. International Journal of Economics and Financial Issues ISSN: 2146-4138 available at http: www.econjournals.com International Journal of Economics and Financial Issues, 2017, 7(4), 645-649. International Journal of Economics and Financial Issues | Vol 7 • Issue 4 • 2017 645 Functional Forms and Oligopolistic Models: An Empirical Analysis David Bouras1*, Troy Frank2, Eric Burgess3 1School of Business, Lincoln University, USA, 2School of Business, Lincoln University, USA, 3School of Business, Lincoln University, USA. *Email: bourasd@lincolnu.edu ABSTRACT This paper attempts to empirically test the sensitivity of the New Empirical Industrial Organization (NEIO) approach to the assumed demand and cost functional forms. Such an approach relies upon oligopolistic models to infer the degree of competition in a market. Unlike prior studies (Genesove and Mullin, 1998; Clay and Troesken, 2003), this paper focuses on the effect of the assumed cost function on the assessment of market power. Using data from the US catfish industry, the empirical results reveal that the NEIO approach is fairly robust to the assumed demand and cost functional forms. Keywords: Demand, Cost, New Empirical Industrial Organization, Market Power JEL Classifications: C51, D24, D40 1. INTRODUCTION The bulwark of the “New Empirical Industrial Organization” method (NEIO, hereafter) is the relation linking the equilibrium price to marginal cost (e.g., Bresnahan, 1982; Genesove and Mullin, 1998); that is, P MC Q dP dQ =       -θ ’ (1) Where, P, Q, and MC, are, respectively, output price, industry output, and marginal cost; and θ is an index of market power capturing various market structures. These include, among others, perfect competition when θ is equal to zero; monopoly power when θ is equal to one; and oligopoly power when θ lies between zero and one. Inferring the degree of competition using equation (1) can be problematic, however. Cost information and demand parameters are not readily available. Hence, empirical estimations appear to be the only way of obtaining the needed cost and demand information. The empirical estimations of the parameters of the cost and demand functions, however, require the selection of functional forms. Such a task can be tedious due to lack of theoretical and empirical foundations regarding the choice of appropriate functional forms. This, in turn, prompted economists to question the robustness of NEIO technique to the choice of the demand functional form. In that context, two studies attempted to evaluate empirically the robustness of the NEIO method: One in the sugar industry by Genesove and Mullin (1998), and the other in the whisky industry by Clay and Troesken (2003). These two studies used the adjusted Lerner index (ALI) as a benchmark and found that the NEIO approach is robust to the assumed demand functional form, especially for small values of the conduct parameter1. Whether the NEIO technique is robust to the assumed cost functional form is an open question that needs to be addressed. In that setting, this paper fills the gap in the extant literature by testing empirically the sensitivity of the NEIO approach to the assumed cost functional form. It also contributes to the existing literature that examined competition in the US catfish industry in two different ways2. First, it addresses the sensitivity of the NEIO methodology to the selection of the cost and demand functional forms. Second, one of the pitfalls of the research papers that investigated competition in the US catfish industry is their assumption that catfish processors sell a homogeneous product. In reality, however, catfish processing plants sell different forms 1 The ALI is given by: θ η= P - MC P       , where η is the elasticity of demand; and θ is the ALI or the conduct parameter. 2 A substantial body of literature looked at competition in the US catfish industry (e.g., Kinnucan and Sullivan, 1986; Kouka, 1995; Bouras and Engle, 2007; Bouras et al., 2010). These studies, however, relied upon a simple functional form for the demand and cost functions. Bouras, et al.: Functional Forms and Oligopolistic Models: An Empirical Analysis International Journal of Economics and Financial Issues | Vol 7 • Issue 4 • 2017646 of processed catfish, including primarily fresh and frozen whole catfish, fillet, nuggets, and steak. To address this issue, we focus exclusively on fresh whole dressed catfish. The paper is organized as follows: The first section provides the conceptual model; the second section contains data and the estimation of the econometric models; the third section reports comparative analyses; the last section concludes the paper. 2. CONCEPTUAL MODEL The starting point of our conceptual model is a catfish processor that converts live catfish into, among others, fresh whole dressed catfish. In that setting, the profit of producing and selling fresh whole dressed catfish can be formulated as: πi = P(Q) × qi – TC (2) Where, πi is the profit of the i th processor; P and qi are, respectively, the price and quantity sold of fresh whole dressed catfish; and TC is the total cost of producing fresh whole dressed catfish, which includes the processing cost and the cost of live catfish. The first- order condition can be expressed as: i + i i i d dP dQ= 0 ==> P q - MC = 0 dq dQ dq  (3) After a few algebraic manipulations, equation (3) can be re- written as: P = MC Q dP dQ ¸- (4) Where  = dQ dq Q i i q      is a measure of market power (Iwata, 1974; Bresnahan, 1982; Lau, 1982). Inferring θ from equation (4) requires the estimation of the parameters of the cost and demand functions. To this end, on the demand side, we use the following demand functional form (e.g., Genesove and Mullin, 1998): Q(P) = β(α−P)γ (5) Where, Q is the quantity sold of fresh whole dressed catfish; and P is the price of fresh whole dressed catfish. We choose the above functional form because it encompasses several forms, including, among others, the log-linear form (α = 0 and γ < 0) and the linear form (γ = 1). For empirical application, we focus on the most commonly used functional forms, that is, the linear and log-linear forms. These functional forms are, respectively, given by: Linear: Q = β(α−P) + µ, (6) Log-linear: ln(Q) = ln(−β) + γln(P) + µ (7) Using the demand function given in equation (5), equation (4), after a few algebraic manipulations, takes the following final form: + + + P = MCγ γ αθ γ     θ       θ . (8) Equation (8) is termed the generalized pricing rule function (for example, Genesove and Mullin, 1998). On the cost side, we decompose marginal cost into two major components: Marginal processing cost and the cost of live catfish. It is known that live catfish is converted in fixed and known proportions into fresh whole dressed catfish. Hence, marginal cost can be formally expressed as: MC = mpc + k × W (9) Where, mpc is marginal processing cost; k is the conversion factor; and W is the price of live catfish. The estimation of marginal cost as given in equation (9) requires knowledge of mpc and k. While the value of the conversion factor, k, can be obtained from extraneous information, the estimation of marginal processing cost, mpc, can be tedious. In the existing NEIO literature, the estimation of marginal processing cost is based on the selection and estimation of a specific functional form. In this paper, and in order to test the sensitivity of NEIO methodology to the selection of the cost functional form, we use three different methods. These methods include: The Generalized Leontief form, the linear form, and the Genoseve and Mullin’s technique (1998). The Generalized Leontief marginal processing cost (e.g., Appelbaum, 1982) is given by: ( )1/2ij i j i=1 j=1 mpc = p pδ∑∑ , (10) Where, p is a vector of input prices. As in Bouras and Engle (2007), we use three input prices: Labor (pL), capital (pK), and energy (pE). Marginal processing cost as given in equation (10) becomes mpc = δEEpE + δKKpK + δLLpL + 2δEK(pEpK) ½ + 2δEL(pEpL) ½ + 2δKL(pEpL) ½ (11) The linear marginal processing cost (e.g., Corts, 1999), on the other hand, can be formulated as: mpc = p 0 i i=1 i +ϕ ϕ∑ . (12) Using the input prices, as previously defined, the linear marginal processing cost becomes: mpc = φ0 + φEpE + φKpK + φLpL (13) Finally, the Genesove and Mullin’s technique (1998) consists of treating marginal processing cost as a parameter and then estimating it along with other model’s parameters. Specifically, marginal cost can be expressed as: MC = mpc + k × W (14) Where, MC is marginal cost; mpc is a parameter representing marginal processing cost; k is the conversion factor; and W is the price of live catfish. 3. DATA AND ESTIMATION PROCEDURE The econometric models include two different demand functional forms, including the log-linear and linear forms; and three Bouras, et al.: Functional Forms and Oligopolistic Models: An Empirical Analysis International Journal of Economics and Financial Issues | Vol 7 • Issue 4 • 2017 647 generalized pricing rule functions. These pricing rule functions are obtained by using three different cost functions, including the linear and generalized Leontief forms, and the Genesove and Mullin’s technique. To empirically estimate the econometric models, we use quarterly data ranging from 1992:II to 2004:IV. The data were collected from various sources. The price of live catfish, and the quantity sold and price of fresh whole dressed catfish were collected from the United States Department of Agriculture; hourly minimum wage was obtained from the United States Department of Labor; average retail electricity price was taken from the United States Department of Energy; and the bank loan rates were compiled from the Federal Reserve Bank of St. Louis. Summary statistics are reported in the Table 1. To estimate the demand functions, we use the non-linear least squares method. Heteroskedasticity and autocorrelation-consistent standard errors are computed using the Newey and West’s technique (1987). Table 2 contains the estimates of the parameters of the log-linear and linear demand functions. Also reported in Table 2 are the estimates for the demand elasticities (η) for the log-linear and linear demand functions3. Having estimated the parameters of the demand functions, the next step is to use them along with an estimate of k, the conversion factor, to estimate the parameters of the generalized pricing rule functions. Our estimate of k is taken from Silva and Dean’s study (2001). According to that study, one pound of live catfish yields 0.62 pounds of fresh whole dressed catfish; that is, for every pound of fresh whole dressed catfish produced 1.61 pounds of live catfish will be required. So, the value of k is 1.61. The estimates of the parameters of the generalized pricing rule functions are presented in Table 3. Of paramount relevance is the index of market power, θ. A casual look at the results shows that 3 The elasticity of demand for the linear form is computed at the mean values for the quantity and price of fresh whole dressed catfish. the point estimates of θ are close to zero. More importantly, and in most cases, the null hypothesis that θ is equal to zero cannot be rejected, implying that the market for fresh whole dressed catfish is competitive. The empirical results, therefore, suggest that the NEIO methodology is fairly robust to the assumed demand and cost functional forms. 4. COMPARATIVE ANALYSES In this section, we compare our estimate for market power; i.e. θ, obtained previously using the NEIO technique with a benchmark for market power. Our benchmark for market power is obtained by using the ALI. This index is given by: ALI= P-MC P η    , (15) Where, P is the price of fresh whole dressed catfish; η is the elasticity of demand for fresh whole dressed catfish; and MC is marginal cost. Substituting (14) into (15) results in: ALI= P mpc k W P - - ×    , (16) Where, mpc is marginal processing cost; k is the conversion factor; and W is the price of live catfish. To compute the ALI, we use our previous estimates for the elasticities of demand for fresh whole dressed catfish, η, for the log-linear and linear demand functions; the price of live catfish; the price of fresh whole dressed catfish; and an estimate for marginal processing cost (mpc). Our proxy for marginal processing cost comes from a study by Lazur (1997). Following that study, the cost of processing one pound of live catfish into fresh whole dressed catfish is $0.44. So, the value of mpc is 0.44. Yearly estimates of adjusted Lerner indices are provided in Table 4 and Figure 1. The average values of the adjusted Lerner indices are 0.040 and 0.046 for the log-linear and linear demand functional forms, respectively. These estimates are close to zero suggesting, once again, that the market for fresh whole dressed catfish is competitive. In addition, the estimates of the adjusted Lerner indices are, to a greater extent, similar to those of the index of market power, θ, obtained previously using the NEIO methodology. It is also interesting to compare our proxy for marginal processing cost with the estimates for marginal processing cost obtained previously using the Genesove and Mullin’s technique (1998). Table 5 contains the 99% confidence interval for marginal processing cost estimated using the Genesove and Mullin’s Table 1: Summary statistics Variable Minimum Maximum Mean±SD Price of live catfish ($/Lb) 0.55 0.80 0.70±0.08 Price of fresh whole dressed catfish ($/Lb) 1.43 1.89 1.72±0.14 Quantity sold of fresh whole dressed catfish (1000 Lbs) 5851.00 9955.00 7520.14±855.17 Bank loan rate (%) 4.00 9.50 6.98±1.78 Electricity price (ȼ/kWh) 4.26 5.56 4.77±0.31 Hourly minimum wage ($/h) 4.25 5.15 4.80±0.42 SD: Standard deviation Table 2: Nonlinear estimates of demand functions Parameter Demand functional form Linear Log-linear α or ln(−β) 5.03* (1.602) 9.17* (0.134) β or γ 2268.74* (1118.571) −0.47** (0.232) Demand elasticity (η) −0.52* (0.256) −0.47** (0.232) R2 (%) 13.80 12.60 Log-likelihood −412.39 42.88 Number of observations 51 51 Standard errors are between parentheses. *,**, and *** represent 1%, 5%, and 10% significance level, respectively Bouras, et al.: Functional Forms and Oligopolistic Models: An Empirical Analysis International Journal of Economics and Financial Issues | Vol 7 • Issue 4 • 2017648 technique (1998) along with the direct measure of marginal processing cost taken from Lazur’s study (1997). The estimates of marginal processing cost using the Genesove and Mullin’s technique (1998) range from $0.42/Lb to $0.66/Lb for the log- linear demand function; and range from $0.45/Lb to $0.92/Lb for the linear demand function. It should be pointed out that the Genesove and Mullin’s technique (1998) performed relatively well, especially when using the parameters taken from the log- linear demand function. 5. CONCLUSION This paper attempts to empirically test the sensitivity of the NEIO technique to the assumed demand and cost functional forms. The focus of the paper is, however, on the effect of the assumed cost function on the estimation of market power. Using data from the US catfish industry, our empirical results reveal that the New Empirical Industrial Organization approach is fairly robust to the assumed demand and cost functional forms. In addition, the empirical results indicate that the Genesove and Mullin’s technique for estimating marginal processing cost performed relatively well particularly when using the parameters taken from the log-linear demand function. Table 3: Nonlinear estimates of generalized pricing rule functions Parameter Demand specification Linear form Log-linear form Cost specification Cost specification GLF LF GMT GLF LF GMT Estimate Estimate φ0 or mpc −0.02 0.69* −0.58* 0.54* (0.391) (0.088) (0.164) (0.046) θ −0.10* −0.11 −0.03 0.01 0.05 0.01 (0.037) (0.093) (0.026) (0.021) (0.044) (0.012) φE or δEE −0.61 0.10* −0.10 0.10* (0.423) (0.019) (0.424) (0.019) φK or δKK 0.08 0.01** 0.13* 0.01** (0.063) (0.006) (0.044) (0.006) φL or δLL −0.46 0.09* −1.14** 0.09* (0.380) (0.021) (0.512) (0.021) δEK 0.02 −0.13** (0.061) (0.049) δEL 0.67*** 1.19** (0.372) (0.446) δKL −010* 0.003 (0.031) (0.065) R2 (%) 95.14 93.84 82.86 93.88 93.84 82.86 Log-likelihood 105.55 99.47 73.38 99.63 99.47 73.38 Number of observations 51 51 51 51 51 51 Standard errors are between parentheses. These are computed using the Newey and West’s technique (1987); GLF: Generalized Leontief form, LF: Linear form, GMT: Genesove and Mullin’s technique, *,**, and *** represent 1%, 5%, and 10% significance level, respectively Table 4: Yearly estimates of ALI Year ALI Mean±SD Linear demand Log-linear demand 1992 0.013±0.009 0.017±0.011 1993 0.014±0.006 0.014±0.005 1994 0.038±0.010 0.033±0.006 1995 0.042±0.005 0.035±0.004 1996 0.050±0.012 0.041±0.006 1997 0.048±0.004 0.042±0.004 1998 0.043±0.009 0.038±0.006 1999 0.046±0.011 0.040±0.008 2000 0.070±0.022 0.055±0.014 2001 0.069±0.005 0.063±0.004 2002 0.034±0.002 0.037±0.005 2003 0.051±0.016 0.049±0.009 2004 0.081±0.013 0.058±0.006 Average 0.046±0.010 0.040±0.007 SD: Standard deviation, ALI: Adjusted Lerner index Table 5: Marginal processing cost: Direct measure versus Genesove and Mullin’s technique Direct measure 99% confidence interval for marginal processing cost/Genesove and Mullin’s technique Linear demand Log-linear demand Lower bound Upper bound Lower bound Upper bound 0.44 0.45 0.92 0.42 0.66 Figure 1: Yearly estimates of adjusted lerner indices Bouras, et al.: Functional Forms and Oligopolistic Models: An Empirical Analysis International Journal of Economics and Financial Issues | Vol 7 • Issue 4 • 2017 649 REFERENCES Appelbaum, E. 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