1 Jon Vislie ECON 4350 (Growth and Investment) Spring 2009 Seminar 2 (wee 9) Problem 1. In the standard neoclassical growth model (the Solow model ), we have, under certain circumstances, a steady state or a stationary solution. For this unique stationary (long run) equilibrium, discuss whether the following statements are true: The higher is the saving rate, the higher is the wage rate The higher is the steady state value of capital per labour, the lower is the rate of return on capital Consumption per head will be higher if the actual saving rate exceeds the one associated with the Golden Rule of Accumulation. In the standard neoclassical growth model, reduced to t () = sft ( ()) nt (), where : = ; capital per labour unit, s (0,1) is an exogenous saving rate, whereas n is X the growth rate in the labour force. f () = F(,1): f () = =, with F (, ) is CRS, and strictly concave, with positive marginal productivities. Given that the Inadaconditions are satisfied, implying the inputs to be essential; F(0, ) = F(, 0) = 0, n then we have lim 0 f ( ) =, lim f ( ) = 0 ; hence we have f (0) >. We then s have: f(0) = 0, f ( ) > 0, f ( ) < 0. Furthermore, = > 0: sf( ) n = 0, with d being asymptotically stable, as = sf ( ) n < 0, as sf () cuts n from d = 0 above. n f( ) The stationary state is a decreasing function of =, because the RHS is s n declining in. et = φ( ), with φ < 0; hence we have s f( ) f ( ) + sf ( ) n = 0 = > 0. We also now that s s s sf ( ) n w = f() f () and r = f () as respectively real wage and rate of return on capital. An illustration below:
2 f () r ( ) n sf () w ( ) What is the impact on real wage in long run equilibrium from a higher saving rate? We have w ( ( s)) = f ( ( s)) ( s) f ( ( s)). We then have: w f ( ) f ( ) f ( ) f ( ) = = > 0. The s s s s s statement is true. dr Because r = f ( ), we find that = f ( ) < 0, due for instance to a higher d saving rate. The statement is true! Golden Rule of Accumulation is related to the saving rate that maximises steady state consumption per head: c() s = (1 s) f();.. st sf( ) = n. For each saving rate there corresponds a steady state value of capital per worer. The problem is then reformulated to be: Ma x f s n s 0,1 ( ( )) ( s). The [ ] { } saving rate solving this problem, s (0,1), is determined implicitly by f ( ( s )) = n, which defines the optimal social rate of discount. Because consumption per head is maximised for s = s, we must have c < c ( s ) for any s ( s,1]. The statement is not true!
3 Problem 2. a) Show that the rate of growth of total output, Xt () = Ft ( (), t ()), can be expressed as Xt () t () t () = n + ε, where t () = nt (), t (): = and where ε : = F is the Xt () t () t () X marginal elasticity of capital, which is positive and less than one. b) Explain the transition towards the stationary state, and show that as long as capital per t () Xt () t () worer is increasing, > > = n. How will the relative income distribution be t () Xt () t () affected during this transition? c) Explain the transition towards a stationary state if initial capital per worer is above its steady state level. a) This follows from straightforward differentiation by taing the log s or ln of Xt ( ) = t ( ) ft ( ( )) ln Xt ( ) = ln t ( ) + ln ft ( ( )), which gives: t () Xt () t () t () t () t () t () t () = + f (()) t = n + F Xt () = n + ε. Hence Xt () t () ft (()) t () t () t () t () Xt () t () we observe that the rate of change in output per worer; n = ε Xt () t () will increase as long as capital per worer is increasing; i.e. as long as t () t () Xt () t () t () > = n. In steady state, we have = = = n. t () t () Xt () t () t () b) In the region where capital per worer is increasing; i.e. as long as t () <, t () t () Xt () we of course have >. What about in this region? From the t () t () Xt () production function we get: () () () () ( () () () Xt t t t t Xt = Ft + Ft = F + F = ε + ε. Xt () X t () X t () t () t () Because F (, ) is CRS, we have ε + ε = 1 and both ε j > 0. Using that t () t () > in this region, we have: t () t () t () t ( t ( Xt () t () t ( t ( t ( ε + ε = > = ε + ε > ε + ε t () t () t () Xt () t () t () t () t () = ( ε + ε ) n = n (The same conclusion can be reached by other means.) When capital per labour is increasing, we have according to our findings above that domestic output per worer will increase. But due to decreasing marginal productivity of capital, or f < 0, output per worer will increase at a slower rate; hence as we move towards the steady state, the rate of increase
4 Xt () t ( t ( Xt ( ) will decline towards = n. Because we also have > when Xt () t () t () Xt () Xt () 1 t () capital per worer is increasing, output per unit capital; =, will t () st () t () decline, as will also the rate of increase in capital. When t, we have t () Xt () t () t () = = = n. Xt () t () t () What about the income distribution during this transition? The relative r income distribution can be considered by the ratio w : Z w = = λ =, where r w λ : = and Z : =. r λ Question: How is Z affected during the transition towards steady state when we start out with (0) <? In other words, how is the factor proportion or capital per worer affected by an increasing λ? (We now that as we move towards the steady state from (0) <, wage will increase whereas rate of return on capital will go down.) We then define a concept called the elasticity of d λ d substitution; defined formally as σ : = El : λ = = ; showing a curvature dλ dλ λ property of an isoquant; the higher (smaller) is σ, the less (more) curved is an isoquant. (A eontief technology has σ = 0.) A one percent increase in the price ratio will lead to a σ percent increase in. We then get: El Z : λ = El : λ Elλ : λ = σ 1. Hence, if we have a production technology with an elasticity of substitution exceeding one, then as we move towards steady state, capital owners will improve their relative standing as their income (r) increases relative to wage income (w). As the factor price ratio λ is increasing, a high elasticity of substitution will lead to large reduction in the use of labour, for fixed output, relative to the use of capital. But output itself will increase. On the other hand if we have no substitution, σ = 0, then as we move towards steady state, Z will go down capital owners will experience a decline in their relative income raning/standing. c) Reverse the line of reasoning above.
5 Problem 3. Explain the transition from one steady state towards another one when the saving rate decreases from a level above the one associated with the Golden Rule of Accumulation towards the Golden Rule saving rate. We start with a saving rate above s, with a too high level of capital per worer, and c < c( s ). The transition starts with a jump upwards in consumption per head at the point in time we reduce the saving rate. Thereafter, as we move downwards towards ( s ), consumption per head will decline, but will never approach the lower value we started from. The initial situation was characterised by oversaving or too high capital accumulation and was dynamically inefficient, because, by lowering the saving rate towards s, consumption pr head can be increased for all generations. The reduction of the saving rate produces a pure gain! (We have a too capital dense or overloaded economy, as a lot of output is used to furnish or provide a growing labour force with a too large amount of capital. Reduction in capital per head yields a Pareto superior consumption path.) Problem 4 Show that the Golden Rule of Accumulation saving rate can be expressed as s ε ( ), where ε is the marginal elasticity of capital. = Suppose that the factors of production are rewarded according to their marginal productivities, and consider the saving rate associated with the Golden Rule of Accumulation. Given that the macro production function has constant returns to scale, show that total consumption is equal to total wage bill, while capital accumulation (saving or investment) is equal to X w ( profits ). (What is then derived as a normative result in the neoclassical growth model is one of the assumptions of the classical growth model!) We have that is characterised by f (( s )) = n. Because f () = F, we can write: n f = F = n ε = n =. Using that for any steady X X X f() X state: sf () = n. Then it directly follows that s = ε ( ), where : = ( s ). If the factors are rewarded according to their marginal productivities, the production function has CRS, and we have maximal consumption per head, as under, we have: C = X = (1 s) X = X εx = (1 ε) X = εx = FX = w, if X F = w. Hence we get = X C = X w = profits. As was assumed by the classical economists.