Review: Solow Model Review: Ramsey Model Endogenous Growth Lecture 17 & 18 Topics in Macroeconomics December 8 & 9, 2008 Lectures 17 & 18 1/29 Topics in Macroeconomics
Outline Review: Solow Model Review: Ramsey Model Review: Solow Model Review: Ramsey Model Ak model: A simple model of endogenous long-run growth Growth rates are constant Growth rates of c, k and y are the same Balanced growth path theorem Social Planner s problem Euler equation Optimal k/h ratio and consumption growth rate Conclusion Lectures 17 & 18 2/29 Topics in Macroeconomics
Review: Solow Model Review: Ramsey Model Review: Solow Model 3 Solow Model Exogenously given savings rate s Production function Assumptions (Lecture 3): A1: Constant returns to scale (CRS) F(λK, λl) = λf(k, L) A2: Marginal products positive and diminishing F K > 0, F L > 0, F KK < 0, F LL < 0 From CRS (A1), we can write F in per capita terms f(k) = F(k, 1). Then (A2) becomes F K > 0 f (k) > 0 F KK < 0 f (k) < 0 For example, f(k) = Ak α with α < 1 Lectures 17 & 18 3/29 Topics in Macroeconomics
Review: Solow Model Review: Ramsey Model Review: Solow Model Results 4 Solow Model Unless there is exogenous technological change (A t+1 = (1 + g)a t, g > 0), the economy converges to a steady state in per capita variables. No long-run growth except due to technological change. Golden Rule steady state level of consumption requires s = α Investments result from choices: the Solow model has nothing to say about savings/investment decisions Ramsey model Lectures 17 & 18 4/29 Topics in Macroeconomics
Review: Solow Model Review: Ramsey Model Review: Ramsey Model 5 Ramsey Model Households choose how much to save and how much to consume dynamics results from this choice. Production function: same as for the Solow model Assumptions (Lecture 3): A1: Constant returns to scale (CRS) A2: Marginal products positive and diminishing From CRS (A1), we can write F in per capita terms f(k) = F(k, 1). Then (A2) becomes F K > 0 f (k) > 0 F KK < 0 f (k) < 0 For example, f(k) = Ak α with α < 1 Lectures 17 & 18 5/29 Topics in Macroeconomics
Review: Solow Model Review: Ramsey Model Review: Ramsey Model Results 6 Ramsey Model Unless there is exogenous technological change (A t+1 = (1 + g)a t, g > 0), the economy converges to a steady state in per capita variables (g = 0). No long-run growth except due to technological change. Modified Golden Rule steady state level of consumption and capital are smaller than Golden Rule. This is due to impatience of households. If government consumption is financed with taxes on capital gains, HH save less and the steady state (c τ k) and (k τ k) are lower than with LS taxes. How does long-run growth occur endogenously? Lectures 17 & 18 6/29 Topics in Macroeconomics
Growth rates are constant Growth rates of c, k and y are the same Balanced growth path theorem A simple model of endogenous long-run growth 7 Ak Model Take Ramsey Model but change 1 assumption: α = 1 Thus the production function becomes: f(k) = Ak (or F(K, L) = AK, i.e. labor does not enter into the production function) That is, f (k) = A > 0 used in Euler equation That is, f (k) = 0. There are no more diminishing marginal returns to capital. It is constant returns to capital alone! How does long-run growth occur endogenously? Lectures 17 & 18 7/29 Topics in Macroeconomics
Growth rates are constant Growth rates of c, k and y are the same Balanced growth path theorem Growth rate of consumption and the Euler equation 8 Ak Model The Euler equation becomes: +1 = [β(f (k t+1 ) + 1 δ)] 1/σ +1 = [β(a + 1 δ)] 1/σ = γ c There is no steady state in this model But clearly, consumption grows at a constant rate FOREVER!!! Lectures 17 & 18 8/29 Topics in Macroeconomics
Growth rates are constant Growth rates of c, k and y are the same Balanced growth path theorem Growth rate of consumption and the Euler equation 9 Ak Model: Assumption 1 +1 = [β(a + 1 δ)] 1/σ = 1 + γ c Assumption 1: γ c > 0 This requires: β(a + 1 δ) > 1 Lectures 17 & 18 9/29 Topics in Macroeconomics
Growth rates are constant Growth rates of c, k and y are the same Balanced growth path theorem Growth rate of capital and the resource constraint 10 Ak Model: The growth rate of capital is constant The resource constraint can be written as: Dividing both sides by k t + k t+1 = (A + 1 δ)k t k t + k t+1 k t = A + 1 δ Lectures 17 & 18 10/29 Topics in Macroeconomics
Growth rates are constant Growth rates of c, k and y are the same Balanced growth path theorem Growth rate of capital and the resource constraint 11 Ak Model: The growth rate of capital is constant k t + k t+1 k t = A + 1 δ Suppose the growth rate of k is increasing over time k t must be decreasing over time violates transversality condition Suppose the growth rate of k is decreasing over time k t must be increasing over time drives k and in turn o 0, cannot be optimal Thus, k t+1 k t = 1 + γ k is constant Lectures 17 & 18 11/29 Topics in Macroeconomics
Growth rates are constant Growth rates of c, k and y are the same Balanced growth path theorem Equal growth rates 12 Ak Model: Equal growth rates γ c = γ k = γ y k t + (1 + γ k ) = A + 1 δ This implies the ratio of consumption to capital is constant Thus, capital and consumption must grow at the same rate Since y t = Ak t, output per capita must be growing at the same rate as well Hence, γ = γ c = γ k = γ y = [β(a + 1 δ)] 1/σ 1 Lectures 17 & 18 12/29 Topics in Macroeconomics
Growth rates are constant Growth rates of c, k and y are the same Balanced growth path theorem Conditions for a BGP to exist 13 Ak Model: Assumption 2 Assumption 2: k t + (1 + γ k ) = A + 1 δ k t + [β(a + 1 δ)] 1/σ = A + 1 δ k t = (A + 1 δ)(1 β 1 σ (A + 1 δ) 1 σ 1 ) β 1 σ (A + 1 δ) 1 σ 1 < 1 Lectures 17 & 18 13/29 Topics in Macroeconomics
Growth rates are constant Growth rates of c, k and y are the same Balanced growth path theorem Conditions for a BGP to exist 14 Ak Model: Theorem Consider the social planner s problem with linear technology f(k) = Ak and CEIS preferences. Suppose (β,σ, A,δ) satisfy β(a + 1 δ) > 1 > β 1 σ (A + 1 δ) 1 σ 1 Then the economy exhibits a balanced growth path where capital, output and consumption all grow at a constant rate given by k t+1 k t = y t+1 y t = +1 = 1 + γ = [β(a + 1 δ)] 1/σ The growth rate is increasing in A and β and it is decreasing in δ and σ. Lectures 17 & 18 14/29 Topics in Macroeconomics
Outline Social Planner s problem Euler equation Optimal k/h ratio and consumption growth rate Review: Solow Model Review: Ramsey Model Ak model: A simple model of endogenous long-run growth Growth rates are constant Growth rates of c, k and y are the same Balanced growth path theorem Social Planner s problem Euler equation Optimal k/h ratio and consumption growth rate Conclusion Lectures 17 & 18 15/29 Topics in Macroeconomics
Social Planner s problem Euler equation Optimal k/h ratio and consumption growth rate Setup: Production function 16 Production function with human capital Y t = F(K t, H t ) = F(K t, h t L t ) F : Neoclassical production function (Lecture 3): A1: Constant returns to scale (CRS) F(λK,λH) = λf(k, H) A2: Marginal products positive and diminishing F K > 0, F H > 0, F KK < 0, F HH < 0 Use CRS, write F in per capita terms F(K,H) L = F(k, h) For example, f(k, h) = Ak α h 1 α with α < 1 H t = h t L t is effective labor. Lectures 17 & 18 16/29 Topics in Macroeconomics
Social Planner s problem Euler equation Optimal k/h ratio and consumption growth rate Setup: Investments and Resource constraint 17 Capital-type specific investment i kt : investment in physical capital i ht : investment in human capital Resource constraint Total output can be used for consumption or investment in either type of capital. + i kt + i ht = F(k t, h t ) Lectures 17 & 18 17/29 Topics in Macroeconomics
Social Planner s problem Euler equation Optimal k/h ratio and consumption growth rate Setup: Investments and depreciation 18 Laws of motion k t+1 = i kt + (1 δ)k t h t+1 = i ht + (1 δ)h t Resource constraint can be written as + k t+1 + h t+1 = F(k t, h t ) + (1 δ)k t + (1 δ)h t Lectures 17 & 18 18/29 Topics in Macroeconomics
Social Planner s problem Euler equation Optimal k/h ratio and consumption growth rate Social Planner s problem 19 With the usual utility function, Social planner s problem can be written as max,(k t+1,h t+1 ) t=0 s.t. β t u( ) t=0 + k t+1 + h t+1 = F(k t, h t ) + (1 δ)k t + (1 δ)h t, for all t = 0, 1, 2,... k 0, h 0 > 0 given Lectures 17 & 18 19/29 Topics in Macroeconomics
Social Planner s problem Euler equation Optimal k/h ratio and consumption growth rate Social Planner s problem 20 Assume u(c) = c1 σ 1 σ Assume F(k, h) = Ak α h 1 α With functional forms, the planner s problem becomes max,(k t+1,h t+1 ) t=0 s.t. + k t+1 + h t+1 t=0 β t c1 σ t 1 σ = Akt α h 1 α t + (1 δ)k t + (1 δ)h t, for all t k 0, h 0 > 0 given Lectures 17 & 18 20/29 Topics in Macroeconomics
Social Planner s problem Euler equation Optimal k/h ratio and consumption growth rate Euler equations 21 Since there are 2 types of capital, we have 2 Euler eqns Euler equation for physical capital +1 = [β(1 + F k (k t+1, h t+1 ) δ)] 1/σ ( ) c 1 α t+1 ht+1 = [β(1 + Aα δ)] 1/σ Euler equation for human capital +1 k t+1 = [β(1 + F h (k t+1, h t+1 ) δ)] 1/σ ( ) c α t+1 kt+1 = [β(1 + A(1 α) δ)] 1/σ h t+1 Lectures 17 & 18 21/29 Topics in Macroeconomics
Social Planner s problem Euler equation Optimal k/h ratio and consumption growth rate Rate of return condition and k/h ratio 22 Combining Euler equations, we get or, Aα F h (k t+1, h t+1 ) = F k (k t+1, h t+1 ) ( ht+1 k t+1 ) 1 α = A(1 α) ( ) α kt+1 h t+1 Hence, the optimal ratio of physical to human capital is given by k t+1 h t+1 = α 1 α Notice that it is constant. Hence, they grow at the same rate. Lectures 17 & 18 22/29 Topics in Macroeconomics
Social Planner s problem Euler equation Optimal k/h ratio and consumption growth rate Growth rate of consumption 23 Using the k/h ratio in either one Euler equation, we find ( kt+1 ) c α t+1 = [β(1 + A(1 α) δ)] 1/σ h t+1 ( ) +1 α α = [β(1 + A(1 α) δ)] 1/σ 1 α +1 = [β(1 + A(1 α) 1 α α α δ)] 1/σ Lectures 17 & 18 23/29 Topics in Macroeconomics
Conclusion Existence of BGP and Growth rates 24 Using the k/h ratio in laws of motion We found k t+1 h t+1 Therefore, = α 1 α Also, Hence, h t+1 = 1 α α k t+1 = 1 α α (i kt + (1 δ)k t ) h t+1 = i ht + (1 δ)h t = i ht + (1 δ) 1 α α k t i ht = 1 α α i kt Lectures 17 & 18 24/29 Topics in Macroeconomics
Conclusion Existence of BGP and Growth rates 25 By redefining the variables as follows, this model is equivalent to an Ak model This is very useful since we know the conditions on parameters for the Ak model so that a BGP exists and we know the properties in terms of growth rates. Let î t = 1 α i kt,  = A(1 α) 1 α α α and ˆkt = 1 α k t Then we can rewrite the resource constraint and laws of motion as follows* + ît = ˆk t ˆk t+1 = ît + (1 δ)ˆk t Lectures 17 & 18 25/29 Topics in Macroeconomics
Conclusion Existence of BGP and Growth rates 26 The social planner s problem becomes max (ˆk t+1, ) t=0 s.t. t=0 β t c1 σ t 1 σ + ˆk t+1 = ˆk t + (1 δ)ˆk t, for all t ˆk 0 > 0 given This is just an Ak model. Using the conditions on parameters from the Theorem... we find Lectures 17 & 18 26/29 Topics in Macroeconomics
Conclusion Balanced growth path theorem for Akh model 27 Akh Model: Theorem Consider the social planner s problem with CRS technology f(k, h) = Ak α h 1 α and CES preferences. Suppose (β,σ, A,δ,α) satisfy β(â + 1 δ) > 1 > β 1 σ ( Â + 1 δ) 1 σ 1 β(a(1 α) 1 α α α + 1 δ) > 1 > β 1 σ (A(1 α) 1 α α α + 1 δ) 1 σ 1 and suppose h 0 = 1 α α k 0. Then the economy exhibits a balanced growth path where capital, output and consumption all grow at a constant rate given by ˆk t+1 ˆk t = k t+1 k t = h t+1 h t = y t+1 y t = +1 = 1 + γ = [β(â + 1 δ)]1/σ Lectures 17 & 18 27/29 Topics in Macroeconomics
Conclusion Concluding remarks 28 This model exhibits long run growth, even though some form of labor is taken into account. This is because the QUALITY of labor is taken into account. This quality, human capital, is accumulable. Therefore the diminishing marginal returns don t kick in. Both factors grow simultaneously and therefore allow the economy to grow FOREVER. Lectures 17 & 18 28/29 Topics in Macroeconomics
Conclusion Think about other production functions 29 1. Capital and land enter the production function, Y = F(K, L) land is not accumulable, it is fixed decreasing marginal products kick in steady state land is accumulable, e.g. US expansion East to West economy expands 2. Capital and population/labor enter the production function, Y = F(K, L) labor is not accumulable, it is fixed decreasing marginal products kick in steady state Y = F(K, N) population is accumulable endogenous fertility models economy expands Lectures 17 & 18 29/29 Topics in Macroeconomics