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

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. Motivation: energy crisis in Pakistan
We realize the suffering that load shedding causes our people. We are painfully
aware of the darkness it spreads, how children study by candlelight, and how
the wheels of industry often stop.
President Asif Ali Zardari's Speech at the Joint Session of Parliament
Islamabad, April 5, 2010
The scale of the present crisis is formidable and requires persistent structural
and pricing reforms in the sector, increased implementation of loss-reduction
programs, and expanded investments. The government recognizes the need for
substantial external assistance to help overcome the energy crisis.
Pakistan Yearly Energy Book 2010

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. 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. 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. Energy related subsidies in Pakistan
(millions of rupees)
2006-07 2007-08 2008-09 2009-10 2010-11 2011-12
WAPDA 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 10,807
Fertilizer 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Main characteristics of New Keynesian
model we are building
The model has production, consumption, international trade and financial markets.
Production
The economy produces output using labor and physical capital. In turn, in order to
utilize the physical capital stock, producers demand energy. As a result, the use of
energy affects the use of the capital stock, in turn affecting the productivity of labor
and, ultimately, the output that the economy can produce.
Reflecting the main components of the energy mix in Pakistan, energy is produced using
oil and gas.
Imperfect (monopolistic) competition characterizes the domestic factor and commodity
markets.
Consumption
Households maximize inter-temporal utility, which depends on consumption, real
balanced held for transaction purposes, and labor effort. Increases in the real interest
rate lead households to postpone consumption over time.

17. Main characteristics of New Keynesian
model we are building
International trade
Exports (imports) are determined by global demand (domestic absorption) and the real
exchange rate.
Financial markets
The 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 monetary
policy via the use of a Taylor rule, by which the Central Bank lifts the interest rate when
output grows above steady-state growth, and/or when domestic inflation exceeds the
steady-state one. The domestic interest rate, the foreign interest rate and the exchange
rate are inter-linked through the uncovered interest parity condition, such that in
equilibrium the expected return of domestic and foreign financial assets coincide.

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. Bibliography
An, 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 the
capital account in African economies”, WEF Working Papers 0037, ESRC World Economy
and 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 of
Monetary Economics, Elsevier, vol. 12(3), pages 383-398, September.
Choudhri, E.U. and H. Malik (2012), “Monetary Policy in Pakistan: A Dynamic Stochastic
General Equilibrium Analysis”, mimeo.
Economist (2011). Lights out. Pakistan energy shortage.

20. Bibliography
Friends_of_Democratic_Pakistan and Energy_Sector_Task_Force (2010). Integrated
Energy Sector Recovery Report & Plan.
Gali, J (2008), ‘Monetary Policy, Inflation and the Business Cycle, An Introduction to the
New 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 DSGE
Model For a Small Open Economy”, Central Bank of Chile, Working Paper 353.
Peiris, S and Saxegaard, M. (2007), “An Estimated DSGE Model for Monetary Policy
Analysis 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 Food
Security”, 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. August
2011. Washington DC: World Bank.

21. Model equations*
µ
Ε0 ∑∞ 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
t
𝑟 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 𝑡
𝑈 � 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+1
𝑈 � � 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. Model equations
𝜂
𝑐𝑡 𝑗 = 𝛿 𝑒𝑒 𝑡 𝑗 + 1− 𝛿 𝑧𝑡 𝑗
1 𝜂−1 1 𝜂−1 𝜂−1
𝜂 𝜂 𝜂 𝜂 (6)
𝜃
𝑧𝑡 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−𝛿
𝑧𝑡 𝑗 = 𝑒𝑒 𝑡 𝑗
𝑝𝑝 𝑡 𝛿
(10)
𝑝𝑝 𝑡 −𝜃 1−𝛾
𝑐𝑐 𝑡 𝑗 = 𝑝𝑝𝑡 𝛾
𝑐𝑐𝑡 𝑗 (11)

23. Model equations
𝑝𝑡 = 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)
εl
𝑙 𝑡 = ∫0 lt j dj
εl −1
1 εl
εl −1
(14)
𝑤 𝑡 𝑗 −εl
𝑙𝑡 𝑗 = 𝑙𝑡
𝑤𝑡
(15)
1
𝑤𝑡 = ∫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. Model equations
Εt ∑∞ 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+2
𝑈 �
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 =0
𝑟𝑡 𝜙 𝑧𝑧 𝜙 𝑒 −1
𝑝𝑡
(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. Model equations
= 𝑧𝑧 𝑡 𝑗
𝑒 𝑧𝑧,𝑡 (𝑗) 𝜒𝑒 𝜙𝑒
𝑘 𝑡−1 (𝑗) 𝜙𝑒
(25)
𝜔𝑒
𝑒𝑒 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,𝑡
𝑜𝑜𝑜 𝑡 1−𝛼 𝑒 𝑝 𝑔𝑔𝑔,𝑡
(27)
𝜔
𝑦𝑦 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 𝑤𝑡
𝑙𝑡 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. Model equations
𝐹 𝑡 𝑠 = 𝑝𝑝 𝑡 𝑠 𝑦𝑦 𝑡 𝑠 − 𝑦𝑦 𝑡 𝑠 𝑚𝑚 𝑡 (31)
εh
𝑦𝑦 𝑡 = ∫0 yht s ds
εh −1
1 εh
εh −1
(32)
𝑝𝑝 𝑡 𝑠 −εh
𝑦𝑦 𝑡 𝑠 = 𝑝𝑝
𝑦𝑦 𝑡 (33)
1
𝑝𝑝 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. 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−1
𝑟 𝑟 ζ �𝜋
(44)
𝜃𝑔
1 𝜃 𝑔 −1 𝜃 𝑔 −1
𝑔𝑡 = 𝛾𝑔 𝑔𝑔𝑡 + 1 − 𝛾𝑔 𝑔𝑔 𝑡
1 𝜃 𝑔 −1
𝜃𝑔 𝜃𝑔 𝜃𝑔 𝜃𝑔
(45)
𝑝𝑝 𝑡 −𝜃 𝑔 1−𝛾 𝑔
𝑔𝑔 𝑡 = 𝑔𝑔𝑡
𝑝𝑝𝑡 𝛾𝑔
(46)

28. Model equations
𝑝 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 𝑡
1+i∗ Ξ 𝑡
(61)
t 𝑝𝑝 𝑡 𝑥 𝑡