The general Solow model

Transcription

1 The general Solow model Back o a closed economy In he basic Solow model: no growh in GDP per worker in seady sae This conradics he empirics for he Wesern world (sylized fac #5) In he general Solow model: Toal facor produciviy, B, is assumed o grow a a consan, exogenous rae (he only change) This implies a seady sae wih balanced growh and a consan, posiive growh rae of GDP per worker The source of long run growh in GDP per worker in his model is exogenous echnological growh No deep, bu: i s no rivial ha he resul is balanced growh in seady sae, reassuring for applicaions ha he model is in accordance wih a fundamenal empirical regulariy Our focus is sill: wha creaes economic progress and prosperiy

2 The micro world of he Solow model is he same as in he basic Solow model, eg: The same objec (a closed economy) The same goods and markes Once again, markes are compeiive wih real prices of, r and w, respecively There is one ype of oupu (one secor model) The same agens: consumers and firms (and governmen), essenially wih he same behaviour, specifically: one represenaive profi maximising firm d d decides and given r and w K L One difference: he producion funcion There is a possibiliy of echnological progress: Y = B K L, 0 < < The full sequence B is exogenous and B > 0 for all Special case is B = B (basic Solow model)

3 The producion funcion wih echnological progress Y wih a given sequence, = B K L B / ( ) Y wih a given sequence, ( A ), A B = K A L Wih a Cobb-Douglas producion funcion i makes no difference wheher we describe echnical progress by a sequence, ( B ), for TFP or by he corresponding sequence,, for labour augmening produciviy A In our case he laer is he mos convenien The exogenous sequence,, is given by: A A = + + g A, g > 0 A = + g A, g > Technical progress comes as manna from heaven (i requires no inpu of producion)

4 Remember he definiions: y Y / L and k K / L Dividing by on boh sides of Y = K A L gives he per capia producion funcion: L From his follows: = y k A ( ) lny lny = lnk lnk + lna lna ( ) ( ) g = g + g g + g y k A k y Growh in y can come from exacly wo sources, and is k he weighed average of g and g wih weighs and y If, as in balanced growh, k / y is consan, hen g = g! g

5 The complee Solow model L = + + n L, L given 0 Parameers:,s, δ,n,g Le g > 0 Sae variables: K,L and A = Y K A L Full model? Yes, given K 0,L0 and A0 he model deermines he full sequences K, L, A, Y, r, w, S K r = K ( A L ) = A L K w = ( ) K L A = ( ) A A L S = sy K given K = S δ + K, K0 A = + + g A, A0 given

6 Noe: = = ( ) ( ) r K K A L Y w L = K A L = Y Tha is: capial s share =, labour s share =, pure profis = 0 Our should sill be around / 3 Also noe: defining effecive labour inpu as Lɶ = A L: ( ɶ ) L ɶ = + + n + g L ɶ + n L ɶ, The model is maemaically equivalen o he basic Solow model wih L ɶ aking he place of L, and nɶ aking he place of n, and wih B =! We could, in principle, ake over he full analysis from he basic Solow model, bu we will neverheless be

7 Analyzing he general Solow model If he model implies convergence o a seady sae wih balanced growh, hen in seady sae k and y mus grow a he same consan rae (recall again ha k / y is consan under balanced growh) Remember also: ( ) g = g + g y k A y k y k A Hence if g = g, hen g = g = g If here is convergence owards a seady sae wih balanced growh, hen in his seady sae k and y will boh grow a he same rae as A, and hence k / A and y / A will be consan Furhermore: from he above menioned equivalence o he basic Solow model, K / Lɶ = K / ( A L ) = k / A and Y / Lɶ = Y / ( A L ) = y / A converge owards consan seady sae values Each of he above observaions suggess analyzing he model in erms of:

8 k K kɶ = and A A L ɶ y Y y = A A L = ɶ 2 From Y K A L we ge y = k ɶ 3 From K K S δ K and S = sy we ge = + K = sy + ( δ + ) K 4 Dividing by A on boh sides gives L = + g n A L k ɶ = sy + ( δ ) k ɶ + ( + n)( + g ) ( ɶ ) 5 Insering yɶ = k ɶ gives he ransiion equaion: k ɶ = ( ) + sk ɶ k + n + g + δ ɶ 6 Subracing k ɶ from boh sides gives he Solow equaion: k ɶ ( ) k ɶ = sk ɶ + n g δ ng k + n + g ɶ

9 Convergence o seady sae: he ransiion diagram The ransiion equaion is: k ɶ = ( ) sk ɶ + k + n + g + δ ɶ I is everywhere increasing and passes hrough (0,0) The slope of he ransiion curve a any k ɶ is: dkɶ dkɶ ( δ ) ( + n)( + g ) s k + + = ɶ We observe: lim dk ɶ Furhermore, 0 / dk ɶ kɶ + = lim dk ɶ / dk ɶ n g δ ng 0 We assume ha kɶ + < > he laer very plausible sabiliy condiion is fulfilled

10 The ransiion equaion mus hen look as follows:

11 Convergence of o he inersecion poin k ɶ k ɶ follows from he diagram Correspondingly: y ɶ y ɶ = k ɶ Some firs conclusions are: In he long run, kɶ k / A and yɶ = y / A converge o consan levels, k ɶ and yɶ, respecively These levels define seady sae In seady sae, k and y boh grow a he same rae as A, ha is, a he rae g and he capial oupu raio, K / Y = k / y, mus be consan

12 The Solow equaion Seady sae k ɶ k ɶ = + sk ɶ ( n + g + δ + ng ) k ɶ ( + n)( + g ) ogeher wih k ɶ = + k ɶ = k ɶ gives: kɶ ( ) s = Using kɶ k / A and yɶ y / A we ge he seady sae growh pahs: s = k A, s y ɶ = s = y A

13 Since c = s y, s = ( ) c A s I also easily follows from r k = ɶ and w ( ) = A kɶ ha r = s and s = ( ) w A There is balanced growh in seady sae: k, y and grow a he same consan rae, g, and is consan There is posiive growh in GDP per capia in seady sae (provided ha g > 0 ) r w

14 Srucural policies for seady sae Oupu per capia and consumpion per capia in seady sae are: ( s y ) and = A0 + g ( ) ( s c = A0 + g s ) Golden rule: he s, ha maximises he enire pah, c Again: s = The elasiciies of y wr s and n g δ are again and / a, respecively + + / ( a) Policy implicaions from seady sae are as in he basic Solow model: encourage savings and conrol populaion growh Bu we have a new parameer, g ( A corresponds o B 0 ) We wan a large g, bu i is no easy o derive policy conclusions wr echnology enhancemen from our model ( g is exogenous)

15 Empirics for seady sae s y A n g ng = + + δ + y = A + s n g δ ng ln ln ln ln Assume ha all counries are in seady sae in 2000! I s hard o ge good daa for A, so make he heroic assumpion ha is he same for all counries in 2000 A Se (plausibly) g + δ i If y00 is GDP per worker in 2000 of counry i, he above equaion suggess he following regression equaion: i i i lny00 = γ 0 + γ lns ln n 0 075, i i wih s and n measured appropriaely (here as averages over ), and where γ = /

16 An OLS esimaion across 86 counries gives: lny = lns ln n 0 075, adj R = 0 55 i i i 00 2 ( se = 0 4)

17 High significance! Large R 2! Even hough we have assumed ha is he same in all counries! A 00 Bu always remember he problem of correlaion vs causaliy Furhermore: he esimaed value of γ is no in accordance wih he heoreical (model-prediced) value of / 2 Or: = = The conclusion is mixed: he figure on he previous slide is impressive, bu he figure s line is much seeper han he model suggess