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Solving the Endogeneity Problem in Empirical Cost Functions: An Application to US Banks

  • Mark G. Lijesen EMAIL logo
Published/Copyright: September 7, 2013
The B.E. Journal of Economic Analysis & Policy
From the journal The B.E. Journal of Economic Analysis & Policy Volume 13 Issue 2

Abstract

Empirical cost functions tend to ignore that firm differences in output levels depend on differences in cost levels and hence suffer from an endogeneity problem. We argue that traditional solutions for the endogeneity problem are insufficient for solving the problem and propose a structural approach to solve the problem. We apply both the traditional and the alternative models to panel data on large banks and found that a hybrid version yields results that are fully compatible with economic theory, whereas the traditional model provides theoretically incorrect in-sample predictions of scale elasticities.

Keywords: costs; firm size; translog; economies of scale; endogeneity

Appendix A: derivation of multi-product model, hybrid model and model including input prices

In this appendix, we extend the base model from the main text to a multi-product model, the hybrid model and a model including input prices.

The multi-product model

We start by rewriting eq. [1] for two products:

[7]
[7]

From the log differential, we may define marginal costs as:

[8]
[8]

and similar for X2,i. Firms equate marginal costs to marginal revenue to maximize profits, that is,

[9]
[9]

and similar for X2,i. We now rewrite eq. [9] to find

[10]
[10]

and

[11]
[11]

Both results may now be substituted into the cost equation [7]:

[12]
[12]

From eq. [12], it is straightforward to derive the more general form:

[13]
[13]

The hybrid model

For the hybrid model, we assume w.l.o.g. that the market for good 1 is homogeneous and the market for good 2 is non-homogeneous. This implies that eq. [10] holds and eq. [11] does not, so we have to estimate γ2 and hence drop subscripts i and t for this parameter. We substitute eq. [10] into eq. [7] to get

[14]
[14]

Input prices

We add prices for two inputs to eq. [7], following normal translog conventions:

[15]
[15]

From the log differential, we may define marginal costs as:

[16]
[16]

and similar for X2,i. Firms equate marginal costs to marginal revenue to maximize profits, that is,

[17]
[17]

and similar for X2,i. We rewrite eq. [17] to find

[18]
[18]

and

[19]
[19]

We substitute both results back into the cost equation and rewrite it to the general form directly to avoid lengthy notation.

[5]
[5]

Combining eq. [5] and a generalized form of eq. [14] then yields the hybrid model with input prices:

[6]
[6]

Appendix B: the case of heterogeneity in θ

In Section 3, we allow cost elasticity parameter γ to vary between firms and over time, while cost parameter θ is assumed to be generic. The model cannot be estimated if both parameters are firm- and time-specific. In this appendix, we take a closer look at how our results would change if we assumed that parameter θ, rather than parameter , is firm- and time-specific. Eq. [1] would have to be reformulated as:

[20]
[20]

Firms again equate marginal costs to marginal revenue to maximize profits:

[21]
[21]

Since we do not observe , we rewrite eq. [21] to find

[21’]
[21’]

We then substitute this result into the cost equation:

[22]
[22]

which we may rewrite to

[23]
[23]

We note that this is very similar to eq. [3] in the main text of the article. The only differences are that some parameters are now divided by 2 and that eq. [23] contains rather than . We estimated this specification on the same data and found results that are very similar to the ones presented in the main text of the article. Unfortunately, the estimated parameter for γ did not have the expected positive sign, implying that this specification does not yield valid results.

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  1. 1

    The endogeneity problem discussed here also arises when estimating production functions. Olley and Pakes (1996) and Levinsohn and Petrin (2003) introduce methods to account for serially correlated unobserved shocks to the production technology. Note that the scope of this article is wider though.

  2. 2

    It can be checked that the entire line of reason holds for oligopoly markets. The assumption of P = MC does not drive the result and is merely used here for clarity.

  3. 3

    The question whether fixed effects estimation provides a sufficient solution for the endogeneity problem is an empirical one and will be assessed in Section 4.

  4. 4

    The alternative assumption, allowing θ to be firm- and time-specific, is discussed in Appendix B.

  5. 5

    Estimating eq. [3] implies that market price P can be viewed as a parameter. This requires that the product under scrutiny is sufficiently homogeneous (so that the law of one price holds) and that variations in price over time are limited within the period of analysis.

  6. 6

    Note that the expression on the right-hand side of eq. [4] boils down to marginal costs divided by average costs, which equals the scale elasticity (see Goh and Yong, 2006).

  7. 7

    The derivation of the multi-product case can be found in Appendix A.

  8. 8

    Again, see Appendix A for the derivation of this function.

  9. 9

    We performed the same analysis using the five-product specification adopted by Hughes and Mester (1998), which yielded similar results.

  10. 10

    A version of the model with dummy variables for all quarters included provided similar results.

  11. 11

    Both other conditions, and do hold for both specifications.

  12. 12

    The study by Feng and Serletis (2010) is very similar, as it uses the same product definition and geographical scope, and their time period lies within that of our study. They estimate a translog production function and find increasing returns to scale, which is similar to finding economies of scale in a cost function.

  13. 13

    Note that we have no test statistic for the significance of scale economies in the adjusted version, but the scatter plot in Figure 2 clearly suggests a significant difference from unity at the mean.

Received: 2012-12-10
Accepted: 2013-08-19
Published Online: 2013-09-07

©2013 by Walter de Gruyter Berlin / Boston

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https://doi.org/10.1515/bejeap-2012-0070
Keywords for this article
costs; firm size; translog; economies of scale; endogeneity

Articles in the same Issue

  1. Masthead
  2. Masthead
  3. Advances
  4. When Does Inter-School Competition Matter? Evidence from the Chilean “Voucher” System
  5. Investment, Dynamic Consistency and the Sectoral Regulator’s Objective
  6. Contributions
  7. Vertical Contracts and Mandatory Universal Distribution
  8. Ticket Pricing and Scalping: A Game Theoretical Approach
  9. Loyalty Discounts
  10. Age, Human Capital, and the Quality of Work: New Evidence from Old Masters
  11. Effects of the Endogenous Scope of Preferentialism on International Goods Trade
  12. Fiscal Decentralization and Environmental Infrastructure in China
  13. Declining Equivalence Scales and Cost of Children: Evidence and Implications for Inequality Measurement
  14. Political Parties, Candidate Selection, and Quality of Government
  15. Professors’ Beauty, Ability, and Teaching Evaluations in Italy
  16. Pass-through of Per Unit and ad Valorem Consumption Taxes: Evidence from Alcoholic Beverages in France
  17. The Tuesday Advantage of Politicians Endorsed by American Newspapers
  18. Topics
  19. The Effects of Medicaid Earnings Limits on Earnings Growth among Poor Workers
  20. Opportunities Denied, Wages Diminished: Using Search Theory to Translate Audit-Pair Study Findings into Wage Differentials
  21. Horizontal Mergers, Firm Heterogeneity, and R&D Investments
  22. Product Differentiation and Consumer Surplus in the Microfinance Industry
  23. The Multitude of Alehouses: The Effects of Alcohol Outlet Density on Highway Safety
  24. Solving the Endogeneity Problem in Empirical Cost Functions: An Application to US Banks
  25. The Internet, News Consumption, and Political Attitudes – Evidence for Sweden
  26. A Cross-Cultural Real-Effort Experiment on Wage-Inequality Information and Performance
  27. Are Students Dropping Out or Simply Dragging Out the College Experience? Persistence at the Six-Year Mark
  28. The Welfare Effects of Location and Quality in Oligopoly
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