THE SHAPE OF PRODUCTION FUNCTIONS AND THE DIRECTION OF TECHNICAL CHANGE CHARLES I JONES This paper views the standard production function in macroeconomics as a reduced form and derives its properties from microfoundations The shape of this production function is governed by the distribution of ideas If that distribution is Pareto then two results obtain the global production function is Cobb Douglas and technical change in the long run is labor augmenting Kortum showed that Pareto distributions are necessary if search based idea models are to exhibit steady state growth Here we show that this same assumption delivers the additional results about the shape of the production function and the direction of technical change I INTRODUCTION Much of macroeconomics and an even larger fraction of the growth literature makes strong assumptions about the shape of the production function and the direction of technical change In particular it is well known that for a neoclassical growth model to exhibit steady state growth either the production function must be Cobb Douglas or technical change must be labor augmenting in the long run But apart from analytic convenience is there any justification for these assumptions Where do production functions come from To take a common example our models frequently specify a relation y f k that determines how much output per worker y can be produced with any quantity of capital per worker k We typically assume that the economy is endowed with this function but consider how we might derive it from deeper microfoundations Suppose that production techniques are ideas that get discovered over time One example of such an idea would be a Leontief technology that says for each unit of labor take k units of capital Follow these instructions omitted and you will get out y units of output The values k and y are parameters of this production technique I am grateful to Daron Acemoglu Susanto Basu Francesco Caselli Harold Cole Xavier Gabaix Douglas Gollin Peter Klenow Jens Krueger Michael Scherer Robert Solow Alwyn Young and participants at numerous seminars for comments Samuel Kortum provided especially useful insights for which I am most appreciative Meredith Beechey Robert Johnson and Dean Scrimgeour supplied excellent research assistance This research is supported by NSF grant SES 0242000 2005 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology The Quarterly Journal of Economics May 2005 517 518 QUARTERLY JOURNAL OF ECONOMICS If one wants to produce with a capital labor ratio very different from k this Leontief technique is not particularly helpful and one needs to discover a new idea appropriate to the higher capital labor ratio 1 Notice that one can replace the Leontief structure with a production technology that exhibits a low elasticity of substitution and this statement remains true to take advantage of a substantially higher capital labor ratio one really needs a new technique targeted at that capital labor ratio One needs a new idea According to this view the standard production function that we write down mapping the entire range of capital labor ratios into output per worker is a reduced form It is not a single technology but rather represents the substitution possibilities across different production techniques The elasticity of substitution for this global production function depends on the extent to which new techniques that are appropriate at higher capitallabor ratios have been discovered That is it depends on the distribution of ideas But from what distribution are ideas drawn Kortum 1997 examined a search model of growth in which ideas are productivity levels that are drawn from a distribution He showed that the only way to get exponential growth in such a model is if ideas are drawn from a Pareto distribution at least in the upper tail This same basic assumption that ideas are drawn from a Pareto distribution yields two additional results in the framework considered here First the global production function is Cobb Douglas Second the optimal choice of the individual production techniques leads technological change to be purely laboraugmenting in the long run In other words an assumption Kortum 1997 suggests we make if we want a model to exhibit steady state growth leads to important predictions about the shape of production functions and the direction of technical change In addition to Kortum 1997 this paper is most closely related to an older paper by Houthakker 1955 1956 and to two recent papers Acemoglu 2003b and Caselli and Coleman 2004 1 This use of appropriate technologies is related to Atkinson and Stiglitz 1969 and Basu and Weil 1998 THE SHAPE OF PRODUCTION FUNCTIONS 519 The way in which these papers fit together will be discussed below 2 Section II of this paper presents a simple baseline model that illustrates all of the main results of this paper In particular that section shows how a specific shape for the technology menu produces a Cobb Douglas production function and labor augmenting technical change Section III develops the full model with richer microfoundations and derives the Cobb Douglas result while Section IV discusses the underlying assumptions and the relationship between this model and Houthakker 1955 1956 Section V develops the implications for the direction of technical change Section VI provides a numerical example of the model and Section VII concludes II A BASELINE MODEL II A Preliminaries Let a particular production technique call it technique i be defined by two parameters a i and b i With this technique output Y can be produced with capital K and labor L according to the local production function associated with technique i 1 Y F b iK a iL We assume that F exhibits an elasticity of substitution less than one between its inputs and constant returns to scale in K and L In addition we make the usual neoclassical assumption that F possesses positive but diminishing marginal products and satisfies the Inada conditions This production function can be rearranged to give 2 Y a iLF so that in per worker terms we have 3 y a iF b iK 1 a iL bi k 1 ai 2 The insight that production techniques underlie what I call the global production function is present in the old reswitching debate see Robinson 1953 The notion that distributions for individual parameters aggregate up to yield a well behaved function is also found in the theory of aggregate demand see Hildenbrand 1983 and Grandmont
View Full Document
Unlocking...