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Neutral or factor saving

innovations?

Hernando Zuleta

Andrés Zambrano

Documentos

CEDE

ISSN 1657-7191 Edición electrónica.

No.

65

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Serie Documentos Cede, 2018-65

ISSN 1657-7191 Edición electrónica.

Diciembre 2018

© 2018, Universidad de los Andes, Facultad de Economía,

CEDE. Calle 19A No. 1 – 37 Este, Bloque W.

Bogotá, D. C., Colombia Teléfonos: 3394949- 3394999,

extensiones 2400, 2049, 2467

infocede@uniandes.edu.co http://economia.uniandes.edu.co

Impreso en Colombia – Printed in Colombia

La serie de Documentos de Trabajo CEDE se circula con propósitos de discusión y divulgación. Los artículos no han sido evaluados por pares ni sujetos a ningún tipo de evaluación formal por parte del equipo de trabajo del CEDE. El contenido de la presente publicación se encuentra protegido por las normas internacionales y nacionales vigentes sobre propiedad intelectual, por tanto su utilización, reproducción, comunicación pública, transformación, distribución, alquiler, préstamo público e importación, total o parcial, en todo o en parte, en formato impreso, digital o en cualquier formato conocido o por conocer, se encuentran prohibidos, y sólo serán lícitos en la medida en que se cuente con la autorización previa y expresa por escrito del autor o titular. Las limitaciones y excepciones al Derecho de Autor, sólo serán aplicables en la medida en que se den dentro de los denominados Usos Honrados (Fair use), estén previa y expresamente

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Universidad de los Andes | Vigilada Mineducación

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1

Neutral

or

factor

saving

innovations?

Hernando Zuleta∞ Andrés Zambrano⁎

Abstract

We present a theoretical argument to identify the conditions under which a firm prefers to invest in factor saving innovations rather than neutral innovations. We prove that incentives to invest in factor saving innovations positively depend on i) total factor productivity and ii)

the scarcity of the factor.

JEL Codes: O11, O30, O41, O47, E01, E25

Keywords: Factor Shares, ProductionFunction, Productivity.

∞Department of Economics, Universidad de Los Andes. email: h.zuleta@uniandes.edu.co

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2

Innovaciones

neutrales

o

ahorradoras

de

factores?

Hernando Zuleta£ Andrés Zambrano§

Resumen

Este documento presenta un argumento teórico para identificar las condiciones bajo las cuales una firma prefiere invertir en innovaciones ahorradoras de factores en lugar de innovaciones neutrales. Los autores muestran que los incentivos para invertir en innovaciones ahorradoras de factores dependen positivamente de i) productividad total de los factores y ii) la escasez del factor.

Códigos JEL: O11, O30, O41, O47, E01, E25

Palabras claves: Participación factorial, Función de Producción, Productividad.

£ Facultad de Economía, Universidad de Los Andes. email: h.zuleta@uniandes.edu.co

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3

1. Introduction

Why should a firm invest in factor saving innovations rather than neutral innovations? To answer this question, we model the behavior of a firm for which the quantity of factors is exogenous. We prove that, under general assumptions, i) the total factor productivity is complementary to biased innovations as long as it is towards a relatively abundant factor and ii) the more scarce is a factor, the higher are the incentives to invest in technologies that can reduce the use of this factor.

This paper belongs to the literature on factor saving innovations and it is closely related to Acemoglu

(2007 and 2010).1 Acemoglu (2007) proves that, if technological change is factor augmenting, then

the increase in the supply of a factor induces technological change relatively biased toward that factor. In a similar way, Acemoglu (2010) shows that labor scarcity encourages technological change if new technologies are strongly labor saving and discourages technological change if new technologies are strongly labor complementary. These two articles are clearly related to our paper. However, Acemoglu does not model the problem of choosing between a factor saving innovation and a neutral innovation. More recently, Seater and Yenokyan (2018) study the decision over factor augmenting technical change and factor eliminating technical change simultaneously, whereas our emphasis is on the simultaneous choice over factor augmenting technical change and neutral technological change.

In the next section, we present a partial equilibrium model where the quantity of factors is exogenous and model the choice of technologies. Finally, we present a possible application of our theoretical results.

2. The Theory

2.1. Basic Assumptions

The production function is characterized in the following way: 𝑌 𝐴𝐹 𝐾, 𝐿, 𝛼 . Where 𝐴 is total

factor productivity, 𝐾 is the stock of capital, 𝐿 is the quantity of labor and 𝛼 is a technological

parameter associated to the relative productivity of capital (labor). Additionally, we denote 𝑘 as

the capital labor ratio.

The function 𝐹 . is continuous and has the following properties:

𝐹 . 0 ; 𝐹 . 0; 𝐹 . 0; 𝐹 . 0; 𝐹 . 0;

𝐹 . 0; 𝐹 . 0; lim

→ 𝐹 . → ∞; 𝐹 , . 0 and lim→ 𝐹 . ∞;

𝜔 𝐾, 𝐿, 𝛼 0 where 𝜔 𝐾, 𝐿, 𝛼 .

. .

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4

The first five assumptions are standard. The last assumptions imply that there are capital using and labor saving innovations. Finally, we assume that this type of innovation has a positive effect on

output if the economy is relatively abundant in capital, namely, if 𝐾 𝐿 then 𝐹 . 0.

2.2. Buying Technologies

Consider a firm with a fixed quantity of factors (𝐾 and 𝐿) and initial technologies 𝐴 and 𝛼 . This firm may allocate resources to increase total factor productivity (𝐴) or to change the relative marginal productivity of factors (𝛼).

Assume that the technologies 𝐴 𝐴 and 𝛼 𝛼 are available in the market and can be bought in

the market at prices 𝑃 and 𝑃. Assume also that once a firm buys a technology it can use it without

additional costs.

Normalizing 𝑃 1, the firm solves the following problem:

max

∈ , , ∈ , 𝐴𝐹 𝐾, 𝐿, 𝛼 𝑃 𝐼 𝑃 𝐼

Given the availability of factors (𝐾, 𝐿) the solution is characterized by:

1) If 𝑃 𝐴 𝐴 𝐹 . , then the firm invests in neutral technological change.

2) If 𝑃 𝐴 𝐹 𝐾, 𝐿, 𝛼 𝐹 𝐾, 𝐿, 𝛼 , then the firm invests in capital using and labor saving

technological change.

Proposition 1: For any price 𝑃 , given A and L, there exists a critical level for the capital stock 𝐾 such that, if the capital stock of the firm is higher than this critical level (𝐾 𝐾 , then the firm invests in biased technological change.

Proof:

1. For any finite 𝑃 there exists a finite number 𝑀 such that 𝑀 𝑃 .

2. 𝐹 , . 0 and lim

→ 𝐹 . ∞. Therefore, 𝐹 𝐾, 𝐿, 𝛼 𝐹 𝐾, 𝐿, 𝛼 0 and

lim

→ 𝐹 𝐾, 𝐿, 𝛼 𝐹 𝐾, 𝐿, 𝛼 ∞ and for any finite number 𝑀 there exists a critical capital

stock 𝐾 such that for any 𝐾 𝐾 it holds that 𝐴 𝐹 𝐾, 𝐿, 𝛼 𝐹 𝐾, 𝐿, 𝛼 𝑀.

3. From 1 and 2 it follows that, given L, 𝑃 𝐴 𝐹 𝐾, 𝐿, 𝛼 𝐹 𝐾, 𝐿, 𝛼 for any 𝐾 𝐾. ■

Corollary 1: For any price 𝑃 , given A and 𝐾, there exists a critical level for labor 𝐿 such that, if the

amount of labor hired by the firm is lower than the critical level (𝐿 𝐿 , then the firm invests in

biased technological change.

Proof:

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5

2. 𝐹 , . 0 and lim

→ 𝐹 . ∞. Therefore, 𝐹 𝐾, 𝐿, 𝛼 𝐹 𝐾, 𝐿, 𝛼 0 and

lim

→ 𝐹 𝐾, 𝐿, 𝛼 𝐹 𝐾, 𝐿, 𝛼 ∞ and for any finite number 𝑀 there exists a critical labor

level 𝐿 such that for any 𝐿 𝐿 it holds that 𝐴 𝐹 𝐾, 𝐿, 𝛼 𝐹 𝐾, 𝐿, 𝛼 𝑀.

3. From 1 and 2 it follows that, given K, 𝑃 𝐴 𝐹 𝐾, 𝐿, 𝛼 𝐹 𝐾, 𝐿, 𝛼 for any 𝐿 𝐿. ■

The scarcity of labor (or the abundance of capital) is given by that the ratio . Therefore, Proposition 1 and corollary 1 imply that the more abundant is the capital the greater are incentives to invest in biased innovations.

Proposition 2: There exists a critical A such that the firm invests in biased technological change as

long as𝐹 𝐾, 𝐿, 𝛼 𝐹 𝐾, 𝐿, 𝛼 0, that is, for a sufficiently high K.

Proof:

Define a small positive number 𝜀 such that 𝐹 𝐾, 𝐿, 𝛼 𝐹 𝐾, 𝐿, 𝛼 𝜀. Now define 𝐴

. If 𝐴 𝐴 then the firm invest in biased innovations. ■

Proposition 3: If 𝐾 𝐿, the decision to invest in neutral technological change depends positively on

𝛼.

If 𝐹 . , then the firm invests in neutral technological change. Now, if 𝐾 𝐿 then

𝐹 . 0 and, therefore, if there exists a 𝛼 such that 𝐹 𝐾, 𝐿, 𝛼 , then for any 𝛼 𝛼

we have 𝐹 . . ■

Proposition 4: Given 𝑃 , the decision to invest in neutral technological change depends positively on the size of the firm.

Proof:

1. For any 𝑃 there exists a finite number M such that 𝑀 𝑃 .

2. Since 𝐹 . is strictly increasing in 𝐾 and 𝐿, if there exists a number 𝜆 0 such that

𝐹 𝜆𝐾, 𝜆𝐿, 𝛼 𝑀, then 𝐹 𝜆𝐾, 𝜆𝐿, 𝛼 𝑃 for any 𝜆 𝜆. ■

Therefore, using 𝜆 as a measure of the size of the firm, if there exists a minimum size such that the

firm chooses to invest in neutral technological change, then any size higher than this one induces the same investment.

Now assume that the firm must choose only one type of innovation. To do that, the firm compares the benefits for choosing each innovation.

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6 Proof:

1. If 𝐴 , ,

, , then the firm chooses biased technological change.

2. Given the properties of the function 𝐹 . , given 𝛼, 𝛼 , 𝐿 and 𝐾, there are two real numbers

M and M’ such that 𝑀 𝐹 𝐾, 𝐿, 𝛼 and 𝑀 𝐹 𝐾, 𝐿, 𝛼 .

3. Define 𝐴 .

4. From 1, 2 and 3 it follows that for any 𝐴 𝐴 the firm chooses biased technological change.■

Until now, we have assumed that technologies are available in the market and firms choose to buy or not. However, many times the firms have an R&D department and produce their own technology.

2.3. Innovations inside the firm.

In this section, we study the production of technologies inside the firm (in-house R&D). Firms choose

the technology 𝐴′, 𝛼′ in order to maximize profits. We assume there is an initial technology 𝐴, 𝛼

and that 𝐴 𝐴 and 𝛼 𝛼, that is, the firm cannot make money by downgrading its technology.

Assume that the cost of a technological improvement has the form 𝐶 𝐴 , 𝛼 0, where 𝐶 . is a

continuous and bounded function with the following conditions:

𝐶 . 0, where 𝐶 𝐴, . 0; 𝐶 . 0

𝐶 . 0, where 𝐶 . , 𝛼 0; 𝐶 . 0

𝐶 . 0.

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In this case, the profit function (𝑃 1) is:

𝜋 𝐴′𝐹 𝐾, 𝐿, 𝛼′ 𝑤𝐿 𝑟𝐾 𝐶 𝐴 , 𝛼 (3)

The optimal decisions satisfy:

𝐴′𝐹 . 𝐶 . 0

𝐹 . 𝐶 . 0

Proposition 5: The size of investment in 𝛼′ depends positively on 𝐾, negatively on 𝐿, and positively on 𝐴 as long as 𝐾 𝐿.

Proof:

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7 0

0

0 if 𝐾 𝐿. ■

Proposition 6: The decision to invest in neutral innovations depends positively on the size of the firms, and positively on 𝛼 as long as 𝐾 𝐿.

Proof:

Use the implicit function theorem to obtain:

0,

0,

0 if 𝐾 𝐿.■

It follows that i) using 𝐾 as a measure of the size of the firms, a higher 𝐾 generates more investment in neutral innovations. ii) Using 𝐿 as a measure of the size of the firms, a higher 𝐿 generates more investment in neutral innovations. There is also a positive complementarity between the neutral innovation and the biased innovation as long is towards using more the relatively abundant factor.

In summary, this theoretical approach suggests that (i) capital abundant and labor scarce firms have

incentives to adopt capital using - labor saving technologies, (ii) big firms have incentives to adopt

neutral technological improvements and (iii) firms with higher total factor productivity have more

incentives to adopt biased technological innovations as long as the factor is relatively abundant.

These results imply that the growth process of firms is characterized by periods of neutral innovations and capital accumulation followed by periods of biased innovations.

3. A possible application

During the second half of the 20th century the labor share was fairly constant (at least in United

States).2 However, in the last decades that share has fallen.3 These facts can be consistent with a

growth dynamic characterized by neutral technological change and capital accumulation in the 20th

century followed by a period of biased innovations in the 21st century.

According to Piketty (2014), the constancy of factor shares during the 20th century was the result of a “historical accident” in the period between 1914 and 1945: the world wars and the depression ended up destroying capital. The destruction of capital changed the relative abundance of factors generating

2 See Cobb and Douglas (1928), and Kaldor (1961).

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incentives for neutral rather than biased innovations. However, after decades of capital accumulation and neutral technological change the incentives for labor saving and capital using innovations appeared again.

References

Alesina, A. Batistti, M. and Zeira J. 2017. Technology, and Labor Regulations: Theory and Evidence. Journal of Economic Growth. 23(1), 41-78.

Acemoglu, D., 2007. Equilibrium Bias of Technology. Econometrica, 75 (5), 1371-1409.

Acemoglu, D., 2010. When Does Labor Scarcity Encourage Innovation?. Journal of Political Economy, 118, (6), 1037-1078

Acemoglu, D. and Restrepo, P. 2016. The Race Between Machine and Man: Implications of Technology for Growth, Factor Shares and Employment. NBER Working Paper No. 22252.

Acemoglu, D. and Restrepo, P 2017. Robots and Jobs: Evidence from US Labor Markets. NBER Working Paper No. 23285.

Cobb, C.W. and Douglas, P.H. 1928. A Theory of Production. The American Economic Review, 18(1), 139-165.

Kaldor, N., 1961. Capital Accumulation and Economic Growth. In FA Lutz and DC Hague, eds., The Theory of Capital, 177-222. New York St, Martin’s Press.

Karabarbounis, L. and Neiman, B. 2013. The Global Decline of the Labor Share. The Quarterly Journal of Economics, 129 (1), 61-103.

Peretto, P., and Seater, J., 2013. Factor-eliminating technical change. Journal of Monetary Economics, 60 (4), 459-473.

Piketty, T., 2014. Capital in the Twenty-First Century, Harvard University Press.

Rodriguez, F. and Jayadef, A. 2013. The Declining Labor Share of Income. Journal of Globalization and Development, 3 (2), 1–18.

Seater, J. and Yenokyan, K. 2018. Factor Augmentation, Factor Elimination, and Economic Growth”. Economic Inquiry, Forthcoming.

Zeira, J. 1998. Workers, Machines and Economic Growth. Quarterly Journal of Economics, 113 (4), 1091-1117.

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Zuleta, H., 2008. Factor Saving Innovations and Factor Income Shares. Review of Economic Dynamics, 11 (4), 836-851.

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