Macroeconomics I, UPF Professor ntonio Ciccone SOUTIONS PS 5, preliminary version 1 The Solow K model with transitional dynamics Consider the following Solow economy: production is determined by Y F (K, ) K + K α 1 α Population grows at rate n, capital depreciates at rate δconsumers save a fraction s of their income Moreover, s >δ + n δk, or, using d dt ( K ) K K and hence The intertemporal resource constraint is given by K sf (K, ), expressed per capita terms: sf() (δ + n) (s δ n) + s α s (δ + n) + sα 1 (1) Clearly, as t (ie ) : s (δ +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, 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)) Y Output per worer, + α, will therefore also approach a constant growth rate that is lower than the initial growth rate For two economies that differ only in their initial levels of capital, the poorer country will enjoy higher initial growth rates of both capital and output per worer 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 stoc per worer - and thus higher output per worer - ad infinitum 1
2 Ramsey-Cass Koopmans and the K model Consider an economy where the production possibilities are described by Y F (K ) K, where is a positive constant Consumption is maximized intertemporally by infinitely-lived consumers, which leads to the well-nown Euler equation: σ[r ρ] c(t) df (K) The capital maret equilibrium implies MP K : dk r + δ r δ 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) σ[r ρ] σ[ δ ρ] : γ c (2) (b) In this case capital growth will never be zero, but in the BGP it will grow at a constant rate This can only be the case if capital grows at the same rate as consumption Formally: t t γ ct t n δ ct t γ + n δ constant t t ct c t Further tae the derivative of the log wrt time (dlog/dt) of the production function to get: y t y t t t (c) from (2) follows that a decrease in ρ leads to an increase in the growth rate of consumption s permanent income does not change, the higher consumption growth can only be generated by consuming less today and more tomorrow Thus we need to increase the savings rate Formally sr s f() f() c f() [f() (n+δ) ] f() f() (n+δ)+ n+δ+γ n+δ+γc n+δ+σ( δ ρ) δsr δρ σ < 0 Thus the lower ρ (ie the less we discount future consumption), the more we will save 2
In case of decreasing returns to capital, we get the same qualitative results (ie a negative effect of the discount rate on the savings rate) However, in this Ramsey-Cass-Koopmans K model there are no transitional dynamics The savings rate immediately jumps to its new permanent level as in the Solow model 3 Growth through nowledge externalities in a Ramsey-Cass-Koopmans model Consider the following production function: Y K α () 1 α Efficiency is determined by the size of the aggregate capital stoc, K Population growth is zero Consumption is determined by the dynamic maximization, leading to the usual Euler equation: σ[r ρ] c(t) ey assumption: K ( 1 α ) with α, > 0, α + < 1 [note that in former versions of this problem there was K, but this results just in the K model (lie 52)] In either scenario, in the BGP we will have c(t) σ[mp K δ ρ] 0 MP K δ + ρ The Scenarios differ in the MPK In the decentralized equilibrium, HHs tae as given and therefore get: MPK Sc1 δy δk αkα 1 () 1 α plugging in the expression for αk α 1 K 1 α, in BGP: αk α 1 K 1 α δ + ρ 3
K Sc1 ( α 1 δ+ρ 1 α ) 1 α The Social Planner instead taes into consideration that depends on the capital stoc Therefore MPK Sc2 δy δk (α + )Kα+ 1 1 α, in BGP: (α + )K α+ 1 1 α δ + ρ K Sc2 ( α+ 1 δ+ρ 1 α ) 1 α K Sc1 < K Sc2 This comes from the fact, that households underestimate the overall productivity of capital, they do not tae into consideration the positive externality of capital on technology (b) Obviously, K Sc1 < K Sc2 depend on This changes if we assume that 1 α ) is determined by a function of captial per capita, eg or K( In the latter case we would get the same results for the capital stoc as in a), with the difference that cancels out 4 ucas (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 worforce input in production can hence be described as uh, where represents total labour force, and h is the level of human capital per capita: Y K α [uh] 1 α (Physical) capital depreciates with rate δ Human capital accumulation is proportional to the amount of time spent in education: ḣ (1 u) h Consumption is determined by dynamic maximization of infinitely-lived households, leading to the usual Euler equation: Output per capita: σ[r ρ] c(t) Y : f() ( ) α K (uh) 1 α α (uh) 1 α 4
The intertemporal resource constraint is, K F (K, ) δk C, or, in per capita terms: f() (δ + n) c where c : C/ (b) Define uh and plug it into the production function to get the usual Solow model The well-nown result is that in the steady state, per-capita capital stoc, consumption and output grow at the rate of, ẏ(t) eg y ỹ ỹconstant y(t) Ȧ(t) (t) a in this case uh, thus Ȧ (uh) uh uḣ+ uh ḣ uh u0 h (1 u) Thus in the BGP, c, y, and h grow at the rate (1-u) Clearly a policy maer can change this fundamental growth rate just by changing u 5 earning-by-doing Consider a standard Solow model without depreciation and without population growth ( δ n 0) The aggregate production function is Y K α () 1 α, α (0, 1) Substitute Y Kα () 1 α α a into s α 1 a to get ( ) s α Both s and are (positive) constants Therefore, the long-run growth rate of capital per effective worer must be zero, ie 0 Hence, the long-run level of is: BGP s The balanced growth path growth rate of income per capita is: ẏ y α s α 1 α BGP Finally, using y ỹ, the long-run level of income per capita is given by: ( s ) α y BGP 5
6 earning-by-doing again Now assume an economy as described in problem 55, but θ < 1 ( K α () 1 α) θ θ 1 θ αθ a Y θ where 0 < where, as usual, K/ The two-dimensional system governing the dynamics of capital per effective worer and technology is given by: ȧ a αθ + (θ 1)a s α 1 a In (, a) space the ȧ a 0 and 0 isoclines yield the following two curves ȧ a 0 a 0 a s α 1 αθ α 1 s 1 θ(1 α) Graphically, the system can be represented as in figure 1 Note, that the economy approaches a steady state just asymptotically, with and a 0 Extension: if instead n > 0, the the isoquants might cross and we can have a solution in finite time 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 of capital respectively labour are used in R&D The production function hence is: Y [(1 a K ) K] α [((1 a ) ] 1 α while new ideas are generated according to the following R&D process: B [a K K] [a ] γ θ where B,, γ, θ are positive constants Consumers save a fraction s of their income; depreciation is equal to zero, population grows at rate n g : B [a KK] [a ] γ θ 1 Hence, taing logs and derivatives For ġk gk ġ g g K + (θ 1)g + γn (3) observe that since δ 0 the capital accumulation equation simplifies to K sy Thus, g K K K sy K, where Y is given above and ġ K g K (α 1)g K + (1 α) g + (1 α)n (4) (b) Isoclines [( + θ) < 1] : ġ 0 g K γn g ġ K g K 0 g K n + g + (1 θ) }{{} >1 In this case we obtain convergence since the slope of the g isocline is smaller than the slope of the g K isocline (by assumption) ġ K gk (c) See figure 2 for a graphical illustration ftbpf35345in22701in0ptpicture2wmfon the balanced growth path, which yields g + γ 1 θ n g ġ ga 7
The equation for g depends only on the structural parameters of the R&D sector (d) ssumptions: + θ 1 and n 0 The isoclines are now: ġ g 0 g K ġ K g K 0 g K g Two possibilities may be distinguished: 1 if (1 θ) 2 if (1 θ) 1 the isoclines are identical (1 θ) }{{} 1 g 1 only point of intersection is the origin 8 Population growth and technological change (Kremer 1993) In a now famous QJE paper, Michael Kremer set out to explain why between 1 Million BC 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 Y t t α t T 1 α where T indicates (a fixed amount of) land, total factor productivity, world population and Y world output (Y/) t ȳ t t α 1 T 1 α Population adjusts such that income per person is equal to ȳ all the time Hence, ȳ a + (α 1)n 0 n a ȳ 1 α 8
Population growth is constant, and the popluation growth rate does not depend on the size of population Ȧ (b) If θ, the rate of population growth is no longer constant and the relationship with total world population at any point in time is: ȳ ȳ t t + (α 1)n t 0 n t θ 1 α t Thus the pop growth rate is proportional to the size of population The total population at any time t can be found by solving the above differential equation which yields t 0(1 α) (1 α) 0 θt Ȧ (c) If θϕ 1, ϕ < 1, the relationship of population growth and population is: ȳ ȳ t t + (α 1)n t 0 n t θ 1 α t ϕ 1 t Since ϕ < 1, population growth is inversely correlated with technology The total population is 0 (1 α) t (1 α) 0 θ t 0 (τ)dτ 9