Growth Theory: Review

Growth Theory: Review Leure 1, Endogenous Growth Economic Policy in Development 2, Part 2 April 20, 2007 Leure 1, Endogenous Growth 1/28 Economic Policy in Development 2, Part 2 Outline Review: Solow Model Review: Ramsey Model Leure 1, Endogenous Growth 2/28 Economic Policy in Development 2, Part 2 Review: Solow Model Review: Ramsey Model Review: Solow Model 3 Review: Solow Model Review: Ramsey Model Review: Solow Model Results 4 Solow Model Exogenously given savings rate s Produion funion Assumptions: A1: Constant returns to scale (CRS) F(λK, λl) = λf(k, L) A2: Marginal produs positive and diminishing FK > 0, FL > 0, FKK < 0, FLL < 0 From CRS (A1), we can write F in per capita terms f(k) = F(k, 1). Then (A2) becomes FK > 0 f (k) > 0 FKK < 0 f (k) < 0 Solow Model Unless there is exogenous technological change (At+1 = (1 + g)at, 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 For example, f(k) = Ak α with α < 1 Leure 1, Endogenous Growth 3/28 Economic Policy in Development 2, Part 2 Leure 1, Endogenous Growth 4/28 Economic Policy in Development 2, Part 2

Review: Solow Model Review: Ramsey Model Review: Ramsey Model 5 Review: Solow Model Review: Ramsey Model Review: Ramsey Model Results 6 Ramsey Model Households choose how much to save and how much to consume dynamics results from this choice. Produion funion: same as for the Solow model Assumptions (Leure 3): A1: Constant returns to scale (CRS) A2: Marginal produs positive and diminishing From CRS (A1), we can write F in per capita terms f(k) = F(k, 1). Then (A2) becomes FK > 0 f (k) > 0 FKK < 0 f (k) < 0 For example, f(k) = Ak α with α < 1 Leure 1, Endogenous Growth 5/28 Economic Policy in Development 2, Part 2 Ramsey Model Unless there is exogenous technological change (At+1 = (1 + g)at, 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? Leure 1, Endogenous Growth 6/28 Economic Policy in Development 2, Part 2 A simple model of endogenous long-run growth 7 Growth rate of consumption and the 8 Ak Model Take Ramsey Model but change 1 assumption: α = 1 Thus the produion funion becomes: f(k) = Ak (or F(K, L) = AK, i.e. labor does not enter into the produion funion) That is, f (k) = A > 0 used in 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? Ak Model The becomes: +1 = [β(f (+1) + 1 δ)] 1/σ +1 = [β(a + 1 δ)] 1/σ = 1 + γc There is no steady state in this model But clearly, consumption grows at a constant rate FOREVER!!! Leure 1, Endogenous Growth 7/28 Economic Policy in Development 2, Part 2 Leure 1, Endogenous Growth 8/28 Economic Policy in Development 2, Part 2

Growth rate of consumption and the 9 Growth rate of capital and the resource constraint 10 Ak Model: Assumption 1 +1 = [β(a + 1 δ)] 1/σ = 1 + γc Assumption 1: γc > 0 This requires: β(a + 1 δ) > 1 Ak Model: The growth rate of capital is constant The resource constraint can be written as: + +1 = (A + 1 δ) Dividing both sides by + +1 = A + 1 δ Leure 1, Endogenous Growth 9/28 Economic Policy in Development 2, Part 2 Leure 1, Endogenous Growth 10/28 Economic Policy in Development 2, Part 2 Growth rate of capital and the resource constraint 11 Equal growth rates 12 Ak Model: The growth rate of capital is constant + +1 = A + 1 δ Suppose the growth rate of k is increasing over time must be decreasing over time violates transversality condition Suppose the growth rate of k is decreasing over time must be increasing over time drives k and in turn c to 0, cannot be optimal Thus, k t+1 = 1 + γk is constant Ak Model: Equal growth rates γc = γk = γy + (1 + γk) = A + 1 δ Thus the ratio of consumption to capital is constant Thus, capital and consumption must grow at the same rate Since yt = A, output per capita must be growing at the same rate as well Thus, γ = γc = γk = γy = [β(a + 1 δ)] 1/σ 1 Leure 1, Endogenous Growth 11/28 Economic Policy in Development 2, Part 2 Leure 1, Endogenous Growth 12/28 Economic Policy in Development 2, Part 2

Conditions for a BGP to exist 13 Conditions for a BGP to exist 14 Ak Model: Assumption 2 + (1 + γk) = A + 1 δ + [β(a + 1 δ)] 1/σ = A + 1 δ = (A + 1 δ)(1 β 1 σ (A + 1 δ) 1 σ 1 ) Assumption 2: β 1 σ (A + 1 δ) 1 σ 1 < 1 Leure 1, Endogenous Growth 13/28 Economic Policy in Development 2, Part 2 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 +1 = yt+1 yt = +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. Leure 1, Endogenous Growth 14/28 Economic Policy in Development 2, Part 2 : Produion funion 15 : Investments and Resource constraint 16 Produion funion with human capital Yt = F(Kt, Ht) = F(Kt, htlt) F : Neoclassical produion funion (Leure 3): A1: Constant returns to scale (CRS) F(λK,λH) = λf(k, H) A2: Marginal produs positive and diminishing FK > 0, FH > 0, FKK < 0, FHH < 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 Capital-type specific investment i : investment in physical capital iht : investment in human capital Resource constraint Total output can be used for consumption or investment in either type of capital. + i + iht = F(, ht) Ht = htlt is effeive labor. Leure 1, Endogenous Growth 15/28 Economic Policy in Development 2, Part 2 Leure 1, Endogenous Growth 16/28 Economic Policy in Development 2, Part 2

: Investments and depreciation 17 Laws of motion +1 = i + (1 δ) = iht + (1 δ)ht Resource constraint can be written as + +1 + = F(, ht) + (1 δ) + (1 δ)ht 18 With the usual utility funion, Social planner s problem can be written as max β t u() (k t+1,) t=0 t=0 s.t. + +1 + = F(, ht) + (1 δ) + (1 δ)ht, for all t = 0, 1, 2,... k0, h0 > 0 given Leure 1, Endogenous Growth 17/28 Economic Policy in Development 2, Part 2 Leure 1, Endogenous Growth 18/28 Economic Policy in Development 2, Part 2 19 s 20 Assume u(c) = c1 σ 1 σ Assume F(k, h) = Ak α h 1 α With funional forms, the planner s problem becomes max (k t+1,) t=0 s.t. + +1 + β t c1 σ t 1 σ t=0 = A α ht 1 α + (1 δ) + (1 δ)ht, for all t k0, h0 > 0 given Since there are 2 types of capital, we have 2 Euler eqns for physical capital +1 = [β(1 + Fk(+1, ) δ)] 1/σ ( ) 1 α +1 = [β(1 + Aα δ)] 1/σ k t+1 for human capital +1 = [β(1 + Fh(+1, ) δ)] 1/σ ( ) α +1 +1 = [β(1 + A(1 α) δ)] 1/σ Leure 1, Endogenous Growth 19/28 Economic Policy in Development 2, Part 2 Leure 1, Endogenous Growth 20/28 Economic Policy in Development 2, Part 2

Rate of return condition and k/h ratio 21 Growth rate of consumption 22 Combining s, we get Using the k/h ratio in either one, we find or, Fh(+1, ) = Fk(+1, ) ( ) 1 α ( ) α +1 Aα = A(1 α) +1 Hence, the optimal ratio of physical to human capital is given by +1 = α 1 α Notice that it is constant. Hence, they grow at the same rate. ( ) α +1 +1 = [β(1 + A(1 α) δ)] 1/σ ( ) +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 Leure 1, Endogenous Growth 21/28 Economic Policy in Development 2, Part 2 Leure 1, Endogenous Growth 22/28 Economic Policy in Development 2, Part 2 Existence of BGP and Growth rates 23 Using the k/h ratio in laws of motion We found k t+1 h t+1 = α 1 α Therefore, Also, Hence, = 1 α α +1 = 1 α (i + (1 δ)) α = iht + (1 δ)ht = iht + (1 δ) 1 α α iht = 1 α α i Existence of BGP and Growth rates 24 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 ît = 1 α i,  = A(1 α)1 α α α and ˆ = 1 α Then we can rewrite the resource constraint and laws of motion as follows* + ît = ˆ ˆ+1 = ît + (1 δ)ˆ Leure 1, Endogenous Growth 23/28 Economic Policy in Development 2, Part 2 Leure 1, Endogenous Growth 24/28 Economic Policy in Development 2, Part 2

Existence of BGP and Growth rates 25 for Akh model 26 The social planner s problem becomes max (k t+1,) t=0 s.t. β t c1 σ t 1 σ t=0 + ˆ+1 = ˆ + (1 δ)ˆ, for all t ˆk0 > 0 given This is just an Ak model. Using the conditions on parameters from the Theorem... we find Akh Model: Theorem Consider the social planner s problem with linear technology f(ˆk) = ˆk and CEIS preferences. Suppose (β,σ, A,δ,α) satisfy β(â + 1 δ) > 1 > β 1 σ (  + 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 h0 = 1 α α k0.then the economy exhibits a balanced growth path where capital, output and consumption all grow at a constant rate given by ˆ+1 ˆ = +1 = ht = yt+1 yt = +1 = 1 + γ = [β(â + 1 δ)]1/σ Leure 1, Endogenous Growth 25/28 Economic Policy in Development 2, Part 2 Leure 1, Endogenous Growth 26/28 Economic Policy in Development 2, Part 2 Concluding remarks 27 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 faors grow simultaneously and therefore allow the economy to grow FOREVER. Think about other produion funions 28 1. Capital and land enter the produion funion, Y = F(K, L) land is not accumulable, it is fixed decreasing marginal produs kick in steady state land is accumulable, e.g. US expansion East to West economy expands 2. Capital and population/labor enter the produion funion, Y = F(K, L) labor is not accumulable, it is fixed decreasing marginal produs kick in steady state Y = F(K, N) population is accumulable endogenous fertility models economy expands Leure 1, Endogenous Growth 27/28 Economic Policy in Development 2, Part 2 Leure 1, Endogenous Growth 28/28 Economic Policy in Development 2, Part 2