Consider the profit margin of selected comparables in the general case when the industry includes two types of uncontrolled (or consolidated) enterprises: (*i*) *innovators* that can earn a temporary or persistent excess profit margin, and (*ii*) *imitators* that are attracted by the excess profit but whose entry in the industry have the effect of eroding the excess profit margin. As a result, the industry of the controlled taxpayer may exhibit over an audit cycle both a common (or equilibrium) and disequilibrium profit margins earned by innovators and their predators. Here is a schema of this competitive technological and marketing treadmill:

##### (0) Innovation(1) → Excess profit margin → Entry of imitators → Erosion of excess profit margin → Innovation(2), etc.

This excess profit seeking among rivals operating in the same or similar industry of the taxpayer can be represented by a simple dynamic system:

##### (1) Y_{t} ≡ α + S_{t}

_{t}

_{t}

where Y* _{t}* denotes the actual profit margin of an enterprise in period

*t*. The variable S

*≥ 0 is the surplus or excess profit margin in period*

_{t}*t*= 1, 2, …,

*T*.

We assume that the current excess profit margin depends on its prior value plus a random error:

##### (2) S_{t} = φ S_{t}_{ – 1 }+ U_{t}

_{t}

_{t}

_{t}

where we expect |φ| < 1 and the random errors are assumed to be well-behaved:

##### (3) U_{t} is NID (0, σ)

_{t}

This means that the residual errors U* _{t}* are normally and independently distributed (NID) with zero mean and a common standard deviation σ. For simplification, we do not use a second subscript denoting the

*i*-th comparable in the industry of the taxpayer.

We substitute (2) into (1) and obtain a reduced-form equation for further consideration:

##### (4) Y_{t} = α + φ S_{t}_{ – 1 }+ U_{t}

_{t}

_{t}

_{t}

The parameters α, φ, and σ can’t be estimated by using regression equation (4) because S_{t}_{ – 1} is not observed. However, we can use equation (4) to consider two alternating cases germane to this innovation(1) **→ **imitation(1) **→** innovation(2) *ad infinitum*** **competitive system. First, during a period in which S* _{t}* > 0 the innovating enterprise can make a positive excess profit margin. Second, after one or more imitators enter the industry, S

*→ 0 (the excess profit margin begins to vanish towards zero), such that over time both the innovators and imitators tend to earn the common profit margin of the industry plus a random disturbance. In oligopolistic industries, S*

_{t}*> 0 may persist over an extended period exceeding a normal transfer pricing audit cycle. See equation (8) below.*

_{t}Given the dynamic characteristics of the combined equations (1), (2) and (3), only in special circumstances when S* _{t}* = 0, and thus φ = 0, or else when the comparables to the controlled taxpayer earn an equilibrium profit margin, do we get the static model prescribed by the OECD and the U.S. transfer pricing regulations:

##### (5) Y_{t} = α + U_{t}

_{t}

_{t}

The difference between (4) and (5) is the middle element φ S_{t}_{ }_{– 1}. This special model (5), assuming that φ = 0, may have limited applicability because facts and circumstances may not support this idealized equilibrium profit margin. Unless we test the results, equation (5) may produce an unreliable profit margin. Equations (1), (2) and (3) provide a dynamic model contrasted to using the static or equilibrium equation (5). See the resulting equation (8) below.

As described above, this competitive ("creative destruction") treadmill keeps on turning because the incumbent enterprises and the potential new entrants are motivated to introduce more technological and marketing innovations to obtain a recurring or persistent excess profit margin above α. For the management of an individual enterprise, the creation of trade secrets leading to S* _{t}* > 0 act like an external coercive force. For tax administration, it's important to test and provide a reliable measure of the excess profit margin. This test can't be done using equation (5).

#### Equilibrium Profit Margin

We can convert equation (4) into a first-order (dynamic) difference equation and determine the equilibrium profit margin of the comparable enterprises. For this purpose, we lag the profit margin expressed in equation (1) one-period, and obtain:

##### (6) Y_{t} _{– 1} = α + S_{t}_{ – 1} ↔ S_{t}_{ – 1} = Y_{t} _{– 1} – α.

_{t}

_{t}

_{t}

_{t}

Next, we substitute (6) into (4) and obtain a first-order dynamic equation that we can estimate using comparable company data:

##### (7) Y_{t} = α + φ (Y_{t} _{– 1} – α) + U_{t}

_{t}

_{t}

_{t}

##### (8) Y_{t} = μ + φ Y_{t} _{– 1} + U_{t},

_{t}

_{t}

_{t}

where μ = α (1 – φ) is the intercept and φ is the slope coefficient of this first-order dynamic equation (8). The cyclical properties of equation (8) are well-known among economists, so we don't need to discuss them here. See e.g. Oded Galor, *Discrete Dynamical Systems* (Springer, 2007), pp. 10-11. We can get more complex and also well-known cyclical dynamics by using a 2nd-order difference equation. However, this exercise is beyond our scope here.

In equilibrium (when innovators and imitators are in stasis), we expect the profit margin earned by the comparables in the industry of the taxpayer to stabilize between two or more adjacent periods, and thus we can assume that Y* _{t}* = Y

_{t}_{ – 1 }= Y. From this hypothetical equilibrium, equation (8) becomes more tractable and is equivalent to the idealized equation (5):

(9) Y = μ + φ Y because the expected value of E(U* _{t}*) = 0 (see equation (3) above).

Therefore, from (9) we obtain:

##### (10) Y − φ Y = (1 – φ) Y = μ,

from which we determine the *equilibrium profit margin* of the selected comparables:

(11) Y = μ / (1 – φ) = α, which is equivalent to (5).

In sum, we can use the first-order dynamic model (8) to estimate the regression parameters α, φ, and σ, and calculate the predicted profit margin of the *N* comparables taking into consideration the probable existence of surplus profit margins. It may be unreliable to assume them away à priori. We can also use the estimated standard error of the regression (which is proportional to σ) to determine an arm’s length range of profit margins for the tested party. This model (8) has been applied by different economists using company-level data in several countries, including Canada, France, Germany, Japan, the UK, and the United States. See Dennis Mueller (ed.), *The Dynamics of Company Profits* (Cambridge University Press, 1990).

Please contact us at RoyaltyStat for your (royalty rates and company financials) comparable data needs, and for expert advice regarding inter-group corporate reorganization and transfer pricing planning and audit defense. As suggested in Dante’s *Purgatorio* (Canto V, 13), “*Vien diento a me, e lascia dir le genti*; *sta come torre ferma, che non crolla*.”

**Ednaldo Silva**(Ph.D.) is founder and managing director of RoyaltyStat. He helped draft the US transfer pricing regulations and developed the comparable profits method called TNNM by the OECD. He can be contacted at: esilva@royaltystat.com

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