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Macroeconomics I, UPF Professor Antonio Ciccone SOLUTIONS PROBLEM SET 5 1 The Solow AK model with transitional dynamics Consider the following Solow economy: production is determined by Y = F (K; L) = AK + K L1 Population grows at rate n, capital depreciates at rate :Consumers save a fraction s of their income. Moreover, sA > + n: (a) The intertemporal resource constraint is given by K_ = sF (K; L) _ _ K L d K (L) = K K, or, using k_ = dt L L L ; expressed per capita terms: k_ and hence = sf (k) = (sA k_ = sA k ( + n)k n)k + sk ( + n) + sk 1 (1) _ Clearly, as t ! 1 (i:e: k ! 1) : kk = sA ( + n) > 0, by assumption. This Solow economy features a constant positive growth rate in the very long run. Because of this perpetual growth dynamics no balanced growth path exists. The result is due to the fact that the returns to capital are decreasing but are never lower than A; even in the very long run. In consequence, net savings are always positive as capital accumulation is always advantageous. (b) Because of decreasing marginal product of capital the growth rate of capital must decrease as the economy accumulates more capital (see (1)). Output per worker, YL = Ak + k , will therefore also approach a constant growth rate that is lower than the initial growth rate. For two economies that di¤er only in their initial levels of capital, the poorer country will enjoy higher initial growth rates of both capital and output per worker than the rich country. Hence, there is convergence in growth rates in the very long-run but not in levels since because of the perpetual growth dynamics the initially richer country will have a higher capital stock per worker - and thus higher output per worker - ad in…nitum. 1 2 Ramsey-Cass Koopmans and the AK model Consider an economy where the production possibilities are described by Y = F (K ) = AK , where A is a positive constant. Consumption is maximized intertemporally by in…nitely-lived consumers, which leads to the well-known Euler equation: c(t) _ = [r c(t) ] (K) (a) The capital market equilibrium implies M P K := dFdK = A = r+ , r = A . The real interest rate is a constant in this model because the marginal product of capital is constant. Thus consumption growth in equilibrium is c(t) _ = [r ] = [A ] =: c : (2) c(t) (b) A constant real interest rate implies a constant consumption - capital ratio and hence a constant consumption - output ratio. The policy function for consumption has the following form: C(t) = cY (t); where c is some positive constant. Hence, in a balanced growth path the growth rate of consumption will be equal to the growth rates of k and y; that is, c = k = y : (c) By (2) an unexpected fall in leaves r = A unchanged and induces an increase in the growth rates of consumption, capital and output in the new balanced growth path. From the capital accumulation equation we see _ K(t) C(t) that K(t) =A due to a fall in must lead to K(t) = . An increase in C(t) a fall in the consumption - capital ratio, K(t) , which is to say that the savings rate has to increase. To see this e¤ect more directly, notice that we can rewrite the capital accumulation equation as _ AK(t) C(t) K(t) = AK(t) AK(t) | {z } _ K(t) K(t) , =: AK(t) K(t) = sA ,s= + ; A s where s is the savings rate. Thus, an increase in the growth rate, , increases the savings rate, as well. The same qualitative results materialise if the production function is subject to decreasing returns to capital. However, in this RamseyCass-Koopmans AK model there are no transitional dynamics. The savings rate immediately jumps to its new permanent level as in the Solow model. 2 3 Growth through knowledge externalities in a Ramsey-Cass-Koopmans model Consider the following production function: Y = K (AL)1 E¢ ciency A is determined by the size of the aggregate capital stock, K. Population growth is zero. Consumption is determined by the dynamic maximization, leading to the usual Euler equation: c(t) _ = [r c(t) ]: (a) In the decentralized equilibrium …rms maximize their pro…ts subject ~ = rt + ; i.e. the externality is not taken into account to costs which yields f 0 (k) by the marginal …rm. The capital accumulation equation for capital per e¤ective labour is unchanged : k~t = f (k~t ) ct k~t = k~t ct k~t : The amount of capital on the balanced growth path is determined by the c_ = 0 locus: ~ f 0 (k) = + , k~ 1 = + (3) 1 1 ) k~BGP = : + The social planner, on the contrary, takes into account the interaction between the size of the capital stock K and A. That is, the planner chooses a capital market equilibrium that re‡ects the total marginal product of capital: K 1 ~ 1 : Y = F (K; AL) = KL1 since A = K and therefore y~ AL = L = kL Thus, @ y~ = L1 : @ k~ which implies f 0 (k) = L1 = + : Hence, 1 k~SP = = L 1 + 1 1 : (4) Comparing the capital stock in the decentralized market equilibrium (3) and the social planner allocation (4) we observe that since 2 (0; 1) ; k~SP > k~BGP : Because …rms do not take into account the link between technology and the 3 capital stock in the decentralized equilibrium there is less capital than in the social planner optimal allocation. (b) Yes, the capital stock in (a) depends on the size of the labour force. The marginal product of capital depends on the size of the labour force and so does the capital stock. If the externality is determined by capital intensity, i.e. A = k, instead, the production function facing the social planner reduces to 1 = K: Hence, y~ = k~ and the marginal product Y = K (AL)1 = K ( K L L) 0 ~ of capital is f (k) = 1: It does not depend on labour, therefore the capital stock is independent of the size of the labour force. 4 Lucas (1988) model of human capital Consider the following economy: individuals spend a fraction (1-u) of their lifetimes in accumulating human capital and the remainder (u) in production. The workforce input in production can hence be described as uLh, where L represents total labour force, and h is the level of human capital per capita: 1 Y = K [uLh] : (Physical) capital depreciates with rate : Human capital accumulation is proportional to the amount of time spent in education: h_ = (1 h u): Consumption is determined by dynamic maximization of in…nitely-lived households, leading to the usual Euler equation: c(t) _ = [r c(t) (a) ]: Output per capita: Y =: f (k) = L K L 1 1 (uh) = k (uh) The intertemporal resource constraint is, K_ = F (K; L) capita terms: k_ = f (k) ( + n) k c : K C; or, in per where c := C=L: K : Then, as in the standard Ramsey-Cass-Koopmans (b) De…ne k~ := uhL ~ model, there exists a balanced growth path for k~ where dk=dt =0: Implying that on a balanced growth path d ~ d (k) = dt dt K uhL = 4 d dt k uh = 0: (5) _ Since h=h = (1 h: Thus, u); in order for (5) to be ful…lled, k must be proportional to k_ h_ = = (1 u): k h Moreover, since the capital market is in equilibrium capital earns its marginal 1 kt = rt + : On a balanced growth path k product: M P K = f 0 (k) = uht and h grow at the same (constant) rate therefore the physical-to-human capital ratio, k=uh; is constant along a balanced growth path. Hence, the real interest rate is constant, implying that consumption, physical and human capital all _ _ grow at the same constant growth rate in equilibrium: c=c _ = k=k = h=h = (1 u): The only way policymakers can in‡uence long-run growth is to increase the accumulation of human capital. 5 Learning-by-doing Consider a standard Solow model without depreciation and without population growth ( = n = 0): The aggregate production function is Y = K (AL)1 ; 2 (0; 1): Substitute 1 A_ Y K (AL) = = = k~ L = a A A A into : k~ = sk~ 1 a k~ to get : k~ = k~ k~ s k~ L : Both s and L are (positive) constants. Therefore, the long-run growth rate of : ~ capital per e¤ective worker must be zero, i.e. kk~ = 0: Hence, the long-run level of k~ is: s k~BGP = : L The balanced growth path growth rate of income per capita is: y_ y = Lk~ = s L1 : BGP Finally, using y = A~ y , the long-run level of income per capita is given by: yBGP = A 5 s L : 6 Learning-by-doing again Now assume an economy as described in problem 5.5, but A_ = Y < 1: where 0 < (a) K (AL)1 A_ = A A =A 1 L k~ =a where, as usual, k~ = K=AL: The two-dimensional system governing the dynamics of capital per e¤ective worker and technology is given by: : a_ a : k~ = = sk~ k In (k; a) space the a_ a = 0 and a_ a k~ +( k k_ k 1 1)a a: = 0 isoclines yield the following two curves = 0)a= = 0 ) a = sk~ 1 (1 ) sk~ 1 : k~ k 1 Graphically, the system can be represented as in …gure 1. The balanced growth path is globally stable as can be seen from the two red lines depicting possible paths of the system. k - isocline a - isocline 6 7 Endogenous growth, ideas and capital Consider the following endogenous growth model: both capital and labour can be used either in production of goods or in research and development. Fractions a K and a L of capital respectively labour are used in R&D. The production function hence is: 1 Y = [(1 aK ) K] [(A(1 aL ) L] while new ideas are generated according to the following R&D process: A_ = B [aK K] [aL L] A where B, ; , are positive constants. Consumers save a fraction s of their income; depreciation is equal to zero, population grows at rate n. (a) A_ = B [aK K] [aL L] A 1 : gA := A Hence, taking logs and derivatives g_ A = gK + ( gA 1)gA + n: (6) K For gg_ K observe that since = 0 the capital accumulation equation simpli…es to _ K _ K = sY: Thus, gK = K = sY K ; where Y is given above and g_ K =( gK (b) 1)gK + (1 ) gA + (1 )n: (7) Isoclines [( + ) < 1] : g_ A gA = n 0 ) gK = + (1 ) gA | {z } >1 g_ K gK = 0 ) gK = n + gA In this case we obtain convergence since the slope of the gA isocline is smaller than the slope of the gK isocline (by assumption). (c) See …gure 2 for a graphical illustration. K which yields On the balanced growth path, gg_ Aa = gg_ K gA = + 1 n: The equation for gA depends only on the structural parameters of the R&D sector. 7 gK gA isocline gK isocline gA (d) Assumptions: + g_ A gA 1 and n = 0: The isoclines are now: = 0 ) gK = (1 ) gA | {z } 1 g_ K gK = 0 ) gK = gA Two possibilities may be distinguished: 8 1. if (1 ) = 1 ) the isoclines are identical. 2. if (1 ) 6= 1 ) only point of intersection is the origin. Population growth and technological change (Kremer 1993) In a now famous QJE paper, Michael Kremer set out to explain why between 1 Million B.C. and 1990, a larger world population was going together with a faster rate of world population growth. His idea was that this could be explained by more people generating more ideas. To show this, he postulated that the aggregate production function during this period was Yt = At Lt T 1 where T indicates (a …xed amount of ) land, A total factor productivity, L world population and Y world output. 8 (a) (Y =L)t = y = At Lt 1 T 1 : Population adjusts such that income per person is equal to y all the time. Hence, : y =a+( y 1)n = 0 ) n = a 1 : Population growth is constant, as well. The level of population at any point in time is Lt = L0 exp(nt) where n = n(a) is given above. _ A (b) If A = L; the rate of population growth is no longer constant and the relationship with total world population at any point in time is: : A_ t y = +( y At 1)nt = 0 ) nt = 1 Lt : The total population at any time t can be found by solving the above di¤erential equation which yields L0 (1 ) Lt = : (1 ) L0 t _ A (c) If A = LA' population is: 1 ; ' < 1; the relationship of population growth and : A_ t y = +( y At 1)nt = 0 ) nt = 1 Lt A' t 1 : Since ' < 1; population growth is inversely correlated with technology. The total population is L0 (1 ) Lt = : Rt (1 ) L0 0 A( )d 9

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