( ) (, ) F K L = F, Y K N N N N. 8. Economic growth 8.1. Production function: Capital as production factor

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1 8. Economic growh 8.. Producion funcion: Capial as producion facor Y = α N Y (, ) = F K N Diminishing marginal produciviy of capial and labor: (, ) F K L F K 2 ( K, L) K 2 (, ) F K L F L 2 ( K, L) L 2 > 0 < 0 > 0 < 0 Consan reurn o scale: xy (, ) = F xk xn Y K N = F, N N N y = f k where oupu per worker is denoed by Y y =, and capial per worker is N denoed by K k =, he laer of which is also called a labor equipmen raio. N

2 2 Profi-maximizaion: KL, ( δ ) max Π= F K, L r + K wl (, ) F K L K (, ) F K L L = r + δ = w ( δ) maxπ = f k r+ k w k f ( k) = r+ δ or r = f k δ

3 3 Cobb-Douglas producion funcion: Y = AK α N y = Ak α α where A is called oal facor produciviy. An increase in A implies he improvemen of produciviy or echnological progress. r + δ = αak L α α capial income share: r + δ Y K = α Noe ha depreciaion coss are included in capial income. wl labor income share: = α Y r + δ = α Ak α capial income share: r + δ y k = α

4 4 Economic growh as a consequence of capial accumulaion Economic growh as a consequence of echnological progress

5 A case wihou any echnological progress i = k + δ k + i : gross fixed invesmen k = k k : ne fixed invesmen + + y = c + i ( δ ) = c + k + k + k = k k = f k c δ k + + c = cf k s = y c ( c) f ( k ) sf ( k ) = = k = sf k δ k +

6 6 a seady sae: δ 0 k = sf k k = sak α = δ k * sa k = δ α

7 Golden rule A fundamenal quesion: An increase in capial unambiguously leads o an increase in oupu. Is i a case for consumpion? A a seady sae or k = 0, consumpion is maximized under he golden rule. max k c= f k δ k f ( k) = δ r = f ( k) δ = 0 αak α = δ α A k = δ α Compare beween α A k = δ α and α A k = δ α

8 8 The aached excel file includes daa on nominal GDP, and he shares by expendiures (household consumpion, governmen consumpion, fixed invesmen, and ne expor) for 35 counries ( in Asia, 3 in Norh America, 2 in Souh America, 6 in Europe, 2 in Africa, and in Oceania). The sample period is beween 2009 and 20. Now, we define saving raes as follows: No min al GDP-Nominal Household Consumpion-Nominal Governmen Consumpion No min al GDP Compue he hree-year average of he above-defined saving rae for each counry, and discuss he sae of capial accumulaion in erms of dynamic efficiency under he assumpion α = 0.4. Noe: Rigorously, a saving rae should be compued from real erms. Given limied daa availabiliy, nominal daa are employed as a firs approximaion purpose.

9 From 2009 o 20 9

10 A case wih echnological progress and populaion growh ( ) ( ) N = + g N = + g N N N 0 ( ) ( ) A = + g A = + g A A A 0 Oupu per effecive worker, and capial per effecive worker y k Y = AN K = AN k + k k ln k + ln k K A N K A N Y C δ K g = A K g N K Y C δ K k = g + g k ( δ ) ( δ ) + A N AN K = y c + g + g k A N = f k c + g + g k A N Golden rule: δ A N f k = + g + g r = f k δ = g + g A N

11 8.5. Growh accoun Y = AK N α ( α) lny = ln A + αln K + N Y Y + lny + lny ( ln A ln A ) α( ln K ln K ) ( α)( ln N ln N ) A K N + α + ( α) A K N + + = + + A Y K N = α ( α) A Y K N y A k = + α y A k + + A y k = α A y k + +

12 2 Alwyn Young repors he following growh accoun resuls of he four Asian counries in his aricle eniled "The yranny of numbers: Confroning he saisical realiies of he Eas Asian growh experience" in The Quarerly Journal of Economics, 995. Commen on his resuls. Annual GDP growh ( Y ) Y Annual TFP growh ( A ) A Hong Kong (966-99) 7.3% 2.3% Singapore ( ) 8.7% 0.2% Korea ( ) 0.3%.7% Taiwan ( ) 9.4% 2.6%

13 Issues on echnological progress Produciviy and oupu Produciviy and unemploymen Technological progress and unemploymen

14 Revisi of consumpion and invesmen by Ramsey model Ramsey model Le us sar wih a simple economy wih y = c+ i in per-worker erms. In he Keynesian framework, given an increase in fixed invesmen, consumpion and oupu increases ogeher hanks o he muliplier effecs. This muliplier effec is so impressive among sudens of inroducory macroeconomics ha for such sudens, capial accumulaion by inensive invesmen is always good news. As he argumen of he golden rule implies, however, aggressive capial accumulaion resuls in an increase in oupu, bu no necessarily in an increase in consumpion. Capial accumulaion may enhance oupu a he sacrifice of consumpion. According o Solow s growh model wih he Cobb- Douglas producion funcion, if he saving rae s is beyond he capial income share α, hen he seady-sae level of capial exceeds he goldenrule capial; ha is, an economy wih a high saving rae may resul in oo much capial wih oo lile consumpion. While Solow s model delivers several imporan implicaions for capial accumulaion, is serious problem is he assumpion ha he saving rae is fixed over ime; he oal oupu is always divided beween consumpion and saving/invesmen a a fixed proporion. In Solow s growh model, wheher he seady sae capial exceeds he golden-rule level depends on how he saving rae is picked up. Hence, we canno discuss how capial is accumulaed efficienly or how over-accumulaion emerges. Then, le us explore how oupu is divided beween consumpion and invesmen over ime using Ramsey model. Pick up he problem of he consumpion allocaion beween ime and + as follows: max u c c, c+ + ρ + u( c ) + subjec o c s w ( r )( w c ) + = + +. A ime preference is inroduced ha ime + uiliy may be compared wih ime uiliy as of ime. From a se of he firs-order condiions for he above uiliy maximizaion problem, we obain + r u ( c) = u ( c+ ) + ρ

15 5 Employing he mahemaical ools wih a paricular form of uiliy funcion u c σ c =, we can derive he following approximaed relaionship. σ c+ c+ c = = σ ( r ρ) c c The above equaion is ofen called he Euler equaion. The inroducion of he Euler equaion immediaely makes he capial accumulaion model exremely rich. Insead of replacing consumpion by c ( s) f ( k ) make consumpion explici. k = k k = f k c δ k + + =, we sill From he profi maximizaion in compeiive markes, we have r = f k δ. Suppose below ha a echnological progress is absen. A he golden rule, g g f k = δ. Thus, if k < k, hen r is posiive, oherwise i is negaive. k = 0 locus: c= f k δ k c = 0 locus: mg f k = ρ+ δ Noe ha ( mg g f k ) f ( k ) mg g > or k < k.

16 Asse pricing ρ= ΔS + +r S = ΔS + +f (k S )-δ S = S -+ [f (k -+ )-δ] S + + lim (+ρ) (+ρ) = When k s = k, S + lim = 0 ( + ρ ) S = = S -+ [f (k -+ )-δ] (+ρ) When k g > k, S + lim ( + ρ ) = ΔS + S =ρ+[δ-f (k )]>ρ S > = S -+ [f (k -+ )-δ] (+ρ)

17 Consumpion a+=a(+r)+w-c a = a + c w r + r a=a0(+r0)+w0-c0 a2=a(+r)+w-c a3=a2(+r2)+w2-c2, a 0 = a -w 0 +c 0 +r 0 a = a 2-w +c +r a 2 = a 3-w 2 +c 2 +r 2 a 0 + =0 w (+r i ) i=0 = =0 c (+r i ) i=0 + lim aa (+r i ) i=0 (6-4) lim a / +r i < 0 i=0 lim a / (+r i ) 0 i=0 a 0 + =0 w (+r i ) i=0 = =0 c (+r i ) i=0

18 8 c c c c c ( r ρ) + + = = σ wih r c = c 0 = ρ, =0 w a 0 + =c 0 (+ρ) + (+ρ) + =0 c 0 =ρ a 0 + =0 w (+ρ) +

19 Consumpion and ne invesmen y = y δk = f k δk (6-23) NDP dy dk NDT = f ( k ) δ NDP = ( δ )( ) ( k ) y y f k k k NDP + + = r k + y = k k + c NDP + y + y = r y c NDP NDP NDP y rc + y NDP NDP + = + r (6-24-) y y r c + y NDP NDP = + r + r c + y NDP NDP = + r + 2 (6-24-2) (6-24-3) y rc NDP NDP = + lim = Π ( + r ) ( + r ) i i= y y NDP = rc + (6-25) = Π ( + ri ) i=

20 20 k k = + y c NDP + i= rc = Π ( + r ) i c = k k + c r r = c = c ( r) r = r c c k k = c c + (6-26) Table : Ne fixed invesmen and fuure consumpion (i) Ten-year average of ne fixed invesmen/real household consumpion (%, 200 o 203 for 200s) (ii) Ten-year average of real household consumpion (billion yen, 200 o 203 for 200s) (iii) Consumpion growh rae from he curren decade o he nex decade (%, 200 o 203 for 200s) 980s 990s 2000s 200 o % 7.9% 4.% -0.5% 206, , , , % 2.7% 5.%???