^{1}

^{2}

In this note, we prove that even if the technology of firms exhibits increasing returns to scale, the Panzar-Rosse statistic in a monopolistic competitive market is still available and has a negative value. Further, we show that the statistic would be greater than unity if firms with increasing-returns-to-scale technology were to choose a saddle point under certain conditions. This implies that the value of greater than unity is not actually observed.

The competitiveness of financial intermediaries has attracted the attention of many economic researchers. Bikker and Haaf [

One of the reasons why attention is given to the financial market might be that many researchers study and discuss whether economic growth and business cycles can be stimulated by the financial markets’ activities^{1}. For instance, Claessens and Leaven [

The method suggested by Panzar and Rosse [

ln R i = β 0 + β 1 ln w 1 i + ⋯ + β K ln w K i + X i γ + ε i ( i = 1 , ⋯ , n )

where R i denotes total revenue, w k i is a factor price of k th input ( k = 1 , ⋯ , K ) , X i is a vector of control variables, and ε i is a disturbance. Since β k indicates the k th-price elasticity of R , the estimated Panzar-Rosse statistic is defined as ψ ^ = ∑ k = 1 K b k where b k denotes the ordinary least squares estimate for β k ^{2}. Panzar and Rosse [

In this note, we consider the Panzar-Rosse statistic in a simple version of a monopolistic competition model. It is natural to study this model, since it is exploited in a large number of economics fields: economic growth, business cycles, monetary economics, spatial economics, and so on. We prove that, in the market, the Panzar-Rosse statistic is negative, and available even when firms have an increasing-returns-to-scale technology. From this point of view, the Panzar-Rosse methodology is superior to others. For example, the method of Hall [

Suppose that there are a single final good, Y , and intermediate goods, y ( i ) for i ∈ [ 0 , 1 ] . A firm which produces the final good has a production technology,

Y = [ ∫ 0 1 y ( i ) λ d i ] 1 / λ (1)

where λ ∈ ( 0 , 1 ) is the elasticity of substitution. The profit of this firm is

Π = P Y − ∫ 0 1 p ( i ) y ( i ) d i (2)

where P denotes the price of the final good and p ( i ) denotes the price of the intermediate good i . The firm maximizes Equation (2) subject to Equation (1). From the first order necessary condition for this problem, the demand for the i ′ th intermediate product is

y ( i ) = [ p ( i ) / P ] 1 / ( λ − 1 ) Y (3)

The firm which produces the i ′ th intermediate good has Cobb-Douglas production technology,

y ( i ) = k ( i ) α l ( i ) β (4)

where α > 0 and β > 0 . Note that we do not assume anything about returns to scale. For simplicity, we assume without loss of generality that these goods are produced using two inputs, i.e., capital stock ( k ( i ) ) and labor ( l ( i ) ) . The profit of the intermediate firm is

π ( i ) = R ( i ) − C ( i ) (5)

where the revenue and cost functions are

R ( i ) = y ( i ) p ( i ) / P , C ( i ) = r ( i ) k ( i ) / P + w ( i ) l ( i ) / P .

The firm producing the i ′ th intermediate good maximizes Equation (5) with respect to k ( i ) and l ( i ) subject to Equations (3) and (4).

Assumption 1. λ ( α + β ) < 1 .

This inequality assures these intermediate firms of positive profits. (See also the second-order conditions shown below.) The technology of the firm may exhibit increasing returns to scale, e.g., α + β > 1 , as long as Assumption 1 holds.

Definition 1. The Panzar-Rosse statistic in the model with two inputs is

ψ = r ˜ ( i ) R * ∂ R * ( i ) ∂ r ˜ ( i ) + w ˜ ( i ) R * ∂ R * ( i ) ∂ w ˜ ( i ) ,

where R * ( i ) denotes the firm’s reduced form revenue function, r ˜ ( i ) ≡ r ( i ) / P , and w ˜ ( i ) ≡ w ( i ) / P .

Substituting (3) and (4) into (5), we have

π ( i ) = k ( i ) λ α l ( i ) λ β Y 1 − λ − r ˜ ( i ) k ( i ) − w ˜ ( i ) l ( i ) (6)

The first-order necessary conditions for this problem are

λ α Y 1 − λ k * ( i ) λ α − 1 l * ( i ) λ β = r ˜ ( i ) , λ β Y 1 − λ k * ( i ) λ α l * ( i ) λ β − 1 = w ˜ ( i ) .

These equations are rewritten as

k * ( i ) = λ 1 1 − γ Y 1 − λ 1 − γ [ α r ˜ ( i ) ] 1 − λ β 1 − γ [ β w ˜ ( i ) ] λ β 1 − γ ,

l * ( i ) = λ 1 1 − γ Y 1 − λ 1 − γ [ α r ˜ ( i ) ] λ α 1 − γ [ β w ˜ ( i ) ] 1 − λ α 1 − γ ,

where γ ≡ λ ( α + β ) . Therefore, we obtain the reduced form revenue function

R * ( i ) = λ γ 1 − γ Y γ ( 1 − λ ) 1 − γ [ α r ˜ ( i ) ] λ α 1 − γ [ β w ˜ ( i ) ] λ β 1 − γ .

Provided Y and P as given, the first derivatives of R * ( i ) with respect to the factor prices are

∂ R * ( i ) ∂ r ˜ ( i ) = − λ α 1 − γ R * ( i ) r ˜ ( i ) , ∂ R * ( i ) ∂ w ˜ ( i ) = − λ β 1 − γ R * ( i ) w ˜ ( i ) .

Hence, we have the Panzar-Rosse statistic,

ψ = − γ / ( 1 − γ ) .

Since 0 < γ < 1 from α , β > 0 and Assumption 1, ψ is less than zero.

Proposition 1. In a monopolistic competitive market under Assumption 1, even if the production technology exhibits increasing returns to scale, the Panzar-Rosse statistic is available, and has a negative value.

It is important to note that Proposition 1 depends critically on Assumption 1. If Assumption 1 is violated, i.e., γ > 1 , then the Panzar-Rosse statistic must be greater than unity. Further, from Equation (6) and α , β > 0 , the technology of the firms exhibits increasing returns to scale. Note, however, that the second-order sufficient condition for maximization of the firms’ problem is that the Hessian is definitely negative, that is,

π k k < 0 : λ α ( λ α − 1 ) Y 1 − λ k ( i ) λ α − 2 l ( i ) λ β < 0 , π l l < 0 : λ β ( λ β − 1 ) Y 1 − λ k ( i ) λ α l ( i ) λ β − 2 < 0 , π k k π l l > π k l 2 : λ ( α + β ) < 1.

So the firms need to satisfy λ α < 1 , λ β < 1 and γ < 1 . Hence, when γ > 1 , we obtain a saddle point from the solution of the first-order conditions. If γ > 1 and if firms with increasing-returns-to-scale technology were to choose a saddle point, then the Panzar-Rosse statistic would be greater than unity. Needless to say, the set-up where each firm chooses a saddle point instead of a maximum is extremely unrealistic. In applications, such a value is not estimated if the regression models that researchers use are appropriately specified. Therefore, once one obtains an estimate that is significantly greater than unity, the robustness of the model must be thoroughly checked.

This note shows that the Panzar-Rosse statistic is available even when production technology exhibits increasing returns to scale, and that it would be greater than unity if the firms were to choose a saddle point. Thus, the statistic actually would not be estimated to be statistically significantly greater than unity.

This work was supported by JSPS KAKENHI Grant Number JP22730260. The authors are grateful to Masayoshi Tsurumi and seminar participants at Hosei University for their comments. Of course, all remaining errors are our responsibility.

Gunji, H. and Yuan, Y. (2017) The Panzar-Rosse Statistic Revisited. Theoretical Economics Letters, 7, 30-34. http://dx.doi.org/10.4236/tel.2017.71004