Network Effect, Business Dynamism and Wage Inequality in a Sharing Economy

Hamid Beladi; Chi-Chur Chao; Pamela Smith

doi:10.1515/bejeap-2022-0177

Article

Network Effect, Business Dynamism and Wage Inequality in a Sharing Economy

Hamid Beladi , Chi-Chur Chao and Pamela Smith

Published/Copyright: December 19, 2022

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From the journal The B.E. Journal of Economic Analysis & Policy Volume 23 Issue 1

Abstract

This paper examines the network effect on income distribution and social welfare in a sharing economy with rural-urban migration. A rise in the urban network effect via an increase of the number of users attracts capital to the urban sector. Urban skilled wage rises but rural unskilled wage falls. Due to rising costs from capital rental and skilled wage, urban firms exit if the demand elasticity of urban goods is large. This business-dynamism effect via firm exit mitigates the skilled-unskilled wage gap in the long run. Furthermore, rural-urban migration is mitigated from business dynamism by relocating capital to the rural sector. Network effects thus serves as a mitigator to cope with urban unemployment. In addition, the role of network effects on the effects of changes in the endowments are examined in the sharing economy.

Keywords: sharing economy; network effect; urban unemployment; business dynamism; wage inequality

JEL Classification: D85; R13; R23

Corresponding author: Hamid Beladi, College of Business, University of Texas at San Antonio, San Antonio, TX 78249, USA, E-mail: Hamid.Beladi@utsa.edu

Acknowledgments

We are indebted to editor and two anonymous reviewers for valuable comments and suggestions. Chi-Chur Chao acknowledges research support from the National Science and Technology Council (NSTC), Taiwan (NSTC 111-2410-H-035-001). The usual caveat applies.

Appendix

A.1 Changes of Equations

Let ^ over a variable denote its percentage change. Following Jones (1965), we totally differentiate (1)–(7) to have the changes of equations, as follows:

(A1) − 1 + 1 / n + η / n x ̂ − ε b θ K X m r ̂ = 1 + η / n n ̂ − ε δ 1 − e / n N ̂ ,

(A2) − θ L Y w ̂ Y − θ K Y r ̂ ,

(A3) μ ̂ = − 1 + μ / μ w ̂ Y ,

(A4) w ̂ S = r ̂ + n ̂ / s S X f − S ̂ / s S X f ,

(A5) 1 + μ λ L X x ̂ + λ L X Y ̂ + 1 + μ s L X + s L Y r ̂ − 1 + μ λ L X + s L Y w ̂ Y = − 1 + μ λ L X n ̂ + L ̂ ,

(A6) λ K X m x ̂ + λ K Y Y ̂ − s K X f + s K X m + s K Y r ̂ + s K Y w ̂ Y + s K X f w ̂ S = − λ K X n ̂ + K ̂ ,

(A7) − 1 − 1 / n x ̂ − ε 1 − b + b θ K X m r ̂ = 1 + ε 1 − b θ S X f / s S X f n ̂ − ε δ N ̂ − ε 1 − b θ S X f / s S X f S ̂ ,

where δ = Np_N/p > 0, b = m/p < 1, ε = −p/p_XX is the price elasticity of demand for good X, λ_ji and θ_jY are the allocative and cost shares of factor j in sector i, and θ j X m represents the variable cost share of factor j in producing good X. Note that the elasticity of factor substitution between skilled labour and capital is defined as σ X f = ff_wr/f_wf_r. Following Jones (1965), the substitution effect in demand for skilled labour is s S X f = σ X f θ K X f , where θ K X f (=rf_r/f) is the cost share of capital in the fixed cost of sector X. Also, we have s_LY = σ_Yθ_KYλ_LY, where σ_Y = gg_wr/g_wg_r.

Moreover, we define η = Xp_XX/p_X and e = −Xp_XN/p_N, where p_XX = 3Nq_XX and p_XN = Xq_XX with q_XX = u_XXX/N. Note that u_XXX measures the rate of change of marginal utility for good X and it is assumed to be not positive, u_XXX ≤ 0, implying that marginal utility decreases and diminishes quickly. Thus, we have η ≥ 0 and e ≥ 0, while η = e = 0 for the quadratic utility function. In addition, we consider that in the presence of network effects, the output of good X and the number N of users exhibit strategic complements by mutually reinforcing each other, i.e., dMR_X/dN = p_N(1 − e/n) > 0, to producers (cf. Bulow, Geanakoplos, and Klemperer 1985), where MR_x [=p(X, N) + xp_X(X, N)] is marginal revenue of good X for producers. This gives e/n < 1.

A.2 Stability

Letting a dot over a variable represent the time derivative (e.g., z ̇ = d z / d t ), the adjustments of the equations in (1)–(7) can be approximated linearly as:

X ̇ Y ̇ w ̇ Y r ̇ n ̇ = J X ̂ Y ̂ w ̂ Y r ̂ n ̂

where the coefficient J matrix is:

− 1 + 1 / n + η / n 0 0 − ε b θ K X m − 1 0 0 − θ L Y − θ K Y 0 1 + μ λ L X λ L Y − s L Y + 1 + μ λ L X s L Y + 1 + μ s L X 1 + μ λ L X λ K X m λ K Y s K Y − s K Y + s K X m λ K X + s K X f / s S X f − 1 − 1 / n 0 0 − ε 1 − b + b θ K X m − 1 + ε 1 − b θ S X f / s S X f

The principal minors of the above coefficient matrix are given by:

Δ 1 = − 1 + 1 / n + η / n < 0 ,

Δ 2 = 0 ,

Δ 3 = − λ L Y θ L Y 1 + 1 / n + η / n < 0 ,

Δ 4 = D = 1 + 1 / n + η / n A + ε b θ L Y θ K X m λ m > 0 ,

Δ 5 = Δ ,

where A = λ_KYs_LY + λ_LYs_KY + λ_LYθ_LY s K X m + (1 + μ)λ_KY(θ_LYs_LX + θ_KYλ_LX). The stability condition requires that the odd principal minors are non-positive and the even principal minors are non-negative. Hence, for stability of the model, we need Δ < 0 and |λ^m| = λ K X m λ_LY − (1 + μ)λ_LXλ_KY > 0. That is, |λ^m| > 0 indicates that good X is marginal capital intensive relative to good Y. This condition in the Harris–Todaro model is obtained in Khan (1980b) and used in Chao and Yu (1992, 1997.

A.3. Comparative-Static Results

By solving (A1)–(A6) with the given number of urban firms, the short-run results are obtained as

r ̂ / N ̂ = ε δ 1 − e / n θ L Y λ m / D > 0 ,

w ̂ Y / N ̂ = − θ K Y / θ L Y r ̂ / N ̂ < 0 ,

w ̂ S / N ̂ = r ̂ / N ̂ > 0 ,

μ ̂ / N ̂ = − 1 + μ / μ w ̂ Y / N ̂ > 0 ,

x ̂ / N ̂ = ε δ 1 − e / n A / D > 0 ,

r ̂ / n ̂ = θ L Y 1 + 1 / n + η / n λ + λ L Y s K X f / s S X f − 1 + η / n λ m / D > 0 ,

w ̂ Y / n ̂ = − θ K Y / θ L Y r ̂ / n ̂ < 0 ,

w ̂ S / n ̂ = r ̂ / n ̂ + 1 / s S X f > 0 ,

μ ̂ / n ̂ = − 1 + μ / μ w ̂ Y / n ̂ > 0 ,

x ̂ / n ̂ = − 1 + η / n A + ε b θ L Y θ K X m λ + λ L Y s K X f / s S X f / D < 0 ,

r ̂ / L ̂ = 1 + 1 / n + η / n λ K Y θ L Y / D > 0 ,

w ̂ Y / L ̂ = − θ K Y / θ L Y r ̂ / L ̂ < 0 ,

w ̂ S / L ̂ = r ̂ / L ̂ > 0 ,

μ ̂ / L ̂ = − 1 + μ / μ w ̂ Y / L ̂ > 0 ,

x ̂ / L ̂ = − ε b θ K X m λ K Y θ L Y / D < 0 ,

r ̂ / K ̂ = − 1 + 1 / n + η / n λ L Y θ L Y / D < 0 ,

w ̂ Y / K ̂ = − θ K Y / θ L Y r ̂ / K ̂ > 0 ,

w ̂ S / K ̂ = r ̂ / K ̂ < 0 ,

μ ̂ / K ̂ = − 1 + μ / μ w ̂ Y / K ̂ < 0 ,

x ̂ / K ̂ = ε b θ K X m λ L Y θ L Y / D > 0 ,

r ̂ / S ̂ = − 1 + 1 / n + η / n λ L Y θ L Y s K X f / s S X f / D < 0 ,

w ̂ Y / S ̂ = − θ K Y / θ L Y r ̂ / S ̂ > 0 ,

w ̂ S / S ̂ = r ̂ / S ̂ − 1 / s S X f < 0 ,

μ ̂ / S ̂ = − 1 + μ / μ w ̂ Y / S ̂ < 0 ,

x ̂ / S ̂ = ε b θ K X m λ L Y θ L Y s K X f / s S X f / D > 0 .

For the long run with firm entry/exit, from (A1)–(A7), we can obtain the effects of the increase in the number of users for the network good on the number of urban firms, factor returns and the urban unemployment ratio:

n ̂ / N ̂ = − ε δ 1 / n 2 + η + e − e / n A − ε θ L Y × 1 − b 1 − e / n − b θ K X m e / n λ m / Δ ≷ 0 ,

r ̂ / N ̂ = − ε δ θ L Y 1 / n 2 + η + e − e / n λ + λ L Y s K X f / s S X f + ε 1 − b θ S X f / s S X f − 1 / n η + e λ m / Δ > 0 ,

x ̂ / N ̂ = − ε δ − 1 / n η + e + ε 1 − b 1 − e / n θ S X f / s S X f A + ε θ L Y 1 − b 1 − e / n − b θ K X m e / n λ + λ L Y s K X f / s S X f / Δ ≷ 0 ,

X ̂ / N ̂ = x ̂ / N ̂ + n ̂ / N ̂ = − ε δ 1 / n 2 − e / n + ε 1 − b 1 − e / n θ S X f / s S X f A + ε θ L Y 1 − b 1 − e / n − b θ K X m e / n λ − λ m + λ L Y s K X f / s S X f / Δ ≷ 0 ,

w ̂ Y / N ̂ = − θ K Y / θ L Y r ̂ / N ̂ < 0 ,

w ̂ S / N ̂ = r ̂ / N ̂ + 1 / s S X f n ̂ / N ̂ ≷ 0 ,

w ̂ S / N ̂ − w ̂ Y / N ̂ = 1 / θ L Y r ̂ / N ̂ + 1 / s S X f n ̂ / N ̂ ≷ 0 ,

μ ̂ / N ̂ = − 1 + μ / μ w ̂ Y / N ̂ > 0 ,

n ̂ / L ̂ = ε λ K Y θ L Y 1 + 1 / n + η / n 1 − b + 1 / n 2 + η b θ K X m / Δ < 0 ,

r ̂ / L ̂ = λ K Y θ L Y ε 1 + 1 / n + η / n 1 − b + 2 + η / n / Δ < 0 ,

w ̂ Y / L ̂ = − θ K Y / θ L Y r ̂ / L ̂ > 0 ,

w ̂ S / L ̂ = r ̂ / L ̂ + 1 / s S X f n ̂ / L ̂ < 0 ,

w ̂ S / L ̂ − w ̂ Y / L ̂ = 1 / θ L Y r ̂ / L ̂ + 1 / s S X f n ̂ / L ̂ < 0 ,

μ ̂ / L ̂ = − 1 + μ / μ w ̂ Y / L ̂ < 0 ,

n ̂ / K ̂ = − ε λ L Y θ L Y 1 + 1 / n + η / n 1 − b + 1 / n 2 + η b θ K X m / Δ > 0 ,

r ̂ / K ̂ = λ L Y θ L Y ε 1 + 1 / n + η / n 1 − b + 1 / n 2 + η / Δ < 0 ,

w ̂ Y / K ̂ = − θ K Y / θ L Y r ̂ / K ̂ > 0 ,

w ̂ S / K ̂ = r ̂ / L ̂ + 1 / s S X f n ̂ / K ̂ ≷ 0 ,

w ̂ S / K ̂ − w ̂ Y / K ̂ = 1 / θ L Y r ̂ / K ̂ + 1 / s S X f n ̂ / K ̂ ≷ 0 ,

μ ̂ / K ̂ = − 1 + μ / μ w ̂ Y / K ̂ < 0 ,

n ̂ / S ̂ = − ε λ L Y θ L Y s K X f / s S X f 1 + 1 / n + η / n 1 − b + 1 / n 2 + n b θ K X m + 1 − b θ S X f / s S X f D / Δ > 0 ,

r ̂ / S ̂ = − θ L Y ε 1 − b θ S X f / s S X f 1 + 1 / n + η + n λ − λ m − 1 / n 2 + η λ L Y s K X f / s S X f / Δ ≷ 0 ,

w ̂ Y / S ̂ = − θ K Y / θ L Y r ̂ / S ̂ ≷ 0 ,

w ̂ S / S ̂ = r ̂ / L ̂ + 1 / s S X f n ̂ / S ̂ − 1 / s S X f ≷ 0 ,

w ̂ S / S ̂ − w ̂ Y / S ̂ = 1 / θ L Y r ̂ / S ̂ + 1 / s S X f n ̂ / S ̂ ≷ 0 ,

μ ̂ / S ̂ = − 1 + μ / μ w ̂ Y / S ̂ ≷ 0 ,

where Δ < 0 by the stability condition. Note that |λ| = λ_KXλ_LY − (1 + μ)λ_LXλ_KY > 0, expressing that good X is average capital intensive relative to good Y, where λ_KX = λ K X f + λ K X m . This gives |λ| − |λ^m| > 0.

References

Beladi, H., and S. Marjit. 1996. “An Analysis of Rural-Urban Migration and Protection.” Canadian Journal of Economics 29: 930–40. https://doi.org/10.2307/136221.Search in Google Scholar

Beladi, H., A. Chakrabarti, and S. Marjit. 2011. “North- South Outsourcing, Immigration, and Killed Wages: Through the Lens of Incomplete Contracts.” Review of Development Economics 15: 417–28. https://doi.org/10.1111/j.1467-9361.2011.00617.x.Search in Google Scholar

Beladi, H., C. C. Chao, M. Ee, and D. Hollas. 2020. “Urban Development, Excessive Entry of Firms and Wage Inequality in Developing Countries.” The World Economy 43: 212–38. https://doi.org/10.1111/twec.12778.Search in Google Scholar

Brander, J. A., and B. J. Spencer. 1985. “Export Subsidies and International Market Share Rivalry.” Journal of International Economics 18: 83–100. https://doi.org/10.1016/0022-1996(85)90006-6.Search in Google Scholar

Bulow, J., J. Geanakoplos, and P. Klemperer. 1985. “Multimarket Oligopoly: Strategic Substitutes and Strategic Complements.” Journal of Political Economy 93: 488–511. https://doi.org/10.1086/261312.Search in Google Scholar

Calvano, E., and M. Polo. 2021. “Market Power, Competition and Innovation in Digital Markets: A Survey.” Information Economics and Policy 54: 100853. https://doi.org/10.1016/j.infoecopol.2020.100853.Search in Google Scholar

Chao, C. C., and E. S. H. Yu. 1992. “Capital Markets, Urban Unemployment and Land.” Journal of Development Economics 38: 407–13. https://doi.org/10.1016/0304-3878(92)90008-w.Search in Google Scholar

Chao, C. C., and E. S. H. Yu. 1997. “Trade Liberalization in Oligopolistic Competition with Unemployment: A General Equilibrium Analysis.” Canadian Journal of Economics 30: 479–96. https://doi.org/10.2307/136352.Search in Google Scholar

Choi, K., and D. Lee. 2017. “Welfare-Improving Vertical Separation with Network Externality.” Economics Letters 151: 115–8. https://doi.org/10.1016/j.econlet.2016.12.024.Search in Google Scholar

Clementi, G. L., and B. Palazzo. 2016. “Entry, Exit, Firm Dynamics, and Aggregate Fluctuations.” American Economic Journal: Macroeconomics 8: 1–41. https://doi.org/10.1257/mac.20150017.Search in Google Scholar

Corden, W. M., and R. Findlay. 1975. “Urban Unemployment, Intersectoral Capital Mobility and Development Policy.” Economica 42: 59–78. https://doi.org/10.2307/2552986.Search in Google Scholar

Dixit, A. K. 1979. “A Model of Duopoly Suggesting a Theory of Entry Barriers.” The Bell Journal of Economics 10: 37–42. https://doi.org/10.2307/3003317.Search in Google Scholar

Fang, Z., L. Huang, and A. Wierman. 2017. “Prices and Subsidies in the Sharing Economy.” In Proceedings of the 26th International Conference on World Wide Web. International World Wide Web Conferences Steering Committee.10.1145/3038912.3052564Search in Google Scholar

Grinols, E. L. 1991. “Unemployment and Foreign Capital: The Relative Opportunity Costs of Domestic Labor and Welfare.” Economica 58: 107–21. https://doi.org/10.2307/2554978.Search in Google Scholar

Harris, J. R., and M. Todaro. 1970. “Migration, Unemployment and Development: A Two-Sector Analysis.” The American Economic Review 60: 126–42.Search in Google Scholar

Hoernig, S. 2012. “Strategic Delegation under Price Competition and Network Effects.” Economics Letters 117: 487–9. https://doi.org/10.1016/j.econlet.2012.06.045.Search in Google Scholar

Jones, R. W. 1965. “The Structure of Simple General Equilibrium Models.” Journal of Political Economy 73: 557–72. https://doi.org/10.1086/259084.Search in Google Scholar

Katz, M. L., and C. Shapiro. 1985. “Network Externalities, Competition and Compatibility.” The American Economic Review 75: 424–40.Search in Google Scholar

Katz, L. F., and A. B. Krueger. 2019. “The Rise and Nature of Alternative Work Arrangements in the United States 1995-2015.” ILR Review 72: 382–416. https://doi.org/10.1177/0019793918820008.Search in Google Scholar

Khan, M. A. 1980a. “The Harris-Todaro Hypothesis and the Heckscher-Ohlin-Samuelson Trade Model: A Synthesis.” Journal of International Economics 10: 527–47. https://doi.org/10.1016/0022-1996(80)90004-5.Search in Google Scholar

Khan, M. A. 1980b. “Dynamic Stability, Wage Subsidies and the Generalized Harris-Todaro Model.” Pakistan Development Review 19: 1–24. https://doi.org/10.30541/v19i1pp.1-24.Search in Google Scholar

Lahiri, S., and Y. Ono. 1995. “The Role of Free Entry in an Oligopolistic Heckscher-Ohlin Model.” International Economic Review 36: 609–24. https://doi.org/10.2307/2527363.Search in Google Scholar

Lahiri, S., and Y. Ono. 2011. “An Oligopolistic Heckscher-Ohlin Model of Foreign Direct Investment.” The Japanese Economic Review 62: 331–47. https://doi.org/10.1111/j.1468-5876.2010.00528.x.Search in Google Scholar

Lee, D. 2016. “How Airbnb Short-Term Rentals Exacerbate Los Angeles’s Affordable Housing Crisis: Analysis and Policy Recommendations.” Harvard Law and Policy Review 229: 234–40.Search in Google Scholar

Mankiw, N. G., and M. D. Whinston. 1986. “Free Entry and Social Inefficiency.” The RAND Journal of Economics 17: 48–58. https://doi.org/10.2307/2555627.Search in Google Scholar

Marjit, S., and H. Beladi. 2002. “The Stolper-Samuelson Theorem in a Wage Differential Framework.” The Japanese Economic Review 53: 177–81.10.1111/1468-5876.00019Search in Google Scholar

Marjit, S., S. Ganguly, and R. Acharyya. 2020. “Minimum Wage, Trade and Unemployment in General Equilibrium.” International Journal of Economic Theory. Online, 24 June 2020. https://doi.org/10.1111/ijet.12264.Search in Google Scholar

Miller, D. 2019. “The Sharing Economy and How it Is Changing Industries.” In The Balance Small Business. Also available at https://www.thebalancesmb.com/the-sharing-economy-and-how-it-changes-industries-4172234 (accessed June 25, 2019).Search in Google Scholar

Ordonez-de-Haro, J., and J. L. Torres. 2019. “Sharing Economy in Macroeconomics: Collaborative Consumption and Dural Goods.” In Malaga Economic Theory Research Center, Working Papers, WP-1. Malaga, Spain: University of Malaga.Search in Google Scholar

Pi, J., and Y. Fan. 2020. “Capital Concentration and Wage Inequality.” The B.E. Journal of Theoretical Economics 20: 20190049. https://doi.org/10.1515/bejte-2019-0049.Search in Google Scholar

Pi, J., and K. Zhang. 2022. “Search Costs and Wage Inequality.” The B.E. Journal of Theoretical Economics 22 (1): 67–104. https://doi.org/10.1515/bejte-2019-0168.Search in Google Scholar

Restuccia, D., and R. Rogerson. 2013. “Misallocation and Productivity.” Review of Economic Dynamics 16: 1–10. https://doi.org/10.1016/j.red.2012.11.003.Search in Google Scholar

Singh, N., and X. Vives. 1984. “Price and Quantity Competition in a Differentiated Duopoly.” The RAND Journal of Economics 15: 546–54. https://doi.org/10.2307/2555525.Search in Google Scholar

Stein, G. M. 2020. “Inequality in the Sharing Economy.” Brooklyn Law Review 85: 787–849.Search in Google Scholar

Weitzman, M. L. 1984. The Share Economy. Cambridge: Harvard University Press.Search in Google Scholar

Yu, E. S. H. 1981. “Regional Factor Specificity, Wage Differential and Resource Allocation.” Regional Science and Urban Economics 11: 69–79. https://doi.org/10.1016/0166-0462(81)90024-7.Search in Google Scholar

Received: 2022-05-09

Revised: 2022-11-08

Accepted: 2022-11-30

Published Online: 2022-12-19

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https://doi.org/10.1515/bejeap-2022-0177

Keywords for this article

sharing economy; network effect; urban unemployment; business dynamism; wage inequality