Chapter 10 Skill biased technological change

Transcription

1 Chapter 10 Skill biased technological change O. Afonso, P. B. Vasconcelos Computational Economics: a concise introduction O. Afonso, P. B. Vasconcelos Computational Economics 1 / 29

2 Overview 1 Introduction 2 Economic model 3 Numerical solution 4 Computational implementation 5 Numerical results and simulation 6 Highlights 7 Main references O. Afonso, P. B. Vasconcelos Computational Economics 2 / 29

3 Introduction Skill biased Technological Change is a shift in technological knowledge production that favours high-skilled labour by increasing its relative productivity. The shift increases the relative demand of high-skilled labour that exceeds the increase in the relative supply, thus increasing the skill premium; it is considered a dynamic setting where: scale effects are removed; the capacity to learn, assimilate and implement advanced technological knowledge is different between types of labour. To better understand the mechanism a standard economic structure in endogenous R&D-growth theory is proposed based on the exposition in Afonso (2006), bearing in mind Acemoglu (2002, 2002b, 2013). By solving the transitional dynamics numerically it is shown that the recent rise of the skill premium arises from the price-channel effect, complemented with a technological knowledge absorption mechanism. MATLAB/Octave is used to solve the model making use of Runge Kutta methods (Dahlquist and Björck (2008), Süli and Mayers (2003)). O. Afonso, P. B. Vasconcelos Computational Economics 3 / 29

4 Economic model Productive side Each final good n [0, 1] is produced by one of two technologies. The L-technology uses low-skilled labour, L, complemented with a continuum of L-intermediate goods indexed by j [0, J]. The H-technology uses high-skilled labour, H, complemented with a continuum of H-intermediate goods indexed by j [J, 1]. The output of n, Y n, at time t is {[ ] J ( ) 1 α Y n (t) = A q k(j,t) x n (k, j, t) dj [(1 n)ll n ] α [ 1 J ] } ( ) 1 α q k(j,t) x n (k, j, t) dj [nhh n ] α. (1) O. Afonso, P. B. Vasconcelos Computational Economics 4 / 29

5 Economic model Productive side A is the exogenous productivity dependent on the quality of institutions. The integrals sum up the contributions of intermediate goods. The quantity of each j, x, is quality-adjusted the constant quality upgrade is q > 1, and k is the highest quality rung at time t. Expressions with exponent α (0, 1) represent the role of the labour inputs. An absolute productivity advantage of H over L is measured by h > l 1. n and (1 n) imply that H is relatively more productive in final goods indexed by higher n values, and vice versa; i.e. a threshold final good exists. O. Afonso, P. B. Vasconcelos Computational Economics 5 / 29

6 Economic model Productive side In the production of final goods, the threshold final good indicates that a shift from one technology to another is advantageous. As a result, there exists complementarity between inputs and substitutability between technologies, which is crucial for analysing the effects of inputs levels on the technological knowledge bias and thus on the wage premium. The production of quality-adjusted intermediate goods occurs under monopolistic competition. There is a mark-up for the firm producing the final quality of intermediate good j (i.e., the leader firm) can capture the entire market. It is considered that firms follow the limit price strategy. The value of the leading-edge patent depends on the profit-yields accrued by the monopolist at each time t and on the duration of the monopoly. The duration depends on the probability of successful R&D, which results in designs (prototypes) to produce a new quality of j that creatively destroys the current leading-edge design. O. Afonso, P. B. Vasconcelos Computational Economics 6 / 29

7 Economic model Productive side At the heart of the Schumpeterian R&D models is the instantaneous probability of successful innovation, pb(k, j, t), which is a Poisson arrival rate in the next quality of j, k(j, t) + 1, which complements m-type labour (m = L if 0 < j J and m = H if J < j 1): pb(k, j, t) = rs(k, j, t) }{{} (i) βq k(j,t) }{{} (ii) ς 1 q k(j,t)( 1 α ) }{{} (iii) m }{{} 1 f (j) where: }{{} (iv) (v) (i) is the flow of resources (in terms of Y ) towards R&D in j at t; (ii) is the positive learning effect arising from past R&D in j at t; (iii) is the adverse effect caused by the increasing complexity of new quality improvements in j at t; (iv) is the adverse effect induced by the market size; (v) is the absolute advantage of H over L to learn, assimilate and implement advanced technological knowledge, the specification for function f (j) being f (j) = { ( H ) σ if 0 j J; i.e. m = L if J j 1; i.e. m = H, where σ = 1 + H L. H+L O. Afonso, P. B. Vasconcelos Computational Economics 7 / 29

8 Economic model Consumption (demand) side A time-invariant number of heterogeneous individuals, a [0, 1], decide the allocation of income between consumption and savings. The infinite horizon lifetime utility of an individual with ability a is the integral of a discounted CIES utility function, U(a, t) = 0 c(a, t) 1 θ 1 e ρt dt, where: (3) 1 θ c(a, t) is the consumption (of Y ) by the individual with ability a, at t; ρ is the homogeneous subjective discount rate; θ is the inverse of the inter-temporal elasticity of substitution. Savings consists of accumulation of financial assets, with return r, in the form of ownership of the firms that produce intermediate goods. Each individual maximises lifetime utility (3), subject to budget constraint; the solution for the consumption path is the standard Euler equation ĉ(a, t) = ĉ(t) = r(t) ρ θ where: ĉ is the growth rate of c, i.e. ĉ = ċ c, and r is the interest rate. O. Afonso, P. B. Vasconcelos Computational Economics 8 / 29 (4)

9 Economic model Equilibrium for given factor levels The economic viability of the two technologies in (1) relies on the relative productivity, h l, and prices of m-type labour: the prices of labour rely on the quantities, H and L; in relative terms, the adjusted productivity of quantity H in production is hh. ll The economic viability also relies on the relative productivity and prices of the intermediate goods: productivity and prices rely on complementarity with each type of labour, H or L, on the embodied technological knowledge and on the mark up; these determinants are summarised on the aggregate quality indexes: Q L (t) J 0 q 1 α k(j,t)( α ) dj and Q H (t) 1 J q 1 α k(j,t)( α The endogenous threshold final good arises from the equilibrium in inputs and thus from the determinants of economic viability of the technologies: n(t) = [ 1 + ) dj. ( ) ] 1/2 1 QH (t)hh. (5) Q L (t)ll O. Afonso, P. B. Vasconcelos Computational Economics 9 / 29

10 Economic model Equilibrium for given factor levels The threshold final good can be related with prices, since on the threshold both an L- and H-technology firm should break even; thus, ( ) α { p H (t) n(t) p L (t) = pl = p, where: n (1 n) α = exp( α)n α 1 n(t) p H = p n n α α. (6) = exp( α)(1 n) Thus, a higher Q H Q L (technological knowledge bias measure) and/or a higher H L imply(ies) a higher fraction of final goods using H-technology (i.e., a smaller threshold final good). This, in turn, implies a low relative price of goods produced with H-technology (price channel affecting the technological knowledge bias). O. Afonso, P. B. Vasconcelos Computational Economics 10 / 29

11 Economic model R&D equilibrium The expected current value of the flow of profits to the monopolist producer j, V (k, j, t), depends on: the profits at each moment in time, Π(k, j, t); the given equilibrium interest rate, r; the expected duration of the flow, which is the expected duration of the successful researchers technological knowledge leadership V (k, j, t) = Π(k, j, t) r(t) + pb(k, j, t). Under free-entry R&D equilibrium the expected returns are equal to resources spent, pb(k, j, t)v (k, j, t) = rs(k, j, t) and the resulting equilibrium m-specific growth rate is Q m (t) = Q { ( ) } m β q 1 [ = [p m (t)a(1 α)] 1 α mf (.) r(t) q ( ] 1 α α ) 1. Q m ζ q (7) O. Afonso, P. B. Vasconcelos Computational Economics 11 / 29

12 Economic model Steady state Since the output has constant returns to scale in Q H and Q L, and macroeconomic aggregates are multiples of Q H and Q L, the constant and unique steady state endogenous growth rate is driven by an endogenous technological knowledge progress g = Q = Ŷ = X = R = Ĉ = ĉ = r ρ. θ The steady state interest rate, r, is obtained by setting the growth rate of consumption equal to the growth rate of technological knowledge. Once obtained the steady state interest rate, the steady state growth rate results from plugging r into the Euler equation (4). In particular, the steady state wage premium, w H w L, is W w H w L = ( ) 1 QH 2 hl Q L lh (8) which is also constant. O. Afonso, P. B. Vasconcelos Computational Economics 12 / 29

13 Transitional dynamics Economic model Having a higher incentive to improve intermediate goods used with a type of labour, there is a technological knowledge bias, affecting wage inequality. The bias ends when Q H and Q L grow at the same rate (steady state). Transitional dynamics, the path of the variable D = Q H Q L, is described by the differential equation, D(t) = Ḋ D = β ( q 1 ζ q { ( h 1 + H ) [ σ H + L ) [A(1 α)] 1 α exp( α) 1 + ( D(t) hh ll ) 1 ] α [ ( 2 l 1 + D(t) hh ll ) 1 ] α } 2. (9) O. Afonso, P. B. Vasconcelos Computational Economics 13 / 29

14 Transitional dynamics Economic model It is interesting to compare the baseline path with the path arising from the increase in the number of skilled workers, H. Due to the increase in H, the technological knowledge absorption effect, f (j), is greater than in the baseline case, affecting positively D. Such bias increases the supply of H-intermediate goods, thereby increasing the number of final goods produced with H-technology and lowering their relative price. The relative prices of final goods produced with H-technology drop continuously towards the steady state levels, implying that D is increasing but at a decreasing rate until it reaches its new higher steady state. O. Afonso, P. B. Vasconcelos Computational Economics 14 / 29

15 Initial value problems Numerical solution A numerical procedure to solve an initial value problem ẏ(t) = f (t, y(t)), y 0 = y(t 0 ), can be built by finding y n, n = 1, 2,..., which approximates y(t n ): y n+1 = y n + hφ(t n, y n ; h), (10) where h stands for the integration step size. This numerical approach gives rise to the class of one-step (or self-starting) method, which uses only data gathered in the current step. For Φ(t n, y n ; h) = f (t n, y n ) the Euler method is recovered. On the other hand, the multistep methods (methods with memory) express the value of y n+1 as a function of y k, k = n r + 1,, n (r-step method). Relation (10) shows an explicit method, being implicit those methods where y n+1 depends implicitly on itself through f : for the one-step case y n+1 = y n + hφ(t n, t n+1, y n, y n+1 ; h). O. Afonso, P. B. Vasconcelos Computational Economics 15 / 29

16 Numerical solution Taylor series based methods If f and its derivatives are well defined, the truncated Taylor series expansion for y(t) in t 0 can be used, y(t) = y(t 0 ) + p k=1 1 k! (t t 0) k y (k) (t 0 ) + O(h p+1 ). From f (j) = d j f and t dt j n = t 0 + nh, n Z +, p 1 y(t 1 ) y 1 = y(t 0 ) + k! hk f (k 1) (t 0, y 0 ) + O(h p+1 ). k=1 Then, the approximate value of y at time t 2, y 2 y(t 2 ), is obtained similarly. An explicit one-step method can be obtained, for a given p, to approximate the solution y n y(t n ), n = 0, 1,... the form (10) y n+1 = y n + hφ(t n, y n ; h) with Φ(t n, y n ; h) = p k=1 1 k! hk f (k 1) (t n, y n ). (11) O. Afonso, P. B. Vasconcelos Computational Economics 16 / 29

17 Numerical solution Taylor series based methods The simplest Taylor type method is obtained for p = 1, Φ(t n, y n ; h) = f (t n, y n ), and it is the (explicit) Euler method y n+1 = y n + hf (t n, y n ); the Euler method is a method of order 1 (O(h)). O. Afonso, P. B. Vasconcelos Computational Economics 17 / 29

18 Runge Kutta methods Numerical solution The Runge Kutta methods (RK) retain the desirable feature of Taylor methods, but they do not require explicit evaluations of the derivatives of f. Runge Kutta methods compute approximations y n with initial values y 0 = y(t 0 ) using the Taylor series expansion (11). Runge in 1875, based on the knowledge of y(t n ), took y(t n + h) y(t n ) + hf ( t n + h 2, y ( t n + h 2 )) and computed y ( tn + h 2 ) using the Euler method with step h 2 : y n+1 = y n + hk 2 (12) k 1 = f (t n, y n ) k 2 = f (t n + h 2, y n + h 2 k 1). This method does not need to evaluate the derivatives of f and it is more accurate than the Euler method. The explicit s-stage Runge Kutta methods generalises this idea. O. Afonso, P. B. Vasconcelos Computational Economics 18 / 29

19 Numerical solution Explicit s stage Runge Kutta methods A Runge Kutta s-stage (RKs) method is obtained by doing s function evaluations per step, giving rise to y n+1 = y n + hφ(t n, y n ; h), Φ(t n, y n ; h) = s w i k i, (13) i=1 where i 1 k i = f t n + hc i, y n + h a i,j k j, c 1 = 0, j=1 for an explicit method and k i = f t n + hc i, y n + h for an implicit one. s a i,j k j j=1 O. Afonso, P. B. Vasconcelos Computational Economics 19 / 29

20 Numerical solution Explicit two stage and 2nd order Runge Kutta methods To build a two stage and 2nd order method, s = p = 2, from (13) the result is y n+1 = y n + hw 1 k 1 + hw 2 k 2 (14) k 1 = f (t n, y n ) k 2 = f (t n + hc 2, y n + ha 2,1 k 1 ). Several solutions can be obtained specifying w 1, w 2, c 1 and a 2,1, including the following. Modified Euler method: (c 2 = 1 2, w 1 = 0, w 2 = 1 and a 2,1 = 1 2 see (12)) [ y n+1 = y n + hf t n h, y n + 1 ] 2 hf (t n, y n ), (15) Improved Euler method: (c 2 = 1, w 1 = 1 2, w 2 = 1 2 and a 2,1 = 1) y n+1 = y n h [f (t n, y n ) + f (t n + h, y n + hf (t n, y n ))]. (16) O. Afonso, P. B. Vasconcelos Computational Economics 20 / 29

21 Numerical solution Explicit four stage and 4th order Runge Kutta methods Higher order methods can be developed; one of the most frequently used methods of the Runge Kutta family is the (classical) 4th order method: 4th order Runge Kutta y n+1 = y n h (k 1 + 2k 2 + 2k 3 + k 4 ) (17) k 1 = f (t n, y n ) ( k 2 = f t n h, y n + 1 ) 2 hk 1 ( k 3 = f t n h, y n + 1 ) 2 hk 2 k 4 = f (t n + h, y n + hk 3 ). The values of k 2 and k 3 are approximations to the derivative ẏ at intermediate points, and the value of Φ(t n, y n ; h) is the weighted average of the k i, i = 1,, 4. O. Afonso, P. B. Vasconcelos Computational Economics 21 / 29

22 Computational implementation Presentation and parameters %% SBTC model % S k i l l Biased Technological Change w i t h o u t scale e f f e c t s % based on : Oscar Afonso, Applied Economics, 38: (2006) % implemented by : P. B. Vasconcelos and O. Afonso disp ( ) ; disp ( SBTC w i t h o u t scale e f f e c t s model ) ; disp ( ) ; % parameters global beta zeta q alpha l h A L H sigma beta = 1. 6 ; zeta = 4. 0 ; q = ; alpha = 0. 7 ; l = 1. 0 ; h = 1. 2 ; A = 1. 5 ; L = 1. 0 ; H = 0. 7 ; sigma = H/ L+1; O. Afonso, P. B. Vasconcelos Computational Economics 22 / 29

23 Computational implementation Solution % solve the ode y0 = 1. 0 ; tspan = [ ] ; step = ; [ t, y ] = my_rk4 tspan, y0, step ) ; % p l o t the s o l u t i o n of the ODE [ ax, h1, h2 ] = p l o t y y ( t, y, t, ( h L / ( l H). y ). ^ 0. 5 ) ; i f ~ exist ( OCTAVE_VERSION, b u i l t i n ) xlabel ( $t$, I n t e r p r e t e r, LaTex ) ; legend ( $D$, $W$ ) ; set ( legend, I n t e r p r e t e r, l a t e x ) ; else xlabel ( t ) ; legend ( D ) ; legend ( D, W ) ; end set ( get ( ax ( 1 ), Ylabel ), S t r i n g, Technological knowledge bias ) set ( get ( ax ( 2 ), Ylabel ), S t r i n g, Wage i n e q u a l i t y ) set ( h2, L i n e S t y l e, ) O. Afonso, P. B. Vasconcelos Computational Economics 23 / 29

24 Computational implementation SBTC function function dydt = ode_sbtc ( ~, y ) % ode equation f o r sbtc model global beta zeta q alpha l h A L H sigma dydt = y ( 1 )... ( beta / zeta ( ( q 1) / q ) (A (1 alpha ) ) ^ ( 1 / alpha ) exp( alpha )... ( ( 1 + (H / ( H+L ) ) ) ^sigma h ( 1 + ( ( h / l ) (H/ L ) y ( 1 ) ) ^( 0.5) ) ^ alpha... l ( 1 + ( ( h / l ) (H/ L ) y ( 1 ) ) ^ 0. 5 ) ^ alpha ) ) ; O. Afonso, P. B. Vasconcelos Computational Economics 24 / 29

25 Computational implementation RK4 method function [ t, y ] = my_rk4 ( f, tspan, y0, h ) %Runge Kutta order 4 method with tspan =[ t0 t f ] and f i x e d % step size to solvebthe IVP y = f ( t, y ), y ( t0 ) =y0 % input : % f : f u n c t i o n to i n t e g r a t e ( f =@( t, y ) ) % tspan : i n t e g r a t i o n i n t e r v a l [ t0, t f i n a l ] % y0 : i n i t i a l c o n d i t i o n at t0 % h : ( constant ) step size % output : % t : s p e c i f i c times used % y : s o l u t i o n evaluated at t t0 = tspan ( 1 ) ; t f = tspan ( 2 ) ; t = t0 : h : t f ; y = zeros ( 1, length ( t ) ) ; y ( 1 ) = y0 ; yn = y0 ; tn = t0 ; for n = 1: length ( t ) 1 k1 = f ( tn, yn ) ; k2 = f ( tn+h / 2, yn+h k1 / 2 ) ; k3 = f ( tn+h / 2, yn+h k2 / 2 ) ; k4 = f ( tn+h, yn+h k3 ) ; yn = yn+h ( k1+2 k2+2 k3+k4 ) / 6 ; y ( n+1) = yn ; tn = tn+h ; end end O. Afonso, P. B. Vasconcelos Computational Economics 25 / 29

26 Numerical results and simulation 15 D W 6 Technological knowledge bias Wage inequality t Path of D and W O. Afonso, P. B. Vasconcelos Computational Economics 26 / 29

27 Numerical results and simulation H increases (decreases) from 0.9 to 1.1 (0.7) D for H = 0.9 ˆD for H = 1.1 D for H = t Path of variable D for different values of H O. Afonso, P. B. Vasconcelos Computational Economics 27 / 29

28 Highlights In the skill biased technological change literature, labour levels affect the direction of technological knowledge, which in turn drives the wage inequality dynamics through the market size channel. In the proposed model, the direction of technological knowledge is analysed in a dynamic setting where the scale effects are removed; the chain of effects is then induced by the price channel. It is assumed that the capacity to learn, assimilate and implement advanced technological knowledge is different between types of labour. In the proposed context, the rise in the skill premium results from the fact that the price channel dominates the market size channel. This chapter continues to introduce numerical methods for the solution of initial value problems, namely, the Runge Kutta family of methods with emphasis on the RK4 method. O. Afonso, P. B. Vasconcelos Computational Economics 28 / 29

29 Main references G. Dahlquist, A. Bjöck. Numerical methods in scientific computing, volume 1. Society for Industrial Mathematics, E. Süli, D. F. Mayers. An introduction to numerical analysis Cambridge University Press, D. Acemoglu Directed technical change. Review of Economic Studies, 69 (4): , D. Acemoglu Technical change, inequality and the labour market. Journal of Economic Literature, 40(1): 7 72, 2002b. D. Acemoglu Patterns of skill premia. Review of Economic Studies, 70(2): , O. Afonso Skill biased technological knowledge without scale effects. Applied Economics, 38(1): 13 21, O. Afonso, P. B. Vasconcelos Computational Economics 29 / 29