Tasks, technology, and factor prices in the neoclassical production sector

Abstract

This paper introduces tasks into the neoclassical production sector. Competitive firms choose the profit-maximizing amounts of factor-specific tasks that determine their factor demands and output supplies. We show that the effect of factor-augmenting technical change on relative and absolute factor prices can be decomposed into a productivity effect and a task-demand effect of opposite sign. These effects appear since the novel task-based approach distinguishes between the demands for tasks and the demands for factors. This perspective provides a new intuition for the emergence of relative and absolute factor biases and the role of the elasticity of substitution.

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Notes

  1. 1.

    The labels “capital” and “labor” are just for convenience. Mathematically, these two factors may represent any pair of distinguishable inputs. Mutatis mutandis, the analytical setup developed here extends readily to more than two inputs each with a corresponding task type. For instance, one may want to replace the single type of tasks performed by labor with two distinct types, one performed by skilled, the other be unskilled labor.

  2. 2.

    To include the CES production function, we dispense with Inada conditions. However, throughout I assume that the \(\lim _{M(i)\rightarrow 0}\partial F\left( M(i),N(i)\right) /\partial M(i)\) and \(\lim _{N(i)\rightarrow 0}\partial F\left( M(i),N(i)\right) /\partial N(i)\) are sufficiently high to deliver an interior solution to the profit-maximization problem of firms.

  3. 3.

    See, e.g., Varian (1992), Chapter 1, for a traditional textbook treatment of “activity analysis.” In my analytical framework a firm’s input requirement set comprises all vectors of tasks that allow for the production of a given amount of output. Therefore, I refer to this set as the task input requirement set.

  4. 4.

    Little would change in Proposition 1 and its proof if we replaced the production function of (2.1) by any homothetic function \({\tilde{F}}:{\mathbb {R}}_+^2\rightarrow {\mathbb {R}}_+\) where \({\tilde{F}}\left( M(i),N(i)\right) =h\left( F\left( M(i),N(i)\right) \right) \) with \(h'\left( \cdot \right)>0>h''\left( \cdot \right) \). In fact, the equilibrium allocation and equilibrium factor price are simply obtained by replacing F by \({\tilde{F}}\).

  5. 5.

    All results of Sect. 3.2.1 go through if we allow for decreasing returns to scale and replace the CES of (3.4) by \({\tilde{F}}\left( M,N\right) =\left[ F\left( M,N\right) \right] ^\nu \), where \(0<\nu <1\). Since \({\tilde{F}}\left( M,N\right) \) is homothetic, Proposition 1 continues to hold (see, Footnote 4).

  6. 6.

    See Irmen (2014) for an analysis of the effect of factor-augmenting technical change on factor prices in the canonical neoclassical production sector.

  7. 7.

    With slight modifications, the results of Sect. 3.2.2 go through under decreasing returns to scale as suggested in Footnote 5. As to the sign of \(dR^*/db\), we have to replace \(\gamma \) on the right hand side of (3.16) by \({\tilde{\gamma }}\equiv \nu \gamma \). The sign of \(dR^*/da\) in (3.15) is preserved as long as \(\sigma <1/(1-\nu )\).

  8. 8.

    Observe that \(F\left( \ln (1+\delta bK),\ln (1+\delta a L)\right) /\delta \) is not homothetic in (bKaL). Therefore, the findings on technical change and factor prices are quite different from and have different intuitions than those stated in the Footnotes 4 - 7 for a CES with decreasing returns to efficient factor supplies.

  9. 9.

    Notice that the ‘number’ of firms will affect aggregate output and factor prices since capital and labor exhibit diminishing returns in the accomplishment of tasks. A larger number of firms implies that each firm operates at a smaller scale. The equilibrium productivity of capital and labor will then be higher and aggregate output increases. In what follows we neglect this complication to facilitate the comparison with the results derived in Sect. 3.

  10. 10.

    Observe that \(d\ln (R/w)/d\ln (b/a)=d\ln (R/w)/d\ln b=-d\ln (R/w)/d\ln a\) since \(d\ln (b/a)=d\ln b-d\ln a\). Therefore, the rule stated in (3.13) also applies to \(d\ln (R/w)/d\ln b\) and with opposite sign to \(d\ln (R/w)/d\ln a\).

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Acknowledgements

This is a revised version of my discussion paper Irmen (2018). I gratefully acknowledge financial assistance under the Inter Mobility Program of the FNR Luxembourg (“Competitive Growth Theory - CGT”). A part of this research was written while I was visiting Brown University in the Spring 2018. I would like to express my gratitude to Brown’s Economics Department for its kind hospitality. Moreover, I would like to thank Oded Galor, Ka-Kit Iong, two anonymous referees, and the editor for helpful comments.

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Appendix: Proofs

Appendix: Proofs

Proof of Proposition 1

Consider firm i’s demand for capital. With (2.5), we have \(K(i)=M(i)/b\). Hence, for \(R>0\), (2.6) delivers \(\int _0^1 K(i)di=\int _0^1 M(i)/b=K\). Hence, \(\int _0^1 M(i)di=bK=M^*\). Similarly, one derives for labor and \(w>0\) that \(\int _0^1 N(i)di=aL=N^*\). Next consider the first-order conditions (3.2). Since the derivatives \(\partial F\left( M(i),N(i)\right) /\partial M(i)\) and \(\partial F\left( M(i),N(i)\right) /\partial N(i)\) are homogeneous of degree zero, these conditions imply \(M(i)/N(i)=M(i')/N(i')\) for all \(i,i'\in [0,1]\) and \(i\ne i'\). Therefore, it holds that \(M(i)/N(i)=\left[ M(i)+M(i')\right] /\left[ N(i)+N(i')\right] \) and

$$\begin{aligned} \left( \frac{M(i)}{N(i)}\right) ^*=\frac{\int _0^1 M(i)di}{\int _0^1 N(i)di}=\frac{bK}{aL}=\frac{M^*}{N^*}. \end{aligned}$$
(6.1)

Accordingly, the equilibrium task intensity, the aggregate equilibrium output, and the equilibrium factor prices are as stated in Proposition 1. Since \(M^*\) and \(N^*\) are positive and finite, we have \(R^*>0\) and \(w^*>0\). Since \(\pi (i)\) is homogenous of degree one in \(\left( M(i),N(i)\right) \), the absolute levels \(M^*(i)\) and \(N(i)^*\) remain indeterminate. \(\square \)

Proof of Proposition 2

In equilibrium (3.6) becomes

$$\begin{aligned} \ln \left( \frac{R^*}{w^*}\right) =\ln \gamma +\ln \left( \frac{b}{a}\right) - \frac{1}{\sigma }\ln \left( \frac{M^*}{N^*}\right) . \end{aligned}$$

In light of (3.8) the proposition follows. \(\square \)

Proof of Proposition 3

We start proving that \(M(i)=M(j)=M^*\) and \(N(i)=N(j)=N^*\) for all \(i,j\in [0,1]\) and \(i\ne j\) holds in equilibrium. Since \(C(i)=C\left( M(i),N(i)\right) \) is strictly convex, profits \(\pi (i)=\pi \left( M(i),N(i)\right) \) are strictly concave. Assume to the contrary that two firms i and j choose \(\left( M(i),N(i)\right) \ne \left( M(j),N(j)\right) \). This is only possible if \(\pi \left( M(i),N(i)\right) =\pi \left( M(j),N(j)\right) \). Now, consider \(\left( M,N\right) =\mu \left( M(i),N(i)\right) +(1-\mu )\left( M(j),N(j)\right) \) for some \(\mu \in (0,1)\). Since profits are strictly concave we have \(\pi \left( M,N\right) >\mu \pi \left( M(i),N(i)\right) +(1-\mu )\pi \left( M(j),N(j)\right) \). Therefore, \(\pi \left( M,N\right) >\pi \left( M(i),N(i)\right) =\pi \left( M(j),N(j)\right) \) which is a contradiction. Hence, for all \(i\in [0,1]\) we must have \(\left( M(i),N(i)\right) =\left( M,N\right) \).

As a consequence, the aggregate demand for capital of all firms is \(\int _0^M e^{\delta m}dm/b=\left( e^{\delta M}-1\right) /(\delta b)\). Hence, for \(R>0\), (2.6) delivers \(M^*\) as indicated in the proposition. Similarly, one derives \(N^*\). The first-order conditions (4.2) deliver \(R^*\) and \(w^*\) as indicated. Since \(F(\cdot )\) has constant returns to scale, \(F\left( M^*,N^*\right) =F\left( \ln (1+\delta bK),\ln (1+\delta a L)\right) /\delta \). \(\square \)

Proof of Proposition 4

The statements of Eq. (4.6) are immediate from (4.5). Computing the respective effects delivers with \(z\equiv \delta b K\)

$$\begin{aligned} \delta M^* \frac{\partial \ln M^*}{\partial \ln b}=\frac{z}{1+z}\quad \text{ and }\quad \frac{1}{\sigma }\frac{\partial \ln M^*}{\partial \ln b}=\frac{1}{\sigma }\left( \frac{z}{1+z}\right) \left( \frac{1}{\ln (1+z)}\right) . \end{aligned}$$
(6.2)

It follows that \({\bar{\sigma }}_b=z/(\ln (1+z))>1\) which is increasing in z, hence, also in \(\delta \). As to labor, let \(x\equiv \delta a L\). Then,

$$\begin{aligned} \delta N^* \frac{\partial \ln N^*}{\partial \ln a}=\frac{x}{1+x}\quad \text{ and }\quad \frac{1}{\sigma }\frac{\partial \ln N^*}{\partial \ln a}=\frac{1}{\sigma }\left( \frac{x}{1+x}\right) \left( \frac{1}{\ln (1+x)}\right) . \end{aligned}$$
(6.3)

It follows that \({\bar{\sigma }}_a=x/(\ln (1+x))>1\) which is increasing in x, hence, also in \(\delta \). \(\square \)

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Irmen, A. Tasks, technology, and factor prices in the neoclassical production sector. J Econ (2020). https://doi.org/10.1007/s00712-020-00705-9

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Keywords

  • Technical change
  • Factor prices
  • Factor-specific tasks
  • Neoclassical production

JEL Classification

  • O33
  • O41