IS-LM Model with Labor
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1 IS-LM Model wth Labor Demand for Labor ( D - assumes frms hre workers to make proft roft - prce x quantty [.e., total revenue] - (labor cost + materals cost + fxed cost [.e., total cost] Value Added - snce we're talkng about GD, we're only worred about value added; that means we can gnore raw materals cost; we also change prce to a "value added" or "net" prce ( to account for the raw materals Labor Decson - fxed costs are rrelevant n the decson; use net prce ( tmes output ( mnus wage (W tmes number of workers (: roft = - W Assumptons - (1 compettve goods and labor markets (.e., can't pck or W ( captal (K s fxed n the short run (3 producton functon relates to : = F(K, (4 producton functon s ncreasng at dmnshng rate (F > 0; F < 0 Hrng Decson - want to maxme F(K, - W - W 1st Order Condton - set dervatve to ero: F - W = 0 nd Order Condton - d ( - W/d = F < 0 (by assumpton Margnal roduct of Labor (ML - F Margnal Revenue of Labor - F Margnal Cost of Labor - W Shftng Curve - three thngs wll shft curve and change hrng decson: (1 W - W moves curve down; hre fewer workers ( - moves curve up; hre more workers W/ or F (3 Technology - tech mprovement ncreases F ; hre more workers ML = Real Wage - rearrange F = W to get ML = Real Wage (w: F = W/ = w Demand Curve - snce frms are wllng to pay the ML, the ML s bascally the demand curve for labor Downward Slopng - d(w//d = d(f /d = F < 0 (for all producton functons wth dmnshng returns Supply of Labor ( - n short run, only so many workers so supply s fxed; besdes, emprcal studes show the supply of labor s nelastc Classcal Model (Long Run - always have full employment; more approprate for long run; descrbes aspects we stll worry about, but doesn't capture t all (that's why we have Keynesan Model Full Employment - f real wage s too hgh (w 1, there s downward pressure on wages to move toward equlbrum n the labor market; eventually everyone s employed 5 Equatons - combnng these 3 new equatons wth the IS-LM model gves us long term equlbrum; unknowns are,, w,, F = w Labor Market = = F(K, Goods Market = C( - T + I( - π e + G Asset Market L(, = M/ ote: Really have 7 equatons & 7 unknowns because C and I are embedded n the model; could have 8 f you make taxes endogenous 1 of 7 W/ or F w 1 w* 1 Unemployment F(K,
2 Get Multplers - take total dfferentals (1 F d + F K dk = dw ( d = d (3 d = F K dk + F d (4 d = C 'd - C 'dt + I 'd - I 'dπ e + dg (5 L d + L d = dm/ - (M/ d Others - f you want dc/d? and di/d?, you'll need dc = C 'd - C 'dt and di = I 'd + I 'dπ e Reduced Form - get one endogenous (unknown varable on the left sde wth all exogenous (known varables on the other d = 1 d (from Eqn (; not a very exctng result, but reale d/d? = 0 for all other exogenous varables. lug ths result nto d from Eqn (3: d = F K dk + F d ; (both > 0 very surprsng result... back to the Solow Model; notce the only thngs that affect output ( n the long run are captal (K and labor (; Fscal olcy - d/dg = 0; makes snce n long run because f economy s at full employment, government purchases don't mpact the number of workers or the captal base; potental output remans constant; f you calculate all the multplers, you can also see that d/dg = -1/I ' and di/dg = -1 (100% crowdng out; d/dt = 0; lowerng taxes doesn't change affect supply (stll only K and that do that; total output doesn't change, but composton of the output changes because consumpton goes up and nvestment goes down; di/dt = C ' (100% crowdng out Monetary olcy - d/dm = 0; you can also calculate di/dm = 0 and d/dm = 0; changng M only affects one of the endogenous varables: ; d/dm = /M > 0; ths means ncreasng money supply just ncreases prces n proporton Summary - n long run, fscal polcy just changes dvson of output between C, I, and G; monetary polcy just changes prce level (nflaton; remember, the only thng that ncreases potental output s captal and labor (and technology mprovement lug d nto Eqn (1: dw = F d + F K dk lug n d nto Eqn (4 and solve for d: d = F K F dk + 1 d + dt + 1 dπe + dg I ' < 0 < 0 < 0 > 0 > 0 lug n d and d nto Eqn (5 and solve for d: (M/ d = dm/ - L d - L d = (M/ d = dm/ - L F K dk - L F L dπ e - L I ' dg d - F K L dk - F L d - L dt - of 7
3 ( L (1 L d = dm + + M ( L FK L dk + M M > 0 < 0 < 0 L L dt + L dπ e + M M M < 0 > 0 > 0 + dg F d + Other Multplers - use dc = C 'd - C 'dt and di = I 'd - I 'dπ e to get other multplers dc = C 'F K dk + C 'F d + -C ' dt > 0 > 0 < 0 di = (1 - C 'F K dk + (1 - C 'F d + C ' dt + -1 dg > 0 > 0 > 0 < 0 1 F(K, Keynesan Model (Short Run, Fxed-Wage - ot always operatng wth equlbrum n all three markets; short run model looks at what happens n reachng equlbrum; usual assumpton s that wage s too hgh to have equlbrum n the labor market ("excess supply" = unemployment... economy s on the demand curve for labor, but not on the supply curve Goods Market - C + I + G - = excess demand; f = 0, we have goods market equlbrum; f > 0, there s excess demand and prces rse; f < 0, there s excess supply and prces fall; a(c + I + G - = d/dt (change n prce wth respect to tme Asset Market - smlar argument; b(l - M/ = d/dt (change n nterest rates wth respect to tme W/ or F Labor Market - smlar argument; c(d - = dw/dt (change n wage wth respect to tme Constants - a, b, and c determne how fast prces, nterest rates, and wages change to brng thngs back to equlbrum; of the three, nterest rates are quckest (almost nstantly; wages are the slowest to respond Smulaton - we could smulate to see the n between effects, but need to know all the equatons (.e., explct functons and results wouldn't be general Stepped Comparatve Statcs - nstead of smulatng, look at an n between statc condton were we get to goods and asset equlbrum, but not labor (snce t's slowest to respond; we look at W beng fxed and drop Eqn ( from the long run model; we also have to use W/ rather than w 4 Equatons - unknowns are,,, F = W/ Labor Market = F(K, Goods Market = C( - T + I( - π e + G Asset Market L(, = M/ ote: Really have 6 equatons & 6 unknowns because C and I are embedded n the model; could have 7 f you make taxes endogenous Get Multplers - take total dfferentals (1 F d + F K dk = dw/ + (W/ d ( d = F K dk + F d (3 d = C 'd - C 'dt + I 'd - I 'dπ e + dg (4 L d + L d = dm/ - (M/ d Work done n Homework W Unemployment 3 of 7
4 ( F FK FK F M F M d = dk + ' ( F L I dw + dt + > 0 < 0 < 0 ( F L ( F L ( F I dπ e + dg + ' dm > 0 > 0 > 0 ( 1 ( F FK FK F M ( 1 F M d = dk + < 0 > 0 d = = F F ( F [( F L F M ] / dw + ( L ( F F M / dt+ dπ e + < 0 > 0 (< 1 M / L ( F ( 1 ( F dg + > 0 < 0 dm F L F L F L I ' F dt + dπ e + dg + < 0 > 0 > 0 > 0 dm + ( L + I ' L ( F FK FK F ( L + I ' L dw + > 0? / ( L + L F M < 0 dk F F K - F K F - ths term s negatve for most producton functons (ncludng all Cobb- Douglas producton functons; the frst term s negatve by the dmnshng returns assumpton (F < 0; f the second (F K s nonnegatve, the dfference s negatve; n order for the dfference to be postve must have F K < F F K /F (whch s negatve 4 of 7
5 Aggregate Supply and Demand Shfts from K, W, Tech Shft from Aggregate Supply ( - frst two equatons for short-run model: F = W/ Labor Market = F(K, equatons and 3 unknowns (,, ; f you graph all combnatons of and that solve the system for a gven level of, you end up wth the curve otental GD (pgd - curve s vertcal at ths pont; doesn't matter how much hgher prce level gets, frms can't supply any more Shfts n - (shft rght f K, W, or Technology advances; pgd shfts rght (along wth f Aggregate Demand (AD - substtuton effect and ncome effect result form change n relatve prces; ths model talks about prce level (.e., all goods so t has no effect on aggregate demand; nstead, look at last two equatons for short-run model: Goods Market = C( - T + I( - π e + G Asset Market L(, = M/ equatons and 3 unknowns (,, ; f you graph all combnatons of and that solve the system, you end up wth the AD curve LM Curve - prce enters n L(, = M/... LM (see graph AD Slope - < 0; get ths result form IS-LM multpler for d/dg 3 LM 3 LM IS Shfts n AD - AD (shft rght f G, T, or π e (.e., IS ; or M (.e., LM 1 AD Fscal olcy - tryng to get back to pgd by G or T (.e., IS Fxed-rce Multpler - assumes s horontal lne; ths gnores change n prce Short-Run Multpler - LM and y ; so wll be less and wll be more than fxed-prce multpler predcted 3 1 Fxed-rce Multpler LM Short-Run Multpler smaller; larger LM IS 1 AD 1 IS 1 3 AD 3 Monetary olcy - smlar result except LM shfts twce (frst LM from M ; then LM from ; smlar result: smaller; smaller (than wth fxed-prce multpler olcy Implcatons - f you do nothng, wages fall and ; get to pgd n deflatonary way; f you use fscal polcy, AD so you get to pgd faster, but n nflatonary way 5 of 7
6 uttng It All Together Short-Run vs. Long-Run F(K, LM F(K, LM IS IS W/ or F 0 W/ or F 0 0 AD 0 0 AD 0 0 Fscal olcy - assume at pgd and look at G Short-Run vs. Long-Run G IS AD LM W LM 1 F(K, G LM 0 1 F(K, 1 G LM 0 LM 1 1 IS IS W/ or F 1 W/ or F 1 = AD 0 W 1 W AD 0 = d/dg - fxed-prce > short-run > long-run = 0 d/dg - fxed-prce < short-run < long-run d/dg - fxed-prce < short-run < long-run Result - although ultmately the value of doesn't change n the long-run, t's composton does change; we're tradng off G for I (.e., f G, then I by same amount n long-run; f usng taxes, T C I by same amount n long-run Monetary olcy - assume at pgd and have M ; same as above except frst shft s LM nstead of IS ; everythng else s smlar d/dm - fxed-prce > short-run > long-run = 0 d/dm - fxed-prce < short-run < long-run = 0 d/dm - fxed-prce < short-run < long-run = /M eutral Money - composton of s unchanged (n terms of C, I, and G, unlke wth fscal polcy 6 of 7
7 M G M G M 1 G 1 Tmng - so when s short-run and long-run? Tme Tme 1 1 Tme Tme Tme Tme 1 1 Tme Tme M Short-run Long-run G Short-run Long-run Fscal olcy vs. Do othng - can use short-run multplers to fgure out exactly how much G t wll take to go from 1 to (by accountng for LM ; f there were no fscal polcy, eventually W & LM and you get to same result; dfference s fscal polcy s faster at expense of hgher whch means less I Monetary olcy - end up wth same as f economy fxed tself, but have hgher ; prevents deflaton that would've occurred naturally Long-Run - d/dg = 0... compare n long-run after G to what would've been wthout G ; t's the same Captal Stock - gnored t n IS-LM (covered n Solow Model; although t takes a long tme to be evdent (very long-run, G decreases growth rate of (less nvestment G G LM G LM 1 G G G IS G G 1 1 AD G 1 Tme Other Areas to Explore - (1 otental Output = "full employment"; but how to we defne ths? ( How do we modfy IS-LM to deal wth ongong nflaton; what about ongong ncrease n money supply? (3 How are expectatons formed? (π e (4 How do IS-LM results change for open economy (5 Consumpton - consdered t a functon of after tax ncome: C( - T... wll study n next secton Tme 7 of 7
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