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Indirect production function and the output effect of public transit subsidies

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Abstract

This paper uses an indirect production function to decompose the effects of subsidies on output into the lump-sum, cost and inefficiency effects. Using 2006 data for U.S. transit systems it estimates an indirect production function and uses the results to calculate these effects. It finds that the lump-sum effects exceed the other effects and that the average total effect of the subsidies is a 4.72% increase in output. The range of the output change shows that in many transit systems the output increases from the subsidies are quite large. The paper suggests that reductions in allocative inefficiencies from the subsidies would result in very large increases in output.

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Notes

  1. For the FY 2009 apportionment formula see, http://www.fta.dot.gov/documents/2009fullyear_-_Table_4_-_sec_5307__Apportioment_Formula.xls.

  2. The formula for capital subsidies can be found in http://www.fta.dot.gov/documents/ARRA_Table_3_-_sec_5307__Apportioment_Formula-HBS.xls.

  3. Even though we have discussed the federal subsidy allocation formula it is noteworthy that we are not modeling the formula but how the funds from the formula are used.

  4. It is possible to argue that the sequence of decisions is the reverse of that described herein. For example, the subsidy decision could be made first followed by a determination of the level of transit output level to be produced. Furthermore, the sequence may be different in contracted services where the transit system determines the service to be provided (output) and specifies the subsidy to be paid to the contractor. Despite these possible differences in sequencing, they do not alter the results in this paper.

  5. For example, in buying buses, the amount of capital subsidies a transit system receives from federal sources depends upon the number bought. At the margin, this subsidy is 0.8(w K K) after suppressing the recent changes that have been made to allow these subsidies to be used for non-capital purposes. Similarly, because the federal share in operating losses is 50% at the margin, the total federal operating subsidies a transit system receives is 0.5(pQ − w L L − w F F) where, pQ is fare revenue and p output price. In both cases the amounts of the subsidies clearly depend upon input levels.

  6. This assumption can also be after-subsidy loss.

  7. These authors showed that this relationship is exact if the production function underlying the cost function is Cobb-Douglas. In another context, Kumbhakar (1997) generalized the relationship between implied and actual cost to situations where the cost function is translog.

  8. These implied prices can be obtained by maximizing output subject to a net cost constraint where net cost is total cost less the amounts of operating subsidies and capital subsidies expended. These subsidies are functions of all inputs. They can also be obtained by minimizing net cost subject to a production function constraint.

  9. If the implied cost function is Cobb-Douglas of the form \( C^{*} = \left( {1/\eta_{0} } \right)^{\theta } w_{L}^{{*\beta_{L} \theta }} w_{K}^{{*\beta_{K} \theta }} w_{F}^{{*\beta_{F} \theta }} Q^{\theta } \) which is homogeneous of degree one in input prices implying that θ(β L  + β K  + β F ) = 1 then the indirect production function is, \( Q = \eta_{0} \left( {C^{*} /w_{L}^{*} } \right)^{{\beta_{L} }} \left( {C^{*} /w_{K}^{*} } \right)^{{\beta_{K} }} \left( {C^{*} /w_{F}^{*} } \right)^{{\beta_{F} }} .\;\;\;\;\;\; \)

  10. The decomposition is most apparent when the underlying technology is characterized as translog, Cobb-Douglas or Lewbel. If, for example, the technology is generalized quadratic, \( Q = \sum\nolimits_{i} {\sum\nolimits_{j} {\alpha_{ij} \left\{ {\left( {{\frac{{C^{*} }}{{w_{i}^{*} }}}} \right)^{\alpha \beta } \left( {{\frac{{C^{*} }}{{w_{j}^{*} }}}} \right)^{{\alpha \left( {1 - \beta } \right)}} } \right\}} }^{1/\beta } \)the decomposition cannot be obtained.

  11. A reviewer suggests that Eq. 9 is from Shephard’s lemma. Our check shows early references attribute it to Roy (1943). Of course it can also be derived from Shephard’s lemma.

  12. Notice that these are the transit systems submitting their annual data to the Federal Transit Administration and that not all transit systems do so. Therefore, they do not represent all the transit systems in the U.S.

  13. Some of the ratios of operating subsidies to capital cost were 100 or higher, and the ratios of capital subsidies to labor cost were in some case greater than 50.

  14. This is comparable to the average 2007 and 2008 new bus price of $424,880 reported by APTA (2008).

  15. Except the ratios of subsidies to input costs each variable is normalized by subtracting its mean from its value. This allows us to calculate allocative distortion for the mean transit system.

  16. This result was obtained after many trials using different starting values. In all cases the values of the coefficients at convergence were very close suggesting a global convergence point had been reached.

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Acknowledgments

I am grateful to the anonymous referees and Prof. R. Sakano of the Department of Economics and Finance, North Carolina A&T State University, Greensboro NC, for their comments.

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  1. Department of Marketing, Transportation and Supply Chain, School of Business and Economics, North Carolina A&T State University, Greensboro, NC, 27411, USA

    K. Obeng

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Correspondence to K. Obeng.

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A longer version of this paper is available on request to the author at obengk@ncat.edu.

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Obeng, K. Indirect production function and the output effect of public transit subsidies. Transportation 38, 191–214 (2011). https://doi.org/10.1007/s11116-010-9296-7

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