Manuelli & Seshadri: Human Capital and the Wealth of Nations

Transcription

1 Manuelli & Seshadri: Human Capital and the Wealth of Nations Econ821 Prof. Lutz Hendricks April 19, / 32

2 Introduction We write out an organized solution to Manuelli and Seshadri (2014). Then we think about how to organize the code for this model. My version of the code is on github. Note: almost all the derivations are straight out of MS / 32

3 A few notes The model solution in the paper is correct only for the case where the schooling, child care and job training inputs are the same good i.e., β = θ, p S = p E = p w The reported results are for the case where β < θ a correction is being worked on The paper contains about a dozen typos, mostly in the proofs. The paper contains a tax rate in some places, which is set to 0 in all computations. We consider a slightly more general model. 3 / 32

4 Model Elements Small open economy (r exogenous) Steady state Demographics: OLG lifetime T N (a,t) = φ (a)e ηt mass of persons aged a in steady state: φ (a) = η e ηa 1 e ηt 4 / 32

5 Model Elements Preferences: agents maximize lifetime earnings. Endowments at birth: h B units of human capital Endowments each period: 1 unit of time can be spent on working or learning 5 / 32

6 Human capital production Phase 1: early childhood h E = h B x v E (1) Phase 2: schooling: ḣ(a) = G s (h(a),x(a)) δ h h(a) (2) starts at a = a 0, duration s (a choice) h(a 0 ) = h E 6 / 32

7 Human capital production Phase 3: job training: ḣ(a) = G w (h(a),n(a),x(a)) δ h h(a) (3) n < 1 starts at a = a 0 + s with h(a 0 + s) from schooling problem 7 / 32

8 Output Production Consumption good: output per worker y c = zf ( k c, h c ) (4) F (κ c,1) = ( k c / h c ) θ (5) h c : human capital per worker devoted to that sector no resource constraint (open economy) numeraire also the x input during the job training phase (p w = 1) Human capital good: the same technology: y s = F h ( ) k s, h s = xs + x e used to produce x s and x e 8 / 32

9 Aggregation Define an aggregation function M (x,a 1,a 2 ) = a2 a 1 φ (a)x(a)da (6) Mass working: R φ = M (1,a 0 + s,r) = φ (a)da (7) 6+s Human capital per worker (labor supply in efficiency units): h = M (h(1 n),a 0 + s,r) φ (8) Labor market clearing: h = h s + h c (9) 9 / 32

10 Aggregation Aggregate goods used to make x s and x e : x s = M (x s,a 0,a 0 + s) (10) x e = φ (6)x E (11) 10 / 32

11 Factor Prices Consumption goods sector: p k (r + δ k ) = zf k (κ c,1) = zθκ θ 1 c (12) w = zf h (κ c,1) = z(1 θ)κ θ c (13) Human capital sector: p k (r + δ k ) = p s F h k (κ s,1) = zβκ β 1 s (14) w = p s F h h (κ s,1) = z(1 β)κ β s (15) Key: These determine all factor and goods prices, given the exogenous r and z. 11 / 32

12 Equilibrium A steady state is defined by these objects: household: x(a),n(a),h(a), x E,h E aggregates: y,y s, x s, x e, h, h s, h c prices: p k,p s,p e,p w These satisfy: household optimization firm first-order conditions definition of h, x s, x e y s = F h ( k s, h s ) = xs + x e h = h c + h s There is no market clearing condition for final goods (small open economy). 12 / 32

13 Household Problem The household solves max p e x e + W (h e ) + e rs V (h s,r s a 0 ) (16) s,x(a),n(a),h(a),x e,h e subject to h(a 0 ) = h E = h B xe v ḣ(a) = G s (h(a),x(a)) δ h h(a); a a 0 + s ḣ(a) = G w (h(a),n(a),x(a)) δ h h(a); a > a 0 + s W (h e ) = s 0 p sx s (t)dt V (h s,t) = T 0 e ra {wh(a)(1 n(a)) p w x(a)}da n(a) 1 13 / 32

14 How to solve this? Backward induction. 1. Solve the job training problem (with n interior). 2. Solve the schooling problem with job training as continuation value. 3. Solve the childcare problem with schooling as continuation value. Computationally, it turns out to be more efficient to solve schooling and child care simultaneously. 14 / 32

15 Job training problem Write this as starting at a = 0 (shifting the age range) T V (h(0),t) = max e ra {wh(a)(1 n(a)) p w x(a)}da (17) 0 ḣ(a) = G w (h(a),n(a),x(a)) δ h h(a) (18) = z h (h(a)n(a)) γ 1 x(a) γ 2 δ h h(a) (19) h(0) given (20) 15 / 32

16 Hamiltonian Γ = wh(1 n) p w x + q[g w (h,n,x) δ h h] (21) FOCs: whn = qγ 1 G w (22) p w x = qγ 2 G w (23) q = rq q{γ 1 G w /h δ h } w(1 n) (24) q T = 0 (25) Implied static condition p w x = whnγ 2 /γ 1 (26) 16 / 32

17 Solution Plug first-order conditions into the law of motion for q: q = (r + δ h )q w (27) Integrate: Let then T q(a) = e +(r+δh)a w e (r+δh)t dt (28) a m(t) = 1 exp((r + δ h )t) (29) q(t) = (1 τ)w m(t) r + δ h (30) Both are functions of the time remaining t. 17 / 32

18 Solution for nh From static FOC: ( ) G w γ2 w γ2 = z h (hn) γ (31) γ 1 p w Sub into FOC for n to obtain n(a)h(a) = [Qm(T a)] 1/(1 γ) (32) Then x(a) follows from the static condition. An interesting feature: job training investment is purely forward looking. It does not depend on current h. 18 / 32

19 Solution for h(a) Plug q(a) and focs into ḣ equation and integrate: with and a h(a) = e δha h(0) + Ce δ ha e δht m(t t) γ/(1 γ) dt (33) 0 ( ) C = z h Q γ/(1 γ) γ2 w γ2 (34) γ 1 p w Q = z hγ 1 γ 2 1 γ γ ( ) 2 2 w γ2 (35) r + δ h p w 19 / 32

20 Value function Integration of wh(1 n) p w x yields: V (h,t) = q(t)h + w 1 γ T Q 1/(1 γ) e rt m(t t) 1/(1 γ) (36) γ / 32

21 Summary: Job training phase Given: s,h s Closed form solutions for the entire problem. Marginal value of human capital at the start: q(t a 0 s), also known. So it is logical to make the solution to this into a general purpose function. See BenPorathContTimeLH class. 21 / 32

22 Schooling The household solves max p e x e + s,x(a),n(a),h(a),x e,h e subject to s h(a 0 ) = h E = h B xe v ḣ(a) = G s (h(a),x(a)) δ h h(a); a a 0 + s h s = h(a 0 + s) Terminal value function from job training. 0 p s x s (t)dt+e rs V (h s,r s a 0 ) (37) 22 / 32

23 Hamiltonian Γ = p s x s + q(g s (h,x s ) δ h h) (38) First order conditions (essentially the same as job training without the condition for n): p s + q G s / x s = 0 (39) q = (r + δ h )q q G s / h (40) q(s) = V/ h (41) In addition: x e solves V 0 = max x e q e h B x v e p e x e (42) where q e = q(a 0 ) = W/ h from schooling problem. 23 / 32

24 Special case in the paper Schooling and job training have the same technologies and input prices. Then the terminal conditions become: 1. The terminal value of h is given by the job training problem: q(a 0 + s) = (1 τ)w m(r a 0 s) r + δ h (43) 2. When job training starts, the agent chooses n = 1 Take the FOC for nh at a = a 0 + s, set n = 1, and you get h(a 0 + s) = [Qm(R a 0 s)] 1/(1 γ) (44) 24 / 32

25 Schooling: Solution (40) implies ( ) x(a) = (z h β 2 /p s ) 1/(1 β 2) q(a)h(a) β 1/(1 β2 ) 1 (45) Therefore g(x s ) = g(q) + β 1g(h) 1 β 2 (46) From laws of motion g(h) = G s /h δ h (47) q = (r + δ h )q β 1 G s /h (48) 25 / 32

26 Schooling: Solution Therefore, x grows at a constant rate: Therefore, g(x s ) = g(q) + β 1g(h) 1 β 2 = r + δ h (1 β 1 ) 1 β 2 (49) q(s)h β 1 s = q E h β 1 E eg(x s)(1 β 2 )s (50) 26 / 32

27 Schooling: Solution Solution is now: x e,h E,q E,h s,s,q s that solve: 1. foc for x E and production function for h E : h E = [h B (vq E /p E ) v ] 1/(1 v) (51) 2. constant growth of qh β 1: (50) 3. foc for x s : (40) 4. equation for q(a 0 + s) from OJT All we need now is a first-order condition for s 27 / 32

28 Optimal s This is a problem with free terminal time (see Leonard and Van Long (1992) p. 245). A necessary condition is or e rs Γ(s) + d ds e rs V (h s,t a 0 s) = 0 (52) p s x s (s)+q(s)(g s (h(s),x s (s)) δ h h(s)) rv (h(s),s)+ V/ s = 0 (53) Intuition... After simplification rv + V/ s = wh [ m (R a 0 s) rm(r a 0 s) ] w 1 γ Q 1/(1 γ) m r + δ h γ 1 (54) 28 / 32

29 Schooling: Summary We have a closed form solution the entire schooling problem easy to solve numerically Key feature: we do not need to solve the job training problem in order to solve the schooling problem all we need to know is the function for q(a 0 + s): (43) 29 / 32

30 Equilibrium Equilibrium prices do not depend on household decisions see equations (9) and following this is because of the small open economy assumption r determines the k/h ratio in production and thus the wage Given prices, we know how to solve the schooling / child care problem this yields h E,q E,x E,s,h(a 0 + s) Given s,h(a 0 + s), we know how to solve the job training problem Now we just need to aggregate to get the equilibrium no need to worry about market clearing b/c of the small open economy 30 / 32

31 What s next Time to think about how to organize the code / 32

32 References I Leonard, D. and N. Van Long (1992): Optimal control theory and static optimization in economics, Cambridge University Press. Manuelli, R. E. and A. Seshadri (2014): Human Capital and the Wealth of Nations, The American Economic Review, 104, / 32