the shape of production functions and the ...chadj/jonesqje2005.pdfthe shape of production functions...

THE SHAPE OF PRODUCTION FUNCTIONS AND THEDIRECTION OF TECHNICAL CHANGE

CHARLES I JONES

This paper views the standard production function in macroeconomics as areduced form and derives its properties from microfoundations The shape of thisproduction function is governed by the distribution of ideas If that distribution isPareto then two results obtain the global production function is Cobb-Douglasand technical change in the long run is labor-augmenting Kortum showed thatPareto distributions are necessary if search-based idea models are to exhibitsteady-state growth Here we show that this same assumption delivers the addi-tional results about the shape of the production function and the direction oftechnical change

I INTRODUCTION

Much of macroeconomicsmdashand an even larger fraction of thegrowth literaturemdashmakes strong assumptions about the shape ofthe production function and the direction of technical change Inparticular it is well-known that for a neoclassical growth modelto exhibit steady-state growth either the production functionmust be Cobb-Douglas or technical change must be labor-aug-menting in the long run But apart from analytic convenience isthere any justification for these assumptions

Where do production functions come from To take a commonexample our models frequently specify a relation y f(k ) thatdetermines how much output per worker y can be produced withany quantity of capital per worker k We typically assume thatthe economy is endowed with this function but consider how wemight derive it from deeper microfoundations

Suppose that production techniques are ideas that get dis-covered over time One example of such an idea would be aLeontief technology that says ldquofor each unit of labor take kunits of capital Follow these instructions [omitted] and you willget out y units of outputrdquo The values k and y are parametersof this production technique

I am grateful to Daron Acemoglu Susanto Basu Francesco Caselli HaroldCole Xavier Gabaix Douglas Gollin Peter Klenow Jens Krueger MichaelScherer Robert Solow Alwyn Young and participants at numerous seminars forcomments Samuel Kortum provided especially useful insights for which I ammost appreciative Meredith Beechey Robert Johnson and Dean Scrimgeoursupplied excellent research assistance This research is supported by NSF grantSES-0242000

copy 2005 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnologyThe Quarterly Journal of Economics May 2005

517

If one wants to produce with a capital-labor ratio very differ-ent from k this Leontief technique is not particularly helpfuland one needs to discover a new idea ldquoappropriaterdquo to the highercapital-labor ratio1 Notice that one can replace the Leontiefstructure with a production technology that exhibits a low elas-ticity of substitution and this statement remains true to takeadvantage of a substantially higher capital-labor ratio one reallyneeds a new technique targeted at that capital-labor ratio Oneneeds a new idea

According to this view the standard production function thatwe write down mapping the entire range of capital-labor ratiosinto output per worker is a reduced form It is not a singletechnology but rather represents the substitution possibilitiesacross different production techniques The elasticity of substitu-tion for this global production function depends on the extent towhich new techniques that are appropriate at higher capital-labor ratios have been discovered That is it depends on thedistribution of ideas

But from what distribution are ideas drawn Kortum [1997]examined a search model of growth in which ideas are productiv-ity levels that are drawn from a distribution He showed that theonly way to get exponential growth in such a model is if ideas aredrawn from a Pareto distribution at least in the upper tail

This same basic assumption that ideas are drawn from aPareto distribution yields two additional results in the frame-work considered here First the global production function isCobb-Douglas Second the optimal choice of the individual pro-duction techniques leads technological change to be purely labor-augmenting in the long run In other words an assumptionKortum [1997] suggests we make if we want a model to exhibitsteady-state growth leads to important predictions about theshape of production functions and the direction of technicalchange

In addition to Kortum [1997] this paper is most closelyrelated to an older paper by Houthakker [1955ndash1956] and to tworecent 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]

518 QUARTERLY JOURNAL OF ECONOMICS

The way in which these papers fit together will be discussedbelow2

Section II of this paper presents a simple baseline model thatillustrates all of the main results of this paper In particular thatsection shows how a specific shape for the technology menu pro-duces a Cobb-Douglas production function and labor-augmentingtechnical change Section III develops the full model with richermicrofoundations and derives the Cobb-Douglas result whileSection IV discusses the underlying assumptions and the rela-tionship between this model and Houthakker [1955ndash1956] Sec-tion V develops the implications for the direction of technicalchange Section VI provides a numerical example of the modeland Section VII concludes

II A BASELINE MODEL

IIA Preliminaries

Let a particular production techniquemdashcall it technique imdashbe defined by two parameters ai and bi With this techniqueoutput Y can be produced with capital K and labor L according tothe local production function associated with technique i

(1) Y FbiKaiL

We assume that F( ) exhibits an elasticity of substitution lessthan one between its inputs and constant returns to scale in Kand L In addition we make the usual neoclassical assumptionthat F possesses positive but diminishing marginal products andsatisfies the Inada conditions

This production function can be rearranged to give

(2) Y aiLF biKaiL

1

so that in per worker terms we have

(3) y aiFbi

aik1

2 The insight that production techniques underlie what I call the globalproduction function is present in the old reswitching debate see Robinson [1953]The notion that distributions for individual parameters aggregate up to yield awell-behaved function is also found in the theory of aggregate demand seeHildenbrand [1983] and Grandmont [1987]

519THE SHAPE OF PRODUCTION FUNCTIONS

where y YL and k KL Now define yi ai and ki aibiThen the production technique can be written as

(4) y yiF kki

1

If we choose our units so that F(11) 1 then we have the niceproperty that k ki implies that y yi Therefore we can thinkof technique i as being indexed by ai and bi or equivalently byki and yi

The shape of the global production function is driven by thedistribution of alternative production techniques rather than bythe shape of the local production function that applies for a singletechnique3 To illustrate this consider the example given in Fig-ure I The circles in this figure denote different production tech-niques that are availablemdashthe set of (kiyi) pairs For a subset ofthese we also plot the local production function y F(bikai)Finally the heavy solid line shows the global production functiongiven by the convex hull of the local production techniques Forany given level of k the global production function shows themaximum amount of output per worker that can be producedusing the set of ideas that are available

The key question wersquod like to answer is this what is theshape of the global production function To make progress wenow turn to a simple baseline model

IIB The Baseline Model

We begin with a simple model really not much more than anexample However this baseline model turns out to be very use-ful it is easy to analyze and captures the essence of the modelwith more detailed microfoundations that is presented in Sec-tion III

At any given point in time a firm has a stock of ideasmdashacollection of local production techniquesmdashfrom which to chooseThis set of production techniques is characterized by the follow-ing technology menu

(5) Hab N

where Ha 0 Hb 0 and N 0 Along this menu there is a

3 Other models in the literature feature a difference between the short-runand long-run elasticities of substitution as opposed to the local-global distinctionmade here These include the putty-clay models of Caballero and Hammour [1998]and Gilchrist and Williams [2000]


trade-off ideas with a high value of b are associated with a lowvalue of a N parameterizes the location of this technology menuand might be thought of as the level of knowledge A higher Nmeans the technology menu supports higher levels of a and bAssociated with any (ab) pair from this technology menu is alocal production function Y F(bKaL) with the propertiesassumed above in equation (1) including an elasticity of substi-tution less than one and constant returns to scale in K and L

The global production function for this firm describes themaximum amount of output the firm can produce from a particu-lar set of inputs when it is free to choose any production tech-nique from the technology menu That is the global productionfunction F(KLN) is defined as

(6) Y FKLN maxba

FbKaL

subject to (5)

FIGURE IAn Example of the Global Production Function

Circles identify distinct production techniques for some of these the localproduction function associated with the technique has been drawn as a dashedline The heavy solid line shows the convex hull of the local production functionsie the global production function


Characterizing the global production function is straightfor-ward Graphically one version of this problem with an interiorsolution is shown in Figure II Algebraically an interior solutionequates the marginal rate of technical substitution along theisoquant to the marginal rate of technical substitution along thetechnology menu We can express this in its elasticity form anduse the fact that the elasticity of production with respect to b isthe same as the elasticity with respect to K to get the followingresult

(7)K

L

b

a

where K(abKL) F1bKY is the capital share L 1 Kis the analogous labor share b (Hb)(bH) is the elasticityof H with respect to b and a is the analogous elasticity withrespect to a The optimal technology choice equates the ratio ofthe capital and labor shares to the ratio of the elasticities of thetechnology menu

In Figure II we drew the technology menu as convex to theorigin Of course we could have drawn the curve as concave or

FIGURE IIThe Direction of Technical Change


linear or we could have drawn it as convex but with a sharpercurvature than the isoquant However it turns out that theconstant elasticity version of the convex curve delivers a particu-larly nice result4 In particular suppose that the technologymenu is given by

(8) Hab ab N 0 0

In this case the elasticity ba is constant so the optimalchoice of the technology levels leads to a first-order condition thatsets the capital share equal to the constant ( )

The constancy of the capital share then leads to two usefuland interesting results First the global production functiontakes a Cobb-Douglas form for any levels of the inputs K and Land any location of the technology menu N the choice of tech-nology leads the elasticity of output with respect to capital andlabor to be constant

In fact it is easy to derive the exact form of the globalproduction function by combining the local-global insights of sub-section IIA with the technology menu For some technique irecall the equivalent ways we have of describing the technique

(9) yi ai

(10) ki aibi

From the technology frontier in equation (8) we know that ai andbi are related by ai

bi N Simple algebra shows that yi and ki

are therefore related by

(11) yi Nki1

That is given the constant elasticity form of the technology fron-tier a plot of the techniques in (ky) space like that in Figure Iyields a Cobb-Douglas production function With this continuousformulation for the frontier the global production function isequal to the technology frontier in (ky) space5 Multiplying by L

4 In this case the assumption that F has an elasticity of substitution lessthan one means that the isoquant curves are more sharply curved than thetechnology menu which has an elasticity of substitution equal to one Thisguarantees an interior solution

5 For this to be true we need the local production techniques to paste upsmoothly with the global production function For example if F is a CES functionwith a capital share parameter 13 (see for example equation (36) below) the globalproduction function is actually proportional to that in equation (12) To make thefactor of proportionality equal to one we need the share parameter 13 to equal( ) so that the factor share at k ki is exactly ( )


to get back to the standard form the global production function isgiven by

(12) Y NKL1

That is we get a Cobb-Douglas production function with constantreturns to scale

The second key result is related to the direction of technicalchange To see this consider embedding this production setup ina standard neoclassical growth model6 The fact that the globalproduction function is Cobb-Douglas implies immediately thatsuch a model will exhibit a balanced growth path with positivegrowth provided N grows exponentially

The balanced growth path result turns out to have a strongimplication for the direction of technical change In particular itimplies that the level of b will be constant along the balancedgrowth path and all growth will occur because a rises over timeTo see this result notice that the first-order condition in equation(7) can be written as

(13)bKF1bKaL

aLF2bKaL

Now let x bKaL Because F exhibits constant returns to scalethe marginal products are homogeneous of degree 0 This meanswe can rewrite equation (13) as

(14)xF1 x1

F2 x1

Since x is the only variable in this equation the optimal choice oftechnology is such that x is constant at all points in time

Finally we now need to show that along a balanced growthpath the only way x bKaL can be constant is if b is constantImportantly recall that output is always produced with somelocal production technique That is

(15) Yt FKtLtNt FbtKtatLt

where bt and at are the optimal choices of the technology levelsBecause F exhibits constant returns we have

6 By this we mean the usual Ramsey-Cass-Koopmans model with isoelasticutility constant population growth and constant growth in N


(16)Yt

atLt FbtKt

atLt1

Since x bKaL must be constant this implies that YaL mustalso be constant And this means that bKY must be constantBut we know that KY is constant along a balanced growth pathin the neoclassical growth model so this implies that b must beconstant as well which completes the proof Moreover the factthat the capital share equals ( ) implies that the level of bis chosen so that the capital share is invariant to the capital-output ratio one of the key results in Acemoglu [2003b]

Of course the result that b must be constant along thebalanced growth path is really just an application of the Steady-State Growth Theorem if a neoclassical growth model exhibitssteady-state growth with constant and positive factor sharesthen either the production function is Cobb-Douglas or technicalchange is labor-augmenting In fact we just proved a version ofthis theorem for the case in which the local production function isnot Cobb-Douglas7

What is the intuition for the result that technical change ispurely labor-augmenting Since the local production function isnot Cobb-Douglas balanced growth requires bKaL to be con-stant so that bK and aL must grow at the same rate In factsince Y F(bKaL) this suggests an alternative interpretationof the word ldquobalancedrdquo in the phrase ldquobalanced growth pathrdquo theeffective inputs bK and aL must be balanced in the sense thatthey grow at the same rate But the only way this can happen isif b is constant For example we know that with b constant Kwill grow at the same rate as aL If b were to grow on top of thisbK would grow faster than aL and growth would be unbalancedThe consequence that would result is that the factor shares wouldtrend to zero and one

In the context of our model it is easy to be confused by thistheorem It is well-known that with Cobb-Douglas production theldquodirectionrdquo of technical change has no meaning capital-augment-ing technical change can always be written as labor-augmentingBut the results just outlined seem to be that production is Cobb-

7 For the proof of the general theorem the classic reference is Uzawa [1961]see also Barro and Sala-i-Martin [1995] for a proof in the special case of factor-augmenting technologies Jones and Scrimgeour [2005] present a formal state-ment of the theorem discuss a version of Uzawarsquos proof and develop intuition inthe general case


Douglas and technical change is labor-augmenting How can thisbe

The key to resolving this confusion is to look back at equation(15) First recall that production always occurs with some localproduction technique F(btKtatLt) Since this local technique hasan elasticity of substitution less than one the direction of tech-nical change is a well-defined concept Our result is that bt isconstant along a balanced growth path so that technical changein the local production function is purely labor-augmenting Sec-ond equation (15) also reminds us of the definition of the globalproduction function F(KLN) It is this global production func-tion that we show to be Cobb-Douglas At any point in time bothldquoviewsrdquo of the production function are possible and it is by takingdifferent points of view that we get our two results

IIC Discussion

We now pause to make some more general remarks about thebaseline model First notice that an alternative way to set up thebaseline model would be to write down the firmrsquos full profitmaximization problem That is in addition to choosing a and bwe could allow the firm to choose K and L taking factor prices asgiven We view the analysis of the global production function asconceptually coming a step before profit maximization The globalproduction function is defined over any combination of K and L ifone desires one can embed this global production function into amodel of how firms choose their inputs For our purposes how-ever all we are assuming about firm behavior is that they operatetheir technology efficiently A helpful analogy might be that onecan write down the cost-minimization problem as a precursor tothe profit-maximization problem8

Second our problem is closely related to the world technologyfrontier problem considered by Caselli and Coleman [2004] Ca-selli and Coleman specialize to CES functions for both F and thetechnology menu H and embed their setup in a profit maximiza-

8 In the context of profit maximization a more formal justification for theglobal production function approach can be given For example the full profitmaximization problem can be written as

maxabKL

FbKaLKaL wL rK subject to Hab N

The global production function approach can be justified by noting that it ischaracterized by the first-order condition associated with the technology choice inthe profit maximization problem


tion problem They are concerned primarily with characterizingthe choices of the technology levels in a cross-country contextrather than over time But the similarity of the setups is inter-esting and suggests some potentially productive avenues forresearch9

Finally notice that the problem here is to choose the levels ofa and b Related problems appear in the literature on the direc-tion of technical change see Kennedy [1964] Samuelson [1965]and Drandakis and Phelps [1966] However in these problemsthe choice variables and the constraints are typically expressed interms of the growth rates of a and b rather than the levels AsAcemoglu [2003a] and others have pointed out this results in anarbitrary optimization problem in the early literature related tomaximizing the growth rate of output

Acemoglu [2003b] recasts the traditional problem in terms ofa two-dimensional version of Romer [1990] with explicit micro-foundations and profit-maximizing firms Under some strongmdashand arguably implausible10mdashconditions on the shape of the ideaproduction functions Acemoglu shows that technical change willbe purely labor-augmenting in the long run and that the long-runcapital share will be invariant to policies that change the capital-output ratio These results are obviously closely related to whatwe have here despite the considerably different approaches of thetwo papers The main differences in terms of the results are that(a) we provide a very different perspective on the conditionsneeded to get technical change to be labor-augmenting and (b)we explicitly bring out the link to a Cobb-Douglas productionfunction11

To sum up the insight from this baseline model is that if thetechnology frontiermdashie the way in which the levels of a and b

9 Caselli and Coleman also contain a helpful discussion of the existence ofinterior versus corner solutions in their setup

10 The production functions for capital-ideas and labor-ideas must be pa-rameterized ldquojust sordquo In particular let N denote the stock of labor-augmentingideas Then the cost of producing new labor-augmenting ideas relative to the costof producing new capital-augmenting ideas must decline at exactly the rate NNPlausible specificationsmdashsuch as one in which the output good itself is the maininput into the production of new ideas (in which case the relative cost of producinglabor- and capital-ideas is constant) or the idea production function employed byJones [1995] to remove scale effects from the growth rate (in which case therelative cost of producing labor-ideas declines with N)mdashlead to a model that doesnot exhibit a steady state with a positive capital share

11 The results here suggest that one might interpret Acemoglursquos setup asproviding a Cobb-Douglas production function in the long run In contrast ourresult delivers Cobb-Douglas production at any point in time


trade offmdashexhibits constant elasticities then the global produc-tion function will be Cobb-Douglas and technological change willbe labor-augmenting in the long run But is there any reason tothink that the technology frontier takes this particular shape

III MICROFOUNDATIONS PARETO DISTRIBUTIONS

The baseline model is straightforward and yields strong pre-dictions However it involves a very particular specification of thetechnology menu It turns out that this specification can be de-rived from a model of ideas with substantially richer microfoun-dations This is the subject of the current section12

IIIA Setup

An idea in this economy is a technique for combining capitaland labor to produce output The production technique associatedwith idea i is F(biKaiL) Because it results in a more tractableproblem that yields analytic results we make the extreme as-sumption that this local production technology is Leontief

(17) Y FbiKaiL min biKaiL

Of course the intuition regarding the global production functionsuggests that it is determined by the distribution of ideas not bythe shape of the local production function In later simulationresults we confirm that the Leontief assumption can be relaxed

A production technique is parameterized by its labor-aug-menting and capital-augmenting parameters ai and bi To derivethe Cobb-Douglas result we make a strong assumption about thedistribution of ideas

ASSUMPTION 1 The parameters describing an idea are drawn fromindependent Pareto distributions

18 Prai a 1 aa

a a 0

12 I owe a large debt to Sam Kortum in this section A previous version ofthis paper contained a much more cumbersome derivation of the Cobb-Douglasresult Kortum in discussing this earlier version at a conference offered a numberof useful comments that simplify the presentation including the Poisson approachthat appears in the Appendix


19 Prbi b 1 bb

b b 0

where 0 0 and 113

With this assumption the joint distribution of ai and bi satisfies

(20) Gba Prbi b ai a bb a

a

We specify this distribution in its complementary form becausethis simplifies some of the equations that follow

Let Yi(KL) F(biKaiL) denote output using technique iThen since F is Leontief the distribution of Yi is given by

21

H y PrYi y PrbiK y aiL y

G yK

yL

KLy

where ab

That is the distribution of Yi is itself Pareto14

IIIB Deriving the Global Production Function

The global production function describes as a function ofinputs the maximum amount of output that can be producedusing any combination of existing production techniques Wehave already made one simplification in our setup by limitingconsideration to Leontief techniques Now we make another byignoring combinations of techniques and allowing only a singletechnique to be used at each point in time Again this is asimplifying assumption that allows for an analytic result but itwill be relaxed later in the numerical simulations

Let N denote the total number of production techniques thatare available and assume that the N ideas are drawn indepen-dently Then we define the global production function

13 This last condition that the sum of the two parameters be greater thanone is needed so that the mean of the Frechet distribution below exists On arelated point recall that for a Pareto distribution the kth moment exists only ifthe shape parameter (eg or ) is larger than k

14 Since bi b and ai a the support for this distribution is y min bKaL


DEFINITION 1 The global production function F(KLN) is given as

(22) FKLN maxi1 N

FbiKaiL

Let Y F(KLN) Since the N draws are independent thedistribution of the global production function satisfies

23PrY y 1 H yN

1 KLyN

Of course as the number of ideas N gets large this probability forany given level of y goes to zero So to get a stable distribution weneed to normalize our random variable somehow in a manneranalogous to that used in the Central Limit Theorem

In this case the right normalization turns out to involve zNwhere

(24) zN NKL1

In particular consider

25

PrY zNy 1 KLzNyN

1 y

N N

Then using the standard result that limN3 (1 xN)N exp(x) for any fixed value of x we have

(26) limN3

PrY zNy expy

for y 0 This distribution is known as a Frechet distribution15

Therefore

(27)Y

NKL1 a

Frechet

The global production function appropriately normalized con-verges asymptotically to a Frechet distribution This means thatas N gets large the production function behaves like

(28) Y NKL1

15 This is a special case of the much more general theory of extreme valuesFor a more general theorem relevant to this case see Theorem 211 of Galambos[1978] as well as Kortum [1997] and Castillo [1988]


where is a random variable drawn from a Frechet distributionwith shape parameter and a scale parameter equal to unity

Here we have derived the Cobb-Douglas result as the num-ber of ideas goes to infinity We will show in the simulations thatthe approximation for a finite number of ideas works well Inaddition the Appendix shows how to obtain the Cobb-Douglasresult with a finite number of ideas under the stronger assump-tion that the arrival of ideas follows a Poisson process

IV DISCUSSION

The result given in equation (28) is one of the main results inthe paper If ideas are drawn from Pareto distributions then theglobal production function takes at least as the number of ideasgets large the Cobb-Douglas form For any given productiontechnique a firm may find it difficult to substitute capital forlabor and vice versa leading the curvature of the productiontechnique to set in quickly However when firms are allowed toswitch between production technologies the global productionfunction depends on the distribution of ideas If that distributionhappens to be a Pareto distribution then the production functionis Cobb-Douglas

We can now make a number of remarks First the exponentin the Cobb-Douglas function depends directly on the parametersof the Pareto search distributions The easier it is to find ideasthat augment a particular factor the lower is the relevant Paretoparameter (eg or ) and the lower is the exponent on thatfactor Intuitively better ideas on average reduce factor sharesbecause the elasticity of substitution is less than one Some ad-ditional remarks follow

IVA Relationship to the Baseline Model

The simple baseline model given at the beginning of thispaper postulated a technology menu and showed that if this menuexhibited a constant elasticity then one could derive a Cobb-Douglas global production function The model with microfoun-dations based on Pareto distributions turns out to deliver a sto-chastic version of this technology menu

In the model the stochastic version of this menu can be seenby considering an isoprobability curve Pr[bi bai a] G(ba) C where C 0 is some constant With the joint Paretodistribution this isoprobability curve is given by


(29) ba C

This isocurve exhibits constant elasticities and shifts up as theprobability C is lowered analogous to an increase in N in thebaseline model

In terms of the baseline model the Pareto distribution there-fore delivers a and b and we get the same form of theglobal production function compare (12) and (28)

IVB Houthakker [1955ndash1956]

The notion that Pareto distributions appropriately ldquokickedrdquocan deliver a Cobb-Douglas production function is a classic resultby Houthakker [1955ndash1956] Houthakker considers a world ofproduction units (eg firms) that produce with Leontief technol-ogies where the Leontief coefficients are distributed across firmsaccording to a Pareto distribution Importantly each firm haslimited capacity so that the only way to expand output is to useadditional firms Houthakker then shows that the aggregate pro-duction function across these units is Cobb-Douglas

The result here obviously builds directly on Houthakkerrsquosinsight that Pareto distributions can generate Cobb-Douglas pro-duction functions The result differs from Houthakkerrsquos in severalways however First Houthakkerrsquos result is an aggregation re-sult Here in contrast the result applies at the level of a singleproduction unit (be it a firm industry or country) Second theLeontief restriction in Houthakkerrsquos paper is important for theresult it allows the aggregation to be a function only of the Paretodistributions Here in contrast the result is really about theshape of the global production function looking across tech-niques The local shape of the production function does not reallymatter This was apparent in the simple baseline model givenearlier and it will be confirmed numerically in Section VI

Finally Houthakkerrsquos result relies on the presence of capac-ity constraints If one wants to expand output one has to addadditional production units essentially of lower ldquoqualityrdquo Be-cause of these capacity constraints his aggregate productionfunction is characterized by decreasing returns to scale In thecontext of an idea model such constraints are undesirable onewould like to allow the firm to take its best idea and use it forevery unit of production That is one would like the setup to


respect the nonrivalry of ideas and the replication argument forconstant returns as is true in the formulation here16

IVC Evidence for Pareto Distributions

The next main comment is that Pareto distributions arecrucial to the result Is there any evidence that ideas follow aPareto distribution

Recall that the defining property of the Pareto distribution isthat the conditional probability Pr[X xX x] for 1 isindependent of x The canonical example of a Pareto distributionis the upper tail of the income distribution Indeed it was thisobservation that led Pareto to formulate the distribution thatbears his name Given that we observe an income larger than xthe probability that it is greater than 11x turns out to be invari-ant to the level of x at least above a certain point For exampleSaez [2001] documents this invariance for the United States in1992 and 1993 for incomes between $100000 and $30 million

Evidence of Pareto distributions has also been found forpatent values profitability citations firm size and stock returnsFirst it is worth noting that many of the tests in this literatureare about whether or not the relevant variable obeys a Paretodistribution That is Pareto serves as a benchmark In terms offindings this literature either supports the Pareto distribution orfinds that it is difficult to distinguish between the Pareto and thelognormal distributions For example Harhoff Scherer andVopel [1997] examine the distribution of the value of patents inGermany and the United States For patents worth more than$500000 or more than 100000 Deutsche Marks a Pareto distri-bution accurately describes patent values although for the entirerange of patent values a lognormal seems to fit better Bertran[2003] finds evidence of a Pareto distribution for ideas by usingpatent citation data to value patents Grabowski [2002] producesa graph of the present discounted value of profits for new chem-ical entities by decile in the pharmaceutical industry for 1990ndash1994 that supports a highly skewed distribution

Lotka [1926] a classic reference on scientific productivity

16 Lagos [2004] embeds the Houthakker formulation in a Mortenson-Pis-sarides search model to provide a theory of total factor productivity differences Inhis setup firms (capital) match with labor and have a match quality that is drawnfrom a Pareto distribution Capital is the quasi-fixed factor so that the setupgenerates constant returns to scale in capital and labor Nevertheless becauseeach unit of capital gets its own Pareto draw a firm cannot expand production byincreasing its size at its best match quality


shows that the distribution of scientific publications per author isPareto This result appears to have stood the test of time across arange of disciplines even in economics as shown by Cox andChung [1991] It also applies to citations to scientific publications[Redner 1998] Huber [1998] looks for this result among inventorsand finds some evidence that the distribution of patents perinventor is also Pareto although the sample is small Otherevidence of Pareto distributions is found by Axtell [2001] for thesize of firms in the United States and by Gabaix et al [2003] forthe upper tail of stock returns Finally somewhat farther afieldPareto distributions are documented by Sornette and Zajdenwe-ber [1999] for world movie revenues and by Chevalier and Gools-bee [2003] for book sales While by no means dispositive thisevidence of Pareto distributions for a wide range of economicvariables that are certainly related to ideas is suggestive

In addition to the direct evidence there are also conceptualreasons to be open to the possibility that ideas are drawn fromPareto distributions First consider Kortum [1997] He formu-lates a growth model where productivity levels (ideas) are drawsfrom a distribution He shows that this model generates steady-state growth only if the distribution has a Pareto upper tail Thatis what the model requires is that the probability of finding anidea that is 5 percent better than the current best idea is invari-ant to the level of productivity embodied in the current best ideaOf course this is almost the very definition of a steady state theprobability of improving economywide productivity by 5 percentcannot depend on the level of productivity This requirement issatisfied only if the upper tail of the distribution is a powerfunction ie only if the upper tail is Pareto

Additional insight into this issue emerges from Gabaix[1999] Whereas Kortum shows that Pareto distributions lead tosteady-state growth Gabaix essentially shows the reverse in hisexplanation of Zipfrsquos Law for the size of cities He assumes thatcity sizes grow at a common exponential rate plus an idiosyn-chratic shock He then shows that this exponential growth gen-erates a Pareto distribution for city sizes17

17 An important additional requirement in the Gabaix paper is that there besome positive lower bound to city sizes that functions as a reflecting barrierOtherwise for example normally distributed random shocks result in a lognormaldistribution of city sizes Alternatively if the length of time that has passed sinceeach city was created is a random variable with an exponential distribution thenno lower bound is needed and one recovers the Pareto result See Mitzenmacher


The papers by Kortum and Gabaix suggest that Pareto dis-tributions and exponential growth are really just two sides of thesame coin The result in the present paper draws out this con-nection further and highlights the additional implication for theshape of production functions Not only are Pareto distributionsnecessary for exponential growth but they also imply that theglobal production function takes a Cobb-Douglas form

V THE DIRECTION OF TECHNICAL CHANGE

The second main result of the paper is related to the directionof technical change It turns out that this same setup whenembedded in a standard neoclassical growth model delivers theresult that technological change is purely labor-augmenting inthe long run That is even though the largest value of bi associ-ated with any idea goes to infinity this Pareto-based growthmodel delivers the result that a(t) grows on average while b(t) isstationary

To see this result we first embed our existing setup in astandard neoclassical growth model The production side of themodel is exactly as specified in Section III Capital accumulates inthe usual way and we assume that the investment rate s is aconstant

(30) Kt1 1 Kt sYt s 01

Finally we assume that the cumulative stock of ideas Ntgrows exogenously at rate g 0

(31) Nt N0egt

As in Jones [1995] and Kortum [1997] one natural interpretationof this assumption is that ideas are produced by researchers sothat g is proportional to population growth18

For this model we have already shown that the global pro-duction function is (either for N large or for finite N using thePoisson approach in the Appendix)

[2003] for a direct discussion of these alternatives as well as Cordoba [2003] andRossi-Hansberg and Wright [2004]

18 For example one could have Nt1 Rt13Nt

where Rt represents thenumber of researchers working in period t In this case if the number of research-ers grows at a constant exponential rate then the growth rate of N converges toa constant that is proportional to this population growth rate


(32) Yt NtKtLt

1t

It is then straightforward to show that the average growth rate ofoutput per worker y in the model in a stationary steady state is19

(34) E logyt1

yt

g

The growth rate of output per worker is proportional to the rateof growth of research effort The factor of proportionality dependsonly on the search parameter of the Pareto distribution for thelabor-augmenting ideas In particular the easier it is to findhigher ai the faster is the average rate of economic growth

The fact that this growth rate depends on but not on isthe first clue that there is something further to explore here if itis easier to find better labor-augmenting ideas the averagegrowth rate is higher but if it is easier to find better capital-augmenting ideas the average growth rate is unaffected

To understand this fact it is helpful to look back at the localproduction function Even though the global production functionis Cobb-Douglas production at some date t always occurs withsome technique i(t)

(35) Yt FbitKtaitLt

Now recall the Steady-State Growth Theorem discussed earlier ifa neoclassical growth model exhibits steady-state growth with anonzero capital share then either the production function isCobb-Douglas or technical change is labor-augmenting In thiscase the (local) production function is not Cobb-Douglas and wedo have a (stationary) steady state Exactly the same proof thatwe gave earlier for the baseline model in subsection IIB appliesThe implication is that technical change must be labor-augment-ing in the long run That is despite the fact that maxi bi 3 ast 3 the time path for bi(t)mdashie the time path of the birsquosassociated with the ideas that are actually usedmdashmust have anaverage growth rate equal to zero in the limit The intuition isalso the same as in the simple baseline model to keep the factor

19 Rewriting the production function in per worker terms one has

(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t

Taking expectations of this equation and equating the growth rates of y and kyields the desired result


shares constant growth must be balanced in the sense that bKand aL must grow at the same rate and the only way this canhappen is if b is stable20

VI SIMULATION RESULTS

We now turn to a full simulation based on the Pareto modelIn addition to providing an illustration of the results we take thisopportunity to relax the Leontief assumption on the local produc-tion function Instead we assume that the local production func-tion takes the CES form

(36) Yt FbiKtaiLt 13biKt 1 13aiLt

1

where 0 so that the elasticity of substitution is 1(1 ) 1 We also allow production units to use two productiontechniques at a time in order to convexify the production setanalogous to the picture given at the beginning of the paper inFigure I

The remainder of the model is as specified before Apart fromthe change to the CES function the production setup is the sameas that given in Section III and the rest of the model follows theconstant saving setup of Section V

We begin by showing that the CES setup still delivers aCobb-Douglas global production function at least on average Forthis result we repeat the following set of steps to obtain 1000capital-output pairs We first set N 500 so that there are 500ideas in each iteration We compute the convex hull of the CESfunctions associated with these ideas to get a global productionfunction21 Next we choose a level of capital per worker k ran-

20 This result leads to an important observation related to extending themodel Recall that with the Pareto assumption b is the smallest value of b thatcan be drawn and similarly a is the smallest value of a that can be drawn Nowconsider allowing these distributions to shift There seems to be no obstacle toallowing for exponential shifts in a over time However increases in b turn outto lower the capital share in the model If b were to rise exponentially the capitalshare would be driven toward zero on average This does not of course mean thatb has never shifted historically only that it should not have exhibited large shiftsduring the recent history when we have observed relatively stable factor sharesAn alternative way in which the distributions may shift out over time is if thecurvature parameters and shift As long as the ratio does not change itmay be possible to allow the mass of the distributions to shift out while keepingthe capital share stable

21 Computing the convex hull of the overlapping CES production functionsis a computationally intensive problem especially when the number of ideas getslarge To simplify we first compute the convex hull of the (kiyi) points Then wecompute the convex hull of the CES functions associated with this limited set of


domly from a uniform distribution between the smallest value ofki and the largest value of ki for the iteration Finally we recordthe output of the global production function associated with thisinput

Following this procedure yields a graph like that shown inFigure III The key parameter values in this simulation are 5 and 25 so that the theory suggests we should expect aCobb-Douglas production function with a capital exponent of 13 As the figure shows the relation between log y andlog k is linear with a slope that is very close to this value

We next consider a simulation run for the full dynamic timepath of the Pareto model Continuing with the parameter choicesalready made we additionally assume that g 10 which im-plies an annual growth rate of 2 percent for output per worker in

points To approximate the CES curve we divide the capital interval into 100equally spaced points

FIGURE IIIThe Cobb-Douglas Result

The figure shows 1000 capital-output combinations from the global productionfunction The parameter values used in the simulation are N 500 5 25 a 1 b 02 and 1


the steady state We simulate this model for 100 years and plotthe results in several figures22 Figure IV shows a subset of themore than 1 million techniques that are discovered over these 100periods In particular we plot only the 300 points with the high-est values of y (these are shown with circles ldquoordquo) Without thistruncation the lower triangle in the figure that is currently blankbut for the ldquoxrdquo markers is filled in as solid black In addition thecapital-output combinations that are actually used in each periodare plotted with an ldquoxrdquo When a single technique is used for alarge number of periods the points trace out the local CES pro-duction function Alternatively if the economy is convexifying byusing two techniques the points trace out a line Finally whenthe economy switches to a new technique the capital-outputcombinations jump upward

Figure V shows output per worker over time plotted on a logscale The average growth rate of output per worker in this

22 Additional parameter values used in the simulation are listed in thecaption to Figure IV

FIGURE IVProduction in the Simulated Economy

Circles indicate ideas the ldquoxrdquo markers indicate capital-output combinationsthat are actually used The model is simulated for 100 periods with N0 50 5 25 g 10 a 1 b 02 k0 25 s 02 05 and 1


particular simulation is 163 percent as compared with the theo-retical value of 2 percent implied by the parameter values givenby g23

A feature of the model readily apparent in Figure V is thatthe economy switches from one production technique to anotherrather infrequently These switches are shown in the graph as thejumps that occur roughly every fifteen years or so Moreoverwhen the jumps occur they are typically quite large

What explains these patterns Recall that matching a Cobb-Douglas exponent on capital of 13 pins down the ratio of butit does not tell us the basic scale of these parameters The studiescited earlier related to patent values scientific productivity andfirm size typically find Pareto parameters that are in the range of05 to 15 We have chosen higher values of 5 and 25 Thefollowing exercise is helpful in thinking about this what is themedian value of a productivity draw conditional on that draw

23 We compute the average growth rate by dropping the first twenty obser-vations (to minimize the effect of initial conditions) and then regressing the log ofoutput per worker on a constant and a time trend

FIGURE VOutput per Worker over Time

See caption to Figure IV


being larger than some value x If is the Pareto parameterthen the answer to this question turns out to be 21x (1 07) x For example if 1 then the median value conditionalon a draw being higher than x is 2x This says that the averageidea that exceeds the frontier exceeds it by 100 percent Thisimplies very large jumps which might be plausible at the microlevel but seem too large at the macro level A value of 5instead gives an average jump of about 14 percent which is stillsomewhat large and which explains the large jumps in Figure VWe could have chosen an even larger Pareto parameter to yieldsmaller and more frequent jumps but this would have placed thevalue further from the range suggested by empirical studies Ifthe goal were to produce a simulation that could match the smallfrequent jumps in the aggregate data with plausible Pareto coef-ficients I suspect one would need a richer model that includesmultiple sectors and firms The jumps at the micro level would belarge and infrequent while aggregation would smooth things outat the macro level This is an interesting direction for furtherresearch24

Figure VI plots the capital share FKKY over time Eventhough the economy grows at a stable average rate the capitalshare exhibits fairly large movements When the economy isusing a single production technique the accumulation of capitalleads the capital share to decline Alternatively when the econ-omy is using two techniques to convexify the production set themarginal product of capital is constant so the capital share risessmoothly

It is interesting to compare the behavior of the capital sharein the Pareto model with the behavior that occurs in the simplebaseline model In the simple model the economy equates thecapital share to a function of the elasticity of the technologymenu If this elasticity is constant then the capital share wouldbe constant over time Here the technology menu exhibits aconstant elasticity on average but the menu is not a smoothcontinuous function Quite the opposite the extreme value natureof this problem means that the frontier is sparse as the exampleback in Figure I suggests This means that the capital share will

24 Gabaix [2004] is related to this point That paper shows that with aPareto distribution of firm sizes and a Pareto parameter less than two idiosyn-chratic shocks are smoothed out at a substantially slower rate than the standardcentral limit theorem suggests


be stationary but that it can move around both as the economyaccumulates capital and as it switches techniques

Figure VII shows the technology choices that occur in thissimulation As in Figure IV the 300 ideas with the highest levelof yi ai are plotted This time however the (aibi) pair corre-sponding to each idea is plotted The graph therefore shows thestochastic version of the technology menu In addition the figureplots with a ldquordquo the idea combinations that are actually used asthe economy grows over time Corresponding to the theoreticalfinding earlier one sees that the level of bi appears stationarywhile the level of ai trends upward On average technologicalchange is labor-augmenting

VII CONCLUSION

This paper provides microfoundations for the standard pro-duction function that serves as a building block for many eco-nomic models An idea is a set of instructions that tells how to

FIGURE VIThe Capital Share over Time



produce with a given collection of inputs It can be used with adifferent mix of inputs but it is not especially effective with thedifferent mix the elasticity of substitution in production is low fora given production technique Instead producing with a differentinput mix typically leads the production unit to switch to a newtechnique This suggests that the shape of the global productionfunction hinges on the distribution of available techniques

Kortum [1997] examined a model in which productivity lev-els are draws from a distribution and showed that only distribu-tions in which the upper tail is a power function are consistentwith exponential growth If one wants a model in which steady-state growth occurs then one needs to build in a Pareto distribu-tion for ideas We show here that this assumption delivers twoadditional results Pareto distributions lead the global productionfunction to take a Cobb-Douglas form and produce a setup wheretechnological change in the local production function is entirelylabor-augmenting in the long run

FIGURE VIITechnology Choices

From more than 1 million ideas generated the 300 with the highest level of aare plotted as circles The figure also plots with a ldquordquo the (aibi) combinations thatare used at each date and links them with a line When two ideas are usedsimultaneously the idea with the higher level of output is plotted See also notesto Figure IV


There are several additional directions for research sug-gested by this approach First our standard ways of introducingskilled and unskilled labor into production involve productionfunctions with an elasticity of substitution bigger than one con-sistent with the observation that unskilled laborrsquos share of in-come seems to be falling25 How can this view be reconciled withthe reasoning here

Second the large declines in the prices of durable investmentgoods are often interpreted as investment-specific technologicalchange That is they are thought of as increases in b rather thanincreases in a26 This is the case in Greenwood Hercowitz andKrusell [1997] and Whelan [2003] and it is also implicitly theway the hedonic pricing of computers works in the NationalIncome and Product Accounts better computers are interpretedas more computers The model in this paper suggests instead thatb might be stationary so there is a tension with this other workOf course it is not at all obvious that better computers areequivalent to more computers Perhaps a better computer is likehaving two people working with a single computer (as in extremeprogramming) In this case better computers might be thought ofas increases in a instead This remains an open question Alter-natively it might be desirable to have microfoundations for aCobb-Douglas production function that permits capital-augment-ing technological change to occur in the steady state

Finally one might ask how the model relates to recent dis-cussions about the behavior of capital shares The literature is insomething of a flux For a long time of course the stylized facthas been that capitalrsquos share is relatively stable This turns out tobe true at the aggregate level for the United States and GreatBritain but it is not true at the disaggregated level in the UnitedStates or in the aggregate for many other countries Rather themore accurate version of the fact appears to be that capitalrsquosshare can exhibit large medium term movements and even trendsover periods longer than twenty years in some countries andindustries27 This paper is somewhat agnostic about factor

25 See Katz and Murphy [1992] and Krusell Ohanian Rios-Rull and Vio-lante [2000] for example

26 This is loose In fact they are thought of as increases in a term thatmultiplies investment in the capital accumulation equation Of course for manypurposes this is like an increase in b

27 The recent papers by Blanchard [1997] Bentolila and Saint-Paul [2003]and Harrison [2003] discuss in detail the facts about capital and labor shares andhow they vary Gollin [2002] is also related that paper argues that in the cross


shares As shown in Figure VI the Pareto model predicts that thecapital share may vary over time while of course the baselinemodel implied a constant capital share However there are manyother determinants of capital shares left out of this model includ-ing aggregation issues and wedges between marginal productsand prices so care should be taken in interpreting the modelalong this particular dimension

APPENDIX AN ALTERNATIVE DERIVATION OF THE COBB-DOUGLAS RESULT

Here we show how to derive the Cobb-Douglas result for afinite number of ideas The key to this stronger result is anassumption common in the growth literature the assumptionthat the discovery of ideas follows a Poisson process28

We now make the research process explicit New ideas forproduction are discovered through research A single researchendeavor yields a number of ideas drawn from a Poisson distri-bution with a parameter normalized to one In expectation theneach research endeavor yields one idea Let N denote the cumu-lative number of research endeavors that have been undertakenThen the number of ideas n that have been discovered as aresult of these N attempts is a random variable drawn from aPoisson distribution with parameter N This additional layer isthe only change to the model in Section III

For a given number of production techniques the globalproduction function F(KLn) is

(37) FKLn maxi0 n1

FbiKaiL

As before let Yi denote production using technique i with a givenamount of capital and labor Then

38 PrYi y PrbiK y aiL y

G yKyL

The output level associated with the global production func-tion is then distributed as

section of countries labor shares are more similar than rough data on employeecompensation as a share of GDP suggest because of the very high levels ofself-employment in many poor countries

28 For example see Aghion and Howitt [1992]


(39) Prmaxi

Yi y 1 G yK yLn

At this point we can use the nice properties of the Poissondistribution to make further progress Recall that n Poisson(N) soas a function of the total number of research attempts N we have

40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL

For a general joint distribution function G this last equationdescribes the distribution of the global production function whencumulative research effort is N29

Now assume as in the main text that the ideas are drawnfrom a joint Pareto distribution so that

(41) PrYi y G yK yL KLy

Combining this result with equation (40) it is straightforward toshow that the distribution of the output that can be producedwith the global production function given inputs of K and L is

(42) Prmax Yi y eNKLy

which is the Frechet distributionFinally taking expectations over this distribution one sees

that expected output given N cumulative research draws andinputs K and L is given by

(43) EY Emax Yi NKL1

where (1 1( )) is a constant that depends on Eulerrsquosfactorial function30

29 See Proposition 21 in Kortum [1997] for this style of reasoning ie for anapproach that uses a Poisson process to get an exact extreme value distributionthat is easy to work with rather than an asymptotic result See also Johnson Kotzand Balakrishnan [1994 pages 11 and 91ndash92]

30 Surprisingly few of the reference books on extreme value theory actuallyreport the mean of the Frechet distribution For a distribution function F( x) exp((( x 13))) Castillo [1988] reports that the mean is 13 (1 1) for 1


One can also use the distribution in equation (42) to write thelevel of output as a random variable

(44) Y NKL1

where is a random variable drawn from a Frechet distributionwith parameter That is we get the same result as inequation (28) but exactly for finite N rather than as an asymp-totic approximation

UNIVERSITY OF CALIFORNIA AT BERKELEY AND NATIONAL BUREAU OF ECONOMIC

RESEARCH

REFERENCES

Acemoglu Daron ldquoFactor Prices and Technical Change From Induced Innovationto Recent Debatesrdquo in P Aghion R Frydman J Stiglitz and M Woodfordeds Knowledge Information and Expectations in Modern Macroeconomics(Princeton NJ Princeton University Press 2003a)

mdashmdash ldquoLabor- and Capital-Augmenting Technical Changerdquo Journal of EuropeanEconomic Association I (2003b) 1ndash37

Aghion Philippe and Peter Howitt ldquoA Model of Growth through Creative De-structionrdquo Econometrica LX (1992) 323ndash351

Atkinson Anthony B and Joseph E Stiglitz ldquoA New View of TechnologicalChangerdquo Economic Journal LXXIX (1969) 573ndash578

Axtell Robert L ldquoZipf Distribution of U S Firm Sizesrdquo Science CCXCIII (2001)1818ndash1820

Barro Robert J and Xavier Sala-i-Martin Economic Growth (New YorkMcGraw-Hill 1995)

Basu Susanto and David N Weil ldquoAppropriate Technology and Growthrdquo Quar-terly Journal of Economics CXIII (1998) 1025ndash1054

Bentolila Samuel and Gilles Saint-Paul ldquoExplaining Movements in the LaborSharerdquo CEMFI mimeo 2003

Bertran Fernando Leiva ldquoPricing Patents through Citationsrdquo University ofRochester mimeo 2003

Blanchard Olivier J ldquoThe Medium Runrdquo Brookings Papers on Economic Activity2 (1997) 89ndash141

Caballero Ricardo J and Mohamad L Hammour ldquoJobless Growth Appropri-ability Factor Substitution and Unemploymentrdquo Carnegie Rochester Con-ference Series on Public Policy XLVIII (1998) 51ndash94

Caselli Francesco and Wilbur John Coleman ldquoThe World Technology FrontierrdquoHarvard University mimeo 2004

Castillo Enrique Extreme Value Theory in Engineering (London Academic Press1988)

Chevalier Judith and Austan Goolsbee ldquoPrice Competition Online Amazonversus Barnes and Noblerdquo Quantitative Marketing and Economics I (2003)203ndash222

Cordoba Juan Carlos ldquoOn the Distribution of City Sizesrdquo Rice Universitymimeo 2003

Cox Raymond and Kee H Chung ldquoPatterns of Research Output and AuthorConcentration in the Economics Literaturerdquo Review of Economics and Sta-tistics LXXIII (1991) 740ndash747

Drandakis E M and Edmund S Phelps ldquoA Model of Induced Invention Growthand Distributionrdquo Economic Journal LXXVI (1966) 823ndash840

Gabaix Xavier ldquoZipfrsquos Law for Cities An Explanationrdquo Quarterly Journal ofEconomics CXIV (1999) 739ndash767


mdashmdash ldquoPower Laws and the Granular Origins of Aggregate Fluctuationsrdquo Massa-chusetts Institute of Technology mimeo 2004

Gabaix Xavier Parameswaran Gopikrishnan Vasiliki Plerou and H EugeneStanley ldquoA Theory of Power Law Distributions in Financial Market Fluctua-tionsrdquo Nature CDXXIII (2003) 267ndash270

Galambos Janos The Asymptotic Theory of Extreme Order Statistics (New YorkJohn Wiley amp Sons 1978)

Gilchrist Simon and John C Williams ldquoPutty Clay and Investment A BusinessCycle Analysisrdquo Journal of Political Economy CVIII (2000) 928ndash960

Gollin Douglas ldquoGetting Income Shares Rightrdquo Journal of Political Economy CX(2002) 458ndash474

Grabowski Henry ldquoPatents and New Product Development in the Pharmaceuti-cal and Biotechnology Industriesrdquo Duke University mimeo 2002

Grandmont Jean-Michel ldquoDistributions of Preferences and the lsquoLaw of DemandrsquordquoEconometrica LV (1987) 155ndash161

Greenwood Jeremy Zvi Hercowitz and Per Krusell ldquoLong-Run Implications ofInvestment-Specific Technological Changerdquo American Economic ReviewLXXXVII (1997) 342ndash362

Harhoff Dietmar Frederic M Scherer and Katrin Vopel ldquoExploring the Tail ofPatented Invention Value Distributionsrdquo WZB Working Paper 97-27 1997

Harrison Ann E ldquoHas Globalization Eroded Laborrsquos Share Some Cross-CountryEvidencerdquo University of California at Berkeley mimeo 2003

Hildenbrand Werner ldquoOn the lsquoLaw of Demandrsquordquo Econometrica LI (1983)997ndash1020

Houthakker Hendrik S ldquoThe Pareto Distribution and the Cobb-Douglas Produc-tion Function in Activity Analysisrdquo Review of Economic Studies XXIII (1955ndash1956) 27ndash31

Huber John C ldquoCumulative Advantage and Success-Breeds-Success The Valueof Time Pattern Analysisrdquo Journal of the American Society for InformationScience XLIX (1998) 471ndash476

Johnson Norman L Samuel Kotz and N Balakrishnan Continuous UnivariateDistributions Volume 2 (New York Wiley Interscience 1994)

Jones Charles I ldquoRampD-Based Models of Economic Growthrdquo Journal of PoliticalEconomy CIII (1995) 759ndash784

Jones Charles I and Dean Scrimgeour ldquoThe Steady-State Growth Theorem AComment on Uzawa (1961)rdquo University of California at Berkeley mimeo2005

Katz Lawrence and Kevin Murphy ldquoChanges in Relative Wages 1963ndash1987Supply and Demand Factorsrdquo Quarterly Journal of Economics CVII (1992)35ndash78

Kennedy Charles M ldquoInduced Bias in Innovation and the Theory of Distribu-tionrdquo Economic Journal LXXIV (1964) 541ndash547

Kortum Samuel S ldquoResearch Patenting and Technological Changerdquo Economet-rica LXV (1997) 1389ndash1419

Krusell Per Lee Ohanian Jose-Victor Rios-Rull and Giovanni Violante ldquoCapi-tal-Skill Complementarity and Inequality A Macroeconomic AnalysisrdquoEconometrica LXVIII (2000) 1029ndash1053

Lagos Ricardo ldquoA Model of TFPrdquo New York University working paper 2004Lotka A J ldquoThe Frequency Distribution of Scientific Productivityrdquo Journal of

the Washington Academy of Sciences XVI (1926) 317ndash323Mitzenmacher Michael ldquoA Brief History of Generative Models for Power Law and

Lognormal Distributionsrdquo Internet Mathematics I (2003) 226ndash251Redner Sidney ldquoHow Popular Is Your Paper An Empirical Study of the Citation

Distributionrdquo European Physical Journal B IV (1998) 131ndash134Robinson Joan ldquoThe Production Function and the Theory of Capitalrdquo Review of

Economic Studies XXI (1953ndash1954) 81ndash106Romer Paul M ldquoEndogenous Technological Changerdquo Journal of Political Econ-

omy XCVIII (1990) S71ndashS102Rossi-Hansberg Esteban and Mark L J Wright ldquoUrban Structure and Growthrdquo

Stanford University mimeo 2004Saez Emmanuel ldquoUsing Elasticities to Derive Optimal Tax Ratesrdquo Review of

Economic Studies LXVIII (2001) 205ndash229


Samuelson Paul A ldquoA Theory of Induced Innovations along Kennedy-WeisackerLinesrdquo Review of Economics and Statistics XLVII (1965) 343ndash356

Sornette Didier and Daniel Zajdenweber ldquoThe Economic Return of ResearchThe Pareto Law and its Implicationsrdquo European Physical Journal B VIII(1999) 653ndash664 httpxxxlanlgovabscondmat9809366

Uzawa Hirofumi ldquoNeutral Inventions and the Stability of Growth EquilibriumrdquoReview of Economic Studies XXVIII (1961) 117ndash124

Whelan Karl ldquoA Two-Sector Approach to Modeling U S NIPA Datardquo Journal ofMoney Credit and Banking XXXV (2003) 627ndash656


If one wants to produce with a capital-labor ratio very differ-ent from k this Leontief technique is not particularly helpfuland one needs to discover a new idea ldquoappropriaterdquo to the highercapital-labor ratio1 Notice that one can replace the Leontiefstructure with a production technology that exhibits a low elas-ticity of substitution and this statement remains true to takeadvantage of a substantially higher capital-labor ratio one reallyneeds a new technique targeted at that capital-labor ratio Oneneeds a new idea

According to this view the standard production function thatwe write down mapping the entire range of capital-labor ratiosinto output per worker is a reduced form It is not a singletechnology but rather represents the substitution possibilitiesacross different production techniques The elasticity of substitu-tion for this global production function depends on the extent towhich new techniques that are appropriate at higher capital-labor ratios have been discovered That is it depends on thedistribution of ideas

But from what distribution are ideas drawn Kortum [1997]examined a search model of growth in which ideas are productiv-ity levels that are drawn from a distribution He showed that theonly way to get exponential growth in such a model is if ideas aredrawn from a Pareto distribution at least in the upper tail

This same basic assumption that ideas are drawn from aPareto distribution yields two additional results in the frame-work considered here First the global production function isCobb-Douglas Second the optimal choice of the individual pro-duction techniques leads technological change to be purely labor-augmenting in the long run In other words an assumptionKortum [1997] suggests we make if we want a model to exhibitsteady-state growth leads to important predictions about theshape of production functions and the direction of technicalchange

In addition to Kortum [1997] this paper is most closelyrelated to an older paper by Houthakker [1955ndash1956] and to tworecent 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]




II A BASELINE MODEL

IIA Preliminaries


(1) Y FbiKaiL



(2) Y aiLF biKaiL

1


(3) y aiFbi

aik1




(4) y yiF kki

1







(5) Hab N






(6) Y FKLN maxba

FbKaL

subject to (5)





(7)K

L

b

a






(8) Hab ab N 0 0




(9) yi ai

(10) ki aibi




(11) yi Nki1






(12) Y NKL1




(13)bKF1bKaL

aLF2bKaL


(14)xF1 x1

F2 x1







(16)Yt

atLt FbtKt

atLt1









IIC Discussion




maxabKL















IIIA Setup






18 Prai a 1 aa

a a 0



19 Prbi b 1 bb

b b 0

where 0 0 and 113



a



21


G yK

yL

KLy

where ab









(22) FKLN maxi1 N

FbiKaiL


23PrY y 1 H yN

1 KLyN



(24) zN NKL1


25

PrY zNy 1 KLzNyN

1 y

N N


(26) limN3

PrY zNy expy


Therefore

(27)Y

NKL1 a

Frechet


(28) Y NKL1





IV DISCUSSION







(29) ba C

























(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES
























































II A BASELINE MODEL

IIA Preliminaries


(1) Y FbiKaiL



(2) Y aiLF biKaiL

1


(3) y aiFbi

aik1




(4) y yiF kki

1







(5) Hab N






(6) Y FKLN maxba

FbKaL

subject to (5)





(7)K

L

b

a






(8) Hab ab N 0 0




(9) yi ai

(10) ki aibi




(11) yi Nki1






(12) Y NKL1




(13)bKF1bKaL

aLF2bKaL


(14)xF1 x1

F2 x1







(16)Yt

atLt FbtKt

atLt1









IIC Discussion




maxabKL















IIIA Setup






18 Prai a 1 aa

a a 0



19 Prbi b 1 bb

b b 0

where 0 0 and 113



a



21


G yK

yL

KLy

where ab









(22) FKLN maxi1 N

FbiKaiL


23PrY y 1 H yN

1 KLyN



(24) zN NKL1


25

PrY zNy 1 KLzNyN

1 y

N N


(26) limN3

PrY zNy expy


Therefore

(27)Y

NKL1 a

Frechet


(28) Y NKL1





IV DISCUSSION







(29) ba C

























(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES























































(4) y yiF kki

1







(5) Hab N






(6) Y FKLN maxba

FbKaL

subject to (5)





(7)K

L

b

a






(8) Hab ab N 0 0




(9) yi ai

(10) ki aibi




(11) yi Nki1






(12) Y NKL1




(13)bKF1bKaL

aLF2bKaL


(14)xF1 x1

F2 x1







(16)Yt

atLt FbtKt

atLt1









IIC Discussion




maxabKL















IIIA Setup






18 Prai a 1 aa

a a 0



19 Prbi b 1 bb

b b 0

where 0 0 and 113



a



21


G yK

yL

KLy

where ab









(22) FKLN maxi1 N

FbiKaiL


23PrY y 1 H yN

1 KLyN



(24) zN NKL1


25

PrY zNy 1 KLzNyN

1 y

N N


(26) limN3

PrY zNy expy


Therefore

(27)Y

NKL1 a

Frechet


(28) Y NKL1





IV DISCUSSION







(29) ba C

























(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES
























































(6) Y FKLN maxba

FbKaL

subject to (5)





(7)K

L

b

a






(8) Hab ab N 0 0




(9) yi ai

(10) ki aibi




(11) yi Nki1






(12) Y NKL1




(13)bKF1bKaL

aLF2bKaL


(14)xF1 x1

F2 x1







(16)Yt

atLt FbtKt

atLt1









IIC Discussion




maxabKL















IIIA Setup






18 Prai a 1 aa

a a 0



19 Prbi b 1 bb

b b 0

where 0 0 and 113



a



21


G yK

yL

KLy

where ab









(22) FKLN maxi1 N

FbiKaiL


23PrY y 1 H yN

1 KLyN



(24) zN NKL1


25

PrY zNy 1 KLzNyN

1 y

N N


(26) limN3

PrY zNy expy


Therefore

(27)Y

NKL1 a

Frechet


(28) Y NKL1





IV DISCUSSION







(29) ba C

























(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES























































(7)K

L

b

a






(8) Hab ab N 0 0




(9) yi ai

(10) ki aibi




(11) yi Nki1






(12) Y NKL1




(13)bKF1bKaL

aLF2bKaL


(14)xF1 x1

F2 x1







(16)Yt

atLt FbtKt

atLt1









IIC Discussion




maxabKL















IIIA Setup






18 Prai a 1 aa

a a 0



19 Prbi b 1 bb

b b 0

where 0 0 and 113



a



21


G yK

yL

KLy

where ab









(22) FKLN maxi1 N

FbiKaiL


23PrY y 1 H yN

1 KLyN



(24) zN NKL1


25

PrY zNy 1 KLzNyN

1 y

N N


(26) limN3

PrY zNy expy


Therefore

(27)Y

NKL1 a

Frechet


(28) Y NKL1





IV DISCUSSION







(29) ba C

























(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES























































(8) Hab ab N 0 0




(9) yi ai

(10) ki aibi




(11) yi Nki1






(12) Y NKL1




(13)bKF1bKaL

aLF2bKaL


(14)xF1 x1

F2 x1







(16)Yt

atLt FbtKt

atLt1









IIC Discussion




maxabKL















IIIA Setup






18 Prai a 1 aa

a a 0



19 Prbi b 1 bb

b b 0

where 0 0 and 113



a



21


G yK

yL

KLy

where ab









(22) FKLN maxi1 N

FbiKaiL


23PrY y 1 H yN

1 KLyN



(24) zN NKL1


25

PrY zNy 1 KLzNyN

1 y

N N


(26) limN3

PrY zNy expy


Therefore

(27)Y

NKL1 a

Frechet


(28) Y NKL1





IV DISCUSSION







(29) ba C

























(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES























































(12) Y NKL1




(13)bKF1bKaL

aLF2bKaL


(14)xF1 x1

F2 x1







(16)Yt

atLt FbtKt

atLt1









IIC Discussion




maxabKL















IIIA Setup






18 Prai a 1 aa

a a 0



19 Prbi b 1 bb

b b 0

where 0 0 and 113



a



21


G yK

yL

KLy

where ab









(22) FKLN maxi1 N

FbiKaiL


23PrY y 1 H yN

1 KLyN



(24) zN NKL1


25

PrY zNy 1 KLzNyN

1 y

N N


(26) limN3

PrY zNy expy


Therefore

(27)Y

NKL1 a

Frechet


(28) Y NKL1





IV DISCUSSION







(29) ba C

























(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES






















































(16)Yt

atLt FbtKt

atLt1









IIC Discussion




maxabKL















IIIA Setup






18 Prai a 1 aa

a a 0



19 Prbi b 1 bb

b b 0

where 0 0 and 113



a



21


G yK

yL

KLy

where ab









(22) FKLN maxi1 N

FbiKaiL


23PrY y 1 H yN

1 KLyN



(24) zN NKL1


25

PrY zNy 1 KLzNyN

1 y

N N


(26) limN3

PrY zNy expy


Therefore

(27)Y

NKL1 a

Frechet


(28) Y NKL1





IV DISCUSSION







(29) ba C

























(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES
























































IIC Discussion




maxabKL















IIIA Setup






18 Prai a 1 aa

a a 0



19 Prbi b 1 bb

b b 0

where 0 0 and 113



a



21


G yK

yL

KLy

where ab









(22) FKLN maxi1 N

FbiKaiL


23PrY y 1 H yN

1 KLyN



(24) zN NKL1


25

PrY zNy 1 KLzNyN

1 y

N N


(26) limN3

PrY zNy expy


Therefore

(27)Y

NKL1 a

Frechet


(28) Y NKL1





IV DISCUSSION







(29) ba C

























(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES

































































IIIA Setup






18 Prai a 1 aa

a a 0



19 Prbi b 1 bb

b b 0

where 0 0 and 113



a



21


G yK

yL

KLy

where ab









(22) FKLN maxi1 N

FbiKaiL


23PrY y 1 H yN

1 KLyN



(24) zN NKL1


25

PrY zNy 1 KLzNyN

1 y

N N


(26) limN3

PrY zNy expy


Therefore

(27)Y

NKL1 a

Frechet


(28) Y NKL1





IV DISCUSSION







(29) ba C

























(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES






















































19 Prbi b 1 bb

b b 0

where 0 0 and 113



a



21


G yK

yL

KLy

where ab









(22) FKLN maxi1 N

FbiKaiL


23PrY y 1 H yN

1 KLyN



(24) zN NKL1


25

PrY zNy 1 KLzNyN

1 y

N N


(26) limN3

PrY zNy expy


Therefore

(27)Y

NKL1 a

Frechet


(28) Y NKL1





IV DISCUSSION







(29) ba C

























(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES























































(22) FKLN maxi1 N

FbiKaiL


23PrY y 1 H yN

1 KLyN



(24) zN NKL1


25

PrY zNy 1 KLzNyN

1 y

N N


(26) limN3

PrY zNy expy


Therefore

(27)Y

NKL1 a

Frechet


(28) Y NKL1





IV DISCUSSION







(29) ba C

























(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES
























































IV DISCUSSION







(29) ba C

























(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES






















































(29) ba C

























(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES







































































(30) Kt1 1 Kt sYt s 01


(31) Nt N0egt







(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES






















































(32) Yt NtKtLt

1t


(34) E logyt1

yt

g




(35) Yt FbitKtaitLt



(33) logyt1

yt

1

logNt1

Nt

log

kt1

kt log

t1

t







1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES


























































1


































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES

















































































VII CONCLUSION






















(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES







































































(37) FKLn maxi0 n1

FbiKaiL



G yKyL





(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES






















































(39) Prmaxi

Yi y 1 G yK yLn


40 Prmax Yi y n0

eNNn

n 1 G yK yLn

eN n0

N1 G yKyLn

n

eN eN1G

eNG yK yL














(44) Y NKL1



RESEARCH

REFERENCES























































(44) Y NKL1



RESEARCH

REFERENCES






















































the shape of production functions and the ...chadj/jonesqje2005.pdfthe shape of production functions...

Documents