MAMS: Review of Mathematical Model Structure

Transcription

1 MAMS: Review of Mathematical Model Structure Hans Lofgren World Bank Presentation for Second Training Workshop of the Project Assessing Development Strategies to Achieve the MDGs in the Arab Region, Tunis, January 21-24, 2008.

2 Introduction Objective of presentation: Provide more details on the different relationships built into MAMS. MAMS: A simultaneous equation system that is square (# of variables = # of equations). MAMS can be split into three modules: 1. Within-period (summarized in figure with CGE/SAM payments) 2. Between-period (endogenous updating of stocks and productivity) 3. MDG-and-education This material is presented in more detail in the file Lofgren-MAMS-Math-Stat-simple posting.doc.

3 Introduction Compared to the full mathematical statement, the following key features have been suppressed: time subscripts (except when time lags are relevant; think of all equations, parameters, and variables as having a time index t); features related to the presence of multiple representative households; think of this version as having a single household.

4 Introduction The following notational conventions apply: Upper-case Latin letters are used for variables. Exogenous variables have a bar on top, endogenous variables do not. Parameters have Greek or lower-case Latin letters. Subscripts refer to set indices. Superscripts are part of the name of the variables or parameter. Lower-case set indices refer to individual elements; upper-case set indices refer to the set itself (here used in summations.

5 Acronyms/abbreviations for functions CES CES* CET CET* LEO LES f constant elasticity of substitution function derived from optimizing subject to a CES function constant elasticity of transformation function derived from optimizing subject to a CET function Leontief function (fixed coefficients) Linear Expenditure System other function (often sum of different terms

6 Set indices for within- and between-period modules a activities agov government activities c commodities f factors flab labor fcap capital fcapgovinf government infrastructure capital i institutions h household gov government row rest of world insng nongovernment (h, row ) insd domestic (h, gov ) t time (year)

7 . Within-period module

8 Stylized Model Structure Factor costs Activities Intermediate input cost Factor Markets Wages & rents Households Domestic private savings Taxes Government Gov t Savings Sav./Inv. Sales Commodity markets Exports Imports Private consumption Transfers Gov t Consumption Foreign Transfers Investment Demand Rest of the World Foreign Savings

9 Commodities (Products) Domestic suppliers decide whether to sell to domestic demanders or export.

10 Output transformation (CET function) QX = (, ) CET QD QE c c c domestic domestic exp- output = CET sales, orts of c of c of c

11 Export-domestic supply ratio QE QD c c PE = CET * c PDc export- export- domestic domestic sales = CET * sales QUANTITY PRICE ratio for c ratio for c

12 Export price (,, ) PE = f pwe te EXR c c c export export export exchange price = f price, tax, rate (LCU ( LCU ) (FCU) rate per FCU)

13 Producer price (, ) PX = f PD PE c c c producer domestic sales export price = f price for domestic, price ( LCU ) output (FCU) (LCU)

14 Activity price PA = f PX (, θ ) a c a, c unit out- producer prices yield put price = f for commodities c, of c of a produced by a for a

15 Value-added price (,, ) PVA = f PA PQ ta a a c a value- prices of rate of output added f, inter-, indirect = price price mediate activity for a for a inputs tax

16 Factor demand (,, ) = * f, a a f a QF CES PVA WF tva employ value- value- ment of *, factor, added CES = added factor f by wage tax price activity a rate

17 Activity level a ( ) f, a QA = CES QF level employment of act- = CES of factors f ivity a by activity a

18 Commodity production QX = f QA (, θ ) c a a, c quan- level of yield tity of activities of c f, out- = a prod - for act- put c ucing c ivity a

19 Intermediate input demand (, ) QINT = LEO QA ica ca, a ca, intermediate level intermediate demand for of, demand = LEO c from activ- coefficient activity a ity a for c in a

20 Stylized Model Structure Factor costs Activities Intermediate input cost Factor Markets Wages & rents Households Domestic private savings Taxes Government Gov t Savings Sav./Inv. Sales Commodity markets Exports Imports Private consumption Transfers Gov t Consumption Foreign Transfers Investment Demand Rest of the World Foreign Savings

21 Factor markets ( ) QF = 1 UERAT QFINS f, a f i, f a A i INS total quantity 1 - unem- sum of all employed = ployment institutional en-of factor f rate dowments of f

22 Labor market with unemployment 5 4 Wage unemployment rate (%) Supply Demand

23 Reservation wage f = (,, ) WFRES f QHPC UERAT CPI reserv- unem- con- household ation f -, ployment, sumer per capita wage for rate for price consumption factor f factor f index = f

24 Wage and unemployment constraints WF f WFRES f reservation wage for wage for factor f factor f UERAT f ueratmin f unemploy minimum ment rate unemployment for factor f rate for factor f

25 Complementary slackness relation for wages and unemployment ( WF WFRES ) f f ( UERAT ueratmin ) = 0 f f Complementary slackness relationship: 1. If wage > reservation wage, then unemployment rate = its minimum. 2. If unemployment rate > its minimum, then wage = reservation wage.

26 Institutional factor income (,, ) YIF = f SHIF WF QF i, f i, f f f, a income of share of employwage institution i inst'on i ment of, of (h, gov or =f in endow-, factor f factor RoW) from ment of in activities a f factor f factor f

27 Stylized Model Structure Factor costs Activities Intermediate input cost Factor Markets Wages & rents Households Domestic private savings Taxes Government Gov t Savings Sav./Inv. Sales Commodity markets Exports Imports Private consumption Transfers Gov t Consumption Foreign Transfers Investment Demand Rest of the World Foreign Savings

28 Household income YI = YIF + YIINT + trnsfr + trnsfr EXR h h, f h h, gov h, row f F income net transfers transfers factor of house- = + interest+ from gov-+ from rest income hold h income ernment of world

29 Household net interest income YIINT = f h gintrat, GDEBT ; h fintrat, FDEBT, EXR h h h net interest net from interest on domestic income of (+) and foreign debt (-): = house- debt levels and interest hold h rates; exchange rate

30 Household savings h h ( ) 1 h HSAV = MPS TINS YI savings savings income of h of house-= rate (net of dirhold h of h ect taxes) h

31 Household consumption expenditure h ( ) EH = MPS (1- TINS ) YI 1 h h consumption household income expenditure = net of direct for household h taxes and savings h

32 Household consumption demand QH = LES( EH, PQ ) ch, h c quantity of household demand from consumphousehold =LES tion ex- h for com- penditure, modity c prices

33 Stylized Model Structure Factor costs Activities Intermediate input cost Factor Markets Wages & rents Households Domestic private savings Taxes Government Gov t Savings Sav./Inv. Sales Commodity markets Exports Imports Private consumption Transfers Gov t Consumption Foreign Transfers Investment Demand Rest of the World Foreign Savings

34 Government revenue TINS h, YIh; ta a, tva a,paa, PVAa, QA ; a YG = f tm c, pwm c, QM c; te c, pwe c, QEc; tq c, PQ c, QQ c; trnsfr gov, row, EXR ; YIF gov,f govern ment revenue =f direct tax activity and VA rates,, tax rates, activity household and VA prices, incomes activity levels import tariff export tax rates, import rates, export, prices (FCU), prices (FCU), import quantities export quantities sales tax transfers from government rates, composite rest of world income,, supply prices (FCU), from and quantities exchange rate factor f

35 Government recurrent expenditure PQ c, QG c; trnsfr h, gov; EG = f gintrat h, GDEBT h, fintrat gov, FDEBT gov, EXR government trans- fers to consump govern- + tion of c, house- ment price of c holds recurrent=f expenditure and foreign debt: debt interest on domestic + levels and interest rates; exchange rate

36 Government consumption demand ( ), ct ct, QG = f QG QGGRW ct,, 1 govern- govern- growth in ment con- ment con- government = f, sumption sumption consump - of c in t of c in t-1 tion of c in t

37 Government savings GSAV = YG EG governgovern- govern ment re-ment re- ment = current current savings expenrevenue ditures

38 Government investment demand DKINS = ifa fcap, agov QFINSgov, fcap, t = f QA depr fcap, t QA agov, t 1 gov, fcap, t agov, t government expected government dem- government investment and for fcap in t+ 1 : stock of fcap demand in t capital- input coefficient, in t; dep- for capital fcap activity level in t and t- 1 reciation rate

39 Price of new capital stock ( ; ), PK = f capcomp PQ fcap c fcap c quantities of price of prices commodities new cap-, of com- per unit of = f c ital stock modities new capital fcap c stock f

40 Government investment value f FCAP PK DKINS = GSAV + GBOR f gov, f h government fixed investment governvalue: sum over product of ment prices and investments savings in different capital goods = + government direct borrowing from household h + GBORMS + FBOR EXR h gov net borrowing of net borrowing from RoW monetary system (net of lending to RoW and in-from household h creases in foreign reserces + + t

41 Household investment value f FCAP PK DKINS = HSAV GBOR f h, f h h fixed investment value for house- government household): sum over prod- = hold direct borrowing uct of prices and investments savings from household h GBORMS + EXR FBOR + h net borrowing of net bor- system rowing monetary from household h from RoW t h

42 Rest-of-World investment value (FDI) f FCAP PK DKINS = EXR fdi f row, f t row fixed investment value for RoW: sum over product of prices and investments = [ FDI]

43 Investment demand by commodity (investment by origin or source) QINV c capcompc, fcap ; = f DKINS, DKINS gov, fcap insng, fcap invest- quantities of gross investments ment commodities c in capital stocks demand = f per unit of, fcap by government for com- new capital and non-government institutions modity c stock fcap

44 Stylized Model Structure Factor costs Activities Intermediate input cost Factor Markets Wages & rents Households Domestic private savings Taxes Government Gov t Savings Sav./Inv. Sales Commodity markets Exports Imports Private consumption Transfers Gov t Consumption Foreign Transfers Investment Demand Rest of the World Foreign Savings

45 Composite commodity demand QQ = QINT + QH + QG + QINV c c, a c, h c c a A h H com- inter- house- govern- fixed posite = mediate+ hold con-+ ment con-+ invest- use sumption sumption ment demand

46 Composite demand (Armington function) QQ = (, ) CES QD QM c c c comp- dom- im- osite = CES estic, ports demand sales of c for c of c

47 Import-domestic demand ratio QM QD c c PM = CES * c PDc import- import- domestic domestic sales = CES * sales QUANTITY PRICE ratio for c ratio for c

48 Import price (,, ) PM = f pwm tm EXR c c c import import exchange, tariff price f = price, rate (LCU rate ( LCU ) (FCU) per FCU)

49 Composite commodity price PQ = f PD, PM, tq ( ) c c c c domestic sales composite import sales price for commodity f, price, = tax domestic price ( LCU ) (LCU) rate output (FCU)

50 Stylized Model Structure Factor costs Activities Intermediate input cost Factor Markets Wages & rents Households Domestic private savings Taxes Government Gov t Savings Sav./Inv. Sales Commodity markets Exports Imports Private consumption Transfers Gov t Consumption Foreign Transfers Investment Demand Rest of the World Foreign Savings

51 c CM import spending i INSD Balance of payments (in FCU) pwm c + QM + c factor income to RoW f F YIF EXR row, f + fintrat FDEBT = pwe QE interest payment export = on foreign debt revenue i i c c c CE trnsfrirow, trnsfrf, row + FBOR + fdi i row i INSD f F i INSD + + transfers from factor borrowing foreign + RoW to domestic+ income + from + direct institutions from RoW RoW investment

52 . Between-Period Module Endogenous updating equations for Stocks (part of factor endowments, foreign debt, domestic debt Total factor productivity (efficiency by activity.

53 Capital stocks by institution QFINS = (1 depr ) QFINS i, fcap, t fcap i, fcap, t 1 + DKINS stock of capital old capital gross type f held by = stock (after + institution i in t depreciation) investment in t-1 i, f, t 1

54 Institutional shares of factor endowments SHIF i, f, t = i I QFINS i, f, t QFINS i, f, t = share of institution i in total endowment of factor f endowment of i divided by total en- dowment ment of f

55 Foreign debt of domestic institutions FDEBT = FDEBT + FBOR fdebtrelief it, it, 1 it, it, foreign debt of dom- foreign foreign foreign estic inst'on i (=insd) = debt in t-1 + borrowing debt relief in t (end of yr) (end of yr) in t in t

56 Domestic government debt GDEBT = GDEBT + GBOR ht, h, t 1 ht, government government = + debt to house- debt to house- hold h in t hold h in t-1 (end of year) (end of year) government borrowing from h in t

57 Efficiency (TFP by activity) α ( α va2 ; ; ;), va = a a gov fcap f QFINS TRDGDP effici- trend for level of oppenness ency efficiency, government, ( trade- = f of act- of act - capital GDP ivity a ivity a stocks f ratio)

58 . MDG and Education Module

59 Educational quality ( ; ) EDUQUAL = f QQ QENR cedu cedu cedu index of educational real enroll- quality at level f services in, = ment in ( cycle) cedu cycle cedu cycle cedu

60 Enrollment by cycle (level) QENR cedu, t enroll ment in cycle cedu in t = f = f ( edu QENR ) cedu, t 1 SHRbehav, cedu, t 1 QENR SHR edu cedu', t 1 behav, cedu, t 1 edu behav, cedu, t 1 continuing old students : enrollment in cedu in t-1; student behavior in t-1 (shares that repeat or pass + continue in cedu),,, SHR,, popg1 new students in cedu in t: enrollment in preceding cycle cedu ' in t- 1; + student behavior in t- 1 ( shares that graduate + continue to cedu) and in t (share of cohort entering 1st grade) t

61 Student behavior (top of nest) SHR ( ) edu = LOG ZEDU behav, cedu behav, cedu share of students or popul- ation with behavior behav (pass, continue to next cycle [among graduates], or enter1st grade) in relation to cycle cedu = LOG intermediate behavior variable ZEDU behav, cedu

62 Student behavior (bottom of nest) EDUQUAL ; WF ; QHPC = cedu flab ZEDUbehav, cedu CE MDGVAL mdg 4; QFINS fcapgovinf intermediate variable education quality; wage for share with behavior incentives; per-capita behav (pass, continue to = CE household consumption; next cycle, enter 1st grade) student health (proxy: MDG4); in relation to cycle cedu infrastructure capital stock

63 Student behavior: graduation from cycle ( edu,, ) SHR = f SHR yrcyc shrgrdcyc edu grdcyc, cedu pass, cedu cedu cedu share of pas - student share length sing students that passes of cycle = f,, graduating its grade cedu from cedu in cedu ( years) base-year share of passing stud - ents graduating from cedu

64 Student behavior: residual indicators ( edu ) behav cedu edu = behav, cedu ', SHR f SHR student share with residual behavior behav (repeater, dropout, exiting graduate, passing student continuing within cycle) in cedu = f shares with related be- havior behav' (passing, graduate who continues to next cycle, passing student graduating from cycle)

65 MDG : primary school completion LABPARTRAT QFINS ; h, flab = f poplab, QENRcedu labor force labor force; population particip = f in labor force age not ation rate enrolled in school

66 Labor endowment by type QFINS h, flab, t = f = endowment endow- of labor ment flab for in t-1, f household rate of hin t retirement edu SHRbehav, cedu, t, QFINSh, flab, t 1,, shrlabentcedu, t, depr flab, t 1 shrlabent2 flab, t educational behavior (shares grad - uating from cycle and exiting,, dropping out); labor force entry shares (among students and non- students reaching labor age)

67 Other MDG indicators (top of nest) MDGVAL mdg = ( ) LOG ZMDG mdg intermediate behavior variabl e value for MDG (4 = under - five mort - tality rate; 5 = maternal mortality rate; = LOG 7 = population shares with improved water and sanitation access ZMDGmdg

68 Other MDG indicators (top of nest) MDGVAL mdg = ( ) LOG ZMDG mdg intermediate behavior variabl e value for MDG (4 = under - five mort - tality rate; 5 = maternal mortality rate; = LOG 7 = population shares with improved water and sanitation access ZMDGmdg

69 Other MDG indicators (bottom of nest) QQ, poptot; QHPC cmdg ZMDG = CE mdg MDGVAL ; QFINS mdg fcapgovinf intermediate variable for MDG indicator (4, 5, 7a or 7b) = CE service level, total population; per- capita household consumption; other MDG indicators (7a and 7b influencing 4 and 5); infrastructure capital stock