Estimation of the marginal cost

When estimating equation (4.10), the problem arises that the marginal cost is not ob-servable, at least not directly. The common practice of using cost indices as proxy vari-ables is criticized by GOLDBERG &KNETTER (1997: 1251) since this procedure can be expected to introduce a bias in the estimation. KNETTER (1989; 1993) therefore uses the fixed country effects of a panel model as a measure of the marginal cost and is thus able to control for it. However, this approach is not applicable to (4.10) since the marginal cost of the competitor appears in the equation too. Furthermore, in Chapter 3 we show that when the first symmetry condition – which is already proposed by KNETTER (1995) – holds, the estimated PTM coefficients are biased toward the average PTM coefficient in the sample. This is because the marginal cost may be equal for all destination coun-tries at the port of export; the influence of the marginal cost on prices is not equal¹², but depends on the degree of PTM. The severity of this bias depends heavily on firm,

11 Pricing in a foreign currency together with price rigidity is a known source of falsely detected PTM.

12 KNETTER (1995) is aware of the destination-specific influence of the marginal cost that the symmetry condition implies, although he does not show the theoretical implications for the estimation of the model.

Instead, KNETTER (1995) proposed a non-linear model of PTM that is based on the symmetry condition.

However, this model has some disadvantages and is actually not followed up in the literature.

uct and market characteristics as well as on sample selection. An infelicitously chosen sample can also introduce a huge amount of noise in the fixed effects. SAGHAIAN &

REED (2004) also criticize the usage of fixed effects since they find strong multicolline-arity between the fixed effects and the exchange rates. Instead, they use wholesale pric-es in the exporting countripric-es. However, the main problems with this procedure are a) such series have to exist and b) in order to be a good measure of the marginal cost they should be uninfluenced by market power. We propose an alternative measure of the marginal cost that is independent of sample selection and market power and is always available when a certain number of destination-specific unit values are available too.

The advantage of the availability of destination-specific export unit values is that they give a lot of information regarding a number of agreements of sale. In each period, there are unit values which contain various levels of markups or quality premiums and there are probably unit values which reflect more or less the marginal cost at the port of ex-port. Furthermore, unit values contain measurement errors as well as expectations about future developments, when they – at least in part – consist of batches whose prices are arranged in long- or medium-term contracts. Essentially, the fixed effects of the Knetter model just take the average of the unit values of the sample in a given period while sim-ultaneously correcting them for the estimated PTM. However, the underlying sample is often small and particular unit value series exhibit a high variation.

The alternative to this procedure that preserves the basic idea of KNETTER (1989; 1993;

1995) – that is, the usage of multiple transactions to estimate a measure of the marginal cost – would be a) to use all unit values available for a given period and b) not to apply a simple average with an artificial correction for the estimated PTM. Instead, the factors moving unit values away from the marginal cost can be expected to exhibit certain dis-tributions. When these distributions can be estimated, an estimation of the marginal cost would be available too. This requires some basic distribution assumptions:

1. Measurement error and expectation effects

Both are assumed to be independently normally distributed with a mean of zero.

This means that the sum of these effects should be normally distributed as well.

2. Markups and quality premiums

Both effects are assumed to reflect positive deviations from the marginal cost.

This implies that the firm does not practice dumping and charges at least mar-ginal cost. Since higher markups should not be as likely as smaller ones, a pos-sible distribution for these effects could be a truncated normal distribution.

Under these assumptions, the i-th observation for an export unit value (in the export country’s currency) can be described as follows:

(4.11) where is a time specific constant and represents a composed error term. For the error components we assume:

( ) (4.12)

( ) (4.13) where and are moreover assumed to be independently distributed (KUMBHAKAR

&LOVELL 2003: 169). The error component displays measurement errors and expec-tation effects (see assumption 1) whereas represents markups and quality premiums (see assumption 2). According to assumption 2, a truncated normal distribution (that is very flexible) is chosen for .

The model displayed in (4.11) is a stochastic frontier model without further explanatory variables. We can therefore call the fitted values of the model – i.e. – the stochastic lower price frontier (SLPF). When the distribution assumptions are fulfilled, the SLPF equals an unbiased estimation of the marginal cost that is hereafter called the “stochas-tic” marginal cost (SMC). The most critical point in this regard concerns the distribution of . If a certain unknown minimum markup exists, the normal distribution of is not truncated at zero but instead at an unknown positive value. This implies that the SLPF lies above the true marginal cost. Since we cannot rule out such a case, one should be cautious to use the SLPF as an absolute measure of the marginal cost for the purpose of comparisons with price series. For the usage in the PTM model, the case is different. The question would be rather whether the SMC series is biased in a way that biases the coefficients of the model. If a possible minimum markup is constant over time, the coefficients in equation (4.10) are not biased. Similarly, a random fluctuation of this minimum markup over time would not bias the coefficients but create a larger standard error. If, however, this minimum markup is a function of the price level, the coefficient of the marginal cost could indeed be biased. The probable direction of this bias would be positive – i.e. against finding of PTM through the marginal cost coeffi-cient. This is due to the fact that the certain minimum markup can be expected to be lower in case of a higher price level. It is further worth mentioning that the distribution assumptions imply that the analyzed good is homogenous in the sense that a certain standard quality with the possibility of quality premiums should exist. However, when

the good is too heterogeneous and there is nothing even like a standard quality, then the approach could be invalid.

Given the distribution assumptions, equation (4.11) can be estimated via maximum like-lihood. It can be estimated with cross sectional data (T = 1) as well as (unbalanced) panel data (T > 1). However, according to (4.12) and (4.13) it is assumed that , and depend on t. This implies that only the estimation with cross sectional data is valid because in the case of panel data, the parameters are estimated independently of t. Of course, when the parameters do not depend on t, the usage of panel data is possible too.

However, in this case the SLPF series lies a constant value below the simple time spe-cific mean series of the original data. This does not represent an improvement in com-parison to the usage of the simple time specific mean as a measure of the marginal cost.

Only the variability of parameters with respect to t allows temporally non-constant de-viations from the simple mean. The parameters can vary for different reasons: With re-gard to , it is conceivable that the measurement error is temporally constant but the expectation effects are probably not. For periods with rather unstable prices, they should be more important than otherwise. Similarly, the distribution parameters of can change – for example, in the case of higher prices only smaller markups are optimal.

In summary, only an estimation of (4.11) with cross sectional data should be valid.

However, this causes a disadvantage with regard to the efficiency of estimation: The observations for a single period (e.g. a month) are usually rather limited and the true parameters , and may only change significantly with an increasing time inter-val. Under these circumstances, the estimation efficiency could be raised by expanding the period specific sample – for example, through the two temporally adjacent periods.

This means that equation (4.11) is estimated as a panel model with T = 3 in a rolling window over the whole time span. As long as the true parameters are constant over the subsequent three time periods, this procedure triples the available observations and rais-es the rais-estimation efficiency. Essentially, the choice of the value for T is a tradeoff be-tween estimation efficiency and the most temporally flexible estimation.