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

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

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).

World energy prices (US dollars per physical unit)

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.

0

20

40

60

80

100

120

140

16020

00M

0120

00M

0720

01M

0120

01M

0720

02M

0120

02M

0720

03M

0120

03M

0720

04M

0120

04M

0720

05M

0120

05M

0720

06M

0120

06M

0720

07M

0120

07M

0720

08M

0120

08M

0720

09M

0120

09M

0720

10M

0120

10M

0720

11M

0120

11M

0720

12M

01

Crude oil, Dubai ($/bbl) Liquefied natural gas, Japan ($/mmbtu)

Net barter terms of trade, export and import value 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

050

100150200250300350400450

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)

Energy related subsidies in Pakistan (millions of rupees)

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.

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 10,807 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 -

Fiscal balance (share of GDP)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

-FY00 FY01 FY02 FY03 FY04 FY05 FY06 FY07 FY08 FY09

Source: Handbook of Statistics of Pakistan

• Recurrent large fiscal deficit, which exceeds 7% of GDP in FY 2008

Money supply (yearly growth)

Source: Broad Money Supply, Handbook of Statistics of Pakistan

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

• Through fiscal deficit monetization, money supply accelerates without interruption during last decade, reaching a 6% rate of growth in 2010

Load shedding in Pakistan

YearNational sales (GWH)

National load shedding (GWh)

Total National Demand (GWh)

Load Shedding (%)

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

Real Gross Domestic Product growth (at market prices of 1999-00)

Source: Handbook of Statistics of Pakistan

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

FY01 FY02 FY03 FY04 FY05 FY06 FY07 FY08 FY09 FY10 P

• 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.

Inflation (GDP deflator % changes)

Source: Handbook of Statistics of Pakistan

0.0

5.0

10.0

15.0

20.0

25.0

FY 00 FY 01 FY 02 FY 03 FY 04 FY 05 FY 06 FY 07 FY 08 FY 09 FY 10

• With expansionary macroeconomic (fiscal and monetary) policies and the supply bottleneck generated by energy, domestic inflation accelerated significantly in last years

Real effective exchange rate

0

20

40

60

80

100

120

14019

9019

9119

9219

9319

9419

9519

9619

9719

9819

9920

0020

0120

0220

0320

0420

0520

0620

0720

0820

0920

10

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

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

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?

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.

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.

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.

• 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

Next steps

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.

Bibliography

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.

Ε0 ∑ βtUt j∞t=0 = Ε0 ∑ βt log ct j − ζhct−1 j + s

µmt jpt

µ− φt

lt j 1+σl1+σl

∞t=0 (1)

𝑚𝑡 𝑗 + 𝑏𝑡 𝑗1+𝑖𝑡

+ et𝑏𝑡∗ 𝑗

1+it∗ Ξ𝑒𝑡𝑏𝑡

∗

𝑝𝑝𝑡𝑝𝑡

= 𝑚𝑡−1 𝑗 + 𝑏𝑡−1 𝑗 + et𝑏𝑡−1∗ 𝑗 + 𝑤𝑡 𝑗 𝑙𝑡 𝑗 +

𝑟𝑡 𝑗 𝑘𝑡𝑠 𝑗 + 𝐹𝑡 𝑗 + 𝑇𝑇𝑡 𝑗 − 𝑝𝑡𝑐𝑡 𝑗 − 𝑝𝑡𝑖𝑡 𝑗 − 𝑝𝑡𝑝𝑡𝑒𝑒𝑧𝑧,𝑡(𝑗) (2) 𝑈�𝑐𝑡 − ℎζ𝛽𝑈�𝑐𝑡+1 = 𝛽 1 + 𝑖𝑡

𝑝𝑡𝑝𝑡+1

𝑈�𝑐𝑡+1 − ℎζ𝛽𝑈�𝑐𝑡+2 (3)

1 + it∗ Ξ 𝑒𝑡𝑏𝑡∗

𝑝𝑝𝑡𝑝𝑡

𝑒𝑡+1𝑒𝑡

= 1 + 𝑖𝑡 (4)

𝑈�𝑐𝑡−ℎζ𝛽𝑈�𝑐𝑡+1

𝑝𝑡− 𝛽 𝑈�𝑐𝑡+1−ℎζ𝛽𝑈�𝑐𝑡+2

𝑝𝑡+1= 𝑠

𝑝𝑡

𝑚𝑡𝑝𝑡

𝜇−1 (5)

Model equations*

* The full description of the model is available in DSGE_Pakistan_Model.docx.

𝑐𝑡 𝑗 = 𝛿1𝜂 𝑒𝑐𝑡 𝑗

𝜂−1𝜂 + 1 − 𝛿

1𝜂 𝑧𝑡 𝑗

𝜂−1𝜂

𝜂𝜂−1

(6)

𝑧𝑡 𝑗 = 𝛾1𝜃 𝑐𝑐𝑡 𝑗

𝜃−1𝜃 + 1 − 𝛾

1𝜃 𝑐ℎ𝑡 𝑗

𝜃−1𝜃

𝜃𝜃−1

(7) 𝑝𝑒𝑐𝑡𝑒𝑐𝑡 𝑗 + 𝑝𝑧𝑡𝑧𝑡 𝑗 (8) 𝑝𝑐𝑡𝑐𝑐𝑡 𝑗 + 𝑝ℎ𝑡𝑐ℎ𝑡 𝑗 (9)

𝑧𝑡 𝑗 = 𝑝𝑧𝑡𝑝𝑝𝑡

−𝜂 1−𝛿𝛿𝑒𝑐𝑡 𝑗 (10)

𝑐ℎ𝑡 𝑗 = 𝑝ℎ𝑡𝑝𝑝𝑡

−𝜃 1−𝛾𝛾𝑐𝑐𝑡 𝑗 (11)

Model equations

𝑝𝑡 = 𝛿𝑝𝑒𝑐𝑡1−𝜂 + 1 − 𝛿 𝑝𝑧𝑡1−𝜂1

1−𝜂 (12)

𝑝𝑧𝑡 = 𝛾𝑝𝑐𝑡1−𝜃 + 1 − 𝛾 𝑝ℎ𝑡

1−𝜃1

1−𝜃 (13)

𝑙𝑡 = ∫ lt jεl−1εl dj

10

εlεl−1

(14)

𝑙𝑡 𝑗 = 𝑤𝑡 𝑗𝑤𝑡

−εl𝑙𝑡 (15)

𝑤𝑡 = ∫ wt j 1−εldj10

11−εl (16)

𝑤𝑡+𝑖 𝑗 = Γw,t

i wt j (17)

Γw,ti = ∏ ζ 1 + πt+j−1

ξl 1 + π�t+j1−ξli

j=1 (18)

Model equations

Εt ∑ 𝛽𝜙𝑙 𝑖𝑙𝑡+𝑖 𝑗 𝑈�𝑙𝑡+𝑖εl

εl−1+ wt+i j

pt+i𝑈�𝑐𝑡+𝑖 − ℎζ𝛽𝑈�𝑐𝑡+1+𝑖∞

𝑖=0 = 0 (19)

𝑘𝑡𝑠(𝑗) = 𝑧𝑧𝑡 𝑗 𝑘𝑡−1(𝑗) (20) 𝑈�𝑐𝑡−ℎζ𝛽𝑈�𝑐𝑡+1𝑈�𝑐𝑡+1−ℎζ𝛽𝑈�𝑐𝑡+2

=

𝛽 𝑟𝑡+1𝑧𝑧𝑡+1𝑝𝑡+1

+ 1 − 𝛿𝑘 − 𝜒𝑧𝑧1+𝜙𝑧𝑧

𝑧𝑧𝑡+11+𝜙𝑧𝑧 − 𝑧𝑧𝑠𝑠1+𝜙𝑧𝑧 − 𝜒𝑒𝜙𝑒𝑧𝑧𝑡+1𝜙𝑒 −

𝑝𝑡+1𝑒 𝜒𝑧𝑧𝑧𝑧𝑡+1

𝜙𝑒

𝜙𝑒

(21) 𝑟𝑡𝑝𝑡− 𝜒𝑧𝑧𝑧𝑧𝑡𝜙𝑧𝑧 − 𝑝𝑡𝑒𝜒𝑒𝑧𝑧𝑡𝜙𝑒−1 = 0 (22)

𝑘𝑡 𝑗 = 𝑖𝑡 𝑗 + 1 − 𝛿𝑘 𝑘𝑡−1 𝑗 − Δzu,t (23) Δzu,t = 𝜒𝑧𝑧

1+𝜙𝑧𝑧𝑧𝑧𝑡 𝑗 1+𝜙𝑧𝑧 − 𝑧𝑧𝑠𝑠1+𝜙𝑧𝑧 𝑘𝑡−1(𝑗) (24)

Model equations

𝑒𝑧𝑧,𝑡(𝑗)𝑘𝑡−1(𝑗)

= 𝜒𝑒𝜙𝑒𝑧𝑧𝑡 𝑗 𝜙𝑒 (25)

𝑒𝑒𝑡 = 𝑎𝑒𝑡 1 − 𝛼𝑒1𝜔𝑒𝑜𝑖𝑙𝑡

𝜔𝑒−1𝜔𝑒 + 𝛼𝑒

1𝜔𝑒𝑒𝑎𝑔𝑡

𝜔𝑒−1𝜔𝑒

𝜔𝑒𝜔𝑒−1

(26) 𝑔𝑔𝑠𝑡𝑝𝑖𝑙𝑡

= 𝛼𝑒1−𝛼𝑒

𝑝𝑜,𝑡𝑝𝑔𝑔𝑔,𝑡

𝜔𝑒 (27)

𝑦ℎ𝑡 𝑔 = 𝑎ℎ𝑡 1 −α1𝜔𝑙𝑡 𝑔

𝜔−1𝜔 + 𝛼

1𝜔𝑘𝑡𝑠(𝑔)

𝜔−1𝜔

𝜔𝜔−1

(28) 𝑘𝑡𝑔(𝑠)𝑙𝑡 𝑠

= 𝛼1−𝛼

𝑤𝑡𝑟𝑡

𝜔 (29)

𝑚𝑐𝑡 𝑔 = 𝑚𝑐𝑡 = 1𝑔ℎ𝑡

1 − 𝛼 𝑤𝑡1−𝜔 + 𝛼𝑝𝑜𝑡1−𝜔1

1−𝜔 (30)

Model equations

𝐹𝑡 𝑔 = 𝑝ℎ𝑡 𝑔 𝑦ℎ𝑡 𝑔 − 𝑦ℎ𝑡 𝑔 𝑚𝑐𝑡 (31)

𝑦ℎ𝑡 = ∫ yht sεh−1εh ds

10

εhεh−1

(32)

𝑦ℎ𝑡 𝑔 = 𝑝ℎ𝑡 𝑠𝑝ℎ

−εh𝑦ℎ𝑡 (33)

𝑝ℎ𝑡 = ∫ pht s 1−εhds10

11−εh (33’)

𝑝ℎ𝑡+𝑖 𝑔 = Γh,t

i pht s (34) Γh,ti = ∏ 1 + πht+i−1 ξh 1 + π�t+i 1−ξhi

j=1 (35) Εt ∑ 𝛽𝜙𝑙 𝑖𝜆𝑧𝑡+𝑖𝑦ℎ𝑡+𝑖 𝑔 1 − εh Γh,t

i 𝑝ℎ𝑡 𝑔 + εh𝑚𝑐𝑡+𝑖∞𝑖=0 = 0 (36)

Model equations

𝑐ℎ𝑡∗ = 𝛾∗ 𝑝ℎ𝑡∗

𝑝𝑝𝑡∗

−𝜂∗

𝑐𝑡∗ (37) 𝑟𝑒𝑟𝑡 = 𝑒𝑡𝑝𝑝𝑡∗

𝑝𝑡 (38) 𝜋𝑡 = 𝑝𝑡

𝑝𝑡−1− 1 (39)

𝑝𝑐𝑡 = 1 + 𝜏𝑐 𝑒𝑡𝑝𝑐𝑡∗ (40) 𝑝𝑜𝑡 = 1 + 𝜏𝑝 𝑒𝑡𝑝𝑜𝑡∗ (41) 𝑝𝑒𝑎𝑔𝑡 = 1 + 𝜏𝑔𝑔𝑠 𝑒𝑡𝑝𝑒𝑎𝑔𝑡∗ (42) 𝑝ℎ𝑡∗ = 𝑝ℎ𝑡

𝑒𝑡 (43)

𝑟𝑡𝑟

= 𝑟𝑡−1𝑟

𝜌𝑟𝑦𝑡

𝑦𝑡−1ζ�

Ψy 1−𝜌𝑟πt𝜋�

Ψπ 1−𝜌𝑟 (44)

𝑒𝑡 = 𝛾𝑔

1𝜃𝑔 𝑒𝑐𝑡

𝜃𝑔−1𝜃𝑔 + 1 − 𝛾𝑔

1𝜃𝑔 𝑒ℎ𝑡

𝜃𝑔−1𝜃𝑔

𝜃𝑔𝜃𝑔−1

(45)

𝑒ℎ𝑡 = 𝑝ℎ𝑡𝑝𝑝𝑡

−𝜃𝑔 1−𝛾𝑔𝛾𝑔

𝑒𝑐𝑡 (46)

Model equations

𝑝𝑡𝑒𝑡 = 𝑡𝑡 + 𝑏𝑡 − 1 + 𝑖𝑡 𝑏𝑡−1 + 𝑚𝑡 − 𝑚𝑡−1 (47) 𝑡𝑐𝑡 = 𝜏𝑐

1+𝜏𝑐𝑝𝑡𝑐𝑡 (48) 𝑡𝑜𝑡 = 𝜏𝑜

1+𝜏𝑜𝑝𝑜𝑡𝑜𝑡 (49) 𝑡𝑒𝑎𝑔𝑡 = 𝜏𝑔

1+𝜏𝑔𝑝𝑒𝑎𝑔𝑡𝑒𝑎𝑔𝑡 (50)

𝑡𝑒𝑖𝑡 = 𝜏𝑒𝑒

1+𝜏𝑒𝑒𝑝𝑒𝑖𝑡𝑒𝑖𝑡 (51) 𝑡𝑒𝑐𝑡 = 𝜏𝑒𝑐

1+𝜏𝑒𝑐𝑝𝑒𝑝𝑡𝑒𝑐𝑡 (52)

𝑡𝑡 = 𝑡𝑐𝑡 + 𝑡𝑜𝑡 + 𝑡𝑒𝑎𝑔𝑡 + 𝑡𝑒𝑐𝑡 + 𝑡𝑒𝑖𝑡 + 𝑎𝑖𝑎𝑡𝑒𝑡 (53) 𝑦ℎ𝑡 = 𝑐ℎ𝑡 + 𝑒ℎ𝑡 + 𝑐ℎ𝑡∗ (54) 𝑒𝑒𝑡 = 𝑒𝑐𝑡 + 𝑒𝑧𝑧,𝑡 (55) 𝑐𝑡 = 𝑐ℎ𝑡 + 𝑐𝑐𝑡 + 𝑒𝑐𝑡 (56) 𝑒𝑡 = 𝑒ℎ𝑡 + 𝑒𝑐𝑡 (57) 𝑝𝑖𝑚𝑡𝑖𝑚𝑡 = 𝑝𝑐𝑡𝑐𝑐𝑡 + 𝑝𝑐𝑡𝑒𝑐𝑡 + 𝑝𝑜𝑡𝑜𝑡 + 𝑝𝑒𝑎𝑔𝑡𝑒𝑎𝑔𝑡 (58) 𝑝𝑖𝑚𝑡 = 𝜅𝑝𝑝𝑐𝑡 + 𝜅𝑝𝑝𝑜𝑡 + 𝜅𝑔𝑝𝑒𝑎𝑔𝑡 (59) 𝑝𝑦𝑡𝑦𝑡 = 𝑝𝑡𝑐𝑡 + 𝑝𝑡𝑒𝑡 + 𝑝𝑡𝑖𝑡 + 𝑝ℎ𝑡𝑐ℎ𝑡∗ − 𝑝𝑖𝑚𝑡𝑖𝑚𝑡 (60)

et𝑏𝑡∗

1+it∗ Ξ𝑒𝑡𝑏𝑡

∗

𝑝𝑝𝑡𝑝𝑡

= et𝑏𝑡−1∗ + 𝑝ℎ𝑡𝑐ℎ𝑡∗ − 𝑝𝑖𝑚𝑡𝑖𝑚𝑡 (61)

Model equations