Perfect Competition and Fixed Costs: The Role of the Ownership Structure

Vincent Boitier

doi:10.1515/bejm-2023-0078

Article

Perfect Competition and Fixed Costs: The Role of the Ownership Structure

Vincent Boitier

Published/Copyright: October 12, 2023

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From the journal The B.E. Journal of Macroeconomics Volume 24 Issue 1

Abstract

It is widely considered that a (perfectly) competitive equilibrium cannot survive to the existence of fixed costs because firms generate losses in equilibrium. In this theoretical and methodological article, I demonstrate that this statement is not valid by developing some counter-examples. In particular, I clearly show that a competitive equilibrium and fixed costs are tenable, depending on the ownership structure of models. I then delimit the role of fixed costs in macroeconomic models. Notably, I find that fixed costs can improve the level of aggregate output in the long run.

Keywords: competitive equilibrium; fixed costs; ownership structure; survival assumption

JEL Classification: E13; E20; B40; B10

Corresponding author: Vincent Boitier, Le Mans University, Le Mans, France, E-mail: vincent.boitier@univ-lemans.fr

Appendix A: Proofs

In this section, I derive the proofs of the article.

A.1 Proof of Proposition 1

A competitive equilibrium exists if and only if households face strictly positive revenues. This means the following:

I = g + 1 h ∫ 0 n Π k d k > 0 ⇔ I = g − n f h > 0 ⇔ g h n > f

by symmetry.□

A.2 Proof of Proposition 2

Steady-state equilibrium In steady state, the following set of relationships is satisfied:

(9) 1 = β α K * α − 1 L * 1 − α + 1 − δ

(10) θ 1 − N * 1 C * = W *

1 β = 1 + r *

R * = α K * α − 1 L * 1 − α

W * = ( 1 − α ) K * α L * − α

(11) C * = K * α L * 1 − α − W * f − δ K *

N * = L * + f

as A* = 1. After simple algebra, equation (9) can be rewritten as follows:

K * L * = α β 1 − ( 1 − δ ) β 1 1 − α > 0

because 1 > (1 − δ)β as 0 < δ < 1 and 0 < β < 1. Noting that α β 1 − ( 1 − δ ) β 1 1 − α = ψ , the following relationships can be simplified as follows:

K * L * = ψ , R * = α ψ α − 1 , W * = ( 1 − α ) ψ α , Π * = − ( 1 − α ) ψ α f , r * = R * − δ = α ψ α − 1 − δ > 0

Then note that the couple (N*, L*) is pinned down by two equations. On the one hand, equation (11) collapses to:

C * L * = ψ α − ( 1 − α ) ψ α f L * − δ ψ

On the other hand, equation (10) yields:

C * = ( 1 − α ) ψ α ( 1 − N * ) θ = ( 1 − α ) ψ α ( 1 − L * − f ) θ

as N* = L* + f. Using the two above equations gives:

L * = ( 1 − α ) 1 + ( θ − 1 ) f ψ α ( 1 − α + θ ) ψ α − δ θ ψ > 0

where it is assumed that (1 − α + θ)ψ ^α > δθψ in line with the standard RBC model. It is also assumed that:

1 + ( θ − 1 ) f > 0

In turn, this leads to:

N * = ( 1 + θ f ) ( 1 − α ) + θ f ψ α − δ θ ψ f ( 1 − α + θ ) ψ α − δ θ ψ > 0

C * = ( ψ α − δ ψ ) L * − ( 1 − α ) ψ α f = ( 1 − α ) 1 + ( θ − 1 ) f ( ψ α − δ ψ ) ψ α ( 1 − α + θ ) ψ α − δ θ ψ − ( 1 − α ) ψ α f

Y * = ψ α L * = ( 1 − α ) 1 + ( θ − 1 ) f ψ 2 α ( 1 − α + θ ) ψ α − δ θ ψ > 0

As a consequence, a competitive equilibrium is unique and exists if and only if the level of consumption is strictly positive in equilibrium. This means the following:

C * > 0 ⇔ ψ α − δ ψ ( 2 − α ) ψ α > f

Note that C* > 0 implies positive net revenues and also implies that 1 − N* > 0.□

Comparison By definition a standard RBC model is obtained when f = 0 that is:

N s = L s = ( 1 − α ) ψ α ( 1 − α + θ ) ψ α − δ θ ψ

C s = ( 1 − α ) ( ψ α − δ ψ ) ψ α ( 1 − α + θ ) ψ α − δ θ ψ

Y s = ( 1 − α ) ψ 2 α ( 1 − α + θ ) ψ α − δ θ ψ

where s is for standard. After simple algebra, it is readily verified that:

N * > N s , L * > L s , Y * > Y s ⇔ θ > 1

Last, I compute and find the following:

C * − C s < 0 ⇔ − ( 1 − α ) ψ * − ( ψ α − δ ψ ) < 0

which is verified as ψ ^α − δψ > 0.

Appendix B: Beyond Constant Marginal Costs

In this article, I follow the literature by focusing on constant marginal costs only. This is because constant marginal costs always imply that the net profit is null in equilibrium, a desirable property under perfect competition. In turn, as the net profit is null, the global profit is necessarily negative. To see that, consider the general profit function net of the fixed cost:

π = p q − C ( q )

where C is a continuous and increasing function. Then, it is readily verified that the optimal policy for firms is given by:

p = C ′ ( q )

Integrating such a pricing rule into the definition of the net profit yields:

π = C ′ ( q ) q − C ( q )

By the Euler theorem, the net profit is null π = 0 if the cost function C is homogeneous of degree 1: C′(Q)q = C(q). Such a property is complied only by constant marginal costs C(q) = cq.

However, note that the model can cope with cost functions that vary with output. In particular, under a general cost function, the global equilibrium profit function is given by:^[7]

Π = π − f = C ′ ( q ) q − C ( q ) − f

Assuming that the fixed cost f is high enough such that Π < 0, the condition, ensuring that a perfect competitive equilibrium exists, becomes the following:

g + n Π h = g + n C ′ ( q ) q − C ( q ) − f h > 0

For example, if costs are quadratic C(q) = q ², I obtain the following: C′(q) = 2q, Π = q ² − f < 0 and g + n ( q 2 − f ) h > 0 .

References

Boitier, V. 2020. “Growth and Ideas in a Perfectly Competitive World.” Structural Change and Economic Dynamics 53: 370–6. https://doi.org/10.1016/j.strueco.2019.06.002.Search in Google Scholar

Boitier, V. 2021. “Growth and Ideas in a Perfectly Competitive World: A Complement.” Economics and Business Letters 10: 157–63. https://doi.org/10.17811/ebl.10.2.2021.157-163.Search in Google Scholar

Boitier, V. 2023. “Ownership Structure and Profit Maximization in General Equilibrium Models with Market Power.” Economics and Business Letters 12: 85–90. https://doi.org/10.17811/ebl.12.1.2023.85-90.Search in Google Scholar

Boitier, V., and A. Payen de La Garanderie. 2022 In preparation. Monopolistic Competition, Free Entry and General Equilibrium: A Reappraisal. Le Mans: Le Mans University.Search in Google Scholar

Brown, D. J. 1991. In Equilibrium Analysis with Nonconvex Technologies, Handbook of Mathematical Economics, edited by W. Hildenbrand, and H. Sonnenschein. Amsterdam: North-Holland.Search in Google Scholar

Cornet, B. 1988. “General Equilibrium Theory and Increasing Returns: Presentation.” Journal of Mathematical Economics 17: 103–18. https://doi.org/10.1016/0304-4068(88)90002-x.Search in Google Scholar

Hairault, J. O. 2000a. Analyse Macro Economique, Tome 1, Manuels Reperes. Paris: La découverte.Search in Google Scholar

Hairault, J. O. 2000b. Analyse Macro Economique, Tome 2, Manuels Reperes. Paris: La découverte.Search in Google Scholar

Hart, O. 1979. “On Shareholder Unanimity in Large Stock Market Economies.” Econometrica 47: 1057–83. https://doi.org/10.2307/1911950.Search in Google Scholar

Hart, O. 1985. “Imperfect Competition in General Equilibrium: An Overview of Recent Work.” In Frontiers of Economics, edited by K. Arrow, and S. Honkapohja. Oxford: Basil Blackwell.Search in Google Scholar

Jones, C. 2005. Growth and Ideas, Handbook of Economic Growth, edited by P. Aghion, and S. Durlauf. Amsterdam: North-Holland.10.1016/S1574-0684(05)01016-6Search in Google Scholar

Jones, C. 2017. Macroeconomics, 4th ed. New York: W-W Norton and Co Inc.Search in Google Scholar

Kaldor, N. 1961. “Capital Accumulation and Economic Growth.” In The Theory of Capital, edited by F. A. Lutz, and D. C. Hague, 177–222. St. Martins Press.10.1007/978-1-349-08452-4_10Search in Google Scholar

King, R., and S. Rebelo. 2000. Resuscitating Real Business Cycles. NBER Working Paper 7534.10.3386/w7534Search in Google Scholar

King, R., C. Plosser, and S. Rebelo. 1988. “Production, Growth and Business Cycles: I. The Basic Neoclassical Model.” Journal of Monetary Economics 21: 195–232. https://doi.org/10.1016/0304-3932(88)90030-x.Search in Google Scholar

Kydland, F., and E. Prescott. 1982. “Time to Build and Aggregate Fluctuations.” Econometrica 50: 1345–70. https://doi.org/10.2307/1913386.Search in Google Scholar

Mas-Colell, A. 1984. “The Profit Motive in the Theory of Monopolistic Competition.” Journal of Economics 4: 119–26.10.1007/978-3-7091-7577-4_5Search in Google Scholar

Pasinetti, L. 1962. “Rate of Profit and Income Distribution in Relation to the Rate of Economic Growth.” The Review of Economic Studies 29: 267–79. https://doi.org/10.2307/2296303.Search in Google Scholar

Romer, P. 1990. “Endogenous Technological Change.” Journal of Political Economy 98: S71–102. https://doi.org/10.1086/261725.Search in Google Scholar

Villar, A. 1999. Equilibrium and Efficiency in Production Economies. Berlin: Springer-Verlag.10.1007/978-3-642-59670-4Search in Google Scholar

Received: 2023-05-16

Accepted: 2023-09-15

Published Online: 2023-10-12

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Articles in the same Issue

https://doi.org/10.1515/bejm-2023-0078

Keywords for this article

competitive equilibrium; fixed costs; ownership structure; survival assumption