Chapter 3 Task 1-4. Growth and Innovation Fridtjof Zimmermann
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1 Chapter 3 Task 1-4 Growth and Innovation Fridtjof Zimmermann
2 Recept on how to derive the Euler-Equation (Keynes-Ramsey-Rule) 1. Construct the Hamiltonian Equation (Lagrange) H c, k, t, μ = U + μ(side Condition) Side Condition = Capital Accumulation Function in SS ሶ ሶ ሶ 2. Take derivatives to receive the FOC ( H ) and the CE ( H c k μ H k = H k = k 3. Take the derivative of the FOC in respect to time (t) 4. Reinsert H k 5. Reinsert the FOC ( H ), which eliminates μ c 6. Solve for c 7. Divide by c
3 Derive the Euler equation of the given utility function or s.t. max c U 0 = න 0 u c e ρ n t dt u c = ln(c) u c = c1 γ 1 γ ሶ k = w + rk c n + δ k k 0 > 0
4 Intertemporal elasticity of substitution g c = σ(r δ ρ) σ = u c Intertemporal elasticity of substitution c u cc Measures the responsiveness of the households to a change in the interest rate. Should they consume more now or in the future?
5 No. 1 Describe the implications whether knowledge is embodied (human capital) or disembodied Embodied in humans Accumulation Learning by Doing Schooling (Investment) Rival in use, nondisclosure Positive private Return Incentive for investing in HC Knowledge Disembodied in knowledge stock Disclosure Positive Spillover Effects Allocation is Pareto inefficient Price System is incomplete Non-Disclosure Private Return Incentive for R&D
6 No. 2 Discuss in detail the so-called AK model. Is there a transitory dynamic? Assumptions: No Population Growth Capital includes physical as well as human capital Perfect Competition Household maximize a Utility Function Same Results as in the Keynes-Ramsey Results g c = σ(r t ρ + δ )
7 No. 2 Discuss in detail the so-called AK model. Is there a transitory dynamic? Difference: Produktion Function and hence the first derivative y = Ak y k = f k = r = A lim k f k = A A is constant over time No diminishing return as in the Cass-Koopmann-Ramsey Model Reminder in the Cass-Koopmann-Ramsey Model: y = Ak α y k = f k = A α k α 1 lim k f k = 0
8 No. 2 Discuss in detail the so-called AK model. Is there a transitory dynamic? y = Ak y k = f k = r = A The Euler Equation is thus: g y = g c = g k = σ(a δ ρ) g y, g c, g k grow as long as A > δ + ρ, in other words A δ > ρ There is no change in A and thus the economy grows indefinitely. No transitionary dynamic No catching up as long as the structural parameters (A, δ, and ρ) are the same
9 No. 3 Explain the Arrow model of knowledge spillovers which results in an AK-Type model. Discuss whether the market-based solution is efficient, and what a social planner would do. Policy interventions require taxing. Why does the type of of tax matter?
10 No. 3 Explain the Arrow model of knowledge spillovers which results in an AK-Type model. Idea: Learning comes through expirience Human Capital is an exogeneous, fall-out part of physical investment Experience in operating physical capital is the human capital part (തk). HC is non-excludable and non-rivalry Small Firms take HC as given
11 No. 3 Explain the Arrow model of knowledge spillovers which results in an AK-Type model. Mathematical Definitions Produktion Function Y = f(k, ഥK, L) Arrow (1962): y = f k, ഥK = ഥK θ k α = N θ തk θ k α θ + α = 1 AK-Model Romer (1986): y = f k, ഥK N = ഥK 1 α k α = N 1 α തk 1 α k α α = 1 AK-Model Hausehold maximize Utility function leading to Euler-equation g c = σ r t ρ + δ
12 No. 3 Explain the Arrow model of knowledge spillovers which results in an AK-Type model. So difference to prior models is again on the supply side. Influence on A always through the Production function max K,N π(k) = N ഥK 1 α k α rk w FOCs π K = α ഥK 1 α K α 1 N 1 α r = 0 π N = (1 α) ഥK 1 α K α N α w = 0 With ഥK = Nk r = αn 1 α w = 1 α kn 1 α
13 No. 3 Discuss whether the market-based solution is efficient, and what a social planner would do. Social Planner FOCs With ഥK = Nk max K,N π(k) = N K K1 α rk w N = N K N α rk w π K = N1 α r = 0 π N = 1 α KN α w = 0 r = N 1 α α
14 No. 3 Discuss whether the market-based solution is efficient, and what a social planner would do. Decentral planning (market solution): g c D = σ αn 1 α ρ + δ Social Planner: g c S = σ N 1 α ρ + δ Conclusion g c D < g c S Decentral Planning is inefficient Pareto Inefficient Does not incorporate social returns of capital by inducing growing human capital Amount of Workforce determines possibilities of an ecomy
15 No. 3 Policy interventions require taxing. Why does the type of of tax matter? How can we assure that we reach the g c S level? Trough taxes and subsidies Goal: Create a positive spillover through subsidizing investment Subsidy: θ g c = σ α(1 + θ )N 1 α ρ + δ Optimal Subsidy: θ = 1 α α
16 No. 3 Policy interventions require taxing. Why does the type of of tax matter? Income Tax: (fair) Lowers marginal returns of the production factors Shift of Consumption into the presence Lower saving Lower capital accumulation Lower steady state growth rate Per capita tax: (unfair) No affect on allocation No affect on steady state growth rate Consumption tax Would affect the decision between leisure and working (not modeled)
17 No. 4 a) Human capital in a two-sector model Decribe the components of the Uzawa-Lucas model where HC is produced in a schooling sector Production mh H y s k 1 m h (1 s) Education c Production Sector Schooling Sector
18 No. 4 a) Human capital in a two-sector model Decribe the components of the Uzawa-Lucas model where HC is produced in a schooling sector Assumptions of the schooling sector: Does not require physical Capital Population Growth is constant (can be changed) Human Capital is a Private Good. h = H N
19 No. 4 b) Human capital in a two-sector model What does the household has to decide about? Production mh h y s k 1 m h (1 s) Households have to decide the value of s and m Education c Production Sector Schooling Sector
20 ሶ ሶ ሶ No. 4 b) Human capital in a two-sector model Describe the two dynamic equations of the model Production Sector Y = C + K ሶ = F K, mh = K α mh 1 α y = k α mh 1 α k = k α mh 1 α c = rk + wmh c Schooling Sector H = A( 1 m H) h = A( 1 m h) m: Fraction of human capital, which is distributed to production (1 m): Fraction of human capital, which is distributed to eduction Decided by the households
21 No. 4 b) Human capital in a two-sector model What are the dynamic variables in the model which describe the state of the economy? Production Sector ሶ k = k α mh 1 α c = rk + wmh c Schooling Sector ሶ h = A( 1 m h) The dynamic variables kሶ and hሶ describe the state of the economy yሶ y = k ሶ k = h ሶ = A 1 m h mሶ m = 0 The economy can grow indefinitely
22 Revision What is the Hamiltonian Equation in the case of the Lucas-Uzawa Model? H c, k, t, μ = U + μ 1 Side Condition of k + μ 2 (Side Condition of h) Which derivative do you have to take? FOC: H c, CE: H k, H μ 1, H m H μ 2
23 Revision g c = σ r ρ Whereas in the steady state: g c = g k = g h = g y = const > 0 g m = 0 Easier representation: z = k h, q = c k Which leads to g q = g z = g m = 0 g z = g k g h g q = g c g k r = y k = α m z 1 α
24 Revision g c = σ αm 1 α z 1 α ρ g k = m 1 α z 1 α q g h = A(1 m) And g m = 1 α A α g z = g k g h g z = m 1 α z 1 α A(1 m) + ma q q g q = g c g k g q = σα 1 m 1 α z 1 α σρ + q
25 Revision Model is three dimensional (q, z, m) To draw it graphically two variables have to be combined (This also makes the linearization easier): x = m z 1 α As in the case of the Cass-Koopmann-Ramsey model we draw a vector field with the unique saddle path. The vector field is constructed over a linearization of the model and the Jacobian matrix.
26 No. 4 c) (i) Transitionary dynamic in the Uzawa-Lucas model What happens in the case of a sudden decline of physical capital? Direct Effect after the shock A decline in k means that there is to less physical capital compared to human capital z = k if k declines z h declines q = c if k declines q k increases x = m 1 α z 1 α if k declines z declines x increases
27 No. 4 c) (i) Transitionary dynamic in the Uzawa-Lucas model What happens in the case of a sudden decline of physical capital? Direct Effect after the shock g c = σ αm 1 α z 1 α ρ negative correlation to z g c becomes larger g k = m 1 α z 1 α q negative correlation to z ; negative correlation to q g k becomes larger The marginal rate of capital increases on the remaining capital Increases capital accumulation and thereby the growth rates. g m = 1 α A α + ma q negative correlation to q g m becomes smaller Incentive to increase the human capital stock, due to lower wages
28 No. 4 c) (ii) Transitionary dynamic in the Uzawa-Lucas model What happens in the case of a sudden outflow of human capital? Direct Effect after the shock A decline in h means that there is too less human capital compared to physical capital z = k if h declines z h increases q = c if h declines q stays k unchanged x = m 1 α z 1 α if h declines z increases x decreases
29 No. 4 c) (ii) Transitionary dynamic in the Uzawa-Lucas model What happens in the case of a sudden outflow of human capital Direct Effect after the shock g c = σ αm 1 α z 1 α ρ negative correlation to z g c becomes smaller g k = m 1 α z 1 α q negative correlation to z g k becomes smaller The marginal rate of capital decreases, due to oversatisfaction of capital Decreases capital accumulation and thereby the growth rates. g m = 1 α A α + ma q no correlation to q g m stays unchanged
30 No. 4c) Indirect Effects of shocks After the shock households will adjust their consume and distribution of human capital (m) This happens directly after the shock Moves back to the saddle path. Convergence over time back to the steady state Convergence back to steady state in the case of physical capital decline
31 No. 4 d) Positive externalities from human capital accumulation What are possible reasons for that? y = k α mh 1 α തh η Externalities are unpriced benefits coming from education Teamwork Motivation Indirect effects Healthy nutrition Better functioning of society
32 No. 4 d) Positive externalities from human capital accumulation What are the implications for effeciency and policy Decentralized y = k α mh 1 α തh η With 1 α + η g c = g k = g y = (A ρ) 1 α g h = A ρ < g y Inefficient People work to much Phisical capital is seen as more valuable than human capital. Higher accumulation of physical assets
33 No. 4 d) Positive externalities from human capital accumulation What are the implications for effeciency and policy Social Planner y = k α mh 1 α തh η With 1 α + η g c = g k = g y = A ρ 1 α g h = A 1 α 1 α + η ρ Optimal growth rates Government needs to redistribute human capital from the working force to the education sector. Change m Needs to tax physical assets (capital) and wages and subsidies education.
34 No. 4 d) Positive externalities from human capital accumulation What are the implications for effeciency and policy ሶ k = 1 τ r rk + 1 τ w wmh θw 1 m h c θ > 0 School fees θ < 0 School grants This leads to g y = g c = g 1 α + η 1 τ w k = A ρ 1 α 1 τ w + θ g h = 1 τ w A ρ 1 τ w + θ With θ τ w 1 ηρ = 1 α + η A + ηρ Optimality depends only on the wage tax, not on interest rate tax
35 End
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