δ (real interest rate) is roughly constant.

Transcription

1 Calibraing he Growh Model Kaldor's sylized facs 1. / L (oupu per worker) exhibis coninual growh. 2. K / L (capial per worker) exhibis coninual growh. 3. r δ (real ineres rae) is roughly consan. 4. K / (capial-oupu raio) is roughly consan. 5. rk /, wl / (facor shares) are roughly consan. 6. There are wide differences in he rae of growh of produciviy across counries. N. Kaldor (1961), Capial Accumulaion and Economic Growh, in F. A. Luz and D. C. Hague, ediors, The Theory of Capial. New ork: S. Marin's Press. 1

2 The growh model max β logc = 0 s.. C + K + 1 K wl + ( r δ ) K C, K 0 K0 = K0 = λ L. L 0 where w = g AK L, 1 α (1 )( ) α α α r = α( g ) AK L. 1 α α 1 1 α We can solve max β logc = 0 1 α α 1 α δ s.. C K (1 ) K ( g ) AK L C, K 0 K0 = K0 = λ L. L 0 2

3 Firs-order condiions: Impose consan growh condiions Simple algebra shows ha β = p C 1 α α 1 1 α α ( δ ) p = 1 p ( g ) AK L + 1 C + K 1 (1 δ ) K = ( g ) AK L. 1 α α 1 α + C / L K / L = g, = g c C / L K / L C+ 1/ L+ 1 K+ 1/ L / L+ 1 = = = g. C / L K / L / L k 3

4 Redefine variables in erms of effecive labor unis L = g L = ( gλ) L : 0 c = C / L = g ( C / L) k = K / L = g ( K / L) log C / L = log g c = log c + log g. Noice ha he balanced growh pah is he seady sae c = c, k = k of he redefined problem max β log c = 0 s.. c + gλk 1 (1 δ) k Ak α + c, k 0 k = K / L

5 The balanced growh pah maches Kaldor s sylized facs (alhough he explanaion for fac 6 is no very ineresing): 1. L = g A K L = g Ak grows a rae g 1. 1 α α α / ( ) ( / ) 2. K / L = g k grows a rae g r δ α g AK L δ αak δ gλ β is consan. 1 α α 1 1 α α 1 = ( ) = = / 1 K = k A is consan. 1 α / / 5. rk / = α, wl / = 1 α are consan. 6. rae of growh of / L is deermined solely by g. 5

6 Calibraion o he U.S. daa I. Firs we inerpre he daa as being observaions of a balanced growh pah and use employmen as he measure of labor inpu. All daa is from he Economic Repor of he Presiden, 2004 (hp:// = Real gross domesic produc (Table B-2) (billions of 2000 dollars) L = Civilian employmen (B-35) (housands of persons) L L g λ , ,778 38, , ,678 47, , ,303 51, , ,793 59, , ,891 71, log ' / L' log / L log g Ak α α = log g Ak = ( ' )log g. g = , λ=

7 = Gross domesic produc (B-1) - proprieors income (B-28) (axes on producion and impors (B-28) subsidies (B-28)) (billions of curren dollars) wl= Compensaion of employees (B-28) (billions of curren dollars) (We disribue proprieors income and indirec business axes proporionally beween labor income and capial income.) GDP proprieors indirec subsidies wl 1 α income axes , , , , , , , , , , α = , α =

8 How good is he assumpion ha we are in a balanced growh pah? /( g L) = ( / L) / g = Ak α should be consan. gl , , , , ,034 Ak α = 38,409 8

9 Real GDP per Worker in he Unied Saes U.S. dollars daa balanced growh pah year 9

10 K 1 (1 + δ ) K = Gross privae domesic invesmen (B-1) + governmen gross invesmen (federal defense, federal nondefense, sae and local) (B-20) (billions of curren dollars) δ K = Consumpion of fixed capial (B-26) (billions of curren dollars) = Gross domesic produc (B-1) (billions of curren dollars) privae governmen K+ 1 (1 δ ) K invesmen invesmen δ K K K K + K δ K , , , , , , , , K + 1 K δ K = , =

11 Calculaion of parameers: K 1 K ( gλ 1) k + = = Ak α Ak α = 38, Ak α , 409 k = = = 93,561 gλ K δ K = k 93, Ak = α 38, 409 = = , δ = = rk =, r = = Ak α 38, 409 A = = = α k 93,561 gλ gλ r δ = = = 1, β = = = β 1+ r δ

12 Summary: β = , δ = , g = , A = , α = , λ = Kaldor s sylized facs (again): 1. L 1960 / = (1.0160) 38, K L 1960 / = (1.0160) 93, r δ = K / = rk / = , wl / = g =

13 A puzzle: Ineres raes on bonds i = Corporae bond yield (Moody's Aaa) (percen per year) (B-73) π = Change in implici GNP deflaor (percen per year) (B-3) Arbirage implies ha i π i π r δ = i π = There is an equiy premium. Unil he 1980s, i was very large. See R. Mehra and E. C. Presco (1985), The Equiy Premium: A Puzzle, Journal of Moneary Economics, 15, E. R. McGraan and E. C. Presco (2000), Is he Sock Marke Overvalued? Federal Reserve Bank of Minneapolis Quarerly Review, 24(4),

14 II. Now we inerpre he daa as being observaions of a balanced growh pah, bu we use oal hours worked as he measure of labor inpu and we pu leisure ino he uiliy funcion. The uiliy funcion is now = 0 ( C + Nh L ) max β γ log (1 γ)log( ) where N is he working-age (16-64) populaion and h is he maximum number for hours available for work per person, aken o be 5200 per year (100 hours per week 52 weeks per year). There is a new firs-order condiion: 1 γ γ w γ γ g AK L Nh L C C C L 1 α α α = = (1 α)( ) = (1 α) 1 γ Nh L = (1 α). γ C L 14

15 N = Populaion (B-34) (housands of persons) L = 52 average oal privae weekly hours (B-47, spliced wih average oal manufacuring weekly hours a 1963) civilian employmen (B-35) (housands of persons) ( L is expressed in billions of hours) = Gross domesic produc (B-1) (billions of curren dollars) C = - K 1 (1 + δ ) K (billions of curren dollars) Nh L C N hours employmen L C L γ , , , , , , , , , , , , , , , , , γ =

16 We need o recalibrae g and λ : L L g λ , , , , , g = , λ= How good is he assumpion ha we are in a balanced growh pah? gl Ak α =

17 Real GDP per Hour Worked in he Unied Saes U.S. dollars daa balanced growh pah year 17

18 0.0847Ak α k = = = gλ Ak α A = = = α k The calibraion of all of he oher parameers says he same. Summary: β = , γ = , δ = , g = , A = , α = , λ =

19 III. Now we inerpre he daa as being observaions, no of a balanced growh pah, bu of a perfec foresigh equilibrium. We calculae a capial sock series using invesmen daa and he cumulaion equaion K = 1 (1 δ ) K + + I. We need o choose a value for K We do so by requiring, more or less arbirarily, ha K K = = 1960 We choose δ so ha δ K / = over he period , is average value in he daa over his period. Ieraing on guesses for K 1959 and δ, we obain K 1959 = 5,632.2 and δ =

20 Suppose insead we choose K 1959 so ha K K K = K and ha we choose δ so ha δ K / = over he period We obain K 1959 = 6,104.1 and δ = The wo series generaed for he capial socks are very similar, especially afer 10 years or so, when he values chosen for K 1959 make less and less difference. The wo series are also similar o he series for he capial sock generaed by he balanced growh pah in he previous calibraion

21 Real Capial Sock in he Unied Saes 30,000 billion 2002 U.S. dollars 25,000 20,000 15,000 10,000 capial sock #2 balanced growh pah capial sock #1 5, year 21

22 To calibrae γ, we coninue o use he firs order condiion 1 γ γ 1 w (1 )( g ) AK L (1 ) γ α α γ = α α = α = Nh L C C L C γ = CL. CL + (1 α) ( Nh L) To calibrae β, we use he firs order condiion β β C 1 1 = = β = + C ( r 1 δ ) C C C r C K ( + 1 δ ) ( α / + 1 δ) 1 1 Using daa, we esimae γ = and β = Summary: β = , γ = , δ = , α =

23 A noe on real invesmen We have cumulaed invesmen o generae a capial sock, where real invesmen is nominal invesmen divided by he implici GDP deflaor. I makes less sense, in he conex of he one-secor growh model, o cumulae a real invesmen series, say ha in Table B2, where real invesmen is nominal invesmen divided by an invesmen deflaor. If we wan o model he impac of changes in he relaive price of invesmen o consumpion (in paricular, he fall in his price) over he period , we could use a wo-secor model in which he budge consrain is ( δ ) C + q K 1 (1 ) K wl + rk + where q is he price of invesmen relaive o consumpion. Depending on he choice of he producion echnologies of he consumpion good and he invesmen good, his model can produce resuls similar o hose produced by he one-secor model ha we are sudying. In his wo-secor model, however, we would aribue some echnical progress o improvemens in echnology in he consumpion good secor and some o improvemens in he invesmen good secor. 23