Lecture'Notes'#4'The'Solow'Development'Model'ECON'452'–'Fall'2016'The'LaDn'American'Economies'Instructor:'Fabrício'd’Almeida'The'Solow'Model'• How'to'evaluate'economic'growth'among'countries?'CrossOsecDonal'and'Dme'• Baseline'model:''Y'='A'Ka''Lb'• Y'='economic'output'(GDP)'• K'='capital''• L'='labor'force'(#'workers)'• A'='technology'(broad'sense)''The'Solow'Model'• Why'a'mulDplicaDve'funcDon?''– CobbODouglas'equaDon'– What'is'the'problem'with'an'addiDve'funcDon?'• Returns'to'scale:'Y'='A'Ka''Lb'a'+'b'<'1:'decreasing'returns'a'+'b'='1:'constant'returns'a'+'b'>'1:'increasing'returns'The'Solow'Model'Y'='A'Kα''L1Oα'E[0.5 Beef +0.5 Wool]=0.5 E[Beef ]+0.5 E[Wool]=0.5 ⇤ 10% + 0.5 ⇤ 2%= 6%Var[0.5 Beef +0.5 Wool]=(0.5)2Var[Beef ] + 2(0.5)(0.5) Cov[Beef, Wool]+(0.5)2Var[Wool]=0.25 ⇤ 0.004 0.5 ⇤ 0.002 + 0.25 ⇤ 0.0016=0.0004Cov[Beef,Wool]=0. 2( 0. 20 0.10)( 0. 00 0.02) + 0.6(0 .10 0.10) (0. 00 0.02) + 0.2( 0. 10 0.02)= 0.002(3)1 Lecture 4: Solow ModelYL= AK↵L1↵L LL!YL= AK↵L1↵L= AK↵L↵= A✓KL◆↵y = Ak↵(4)DFCi,t=8<:1,0,2• How'to'compare'countries'in'a'per'capita'basis'instead?!'Why'to'do'so?''y'k'The'Solow'Model'• Output'per'capita'is'composed'of'consumpDon'per'capita''and'investment'per'capita:'• Investment'per'capita'is'defined'as'the'savings'rate'Dmes'the'output'per'capita:''• The'change'on'capital'per'capita'is'investment'(p.c.)'minus'the'depleDon'of'capital'from'the'depreciaDon'rate'and'the'rate'of'growth'of'populaDon:'y = c + ii = syk = i ( + n)k= sy ( + n)k= sAk↵ ( + n)k(5)DFCi,t=8<:1,0,3y = c + ii = syk = i ( + n)k= sy ( + n)k= sAk↵ ( + n) k(5)DFCi,t=8<:1,0,3y = c + ii = syk = i ( + n) k= sy ( + n)k= sAk↵ ( + n) k(5)DFCi,t=8<:1,0,3• Therefore:''The'Solow'Model'y = c + ii = syk = i ( + n)k= sy ( + n)k= sAk↵ ( + n)k(5)DFCi,t=8<:1,0,3y'k'Source'of'Capital''DepleDon'of'Capital''y = c + ii = syk = i ( + n)k= sy ( + n)k= sAk↵ ( + n)ki = sAk↵( + n)k(5)DFCi,t=8<:1,0,3y = c + ii = syk = i ( + n)k= sy ( + n)k= sAk↵ ( + n)ki = sAk↵( + n)k(5)DFCi,t=8<:1,0,3E[0.5 Beef +0.5 Wool]=0.5 E[Beef ]+0.5 E[Wool]=0.5 ⇤ 10% + 0.5 ⇤ 2%= 6%Var[0.5 Beef +0.5 Wool]=(0.5)2Var[Beef ] + 2(0.5)(0.5) Cov[Beef,Wool]+(0.5)2Var[Wool]=0.25 ⇤ 0.004 0.5 ⇤ 0.002 + 0.25 ⇤ 0.0016=0.0004Cov[Beef, Wool]=0.2(0.20 0.10)(0.00 0.02) + 0.6(0.10 0.10)(0.00 0.02) + 0.2(0.10 0.02)= 0.002(3)1 Lecture 4: Solow ModelYL= AK↵L1↵L LL!YL= AK↵L1↵L= AK↵L↵= A✓KL◆↵y = Ak↵(4)2k*'>''>''>''>''>'<'''<''''<'• Marginal'producDvity'of'capital'(MPk):'decreasing'in'k'• MoDon'of'capital:'– 'Δk'>'0':'growth'in'GDP'per'capita'– 'Δk'='0':'equilibrium'(steady'state:'k*)'– 'Δk'<'0':'GDP'per'capital'decreases''• How'the'economy'changes'from'A,'s,'n'and'δ'?''– Play'with'the'graph'– Calculate'the'steady'state'''The'Solow'Model'• The'steady'state:'• Therefore:'The'Solow'Model'y = c + ii = syk = i ( + n)k= sy ( + n)k= sAk↵ ( + n)ki = sAk↵( + n)k(5)k =00=i ( + n)k0=sAk↵ ( + n)k( + n)k =sAk↵k1↵=sA + nk⇤=✓sA + n◆11↵(6)DFCi,t=8<:1,0,3y = c + ii = syk = i ( + n)k= sy ( + n)k= sAk↵ ( + n)ki = sAk↵( + n)k(5)k =00=i ( + n)k0=sAk↵ ( + n)k( + n)k =s Ak↵k1↵=sA + nk⇤=✓sA + n◆11↵(6)DFCi,t=8<:1,0,3• How'about'the'GDP'per'capita'in'the'equilibriu m? '• (1)'• (2)'• So,'(2)'in'(1)'…'The'Solow'Model'y⇤= A(k⇤)↵0=i ( + n)k0=sAk↵ ( + n)k( + n)k =sAk↵k1↵=sA + nk⇤=✓sA + n◆11↵(7)DFCi,t=8<:1,0,4y = c + ii = syk = i ( + n)k= sy ( + n)k= sAk↵ ( + n)ki = sAk↵( + n)k(5)k =00=i ( + n)k0=sAk↵ ( + n)k( + n)k =sAk↵k1↵=sA + nk⇤=✓sA + n◆11↵(6)3y = c + ii = syk = i ( + n)k= sy ( + n)k= sAk↵ ( + n)ki = sAk↵( + n) k(5)k =00=i ( + n)k0=sAk↵ ( + n)k( + n) k =s Ak↵k1↵=sA + nk⇤=✓sA + n◆11↵(6)y⇤= A(k⇤)↵y⇤=A✓sA + n◆↵1↵(7)DFCi,t=8<:1,0,3• From'the'equilibrium'point:''O GDP'per'capita'increases'on:'O Technology:'A'O S avings'rate:'s'O GDP'per'capita'decreases'on:'O DepreciaDon 'rate:'δ''O Pop .'Growth'rate:'n'The'Solow'Model'y = c + ii = syk = i ( + n)k= sy ( + n)k= sAk↵ ( + n)ki = sAk↵( + n) k(5)k =00=i ( + n)k0=sAk↵ ( + n)k( + n) k =s Ak↵k1↵=sA + nk⇤=✓sA + n◆11↵(6)y⇤= A(k⇤)↵y⇤=A✓sA + n◆↵1↵(7)DFCi,t=8<:1,0,3• Empirical'evidence?'• TFP'• QuesDons:'1. How'LaDn'America'compare'to'the'U.S.'or'East'Asia'in'each'of'the'predicDons'from'the'Solow'model?'2. Which'of'the'parameter(s)'from'the'Solow'model'have'changed'the'most'for'Lat.'Am.'in'the'previous'century?'3. Explain'the'difference'between'converging'to'the'steady'state'and'changing'the'steady'state'4. Give'example'of'countries'today'or'in'the'past'with'high'divergence'between'y'and'y*?'Relate'to'their'economic'growth'rate'5.
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