NATURAL CYCLES OF IMITATION AND INTELLECTUAL PROPERTY INFRINGEMENTS

Mark Carlson and Pawel Dyczewski

University of California at Berkeley

Abstract

Reducing intellectual piracy is a major concern of the developed world. Most of the economic literature focuses on using stronger laws and enforcement techniques to accomplish this. We believe that there may be other forces, especially in the developing world, that are instrumental in reducing intellectual piracy. One stylized fact to consider is that the piracy rates for some developing countries were found to decline despite no increase in protection or enforcement. For example a study conducted by the International Intellectual Property Alliance (IIPA) found that the piracy rates for computer business applications software in Poland for years 1995-1997 were 75%, 73% and 61% respectively. This decrease in the piracy rates occurred despite the IIPA’s finding that there was no increase in the enforcement of Polish copyright laws.

In this paper, we introduce a model where firms of developing nations start with a large profit incentive to copy the products of the developed nation. Copying is simply cheaper than innovating. Nevertheless copying does have some drawbacks. The pirating firm is restricted to the domestic market. The longer the developing countries’ firms continue to copy, the more difficult it is to copy more advanced goods. Profits from copying thus decline quickly. At some point, the returns to selling a new good to the world market becomes large enough that it overwhelms the remaining incentives to copy and the firm shifts to research and development. This reasoning explains why piracy is significantly smaller in the developed nations. Piracy is only really advantageous if the pool of goods that can be copied is large enough to allow the imitating firm to copy many goods each period, for several periods. Firms in the developed nations, who are on the technological frontier, have only a few goods they could copy. The profits they could make by copying these goods are significantly smaller than those they could make by engaging in innovation.

The model we use follows Romer (1990) with various modifications.

The model has two countries – a developed "North" and a developing "South". We are concerned with the number of different goods these two countries know how to construct. We refer to the number of goods in the North as A_N and the number in the South as A_S. In order to manufacture a good a country needs to have instructions about how to make that good – a "blueprint". This does not require that the country actually have the knowledge to produce the blueprint from scratch, stealing someone else’s blueprint is a perfectly acceptable way of learning how to produce a good. Our initial condition is that A_S is a subset of A_N.

In each country the setup is essentially the same. There is a final goods industry that produces output according to the function:

Where A can be either A_S or A_N, L is labor, and x(i) are capital goods. We assume b <1. Since output is a single good, we treat this sector as if it consisted of a single representative price taking firm.

There is also an intermediate goods sector that produces capital goods. Each firm (i) in the capital goods sector produces a unique, non-depreciating capital good, which it rents to the final output sector. Capital goods are produced using some fraction of output:

Where h _i is significantly less than one. We assume that h _i is the same for all intermediate firms.

There is also a research sector. The research sector is capable of producing blueprints for goods that are not yet in production. In each country there are J firms that are capable of producing new goods. These firms employ a fixed amount of labor. The firms are however allowed to allocate this labor toward a number of different projects. Each project costs some amount(d ). There are diminishing returns to spreading out the labor supply over a number of research projects. We let v_ji denote the number of attempts that firm j makes to learn how to produce a new good. The number of successes is taken to be ln(v_ji) +1.

If one country has a blueprint for a good and the other country does not then the research sector is able to attempt to imitate or copy that good. Ease of copying depends on how many goods the other country knows how to produce relative to yours. One can think about this in two ways. One possibility is that goods that are closer to the technological frontier are harder to reverse engineer and require more knowledge and skill on the part of the imitator to reproduce. Another possibility is that one does not want to copy every good, only goods that will sell well in your country. Thus the more goods you have to choose from, the easier it is to find one that will sell. The firm hires the same amount of labor as if it were innovating. The firm can again distribute the number of workers amongst a variety of projects each costing some amount (f ). We assume that each research firm can engage in imitating as many goods as it wants, though there are again diminishing returns to number of attempts. Cost will also depend on the knowledge ratio (A_N/A_S). In addition we assume that some firms are better than copying than others. For this reason we introduce g which is a constant that varies by firm and is "distributed" uniformly over the interval [1,2]. We let the number of projects that the firm opt for to be v_jc and the number of success to be ln(v_jc) +1.

Interactions Amongst Actors: We assume that both countries strictly enforces domestic patent laws for blueprints. This implies two things. First a country will not allow its capital-producing firms to purchase a copied blueprint if the original blueprint was produced domestically. Second it implies that capital goods that are produced

with copied blueprints are not imported if the original blueprint was produced

domestically. We also assume that the North enforces international patent laws, which means that its research firms are barred from copying blueprints from the South.

There is no collusion between any firms.

The intermediate good producers rent their capital goods to every final goods producer they can. If the capital good was produced from an innovated blueprint then they can rent it to final goods producers in both countries. If the capital good was from a

copied blueprint, which only occurs in the South, then it can only be rented in the domestic market. If the blueprint is copied then the resulting capital good is rented at just under the price it would be rented by the firm that bought the original blueprint. The firms producing capital goods command a monopoly on those goods and thus charge monopoly prices to final goods producers.

Firms developing knowledge sell it to the capital goods producing firms. Research firms realize that they are the only ones who are able to sell a particular idea to a capital-producing firm. Research firms also realize that the capital producing firms have a monopoly on the capital goods, so the knowledge producing firms charge a price equal to the monopoly rents the capital firms would realize for use of a blueprint.

Innovated blueprints have an infinite patent protection. Copied blueprints have no protection. Once a good has been copied by one research firm, it may

be costlessly copied by another research firm the next period, which it can then sell to another capital goods firm. The two capital goods firms then compete using Bertrand competition and force profits to zero. Thus, the capital firm

buying the first copied blueprint is only able to receive monopoly profits for one period. This is then reflected in the price it pays to the research firm it buys the copied blueprint from.

Research firms are restricted to either copying or innovating.

Wages and population are fixed.

We begin by looking at the profits that the capital goods producing firm is able to realize. In this one sector economy one unit of intermediate goods is produced from h units of output goods. The intermediate firm must "borrow" the output goods from the general public at rate r.

We start deriving the monopoly profits for the capital goods firm by looking at the monopoly profit function for the firm:

We take the derivative of profits with respect to output. We solve for price of output in terms of the elasticity of price of the intermediate good with respect to output of the intermediate good (e ). Doing this we get:

We then find this elasticity by minimizing the cost function of the final goods producing firm with respect to x_j

subject to

Noting that the the final goods producer takes p(x_j) as given. We solve this result for p(x_j) and find that it equals:

Using this equation we are able to solve for the elasticity above. Once we do this and substitute into the price equation we find that the price that the intermediate firm will charge for rental of its intermediate good is

The price of capital good is then fixed. Profits then vary only by the amount of the good that one is able to sell. This is determined by size of the final goods market and is presumed known by all firms in the economy. We make the assumption that the size of the market is the same in both countries. Thus firms selling copied goods make half the profit of a firm selling an innovated good each period.

One period profit for a copied capital good is then:

Profits from innovated blueprints are then more complex. We look at the developing country first. A capital good producing firm here is able to sell its goods to both countries forever. Thus we find profits to be:

Profits in the developing country depend on how long the capital goods firm is able to sell its product in the developing country. We call this time T. Using some inverse sampling with replacement theory we could derive an expression for this, but for now we are content to leave it unknown. We understand that T increases as the amount of research in the developing country increases and T decreases as the amount of copying in the developing country increases. Thus profit is:

Research firms that are engaged in copying will charge a price (p_c) =p _c. They must pay their workers the going wage (w). They face a profit function:

where the firm chooses v_jc. Taking the first order conditions and solving we find that the optimal choice of v_jc is:

Firms that are engaged in innovation are able to charge a price (p_i) =p _i. They face a profit function:

And their optimal choice of v_ji is:

It is possible that to solve the model for a steady state in which research occurs in both countries and there is a steady state growth rate. This we leave for somebody else, rather we are interested in the transition dynamics that involve shifting between imitation and innovation.

Our first observation is that the profits to the research firm involved in innovation in the developing country are time invariant.

Our second observation is that as the ratio (A_S/A_N) increases, profits to the imitating research firm also decline. For this section, unless noted, we let A=(A_S/A_N).

We start with the profit function and expand it to exogenous variables:

Take the derivative with respect to A:

And find profits to copying are always decreasing as the South catches up in blueprints. Now we simply need to show that the south successfully researches more blueprints each period.

First we need to find the ratio of blueprints A where firms switch from

copying to innovating. We start by setting the profits equal so the firm is indifferent between activities.