Solution to Homework 2 - Exogeneous Growth Models

Transcription

1 Solution to Homework 2 - Exogeneous Growth Models ECO-3211 Macroeconomia Aplicada (Applied Macroeconomics Question 1: Solow Model with a Fixed Factor 1 The law of motion for capital in the Solow economy is given by K(t = sy(t δk(t Let k = K/L be the capital per worker Then substituting for output per worker, y = Y/L, using the aggregate production function, in the law of motion above gives k(t k(t = K(t K(t = sk(tα 1 z 1 α β δ, where z = Z/L is land per worker Note that k(t/k(t = K(t/K(t because there is no population growth Then, in steady-state k(t = 0, which gives the following expression for the steady-state k and y: ( sz k 1 α β 1/(1 α =, δ ( s y = δ α/(1 αz (1 α β/(1 α The steady-state is unique and stable because for k > k, sk(t α 1 z 1 α β δ < 0 (the first term is decreasing in k, and for k < k, sk(t α 1 z 1 α β δ > 0 In other words, k(t > 0 for k (0,k and k(t < 0 for k (k, Thus, if the economy starts from an initial capital stock k(0 < k, it will experience in an increase in k till it reaches k, and if the the economy starts from an initial capital stock k(0 > k, it will experience in a decrease in k till it reaches k Rahul Giri Contact Address: Centro de Investigacion Economica, Instituto Tecnologico Autonomo de Mexico (ITAM rahulgiri@itammx 1

2 2 Now, with population growth land per worker, z = Z/L is going to decline at the rate at which population is growing, ie ż(t = nz(t, which implies that z(t = z(0e nt Therefore, lim t z(t = 0 Since y(t = k(t α z(t 1 α β, lim t z(t = 0 implies that lim t y(t = 0, which in turn implies that lim t k(t = 0 Thus, with population growth, there is only one stead-state, which is given by k = z = y = 0 Intuitively, growth in population reduces the amount of land per worker, thereby reducing the output per worker, which ultimately tends to zero since land is an essential factor of production Since the per capita variables tend to zero, it must be that aggregate variables, K and Y, grow over time, though at a rate slower than the population growth rate The return to land is given by p z (t = (1 α βl(t β K(t α Z α β Since L grows at rate n, K also grows, at a rate slower than n, and Z is constant, it implies that lim t p z (t = The intuition is that as population grows land becomes a scarce factor, and since it is essential for production, the return to land goes to The wage rate is given by w(t = βl(t β 1 K(t α Z 1 α β = βk(t α z(t 1 α β Since lim t k(t = lim t z(t = 0, wage rate tends to zero as t Intuitively, as labor becomes more and more abundant, it marginal product falls, and its falls to zero as t 3 The results we found in the previous part hold even when s = 1, ie individuals save all their income Thus, endogenizing savings rate is not going to offset the effect of diminishing returns However, if the growth rate of population was made to depend on the output per capita, then we could have a steady-state with positive y and k The idea is that higher output per capita implies a healthier population, and hence people 2

3 live longer and have more children This is the idea proposed by Malthus L(t L(t = n(y(t, where n (y > 0, lim y n(y = n > 0 and lim y n(y = n < 0 Thus, population growth is positive function of output per capita If output per capita is very high population growth is also very high, If output per capita is very low, population growth may be negative, implying population may shrink There exists a y = y such that n(y = 0 Such a formulation is also used in Parente and Prescott (2005 Question 2: Solow Model with a Different Price of Investment 1 Suppose the population growth rate is n Define capital per worker as k = K/L, and therefore output per worker based on our general production function is given by f(k = F(k,1 The law of motion for k is k(t = sq(tf (k(t (δ +nk(t Suppose there exists a BGP (balanced growth path in which k grows at rate g k 0 Then k(t = k(0e g kt Furthermore, q(t = q(0e γ Kt Using these in the law of motion for k implies g k k(0e g kt = sq(0e γ Kt f ( k(0e g kt (δ +nk(0e g kt, which simplifies to k(0 sq(0 (g k +δ +ne (g k γ K t = f ( k(0e g kt If g k = 0, then as t the left hand side goes to zero whereas the right hand side is constant Therefore, it must be that g k is positive But with positive g k the argument of f(, on the right hand side, goes to infinity Thus we can solve for f(k for any k [k(0, For any k, it is true that k = k(0e g kt, which implies that t = ln(k/k(0/g k Substituting this in the previous equation gives [ ] k(0 f(k = sq(0 (g 1 k +δ +n k (g k γ K /g k, k [k(0, (k(0 (g k γ K /g k 3

4 The term in the big square brackets is a constant Denote it by C Then which implies that f(k = Ck (g k γ K /g k, k [k(0,, F(K,L = Lf(k = CK (g k γ K /g k L γ K/g k, which is a Cobb-Douglas production function The intuition here is related to Uzawa s theorem which states that for the existence of a BGP, technological progress must be labor augmenting asymptotically In our model, here, technological progress is capital-augmenting or Hicks-neutral Thus, there is an inconsistency But, when the production function is Cobb-Douglas, its property of unitary elasticity of substitution between factors ensures that all kinds of technological progress are equivalent Thus, in this model only the Cobb-Douglas production function is consistent with the existence of a BGP 2 With a Cobb-Douglas production function, the law of motion for k is given by k(t k(t = sq(tk(tα 1 (δ +n On a BGP the left hand side is constant This implies that the right hand side is also constant, ie q(tk(t α 1 is not growing, which means that Then, since y(t = k(t α, on the BGP d(q(tk(t α 1 /dt q(tk(t α 1 = 0, q(t q(t +(α 1 k(t k(t = 0, g k = k(t k(t = q(t/q(t 1 α = γ K 1 α g y = ẏ(t y(t = α k(t k(t = α 1 α γ K Substituting the expression for g k in the original law of motion for k gives γ K 1 α +δ +n = sq(tk(tα 1, 4

5 k = [ k(t q(t = 1/(1 α γ K 1 α ] 1/(1 α s (1 +δ +n This is the steady-state value of the normalized capital-labor ratio Note that, off the steady-state, the growth rate of k is given by g ( k(t = k(t k(t γ K 1 α, g ( k(t = sq(tk(t α 1 (δ +n γ K 1 α, ] α 1 1/(1 α γ K g ( k(t = sq(t [ k(tq(t (δ +n 1 α, g ( k(t s k(t α 1 δ n γ K 1 α In steady-state g( k = 0 Since g( k is decreasing in k it implies that g ( k(t > 0 if k(t < k, g ( k(t < 0 if k(t > k This means that the steady-state is globally stable 3 On a BGP, the K/Y ratio must be constant But in this case g k = γ K /(1 α and g y = αg k, which means that aggregate output Y grows at slower rate than aggregate capital K Therefore, K/Y is not constant, which violates the Kaldor facts, and hence the definition of BGP But, in this model the price of consumption good is not the same as the price of the investment good Therefore, K and Y are not expressed in the same units Instead of considering K/Y we should consider (K/q/Y since q is the inverse of the price of capital in terms of the final good The growth rate of K/q is d(k(t/q(t/dt K(t/q(t = K(t K(t q(t q(t = g k +n γ K, d(k(t/q(t/dt K(t/q(t = α 1 α γ K +n = g y +n = Ẏ(t Y(t Hence K(t/q(t grows at the same rate as Y(t, thereby implying that (K/q/Y is constant on this BGP Question 3: Neoclassical Growth Model with Subsistence 5

6 1 The utility function captures the idea of a subsistence level of consumption, γ γ is the minimum level of consumption that consumer must have every period 2 Define effective capital-labor ratio as k = K/AL Then, a competitive equilibrium consistsofallocationsofconsumptionandeffectivecapital-laborratio[c(t,k(t] t=0 and of sequences of wages and interest rates [w(t,r(t] t=0 such that consumers maximize utility taking prices as given, firms maximize profits taking prices as given and markets clear 3 Maximizing the current value Hamiltonian for the consumer yields the the consumption Euler equation Ċ(t C(t = 1 ǫ u (C(t (r(t ρ, where ǫ u (C(t is the inverse of the intertemporal elasticity of substitution, and is given by ǫ u (C(t = u (C(tC(t u (C(t C(t = θ C(t γ Notice that if γ = 0 then ǫ u (C(t = θ, which is the usual case with CRRA preferences Thus, with a subsistence level of consumption, the intertemporal elasticity of substitution is no longer constant; instead it depends on the level of consumption The interest rate is given by the marginal product of capital net of depreciation, ie r(t = F K (K(t,A(tL(t δ = f (k(t δ Therefore, the consumption Euler equation becomes Ċ(t C(t = 1 ǫ u (C(t (f (k(t δ ρ The law of motion for capital per unit of effective labor, k, is given by k(t = f (k(t C(t (δ +gk(t A(t If we define consumption in units of effective labor, c = CL/AL = C/A, then the law of motion for capital and the consumption Euler equation are given by k(t = f (k(t c(t (δ +gk(t ċ(t c(t = 1 ǫ u (c(t (f (k(t δ ρ g, 6

7 where ǫ u (c(t = θ c(t c(t γ A(t Notice the dependence of the intertemporal elasticity of substitution on the level of technology in period t Thus, even though we have expressed our system of differential equations in terms of effective labor, there is still explicit dependence on A(t Again it is due to γ > 0 The consumption Euler equation and the law of motion for capital along with evolution of A(t and the transversality condition characterize the equilibrium The equation for the evolution of A(t and the transversality condition is given by A(t = ga(t, The factor prices are given by lim t e ρt µ(tk(t = 0 r(t = f (k(t δ, w(t = f (k(t f (k(tk(t Suppose there exists a BGP for k(t = k Then it must be that k(t = 0, so that f (k (δ +gk = c(t = C(t A(t Sincethelefthandsideisconstant, itimpliesthatc(t/a(thastobeconstant, orthat consumption per capita, C(t, grows at rate g Then the consumption Euler equation would imply that g = 1 θ C(t γ C(t (f (k δ ρ But this leads to a contradiction since the left hand side is constant but the right hand side changes over time as C(t grows at rate g Thus, the BGP does not exist in this economy The intuition is the consumer s intertemporal elasticity of substitution is not constant Instead it is decreasing in C(t, which means the consumption growth will be increasing in the level of consumption 7

8 4 The first order condition from the Hamiltonian gives µ(t[f (k(t δ g] = ρµ(t µ(t, µ(t = [f (k(t δ g ρ]µ(t, ( t µ(t = µ(0exp (f (k(s δ g ρds Substituting this in the transversality condition gives ( t lim µ(0exp (f (k(s δ gds k(t = 0 t Now, asymptotically lim t ǫ u (C(t = θ This implies that 0 0 Ċ(t lim t C(t = g = 1 θ (f (k δ ρ Using this equation to substitute for f (k in the transversality condition yields µ(0k lim t exp[((1 θg ρt] = 0, which can be satisfied only if ρ > (1 θg This is the required parametric restriction for the transversality condition to be satisfied 5 From the previous part it is clear that this economy has a BGP asymptotically, and this BGP is exactly that of the economy with γ = 0 Since there is no population growth both, aggregate and per capita variables, grow at rate g To see this we go back to the consumption Euler equation in terms of effective consumption where ċ(t c(t = 1 ǫ u (c(t (f (k(t δ ρ, ǫ u (c(t = θ c(t c(t γ A(t NotethatasA(tgrowsovertime,thetermγ/A(tgoestozero,andhencelim t ǫ u (c(t = θ, which is the case with standard CRRA preferences However, along a transition path the dynamics are different To see this, define x(t = C(t γ and x(t = x(t/a(t Then the consumption Euler equation in terms of this redefined variable becomes d x(t/dt x(t = 1 θ (f (k(t δ ρ θg, 8

9 which is identical to the one we see in case of our model with standard CRRA preferences 1 However, the law of motion for capital is no longer the same k(t = f (k(t x(t (δ +gk(t A(t 1 γ This equation has an extra term, A(t 1 γ, which is absent in the law of motion of our standard model Therefore, the k(t = 0 locus is given by x(t = f (k(t (δ +gk(t A(t 1 γ This locus shifts up over time as A(t 1 γ decreases with the growth in A(t As a result asymptotically this economy moves to the steady state of the economy with γ = 0 But, at every t, as the k(t = 0 locus shifts up, so does the saddle path Also, in every period t the saddle path is stable Question 4: Neoclassical Model with Overlapping Generations 1 Acompetitiveequilibriumisasequenceofaggregatecapitalandconsumption[K(t,C 1 (t,c 2 (t] t=0 and sequence of prices [w(t,r(t] t=0 such that households maximize utility given prices, and firms maximize profits given prices and markets clear This translates into max {C 1 (t,c 2 (t+1} logc 1 (t+βlogc 2 (t+1, st C 1 (t+s(t w(t, and C 2 (t+1 R(t+1S(t, 1 lnx(t = ln(c(t γ Taking the derivative wrt time gives us ẋ(t x(t = C(t Ċ(t C(t γ C(t Substituting for Ċ(t C(t, multiplying and dividing the right hand side by θ and using the expression for ǫ u(c(t gives: Since x(t = x(t/a(t, it implies that d x(t/dt x(t d x(t/dt x(t ẋ(t x(t = 1 θ (f (k(t δ ρ = ẋ(t x(t Ȧ(t A(t = 1 θ (f (k(t δ ρ θg Since technology grows at rate g we get 9

10 and R(t = αa(tk(t α 1 L(t 1 α, and finally markets must clear w(t = (1 αa(tk(t α L(t α, K(t+1 = L(tS(t 2 Define k K/L, c = C/L, s = S/L as per capita variables and L(t is the size of generation t Then R(t = αa(tk(t α 1, w(t = (1 αa(tk(t α and k(t + 1 = s(t/(1+n Following the class notes we have that and c 1 (t = 1 β w(t, and s(t = 1+β 1+β w(t, k(t+1 = β(1 αa(tk(tα (1+n(1+β Define capital per unit of effective labor as k = k/a 1/(1 α Then k(t+1 = α β(1 α ( k(t (1+n(1+β(1+g 1/(1 α Then in steady-state k(t+1 = k(t = k, which is given by [ k = β(1 α ] 1/(1 α (1+n(1+β(1+g 1/(1 α In the steady-state the capital labor ratio k grows by a factor of (1 + g 1/(1 α The steady state is stable - you can check that off the steady-state ( k(t+1/ k(t 1 > 0 if k(t < k and ( k(t+1/ k(t 1 < 0 if k(t > k Substituting k = ka 1/(1 α in the equation for interest rate gives us the steady-state interest rate ( k α 1 R α(1+n(1+β(1+g 1/(1 α = α = β(1 α Thus, the interest rate is constant in steady-state Similarly, the wage rate is given by αa(t 1/(1 α w t = (1 α ( k, 10

11 which implies that wage rate grows by the same factor as the capital labor ratio Similarly, consumption per capita of each generation is c 1 (t = 1 1+β w(t, and c 2(t = β 1+β w(tr Therefore, in steady-state the consumption per capita of the young and the old grows by the same factor as capital labor ratio Finally output per capita is given by αa(t y(t = A(tk(t α 1/(1 α = ( k, which means that output also grows by the same factor as capital labor ratio 3 Given a A(0 and k(0, a higher g implies higher k at every t 2 Also, an increase in g reduces the steady-state effective capital labor ratio, k It increases w(t at every t 2 since w(t is positive function of k and A Lastly, R, c 1 and c 2 are all higher at every t 2 4 If β > β then β /(1+β > β/(1+β Thus, if β increases at t = 1 then k(t will be higher for every t 2 Intuitively, a higher β implies that consumer attaches higher weight to future consumption, and therefore it induces individuals to save more thereby increasing capital labor ratio at every t A higher β also means a higher steady-state effective capital labor ratio, k Substituting the expression for k in the expression for c 1 (t gives c 1 (t = A(t 1/(1 α Ωβ α/(1 α (1+β 1/(1 α, where Ω is constant that does not depend on β The expression shows that the effect of an increase in β is ambiguous The intuition is that a higher β induces individuals to save more and therefore reduces c 1 (t, but higher savings imply higher capital stock and higher wages which increases c 1 (t The net effect is ambiguous Similar exercise for c 2 (t gives [ ] α/(1 α β c 2 (t = A(t 1/(1 α Ω 1+β This expression shows that c 2 (t increases due to a higher β This is due to higher savings, which is what the old consume This is despite the decrease in R(t+1 as a result of higher β (due to higher savings Wage rate increases due to higher capital 11