004 57,467 520 57,986 0.9% 2005 61,247 265 61,512 0.4% 2006 67,608 1,208 68,815 1.8% 2007 71,947 2,040 73,982 2.8% 2008 72,518 12,578 85,096 14.8% 2009 69,668 18,222 87,890 20.7% 2010 73,595 21,821 95,238 22.9%• Increasing load shedding from 2005, with alarming level of 22.9% in 2010
  10. 10. Real Gross Domestic Product growth (at market prices of 1999-00) 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 FY01 FY02 FY03 FY04 FY05 FY06 FY07 FY08 FY09 FY10 P Source: Handbook of Statistics of Pakistan• GDP growth was relatively low during most of the last decade considering population and labor supply growth due to contemporaneous population dividend, with energy bottleneck presumably having an important role.
  11. 11. Inflation (GDP deflator % changes) 25.0 20.0 15.0 10.0 5.0 0.0 FY 00 FY 01 FY 02 FY 03 FY 04 FY 05 FY 06 FY 07 FY 08 FY 09 FY 10 Source: Handbook of Statistics of Pakistan• With expansionary macroeconomic (fiscal and monetary) policies and the supply bottleneck generated by energy, domestic inflation accelerated significantly in last years
  12. 12. Real effective exchange rate 140 120 100 80 60 40 20 0 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Source: International Financial Statistics (IFS) - IMF• As domestic inflation systematically exceeded the sum of nominal devaluations and foreign inflation, the country’s real effective exchange rate deteriorated significantly during the last twenty years
  13. 13. Current account balance Source: Handbook of Statistics of Pakistan• Low competitiveness leads Pakistan to face a recurrent trade deficit during the last 30 years.• To a large extent, remittances helped in reducing the associated current account deficit.• Trade balance deteriorates in parallel to the energy crisis
  14. 14. Central questions• How are the increases in world energy prices affecting growth, employment and inflation in Pakistan? Which are the major transmission channels?• How are the energy-related domestic subsidies affecting domestic growth, employment and inflation? Which are the major transmission channels?• How would an increase in the supply of energy impact growth, employment and inflation in the country? Which are the major transmission channels?
  15. 15. Building a New Keynesian model• To answer these questions, we need a model that, capturing the channels by which energy-related shocks and macroeconomic policies affect the domestic economy, is able to shows us how the domestic economy is affected over time by changes in world energy prices, domestic energy subsidies, and domestic energy production.• We frame the questions into the New Keynesian paradigm as: – it captures the short-run interaction between macroeconomic policy, inflation, and the business cycle, accounting for agents inter-temporal planning and expectations. – it allows us to look into how the economy responds over time to a set of relevant impulses through the analysis of impulse-response functions.
  16. 16. Main characteristics of New Keynesian model we are buildingThe model has production, consumption, international trade and financial markets.ProductionThe economy produces output using labor and physical capital. In turn, in order toutilize the physical capital stock, producers demand energy. As a result, the use ofenergy affects the use of the capital stock, in turn affecting the productivity of laborand, ultimately, the output that the economy can produce.Reflecting the main components of the energy mix in Pakistan, energy is produced usingoil and gas.Imperfect (monopolistic) competition characterizes the domestic factor and commoditymarkets.ConsumptionHouseholds maximize inter-temporal utility, which depends on consumption, realbalanced held for transaction purposes, and labor effort. Increases in the real interestrate lead households to postpone consumption over time.
  17. 17. Main characteristics of New Keynesian model we are buildingInternational tradeExports (imports) are determined by global demand (domestic absorption) and the realexchange rate.Financial marketsThe model has explicit markets for money, domestic bonds and foreign bonds.Households allocate their portfolio among a set of financial and physical assets (money,domestic and foreign bonds and capital stock), and rent capital services to firms.As usual in New Keynesian models, the real interest rate is determined by monetarypolicy via the use of a Taylor rule, by which the Central Bank lifts the interest rate whenoutput grows above steady-state growth, and/or when domestic inflation exceeds thesteady-state one. The domestic interest rate, the foreign interest rate and the exchangerate are inter-linked through the uncovered interest parity condition, such that inequilibrium the expected return of domestic and foreign financial assets coincide.
  18. 18. Next steps• Calibrate the model based on macroeconomic data and information specific to the energy sector (Handbook of Statistics of Pakistan, Pakistan Energy Book 2011), using among other data subsidy data that the Pakistan Bureau of Statistics provided us with.• Implement simulations regarding world energy prices, domestic energy subsidies and productivity in the domestic energy sector.• Generate and analyze impulse-response functions that show us the time-path by which the economy returns to its steady-state equilibrium path after relevant shocks.• Get feedback and fine-tune the analysis
  19. 19. BibliographyAn, S and Kang, H. (2011), “Oil Shocks in a DSGE Model for the Korean Economy”,Chapter in NBER book Commodity Prices and Markets, East Asia Seminar on Economics,Volume 20 (2011), Takatoshi Ito and Andrew K. Rose, editors (p. 295 - 321).Adam, C. O’Connell, S and Buffie, E. (2008), “Aid volatility, monetary policy rules and thecapital account in African economies”, WEF Working Papers 0037, ESRC World Economyand Finance Research Programme, Birkbeck, University of London.Almeida, V. (2009), “Bayesian Estimation of a DSGE Model for the Portuguese Economy”,Bank of Portugal Working Papers Series No. 14/2009.Calvo, G. (1983), “Staggered Prices in a Utility-Maximizing Framework”, Journal ofMonetary Economics, Elsevier, vol. 12(3), pages 383-398, September.Choudhri, E.U. and H. Malik (2012), “Monetary Policy in Pakistan: A Dynamic StochasticGeneral Equilibrium Analysis”, mimeo.Economist (2011). Lights out. Pakistan energy shortage.
  20. 20. BibliographyFriends_of_Democratic_Pakistan and Energy_Sector_Task_Force (2010). IntegratedEnergy Sector Recovery Report & Plan.Gali, J (2008), ‘Monetary Policy, Inflation and the Business Cycle, An Introduction to theNew Keynesian Framework’, Princeton University Press.IMF (2010). Pakistan: Poverty Reduction Strategy Paper.Medina, JP and Soto, C. (2005), “Oil shocks and Monetary Policy in an Estimated DSGEModel For a Small Open Economy”, Central Bank of Chile, Working Paper 353.Peiris, S and Saxegaard, M. (2007), “An Estimated DSGE Model for Monetary PolicyAnalysis in Low-Income Countries”, IMF Working Paper 07/282. Washington DC:International Monetary Fund.World Bank. (2011a), “Responding to Global Food Price Volatility and Its Impact on FoodSecurity”, Document for the Development Committee, DC2011-002, April 4, 2011,Washington DC: World Bank.World Bank. (2011b), “Food Price Watch”, Poverty reduction and Equity Group. August2011. Washington DC: World Bank.
  21. 21. Model equations* µΕ0 ∑∞ βt Ut j = Ε0 ∑∞ βt log ct j − ζhct−1 j + − φt 1+σ s mt j lt j 1+σl t=0 t=0 µ pt l (1) 𝑚𝑡 𝑗 + + = 𝑚 𝑡−1 𝑗 + 𝑏 𝑡−1 𝑗 + et 𝑏∗ 𝑡−1 𝑗 + 𝑤 𝑡 𝑗 𝑙 𝑡 𝑗 + 𝑏𝑡 𝑗 et 𝑏 ∗ 𝑗 𝑡 1+𝑖 𝑡 1+i∗ Ξ 𝑒 𝑡 𝑏∗𝑡 t𝑟 𝑡 𝑗 𝑘 𝑡𝑠 𝑗 + 𝐹 𝑡 𝑗 + 𝑇𝑇 𝑡 𝑗 − 𝑝 𝑡 𝑐 𝑡 𝑗 − 𝑝 𝑡 𝑖 𝑡 𝑗 − 𝑝 𝑡 𝑝 𝑡𝑒 𝑒 𝑧𝑧,𝑡 (𝑗) (2) 𝑝𝑝 𝑡 𝑥 𝑡� 𝑐 𝑡 − ℎζ𝛽𝑈 𝑐 𝑡+1 = 𝛽 1 + 𝑖 𝑡𝑈 � 𝑝𝑡 � 𝑐 𝑡+1 − ℎζ𝛽𝑈 𝑐 𝑡+2 𝑈 � 𝑝 𝑡+1 (3) 1+ i∗ Ξ = 1 + 𝑖𝑡 𝑒 𝑡 𝑏∗ 𝑡 𝑒 𝑡+1 t 𝑝𝑝 𝑡 𝑥 𝑡 𝑒𝑡 (4) 𝜇−1 − 𝛽 =� 𝑐 𝑡 −ℎζ𝛽𝑈 𝑐 𝑡+1𝑈 � � 𝑐 𝑡+1 −ℎζ𝛽𝑈 𝑐 𝑡+2 𝑈 � 𝑠 𝑚𝑡 𝑝𝑡 𝑝 𝑡+1 𝑝𝑡 𝑝𝑡 (5)* The full description of the model is available in DSGE_Pakistan_Model.docx.
  22. 22. Model equations 𝜂𝑐𝑡 𝑗 = 𝛿 𝑒𝑒 𝑡 𝑗 + 1− 𝛿 𝑧𝑡 𝑗 1 𝜂−1 1 𝜂−1 𝜂−1 𝜂 𝜂 𝜂 𝜂 (6) 𝜃𝑧𝑡 𝑗 = 𝛾 𝑐𝑐𝑡 𝑗 + 1− 𝛾 𝑐𝑐 𝑡 𝑗 1 𝜃−1 1 𝜃−1 𝜃−1 𝜃 𝜃 𝜃 𝜃 (7)𝑝𝑝𝑝 𝑡 𝑒𝑒 𝑡 𝑗 + 𝑝𝑝 𝑡 𝑧 𝑡 𝑗 (8)𝑝𝑝𝑡 𝑐𝑐𝑡 𝑗 + 𝑝𝑝 𝑡 𝑐𝑐 𝑡 𝑗 (9) 𝑝𝑝 𝑡 −𝜂 1−𝛿𝑧𝑡 𝑗 = 𝑒𝑒 𝑡 𝑗 𝑝𝑝 𝑡 𝛿 (10) 𝑝𝑝 𝑡 −𝜃 1−𝛾𝑐𝑐 𝑡 𝑗 = 𝑝𝑝𝑡 𝛾 𝑐𝑐𝑡 𝑗 (11)
  23. 23. Model equations𝑝𝑡 = 𝛿𝑝𝑝𝑝 𝑡 + 1 − 𝛿 𝑝𝑝 𝑡 1 1−𝜂 1−𝜂 1−𝜂 (12)𝑝𝑝 𝑡 = 𝛾𝑝𝑝𝑡 + 1 − 𝛾 𝑝𝑝 𝑡 1 1−𝜃 1−𝜃 1−𝜃 (13) εl𝑙 𝑡 = ∫0 lt j dj εl −1 1 εl εl −1 (14) 𝑤 𝑡 𝑗 −εl𝑙𝑡 𝑗 = 𝑙𝑡 𝑤𝑡 (15) 1𝑤𝑡 = ∫0 wt j 1−εl d 1 1−εl j (16)𝑤 𝑡+𝑖 𝑗 = Γw,t wt j i (17)Γw,t = ∏i ζ 1 + πt+j−1 1 + πt+j � i ξl 1−ξl j=1 (18)
  24. 24. Model equationsΕt ∑∞ 𝛽𝜙 𝑙 𝑖 𝑙 𝑡+𝑖 𝑗 𝑖=0 � 𝑙 𝑡+𝑖 εl + wt+i 𝑈 j � 𝑐 𝑡+𝑖 − ℎζ𝛽𝑈 𝑐 𝑡+1+𝑖 𝑈 � =0 ε −1 l pt+i (19)𝑘 𝑡𝑠 (𝑗) = 𝑧𝑧 𝑡 𝑗 𝑘 𝑡−1 (𝑗) (20) = � 𝑐 𝑡 −ℎζ𝛽𝑈 𝑐 𝑡+1 𝑈 �� 𝑐 𝑡+1 −ℎζ𝛽𝑈 𝑐 𝑡+2𝑈 �𝛽 + 1− 𝛿 𝑘 − 𝜒 𝑧𝑧 𝑧𝑧 𝑡+11+𝜙 𝑧𝑧 − 𝑧𝑧 𝑠𝑠 1+𝜙 𝑧𝑧 − 𝑧𝑧 𝑡+1 − 𝑝 𝑡+1 𝜒 𝑧𝑧 𝑧𝑧 𝑡+1 𝜙 𝑟 𝑡+1 𝑧𝑧 𝑡+1 𝜒𝑒 𝜙𝑒 𝑒 𝑒 𝑝 𝑡+1 1+𝜙 𝑧𝑧 𝜙𝑒 𝜙𝑒(21) − 𝜒 𝑧𝑧 𝑧𝑧 𝑡 − 𝑝 𝑡𝑒 𝜒 𝑒 𝑧𝑧 𝑡 =0𝑟𝑡 𝜙 𝑧𝑧 𝜙 𝑒 −1𝑝𝑡 (22)𝑘 𝑡 𝑗 = 𝑖 𝑡 𝑗 + 1 − 𝛿 𝑘 𝑘 𝑡−1 𝑗 − Δzu,t (23)Δzu,t = 1+𝜙 𝑧𝑧 𝑡 𝑗 − 𝑧𝑧 𝑠𝑠 1+𝜙 𝑧𝑧 𝑘 𝑡−1 (𝑗) 𝑧𝑧 𝜒 1+𝜙 𝑧𝑧 𝑧𝑧 (24)
  25. 25. Model equations = 𝑧𝑧 𝑡 𝑗 𝑒 𝑧𝑧,𝑡 (𝑗) 𝜒𝑒 𝜙𝑒 𝑘 𝑡−1 (𝑗) 𝜙𝑒 (25) 𝜔𝑒𝑒𝑒 𝑡 = 𝑎𝑎 𝑡 1 − 𝛼 𝑒 𝑜𝑜𝑜 𝑡 + 𝛼𝑒 𝑔𝑔𝑔 𝑡 1 𝜔 𝑒 −1 1 𝜔 𝑒 −1 𝜔 𝑒 −1 𝜔𝑒 𝜔𝑒 𝜔𝑒 𝜔𝑒 (26) 𝜔𝑒 =𝑔𝑔𝑔 𝑡 𝛼𝑒 𝑝 𝑜,𝑡𝑜𝑜𝑜 𝑡 1−𝛼 𝑒 𝑝 𝑔𝑔𝑔,𝑡 (27) 𝜔𝑦𝑦 𝑡 𝑠 = 𝑎𝑎 𝑡 1 −α 𝑙𝑡 𝑠 + 𝛼 𝑘 𝑡𝑠 (𝑠) 1 𝜔−1 1 𝜔−1 𝜔−1 𝜔 𝜔 𝜔 𝜔 (28) 𝜔 =𝑘 𝑡𝑠 (𝑠) 𝛼 𝑤𝑡𝑙𝑡 𝑠 1−𝛼 𝑟𝑡 (29) 𝑚𝑚 𝑡 𝑠 = 𝑚𝑚 𝑡 = 1 − 𝛼 𝑤𝑡 + 𝛼𝑝𝑝 𝑡 1−𝜔 1−𝜔 1 1−𝜔 1 𝑎𝑎 𝑡 (30)
  26. 26. Model equations 𝐹 𝑡 𝑠 = 𝑝𝑝 𝑡 𝑠 𝑦𝑦 𝑡 𝑠 − 𝑦𝑦 𝑡 𝑠 𝑚𝑚 𝑡 (31) εh𝑦𝑦 𝑡 = ∫0 yht s ds εh −1 1 εh εh −1 (32) 𝑝𝑝 𝑡 𝑠 −εh𝑦𝑦 𝑡 𝑠 = 𝑝𝑝 𝑦𝑦 𝑡 (33) 1𝑝𝑝 𝑡 = ∫0 pht s 1−εh d 1 1−εh s (33’)𝑝𝑝 𝑡+𝑖 𝑠 = Γh,t pht s i (34)Γh,t = ∏i 1 + πht+i−1 i j=1 ξh 1 + πt+i � 1−ξh (35)Εt ∑∞ 𝛽𝜙 𝑙 𝑖=0 𝑖 𝜆 𝑢 𝑡+𝑖 𝑦𝑦 𝑡+𝑖 𝑠 1 − εh Γh,t 𝑝𝑝 𝑡 𝑠 + εh 𝑚𝑚 𝑡+𝑖 = 0 i (36)
  27. 27. Model equations −𝜂∗𝑐𝑐∗ = 𝛾∗ 𝑐∗ 𝑟𝑟𝑟 𝑡 = 𝑒𝑡 𝑝 𝜋𝑡 = − 1 (39) 𝑝𝑝∗𝑡 𝑝𝑝𝑡∗ 𝑝𝑡 𝑡 𝑝𝑝𝑡 ∗ 𝑡 𝑝 𝑡−1 𝑡 (37) (38)𝑝𝑝𝑡 = 1 + 𝜏 𝑐 𝑒 𝑡 𝑝𝑝𝑡∗ (40) 𝑝𝑝 𝑡 = 1 + 𝜏 𝑜 𝑒 𝑡 𝑝𝑝∗ 𝑡 (41)𝑝𝑝𝑝𝑝 𝑡 = 1 + 𝜏 𝑔𝑔𝑔 𝑒 𝑡 𝑝𝑝𝑝𝑝 ∗ 𝑝𝑝∗ = 𝑝𝑝 𝑡 𝑡 (42) 𝑡 𝑒𝑡 (43) Ψy 1−𝜌 𝑟 � 𝜌𝑟 πt Ψπ 1−𝜌 𝑟 𝑦𝑡 =𝑟𝑡 𝑟 𝑡−1 𝑦 𝑡−1𝑟 𝑟 ζ �𝜋 (44) 𝜃𝑔 1 𝜃 𝑔 −1 𝜃 𝑔 −1𝑔𝑡 = 𝛾𝑔 𝑔𝑔𝑡 + 1 − 𝛾𝑔 𝑔𝑔 𝑡 1 𝜃 𝑔 −1 𝜃𝑔 𝜃𝑔 𝜃𝑔 𝜃𝑔 (45) 𝑝𝑝 𝑡 −𝜃 𝑔 1−𝛾 𝑔𝑔𝑔 𝑡 = 𝑔𝑔𝑡 𝑝𝑝𝑡 𝛾𝑔 (46)
  28. 28. Model equations𝑝 𝑡 𝑔 𝑡 = 𝑡 𝑡 + 𝑏 𝑡 − 1 + 𝑖 𝑡 𝑏 𝑡−1 + 𝑚 𝑡 − 𝑚 𝑡−1 (47)𝑡𝑡 𝑡 = 𝑐 𝑝 𝑡 𝑐 𝑡 (48) 𝑡𝑡 𝑡 = 1+𝜏 𝑝𝑝 𝑡 𝑜 𝑡 (49) 𝑡𝑡𝑡𝑡 𝑡 = 1+𝜏 𝑝𝑝𝑝𝑝 𝑡 𝑔𝑔𝑔 𝑡 𝜏 𝜏 𝑜 𝜏𝑔 1+𝜏 𝑐 𝑜 𝑔 (50)𝑡𝑡𝑡 𝑡 = 𝑒𝑒 𝑝𝑝𝑝 𝑡 𝑒𝑒 𝑡 (51) 𝑡𝑡𝑡 𝑡 = 1+𝜏 𝑝𝑝𝑝 𝑡 𝑒𝑒 𝑡 𝜏 𝑒𝑒𝜏 1+𝜏 𝑒𝑒 𝑒𝑒 (52)𝑡 𝑡 = 𝑡𝑡 𝑡 + 𝑡𝑡 𝑡 + 𝑡𝑡𝑡𝑡 𝑡 + 𝑡𝑡𝑡 𝑡 + 𝑡𝑡𝑡 𝑡 + 𝑎𝑎𝑎 𝑡 𝑒 𝑡 (53)𝑦𝑦 𝑡 = 𝑐𝑐 𝑡 + 𝑔𝑔 𝑡 + 𝑐𝑐∗ 𝑡 (54) 𝑒𝑒 𝑡 = 𝑒𝑒 𝑡 + 𝑒 𝑧𝑧,𝑡 (55) 𝑐 𝑡 = 𝑐𝑐 𝑡 + 𝑐𝑐𝑡 + 𝑒𝑒 𝑡 (56)𝑔 𝑡 = 𝑔𝑔 𝑡 + 𝑔𝑔𝑡 (57) 𝑝𝑝𝑝 𝑡 𝑖𝑖 𝑡 = 𝑝𝑓𝑡 𝑐𝑐𝑡 + 𝑝𝑓𝑡 𝑔𝑔𝑡 + 𝑝𝑝 𝑡 𝑜 𝑡 + 𝑝𝑝𝑝𝑝 𝑡 𝑔𝑔𝑔 𝑡 (58)𝑝𝑝𝑝 𝑡 = 𝜅 𝑓 𝑝𝑓𝑡 + 𝜅 𝑜 𝑝𝑝 𝑡 + 𝜅 𝑔 𝑝𝑝𝑝𝑝 𝑡 (59)𝑝𝑝 𝑡 𝑦 𝑡 = 𝑝 𝑡 𝑐 𝑡 + 𝑝 𝑡 𝑔 𝑡 + 𝑝 𝑡 𝑖 𝑡 + 𝑝𝑝 𝑡 𝑐𝑐∗ − 𝑝𝑝𝑝 𝑡 𝑖𝑖 𝑡 𝑡 (60) = et 𝑏 ∗ + 𝑝𝑝 𝑡 𝑐𝑐∗ − 𝑝𝑝𝑝 𝑡 𝑖𝑖 𝑡 et 𝑏 ∗ 𝑡 𝑒 𝑡 𝑏∗ 𝑡−1 𝑡1+i∗ Ξ 𝑡 (61) t 𝑝𝑝 𝑡 𝑥 𝑡
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