Recitation #4: Solving Endogenous Growth Models

Transcription

1 Recitation #4: Solving Endogenous Growth Models Ludwig Straub MIT December 2016 Ludwig Straub (MIT) December / 42

2 Logistics Problem Set 4 due Monday at noon Final recitation next week Ludwig Straub (MIT) December / 42

3 Outline for today A roadmap for solving endogenous growth models Knowledge spillovers Case where φ = 1 and n = 0 Case where φ < 1 and n > 0 Schumpeterian growth model Final remarks More examples Simple NGM with population growth Lab Equipment model Problem Set 4, Question 4 Ludwig Straub (MIT) December / 42

4 A roadmap for solving endogenous growth models Section 1 A roadmap for solving endogenous growth models Ludwig Straub (MIT) December / 42

5 A roadmap for solving endogenous growth models Five steps Identify the state variable(s) X of the model. Solve the production side of the model conditional on the state variables, possibly up to an undetermined variable f. Write down the four equations. BGP: Assume constant growth rates & solve the four equations for g and r. Trans. dynamics: Exist if const. growth rate solution only holds if a condition on X s holds t Ludwig Straub (MIT) December / 42

6 A roadmap for solving endogenous growth models The four equations Law of motion of the state variable(s) Ẋ = H(X, f, C ) Value(s) of innovation rv = π(x, f) + V Free entry equation(s) Euler equation G (X, V ) = 0 1 C = C (r ρ) θ Ludwig Straub (MIT) December / 42

7 A roadmap for solving endogenous growth models Side remark on fourth step When solving for BGP, often useful to split equations into two blocks: First three equations: Demand for funds r = r d (g ) given r, how much g will be generated? Euler equation: Supply of funds r = r s (g ) = ρ + θg given g, how much r do households charge? Ludwig Straub (MIT) December / 42

8 Knowledge spillovers Section 2 Knowledge spillovers Ludwig Straub (MIT) December / 42

9 Knowledge spillovers Model Output 1 N 1 β β ν Y = x dν L 1 β E 0 Intermediate goods produced at marg cost ψ = 1 β Labor L is either employed (L E ) or does research (L R ), and grows at rate n Competitive production of ideas N = ηn φ L R with L R paid marginal product (if L R > 0) w = ηn φ V Ludwig Straub (MIT) December / 42

10 Knowledge spillovers First step State variables are L and N. Given those, solve the model as much as possible second step. Ludwig Straub (MIT) December / 42

11 Knowledge spillovers Second step: Intermediates Demand for intermediate goods Y β β 1/β E p ν = = x ν L x ν = L E p ν x ν Thus and profits are β 1 p ν = ψ 1 x ν = L E β 1 1 ' -v " 1 1 β 1 π = 1 ψx ν = βl E 1 β Ludwig Straub (MIT) December / 42

12 Knowledge spillovers Second step: Aggregating This gives output and wages 1 Y = NL E 1 β Y β w = = N L E 1 β This determines the economy given state variables N, L up to the fraction of employed workers f L E /L Ludwig Straub (MIT) December / 42

13 Knowledge spillovers Third step: Four equations Law of motion of the state variables N = ηn φ (1 f)l L = nl Value of innovation Free entry equation rv = βfl + V β N = w = ηn φ V 1 β Euler equation 1 C = C (r ρ) θ Ludwig Straub (MIT) December / 42

14 Knowledge spillovers Case where φ = 1 and n = 0 Subsection 1 Case where φ = 1 and n = 0 Ludwig Straub (MIT) December / 42

15 Knowledge spillovers Case where φ = 1 and n = 0 Fifth step: BGP for φ = 1 Have L = const. Set N = g N N. Then first three equations: g N = η(1 f)l rv = βfl V = η 1 β fl 1 β which implies demand for funds Euler: supply of funds r = (1 β)ηlf = (1 β) (ηl g ) r = ρ + θg Can combine the two to get r and g Ludwig Straub (MIT) December / 42

16 Knowledge spillovers Case where φ = 1 and n = 0 Sixth step: Transitional dynamics for φ = 1? Transitional dynamics? did not find any conditions on state variables N, L here. No transitional dynamics! Ludwig Straub (MIT) December / 42

17 Knowledge spillovers Case where φ < 1 and n > 0 Subsection 2 Case where φ < 1 and n > 0 Ludwig Straub (MIT) December / 42

18 Knowledge spillovers Case where φ < 1 and n > 0 Fifth step: BGP for φ < 1 Already have L = nl. Set N = g N N. Then first three equations imply with derivative and thus g N = ηn φ 1 (1 f)l β V = η 1 N 1 φ 1 β V = (1 φ)gn V π V βfl r = + = + (1 φ)g β N V V η 1 N 1 φ 1 β Note: Per capita consumption grows at rate g C = g Y n = g N g. Vertical demand curve for funds. Euler r and g Ludwig Straub (MIT) December / 42

19 Knowledge spillovers Case where φ < 1 and n > 0 Sixth step: Transitional dynamics for φ < 1 Other conditions for BGP: r = g = ηn φ 1 (1 f)l β 1 β βfl + (1 φ)g η 1 N 1 φ These two equations together give you f and a condition involving N and L r n f = r n + (1 β)g Transitional dynamics! ηn φ 1 L = r n + g 1 β Ludwig Straub (MIT) December / 42

20 Schumpeterian growth model Section 3 Schumpeterian growth model Ludwig Straub (MIT) December / 42

21 Schumpeterian growth model Model Output β Y = q ν x ν dνl β 1 β 0 Intermediate good produced at marg cost ψq ν = (1 β)q ν Quality ladder: q ν (t) = λ n ν (t) q ν (0) Competitive production of better quality: flow rate z ν of success with free entry z ν = η Z ν q ν η λvν = 1 q ν (holds if λ suff. large drastic regime ) Ludwig Straub (MIT) December / 42

22 Schumpeterian growth model First and second step State variables: q ν. But: Will see that only Q Given q ν, solve for the production side: Demand for intermediates 1 0 q ν dν matters! 1/β 1/β x ν = q ν p ν L hence the optimal price is 1 p ν = ψq ν q ν, 1 β quantities are and profits are x ν = L, π ν = βq ν L Ludwig Straub (MIT) December / 42

23 Schumpeterian growth model Second step (cont d) Aggregating things up Y = q ν dν L QL 1 β 0 1 β Y β w = = Q L 1 β These only depend on Q (rather than q ν ) Law of motion of Q: Q t+dt = zdt λq + (1 zdt) Q Q = z(λ 1)Q where z η Z ν dν/q Ludwig Straub (MIT) December / 42

24 Schumpeterian growth model Third step: Four equations (1) Valuation and free entry: Valuation rv ν = π ν + z ν (0 V ν ) + V ν η λvν = 1 q ν Notice that V ν scales in q ν. Define v V ν. Gives q ν rv = βl + z(0 v ) + v ηλv = 1 This is why we re okay using Q instead of the q ν s in the four equations! Ludwig Straub (MIT) December / 42

25 Schumpeterian growth model Third step: Four equations (2) Law of motion Valuation Free entry Euler: Q = (λ 1)zQ rv = βl + z(0 v ) + v ηλv = 1 C = C 1 (r ρ) θ Ludwig Straub (MIT) December / 42

26 Schumpeterian growth model Fourth step: BGP Assume constant growth rates then: g Q = (λ 1)z Valuation + free entry r = ηλβl z Demand for funds g r = ηλβl λ 1 Supply of funds is standard r = ρ + θg Together r, g Ludwig Straub (MIT) December / 42

27 Schumpeterian growth model Fifth step: Transitional dynamics? We did not end up with any conditions on state variables no transitional dynamics Ludwig Straub (MIT) December / 42

28 Final remarks Section 4 Final remarks Ludwig Straub (MIT) December / 42

29 Final remarks Remarks Link between g X and g C is provided by per capita output y = Y (X )/L that we solve for in second step If you want to find C (0), add the resource constraint to the four equations In all these models, parameters need to ensure TVC and positive growth r > g Y g > 0 Today: focus on cases where there exists an exact BGP (rather than only asymptotic BGP) could extend method to allow for asymptotic BGP Ludwig Straub (MIT) December / 42

30 More examples Section 5 More examples Ludwig Straub (MIT) December / 42

31 More examples Simple NGM with population growth Subsection 1 Simple NGM with population growth Ludwig Straub (MIT) December / 42

32 More examples Simple NGM with population growth First and second step State variables are K, L. Given K, L ouput is given by Y = F (K, L) Ludwig Straub (MIT) December / 42

33 More examples Simple NGM with population growth Third step: Four equations Laws of motion Valuation Free entry Euler K = F (K, L) δk C L = nl rv = F K (K, L) δ + V V = 1 C = C 1 (r ρ) θ Ludwig Straub (MIT) December / 42

34 More examples Simple NGM with population growth Fourth and fifth step: BGP Assuming constant growth rates: g K = F (1, L/K ) δ C /K which means n = g L = g K giving demand for funds Supply of funds is standard In addition: For BGP need g = n r = ρ + θg r = F K (K, L) δ which is a restriction of the two state variables K and L! transitional dynamics! Ludwig Straub (MIT) December / 42

35 More examples Lab Equipment model Subsection 2 Lab Equipment model Ludwig Straub (MIT) December / 42

36 More examples Lab Equipment model First and second step State variable N Solving the production side gives (see lecture notes) Y = 1 NL 1 β X interm = (1 β)nl β w = N 1 β π = βl Ludwig Straub (MIT) December / 42

37 More examples Lab Equipment model Third step: Four equations Law of motion Value of innovation Free entry Euler N = ηz (= η(y X C ) rv = βl + V C ηv = 1 1 = C (r ρ) θ Ludwig Straub (MIT) December / 42

38 More examples Lab Equipment model Fourth step: BGP Assuming constant growth rates: g N N = ηz Value of innovation + free entry demand for funds Supply of funds is standard r = ηβl r = ρ + θg No condition on values of state variable N no transitional dynamics! Ludwig Straub (MIT) December / 42

39 More examples Problem Set 4, Question 4 Subsection 3 Problem Set 4, Question 4 Ludwig Straub (MIT) December / 42

40 More examples Problem Set 4, Question 4 First and second step State variables N and L Solving the production side gives Denote f = L E /L Y = N (1 β)/β L E w = βn (1 β)/β Y π = (1 β) N Ludwig Straub (MIT) December / 42

41 More examples Problem Set 4, Question 4 Third step: Four equations Laws of motion of N and L N = ηn φ (1 f)l L = nl Value of innovation Free entry of labor Y rv = (1 β) N + V βn (1 β)/β = w = ηn φ V Euler 1 C = C (r ρ) θ Rest is similar to knowledge spillover section above! Ludwig Straub (MIT) December / 42

42 MIT OpenCourseWare Economic Growth Fall 2016 For information about citing these materials or our Terms of Use, visit: