^{†}

The authors thank three anonymous referees and Ulrich Berger for many helpful suggestions. Any remaining errors are our own.

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Using the political-economic history of the development of telephony during the 1870s as a backdrop, this paper studies a two-player Tullock contest that includes both research effort (R&D) and legal effort (

This paper provides a two-player game-theoretic model of a Tullock [

There are a number of articles in the literature examining the influence of two or more types of effort on the prize value. One of these is Epstein, Nitzan and Schwarz [

With regard to the literature on the influence of effort on the value of the prize, it is helpful to recall that firms in the Tullock [

As per the existence of the strategic equivalence between patent races and rent seeking, this result is reached given that the unique Nash equilibrium of the rent-seeking contest (game) is also the unique Nash equilibrium of the patent-race game developed herein. Thus, this study supports and builds upon prior work by Baye and Hoppe [

The political-economic history of the development of telephony during the 1870s provides a nice example for displaying the two-dimensionality of effort that is the centerpiece of our model. Thus, a brief recounting of the history of telephony appears in the next section. It is followed in

As Lemley [

In 1874, Gray filed his first patent application for a series of transmitters that tuned to different pitches, and he demonstrated his instruments to the Western Union Telegraph Company [

It was not until early 1876 that Bell filed a patent on a telephone. Although Bell’s lawyers asked patent office officials for special, expeditious treatment of Bell’s application, those same officials mandated a competition between Bell and Gray, who had filed a caveat with the patent office for an electric speaking telephone [

It was at this point that Bell’s lawyers took the liberty of informing Gray of the patent office’s reversal. Gray, however, did not understand the circumstances as they were explained to him by Bell’s attorneys, and he remained under the impression that he still had 90 days to file his own patent [

Meanwhile, Dolbear was working to improve his permanent magnet telephone, which, in the fall of 1876, was fit for a patent application. Before applying for a patent, Dolbear took his device to a machine shop to have the rods re-magnetized [

In the following model we analyze the role of two different effort levels, research effort and legal effort, in two equivalent games, a rent-seeking contest and a patent-race game. Bell’s efforts to win the patent race in telephony and curb current and future competition through litigation can be rationalized by our model. In order to keep the model relatively simple, it is assumed that Bell and his company, Bell Telephone Company, have just one competitor at a time, let us say, Western Union Telegraph Company. (Given the brief political-economic history of telephony presented in the previous section of this study, these were the two companies that were vying for the eventual legal monopoly in telephony). Let _{B}_{W}_{B}_{W}_{B}_{W}

The probability of player _{i}_{B}_{W}_{i}_{B}_{W}_{B}_{W}_{B}_{W}_{i}_{B}_{W}_{i}_{i}_{i}_{i}_{i}_{i}

As shown above, spillovers come from litigation (

We assume a concave prize function _{1}(_{B}_{W}_{2}(_{B}_{W}_{11}(_{B}_{W}_{22}(_{B}_{W}_{12}(_{B}_{W}

We adopt the Tullock [_{B}_{W}_{i}_{B}_{W}

Introducing the Tullock [

In order to obtain crisp and explicit solutions to illustrate the workings of the model, we assume that function _{B}_{W}_{B}_{W}

The unique symmetric Nash equilibrium of the above rent-seeking contest (game) is given by,

The main results of the model are given by the comparative statics of this unique, symmetric Nash equilibrium. First, note that research effort grows with legal effort, (d

Also, from the comparative statics analysis of the Nash equilibrium it is easy to show that an increase in judicial costs

One may inquire whether these appealing qualitative results come from the arbitrary choice of Equation (7) and whether they would persist if other specifications were chosen. Note that the function _{B}_{W}_{B}_{W}

In what follows we use the seminal results obtained by Baye and Hoppe [_{i}_{B}_{W}

_{B}_{W}

Note that in the limit when the interest rate goes to zero we have,

In Equation (13) there is a strategic equivalence with our rent-seeking contest and the patent-race game. It is important to stress that this result holds true only in the case of a zero interest rate—that is, when firms have no time preference in the patent-race game. The consequence of the strategic equivalence is that the unique Nash equilibrium of the rent-seeking contest given by Equations (8) and (9) is also the unique Nash equilibrium of the patent-race game.

This paper studies a Tullock [

_{11}(_{B}_{W}

Assuming a symmetric equilibrium, where _{B}_{W}_{B}_{W}_{1}(_{B}_{W}_{2}(_{B}_{W}

Assume a non-separable function _{B}_{W}_{B}_{W}^{a}, where 0 <

In the symmetric equilibrium, where _{B}_{W}_{B}_{W}

It is easy to see that for 0 <

The second order conditions are fulfilled, as long as:

And, the condition

Assuming a symmetric equilibrium: _{B}_{W}_{B}_{W}

As in _{1}(_{B}_{W}^{a−1} = 2