# Problem Set #1: Exogenous Growth Models

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1 University of Warwick EC9A2 Advanced Macroeconomic Analysis Problem Set #1: Exogenous Growth Models Jorge F. Chavez December 3, 2012 Question 1 Production is given by: Y t F (K t, L t ) = AK α t L 1 α t where L t+1 = (1 + n)l t and α (0, 1). a. Show that F exhibits a constant return to scale technology Solution. This implies showing that F : R 2 R is homogeneous of degree 1. That s it, we need to show that for λ > 0, F (λk t, λl t ) = λf (K t, L t ). Then: F (λk t, λl t ) = A(λK t ) α (λl t ) 1 α = λak α t L 1 α t = λf (K t, L t ) b. Express output as a function of the capital labor ratio k t = K t /L t. Solution. Use the fact that F is homogeneous of degree 1: ( ) α F (K t, L t ) = AKt α L 1 α Kt t = L t A = L t Akt α ; L t c. Find the wage rate per worker and the rental rate per capital. Solution. Profit maximization (either by a social planner or by the representative firm in the economy) yields: w t = F (K t, L t ) L t R t = F (K t, L t ) K t = (1 α) AK α t L α t = (1 α) Ak α t = αakt α 1 L 1 α t = αakt α 1 Note that this assumes that the good acts here as a numeraire (its price is set to 1). 1

2 d. Find the dynamical system (describing the evolution of k t over time) under the assumption that the saving rate is s (0, 1) and the depreciation rate is δ (0, 1]. Solution. We need to characterize the path of {k t } t=0 given s (0, 1) and δ (0, 1]. That is, we need to find the non-linear difference equation k t+1 = ϕ(k t ). Start from the law of motion of the aggregate stock of capital: K t+1 = sy t + (1 δ)k t Put everything in per-worker terms by dividing both sides by L t+1 : k t+1 = K t+1 L t+1 = s Y t L t + (1 δ) K t L t L t+1 L t L t L t+1 = sakα t + (1 δ) k t 1 + n ϕ (k t ) (1) where I am using y t = f (k t ) = Ak α t. Sometimes it useful to re-express this condition in terms of k t+1. To do so subtract k t from both sides of (1) to get: k t+1 = sakα t (n + δ) k t 1 + n e. What is the growth rate of k t, γ kt (k t+1 k t ) /k t? Solution. γ kt = k t+1 k t k t = sakα 1 t (δ + n) 1 + n f. Find the steady state level of the stock of capital per worker k, income per worker y and consumption per capita c. Solution. In steady-state kt+1 ss = kt ss = k, which implies that γ k = 0. Then from the expression for γ kt : k = ( ) 1/(1 α) sa δ + n Income per worker is ( ) α y = f (k) = Ak α = A 1 1 α s 1 α δ + n Finally, to get consumption per worker in steady state note that in general y t = c t + i t, i t = sy t and c t = (1 s)y t. Then, in steady state: ( ) α c = (1 s) y = (1 s) A 1 1 α s 1 α δ + n Jorge F. Chávez 2

3 g. What is the Golden Rule value of k? (k in steady state s.th. the consumption in steady state is maximized?) Solution. The Golden Rule value of k t is the stock of capital per worker that maximizes consumption in steady-state. Recall that in general: c t = (1 s)y t = f(k t ) sf(k t ) Note that there is no way to maximize consumption in all states as consumption is a function of f(k t ) which is unbounded (we can only get the corner solution k t = 0 if we look for a stationary point). However, the steady-state condition is: sf (k) }{{} Investment per capita = (n + δ) k }{{} Part of k that is lost because of pop. growth & depreciation (2) which is true for any steady-state (any setting in which k ss t+1 = k ss t = k). Then because c t = f(k t ) sf(k t ), in steady state: c = f(k) (n + δ)k which now will accept an interior solution. The FONC for the maximization problem is: c k = f (k) (n + δ) = 0 f ( k GR) = n + δ For y = Ak α we can get a closed form solution for k GR : f ( k GR) = αa ( k GR) α 1 = n + δ k GR = ( ) 1/(1 α) αa n + δ (3) The concept of the Golden Rule for the case of the Solow model is illustrated in figure 1. There you can see three alternative saving rates, and you can visualize the Golden Rule condition f ( k GR) = n + δ: the slope of f( ) must be equal to the slope of the (n + δ)k line. h. What saving rate is needed to yield the Golden Rule? Solution. Comparing equation (3) with the expression for capital in steady state it is straightforward to see that the saving rate that allows the economy to reach k GR is s GR = α. Alternatively recall that f (k GR ) = n + δ and k t f (k t )/f(k t ) = α. Then: k GR f ( k GR) f (k GR ) = kgr (n + δ) f (k GR ) = α But then the Golden Rule level of capital also satisfies the more general condition for all steady-states. Combining that condition (2) with the above one we can see that s GR = α. i. Find the elasticity of y with respect to s (in steady-state). Can observed differences in saving rates explain the observed differences in income per-capita across the world? Jorge F. Chávez 3

4 Figure 1: The Golden Rule f(k ss ) c 1 (n+δ)k ss c 2 c GR s 2 f(k ss ) s 1 f(k ss ) s GR f(k GR ) k 2 k GR k 1 k ss Solution. Recall that: ( ) α/(1 α) sa y = Ak α = A n + δ Taking logs 1 : ln y = α 1 α ln A + α ln s α }{{} ε ys Suppose that α = 1/3 as it is usually found in empirical studies and there is a difference in saving rates of 3 times across countries (300%). Then the elasticity will be α/(1 α) = 0.5 which implies that the difference in y according to this model (with Cobb-Douglas technology) should be % = 150%. However the observed difference is nearly 20 times (this comes from the lecture notes). j. Find the dynamical system describing the evolution of y t under the assumption of full depreciation δ = 1. Solution. With δ = 1 the law of motion of the stock of capital per worker is just: 1 To see this: k t+1 = sakα t 1 + n log y log y log x = y y x log x x y y x x = % y % x Jorge F. Chávez 4

5 Note that the numerator is investment in per capita terms. Then, this expression says that the new capital stock (which is completely renewed each period) is lower than investment per worker, due to population growth. 2 We want to characterize {y t } t=0 by analyzing a non-linear difference equation y t+1 = φ(y t ). Recall that y t = Akt α. Then: ( ) sak y t+1 = Akt+1 α α α ( ) α = A t syt = A φ (y t ) 1 + n 1 + n k. Suppose you estimate a regression in which ln ( yt+1 i t) /yi is on the left hand side (i.e. you estimate the log of γy i t + 1 across countries, where i indexes countries) and ln s i, ln ( 1 + n i) and ln yt i are on the right hand side. According to the dynamical system you defined in question j, what would be the coefficients on your explanatory variables? How would you interpret these coefficients? Is there β convergence? How would you interpret the constant term? Solution. Consider the regression: ( y i ) ln t+1 = ln A + γ 1 ln s i + γ 2 ln ( 1 + n i) + γ 3 ln yt i + ε it y i t From the model we can get an estimable form by taking logs to the law of motion of y t+1 when δ = 0: ln y i t+1 = ln A + α ln s α ln (1 + n) + α ln y i t Then by subtracting lny i t from both sides of the equation above we can see that γ 0 = ln A, γ 1 = α, γ 2 = α and γ 3 = (1 α). 2 Investment occurs in t while the new capital stock will be readily available in t + 1. Jorge F. Chávez 5

6 Question 2 Output per worker is an increasing and concave function of capital per worker, given by: y t = f(k t ). Output is divided between labor income and capital income according to their marginal productivity. Namely, f (k t ) [ f (k t ) f (k t ) k t ] + f (k t ) k t = w t + r t k t Suppose that the rate of saving from wage income is s w [0, 1] and the rate of saving from capital income is s r [0, 1]. Therefore, total saving are given by s t = s w [ f (k t ) f (k t ) k t ] + sr f (k t ) k t Population and technology are constant and the rate of capital depreciation δ (0, 1) a. Derive the dynamical system governing the evolution of capital per capita: k t+1 = ϕ(k t ). Solution. Recall that the law of motion of the stock of capital per capita comes from the law of motion of the aggregate stock per capita: K t+1 = I t + (1 δ)k t where I t denotes aggregate investment. Because there are two distinct saving rates, we can think of aggregate savings to be determined by an average saving rate ( s) which is a weighted sum of s w and s r : K t+1 = sy t + (1 δ)k t Now, because there is no population growth, if we divide everything by L we get: k t+1 = s t + (1 δ)k t where s t = sf(k t ) = s w [f(k t ) f (k t )k t ] + s r f (k t )k t. Then: k t+1 = s w [f(k t ) f (k t )k t ] + s r f (k t )k t + (1 δ)k t = s w f(k t ) + (s r s w ) f (k t )k t ϕ(k t ) Note that if s w = s r, then the expression for φ(k t ) is the same we had before with a single saving rate. b. Suppose there exist a range of k t where f (k) = 0 (the third derivative is zero). Find a condition on the saving rates s w and s r such that the dynamical system k t+1 = ϕ(k t ) is convex (ϕ (k t ) > 0 in the range of f (k t ) = 0. Solution. First let s get the derivatives: ϕ (k t ) = s w f (k t ) + (s r s w ) [f (k t ) k t + f (k t )] + (1 δ) ϕ (k t ) = s w f (k t ) + (s r s w ) [f (k t ) k t + f (k t )] Let k be the maximum stock of capital per worker that can be attained by economy and assume that Jorge F. Chávez 6

7 k 1 k 2 [0, k] such that k t [k 1, k 2 ], f (k t ) = 0. Then, for those values of k t, using (4): ϕ (k t ) kt [k 1,k 2] = s w f (k t ) + (s r s w ) f (k t ) = (2s r s w ) f (k t ) Therefore, for ϕ(k t ) > 0 we need 2s r < s w and strict concavity in the range [k 1, k 2 ] (concavity is not enough) of f. 3 Suppose now that s r = 0 and f(k t ) = ln (1 + k t ) c. Derive the dynamical system k t+1 = ϕ(k t ). Solution. Now savings come only from wage income: k t+1 = s w f (k t ) s w f (k t ) k t + (1 δ) k t ϕ(k t ) Replacing the functional form for f( ) we get that f (k t ) = 1/(1 + k t ). Then: k t+1 = s w log (1 + k t ) s w + (1 δ) k t ϕ (k t ) 1 + k t k t d. Find ϕ (k t ), kt 0 ϕ (k t ), kt + ϕ (k t ), and ϕ (k t ). Solution. (i) First derivative ϕ (k t ) = = ( ) s w s w (1 + k t ) k t 1 + k t (1 + k t ) 2 + (1 δ) s w k t 2 + (1 δ) (1 + k t ) (ii) Limit when k t 0 k t 0 ϕ (k t ) = 1 δ (iii) Limit when k t + s w k t k t + ϕ (k t ) = k t (1 δ) (1 + k t ) = 1 δ Note that the it of the first term of the RHS is not undetermined (no need to apply L Hospital rule) 3 Strict concavity implies that f (k t) < 0 Jorge F. Chávez 7

8 (iv) Second derivative: ϕ (k t ) = s w [ (1 + k t ) 2 2 (1 + k t ) k t (1 + k t ) 4 [ ] = s w 1 + 2k t + kt 2 2k t 2kt 2 (1 + k t ) 4 ( ) = s w 1 kt 2 (1 + k t ) 4 e. Is the trivial steady state, k = 0, locally stable? Explain. Solution. To analyze stability of the trivial steady-state (k = 0), we need to evaluate the it of the first derivative of ϕ (k t ): k ϕ (k t ) = (1 δ) (0, 1) t 0 Therefore k = 0 is locally stable. Remark 1. Recall that for a non-linear difference equation like x t+1 = g(x t ) with fixed-point or steady state x ss4. Then: If f (x ss ) < 1 then x ss is locally asymptotically stable. If f (x ss ) > 1 then x ss is locally asymptotically instable. Remark 2. Recall that in the standard case: k t+1 = sf(k t ) + (1 δ)k t Then, to analyze the stability of k = 0 we need: k = 0 ϕ (0) = s f (0) + (1 δ) > 1 }{{} + which implies that in that case k = 0 is unstable. ] f. Find the range of k t in which the dynamical system is strictly convex. Solution. We want to find k 1 and k 2 such that k t [k 1, k 2 ], ϕ (k t ) > 0. Recall that: ϕ (k t ) = s w 1 k2 t (1 + k t ) 4 Now, setting ϕ (k t ) > 0 1 k 2 t > 0 k 2 t < 1 k t < 1 or k t > 1. But k t is the stock of physical capital per worker, so in principle k t 0. Thus, because ϕ (0) > 0, the range we were looking for is [0, 1]. g. Show that for δ = 0 a non-trivial steady state level of k does not exist (that is, explain why there exists no k > 0 such that k = ϕ( k)). Find the growth rate of k t (i.e., k t+1 /k t 1) as k t for δ = 0. 4 That is x ss = g(x ss ) Jorge F. Chávez 8

9 Solution. WTS if δ = 0 then a non-trivial steady state k > 0 for {k t }. Note that with δ = 0: k t 0 ϕ (k t ) = k t + ϕ (k t ) = 1 which means that this system does not have an interior steady-state. More formally: Claim 1. There is no non-zero fixed point for ϕ(k t ) (that is k > 0 s.th. k = ϕ(k)). Proof. Suppose not: k > 0 s.th k = ϕ(k). Then: ϕ (k t ) δ=0 = s w log (1 + k t ) s w + k t 1 + k t k t Then k > 0 must satisfy: k = ϕ (k) δ=0 = s w log (1 + k) sw k + k + k k Manipulating this expression a little bit we get to an expression that k > 0 must satisfy: log (1 + k) (1 + k) = k 1 + k = exp ( k ) 1 + k (4) Think about slopes. The LHS is a linear function with a constant slope equal( to 1. ) The RHS is an exponential function with slope < 1 for all values of k 0. 5 For 1 + k = exp to be true, the two functions must intersect at some point k > 0. They only intersect (in fact they are only tangent) at k = 0. This is a contradiction, which means that our starting premise was wrong. Q.E.D Finally, the growth rate of k t is: k 1+k γ kt = k t+1 k t log k t+1 log k t (5) k [ t log (1 + = s w kt ) 1 ] (6) k t 1 + k t 5 To see this note that the slope is ( ) k exp 1+k 1 = k (1 + k) 2 } {{ } <1 ( k exp ) < k } {{ } <1 Jorge F. Chávez 9

10 where the second equality comes from replacing k t+1 with the expression for ϕ(k t ). Taking its: [ log (1 + γ kt ) k t = k t k t sw 1 ] k t 1 + k t [ ] = s w log (1 + k t ) 1 k t k t k t 1 + k t [ ] 1 = s w 1+k t 1 k t 1 k t 1 + k t = 0 where again, the second-to-last equality applies L Hospital rule for undetermined its. Jorge F. Chávez 10

11 Question 3 Production if given by: Y t F (K t, L t ) = Kt α (B t L t ) 1 α where L t+1 = (1 + n)l t, B t+1 = (1 + g)b t and α (0, 1). a. Find the dynamical system describing the evolution of k t = K t / (B t L t ) (that is the stock of capital per-effective worker). Find the steady state level of k t. What is the growth rate of output per worker y t = Y t /L t in the steady state? What is the growth rate of capital per worker K t /L t = B t k t in the steady state? Solution. i. We want to find the non-linear difference equation k t+1 = φ(k t ) for k t K t /(B t L t ) Start from the law of motion of the aggregate stock of capital K t+1 = I t + (1 δ) K t. Recall that this is a closed economy and that there is a constant (and exogenous) saving rate s: S t = I t = sy t Putting everything in per-effective worker terms (by dividing by B t L t ): K t+1 = s F (K t, L t ) B t+1 L t+1 B t L t We get the non-linear difference equation: k t+1 = skα t + (1 δ) k t (1 + g) (1 + n) φ (k t) B t L t + (1 δ) K t B t L t B t+1 L t+1 B t L B t+1 L t+1 As before it is useful to also express the law of motion of k t in terms of k t+1 k t+1 k t. To do this subtract k t from both sides of φ(k t ): k t+1 = skα t + (1 δ) k t (1 + n + g + gn 1 + δ) k t (1 + g) (1 + n) = skα t (n + g + gn + δ) k t (1 + g) (1 + n) ii. Steady state for k t In steady-state kt+1 ss = kt ss = k. Then: k ss t = 0 sk α (n + g + gn + δ) k = 0 Solving for k: [ s k = (n + g + gn + δ) ] 1/(1 α) To check stability of the steady-state we need to check whether k t φ (k t ) = 1 δ (1 + g) (1 + n) < 1 k t φ (k t ) is less than unity: Jorge F. Chávez 11

12 Therefore, the steady-state k is stable. iii. Growth rate of output per worker y t = Y t /L t in steady state Note that: Y t = Kα t (B t L t ) 1 α ( ) α ( ) α Kt = B 1 α Kt t = B L t L t L t B t L t = kt α B t t Now, once the economy reaches an steady-state y ss t = k α B t Then, we can see: yt+1 ss yt ss 1 = B t+1 1 = g B t Alternatively we can take logs to y ss t = k α B t : log yt+1 ss log yt ss = log B t+1 log B t g iv. Growth rate of capital per worker K t /L t = B t k t, where k t = K t /(B t L t ). Same as above. First define the per-capita stock of capital k t = K t /L t : K t L t k t = k t B t Then get the growth rates directly or by the logarithm approximation: log k t+1 log k t = log B t+1 log B t g b. According to Solow s growth accounting, the growth rate in Total Factor Productivity (TFP) is calculated by: A t A t = y t y t α k t k t where y t Y t /L t, k t K t /L t, α is the elasticity of y t with respect to k t (which is equal to the share of capital in a competitive economy). Find TFP growth rate A t /A t in the steady state according to the model in part a. Solution. According to part (a) both y t = Y t /L t and k t = K t /L t will grow at a rate g in steady-state. Therefore: A t A t = y t y t = g αg α k t k t Jorge F. Chávez 12

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