Lecture 5: The neoclassical growth model

Transcription

1 THE UNIVERSITY OF SOUTHAMPTON Paul Klein Office: Murray Building, URL: Economics 3010 Topics in Macroeconomics 3 Autumn 2010 Lecture 5: The neoclassical growth model 1 The environment People are endowed with labour, not goods. Goods are produced by combining labour (L) with foregone consumption, which we call capital (K). Formally, we write Y (t) = F (γ(t) L(t), ) (1) where γ(t) is (labour) productivity at t. We assume that productivity growth is labour-augmenting. If F is Cobb-Douglas, we can equivalently assume (like M & W) that there is growth in total factor productivity (TFP). Endowments of labour are described using the following notation. h t = [ h t (t), h t (t + 1)] People do not care about leisure, so they will spend all available time working. Thus the total amount of labour at date t is N(t) L(t) = h t (t) dh + N(t 1) h t 1(t) dh

2 A feasible allocation is defined as in the environment with storage, i.e. if aggregate consumption C(t) can be realized with some non-negative sequence of aggregate capital stocks where K(0) is given. Formally, an allocation is feasible if there is a sequence () such that 0 and C(t) + K(t + 1) Y (t). This formula looks as if there is full (100 %) depreciation. However, we can, if we want, build any desired rate of depreciation into F. 2 Competitive equilibrium In a competitive economy, factors of production are paid their marginal products. Thus the wage rate w(t) is the marginal product of labor (MPL) and the rental rate R(t) is the marginal product of capital (MPK). Formally, w(t) = MP L = L(t) [F (γ(t) L(t), )] = γ(t) F H(γ(t) L(t), ) where F H denotes the derivative with respect to the first argument and R(t) = MP K = F K (γ(t) L(t), ). Notice that these prices depend on aggregate labour and capital. Individual budget constraints are given by c h t (t) w(t) h t (t) l h (t) k h (t + 1) and c h t (t + 1) w(t + 1) h t (t + 1) + r(t)l h (t) + R(t + 1)k h (t + 1). Can the rental rate possibly be different from the interest rate? Well, a complication here is that we should expect R(t + 1) = r(t) 2

3 so the timing is shifted. Moreover, it would seem that K(t + 1) = 0 is a possibility, in which case we cannot rule out that r(t) > R(t + 1). To avoid this, we assume lim F K(H, K) =. K 0 It follows that r(t) = R(t + 1) in any competitive equilibrium. We will typically assume that technology is Cobb-Douglas, i.e. F (H, K) = K θ H 1 θ. This means that the rental rate is ( γ(t)l(t) R(t) = θ ) 1 θ and the wage rate is ( ) θ ( ) θ w(t) = (1 θ)γ(t) 1 θ = (1 θ)γ(t). L(t) γ(t)l(t) Notice that w(t)l(t) + R(t) = (1 θ)γ(t) 1 θ ( L(t) ) θ ( ) 1 θ γ(t)l(t) L(t) + θ = Y (t) so that if factors are paid their marginal products, income equals output. This is true more generally so long as F exhibits constant returns to scale. We can prove this result for an arbitrary function that exhibits constant returns to scale, or linear homogeneity as it is sometimes called. For this purpose, let x be a vector, i.e. x = and let f(x) be a function that takes such a vector and transforms it into a single number. We write f : R n R. Moreover, let tx we the vector that results from 3 x 1 x 2. x n

4 multiplying each element of x by the number t. That is, tx 1 tx tx = 2.. tx n Now suppose that f is linearly homogeneous, i.e. that, for all x, What we want to show is that n k=1 f(tx) = tf(x). f(x) x k = f(x). x k To do that, we differentiate f(tx) with respect to t. We get, ignoring for the moment that f is linearly homogeneous, d dt f(tx) = n k=1 f(tx) x k x k. On the other hand, f(tx) = tf(x) so it must be that It follows that d f(tx) = f(x). dt f(x) = Set t = 1 and the proof is finished. n k=1 f(tx) x k x k Another useful property of a constant-returns-to-scale production function is that we can rewrite equation (1) on intensive form. Here s what this means. Define y(t) = Y (t) γ(t)l(t) and k(t) = γ(t)l(t) 4

5 Then (why?) y(t) = F (k(t), 1) = f(k(t)). Notice that, conveniently enough, (if we define k = K γl ) MP K = F K (γl, K) = f k (k) and, perhaps slightly less conveniently, MP L = L [F (γl, K)] = γ [f(k) f k(k) k]. For example, if technology is Cobb-Douglas, then f(k) = k θ, MP L = γ(1 θ)k θ and MP K = θk θ 1. We want to investigate the long-run behaviour of the economy under competitive equilibrium. To do that, we make some further simplifying assumptions. Assume that productivity grows at rate g, i.e. γ(t + 1) = (1 + g)γ(t) where we normalize γ(0) = 1, and that the population grows at rate n, i.e. N(t + 1) = (1 + n)n(t) where we again normalize N(0) = 1. Finally, we assume that all young people are endowed with one unit of labour, that all old people are endowed with no labour at all and that preferences are Cobb-Douglas, i.e. u h t = ln c h t (t) + β ln c h t (t + 1). 5

6 Given that old people are not endowed with any labour and preferences are Cobb- Douglas, the aggregate savings function becomes particularly simple. In particular, savings is independent of the rental rate. We have K(t + 1) = β 1 + β w(t)n(t) = β 1 + β (1 θ)[n(t)γ(t)]1 θ Kt θ. Notice that it doesn t matter for aggregate behaviour exactly how skills are distributed among the young. We now translate this into lower case (intensive) terms. We get N(t + 1)γ(t + 1)k(t + 1) = N(t)γ(t) 1 + β kθ t. With our simplifying assumption about growth rates, this becomes k(t + 1) = (1 + β)(1 + n)(1 + g) kθ t. For a given k(0), this gives us k(1), k(2) etc. What happens in the long run? Does k(t) tend to a limit that is independent of k(0)? k(0) > 0. This limit solves and hence k = [ k = (1 + β)(1 + n)(1 + g) (k ) θ (1 + β)(1 + n)(1 + g) The answer is yes, so long as ] 1 1 θ In a diagram one can illustrate that this stationary level of capital is stable in the sense that k(t) will tend to return there if perturbed from it. This model is a lot like the Solow model. The only difference is that the savings rate is endogenously determined. But it is still constant. And the limiting growth rate of capital and output is (1 + g)(1 + n) independently of what the savings rate is. 6